Cours de M2: Star products, deformation
quantization and Toeplitz operators
L. Boutet de Monvel
(etat provisoire - ne pas diffuser)
Dans ce cours nous decrivons la theorie des star-produits, les methodesqu’elle utilise, ainsi que les exemples les plus utiles. Les star produits ont eteinventes (cf. en particulier dans [6]) pour decrire comment une algebre commu-tative, par exemple l’algebre des fonctions differentiables sur une variete, ou unalgebre “d’observables” de la physique classique, se deforme en une algebre noncommutative. Ils servent a decrire comment la mecanique classique hamiltoni-enne est limite de la mecanique quantique (analyse semi-classique). Le calcul desoperateurs pseudo-differentiels a ete developpe a partir de 1965 par de nombreuxauteurs : il donne lieu a un calcul asymptotique “algebrique” essentiellementidentique a celui des star-produits (operateurs pseudo-differentiels, operateursde Toeplitz, analyse microlocale). Ce calcul fournit des solutions asymptotiques,par exemple des developpements asymptotiques aux hautes frequences de solu-tions d’equations aux derivees partielles; il explique bien par exemple commentl’optique geometrique est limite de l’optique ondulatoire. Dans ce calcul le roledu petit parametre de deformation est joue par la taille d’une petite longueurd’onde (inverse d’une haute frequence); la principale difference est qu’il n’y aplus de “petit parametre” de deformation qui commute avec toutes les autresoperations, comme c’est le cas pour des star-produits.
A une star-algebre est toujours associe un crochet de Poisson, qui decritla limite de la loi des commutateurs (dans le cas d’une deformation: f, g =ddt (f ∗t g − g ∗t f)|t=0). Un des problemes classiques de cette theorie est declassifier, isomorphisme pres, les star-produits. Ce probleme a ete resolu parM. De Wilde M. et P. Lecomte [38] dans le cas de la deformation d’un crochetde Poisson symplectique (par V. Guillemin et moi-meme [27] dans le cadre“Toeplitz” indique ci-dessus), et par M. Kontsevich [91] dans le cas general.Dans le cas sympectique B.V. Fedosov [60] a donne une solution tres elegante,qui est celle que nous suivrons ici.
Dans la deuxieme partie du cours, nous illustrerons cette theorie et la theoriedes operateurs de Toeplitz.
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Keywords: star-products, deformation quantization, symplectic geometry, con-tact manifolds, CR geometry, Toeplitz operators, residual trace.
Mathematics Subject Classification (2000): 16S32, 16S80, 32A25, 32V05, 35S05,53C05, 53D10, 53D55, 58J40.
Contents
1 Introduction 6
2 Star Algebras 82.1 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Star products and star algebras . . . . . . . . . . . . . . . . . . . 102.3 Poisson bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Reminder of differential calculus notations . . . . . . . . . 112.3.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 Symplectic cones and contact manifolds . . . . . . . . . . 152.3.4 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Models and examples 193.1 Moyal star product . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Star-product defined by a formal group law . . . . . . . . . . . . 213.3 Formal pseudo-differential operators . . . . . . . . . . . . . . . . 223.4 Pseudo-differential operators . . . . . . . . . . . . . . . . . . . . 233.5 Semi-classical operators . . . . . . . . . . . . . . . . . . . . . . . 253.6 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Homomorphisms, automorphisms 284.1 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Subprincipal Symbol . . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Automorphisms of symplectic deformation algebras . . . . . . . . 324.6 Automorphisms of symplectic algebras . . . . . . . . . . . . . . . 334.7 Automorphisms preserving a subprincipal symbol or an involution 354.8 Fourier integral operators . . . . . . . . . . . . . . . . . . . . . . 35
4.8.1 As functional operators. . . . . . . . . . . . . . . . . . . 35
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5 Classification 375.1 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . 375.2 Non commutative cohomology . . . . . . . . . . . . . . . . . . . . 385.3 Symplectic algebras are locally isomorphic . . . . . . . . . . . . . 405.4 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5 Classification of symplectic algebras . . . . . . . . . . . . . . . . 425.6 Classification of symplectic deformation algebras . . . . . . . . . 435.7 Algebras of pseudo-differential type . . . . . . . . . . . . . . . . . 44
6 Fedosov Connections 486.1 Valuations and relative tangent algebra W . . . . . . . . . . . . . 486.2 Automorphisms and Derivations of W . . . . . . . . . . . . . . . 496.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Vector Fields with Coefficients in W . . . . . . . . . . . . . . . . 516.5 Fedosov Connections . . . . . . . . . . . . . . . . . . . . . . . . . 526.6 Fedosov curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 536.7 Base-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7 Traces 567.1 Residual integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.2 Trace for Moyal products . . . . . . . . . . . . . . . . . . . . . . 567.3 Canonical trace on symplectic algebras. . . . . . . . . . . . . . . 577.4 Trace for deformation algebras . . . . . . . . . . . . . . . . . . . 58
8 Vanishing of the Logarithmic Trace. 598.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Adapted Fourier Integral Operators . . . . . . . . . . . . . . . . 608.3 Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618.4 Generalized Szego projectors . . . . . . . . . . . . . . . . . . . . 628.5 Residual trace and logarithmic trace . . . . . . . . . . . . . . . . 638.6 Trace on a Toeplitz algebra A and on EndA(M) . . . . . . . . . 658.7 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9 Asymptotic equivariant indexof Toeplitz operators. 699.1 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.1.1 Microlocal model . . . . . . . . . . . . . . . . . . . . . . . 699.1.2 Generalized Szego projectors . . . . . . . . . . . . . . . . 709.1.3 Holomorphic case . . . . . . . . . . . . . . . . . . . . . . . 71
9.2 Equivariant trace and index . . . . . . . . . . . . . . . . . . . . . 719.2.1 Equivariant Toeplitz algebra . . . . . . . . . . . . . . . . 719.2.2 Equivariant trace . . . . . . . . . . . . . . . . . . . . . . . 729.2.3 Equivariant index . . . . . . . . . . . . . . . . . . . . . . 749.2.4 Asymptotic index . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 K-theory and embedding . . . . . . . . . . . . . . . . . . . . . . . 769.3.1 A short digression on Toeplitz algebras and modules . . . 76
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9.3.2 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 Asymptotic equivariant index :Atiyah-Weinstein index formula. 8010.1 Equivariant trace and index . . . . . . . . . . . . . . . . . . . . . 81
10.1.1 Equivariant Toeplitz Operators. . . . . . . . . . . . . . . . 8110.1.2 G-trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8210.1.3 G index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.2 K-theory and embedding . . . . . . . . . . . . . . . . . . . . . . . 8510.2.1 A short digression on Toeplitz algebras . . . . . . . . . . 8510.2.2 Asymptotic trace and index . . . . . . . . . . . . . . . . . 8610.2.3 E-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2.4 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 88
10.3 Relative index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.3.1 Holomorphic setting . . . . . . . . . . . . . . . . . . . . . 9110.3.2 Collar isomorphisms . . . . . . . . . . . . . . . . . . . . . 9410.3.3 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.3.4 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.4.1 Contact isomorphisms and base symplectomorphisms . . . 9710.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9810.4.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 99
11 Complex Star Algebras 10111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 Star Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2.1 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10311.2.2 Star Products on a Real or Complex Cone. . . . . . . . . 10411.2.3 Associated Poisson bracket . . . . . . . . . . . . . . . . . 105
11.3 Pseudo-differential Algebras . . . . . . . . . . . . . . . . . . . . . 10611.3.1 E-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 10611.3.2 Differential Operators and D-algebras . . . . . . . . . . . 10811.3.3 Automorphisms and Symbols of Automorphisms . . . . . 10811.3.4 Non Commutative Cohomology Classes . . . . . . . . . . 10911.3.5 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.3.6 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . 112
11.4 E-Algebras on T ∗X, dimX ≥ 2 . . . . . . . . . . . . . . . . . . . 11411.4.1 General Results. . . . . . . . . . . . . . . . . . . . . . . . 11411.4.2 The case dimX ≥ 3 . . . . . . . . . . . . . . . . . . . . . 11511.4.3 The case dimX = 2 . . . . . . . . . . . . . . . . . . . . . 115
11.5 E-Algebras over Curves (dimX = 1) . . . . . . . . . . . . . . . . 11811.5.1 Open curves . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.5.2 Curves of genus g ≥ 2 . . . . . . . . . . . . . . . . . . . . 11911.5.3 Curves of genus 1 . . . . . . . . . . . . . . . . . . . . . . . 11911.5.4 The projective line . . . . . . . . . . . . . . . . . . . . . . 124
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12 Related symplectic star algebras 12712.1 Geometric quantization . . . . . . . . . . . . . . . . . . . . . . . 12712.2 Homomorphisms between Star Algebras . . . . . . . . . . . . . . 12812.3 Action of a Compact Group . . . . . . . . . . . . . . . . . . . . . 12812.4 Circle Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12912.5 Elliptic Circle Action . . . . . . . . . . . . . . . . . . . . . . . . . 13012.6 End of description . . . . . . . . . . . . . . . . . . . . . . . . . . 132
13 Toeplitz operators and asymptoticequivariant index 13513.1 Szego projectors, Toeplitz operators . . . . . . . . . . . . . . . . 135
13.1.1 Example 1: Microlocal model . . . . . . . . . . . . . . . . 13513.2 Example 2 : holomorphic model . . . . . . . . . . . . . . . . . . . 13613.3 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13613.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.5 Equivariant Toeplitz algebra . . . . . . . . . . . . . . . . . . . . . 13813.6 Equivariant trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.7 Equivariant index . . . . . . . . . . . . . . . . . . . . . . . . . . . 14013.8 Asymptotic index . . . . . . . . . . . . . . . . . . . . . . . . . . . 14013.9 K-theory and embedding . . . . . . . . . . . . . . . . . . . . . . . 14213.10Embedding and transfer . . . . . . . . . . . . . . . . . . . . . . . 14413.11Relative index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14613.12Enlargement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14613.13Collar isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 14813.14Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14813.15Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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1 Introduction
Deformation algebras and star products were introduced in [6], so as the mainproblems they pose; a closely related formulation was given by F.A. Berezin[10]. Typically a deformation algebra is given by the following data :
- an initial algebra A, usually A = C∞(X), the algebra of smooth functionson a manifold X; more generally A could be the algebra of holomorphic(resp. algebraic) functions on a holomorphic (resp. algebraic) space X.
- A formal multiplication law B =∑
~nBn where for each integer n, Bn :(f, g) 7→ Bn(f, g) is a C-bilinear map : A×A→ A (later on we will alwayssuppose that the Bn are bidifferential operators)
- a formal family A~ of algebras depending on a formal parameter ~ : itselements are formal power series f =
∑fn~n ∈ A[[~]]; the multiplication
law (f, g) 7→ f ∗ g = B(f, g) is a C[[~]]-bilinear operator B =∑
~nBn :A[[~]] × A[[~]] → A[[~]] (the Bn are C-bilinear : A × A → A, if f =∑
~pfp, g =∑
~qgq, B(f, g) =∑
~n+p+qBn(fp, gq).
The product B must satisfy the following conditions :
i) for ~ = 0 we get the initial law : B0(f, g) = fg
ii) the law is associative : (f ∗ g) ∗ h = f ∗ (g ∗ h), i.e. for all n,∑p+q=n
Bp(Bq(f, g), h)−Bp(f,Bq(g, h)) = 0.
iii) the multiplicative unit is 1 : 1∗f = f ∗1 = f i.e. Bn(1, f) = Bn(f, 1) = Oif n > 0 (the bidifferential operator Bn has no term of order 0 for n > 0).
(The third condition could be omitted; in fact the first two conditions implythat there is a unit, i.e. a formal series a = 1 +O(~) such that a = a ∗ a, and asuitable linear bijection f 7→ f +
∑∞1 will give an equivalent product for which
the unit is 1).
Here we will only deform algebras of functions on manifolds, and we willalways suppose that the Bn are bidifferential (local) operators, i.e. in localcoordinates of the form Bn(f, g) =
∑an,α,β∂
αf∂βg.
In physics ~ usually denotes the Planck constant (an action). We have keptthis notation although here (and in all the sequel) ~ will usually denote a smallformal parameter. In WKB asymptotic expansions from quantum physics, thesmall parameter is the ratio of the Planck constant to actions at the scale orthe phenomenon described, which is very small for phenomenons well abovethe atomic scale. In geometrical optics, one encounters asymptotic expansions,where ~ measures the size of a small wavelength, the inverse of a large frequency:this is very small for optic waves, which explains that geometric optics is relevantat our scale; much less for radio waves and not at all for acoustic waves.
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The calculus of pseudo-differential operators enters essentially in the samedescription, except there is no longer a small central parameter ~. In order toinclude it we will slightly broaden the description above, replacing formal seriesin the “small parameter” ~ by more general asymptotic expansions with respectto a large quantity - e.g. the size of a “large frequency”, playing the role of ~−1.
Remark 1 It is natural to ask if a star algebra corresponds to a true (notformal) family of algebras, i.e. the formal symbols
∑fk~k (or at least some
of them) represent the asymptotics for ~ → 0 of functions (or distributions)f(x, h) with a product law. This is often the case, although one usually does notexpect convergent series, and it is completely unusual that a symbol determinesuniquely the corresponding distribution (e.g. the trivial deformation productcorresponds to the usual commutative product of smooth functions f(x, h), butthe Taylor series of f only determines f up to functions vanishing of infiniteorder for ~ = 0)
The last sections are extracted from recent articles of the author.
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2 Star Algebras.
2.1 Cones
Definition 1 A real cone is a C∞ principal bundle Σ with group R×+. The basisis X = XΣ = Σ/R×+.
The trivial cone with basis X is X×R×+ with the action of R×+ (homotheties)given by λ.(x, r) = (x, λr).
Any point of X has a neighborhood U such that Σ|U = p−1(U) is isomorphicto the trivial cone U × R×+ (p denotes the projection map Σ → U). If X isparacompact, any cone Σ with basis X is trivial, i.e. isomorphic to the trivialcone (it has a section), but the product structure is not part of the data, only thefibration. In the sequel we will only consider paracompact cones and manifolds
if Dω is the long real line, T ∗Dω−(its zero section) is a non trivial real cone.
In some instances it will be convenient to use the complex line bundle Σcextending Σ :
Σc = Σ×R×+C× (1)
Homogeneous functions of integral degree f ∈ O(n) or formal series f ∈ Oobviously extend to Σc.
For complex or algebraic geometry, one need a slightly broader definition : the
basis X is a complex manifold, resp. an algebraic variety over a field k; a cone Σ is a
line bundle over X, not necessarily trivial, deprived of its zero section. Since we will be
using differential operators, X will usually be supposed smooth (without singularities)
and k of characteristic zero.
On a cone Σ the group of homotheties has an infinitesimal generator : thisis the radial vector field ρ corresponding to the derivation Lρ) such that
Lρf(x) =∂
∂λf(λx)|λ=1 (ρ = r
∂
∂rin any trivialisation Σ = X × R+). (2)
Definition 2 (i) We denote O(m) the sheaf of homogeneous functions of degreem on Σ, O =
⊕O(m) (f ∈ O(m) ⇐⇒ ρf = mf).
(ii) We denote O the sheaf on XΣ of formal symbols :
f ∈ O if f =∑m≤m0
fm with fm ∈ O(m), m an integer, m→ −∞ (3)
Elements f ∈ O are asymptotic expansions for ξ → ∞ on Σ. We will usuallyrefer to them as “total symbols”. If f ∈ O is of degree m, its symbol (orprincipal symbol to avoid confusions) is its leading term σm(f) = fm; we willusually write σ(f) if there is no ambiguity.
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O(m) is a vector space, and O is a graded commutative algebra where theproduct is defined by the natural multiplication maps O(m)⊗O(n)→ O(m+n).O is equipped with a canonical decreasing filtration Om (elements of degree
≤ m); the graded associated sheaf is gr O =⊕Om/Om−1 =
⊕O(m)
O(m),O and O are sheaves (of vector spaces or algebras) on the basis X.
We recall that “F is a sheaf on X” means that F(U) is well defined forany open subset U of X (↔ open sub-cone of Σ) (it is a set, a group, a vectorspace, an algebra, or or an object of some category); the restriction maps ρUV :O(V )→ P(U) are defined for U ⊂ V , with ρUW = ρUV ρVW if U ⊂ V ⊂W , andfor any covering U =
⋃Uj , the product of restriction map O(U) →
∏O(Uj)
identifiesO(U) with the equalizer (kernel) of the two maps∏O(Uj)⇒
∏O(Ui∩
Uj) : in other words two sections of O which are equal in some neighborhoodof each point of U are equal, and a family fi ∈ O(Ui) which agree in eachintersection Ui ∩ Uj can be patched together to produce a global section f ∈O(U). If p : E → X is a continuous map, the sheaf of sections is F(U) =Γ(X,U) =the continuous sections f → E (functions such that pf = Id U is asheaf. Any sheaf of sets is isomorphic to some sheaf of sections, with p etale(locally an isomorphism) (cf. [70]). Elements of F(U) are usually called sectionsof F (over U), and F(U) is often denoted Γ(U,F).
Definition 3 (iii) For any integer k ≥ 1 we denote Dk the sheaf (on XΣ) offormal k-differential operators : P (f1, . . . , fk) =
∑m≤m0
Pm(f1, . . . , fk) withPm a k-linear differential operator homogeneous of degree m with respect tohomotheties, m an integer, m→ −∞. For k = 1 we just write D.
Locally we may choose homogeneous local coordinates xj on Σ : Pm(f1, . . . , fk)is a sum of homogeneous monomials
aα(x) ∂α11 . . . ∂αnn
i.e. aα is homogeneous of degree m+∑αkdeg (xk) (in the notation above α is
a multi-index : α = (α1, . . . , αn) ∈ Nn, ∂α = ∂α1 . . . ∂αn)Two cases will be useful :1) x1, . . . , xn−1 are homogeneous of degree 0 and are local coordinates on
the basis XΣ, xn is homogeneous of degree 1 or −1;2) the xj are all of degree 1
2 . There is no restriction on the order of Pm.
Below “degree” will always refer to the degree w.r. to homotheties; thusif P ∈ Dk each term Pm of degree m is of finite order, although the resultinginfinite sum P may be of infinite order.
We will denoteD× ⊂ D (4)
the sheaf of invertible formal differential operators : P =∑Pk ∈ D× is invert-
ible iff its leading term σ(P ) = Pm0 is invertible, i.e. Pm0 is of order 0, themultiplication by a non-vanishing function homogeneous of degree m0.
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Definition 4 (iv) We denote by D×0 the sub-sheaf of those invertible P suchthat P (1) = 1, i.e. P is of degree 0, P0 = 1, Pm(1) = 0 if m < 0.
2.2 Star products and star algebras
Definition 5 Let Σ be a cone. A star product on Σ (or on the basis X) isa formal bilinear product B =
∑Bn ∈ D2 (i.e. for each integer n, Bn is
a bidifferential operator, homogeneous of degree −n), defining a product law :O × O → O - (f, g) 7→ f ∗ g = B(f, g), such that
i) B0(f, g) = fg (or B0 = 1, B is a deformation of the usual product)
ii) the law is associative : (f ∗ g) ∗h = f ∗ (f ∗h), i.e. B(B⊗ 1) = B(1⊗B),or for all n,
∑p+q=nBp(Bq ⊗ 1− 1⊗Bq) = 0.
iii) the multiplicative unit is 1 : 1 ∗ f = f ∗ 1 = f i.e. Bn(1, f) = Bn(f, 1) = 0if n > 0 (the bidifferential operator Bn has no term of order 0)
(Here again the first two conditions imply that there is a unit u with leadingterm u0 = 1, and we can fix this unit equal to 1, e.g. replacing B by B′(f, g) =u−1B(uf, ug)).
Example 1 A typical example of star product is the Leibniz rule for the com-position of differential operators on an open set U ⊂ E, E a vector space (e.g.E = Rn) : a differential operator P = P (x, ∂) =
∑aα(x)∂α, with aα ∈ C∞(U)
is characterized by its total symbol
p(x, ξ) = e−x.ξP (ex.ξ) =∑
aα(x)ξα
which is a function, polynomial in ξ, on the cotangent bundle T ∗U = U×E∗; thecone is Σ = T •U = T ∗U−the zero sectionξ = 0, with x, resp. ξ homogeneousof degree 0, resp. 1.
If P,Q are two operators, the total symbol r of R = P Q is given by Leibniz’rule :
r(x, ξ) =∑ 1
α!∂αξ p ∂
αx q
This formula still makes sense as a formal series if p, q ∈ O(Σ) are symbols(x, ξ of degree 0 resp. 1) : this is the composition law for pseudo-differentialoperators (see below §3.4.
Definition 6 A star-algebra is a sheaf of unitary associative algebras on thebasis XΣ, locally isomorphic to O equipped with a star-product, where the struc-tural patching sheaf of groups is D×0 .
If A is a star algebra, we can pick local frames (i.e. an isomorphism A(U)→O(U) over a small open set U); in such a frame the product is given by abidifferential operator : f ∗ g = B(f, g), with B ∈ D2 (B =
∑Bk(f, g) with Bk
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a bidifferential operator homogeneous of degree −k → −∞, B0 = 1). Transitionmaps from one frame to another are given by invertible operators P ∈ D×0 .
Since any P ∈ D×0 respects the natural filtration of O, a star-algebra A isequipped with a natural filtration. We will denote Am the set of elements ofdegree ≤ m (they form a sheaf). Since the leading term of any P ∈ D×0 is 1, theleading term of any a ∈ A is well defined; if a ∈ Am we denote it σm(a) ∈ O(m)(or σ(a) if there is no confusion) and call it the principal symbol of a. Thesymbol map : grA =
⊕Am/Am−1 → gr O =
⊕O(m) is an isomorphism of
graded algebras : σm+n(ab) = σm(a)σn(b)).
A total symbol on A is a formal differential operator σtot : A → O of degree0 such that σtot(1) = 1. Total symbols exist locally by definition, hence alsoglobally ; if Xi is an open covering of the basis X, and σi a total symbol onXi, then σtot =
∑φiσi is well defined and is a total symbol, for φi a smooth
partition of 1 subordinate to the covering. This does not work for algebraicor holomorphic star algebras. In fact the algebra of differential (or pseudo-differential) operators on the complex projective line P1(C) does not have atotal symbol.
Example 2 differential operators or pseudo-differential operators on a manifoldform a star algebra, for which there is no preferred total symbol (see below §3.4).
The distinction between star-products and star-algebras is not really essen-tial on real manifolds, where a star-algebra is always isomorphic to O equippedwith a star-product because a “total symbol” i.e. a global isomorphism P :A→ O with P ∈ D locally always exists (locally, by definition, and on a realmanifolds these can be patched together using a partition of unity); however, asmentioned in [23] this is no longer true over holomorphic manifolds : the mostusual star-algebras such as the algebra of pseudo-differential or semi-classicalpseudo differential operators on a manifold are not equipped with a canonicaltotal symbol. For functorial manipulations it is more convenient to deal withstar-algebras rather than star-products.
2.3 Poisson bracket
2.3.1 Reminder of differential calculus notations
If X is a manifold, we denote TX the tangent bundle, T ∗X the cotangentbundle. We will often use the cotangent bundle deprived of its zero section,which we denote T •X.
The sections of the exterior algebra∧T ∗X =
⊕∧kTX form the sheaf of
differential forms Ω =∑
Ωk. This is a graded anti-commutative algebra, onwhich several canonical operations are defined :
If ξ is a vector field, it defines a derivation Lξ of the algebra OX of functionson X. More generally the Lie derivative Lξu is defined for any differential
11
object u which is functorially defined, e.g. differential forms, tensors, differentialoperators etc. : Lξu = d
dte−tξ∗ u|t = 0 where etξ denotes the germ of group with
infinitesimal generator ξ (i.e. t 7→ etξx is the solution of the differential equationddtx(t) = ξ, x(0) = x), and ∗ denotes the push-forward.
The exterior derivation d is the unique anti-derivation of degree 1 such thatd2 = 0 and for any function f ∈ Ω0, 〈df, ξ〉 = Lξ(f) (anti-derivation means :d(ab) = da b+ (−1)daa db, where x denotes the degree of x).
If ξ is a vector field, Iξ denotes the anti-derivation of degree −1 such thatIξ(ω) = 〈ξ, ω〉 for ω ∈ Ω1 (also noted ξyω).
On Ω, Lξ is the unique derivation (of degree 0) such that [d, Lξ] = 0 andLξ(f) = 〈df, ξ〉 for f a function. and we have
Lξ = [d, Iξ]
(in a graded anti-commutative algebra [..] denotes the super bracket : [a, b] =ab− (−1)dadbba with dx the degree of x - here [d, Iξ] = dIξ + Iξd).
The exterior derivative d is (locally) exact, i.e. if dω = 0 then any pointhas a neighborhood in which ω is a derivative (ω = dµ). On a cone we willneed homogeneous primitives: if ω is closed, homogeneous of degree k 6= 0, itis globally exact: ω = 1
kLρω = 1kdIρω. Let us choose r > 0 homogeneous of
degree 1 (⇔ a trivialization of Σ): ρ = r∂r. Any form ω ∈ Ωk, homogeneous ofdegree 0 is ω = dr
r µ+ ν with µ, ν pull-backs of forms on the basis X (of degreek − 1 resp. k). Ω is closed, resp. exact iff µ and ν are. In particular for k = 1,ω is closed iff ν is closed and µ is a locally constant function; it is locally exactiff µ = 0.
The bracket [ξ, η] of two vector fields is defined by L[ξ,η] = [Lξ, Lη] (i.e.the corresponding first order operator is the commutator of Lξ and Lη). Ithas a natural extension to the algebra of multivectors (sections of
∧TX): the
Schouten-Nijenhuis (super)bracket ([106, 109]). The extension extension is de-fined as follows : first I : ξ → Iξ as extend as a homorphism of algebras∧T → L(Ω) (Iξ1..ξk = Iξ1 . . . Iξk , so 〈ξ1 . . . ξk, ω〉 = Iξk . . . Iξ1(ω) if ω ∈ Ωk):
the Schouten-Nijenhuis bracket is characterized by
I[ξ,η] = [Iξ[d, Iη]]
(one the right hand side [.[.]] denotes again the super bracket). For instance ifc, c′ are two bivectors (identified with antisymmetric bidifferential operators),the tridifferential operator defined by [c, c′] is
[c, c′](f, g, h) = c(c′(f, g), h) + c(c′(g, h), f) + c(c′(h, f), g) ++ c′(c(f, g), h) + c′(c(g, h), f) + c′(c(h, f), g)
2.3.2 Poisson brackets
Definition 7 A Poisson bracket on a manifold X is a bilinear map (f, g) 7→c(f, g) on C∞(X) (often denoted c(f, g) = f, g) such that
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(i) c(f, f) = 0 (c is skew symmetric, f, g = −g, f)(ii) c is a derivation w.r. to g (or f) : f, gh = f, gh+ gf, h(iii) c satisfies the Jacobi identity : f, g, h+g, h, f+h, f, g = 0
An equivalent definition is that c is a bivector (locally =∑cij∂i∂j such
[c, c] = 0 where [c, c] is the Nijenhuis-Schouten bracket.If f is a smooth function, the derivation = vector field Hf (g) = f, g is
called the hamitonian vector field of f . Condition (iii) is equivalent to
[Hf , Hg] = Hf,g
equivalently, that .. is invariant by Hf : Hf (g, h = Hfg, h+ g,Hfh.
A Poisson bracket on a manifold X is called symplectic if its matrix is invert-ible, i.e. locally c =
∑cij∂i ∧ ∂j with (cij) an invertible antisymmetric matrix
with smooth coefficients. Equivalently c defines an antisymmetric isomorphismTX → T ∗X. The inverse then corresponds to a symplectic 2-form according tothe following definition:
We will call symplectic star algebra a star algebra on a cone Σ with sym-plectic Poisson bracket. σ is then even dimensional. Typical example aree thepseudo-differential algebras, or the Toeplitz algebras mod. smoothing operators(see below).
If A is a deformation algebra on a manifold X, its Poisson bracket on thecone Σ = X × R+ is of the form ..Σ = ~..X with .. a Poisson bracketon X. We will say that it is symplectic is ..X is symplectic. X is then evendimensional (and dim Σ is odd). A typical example is the algebra of semiclassicalpseudo-differential operatos (see below).
Definition 8 A symplectic manifold is a manifold X equipped with a 2-formω ∈ Ω2(X) such that
(i) ω is invertible, i.e. locally ω =∑aijdxidxj where the matrix (aij) is
skew-symmeric, invertible (this implies that dimX is even).(ii) dω = 0
One readily checks that the condition dω = 0 is equivalent to condition (iii)above (Jacobi identity). Thus a symplectic manifold is the same thing as amanifold equipped with an invertible Poisson bracket. In mechanics Poissonbrackets are not always symplectic, especially if they depend on parameters
A Poison cone is a cone equipped with a Poisson bracket homogeneous ofdegree −1. A symplectic cone is a Poisson Σ cone whose Poisson bracket isinvertible, Σ is equipped with a symplectic form ω homogeneous of degree 1.Such a form is always exact: if ρ is the radial vector (infinitesimal generatorof homotheties), we have ω = Lρω = dIρω. The Liouville form λ = Iρω is theunique primitive of ω which is orthogonal to ρ.
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Example 3 Let X be a manifold. The Liouville form λ on T ∗X is the tautolog-ical form: for any smooth function f , df = f∗λ where f is the section X → T ∗Xdefined by df , ∗ denotes the pull-back; in any system of local coordinatesλ =
∑ξjdxj . The canonical symplectic form of T ∗X is ω = dλ =
∑dξjdxj .
The corresponding Poisson bracket is the canonical Poisson bracket of T ∗X :
λ =∑
ξjdxj , ω =∑
dξjdxj , f, g =∑ ∂f
∂ξj
∂g
∂xj− ∂f
∂xj
∂g
∂ξj(5)
Unless otherwise stated, we will always suppose that the Poisson bracketor symplectic form are real. In PDE theory they are pure imaginary, but thismakes only insignificant differences for the calculus. However many of the state-ments and constructions below work with minor modifications, replacing Σ byits complexification Σc.
Exercise A multivector p (section of∧TX) can be viewed as a super-
function p(x, ξ on T ∗X (its symbol), super meaning that the local cotangentcoordinates ξj anti-commute (this is tautological. On the algebra of these func-tions we dispose locally of the derivations ∂xj (of degree 0), ∂ξj , which is ananti-derivation of degree −1. The symbol of the Schouten-Nijenhuis bracket is
[a, b] =∑
∂ξja ∂xj b− (−1)(a−1)(b−1)∂ξj b ∂xja
A Poisson bracket c is an antisymmetric map TX → T ∗X and defines ahomomorphism of graded antisymmetric algebras C :
∧TX →
∧T ∗X, and
Ω → multivectors. Then we have C(dω) = [c, Cω] (both are C derivations,which coincide on functions, and 1-forms).
Note that C−1 is homogeneous of degree 1: it takes multivectors homoge-neous of degree k to forma of degree k+1. We may translate the local exactnessof the exterior differential d:
Lemma 9 if X is a multivector homogeneous of degree k =1 such that [c,X] =0, there exists locally Y of degree k + 1 such that X = [c, Y ]
The same holds on a symplectic manifold, without degree conditions.
A theorem of Liouville states :
Theorem 10 Two symplectic manifolds, resp. symplectic cones of the samedimension are locally isomorphic
We briefly recall the proof : first recall that two invertible skew symmetricinvertible bilinear forms on a vector space are always isomophic (in particularthey are isomorphic to the canonical form c((x, y), (x′, y′) = x.y′−x′.y on V ∗×V- this is true for any ground field).
Let X be a symplectic of dimension n, and (cjk) an invertible antisymmetricmatrix. We construct functions xj in a neighborhood of a given point (origin)as follows: x1 is chosen arbitrarily with dx1 6= 0; Then we construct inductively
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xk so that Hxj (xk) = cjk for j < k, and the Hamiltonian fields Hxj are linearlyindependent. This is possible because the induction hypothesis implies that theHxj commute ([Hxj , Hxk = Hcjk = 0) ahd the Frobenius integrability conditionis satisfied. The independence condition is ensured by fixing appropriate initialconditions, e.g. fixing the values of the Hamiltonian fields (i.e. the dxj at theorigin).
This works in the same manner on a symplectic cone; in the inductive con-struction, one also asks that Hxj , j < n be independent of the radial vector ρ toensure that there is a homogeneous solution. Thus a symplectic cone is alwayslocally isomorphic to the cotangent bundle T •Rn; so as symplectic manifold(without the homogeneity condition)..
2.3.3 Symplectic cones and contact manifolds
A contact form on a manifold X of dimension 2n − 1 is a 1-form λ such thatλ(dλ)n−1 is a volume form. Liouville’s theorem also states that two contactforms of the same dimension are locally isomorphic (it is proved in the samemanner). A 1-form λ on X is a contact form iff the cone Σ = R+λ ⊂ T •X (thepositive multiples of λ- is symplectic.
Let us call call contact morphism u : X → X ′ a smooth map such that λ isa positive multiple of u∗λ′; likewise a symplectic morphism Σ→ Σ′ is a smoothhomogeneous map such that ωΣ = u∗(ωΣ′ . (Such a map is not necessarily anisomorphism but it is necessarily an immersion: its derivative is injective at eachpoint of X (in particular if such a map exists we have dimX ≤ dimX ′ (resp.dim Σ ≤ dim Σ′). If X,X ′ is the basis of Σ,Σ′, a contact map X → X ′ has aunique symplectic lifting Σ → Σ′, and conversely. In other words the functorwhich to a symplectic cone (resp. map) assigns its basis (resp. the base map) isan equivalence. When dealing with symplectic cones, it will often be practicalto replace it by its contact basis, and conversely.
2.3.4 Commutators
Let A be a star algebra on Σ. As mentioned above A has a canonical filtrationcoming from the filtration of O by homogeneity degrees, and there is a canonicalisomorphism :
grA ' gr O =⊕O(m)
because the patching sheaf of groups D×− induces the identity on gr O).
The leading term of the commutator law :
f, g = B1(f, g)−B1(g, f)
is a Poisson bracket on Σ, homogeneous of degree −1, and grA is a Poissonalgebra (i.e. equipped with a Lie bracket which is a bidfferentiation w.r. to themultiplication).
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Conditions (ii) and (iii) immediately follow from the fact that on any algebrathe bracket [a, b] = ab − ba is a Lie bracket: ad a : b 7→ [a,b] is a derivation([a, bc] = [a, b]c + b[a, c]), and the Jacobi identity holds: [a, [b, c]] = [[a, b], c] +[b, [a, c]] (equivalent to [ad a, ad b] = ad [a, b]).
2.4 Functional calculus
Let A be a star algebra, a ∈ Af a smooth function. We want to define f(a).E.g. If f is a poynomial f(T ) =
∑fkT
k, then f(a) =∑fka
k is defined for anya ∈ A. If f =
∑∞0 fkT
k a formal series, f(a) =∑fka
k is also well defined ifdeg a < 0.
The requirements are the following:
• f 7→ f(a) must be a homomorphism of algebras, i.e. (f + g)(a) = f(a) +g(a), fg(a) = f(a) ∗ g(a);
• if f(T ) =∑fkT
k is a polynomial, then f(a) =∑fka
k Plus some conti-nuity condtions which will be made precise later.
If f is holomorphic, we define f(a) in a neighborhood of a point x ∈ X as
12iπ
∫γf(t)(t− a)−1dt, (6)
where γ is any small path/loop of index 1 around the value σa(x). This does notdepend on the choice of the loop, so these definitions patch together to definef(a); f should be holomorphic in a neighborhood of each value of σa.
For any y close to x, we can rewrite this integral as
f(a) =1
2iπ
∫f(t)
∑k
(t− a0(y)−k−1(a− a0(y))kdt =∑ 1
k!f (k)(y)(a− a(y))k
(a0(y) is a number so t− a0(y) and a− a0(y) commute).We denote IyA0 the ideal of elements a ∈ A0 whose principal symbol vanish
at y. We will limit to the case where A is defined by a star product B ofpseudo-differential type, i.e. B =
∑Bn where the n-th component Bn is of
order ≤ (n, n) (see below - where we will also see that this is always possible).Then a =
∑k≤0 ∈ Imy if for each k, ak vanishes of order ≥ m − 2k (for a
symplectic algebra, the converse is true).Then the sum
∑1k!f
(k)(y)(a−a(y))ky=x the k-th term (a−a(y))ky=x is i factof degree ≤ −k2 .
So f(a) is again well defined if a0 is real, and f is a smooth function of areal variable (more generally if f is smooth on the range of a0 and holomorphicinside). We have f(a)− f(b) ∈ ImY if a− b ∈ ImY ; so the “Taylor series” of f(a)at a point y only depends on that of a (and depends continuously on it).
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2.5 Comments
Let X be a Poisson manifold, i.e. a manifold X equipped with a Poisson bracketf, g (see below). A deformation quantization of X is a unitary associativeproduct law f ∗ g =
∑hnBn(f, g) on the space of formal series f =
∑hnfn ∈
C∞(X)[[h]], where each Bn is a bilinear differential operator on X, B0(f, g) =fg, the unit is 1 (Bn has no “term of order 0” if n > 0), and the leading term inthe commutator law is given by the Poisson bracket of X : B1(f, g)−B1(g, f) =f, g.
The corresponding cone is X ×R×+ equipped with the Poisson bracket hcX ,cX the Poisson bracket of X; the canonical coordinate of R×+ is 1
h , homogeneousof degree 1 (in asymptotic analysis it corresponds to a large frequency). Wewill also refer to these as deformation, or “semi-classical” star-products or star-algebras.
In a deformation quantizations (or semi-classical algebra), the deformationparameter h (the “Planck constant”, or its inverse h−1) is central and part ofthe data; for these we only allow automorphisms which preserve h.
Existence of star algebras with a given Poisson bracket, and classification ofthese, was an important problem of the theory.
Existence of a global star-algebra on a real symplectic cone Σ was proved byV. Guillemin and myself in [27] (see also [19]).
Existence of deformations of symplectic Poisson brackets (symplectic semi-classical star algebras) were proved by M. De Wilde and P. Lecomte ([38, 40]),who also described their classification. For these a very elegant method wasdevelopped by B.V. Fedosov [60].
In [91] M. Kontsevitch proved that any Poisson bracket comes from a star-product in the real semiclassical case. More precisely he proves that there isa one to one correspondence between isomorphic classes of star-products andisomorphic classes of formal families of Poisson brackets depending on the “smallparameter” h. His result extends, without changing a word, to star-productson a real cone with the definition above ; families of Poisson brackets should bereplaced by formal Poisson brackets (with coefficients in O) on Σ :
c =∑k≤−1
cm (with cm homogeneous of degree m). (7)
The main point of Kontsevitch’s construction is a formula giving a starproduct from a Poisson bracket on an affine space; this also works on the complexaffine spaces, and adapts easily to the conic setting. Going from a local existenceor classification result to a global one requires some gluing argument. Below wewill follow the method of B.G. Fedosov (loc. cit.), which adapts easily to the realconic framework. The gluing argument uses real constructions, such partitionsof unity, global connections, or tubular neighborhoods of subvarieties, which do
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not exist in the holomorphic or algebraic category: for these it is not knownif a global star product with given Poisson bracket in general exists, even inthe symplectic case, nor what the classification of such algebras looks like; seehowever [88], where it is shown that even if E may not exist, the category ofsheaves of E-modules is can be defined as a stack.
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3 Models and examples.
3.1 Moyal star product
A graded vector space is a vector space equipped with a direct sum decompo-sition E =
⊕En(n ∈ R) (for most purposes R could be replaced by anything).
Equivalently E is equipped with a degree operator d, a diagonalisable linear op-erator: En = ker (d−n), the n-eigenspace; equivalently E is equipped with a one-parameter group of linear transformations (action of R+): λ(
∑xn) =
∑λnxn.
In algebraic cases the grading will be integral (e.g.⊕O(m) above, and
homotheties define an action of the multiplicative group (z.x = zdx for z ∈ C×resp. z ∈ k×, k the ground field).
If E is a graded vector space, Σ = E − E0 is a cone, with the actionof R+ above (one must exclude E0 to get a free action of R+). The spacesO(m)(Σ), O(Σ) are well defined as above.
(Note that the grading of O =LO(m) is integral, by definition. The grading of
E is not necessarily so; occasionally it will be convenient to choose “ocal coordinates”
homogeneous of non integral degree - in particular of degree 12)
The Moyal star product is a typical example of star product s : let E bea graded vector space. Let b =
∑bij∂i∂j ∈ V ⊗ V a 2-tensor of degree −1
(or of integral degree ≤ −1), defining a bidifferential operator with constantcoefficients
b(f, g) =∑
bij ∂if ∂jg.
The Moyal product defined by b lives on Σ = E − E0 :
f ∗b g = eb(∂ξ,∂η) f(ξ)g(η) |η=ξ =∑ 1
n!bn(∂ξ, ∂η)f(ξ)g(η) |η=ξ (8)
the formal sum is well defined since b is of negative degree; it is a star producton V : associativity follows from the bilinearity of b: we have
b(u+ v, w) + b(u, v) = b(u, v + w) + b(v, w),
which implies
(f ∗ g) ∗ h = eb(∂ξ,∂η)e12 b(∂ξ,∂η,∂ζ) f(ξ)g(η)h(ζ) |η=ζ=ξ = · · · = f ∗ (g ∗ h)
The corresponding Poisson bracket is the top-order part (of degree −1) ofthe antisymmetrization of b :
f, g(ξ) = (b(∂ξ, ∂η)− b(∂η, ∂ξ)) f(ξ)g(η) |η=ξ = b(f, g)− b(g, f) (9)
For example the normal law (Leibniz rule, example 1) for differential oper-ators on an open set U ⊂ Rn is a typical example of Moyal star product :
f ∗ g =∑ 1
α!∂
(α)ξ f∂(α)
x g = f ∗bn g
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with bn(f, g) =∑∂ξkf∂xkg − ∂xkf∂ξkg. Its Poisson bracket is the canonical
Poisson bracket of T ∗U : f, g = fξ.gx − fx.gξ (cf. 5).
Proposition 11 Two bidifferential operators b, b′ with the same antisymmetricpart yield isomorphic Moyal products.
Indeed if p = b′−b is symmetric, we have (b′−b)(u, v) = 12 (p(u+v)−p(u)−p(v))
i.e.12p(u+ v) + b(u, v) = b′(u, v) +
12p(u) +
12p(v),
The second order operator P = p(∂ξ, ∂ξ) is homogeneous of degree −1, as b andb′, so the formal sum
e12P f =
∑ 2k!
−kP kf
is well defined. We have
e12P (∂ξ+∂η)eb(∂ξ,∂η) = eb
′(∂ξ,∂ηe12P (∂ξ)e
12P (∂η),
from which immediately follows the equality
e12P (f ∗b g) = (e
12P f) ∗b′ (e
12P g)
It is often convenient to use the equivalent but more symmetric Weyl cal-culus: this is the Moyal product corresponding to the antisymmetrisation ofcn:
bw(f, g) =12
(∑
∂ξkf∂xkg − ∂xkf∂ξkg)
f ∗w g = e12 (∂ξ.∂y−∂η∂x)f(x, ξ)g(y, η)|(y,η)=(x,ξ)
On a graded vector space where all coordinates are of degree 12 this allows more
obvious symmetries, in particular the product is invariant by the full symplecticgroup.
We will see later on that all star products with a real (or pure imaginary)symplectic Poisson bracket are locally isomorphic, in particular they are locallyisomorphic to the pseudo-differential star product (normal law), or to the Weylstar product.
For deformation star products the canonical Moyal model is the following :let X be a vector space (no grading - coordinates xk all of degree 0) equippedwith bidifferential operator b =
∑bjk∂j ⊗ ∂k with constant coefficients. The
star-product, with central formal deformation variable ~ is :
f ∗ g(ξ) = e~c(∂ξ,∂η)f(ξ)g(η)|η=ξ (10)
Here again the corresponding Poisson bracket is ~(b(f, g)−b(g, f)); and two bid-ifferential operators with the same antisymmetric part yield isomorphic Moyalproducts.
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3.2 Star-product defined by a formal group law
Let G be a Lie group. Then the left invariant pseudo-differential operators on G(see below) obviously form a sub-algebra of the algebra of all pseudo-differentialoperators. For these the symbol (principal or total) is completely determinedby its restriction to the cotangent fiber at the origin, and produce a star algebraon the dual of the Lie algebra G of G.
Here we will produce here an “explicit” formula for this star-product
Let V be a finite dimensional vector space equipped with a formal grouplaw :
x y = x+ y + c(x, y) (11)
where c is a formal power series in the variables x = (x1, ..xn) y = (y1, ..yn) in V .Associativity (xy)z = x(yz) implies that the error term c(x, y) = xy−x−yis O(|x||y|). The group law defines a Lie algebra structure on V .
Let V ∗ be the dual of V . If F is a distribution on V with support the origin,its Fourier-Laplace transform is the polynomial
f(ξ) = 〈F, ex.ξ〉 (so that F = f(−∂x) δ) (12)
If F and G are two such distribution, the convolution product F ∗G is definedby
〈F ∗G, u〉 = 〈F (y) G(z), u(y z)〉 = f(∂y) g(∂x) u(y z) |y=z=0 (13)
This is defined for u a smooth function on V , but only depends on the Taylorseries of u at the origin and makes sense for u ∈ C[[V ∗]], an arbitrary formalseries.
In particular the Fourier transform of F ∗G, which we will still denote f ∗ g(with f, g the Fourier transforms of F,G) is
f ∗ g(ξ) = 〈F (y) G(z), eξ.(yz)〉 = f(∂y) g(∂x) exp ξ.(y z) |y=z=0 (14)= exp ξ. (∂η ∂ζ) f(η) g(ζ) |η=ζ=0
Let us rewrite ∂η ∂ζ = ∂η+∂ζ +c(∂η, ∂ζ). Taking into account the fact thatall these differential operators commute and that we have eξ. ∂u f(u) |u=0 = f(ξ)(Taylor formula), we get
f ∗ g (ξ) = eξ.c(∂η,∂ζ)f(η) g(ζ) |η=ζ=ξ (15)
Now c is of order ≥ 2 , so this final expression still makes sense for f, g ∈ Osymbols , ie. f =
∑fm−k, g =
∑gm−k formal series of homogeneous functions
of degree → −∞ (ξ. c(∂η, ∂ζ) is of degree ≤ −1). This is the star productassociated to our group law. The cone is Σ = V ∗ − 0, the Poisson bracket isthe standard Poisson bracket of V ∗ (if f, g are smooth functions on V ∗, ξ ∈ V ∗,
21
the derivatives df(ξ), dg(ξ) belong to the Lie algebra V : we have f, g(ξ) =〈[df(ξ), dg(ξ)], ξ〉).
Isomorphic group laws give rise to equivalent star-products, i.e. up to isomor-phism the star product above only depends on the Lie algebra V , or equivalentlyon the Poisson bracket of V ∗. A canonical construction consists in choosing thegroup law given by the Campbell-Hausdorff formula.
One can also replace the group law by the one deduced by homothety :
eλ−1ξ.c(λ ∂η, λ ∂ζ)..|η=ζ=ξ.
For the standard Fourier transform of real analysis λ = ±i)
Exercise - one recovers the law for semiclassical pseudo-differential operators,using the (formal) Heisenberg group or more generally Moyal laws :
u v = u+ v + q(u, v) e (16)
with q a bilinear form on V, e ∈ V a central vector (the Heisenberg groupcorresponds to the case where q is non-degenerate mod. e).
3.3 Formal pseudo-differential operators
Let U be an open set of Rn. We mentioned above that the composition ofdifferential operators (Leibniz rule, example 1)
p ∗ q(x, ξ) =∑ 1
α!∂αξ p ∂
αx q = e∂ξ.∂yf(x, ξ)g(y, η)|y=x
η=ξ
is a star product, which extends to symbols p, q ∈ O(Σ) (Σ = T •U = U × Rnminus its zero section U ×0, with x, ξ of degree 0 resp. 1). We will denote EUthis algebra. If X is a manifold, Xj an atlas of X i.e. a covering of X by opensets isomorphic to open subsets of some numeric space Rn, the EXj glue togethercanonically (see below) and produce the pseudo-differential algebra EX ; for thisagain the cone is T •X, and the Poisson bracket is the canonical one.
An oscillatory asymptotic expansion with phase φ is an expression of theform
A = eλφ(x)a(λ, x) with a =∑λm−kam−k(x)
where λ is a formal parameter; a(λ, ξ) is a symbol in O(R+ × U
Differential operators act on such expansions: if P = p(x, d) is a differentialoperator, we have P (eφa) = eφPφ(a) with
Pφ = e−λφPeλφ = p(x, d+ dφ) =∑
pα(x) (d+ dφ(x))α
(caution: d and dφ do not commute).
22
Proposition 12 We have Pφ ∈ D(U × R+), i.e. Pφ has formal expansionPφ =
∑λm−kPk where the Pk are differential operators (of order ≤ 2k) on U
(m is the order of P ).
Let φ2(x, y) be defined by φ(x) = φ(y) + (x− y).φ′(y) + φ2(x, y): for any fixedy we have Pφ = eλ−φ2P (x, d+ λφ′(y))eλφ2 , hence
Pφ(a) =∑ 1
γ!∂γξ p(x, λφ
′(y)) ∂γx(eλφ2a)|y=x (17)
Now φ2 vanishes of order ≥ 2 for y = x so ∂γ(eλφ2a)|y=x is a polynomial ofdegree ≤ |γ|2 , and ∂γξ p(x, λφ
′(y))∂γx(eλφ2a)|y=x is a symbol of degree ≤ m− |γ|2 .
If dφ 6= 0 the sum above is still defined and converges as a formal series whenp is symbol, thus defining the action of formal pseudo-differential operators onasymptotic expansions.
If y = χ(x) is a local change of coordinates (i.e. the derivative ψ′ is invertible,the total symbol of a differential operator P = p(x, d) in the new coordinates yis p′(y, η) = e−y.ηP (ey.η): this is still well defined if P is a formal differentialoperator, so the sheaf of formal differential operators on an arbitrary manifold Xis well defined as announced; they can be viewed as operators on all asymptoticexpansions as above, with dφ 6= 0.
The oscillating asymptotic expansions corresponding to φ form a sheaf ofEX -modules, with support the cone Λφ = R+dφ ⊂ T •X (if φ is complexvalued this is still well defined, but Mφ rather lives on the complex cotangentbundle T ∗X ⊗ C)
3.4 Pseudo-differential operators
The theory of pseudo-differential operators, which gives main example of sym-plectic star algebra, was developed since the 1960’s in particular by J.J. Kohn,L. Nirenberg, L. Hormander, and widely used by many others. We describenow how it works and gives rise to the star algebra above. Results are statedwithout proofs, and we send back to the literature for further details.
Pseudo-differential makes use of exponentials eix.ξ with pure imaginary ex-ponent, because they are the only bounded ones; they are those which are enterin the Fourier transformation. Below we have followed custom and put a signi =√−1 in the exponents; an even better solution is to say that calculus lives
on the set of pure imaginary covectors. This makes no significant difference forthe formal calculus (so we omit the sign i for formal calculus), but of course inreal analysis it is essential.
Let U be an open set of Rn. A symbol of degree m on T ∗U = U × Rn is asmooth function on admitting for ξ →∞ an asymptotic expansion
p(x, ξ) ∼∞∑k=0
pm−k(x, ξ)
23
with pm−k smooth, homogeneous of degree m− k w.r. to ξ.(∼means that p−
∑m−k>−N pk has the same size as a homogeneous function
of degree −N , and likewise for any derivative
∂αx ∂βξ (p−
∑k≥−N
pk) = O(|ξ|−N−β) for ξ →∞).
Above k is an integer; m should be an integer for compatibility with the defi-nition above, but occasionally it may be any complex number. There are manyother useful and more general classes of symbols which are “micro-local”; themain point is that high order derivatives of p should be integrable w.r. to ξ.Here we have described the minimal class for which symbolic calculus works andwhich are obviously linked with star algebras.
The pseudo-differential P = p(x,D) is defined by the formula
p(x,D)f = (2π)−n∫eix.ξp(x, ξ)f(ξ)dξ
where f is the Fourier transform of f - on Rn :
f(ξ) =∫e−ix.ξf(x)dx f(x) = (2π)−n
∫eix.ξ f(ξ)dξ
For instance if p(x, ξ) =∑aα(x)ξα is a polynomial in ξ, p(x,D) is the
differential operator∑aα(x)Dα, with D = 1
i ∂x.
A pseudo-differential operator P acts continuously from the space C∞0 ofsmooth functions to C∞: it extends to distributions, and diminishes singularsupports : if f is a distribution, Pf is C∞ in any open set where f is.
The Schwartz-kernel of P is the distribution P such that
Pf(x) =∫P (x, y)f(y)dy
It is the inverse Fourier transform of p : P = (2π)−n∫ei(x−y),ξp(x, ξ)dξ. (the
integral only converges in distribution sense). It is always a smooth functionoutside of the diagonal x = y.
We will write P ∼ Q if P −Q has a smooth Schwartz-kernel (equivalently;P − Q extends as a continuous linear operator from the space of distributionsto the space of smooth functions). P is always equivalent to a “proper” pseudo-differential operator (i.e. the two projections from the support of P to U areproper maps; if P is proper, it acts on the space functions or distributions withcompact support and extends to all distributions (for any compact set K ⊂ Uthere exists another one L such that supp f ⊂ K ⇒ suppPf ⊂ L and f = 0out of L⇒ g = 0 out of K).
The total symbol of P = p(x,D) is the asymptotic formal sum σ(P ) =∑pm−k; it belongs to O(Σ), with Σ = T •U = U×Rn (in the sense of definition
2), where the variables xj , resp. ξj are homogeneous of degree 0 resp. 1.
24
σ(P ) completely defines P mod. smoothing operators (P ∼ 0 ⇔ σ(P ) = 0,i.e. p(x, ξ) if of rapid decrease for ξ →∞ so as all its derivatives - we will againwrite p ∼ 0). In fact we have the asymptotic formula (exact for differentialoperators):
p(x, ξ) ∼ e−ix.ξP (eix.ξ)
Proper pseudo-differential operators form an algebra. The total symbol of aproduct is again given asymptotically by the Leibniz rule (exact for differentialoperators), same as above (except for the i sign): if P = p(x,D), Q = q(x,D)then P Q = R(x,D) with
σ(P Q) =∑ i
α!
−α∂αξ σ(P )∂αx σ(Q)
The set of pseudo differential operators is invariant by diffeomorphisms, sopseudo-differential operators on a manifold X are well defined. Modulo smooth-ing operators, they form a star algebra on the cotangent cone T •X = T ∗Xdeprived of its zero section, isomorphic to the forma star algebra above: theprincipal symbol (leading term) is well defined as a homogeneous function onT •X, although there is no canonical or preferred total symbol.
There is an analogue of the oscillatory asymptotic expansion result: if P is apseudo-differential operator and a(x, λ) a symbol, we have P (eλiφa) ∼ eλiφPφ(a)i.e. the difference between the left and right hand sides is of rapid decrease forλ→∞. This requires that φ be real. It still works if Imφ is ≥ 0 (eλiφ bounded)with a suitable modification of Pφ (which is at first only defined where dφ isreal).
3.5 Semi-classical operators
Semi-classical analysis produces an important example of deformation algebra,which is closely linked to the preceding one. Let X be a smooth real manifold -at first an open set of Rn. A semiclassical differential operator is a differentialoperator depending on a formal parameter ~:
P = P (x, ~D) =∑k≥m
ak,α(x)~k(~D)α
~ is a formal small parameter; in real analysis D = 1i ∂x. For a differential
operator there are only finitely many terms; the degree of P is m (we allowm < 0).
The total symbol is the symbol on T ∗X × R+ :
p = p(~, x, ξ) =∑
akα~kξα = e−i~x.ξP (e
i~x.ξ)
The total symbol of P Q is again given by the Leibniz rule (or Moyal formula) :
p ∗ q =∑ i−γ
γ!∂ξp∂
γxq = exp(−i∂ξ∂y)p(~, x, ξ)q(~, y, η)|y=x
η=ξ
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for any smooth f or formal series∑
~kfk we have
P (f) =∑ i
α!
−|α|~|α|∂αξ ∂αx f
and more generally an oscillatory asymptotic expansion:
P (ei~φa) = e
i~φPφa (a =
∑~kak a formal series)
This is exact is P is a differential operator, and still well defined if p is a symbol,φ real.
As above Pφ = e−i~φPe
i~φ is a formal operator on X × R+, which only
depends on the Taylor expansion of p along the section dφ.
For real analysis one needs a link between these formal expansions and truefunctions : this goes as follows: a symbol on X × R is a smooth functiona(x, ~) (the corresponding formal symbol is just the Taylor expansion along~ = 0; one could allow a pole of finite order for ~ = 0). The correspondingoscillatory object is e
i~φa, and the Taylor expansion of a gives good bounds if φ
is real (or Imφ ≥ 0 so that the exponential is bounded). One can define semi-classical pseudo-differential operators (but for many problems in PDE theory,consideration of the formal symbols is enough).
Some problems that are hard or impossible to solve exactly may becomemuch easier to solve for asymptotic solutions. For example : with P = 1− ~2∆the problem P (e
i~φa = e
i~φb i.e. Pφa = b is very easy, because the leading term
(principal symbol) of Pφ is 1 + |dφ|2 6= 0, so Pφ is invertible and we disposeof explicit (if not short) formulas to write the inverse and solve the problem(ARRANGER)
As above semi-classical operators allow changes of coordinates and are stillwell defined on manifolds.
In fact the can be identified with a sub-algebra of the preceding section asfollows: let Y = X×R, and denote (abusively) (x, t) the variables of Y , Ξ, τ thedual cotangent variables. Then Asc is isomorphic to the open half-space τ > 0of T ∗Y .
3.6 Toeplitz operators
Toeplitz algebras give an example of a symplectic star algebra on a symplecticcone which is not a cotangent bundle. We describe here the standard example,which comes from the theory of several complex variables. We will see laterthat a canonical Toeplitz algebra (i.e. well defined, up to isomorphism) can bedefined on any symplectic cone (cf.[27]).
Let U ⊂ Cn be a complex domain with smooth boundary X (more generallyU could be a Stein space with isolated singularities away fro its boundary). U is
26
strictly pseudo-convex if it can be defined (locally or globally) by an inequalityu < 0 with u a smooth function such that du 6= 0 on X, and ∂∂u 0 (i.e. thematrix (∂zp∂zq is hermitian 0), e.g. U is strictly convex.
The boundary value fX of holomorphic function on U is well defined if f iscontinuous on the closure U = U ∪X, they are stil defined as a distribution if fis of moderate growth (f = O(|u|−N for some N - as for any harmonic function1.
If U is strictly pseudo-convex, the boundary values of holomorphic functionsare (locally or globally) exactly those that satisfy the tangent Cauchy-Riemannequations ∂bg = 0. ∂ denotes the antiholomorphic part of d: ∂f =
∑∂zjdzj ; ∂b
is the induced system on X it defines: ∂bfX = ∂f mod. u, ∂u. Let us denoteOs(X) the space of holomorphic boundary values which lie in the Sobolev spaceHs(X). It is known (cf. [80]) that the micro-singularities of boundary valuesof holomorphic functions (f ∈ ker ∂b i.e.f ∈ Os(X) for some s) lie on the half-line bundle Σ ⊂ T •X of positive multiples of λ = −i∂u|X ; this is real since(∂u+ ∂u)|X = 0, strict pseudo-convexity implies that λ = −i∂u|X is a contactform, and Σ is symplectic.
The Szego projector S is the orthogonal projector L2(X) → O0(X) (thedefinition requires choosing a smooth density on X to define the L2 norm). Itwas proved in [30] that it is a Fourier integral operator with complex canonicalrelation, well behaved with respect to pseudo-differential operators, in particularit is continuous Hs(X)→ Os(X) for all s.
The canonical relation of S is a complex (formal) Lagrangian manifold C ⊂T •X × T •X0; it is the outflow of ∂b × ∂b out of Id Σ.
Toeplitz operators are the operators of the form f 7→ TP (f) = SPS(f)with P a pseudo-differential operator on X. Equivalently they are the Fourierintegral operators A with canonical relation C such that A = SAS.
We will say that A is of degree ≤ m if if is of degree ≤ m as a Fourier integraloperator (⇔ A = TP with P of degree ≤ m); then A is continuous Os → Os−m.
Toeplitz operators form an algebra. Mod. smoothing operators (operatorsof degree −∞) they are localized, and they define a symplectic star algebra overΣ, locally isomorphic to the pseudo-differential albebra in n real variables. Theprincipal symbbol of A = TP is σ(P )Σ and we have
Σ(AB) = σ(A)σ(B), σ[A,B] = −i[σ(A), σ(B)Σ
1without the growth condition, the boundary value would be defined as a hyperfunction ifX is real analytic
27
4 Homomorphisms, automorphisms.
4.1 Morphisms
Let A,A′ be two star algebras, Σ,Σ′ the corresponding cones, with base X,X ′,and Poisson brackets c, c′. A homomorphism U : A → A′ is a linear map whichpreserves the products, and the filtrations (U(ab) = U(aU(b), degUf ≤ deg f)
The symbol map grU : grA → grB, is then a Poisson algebra homomor-phism, i.e. it preserves products and Poisson brackets. Its restriction to O(0)corresponds to a smooth map uX : X ′ → X (becauseX is the maximal spectrumof O(0) ∼ C∞(X) : grUf = f uX if f is of degree 0). U is a homomorphismof sheaves of algebras A → A′ over the smooth map uX : X ′ → X.
grU itself comes from a smooth homogeneous map u : Σ′c → Σc, becausethe spectrum of O =
⊕O(m) is the complexified cone Σc (grUf = f u). u is
compatible with the Poisson brackets : c′(f u, g u) = c(f, g) u; equivalentlythe Poisson bracket c is projectable and u∗c = c′, or the graph of u in (Σ×Σop)cis isotropic. Σ′op denotes the opposite cone, i.e. Σ′ equipped with −c′. (Σc isthe line complexification of Σ). 2
Remark 2 u is a map Σ′ → Σ iff grU is real positive, i.e. σ(Uf) ≥ 0 ifσ(f) ≥ 0. This is what happens for Fourier integral transformations. If grUie real, it is either positive or negative (i.e. σ(Uf) ≤ 0 is f is of degree 1 andσ(f) ≥ 0; then u(Σ ⊂ −Σ or equivalently u defines a map Σ′ → Σop, as forinvolutions below.
Note that the fact that grU is real is not directly related to the fact thatthe Poisson bracket is real; the Poisson bracket of pseudo-differential theory ispure imaginary.
Example A typical example of homomorphism is the formula of change of coordi-
nates for the total symbol of pseudo-differential operators or semi-classical PDO that
can be derived from (17)
If B,B′ are two star algebras on cones Σ,Σ′, the star algebra B ⊗ B′ onΣ×Σ′ is well defined (locally the product law is the exterior product of the twobidifferential operators, B ⊗ B′(f, g) = BxB
′x′(f(x, x′)g(x, x′); which obviously
glue together.There are canonical injective homomorphisms U resp. U ′ : A resp. A′ →
A⊗A′ whose range commute, and whose geometric support are the projectionsfrom Σ × Σ′ to Σ resp. Σ′ (f 7→ f ⊗ 1 resp. 1 ⊗ f ; if we are dealing with starproducts, the pull-back by either projection). The exterior product A ⊗ A′ iswell defined (up to unique isomorphism) by this data. Its Poisson bracket isc+ c′ (exterior sum of two bidifferential operators).
Note that the basis of Σ× Σ′ is not X ×X ′).
2In fact the definition above should be completed by the condition that locally U is de-scribed locally by formal differential operators: the total symbol is given by (U)f(x′) = (Pf)uwith P a suitable formal differential operator on X; this is automatic for the symplectic caseswe usually consider
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If U is a homomorphismA → A′ it defines a monogenous (A′,Aop)-bimoduleMu with one generator e and relations ea = Ua e. The support of Mu is thegraph of U . It is free, of rank 1 on A′. It will sometimes be convenient to viewhomomorphisms via such bimodules.
4.2 Automorphisms
An automorphism of a star product is by definition a formal differential oper-ator U ∈ D which preserves the star-product, the filtration and the principalsymbols, so U ∈ D×0 , grU = Id , and the corresponding Poisson map is u = Id Σ.Automorphisms are local operators : the germ of Uf at a point only dependson the germ of f at this point 3 so the definition extends immediately :
Definition 13 Let A be a star-algebra : an automorphism U of A is a lin-ear sheaf automorphism which preserves the star-product and the filtration (thesecond condition is in fact automatic)
U is a homomorphism A → A and the associated Poisson map is Id Σ.
Since U = 1 mod. operators of negative degree, LogU is well defined:
LogU =∞∑1
(−1)k
k−1
(U − 1)k
it is a derivation, i.e. d(f ∗ g) = df ∗ g + f ∗ dg. U 7→ D = LogU is a bijectionfrom the set of automorphisms to the set of derivations of degree ≤ −1; thereciprocal is D 7→ eD =
∑1n!D
n.
Any other series of (U − 1) also makes sense, in particular the fractionalpowers Us, s ∈ C :
Us = exp(sLogU) =∑(
s
k
)(U − 1)k
which form a one parameter group of automorphisms.
Remark 3 If U is an automorphism of a star product, a =∑ak, the homoge-
neous components of Usa are polynomials of s, as the binomial coefficients.
We have tr (AdA)sP = tr (P ) for all s ∈ Z, hence also for all s ∈ C.
Inner derivations are those of the form ad a : f 7→ [a, f ] = a ∗ f − f ∗ a;inner automorphisms are those of the form Ad a : f 7→ a ∗ f ∗ a−1. We haveAdea = exp ad a, LogAd a = ad log a.
3there may also exist global linear operators preserving products which are not local, butwe will never consider these
29
Let A be a deformation algebra. We denote A× the sheaf of its invertibleelements. Inner automorphisms do not form a sheaf, but they generate a sheafof groups, isomorphic to the quotient sheaf A×/ZA where ZA is the center ofA×.
The automorphisms of a semi-classical algebra B preserve ~, by definition.We will denote Aut ~(B) the group of such automorphisms.
Since ~ is central we have Ad ~ma = Ad a for all m, and any inner au-tomorphism is of the form Ad a with a of degree 0 (a =
∑∞0 ~mam with a0
invertible).Inner automorphisms Ad a with a of integral degree 6= 0 also exist, but for
these Ad as is not, even locally, an inner automorphism. If U = Ad a is aninner automorphism, D = LogU is locally an inner derivation D = ad b withb = Log a, a of degree 0.
4.3 Involutions
If A is a star algebra, the opposite algebra Aop is the same sheaf, with thesymmetric product law (for a star product; Bop(f, g) = B(g, f). It is obviouslyalso a star algebra, with the opposite Poisson bracket [f, g]op = [g, f ] = −[f, g].The corresponding cone is Σop, i.e. Σ with the opposite Poisson bracket.
Definition 14 An anti-involution of a star product is a formal differential op-erator J ∈ D×0 , such such that
(i) J(a ∗ b) = J(b) ∗ J(a)(ii)gr Ja(ξ) = a(−ξ) (i.e. σm(Ja) = (−1)mσm(a) if a is of degree m).This definition extends to star algebras: an anti-involution J of A is a sheaf
isomorphism (over the basis X) A → Aop which is locally an anti-involution ofstar products.
The corresponding base map is jX = IdX , i.e. J preserves principal symbolsof degree 0. The corresponding cone map is the antipodal map j : Σc → Σopc(j(ξ) = −ξ).
gr J(∑
fk) =∑
(−1)nfk if fk is of degree k
j reverses the Poisson bracket c. In fact condition (i) implies J [a, b] = [Jb, Ja] =−[Ja, Jb] so jc = −c. Since c is homogeneous of degree −1 this automaticallyimplies that j is the antipodal map if c does not vanish. We have made it partof the definition in all cases; with this definition the identity map is never ananti-involution.
For an anti-involution on a deformation algebra we require J(~) = −~.
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Examples
1. On the Moyal model (§3.1) defined by an antisymmetric bidifferentialoperator b(∂ξ, ∂η) (half the Poisson bracket), there is a canonical anti-involution: f(ξ) 7→ Jf = f(−ξ) =
∑(−1)kfk. Indeed the star product
defined by b is f ∗b g = eb(∂ξ,∂η)f(ξ)g(η)η=ξ and since b is is homogeneousof degree −1, its image by the antipodal map ξ 7→ −ξ is −b = b(∂η, ∂ξ),which implies J(f ∗b g) = Jg ∗ bJf
2. Likewise for the Moyal model deformation star product (10) with anti-symmetric b, there is a canonical involution : a(x, ~) 7→ a(x,−~).
3. On the algebra of differential operators on Rn there is a canonical involu-tion P (x, ∂x) =
∑aα(x)∂α 7→tP =
∑(−1)α∂α aα(x).
Proposition 15 (i) Any two anti-involutions on A are conjugate (globally).(ii) If A has an anti-automorphism, it also has an anti-involution.
Proof : (i) if J, J ′ are two anti-involutions, U = J ′J is an automorphism andJ ′ = UJ = JU−1, so J ′ = U1/2JU−1/2.
(ii) If A is an anti-automorphism, A2 is an automorphism which commuteswith A, so as the automorphism (A2)−
12 , and J = (A2)−
12A is an anti-involution.
4.4 Subprincipal Symbol
The subprincipal symbol of a pseudo-differential operator P on Rn with totalsymbol p(x, ξ) =
∑pm−k(x, ξ) is subP = pm−1 − 1
2i
∑∂ξj∂xjpm. Together
with the principal symbol, it determines P up to order m− 2.
A symbol map of order 2 on a star algebra is a total symbol P : A → Osuch that for a ∈ Ap, b ∈ Aq we have
(i) P (a ∗ b) = P (a)P (b) +12P (a), P (b) mod. terms of degree ≤ p+ q − 2
(ii) P ([a, b]) = P (a), P (b) mod. terms of degree ≤ p+ q − 3
This only concerns P mod. operators of degree ≤ −2, which we can forget inwhat follows.
If a ∈ Am the leading term of P (a) is the principal symbol σm(a) ∈ O(m).The next term is the subprincipal symbol subm(a) ∈ O(m−1) (associated to P ).Together with the principal, it determines a mod. Am−2. Subpricipal symbolsare characterized by the fact that for all m they define maps Am → O(m − 1)(component of a symbol map as above) such that subm(a) = 0 if a ∈ Am−2,subm(a) = σm−1(a) if a ∈ Am−1. Conditions (i), (ii) above can be rewritten :
(i)bis if a, b are of degree m,m′, we have
subm+m′(a∗b) = sub (a)σ(b)+σ(a)sub (b)+12σ(a), σ(b) ∈ O(m+m′−1)
31
(ii)bis if a, b are of degree m,m′, we have
subm+m′−2[a, b] = sub (a), σ(b)+ σ(a), sub (b) ∈ O(m+m′ − 2)
The subprincipal symbol of pseudo-differential operators mentioned abovesatisfies these conditions (in fact it is invariant by diffeomorpfism for differentialoperators acting on half desnsities - for which the trnsformation law by χ isχ∗, 12P = (Ad
chi′|) 12χ∗P )
The Moyal star product f ∗b g with b with antisymmetric b has a canonicalsubprincipal symbol : f =
∑fm−k ∈ Am 7→ sub f = fm−1. Indeed the product
law is B =∑Bk with Bk(f, g) = 2−k
k! (b(∂ξ, ∂η)k(f(ξ)g(η))η=ξ =; the bracketlaw is [f, g] = 2b(f, g) + B2(f, g) − B2(g, f) + . . . . If b is antisymmetric, thePoisson bracket is .. = b and B2 is symmetric, so for f ∈ Ap, g ∈ Aq we havef ∗ g = fg + 1
2f, g mod. Ap+q−2, [f, g] = f, g mod. Ap+q−3.
Proposition 16 If P,Q are two symbol maps of order 2, P − Q is of degree≤ 0 and its principal part δ = σ−1(P −Q) is a Poisson vector field, i.e.
δ(fg) = δ(f)g + fδ(g), δ(f, g = δ(f), g+ f, δ(g)
the first equality follows immediately from (i), and the second from (ii).
Remark 4 A map satisfying (i) exists on any star algebra (cf. below ref).Equivalently any star product is equivalent to a star product of the form f ∗g =fg + 1
2f, g + . . . ). But this is often not sufficient for global computationsand does not deserve the name “subprincipal symbol”. We will see that onsymplectic algebras two subprincipal symbols are always conjugate; for thiscondition (i) alone is not enough.
Subprincipal symbols, satisfying (ii), do not exist on all star algebras (evensymplectic); in particular they do not always exist on the canonical Toeplitzalgebra constructed in [27].
4.5 Automorphisms of symplectic deformation algebras
Theorem 17 Derivations and automorphisms of symplectic deformation alge-bras are locally inner derivations resp. automorphisms.
Since Adea = exp ad a and LogAd a = ad log a, the two statements are equiva-lent.
If D is a derivation of degree m, its symbol is of the form ~−md with δPoisson vector field on the basis X, i.e. a vector field (δ(fg) = δf g + f δg)such that δf, g = δf, g + f, δg. If X is symplectic, this means that the1-form ω corresponding to δ is closed (dω = 0), so locally there exists a functionφ such that dφ : ω,Hφ = δ; then for any a ∈ A with symbol ~)mφ, D − ad a
32
is of lower degree. By induction we get a (as a convergent formal series) suchthat D = ad a.
Thus the sheaf AutA coincides with the sheaf IntA of automorhisms wichare locally inner inner automorphism of the form Ad a with a ∈ A×0 , the sheafof invertible elements of degree 0.
Lemma 18 The center of A×0 is C[[~]]× (its elements are the formal series∑∞0 ak~k with ak locally constant, a0 6= 0)
The center of A is the algebra of formal series of ~ alone (of the form∑k≥k0
ak~k with ak locally constant). Indeed if f =∑k≥k0
~kfk is central,and fk is constant for k < m, we have fm, g = 0 for all g, which implies thatfm is (locally) constant since the Poisson bracket is symplectic.
Thus we have an exact sequence of sheaves
0→ C[[~]]× → A×0 → AutA → 0 (18)
It will be convenient to modify this, using the sheaf A×0 whose sections arepairs ϕ, f with f ∈ A×0 invertible, ϕ ∈ O(0); eϕ = σ(f) (logarithms of sectionsf ∈ A×0 ) : we have an exact sequence of sheaves :
0→ C[[~]]→ A×0 → AutA → 0 (19)
Although A×0 is not commutative, it is a soft sheaf (it has“partitions ofunity”, and its cohomology is essentially trivial (cf. below)
4.6 Automorphisms of symplectic algebras
Let Σ be a cone, X its basis. We denote H∗hom(Σ) the De Rham cohomology ofhomogeneous forms: Hk
hom(Σ,C) is the space of closed k-forms with coefficientsin O mod. exact forms.
Lemma 19 Hkhom(Σ,C) is canonically isomorphic to Hk(X,C)⊕Hk−1(X,C).
Proof : if ω =∑k≤k0
ωk with ω homogenous of degree k, we have dω =∑dωk,
so ω is closed resp. exact iff each ωk is so.If ρ is the radial vector, we have Lρω = (dIρ + Iρd)ω =
∑kωk. So if ω is
closed, it is cohomologous to ω0 = ω − dIρ∑k 6=0
1kωk
Let ω ∈ Ωk be homogeneous of degree 0: ω = µ + drr ν where µ, ν are pull
backs of forms of degree k rep. k − 1 on X (r denotes a “vertical” coordinate,i.e. a smooth positive function homogeneous of degree 1). Then dω = dµ− dr
r dνso ω is closed, resp. exact iff both µ and ν are, hence the lemma.
We now turn to automorphisms and derivations of symplectic algebras. LetA be a symplectic star algebra on Σ, with symplectic Poisson bracket. We
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denote A×0 the sheaf of invertible elements of degree 0 (i.e. sections a of degree0 such that σ0(a) never vanishes). We denote A×− ⊂ A×0 the sheaf of elementswith symbol 1 (deg (a− 1) < 0).
If D is a derivation of degree m of A, its symbol (leading term) δ = σm(D)is a Poisson derivation δ of degree m, i.e. a vector field such that [c, δ] = 0.These form a sheaf (on the basis X) isomorphic to the sheaf of closed 1-formshomogeneous of degree m+ 1.
If U is an automorphism, we define its symbol as the symbol of LogU :σ(U) ∈ ω0
Proposition 20 Let A be a symplectic star algebra. (i) A derivation of degree< −1 is an inner derivation. A automorhism U such that U − 1 is of degree< −1 is an inner automorphism : there exists a unique a ∈ A of degree ≤ −1such that U = ad a.
(ii) Any automorphism is locally of the form U = (Ad a)sAdP with P in-vertible of degree 0, a a fixed elliptic element of degree 1, s a constant, Anyautomorphism U such that U − Id is of degree < −1 is an inner automorphism.
(iii) We have an exact sequence of sheaves
0→ A×− → AutA → ω0
(iv) Any section of ωm+1 is the symbol of a global derivation. Any sectionof ω0 is the symbol of an automorphism.
Let δ be a Poisson derivation homogeneous of degree m. Then the corre-sponding 1-form ω is closed, homogeneous of degree 6= 0 hence exact: ω = dφwith φ homogeneous of degree m+ 1, and δ = Hφ. Then δ is the leading termof ad a if σ(a) = φ.
Thus if D is a derivation of degree m < −1 we can construct by successiveapproximations a ∈ A such that D = ad a, and if U − 1 is of degree < −1, thenso is D = LogU . hence (i). Note that if A is symplectic, its center is reducedto the constants C, so the element a ∈ A−1 such that U = Ad a is unique.
If m = −1, the closed 1-form ω of degree 0 corresponding to δ is of the formdµ+ sdrr with s a constant, µ closed of degree 0. Then locally µ has a primitiveφ and δ is the symbol of Ad a(Ad b)s if σ(a) = eφ, σ(b) = r (b is of degree 1,(Ad b)s is well defined). Hence (ii) and (iii).
The last assertion follows from the fact that the sheaf A×− is soft. Hereis a softened proof: the closed form µ above has a multi-valued primitive φ(a function on the simply connected cover X of X) whose branches differ byconstants. We may suppose that A defined by a star product. Let a be thesection with total symbol φ: although a is only defined on X, ad a is welldefined, and δ is the leading term of ad a+ sLog Ad b.
If U is an automorphism, its symbol is the closed 1-form ω homogeneous ofdegree 0 corresponding to δ = σ−1(LogU). The exponent of U is the coefficients = ρyω ∈ C of the vertical part sdrr (if X is not connected, it is a locallyconstant function)number s in 20
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4.7 Automorphisms preserving a subprincipal symbol oran involution
Proposition 21 Let A be a symplectic sub-algebra1) An automorphism (resp derivation) preserving a subprincipal symbol is aninner automorphism (resp. inner derivation).2) An automorphism U which preserves an anti-involution J is of the form AdPwith P ∈ A×−, JP = P−1; in particular its symbol is 0. A derivation preservingJ is an inner deerivation ad a with Ja = −a.3) Two subprincipal symbols are conjugate.
Proof 1) If U preserves a subprincipal symbol, then obviously U − Id is ofdegree < −1 so by proposition 20 U is an inner automorphism, of the formAd (1 + P ) with degP ≤ −1.
2) Let J . If U is an automorphism preserving J , it is an inner automorphismof the form Ad a with σ(a) = 1. The condition U = JUJ i.e. Ad a = Ad Ja−1
implies Ja = ca−1 with c a constant, so c = 1 since σ(a) = 1.
3) If sub ′ = sub + δ are two subprincipal symbols, δ is a Poisson vector field(of degree −1) so it is the leading term of of a derivation D, and sub ′ = sub eD.
Note that 1) and 2) are incorrect for deformation algebras. 3) holds fordeformation algebras, because the Poisson derivation, difference between twosubprincipal symbols, is still the leadig term of a star derivation os degree −1(not necessarily inner); but is less useful because as we will see later (ref) twosymplectic algebras possessing an involution or a subprincipal symbol are iso-morphic, but this is not the case for deformation algebras. We will see laterthat the base point algebra from Fedosov’s construction in involutive.
4.8 Fourier integral operators
4.8.1 As functional operators.
1) Fourier integral operators, and more generally Fourier integral distributions,were precisely introduced and described by L. Hormander [82, 46], M.Sato,T.Kawai, M. Kashiwara [108, 86]. A first, somewhat less precise definition,was proposed by V.I. Maslov [99], and J. Egorov [48] showed how they forciblyintroduce symplectic geometry in the picture.
In the main case, a Fourier F integral operator is associated to a homoge-neous symplectic map φ : T ∗X → T ∗Y (where X,Y are manifolds - the zerosections should be removed). They act on distributions and preserve microsup-ports (wavefront sets): if f is a distribution (or a microfunction) on X, Ff is adistribution on Y , and SS(Ff) ⊂ φ(SS(f)).
There is a notion of ”elliptic” Fourier integral operator (the ”principal sym-bol” must be invertible, as for pseudodifferential operators). If F is ellipticis has a parametrix, i.e. a Fourier integral operator G, associated to φ−1, in-verse to F mod smoothing operators (i.e. FG − Id and GF − Id have smoothSchwartz-kernels).
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When this is the case, if P is a pseudodifferential operator, its pushforwardQ = FPG is also a pseudodifferential operator, with principal symbol the push-forward σ(Q) = σ(P ) φ−1 (Egorov’s theorem).
2) Because pseudodifferential operators and Fourier integral operators pre-serve microsupports, mod smoothing operators they are local on the cotangentbundles (deprived of zero sections). This makes it possible to localize PDEproblems on the cotangent bundle.
Analogues or generalizations of Fourier integral transformations mod smoo-thing operators (or transformations defined by (EX − EY )-bimodules) are usedto describe isomorphisms between symplectic algebras.
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5 Classification.
5.1 Hochschild cohomology
We will use repeatedly some elements of Hochschild cohomology.If P ∈ Dk wedefine δP ∈ Dk+1:
δP (f0, . . . , fk) = f0P (f1, . . . , fk − P (f0f1, f2, . . . fk) . . .
+ (−1)k−1P (f0, . . . ; fk−1fk) + (−1)kP (f0, .., fk)fk+1 (20)
e.g. if B is a star product and P ∈ D1 is of negative degree, the top order termof eP∗ B−B is −δP , i.e. ePB(e−P f, e−P g)−B(f, g) = P (fg)−−P (f)gfP (g)+terms of degree < −N
R ∈ D2 (k = 1) is of negative degree, δR = 0 is the leading (linear) term inthe condition ensuring that B +R is associative).
Theorem 22 1) δ is a differential on⊕Dk i.e. δ2 = 0. We have δP = 0 if
P is of order 1, in particular if it is the antisymmetric k-differential operatordefined by a multi-vector.
2) The canonical injection∧TX →
∑Dk, taking a k-multi-vector to the
antisymmetric k-differential operator it defines, is a quasi-isomorphism, i.e. itinduces isomorphisms on the cohomology ker δ/Im δ.
proof: for the general case we refer to the literature [40]. We will only needthe cases k = 1, 2, for which we give a short, although not very illuminatingproof
If k = 1 δP = 0 i.e. P (f, g) = fP (g)−P (fg) +P (f)g = 0 for all f, g meanstat P is a derivation (vector field).
Let k = 2. If P is of total order N , and PN (ξ, η) is its symbol, a homogeneouspolynomial of degree N in ξ, η with smooth coefficients, δP = 0 implies
PN (η, ζ)− PN (ξ + η, ζ)−+PN (ξ, η + ζ)−−PN (ξ, η) = 0 (21)
Taking three successive derivatives with respect to ξ, ζ, η gives, for η = 0
(∂ξi − ∂ζi)∂ξj∂ζk = 0 (22)
We may dismiss the case N = 1 (P = C = δC, C a constant), and N = 1(δP = 0⇒ P = 0).
If N = 2, PN is of the form PN (ξ, η) = Aξ.ξ + Bξ.η + Cη.η with A,Csymmetric matrices. (22) gives −Aξ.(ξ+2η)+Cζ.(ζ+2η) = 0 so A = C = 0: P2
is bilinear. If P2 is bilinear symmetric, we have P2 = 12δQ with Q(ξ) = P (ξ, ξ).
Note that δQ is always symmetric, so a P2 bilinear antisymmetric is not acoboundary.
(22) implies that the pij = ∂ξ∂etajp are polynomials of ξ + η; the 1-forms∑i Pijdξj are closed, so the primitives are of the form ∂ξjp = pj(ξ + η)− qj(η),
i.e. dηp = ω(ξ + η)− ω′(η).
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Then we have dηω(ξ + η) = dηω(η) which implies that both ω and ω′ haveconstant coefficients; if N > 2 they vanish. Integrating one last time withrespect to η shows that PN is of the form a(η) − b(ξ + η) + c(ξ), and (21)requires a = b = c i.e. PN = δa.
By induction on N we get theorem 22 (for k=2).
5.2 Non commutative cohomology
In this section we recall the elementary resuls of noncommutative cohomologythat we will use (for more information see [69]). Let X a manifold (or a para-compact topological space) and G a sheaf of groups on X (e.g. the sheaf ofcontinuous functions with values in a topological group G or the sheaf of sec-tions of a group bundle on X). We denote H0(X,G) = G(X,G) the set of globalsections of G over X : this is a group.
We denote H1(X,G) the set of equivalence classes of cocycles
uij ∈ G(Xi ∩Xj ,G) such that uijujk = uik
associated to open coverings X =⋃Xi ; two cocycles are equivalent if, after
a suitable refinement of the covering, we have uij = uiu′iju−1j for some family
ui ∈ G(Xi,G).H1(X,G) classifies the set of isomorphy classes of G principal homogeneous
right G torsors, i.e. sheaves P on X, equipped with a right action of G, locallyisomorphic to G considered as a right G-sheaf : if P is a torsor, it is locally trivial,i.e. there exists an open covering X =
⋃Xj and isomorphisms Uj : P → G
over Xj . The transition map Uij is an isomorphism of G → G over Xi ∩ Xj ,commuting with right multiplications: Uijf = uij .f where uij ∈ G(Xi ∩ Xj)is the section Uij(1). The family (uij is a cocycle (with coefficients in G) , i.e.uik = uijujk over Xi ∩ Xj ∩ Xk. Changing the local trivialisations (Vif =viUif with vi ∈ G(Xi) leads to the new equivalent cocycle vij = Viuijv
−1j , and
H1(X,G) is the inductive limit of the sets of equivalent classes of cocycles forfiner and finer coverings.
A torsor is also used to twist left G sheaves : if E is such a sheaf, i.e. thereis a sheaf-group action G × E → E (for each open set U a group action G(U)×E(U)→ E(U) compatible with the sheaf restriction maps), the twisted sheaf isEP = P ×G E . In particular the sheaf of automorphisms of P is isomorphic tothe twisted group GP = P ×Ad G G which acts on the left on EP .
Proposition 23 Let0→ A u
→Bv→C → 0
be an exact sequence of sheaves of groups on X, with A normal in B. Thenthere is a exact cohomology sequence ;
0→ H0(X,A)→ H0(X,B)→ H0(X,C)→ H1(X,A)→→ H1(X,B)→ H1(X,C) (23)
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Th second and fith arrows are defined bt u, the third and sixth by v, the middlearrow (coboundary) is defined below. The sequence is exact in the followingsense :
i) it is exact at the first three places (the H0 are groups); the H1 are pointedsets. and (obviously) the range of one map maps to the next basepoint.
ii) The group H0(X,C) acts on the set H1(X,A), and its orbits are the fibersof the map H1(X,A)→ H1(X,B). The action is given by c · (aij) = (biaijb−1
j )if c is a global section of C, and bi ∈ B(Xi) a lifting of c to B over a fine enoughcovering Xi (we have biaijb−1
j ∈ A since bi = bj mod. A and A is normal in B).Two cocycle (aij , (a′ij) have the same image in H1(X,B) iff a′ij = biaijb
−1j for
some family bj ∈ B(Xj) (if the covering (Xj) is fine enough); then the cj = v(bj)patch together into a section c ∈ C(X).
iii) If β ∈ H1(X,B) maps to the base point, it can be defined by a cocyclebij which maps to the trivial cocycle cij = 1, i.e. bij ∈ A
More generally any element β ∈ H1(X,B) defines a torsor Pβ and a twistedsheaf of groups Bβ = IsomB(Pβ , Pβ = Pβ×AdBB, a normal subsheaf of groupsAβ = Pβ ×AdB A ⊂ Bβ , and a quotient Cβ = Bβ/Aβ = Pβ ×AdB C. One seesimmedialely that two elements β, β′ ∈ H1(X,C) iff the Bβ-torsor Isom (Pβ′ , Pβmaps to the base-point of H2(X,Cβ (the trivial torsor). The fiber of the mapH1(X,B)→ H1(X,C) is the image of H1(X,Aβ) in H1(X,C).
A sheaf of groups G on X is soft if any germ of section over a closed setF ⊂ X is the germ of a global section, i.e. for any section s ∈ G(U), with Uan open neighborhood of F , there exists a global section s′ ∈ G(X) such thats = s′ in some neighborhood V , F ⊂ V ⊂ U). G is flabby if any section onany open set extends to the whole of X. Soft is a weaker condition than beingflabby, but occurs often in real analysis - e.g. the sheaf of continuous or smoothsections of a vector bundle is soft, because there are partitions of 1, but it is notflabby unless X is discrete.
Proposition 24 If G is soft, then H1(X,G) = 1
In other words if G is soft, any G-torsor P has a global section, all 2-cocyclesare equivalent to the trivial cocycle uij = 1. In fact a locally soft sheaf as Pis soft : indeed let (Yi)i∈I be a locally finite closed coverig on X such that foreach i, P is trivial, hence soft, in some open neighborhood of Yi. Then is s is agerm of section along a closed set F , it extends as a germ to F ∪ Yi; since (Yi)is locally finite, F ∪
⋃j∈J Yj is closed for any subset J ∈ I, so by Zorn’s lemma
s extends to the whole of X.
In this paper the noncommutative cohomology sequence stops there, andwe will not use higher cohomology objects Hj , j ≥ 2 whose definition is moreelaborate. Exact sequences concerning non commutative cohomology as abovewere introduced by J. Frenkel [65, 66]. See also [69].
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However, the sheaves we will work with, such G = AutA,A, A×0 have acompleted descending filtration for which grG is commutative, and possiblysoft. This will make it possible to push the exact sequence (23) a little further.
If A,B,C are commutative, the higher cohomology groups Hk, k ≥ 0 arewell defined commutative groups: e.g. Hk(X,G) is the inductive limit forfiner and finer coverings of the set of cocycle (k-cochains f = (fi0,...,ik ∈G(Xi0,...,ik) such that the (k + 1)-cochain df vanishes, with dfi0, . . . , ik+1 =∑
(−1)jfi0, . . . , ij , . . . , ik+1, mod. coboundaries dg). For more details we referto the literature f, e.g. [11, 12, 69, 70].
We will use the long cohomology exact sequence in that case, up to order 2:if 0 → A → B → C → 0 is an exact sequence of commutative group sheaves,there is a long exact sequence of homology groups:
0→ H0(X,A)→ H0(X,B)→ H0(X,C)→ H1(X,A)→ H1(X,B)→ (24)
→H1(X,C)→ H2(X,A)→ H2(X,B)→ H2(X,C)→ H3(X,A)→ . . .
(exact means that the range of on arrow is the kernel of the next).
5.3 Symplectic algebras are locally isomorphic
Theorem 25 Let Σ be a symplectic cone with Poisson bracket c. Two staralgebras over Σ (with this same Poisson bracket c are locally isomorphic. Thesame holds for deformation algebras over a symplectic manifold.
We will use the following result (which holds for any star algebra):
Lemma 26 Any star product is equivalent to a star product of the form f ∗g =fg + 1
2f, g+ . . . 4
Proof: let Bf, g) = fg + B1(f, g) + . . . be a star product. Associativity writesB(B ⊗ 1)−B(1⊗B); there is no term of degree 0, an the term of degree −1 is
B0(B1(f, g),h) +B1(B0(f, g), h)−B0(f,B1(g, h))−B1(f,B0(g, h)) =B1(f, g)h+B1(fg, h)− fB1(g, h)−B1(f, gh) = 0
Thus δB1 = 0: B1 is a Hochschild cocycle, so there exists a differential operatorP homogeneous of degree −1 such that B1−δP = C is a bivector (antisymmetricbidifferential operator of order (1, 1)). Recall that eP∗ B is defined by eP∗ B(f, g) =eP (ep−Pf, ep−Pg): we get eP∗ B = B0 + B′1 + . . . with B′1 = B1 − δP = C.Since the Poisson bracket has not changed we have necessarily C(f, g) = 1
2f, g.
Proof of the theorem : Suppose we have two star products B,B′ with thesame symplectic Poisson bracket c. By Lemma26 we may suppose that both
4Much of the literature requires B1 = 12.. in the definition star products. We did not
do it here because it is not true for very natural laws such as the normal composition law ofdifferential operators.
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begin by f ∗ g = fg + 12f, g. . . . , so B′ = B + R with R of degree N ≤ −2
(R ∈ D2). We will show that on any disk-neighborhood V of a point the twoproducts are equivalent. (the exact condition is H2(V,C) = 0 for deformationalgebras, H2
( V,C) +H1(V,C) = 0 for symplectic algebras).We will use repeatedly (inductivly) the following result;
Lemma 27 Let B be a star product, and P a formal differential operator ofdegree −N . Then
(i) The leading term of ep∗B −B (of degree −N) is −δPN .
(ii) If P is a vector field, the leadig term for the error in the bracket law,(eP − 1)(B −tB), of degree −N − 1, is −[C,P ] (the Nijenhuis-Schoutenbracket - C denotes the Poisson bracket of B, of deree −1).
Proof: we have eP (f, g) = B(f, g)+PB(f, g)−B(Pf, g)−B(f, Pg)+ lower orderterms. The leading term in the error, of degree −N , is P (fg)− P (f)g − fP (g)as announced.
This vanishes if P is a vector field. In this case eP behaves as a diffeomor-phism; we have for the leading term of commutators we get eP ([e−P f, e−P g])−[f, g] = PN ([f, g)− PNf, g − f, PNg i.e. the error is [PN , C] = −[C,PN ].
We will kill the leading part RN of the error R in two steps.We first kill the antisymmetric part. We put β(f, g) = B(f, g) − B(g, f)),
ρ(f, g) = R(f, g) − R(g, f). ρ is an antisymmetric cocycle, so is is a bi-vector(bidifferential operator of order 1).The Jacobi identity for brackets is
∑3(β +
ρ)(f ; (β + ρ)(g, h) = 0 where∑
3 is the sum over cyclic permutations of f, g, h.The leading part for the error is then∑
3
f, ρN (g, h)+ ρN (f, g, h = 0
i.e. [C, ρN ] = 0.Now if the Poisson bracket C is symplectic, it is isomorphic to the exterior
differential d. So since ρN is closed ([C, ρn] = 0) there exists a vector field PN−1
homogeneous of degree 1−N < 0 such that ρN = [C,PN−1].If we set B” = e
PN−1∗ B′−B the leading part R”N of B”−B is now symmetric.
By theorem22 it is of the form δP for some formal differential operator of degree−N , and eP∗ B” − B is of degree ≤ −N − 1. By induction, we get an operatorP =
∏Pk with degPk → −∞ such that eP∗ B
′ = B.
Remark If the Poisson bracket is real (or pure imaginary) there exist canon-ical local homogeneous coordinates, i.e. homogeneous functions xj - of degree 0or 1, or all of degree 1
2 , and cij = xi, xj is constant. Then it is easy to constructby successive approximations elements Xj such that σ(Xj) = xj , [Xi, Xj ] = cij(or ~cij in the deformatioon case). One recovers theorem5.3 again by sucessiveapproximation, or using the functional calculus of section2.4.
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5.4 Classification
Theorem 5.3 helps for the classifiction of symplectic algebras (or deformationalgebras): let Σ be a symplectic cone with basis X (resp. X a symplecticmanifold). We denote Alg Σ or AlgX the set of equivalence classes of staralgebras (resp. deformation algebras) with the given Poisson bracket.
If A,B ∈ AlgX they are A,B locally isomorphic, so If Xi is a sufficiently finecovering of X there exist isomorphisms Ui : B → A over Xi defining a 2-cocycleUij = UiU
−1j with coefficients in AutA. We will denote δ(B,A) the element
of H1(X,AutA) it defines.(the corresponding torsor is the sheaf Isom (B,A) oflocal isomorphisms from B to A; conversely the algebra defined by a torsor P isthe sheaf of A-automorphisms of P ). Clearly two algebras B,B′ are isomorphiciff δ(B′,B′) = 0, and this is equivalent to δ(B′,A) = δ(B,A).
Theorem 28 If A is a symplectic algebra, the map B 7→ δ(B,A) is a bijectionAlgX → H1(X,AutA)
Note that this map depends on A. We will see below that H1(X,AutA) isa commutative group, which acts freely on AlgX , and the element δ(B′,B) isreally the difference of the classes of B and A.
The theorem does not ensure the existence of an algebra A; this will beproved later, together with Fedosov’s very elegant description of the classifica-tion.
Let us push our analysis a little further. We make two separate cases.
5.5 Classification of symplectic algebras
Proposition 29 Let A be a star algebra on a symplectic cone Σ: then there isa canonical bijection chA : Alg Σ = H1(X,AutA)→ H2
hom(Σ).
Proof : we have seen (proposition20 that there is an exat sequence
0→ A×− → AutA → ω0 → 0,
whereA×− is the sheaf of invertible elements with principal symbol 1, ω0 the sheafof closed forms homogeneous of degree 0. We have H1(X,ω0) = H2
hom(Σ) (bydefinition) and this in turn is isomorphic to ∼ H2(X,C)⊕H1(X,C) (proposionHhom). The map H1(X,AutA) → H1(X,ω0) comes from the cohomologyexact sequence (prop. 23; it is a bijection because A×− is a soft sheaf.
Here is a more detailed proof.The map is injective (prop. 23); in fact if two algebras B,B′ have the cocycles
the same image in H1(ω0) the image of δ(B′,B) vanishes, i.e. B′ can be definedby a cocyle with coefficients in B×0 . Now B×0 is soft, so B and B′ are equivalent.
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The map is onto: let αij be a 2- cocycle with coefficients in ω0. This can belifted to a family Uij of automorphisms. Set Uijk = UijUjkUki: Uijk − 1 is ofdegree < −1 since αij (its symbol is αij+αjk+αki = 0). Suppose it is of degree−N − 1 < −1; then the symbols αijk of Uijk are closed 1-forms homogeneousof degree −N , and form a 2 cocycle (we have UijkUiklU−1
ijl Uij = UjklU−1ij = 1
so αijk −αijl+αikl −αjkl = 0). Now we have seen that homogeneous forms ofdegree 6= 0 have no cohomology, so αijk is a coboundary: αijk = βij +βjk−βki.If uij is a family of automorphisms with symbol βij (uij = Ad (1 + aij) whereaij is of degree −N and dσ−N (aij) = βij , the U ′ij = uijUij patch togethertoorder −N − 2 (i.e. U ′ijU
′jkU
′ik−1 − 1 is of degree ≤ −N − 2). By succesive
approximations we finally get a cocycle Uij with symbol αij .
Complement
Proposition 30 Two syplectic star algebras possessing a subprincipal symbol(resp. an anti-involution) are isomorphic.
If A,B are two symplectic star algebras possessing subprincipal symbols, itfollows from proposition21 that they are locally isomorphic, and that the sheafIsom sub(A,B) of local isomorphisms preserving sub is a torsor under A×−. Thisis soft, because A×− is soft, so it has a global section, i.e.there exists a globalisomorphism A → B preserving sub .
This applies also to anti-involutive algebras : such algebras possess a subso they are isomorphic, and two anti-involutions are always globally conjugate(proposition15). (note that the sheaf Aut JA of local autoisomorphisms preserv-ing an antiinvolution J is soft : for such an automorphism U there is a uniquea ∈ A−1 such that U = Ad ea, Ja = −a, and these obviously for a soft sheaf).
5.6 Classification of symplectic deformation algebras
Proposition 31 Let A be a deformation algebra on a symplectic manifold X.There is a canonical bijection chA : AlgX → H2(X,C[[~]]).
Proof : we have seen (theorem17) that any automorphism of A is locally inner(AutA = IntA), and there is an exact sequence (19)
0→ C[[~]]→ A×0 → AutA → 0
where A×0 is the sheaf whose sections are pairs ϕ, f with f ∈ A×0 invertible,ϕ ∈ O(0); eϕ = σ(f) (“logarithms” of invertible elements of A0).
Because C[[~]] is commutative, the “cohomology exact sequence” (23) ex-tends one step further, and the resulting map ch : AlgX → H2(X,C[[~]]) is abijection because A×0 is a soft sheaf.
Since we did not define the homology objects of order ≥ 2, here is a moredetailed proof.
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First if αij = Ad aij is a cocyle with coefficients in AutA, the aij ∈ A×0 patchtogether mod. “constants”: aijajk = ecijkaik. The family (cijk) is a cocycle withcoefficients in C[[~]]. This defines the map ch : H1(X,AutA)→ H2(X,C[[~]]).
The map ch is injective : If two cocycles Ad (α = (aij),Ad (α′ = (a′ij))have equivalent images cijk, c′ijk, the difference c′ijk − cijk is a coboundary :c′ijk − cijk = cij + cjk − cik for some family cij (for a sufficiently fine cover-ing). Replacing ; then, with αij = (cij , ecij ), we have (αijaij) is a cocycle withcoefficients in A×0 .
Remark 5 Let A be a symplectic deformation algebra. Aut ~A is the sheaf ofautomorpisms. It is useful to compare it to the sheaf AutA af all star algebraautomorphisms. If U is such an automorphism (not necessarily preserving ~),it still preserves the center C[[~]], and its restriction to C[[~]] is tangent to theidentity : we have U(~) = ~ +O(~2), because U preserves symbols. We denoteAut 0(C[[~]]) the group of such automorphisms. Any u ∈ Aut 0C[[~]] is of theform eδ where δ = Logu is a derivation of the form δ = φ(~)~∂~.
In fact there is an exact sequence of sheaves
0→ Aut ~B → AutA → Aut 0C[[~]]→ 0 (25)
i.e. any automorphism u ∈ Aut 0C[[~]] (such that u(~) = ~+O(~2) can be lifted,at least locally. Indeed u∗A is another symplectic deformation algebra and itis locally isomorphic to A (we will see below (section6) that if the symplecticform of X is not exact, the push-forward u∗A is never globally isomorphic to Aif u 6= Id ).
Similarily we have there is an exact sequence for derivations of degree 0 :
0→ Der ~A → DerA → Der C[[~]] .
5.7 Algebras of pseudo-differential type
1. Let X be a cone. A pseudodifferential operator on X (or operator of pseudo-differential type)is a formal operator P =
∑Pm−k where the componant Pm−k
homogeneous of degree m− k is of order ≤ k.In a local coordinate system x1, . . . xn homogeneous of degree 1, P is of ψD
type iff P =∑aα(x)∂αx where the aα are all symbols of degree ≤ m (aα∂α is
then of degree ≤ m− |α| and the sum converges in D).The total symbol of P in this coordinate system is
p(x, ξ) =∑
aα(x)ξα = e−x.ξP (ex.ξ)
This is a formal power series in ξ with coefficients in O. The principal symbolis
σmP = pm(x, ξ) =∑
σm(aα)ξα
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Pseudo-differential operators obviously form a subalgebra EX ⊂ DX (a sub-sheaf).The total symbol of a product is given by Leibniz’ rule; in particularσm+m′(P Q) = σm(P )σm(Q).
The total symbol of P depends on the choice of a coordinate system (notthe fact it is of pseudo-differentiel type); the principal symbol does not, it is asmooth function on T ∗X, or rather a jet of infinite order along the zero sectionξ = 0, homogeneous of degree m if we define ξ as homogeneous of degree 0.EX is a star algebra (the jet of infinite order of -) on the cone Σ = T ∗X wherehomotheties are given by λ(x, ξ) = (λx, ξ)(= λhλ∗(x, ξ)) (in the last notationhλ is the homothety x 7→ λx, hλ∗ its extension (pushforwrd) to covectors)
2. Pseudo-differential star productsLet Σ be a cone. We will say that a formal differential operator P ∈ D is of
pseudo-differential type (or that it is a pseudo-differential operator) of degreem if P =
∑Pm−k with Pm−k homogeneous of degree m− k and of order ≤ k.
The deformations translation is P =∑
~k−mPk with Pk of order ≤ k.These operators obviously form an algebra EΣ (a sheaf).
If Σ = Rn − 0 with all x-variables homogeneous of degree 1, this meansthat
P =∑
aα(x)∂α with aα ∈ Om, a symbol of degree ≤ m
The total symbol of P is∑aα(x)ξα, which is a jet of infinite order (formal
Taylor series) along ξ = 0 in T ∗Σ; the principal symbol is the leading part,homogeneous of degree m (counting that xi is of degree 1, ξi of degree 0).
The product is again given by the Leibniz rule, hence the name. (This islocally isomorphic to the algebra of jets of standard pseudo-differential operatorson Rn along the x = 0 fiber, except we have exchanged the x and ξ variables.)
Definition 32 We will say that a star product B is of pseudo-differential typeif for any symbol a, (of degree m) the left mutliplication operator La : b 7→ a ∗ bis of pseudo-differential type (of degree m).
Then the map a 7→ La is an embedding A → E . Recall that such an embeddinghas a (formal) geometric support u; this is a (homogeneous) Poisson map T ∗Σ→Σ (or rather T ∗c Σ→ Σc).
Theorem 33 1) Any star product is equivalent to a pseudo-differential starproduct. Equivalently any star algebra on Σ can be embedded in EΣ
2) Two such embeddings are conjugate by an isomorphism. More preciselytwo embeddings with the same geometric support are conjugate trough a P ∈ E,and this is unique if we prescribe P (1) = 1.
Before we prove the theorem, we need some geometric preparation.Lemma? If X is a symplectic manifold, F a foliation, we denote OF the sheaf
of smooth functions constant on the leaves of F . F is a Poisson foliation if OF isa Poisson algebra (i.e. locally f, g is constant on te leaves if f and g are. Thenthere is an orthogonal foliation F⊥ : its tangent distribution is TF⊥, spanned
45
by the Hamiltonian fields Hf , f ∈ F (this satisfies the Frobenius integrabilitycondition since [Hh, Hg] = Hf,g and f, g ∈ F . OF⊥ is the commutator ofOF , i.e. the sheaf of functions φ such that (locally) f, φ = 0 for all f ∈ OF .
The level sets of a map u as above define a Poisson foliation (they aresmooth because u is a fibration). The orthogonal foliation is in fact a fibration(the tangent spaces of fibers of u are linear complements of TX ⊂ T (T ∗X) sotheir orthogonals are linear complements of T (T ∗X)⊥ = T (T ∗X). It defines aPoisson map u⊥ : T ∗X → X
Lemma 34 Let X be a Poisson manifold. Then there exists a (formal) Poissonmap u : T ∗X → X along the zero section (T ∗X is equipped with its usualsymplectic structure).
If u1, u2 are two such maps, there is a unique isomorphism u of T ∗X suchthat u2 = u1 u, u|X = Id
Proof 1) by successive approximationsor: outflow of the Hamiltonian of a suitable “phase function”2) there is a unique map u such that u|X = Id which takes the Hamiltonian
of f u1 (and its flow) to that of f u2 for f a smooth function on X (the flowof the Hamiltonians lie in the orthogonal foliations); this map u is symplectic.
Note
Proof of theorem33 Let A be a star algebra with Poisson product C. We willsay that a “differential operator” U : A → E (locally of the form f 7→ P (f u)is a homomorphism of order N if R(f, g) = UfUg − U(f ∗ g) is of degree≤ deg f + deg g −N (R = B(U ⊗ U)− UB of degree ≤ −N).
R can be viewed as the restriction of a formal differential on E to OF (thefunctions constant along the fibers of u).
For N = 1 this just means Uf = f u+ terms of lower degree. Lemma 34shows that we can choose u so that it preserves Poisson brackets, so an thereexists an embedding of order N = 2, because both the product law of A andthat of E are equivalent to a law if the form f ∗ g = fg + 1
2f, g).Associativity of both products gives
UfUgUh− U(f ∗ g ∗ h) = UfR(g, h) +R(f, g ∗ h) = R(f, g)Uh+R(f ∗ g, h
for the leading term (homogeneous of degree −N this gives
fRN (g, h)−RN (fg, h) +RN (f, gh)−RN (f, g)h = 0 (26)
i.e. RN is a Hochschild cocycle.The commutator error is Γ(f, g) = R(f, g) − R(g, f). For its leading term
we get ∑3
ΓN (f, g, h+ f,ΓN (g, h) = 0 (27)
(∑
3 denotes the sum over cyclic permutations of f, g, h).
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Suppose U is an embedding of order N ≥ 2. We construct an embedding U ′
of order N + 1, such that U ′ − U is of degree ≤ −N + 1 in two steps.We first get rid of the antisymmetric part of R : (26) shows that ΓN is an
antisymmetric Hochschild cocyle, i.e. a bivector, and (27) means that we have[c,RN ] = 0, with c the Poisson bracket of T ∗ Σ). Since the formal leaves ofu are contractible, there exists a unique vector field p homogeneous of degree−N + 1 (vanishing on X) such that ΓN = [c, p]. Then U” = (1 + p)U is anotherembedding of order N , and for this the error RN” is symmetric.
Suppose finally that RN is symmetric : then by 22 there exists a differentialoperator P homogeneous of degree −N (not necessarily of pseudo-differentialtype) P such that ΓN = δP i.e. ΓN (f, g) = fP (g − −P (fg) + P (f)g. ThenU ′ = (1 + P )U” is an embedding of order N + 1 (R′ is of degree ≤ −N − 1)/
Thus by succesive approximations we get an embedding U : A → E .
Uniqueness. Let U1, U2 be two embeddings, with geometrical supports u1, u2.By theorem 33 there exists a (unique) (formal) symplectic map u taking u1 tou2. Now there exists (4.8) a invertible “Fourier integral operator” F with canon-ical relation the graph if u, so FU2F
−1 has the same geometric support as U1
(Note: formally Fourier integral operators as above are of the form eP whereP ∈ E is of degree 1, and the symbol σ1(P ) vanishes of order 2 on Σ (i.e. forξ = 0, so that the flow of HP fixes Σ)).
Suppose now that U1 and U2 have the same geometric support u, and thatU1 − U2 is of degree −N (N ≥ 1). Then the error term r =. Ite leading termrN os a Poisson vector field, and there exists a unique φ homogeneous of degree−N − 1, vanishing for ξ = 9, such that rN = Hφ. Then p ∈ E , σ−N+1(p) = φ,epU1e
−p − U2 is of degree −N − 1. By successive approximations we get p (ofdegree ≤ 0 such that U2 = epU1e
−p.
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6 Fedosov Connections.
In [60], Fedosov gave a very elegant and geometric description for the existenceand classification of semi-classical algebras : a deformation algebra B over areal symplectic manifold X can be embedded as the subalgebra of flat sectionsof a suitable “Fedosov connection” ad∇ in a “universal” star algebra W overthe tangent bundle TX (the definitions are recalled below); the curvature of ∇is a closed central 2-form with leading term h−1ωX , whose cohomology classdetermines B up to isomorphism; in particular this construction singles out abase-point (the algebra whose curvature is exactly h−1ωX). This descriptioncan easily be adapted to symplectic algebras (more generally to Poisson conesof constant rank [20]). We will use this adapted description, and recall here howit works. In fact we give here a direct proof of the embedding theorem.
6.1 Valuations and relative tangent algebra WLet Σ be a Poisson cone. On the tangent bundle TΣ each fiber TxΣ inheritsof a Poisson bracket cx with constant coefficients; there collection is a Poissonbracket cT which is homogeneous of degree −1 with respect to homotheties.
We will denote W the sheaf generated by homogeneous functions on on TΣthat are polynomial in the fibres: if we choose homogeneous local coordinatesx = (x1, . . . , xn) on Σ and ξ = (ξ1, . . . , ξn) denote the corresponding tangentcoordinates, a section of W is locally a finite sum of homogeneous monomialsaα(x)ξα. The degree of such a monomial is
deg aα ξα = deg aα +∑
αkdeg ξk (28)
The degree valuation is
p(f) = inf(−deg fα) if f is sum of homogeneous monomials fα (29)
We define the weight (or order) of a homogeneous monomial aα(x)ξα as
w(aαξα) = −deg (aαξα) +|α|2
(30)
The order valuation is
w(f) = inf w(fα) if f is sum of homogeneous monomials fα (31)
Thus W is equipped with two valuations p ≤ w.
The Poisson brackets cx define fiberwise a Weyl product on W:
f ∗ g (x, ξ) = exp12cx(∂ξ, ∂η) f(x, ξ) g(x, η) |η=ξ (32)
Since we will be dealing simultaneously with several product laws, we will adoptthe following notations, to avoid confusions if need be:
∗ or ∗W , ∗A for the star-product, × (or no sign) for the usual product(33)
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Both p and w are valuations for this product, i.e.
p(f ∗ g) ≥ p(f) + p(g), w(f ∗ g) ≥ w(f) + w(g) (34)
The graded algebra gr pW =⊕
k(Wp≥k/Wp>k) is commutative (its productis the usual product since p([f, g]) ≥ p(f) + p(g) + 1).
The graded algebra grwW is not commutative; in fact the star-product ∗ ishomogeneous for w, i.e. if f, g are sums of monomials of pure weight k resp. k′,then f ∗ g and [f, g] are sums of monomials of pure weight k + k′.
Definition 35 We denote WΣ (or W if there is no confusion) the completionof W for the valuation w.
Note that W is a set of formal functions on TΣ, equipped with a star-product.W is obviously functorial, i.e. if u : Σ→ Σ′ is a morphism of Poisson cones
(a smooth homogeneous map Σ→ Σ′ compatible with the Poisson brackets, i.e.u∗cΣ = cΣ′ or f u, g u = f, g u for any smooth functions f, g on Σ′)then u∗ : WΣ′ → WΣ (f 7→ f Tu) is a homomorphism af algebras, both forthe star-product and the ordinary product.
For deformation algebras on X, we will rather use, as Fedosov, the comple-tion Wh = Wh(X) for the valuation w of the sheaf on X of algebras, whosesections are polynomials f =
∑fk,α(x)hkξα; (the valuations p, w are defined as
above, (p(h) = 1, functions on X are homogeneous of degree 0 (p = 0)). Wh iscanonically a quotient of WΓ: a 7→ h−1, α 7→ 0. 5
6.2 Automorphisms and Derivations of WLet Σ be a symplectic cone (resp. X a symplectic manifold).
In this section we examine derivations and automorphisms of WΣ (resp.Wh). We only consider automorphisms U such that w(U − Id ) > 0.
Any derivation or automorphism preserves the center of W (i.e. OΣ, resp.OX((h))). We will say that D resp. U is an O-derivation (resp. automorphism)if it fixes the center (i.e. D or U − Id vanishes on the center).
Proposition 36 All O-derivations, resp. automorphisms are inner derivations(resp. inner automorphisms). More precisely, if D is an O-derivation, resp. Uan O-automorphism, there exists a unique section d resp. u ∈ W such thatd = 0, resp. u = 1 for ξ = 0, and D = ad d resp. U = Adu (i.e. Df = [d, f ],Uf = u ∗ f ∗ u−1.
5In the deformation case (X a Poisson manifold), the relevant cone is Γ = X×R×+ equipped
with the Poisson bracket hcX with h = a−1, a the canonical coordinate of R×+. We have
TΓ = TX × TR×+; the center of cWΓ is the commutative algebra, completion of C[a, α] for
the order valuation w, where (a, α) are the canonical coordinates of TR×+ (a = h−1, α the
corresponding tangent coordinate; w(a) = −1, w(α) = − 12
).
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Proof : this is obvious on the local model (Σ or X a graded vector space),where all O-derivations (resp. automorphisms) are inner, because ad d = ad (d−d(x, 0)),AdU = Ad U
U(x,0) . The uniqueness property just reflects the fact that
the center is OS resp. OX((h)). The global statement follows (uniquenessensures that objects constructed on a covering patch together).
Note that if U is an O-automorphism (w(U − Id ) > 0), D = LogU is welldefined, w(D) > 0, so D = ad d with some d such that w(d) > 0 and U = Aduwith u = ed; however d(x, 0) = 0 is not equivalent to ed(x, 0) = 1 (the set ofsections which vanish for ξ = 0 is not an ideal for ∗).
An immediate consequence of this proposition is that the sheaf (on Y = BΣor X) of O-derivations or O-automorphisms is soft .
6.3 Embeddings
Let A be a symplectic algebra over Σ (resp. B a deformation algebra over X).We will say that a homomorphism of algebras A → W (resp. B → Wh) is agood embedding if it respects the weight w (w(u) ≥ 0 i.e. w(uf) ≥ p(f)) andlocally, in any set of homogeneous coordinates xj we have u(xj) = xj + ξj + rjwith w(rj) ≥ p(xj) + 1 (in the deformation case we require u(h) = h). Thesecond condition does not depend on the choice of homogeneous coordinates.
Theorem 37 1) For any symplectic algebra A (resp. deformation algebra B)there exists a good embedding A → WΣ (resp. B → Wh).
2) If u1, u2 are two good embeddings there exists a unique O-automorphism Usuch that u2 = Uu1 (two good embeddings are conjugate).
Proof : locally the theorem is immediate: if (xj) is a set of local homogeneoussymplectic coordinates (xi, xj = cij = constant), a model good embeddingis f(x) 7→ uf = f(x + ξ). As mentioned it is often convenient to choose thexj , ξj homogeneous of degree 1
2 ; we may then use them only as intermediatetools since the monomials of WH must be homogeneous of integral degree.
If v is another good embedding, v(xj) = xj + ηj , we have [ηi, ηj ] = [ξi, ξj ] =cij because the xj are central in W, so by the universal property of the Weylalgebra there exists a unique algebra homomorphism U such that Uξj = ηj(Uf = f if f is central). Since w(ηj − ξj) > w(ξj), U respects the valuation w
and its unique continuous extension to W is bijective.This unique conjugacy statement is obviously also global.
In the general case there exists an open covering Σi of Σ (resp. Xi of X)and for each i a good embedding ui : A → W over Σi (resp. ...). For each pair(i, j) there is a unique O-automorphism Uij over Σi ∩Σj such that ui = Uijuj ,and this is a cocycle (UijUjk = Uik because UijUjkuk = Uikuk = ui). Sincethe sheaf of O-automorphisms (or derivations) is soft, (Uij) is a coboundary:UiUij = Uj for a suitable family (Ui) so that the Uiui patch together to producea global good embedding.
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Remark 6 Another way to state this result is the following: the completion ofO ⊗ A is isomorphic to W; the valuation used to define the completion is theunique valuation such that w(f ⊗ g) = p(f) + p(g), w(1⊗ f − f ⊗ 1) = p(f) + 1
2if 1⊗ f − f ⊗ 1 6= 0 i.e. f is not a constant.
Remark 7 In the above construction we did not require that the embedding upreserve the homogeneity valuation p. It is easy to show that there also existsa good embedding u which preserves p (see §4.5 below). When this is the casewe have σ(uf) = σ(f) u where u is a smooth homogeneous map TΣ → Σcompatible with the Poisson brackets, and tangent to the map (x, ξ) 7→ x + ξalong the zero-section of TΣ. It is still true that two such embeddings u1, u2
are conjugate: u2 = Uu1 with U an O-automorphism (w(U − Id ) > 0), but Ucannot preserve the homogeneity valuation p if the geometric maps u1, u2 arenot equal.
6.4 Vector Fields with Coefficients in WTo describe the other derivations or automorphisms, it is convenient to choosea homogeneous symplectic connection ∇s on Σ (resp. X). We can choose ithomogeneous of degree 0 (with respect to homotheties) and torsionless, but anysymplectic connection with coefficients in O0 ⊗ sp (degree ≤ 0) will do. ∇s
extends canonically to W, and if V is a vector field on Σ (homogeneous or withcoefficients in O), ∇sV is a derivation, both for the star-product and the usualproduct, (it respects both valuations p, w if V is homogeneous of degree 0).
We identify the Lie algebra of sections of sp(TΣ) with the sub-Lie algebra W2
of WΣ of homogeneous polynomials of degree 2 with respect to ξ: γ =∑γjkξjξk
(acting by ad : ad γf = [γ, f ]). Thus locally, if we choose homogeneous sym-plectic coordinates xj (xi, xj = cij = cst), we have
∇sf(x, ξ) =∑
dxj(∂f
∂xj+∑
[γijkξjξk, f ] ) (35)
with γ =∑dxi γijk(x)ξjξk homogeneous of degree 1 (ad γ of degree 0).
If D is a derivation of WΣ, its restriction to the center is a derivation of OΣ
i.e. a vector field V on Σ with coefficients in O, so we have D = ∇sV + ad d forsome sections d ∈ W.
It is useful to introduce, as Fedosov, “derivations with coefficients in W”.These are well defined because the symplectic Lie algebra sp is identified with thesubalgebra W2 ⊂ W. They form a Lie algebra VectW which is an extension ofthe Lie algebra Der cW of derivations of W. Its sections can be written ∇sV + d,where the rule for changes of coordinates for ∇sV is the rule for vectors withcoefficients in the Lie algebra sp(TΣ) = W2 ⊂ W. The bracket is
[∇sV1+ d1 , ∇sV2
+ d2] = [∇sV1,∇sV2
] +∇sV1(d2)−∇sV2
(d1) + [d1, d2] (36)
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where the first bracket is given by the rule for connections with coefficients insp = W2:
[∇sV1,∇sV2
]−∇s[V1,V2] = Rs(V1 ∧ V2) ∈W2 (37)
If V is a derivation with coefficients in W we denote adV the derivation itdefines. We have adV = 0 if and only if V is central (of the form ∇s0 +f, f ∈ O),and any derivation D ∈ Der W has a unique representative D = ad (∇sV + f)with f(x, 0) = 0.
6.5 Fedosov Connections
Let ΩΣ (resp ΩX) denote the exterior algebra of differential forms on Σ (resp.X) with coefficients in O (resp O((h))).
As for derivations we define connections with coefficients in W: such a con-nection is of the form ∇ = ∇s + γ with γ a 1-form with coefficients in W. Itacts on forms with coefficients in W by
ad∇f = ∇s(f) + [γ, f ]
The curvature is R = ∇2, identified with a 2-form with coefficients in W. Wehave (ad∇)2 = adR so ad∇ is flat (integrable) if and only if the curvature Ris central, i.e. a 2-form with coefficients in O (independent of ξ). We will thensay, as Fedosov, that ∇ is abelian.
Definition 38 1) Let (xj) be local coordinates on Σ, (ξj) the correspondingtangent coordinates. We denote ξ∨j the dual linear fiber coordinates on TΣ, suchthat [ξ∨i , ξj ] = δij.
2) We denote τ the 1-form with coefficients in W, dual to the canonical formof Σ with coefficients in TΣ (resp. TX):
τ =∑
dxjξ∨j (resp. τ =
1h
∑dxjξ
∨j ) (38)
τ is the unique 1-form with linear coefficients (in ξ) such that for any differentialform f with coefficients in W:
[τ, f ] =∑
dxj∂f
∂ξj(39)
We have p(τ) = −1, w(τ) = − 12 (τ is homogeneous of degree 1 and its coefficients
vanish of order 1 on the zero section). Also τ2 = 12 [τ, τ ] is the symplectic form:
τ2 = ωΣ (resp. h−1ωX) (40)
Definition 39 A Fedosov connection is a connection ∇ with coefficients in Wof the form
∇ = ∇s − τ + γ with w(γ) ≥ 0.
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Theorem 40 Let A be a symplectic algebra on Σ (resp. B a deformation alge-bra on X). Then1) For any good embedding u : A → W (resp. ...) there exists a unique abelianFedosov connection ∇ such that ∇u = 0. This is unique up to a central form.2) Conversely if ∇ is an abelian Fedosov connection, ker∇ ⊂ W is a symplectic(resp. deformation) algebra.
1) Locally, let us choose symplectic coordinates (e.g. homogeneous of degree12 ). The standard local embedding u0 : f(x) 7→ u0f = f(x + ξ) is killed by∇0 = dx − τ , and obviously this is the only Fedosov connection which killsit, with coefficients vanishing on the zero-section ξ = 0. If u is anothergood embedding, it is conjugate to u0 : u = AdU−1u0 so it is killed by theFedosov connection ∇ = ∇0 + γ with γ = U−1∇0(U); we have w(γ) > 0 sincew(U − Id ) > 0. Note that γ may not vanish for ξ = 0, but we can replace it byγ − γ(x, 0).
2) The converse is immediate: if∇ is an abelian Fedosov connection, the mapf ∈ ker ad∇ 7→ f(x, 0) is clearly one to one (for each f ∈ O there exists a uniquef ∈ W such ∇f = 0, f(x, 0) = f(x) because ∇ is “transversal” to the zero-section; further since∇ is a Fedosov connection, we have locally w(f−f(x+ξ) ≥w(f + 1)).
(The proofs for the deformation case are the same).
6.6 Fedosov curvature
Fedosov connections provide a one to one correspondence between symplecticor deformation algebras and 2-forms on Σ (resp. X):
Theorem 41 (Fedosov) 1) Any closed 2-form R = ωΣ+r on Σ with coefficientsin O (resp. ωX + r)..) is the curvature of an abelian Fedosov connection.2) Two abelian connections have the same curvature if and only if they areconjugate. They define isomorphic algebras if and only if their curvatures differby an exact form.
Proof : 1) For the sake of completeness, we repeat the proof of Fedosov’s, bysuccessive approximations, improving the weight w of the error: if∇ is a Fedosovconnection, the leading term (term of lowest weight w) of ad∇ is
ad τ : f 7→∑
dxj∂f
∂ξj(of weight − 1
2)
If ∇2 = R + ρN with w(ρN ) = N2 ≥ 0, one looks for a 1-form γN+1 such that
w(∇+ γN+1)2 −R) ≥ N+12 , i.e.
[τ , γN+1] = ρN + (w ≥ N) (41)
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Let Iξ denote the interior product: Iξf = ξj∂xjyf (the usual product, not thestar-product of W). Then if f =
∑dxi1 ...dxikfα(x) ξα we have
Iξad τ + ad τIξf = Lξf =∑
(|α|+ k)dxi1 ...dxikfα ξα
Since we have w([τ, ρN ]) > N−12 , equation 41 has a solution (in fact it has a
unique solution of pure weight N+12 killed by Iξ). By successive approximations
we get an exact solution - in fact there is a unique and canonical solution γ suchthat Iξγ = 0 (which implies [Iξ,∇] = 0).
Remark 8 The denomination “canonical” is abusive, because it depends infact on the choice of a torsionless homogeneous symplectic connection ∇s onΣ to begin with. In ambiguous cases we will refer to “the canonical Fedosovconnection associated with ∇s”.
2) Obviously if ∇1 = U∇U−1 and R = ∇2 is central, we have ∇21 = URU−1 =
R. The converse (i.e. the existence of U such that ∇1 = U∇U−1 if ∇ and∇1 have the same central curvature) is proved by successive approximation asabove.
Note that ker ad∇ = ker ad∇1 means that ∇1 = ∇ + γ with γ central,w(γ) ≥ 0; this implies R1 = R+ dγ, hence the last assertion of 2).
Remark 9 The canonical solution is in fact of weight p(ad∇) ≥ 0 (p(∇) ≥−1), so that the corresponding embedding u is associated to a geometrical mapu : TΣ→ Σ, preserving Poisson brackets and tangent to the map (x, ξ) 7→ x+ξ.
Remark 10 If A is a symplectic or deformation algebra, an embedding u de-fines a total symbol σU (f) = uf(x, 0) which is an isomorphism A → O (locallyin D×0 ), and a star-product Bu (σu(f ∗
Ag) = Bu(σuf, σug)). If we replace u by
Uu, U an automorphism of W, σu is replaced by PU,uσu for some well definedasymptotic operator PU,u ∈ D×0 (PUV,u = PU,V uPV,u), and Bu is replaced byPU,uBu ((PB)(f, g) = P B(P−1f, P−1g)).
The set of transition operators PU,u is a strict subset of D×0 ; for example wehave PU,u = 1 if U is even, i.e. U = 1 +
∑uα(x)ξα, α > 0 even). Likewise
the set of all total symbols σu is a strict subset of the set D×0 σu of all possibletotal symbols equivalent to σu, and the set of all corresponding star-productsBu is a strict subset of the set of all star-products equivalent to a given one;for instance one always has Bu(f, g) = fg + 1
2g, g + · · · . I will not describefurther here this special class of star-products. In fact I do not know if it hasbeen distinguished before.
6.7 Base-point
Fedosov’s construction gives a canonical base-point in the set of star-algebras,viz. the algebra corresponding to a connection with “trivial curvature” R = ωΣ
(resp. R = h−1ωX).
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Note that the Weyl algebra W has a canonical involution (and a sub-principalsymbol, fiberwise). The algebra A∇ is a sub-involutive algebra of W if ∇ re-spects the involution. If this is the case the curvature R also respects the involu-tion, i.e. it is odd (an odd power series h−1ωX +
∑h2k+1ωk in the deformation
case, R ∼ 0 in the symplectic case). Conversely if R is odd, Fedosov’s construc-tion obviously yields a connection ∇ which respects the involution. Thus thebase-point is involutive; in the symplectic case, it is the unique involutive alge-bra; in the deformation case we have seen that there are many other nontrivialinvolutive algebras.
For further use let us also note the following result: let Gh denote the groupof symbol preserving automorphisms of C((h)) (i.e. automorphisms U of C((h))with its standard commutative algebra structure, such that w(U(h) − h) > 0).Gh obviously acts on the set of deformation algebras on X (replacing h byU(h)).
Proposition 42 If X is compact, the action of Gh on h-Alg (X) is free
On any symplectic manifold X, a symbol-preserving isomorphism U pre-serves the center, hence induces an automorphism U ∈ Gh. Recall that thelocal model for deformation algebras is the algebra B0 of (jets along ξ = 0of) pseudodifferential operators on Rn × R which commute with h−1 = ∂
∂t
(X = T ∗Rn, ξ = h ∂∂x ). Clearly any U ∈ Gh lifts to AutB0 (e.g. U such that
U(x) = x, U( ∂∂x ) = ∂
∂x , U(t) such that [U( ∂∂t ), U(t)] = 1).The proposition states that if X is compact, an isomorphism which preserves
symbols, but not necessarily h, in fact fixes h.
Proof : if U ∈ Gh, it acts on Wh. If R is the Fedosov curvature of a deformationalgebra B, the curvature of U(B) is U(R). Now the leading term of R is h−1ωX .If U 6= 0 the leading term of U−1 is c hk ∂
∂h for some integer k ≥ 2 and constantc 6= 0, so that the leading term of U(R) − R is −c hk−2ωX ; this is 6= 0 if X iscompact (ωX is not a coboundary).
Therefore if U ∈ Gh and UB is isomorphic to B, then U(h) = h i.e. U = Id .Equivalently if U is a symbol-preserving isomorphism of B, it fixes the center(U(h) = h).
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7 Traces.
If A is an algebra, a trace on A is a linear form T : A → C such that T (ab) =T (ba) for any a, b ∈ A. If IA is a two sided ideal, a trace (relative to I) isa linear form I → C such that T (ab) = T (ba) for any a ∈ I, b ∈ A. Fordeformation algebras we will use an obvious extension: an ~-trace is an ~-linearmap A → C((~)) such that T (ab) = T (ba) for any a, b ∈ A (there is also anotion of trace relative to an ideal).
7.1 Residual integral.
Let Σ be a oriented cone of dimension n, and µ =∑µk an n-form (density)
with coefficients in O and compact conic support.
Definition 43 The residual integral∫ res
µ is defined as∫ res
µ =∫X
ρyµ0
where µ0 is the component homogeneous of degree 0 of µ, ρ denotes the radialvector field; the contraction ρyµ0 is the pull-back of a (n− 1)-form on X
It is also the contour integral 12iπ
∑∫γµk (γ = X × S1 ⊂ Σc).
It behaves like an integral :- if µ is an n-form with coefficients in O, µ is exact, i.e. µ = dν with ν a
(n−1)-form with compact conic support iff∫ res
dµ = 0 (indeed if µ is of degreek 6= 0 it is exact: we have µ = 1
kdIρµ; if µ is of degree 0, we have µ = drr ν with
ν = ρyµ. then µ is exact iff ν is, i.e.∫Xν =
∫ resΣ
µ = 0).- in particular
∫ resLvµ = 0 for any vector fiels with coefficients in O, so we
may integrate by parts:∫ res
µLvf = −∫ res
Lvµ.f .
7.2 Trace for Moyal products
Let E be a graded vector space, Σ = E−E0. We equip E with a volume elementdv = c dx1 . . . dxn (this is homogeneous of degree
∑degxi). Let b be bilinear
form with constant coefficients, homogeneous of degree −1 on E, and A be thestar algebra associated to the the Moyal star product ∗b on Σ.
Proposition 44 The residual integral T (a) =∑res
fdv is a trace on the idealof elements with compact conic support.
Proof: we have f ∗ g =∑
1k!b
k(∂ξ, ∂η)f(ξ)g(η)|η=ξ, so
[f, g] =∑ 1
k!(bk(∂ξ, ∂η)− bk(∂η, ∂ξ))f(ξ)g(η)|η=ξ
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Integrating by parts we get∫ res
[bk(∂ξ, ∂η)f(ξ)g(η)]|η=ξ =∫ res
[bk(∂ξ,−∂ξ)f(ξ)]g(η)|η=ξ
Now we have b(−u, u) = b(u,−u) so the two integrals cancel.Note that if b is antisymmetric we even have bk(∂ξ,−∂ξ) = 0 so all terms
vanish for k > 0 and∫ res
f ∗ g =∫ res
fg: then we have T (a ∗ b) = T (ab).
7.3 Canonical trace on symplectic algebras.
Proposition 45 Suppose that c = b −t b is a symplectic Poisson bracket withconstant coefficient, homogeneous of degree −1 (dimE = 2n). Then the traceT (a) above is, up to a constant factor, the unique race on A.
Proof: we show that the set of sums of brackets in A is the kernel of T , a sub-vector space of codimension 1. Let ξk be a hommogeneous basis of coordinates,ξk the dual basis i.e. ξj , ξk = δjk. If a has compact conic support, a dv isexact iff T (a) =
∫ resadv = 0. Then there exist aj with compact conic support
such that a dv = d∑
(−1)j−1ajdξ1..dξj ..dξn i.e. a =∑L∂jaj =
∑[ξj , aj ].
Definition 46 With notations as above, the canonical residual trace on A is :
tr resa = (2π)−n∫res
aωn
n!
where ω is the symplectic form inverse of the Poisson bracket c (ω is of degree1, dv of degre n, the density is dv = ωn
n! ).
Example: let E have canonical homogeneous coordinates ξk, xk (ξp, ξq =xp, xq = ξp, xq − δpq = 0, ω =
∑dξjdxj).
The Weyl product corresponds to the case b = 12
∑∂ξj∧∂xj (the coordinates
ξj , xj are all of degree 12 ). We have trresa ∗w b = trresab
The normal product corresponds to the case b =∑∂ξj ⊗ ∂xj (ξ of degree
1, x of degree 0). We have trres[a, b] = 0 (but not trresanb = trresab).
Theorem 47 tr res is invariant by all isomorphisms of star algebras
Proof: let U, V be open subcones of E −E0 and U : A|U → A|V an isomor-phism. Since trres is unique except for a constant factor, we have U∗trres =ctrres. Now the geometric support u of U is a homogeneous symplectic mapU → V ; it treserves ω and ωn
n! , so also the leading term of trres (trresa =∫ resσ−n(a)ω
n
n! if a is of degree −n), hence c = 1.It follows that any symplectic star algebra A possesses a canonical (residual)
trace (Σ is a union of open subcones Σk isomorphic to open subcones of sym-plectic graded vector spaces as above; then the traces on A|Σk are well definedand glue together).
We complete remark3 as follows:
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Remark 11 If a =∑ak ∈ A and U is an automorphism of A, trUsP is a
polynomial of s (of degree ≤ n+ deg a).
7.4 Trace for deformation algebras
Since deformation algebras are C((~))-algebras, it is natural to study traces withvalues in C((~)). Most of what was said above can be repeated:
if X is a vector space equipped with a bidifferential operator b with constantcoefficients, a =
∑ak(x)hk, dv a volume element (with constant coefficients),
the integral
T (a) =∫X
adv =∑
~k∫X
akdv ∈ C((~))
is a trace on the deformation algebra A corresponding to the Moyal product∗~b, i.e. T [a, b] = 0 (T (a ∗ b) = T (ab) if b is antisummetric).
If the Poisson bracket c = b −t b is symplectic, the trace is unique up toa “constant” factor, i.e. an invertible element of C((~)), and we define thecanonical trace as
tr(a) = (2π~)−n∫aωn
n!
where ω is the symplectic form inverse of c. (We will sometimes use the residualtrace, with values in C: this is the coefficient of ~n in tr a; it agrees withthe residual trace for Toeplitz operators when A can be nicely embedded in aTpeplitz algebra).
The analogue of proposition 45 still holds, but the proof above no longerworks because a trace is not determined by its leading term (no more than aninvertible element of C((~))). We repeat Fedosov’s proof ([61]: first note that thetheorem is local on X, so as the integral defining the trace. tr is also invariantby inner automorphisms (tr (aba−1 − b) = tr [ab, a−1] = 0], so it is invariant byall automorphisms (which are locally inner). The Weyl deformation product isobviously invariant by affine symplectic isomorphisms, so as the trace.
We now turn to general isomorphisms: if X,Y are open subsets of a symplec-tic vector space E and U : AX → AY an isomorphisms of the two deformationWeyl algebras, the geometric support u : Y → X is is a symplectic isomorphism.Let y0 be a point of Y , x0 = u(y0). The affine map v tangent to u preseves thestar product and tr . Replacing U by v−1 U we are reduced to the case whereu is tangent ti the identity map at ξ0
Then one can choose a smooth family ut of symplectic diffeomorphisms de-fined in a fixed neighborhood of ξ0 so that u0 = Id , u1 = u. This in turn canbe lifted in a smooth family Ut of isomorphisms : u−1
tdudt is a smooth family of
symplectic vector fields, which can be lifted to a smooth family of derivationsof degree 0 : Dt = ad ~−1at (a0 = 0); this in turn can be integrated to a smoothfamily of isomorphisms Ut = Ad eat above ut, with U0 = Id (dUtdt = DtUt).Then d
dt trUta = DtUta = 0 so trUta = tr a. Finally trUa = trU1a sinceU−1U1 is an automorphism (its geometric support is the identity map).
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8 Vanishing of the Logarithmic Trace.
In this section we show that the logarithmic trace of Szego projectors introducedby K. Hirachi [78] for CR structures and extended in [25] to contact structuresvanishes identically.6
In [78] K. Hirachi showed that the logarithmic trace of the Szego projectoris an invariant of the CR structure. In [25] I showed that it is also definedfor generalized Szego projectors associated to a contact structure (definitionsrecalled below, sect.8.4), that it is a contact invariant, and that it vanishes if thebase manifold is a 3-sphere, with arbitrary contact structure (not necessarily thecanonical one). Here we show that it always vanishes. For this use the fact thatthis logarithmic trace is the residual trace of the identity (definitions recalledbelow, sect.8.5), and show that it always vanishes, because the Toeplitz algebraassociated to a contact structure can be embedded in the Toeplitz algebra of asphere, where the identity maps of all ‘good’ Toeplitz modules have zero residualtrace.
8.1 Notations
We first recall the notions that we will use. Most of the material below in §1-5is not new; we have just recalled briefly the definitions and useful properties,and send back to the literature for further details (cf. [82, 83, 102, 86]).
If X is a smooth manifold we denote T •X ⊂ T ∗X the set of non-zerocovectors. A complex subspace Z corresponds to an ideal IZ ⊂ C∞(T •X,C)). Zis conic (homogeneous) if it is generated by homogeneous functions. It is smoothif IZ is locally generated by k = codimZ functions with linearly independentderivatives. If Z is smooth, it is involutive if IZ is stable by the Poisson bracket(in local coordinates f, g =
∑ ∂f∂ξj
∂g∂xj− ∂f
∂xj
∂g∂ξj
); it is 0 if locally IZ hasgenerators ui, vj (1 ≤ i ≤ p, 1 ≤ j ≤ q, p + q =codimZ) such that the vj arereal, ui complex, and the matrix 1
i (uk, ul) is hermitian 0. The real partZR is then a smooth real submanifold of T •X, whose ideal is generated by theRe ui, Im ui, vj. If I is 0, it is exactly determined by its formal germ (Taylorexpansion) along the set of real points of ZR.
A Fourier integral operator (FIO) from Y to X, is a linear operator fromfunctions or distributions on Y to same on X defined as a locally finite sum ofoscillating integrals
f 7→ Ff(x) =∫eiφ(x,y,θ)a(x, y, θ)f(y) dθdy ,
where φ is a phase function (homogeneous w.r. to θ), a a symbol function.Here we will only consider regular symbols, i.e. which are asymptotic sums
6arXiv:math.AP/0604166 v1; Proceedings of the Conference in honor of T. Kawai, SpringerVerlag, 2006
59
a ∼∑k≥0 am−k where am−k is homogeneous of degree m− k, k an integer. m
could be any complex number. There is also a notion of vector FIO, acting onsections of vector bundles, which we will not use here.
The canonical relation of F is the image of the critical set of φ (dθφ = 0) bythe differential map (x, y, θ) 7→ (dxφ,−dyφ) - this is always assumed to be animmersion from the critical set onto a Lagrangian sub-manifold of T •X×T •Y 0
(the sign 0 means that we have reversed the sign of the canonical symplecticform; likewise if A is a ring, A′ denotes the opposite ring). We will also use FIOwith complex positive phase function: then the canonical relation is defined byits ideal (the set of complex functions u(x, ξ, y, η which lie in the ideal generatedby the coefficients of dθφ, ξ− dxφ, η+ dyφ which do not depend on θ); it shouldnot be confused with its set of real points.
Following Hormander [82, 83], the degree of F is defined as
degF = deg (adθ)− 14
(nx + ny + 2nθ),
with nx, ny, nθ the dimensions of the x, y, θ-spaces, deg (adθ) the degree of thedifferential form adθ = deg a+
∑νk (νk = deg θk, usually νk = 1 but could be
any real number - not all 0); this only depends on F and not on its representationby oscillating integrals). In what follows we will always require that the degreebe an integer (which implies m ∈ Z/4).
8.2 Adapted Fourier Integral Operators
The Toeplitz operators and Toeplitz algebras used here, associated to a CR ora contact structure, were introduced and studied in [17, 27], using the analysisof the singularity of the Szego kernel (cf. [15, 87]), or in a weaker form, the“Hermite calculus” of [14, 71]. The terminology “adapted” is taken from [27]:lacking anything better I have kept it.
For k = 1, 2, let Σk ⊂ T •Xk be smooth symplectic sub-cones of T •Xk andu : Σ1 → Σ2 an isomorphism.
Definition 48 A Fourier integral operator A is adapted to u if its canonicalrelation C is complex 0, with real part the graph of u. It is elliptic if itsprincipal symbol does not vanish (on Re C).
As above, a conic complex Lagrangian sub-manifold Λ of T •X is 0 ifits defining ideal IΛ is locally generated by n = dimX homogeneous func-tions u1, . . . , un(n = dimX) with independent derivatives, u1, . . . , uk complex,uk+1, . . . , un real for some k (1 ≤ k ≤ n), and the matrix ( 1
i up, uq)1≤p,q≤k ishermitian 0; equivalently; the intersection C ∩ C is clean, and on the tangentbundle the hermitian form 1
iω(U, V ) is positive, with kernel the complexificationof the tangent space of Re C.
Pseudo-differential operators are a special case of adapted FIO (X1 = X2 =X,u = IdT•X); so are Toeplitz operators on a contact manifold (see below).
Adapted FIO always exist (cf [27]), more precisely
60
Proposition 49 For any symplectic isomorphism u as above, there exists anelliptic FIO adapted to u.
In fact if Λ is a complex 0 Lagrangian sub-manifold of T •X, in particularif it is real, it can always be defined by a global phase function with positiveimaginary part (Imφ % dist (.,ΛR)2) living on T •X: it is easy to see that suchphase functions exist locally, and the positivity condition makes it possible toglue things together using a homogeneous partition of the unity. Once one haschosen a global phase function, it is obviously always possible to choose anelliptic symbol - of any prescribed degree (cf. also [27])7. Note that ellipticonly means that the top symbol is invertible on the real part of the canonicalrelation, not that the operator is invertible mod. smoothing operators (for thisthe canonical relation must be real: X1, X2 have the same dimension, Σk =T •Xk and C is the graph of an isomorphism).
8.3 Model Example
Here is a generic example of adapted FIO: let X1, X2, Z be three vector spaces
Σk = TXkXk × T •Z ⊂ T •(Xk × Z) (k = 1, 2, TXkXk the zero section),
u the identity map IdT•Z : Σ1 → Σ2 . (42)
If C 0 is a complex canonical relation with real part the graph of u, thecomplex formal germ along Σ of the restriction to C of the projection map(x, ξ, z, ζ, z′, ζ ′, y, η) 7→ (x, z, ζ ′, y) is an isomorphism (the dimensions are right,and it is an immersion: if v = (0, ξ, 0, ζ, z′, 0, 0, η) is a complex vector with zeroprojection, it is orthogonal to v (because these vectors form a real Lagrangianspace), so if it is tangent to C ′, it is tangent to the real part, i.e. the diagonalof T •Z, and this obviously implies v = 0).
So we can choose the phase function as
φ =< z − z′, ζ ′ > +iq(x, z, ζ ′, y)
where q is smooth complex function of x, z, ζ ′, y alone, homogeneous of degree1 w.r. to ζ ′, vanishing of order 2 for x = y = 0, and Re q % (x2 + y2) |ζ ′| (it iseasy to check that conversely any such phase function corresponds to a positiveadapted canonical relation as above). The operator is
Ff(x, z) =∫ei<z−z
′,ζ′>−q(x,z,ζ′,y)a(x, z, ζ ′, y)f(z′, y)dζ ′dz′dy , (43)
with a a symbol as above.7the intrinsic differential-geometric description of the symbol is elaborate: it is a section of a
line bundle whose definition incorporates half densities and the Maslov index or an elaborationof this in the case of complex canonical relations. However on real manifolds this line bundleis always topologically trivial
61
Since any symplectic sub-cone of a cotangent manifold is always locallyequivalent to T •Z ⊂ T •(X × Z), the model above is universal i.e. any adaptedFIO is micro-locally equivalent to F1 A F2 where F1, F2 are elliptic invertibleFIO with real canonical relations, graphs of local symplectic isomorphisms, andF is as the model above.
For adapted FIO the Hormander degree coincides with the degree in the scaleof Sobolev spaces, i.e. if F is of degree s it is continuous Hm(Y )→ Hm−Re s(X);this is easily seen on the model example above (F is L2 continuous if its degreeis 0 i.e. a is of degree − 1
4 (nx + ny)). (This not true for general FIOs - in factfor a FIO with a real canonical relation C, this is only true if C is locally thegraph of a symplectic isomorphism.)
The following result also immediately follows from the positivity condition:
Proposition 50 Let X1, X2, X3 be three manifolds, Σk ⊂ T •Xk symplecticsub-cones, u resp. v a homogeneous symplectic isomorphism X1 → X2 resp.X2 → X3, F,G FIO (with compact support) adapted to u, v. Then G F isadapted to v u; its canonical relation is transversally defined and positive. Itis elliptic if F and G are elliptic.
This is mentioned in [27]; the crux of the matter is that if Q(y) is a quadraticform with 0 real part, the integral
∫e−|ξ|Q(y)dy does not vanish: it is an
elliptic symbol of degree − 12ny, equal to disc(Q
π )−12 |ξ|− 1
2 ny .
8.4 Generalized Szego projectors
These were called “Toeplitz projectors” in [24, 25]. C. Epstein suggested thepresent name, which is better. References: [30, 17, 27, 21].
Definition 51 Let X be a manifold, Σ ⊂ T •X a symplectic sub-cone. A gen-eralized Szego projector (associated to Σ) is an elliptic FIO S adapted to IdΣ
which is a projector (S2 = S)
(Note that “elliptic” (or “of degree 0”) is part of the definition; otherwisethere exist many non-elliptic projectors, of degree > 0 as FIOs). The case weare most interested in is the case where Σ is the half line bundle correspondingto a contact structure on X (i.e. the set of positive multiples of the contactform). But everything works as well in the slightly more general setting above.
We will not require here that S be an orthogonal projector; this makes senseanyway only once one has chosen a smooth density to define L2-norms.
If S is a generalized Szego projector, its canonical relation C ⊂ T •X × T •Xis idempotent, positive, and can be described as follows: the first projection is acomplex positive involutive manifold Z+ with real part Σ; the second projectionis a complex negative manifold Z− with real part Σ (Z− = Z+ if S is selfad-joint). The characteristic foliations define fibrations Z± → Σ (the fibers are thecharacteristic leaves; they have each only one real point so are “contractible”
62
(they vanish immediately in imaginary domain), and there is no topologicalproblem for them to build a fibration). Finally we have C = Z+ ×Σ Z−
Generalized Szego projectors always exist, so as orthogonal ones (cf. [27,21]). As mentioned in [25], generalized Szego projectors mod. smoothing oper-ators form a soft sheaf on Σ, i.e. any such projector defined near a closed conicsubset of T •X or Σ is the restriction of a globally defined such projector.
8.5 Residual trace and logarithmic trace
The residual trace was introduced by M. Wodzicki [116]. It was extended toToeplitz operators and suitable Fourier integral operators by V. Guillemin [73](cf. also [74, 117]). It is related to the first example of ‘exotic’ trace given by J.Dixmier [43].
Let C be a canonical relation in T •X × T •X. A family As(s ∈ C) of FIOsof degree s belonging to C is holomorphic if s 7→ (∆)−
s2As is a holomorphic
map from C to FIO of fixed degree (in the obvious sense). If As has compactsupport, the trace trAS is then well defined and depends holomorphically on sif Re s is small enough (As is then of trace class). Often, e.g. if the canonicalrelation is real analytic, this will extend as a meromorphic function of s on thewhole complex plane, but this is not very easy to use because the poles are hardto locate and usually not simple poles.
Proposition 52 If C is adapted to the identity IdΣ, with Σ ⊂ T •X a symplecticsub-cone, and As a holomorphic family, as above, then trAs has at most simplepoles are the points s = −n − degA0 + k, k ≥ 0 an integer, n = 1
2dim Σ (thedegree degA0 is defined as above)
Proof: this is obviously true is As is of degree −∞ (there is no pole at all).In general we can write As as a sum of FIO with small micro-support (modsmoothing operators), and a canonical transformation reduces us to the modelcase, where result is immediate.
Definition 53 If A is a FIO adapted to C, the residual trace trresA is theresidue at s = 0 of any holomorphic family As as above, with A = A0.
This does not depend on the choice of a family As: indeed if A0 = 0, thefamily As is divisible by s i.e. As = sBs where Bs is another holomorphicfamily, and since trBs has only simple poles, trAs has no pole at all at s = 0.
Proposition 54 The residual trace is a trace, i.e. if A and B are adaptedFourier integral operators, we have trresAB = trresBA.
Indeed with the notations above trABs and trBsA are well defined andequal for Re s small, so their meromorphic extensions and poles coincide.
Logarithmic trace (contact case)
63
Let Σ ⊂ T •X be a symplectic half-line bundle, defining a contact structureon X. A complex canonical relation C 0 adapted to IdΣ is always the conor-mal bundle of a complex hypersurface Y of X ×X, with real part the diagonal(rather the positive half)8, so if A is a FIO adapted to IdΣ, its Schwartz kernelcan be defined by a one dimensional Fourier integral:
A(x, y) =∫ ∞
0
e−Tφ(x,y)a(x, y, T ) dT ,
with φ = 0 an equation of the hypersurface Y , φ = 0 on the diagonal, Reφ ≥cst dist(.,diag)2, and a is a symbol: a ∼
∑k≤N ak(x, y)T k−1 (N = deg A).
Its singularity has a typically holonomic form:
f(x, y)(φ+ 0)−N + g(x, y) Log (1
φ+ 0) , (44)
with f, g smooth functions on X ×X, and in particular g(x, x) = a0(x, x).
Proposition 55 With notations as above, the residual trace of A is the traceof the logarithmic coefficient:
trresA =∫X
g(x, x) . (45)
An obvious holomorphic family extending A (mod. a smoothing operator) isthe family As with Schwartz kernel
As(x, y) =∫ ∞
1
e−Tφ(x,y)a(x, y, T )T sdT .
Since as(x, x, T ) ∼∑k≤N T
s+k−1ak(x, x) and φ(x, x) = 0, we get
As(x, x) ∼∑ ak(x, x)
s+ k,
with an obvious notation: the meromorphic extension of the trace just has justsimple poles at each integer j ≥ −N , with residue
∫Xa−j(x, x). In particular
the residue for s = 0 is the logarithmic trace.
The residual trace is also equal to the logarithmic trace in the case of pseudo-differential operators, or in the model case. In general the residual trace is welldefined, but I do not know if the logarithmic coefficient can be reasonably definede.g. if the projection Σ→ X is not of constant rank. For the equality with theresidual trace, and for theorem 58 below, the sign is important: the logarithmictrace is the integral of the coefficient of Log 1
φ , not the opposite.
8indeed a complex vertical vector v is as before orthogonal to v; if it is tangent to C at apoint of CR = diag Σ, it is tangent to the real part CR = diag Σ since C 0, but this impliesthat v is the radial vector, because the radial vector is the only vertical vector tangent to Σ.So the projection C → X ×X is of maximal rank 2n− 1 and the image is a hypersurface.
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8.6 Trace on a Toeplitz algebra A and on End A(M)
If S is a generalized Szego projector associated to Σ ⊂ T •X. The correspondingToeplitz operators are the Fourier integral operators of the form TP = SPS, Pa pseudo-differential operator (equivalently, the set of Fourier integral operatorsA with the same canonical relation, such that A = SAS). They form an algebraA on which the residual trace is a trace: trresAB = trresBA. Mod. smoothingoperators, this can be localized, and the Toeplitz algebra AΣ is this quotient; itis a sheaf on Σ (or rather on the basis).
Proposition 56 The Toeplitz algebra, so as the residual trace of S only dependon Σ and not on the embedding Σk ⊂ T •X
Indeed if Σ→ T •X ′ is another embedding, S′ a corresponding Szego projector,it follows from prop. 49 that there exist elliptic adapted FIO F, F ′ from Xto X ′ resp. X ′ to X such that F = S′FS, F ′ = SF ′S′, FF ′ ∼ S, F ′F ∼ Sso S, S′ have the same residual trace, and A 7→ FAF ′ is an isomorphism ofthe two Toeplitz algebras. In the lemma wa could as well embed Σ in anothersymplectic cone endowed with a Toeplitz structure.
The definition of the residual trace extends in an obvious way to EndA(M)when M is a free A-module (EndA(M) is isomorphic to a matrix algebra withcoefficients inA0, where trres(aij) =
∑trresaii is obviously a trace, independent
of the choice of a basis of M). It extends further to the case where M is locallyfree (a direct summand of a free module), and to the case where M admits(locally or globally) a finite locally free resolution: if
0→ LNd−→ . . . L1 d−→ L0 → 0
is such a resolution, i.e. a complex of are locally free A-modules Lj , exact indegree 6= 0, with a given isomorphism ε : L0/dL1 → M : Then EndA(M) isisomorphic to Rhom0(L,L), i.e. any a ∈ EndA(M) extends as morphism a ofcomplexes of L (a = (aj), aj ∈ EndA(Lj)) (if a has compact support, a canbe chosen with compact support). Any two such extensions a, a′ differ by asuper-commutator [d, s] = ds+ sd (s, of degree 1, can be chosen with compactsupport if a− a′ has compact support). The super-trace
supertrres(a) =∑
(−1)jtrres(aj)
is then well defined; it only depends on a, because the super-trace of a super-commutator [s, d] vanishes: this defines the trace in End (M). Below we willuse “good” modules, i.e. which have a global finite locally free resolution forwhich this is already seen on the principal symbols (i.e. M and the Lj areequipped with good filtrations for which gr d is a locally free resolution of grM(in the analytic setting this always exists if M is coherent and has a global goodfiltration, and the base manifold is projective).
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Alternative description of the residual trace
Let S be a generalized Szego projector associated to a symplectic cone Σ ⊂T •X. Then the left annihilator of S is a 0 ideal I in the pseudo-differentialalgebra of X; its characteristic set is the first projection Σ+ of the complexcanonical relation of S; as mentioned earlier it is involutive 0 with real partΣ. (there is a symmetric statement for the right annihilator).
Proposition 57 Let M be the EX-module M = E/I. Then the Toeplitz algebrais canonically isomorphic to End E(M).
Proof: let eM be the image of 1 ∈ E (it is a generator of M). It is elementarythat End E(M) is identified with opposite algebra ([E : I]/I)0 where [E : I]denotes the set of P ∈ E such that IP ⊂ I (to P corresponds the endomorphismaP such that aP (e) = Pe). It is also immediate that the map u which to aPassigns the Toeplitz operator TP = SPS = PS is an isomorphism (both algebrashave a complete filtration by degrees, and the associated graded algebra in bothcases is the algebra of symbols on Σ); clearly u is a homomorphism of algebras,of degree ≤ 0, and gru = Id.
Now M is certainly “good” in the sense above: it is locally defined by trans-verse equations and has, locally, a resolution whose symbol is a Koszul complex.So the residual trace is well defined on End E(M). Since the trace on an algebraof Toeplitz type is unique up to a constant factor, there exists a constant Csuch that
trresaP = C trresTP . (46)
Below we only only need C 6= 0; however with the conventions above we have:
Theorem 58 The constant C above is equal to one (C = 1).
Proof: to the resolution of M above corresponds a complex of pseudo-differentialoperators
0→ C∞(x) D−→ C∞(X,E1)→ . . .D−→ C∞(X,EN )→ 0 , (47)
exact in degree > 0 and whose homology in degree 0 is the range of S (mod.smoothing operators), i.e. there exists a micro-local operator E on (Ek) suchthat DE + ED ∼ 1− S (cf. [14]; E is a pseudo-differential operator of type 1
2 ,not a “classical” pseudo-differential operator, but it preserves micro-supports).
It is elementary that one can modify D, E, and if need be S, by smoothingoperators so that (47) is exact (on global sections) in degree 6= 0, and kerD0
is the range of S. Then if as = (ak, s) is a holomorphic family of pseudo-differential homomorphisms of degree s, and Tas is the Toeplitz operator Tas =a0,s|ker S , we have trTas =
∑(−1)ktr ak,s for Re s 0 hence also equality for
the meromorphic extensions and residues.
Here is an alternative proof (slightly more in the spirit of the paper becauseit really uses operators mod. C∞ rather than true operators). Notice first thattheorem 58 is (micro) local, and since locally all bundles are trivial, we can
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reason by induction on codim Σ. Thus it is enough to check the formula for oneexample, where codim Σ = 2. Note also that above we had embedded in thealgebra of pseudo-differential operators on a manifold, but we could just as wellembed in another Toeplitz algebra.
We choose Σ corresponding to the standard contact (CR) sphere S2n−1 ofCn, embedded as the diameter z1 = 0 in the sphere of Cn+1
The Toeplitz space Hn+1 is the space of holomorphic functions in the unitball of Cn+1 (more correctly: their restrictions to the sphere); we choose Hn
the subspace of functions independent of z1. There is an obvious resolution:0 → Hn+1 → Hn+1 → 0 (Hn = ker ∂1). We choose on Hn the operatora, restriction of ρ−n, with ρ =
∑zj∂zj (this is the simplest operator with a
nonzero residual - our convention is that ρσ kills constant functions for all σ).
Lemma 59 On the sphere S2n−1 the residual trace of the Toeplitz operator ρ−n
is trresρ−n = 1(n−1)! .
Proof: the standard Szego kernel is S(z, w) = 1vol S2n−1
(1 − z.w)−n. Now wehave the obvious identity
ρ(ρ+ 1) . . . (ρ+ n− 1) Log1
1− z.w= (n− 1)!((1− zw)−n − 1) ,
so that the leading coefficient of the logarithmic part of ρ−n is 1vol S2n−1(n−1)! ,
whose integral over the sphere is 1(n−1)! .
On the sphere S2n+1 we have ∂1ρ = (ρ+ 1)∂1 so we choose for (a) the pair(a0 = ρ−n, a1 = (ρ+ 1)−n). Since terms of degree < −n− 1 do not contributeon S2n+1, the super-trace is
supertrres(a) = trres(ρ−n − (ρ+ 1)−n) = trres(nρ−n−1) =n
n!= trresa .
8.7 Embedding
If Σ1,Σ2 are two symplectic cones, with contact basis X1, X2, symplectic em-beddings Σ1 → Σ2 exactly correspond to contact embeddings X1 → X2, i.e.an embedding u : X1 → X2 such that the inverse image u∗(λ2) is a positivemultiple of the contact form λ1 (the corresponding symplectic map take thesection u∗λ2 of T •X1 to the section λ2 of T •X2. With this in mind we have
Lemma 60 If X is a compact oriented contact manifold, it can be embedded inthe standard contact sphere.
Proof: the standard contact (2N − 1)-sphere of radius R has coordinatesxj , yj (1 ≤ j ≤ N ,
∑x2j + y2
j = R2) and contact form λ =∑xjdyj − yjdxj (or
a positive multiple of this). If X is a compact contact manifold, its contact formcan always be written 2
∑m2 xjdyj for some suitable choice of smooth functions
xj , yj , or just as well∑m
1 xjdyj − yjdxj , setting for instance x1 = 1, y1 =
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∑m2 xjyj (m may be larger than the dimension). Adding suitably many other
pairs (xj , yj) with yj = 0, xN = (R2 −∑x2j + y2
j )12 , for R large enough, we get
an embedding in a contact sphere of radius R.
Theorem 61 For any generalized Szego projector Σ associated to a symplecticcone with compact basis, we have trresS = 0. In particular if Σ corresponds toa contact structure on a compact manifold, the logarithmic trace vanishes.
By lemma 60 we can suppose that Σ is embedded in the symplectic cone ofa standard odd contact sphere. Let B be the canonical Toeplitz algebra on thesphere: then by prop. 57, the Toeplitz algebra A of S is isomorphic to EndB(M)where M is a suitable good B module. Now on the sphere any good locally freeB-module is free (any complex vector bundle on an odd sphere is trivial), andthe Szego projector has no logarithmic term, so and trres1M = 0, for any freehence also for any good B-module M .
This result is rather negative since it means that the logarithmic trace cannotdefine new invariants distinguishing CR or contact manifolds. Note however thatthat it is not completely trivial: it holds for the Toeplitz algebras associatedto a CR or contact structure, as constructed in [27], but a contact manifoldcarries many other star algebras which are locally isomorphic to the Toeplitzalgebra (I showed in [24] how Fedosov’s classification of star products [60] canbe adapted to classify these algebras). Any such algebra A carries a canonicaltrace, because the residual trace is invariant by all isomorphisms, so that localtraces glue together. If the contact basis is compact, the trace trres1A is welldefined, but there are easy examples showing that it is not always zero.
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9 Asymptotic equivariant indexof Toeplitz operators.
In this section we describe the asymptotic equivariant trace and index of Toeplitzoperators invariant under the action of a compact group G. This theory is anavatar of M.F. Atiyah’s index theory for relatively elliptic pseudodifferential op-erators [5] on a G-manifold. Atiyah’s theory does not apply directly to Toeplitzoperators on a contact manifold, because the function space on which they act(Toeplitz space) is only defined up to a space of finite dimension from sym-bolic calculus, so the absolute index or trace do not make much sense. TheG-asymptotic trace and index are weaker forms (Atiyah’s trace or index is adistribution on G, the asymptotic trace or index is its singularity). The advan-tage of the asymptotic index is that it is well defined for Toeplitz operators,whereas the “absolute” index is not, and it still contains useful information. Wehave recently used it with E. Leichtnam, X. Tang, A. Weinstein [28], to give anew “simple” proof of the Atiyah-Weinstein conjecture. We refer to loc. cit.for further details about this formula, for which a proof was recently given byC. Epstein [54], using “Heisenberg pseudodifferential calculus”.
9.1 Toeplitz operators
In this section we recall the mechanism of generalized Szego projectors andToeplitz operators. We refer to [27, 21, 26] for more details.
As in [27, 21, 26], we call symplectic cone a smooth (paracompact) manifoldwhich is a principal R×+ bundle, equipped with a symplectic form ω homogeneousof degree 1. The Liouville form is its horizontal primitive λ = ρyω (ω =dλ), where ρ denotes the radial (Euler) vector field, infinitesimal generator ofhomotheties. The basis X = Σ/R×+ is an oriented contact manifold; its contactform λX (any pull-back of λ by a smooth section) is defined up to a smoothpositive factor, and Σ is canonically identified with the set of positive multiplesof λX in T ∗X.
9.1.1 Microlocal model
We first describe the microlocal model for generalized Szego projectors givenin [14]. Let (x, y) = (x1, . . . , xp, y1, . . . , yq) denote the variable in Rp+q. Weconsider the system of pseudodifferential operators D = (Dj) with
Dj = ∂yj + |Dx|yj (j = 1, . . . , q)
The Dj commute; the complex involutive variety charD is defined by the com-plex equations ηj − i|ξ|yj = 0; it is 0, in the sense of [102, 101]. Its real partis the symplectic manifold Σ : ηj = yj = 0.
The kernel of D in L2 is the range of the Hermite operator H (in the senseof [14]) defined by its partial Fourier transform:
f ∈ L2(Rp) 7→ Hf with FxHf(ξ, y) = (π−1|ξ|)q4 e−
12 |ξ|y
2f(ξ)
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The orthogonal projector on kerD is S = HH∗:
f 7→ (2π)−p∫
R2p+qei(<x−x
′,ξ>+i 12 (y2+y′2))(π−1|ξ|)
q2 f(x′, y′)dx′dy′dξ
It is a Fourier integral operator, so as H; its complex canonical relation is 0,with real part the graph of Id Σ (Fourier integral operators are described in[82], Fourier integral operators with complex canonical relation are described in[102, 101]).
9.1.2 Generalized Szego projectors
Let M be a compact manifold, and Σ ⊂ T •M a symplectic subcone (T •Mdenotes T ∗M deprived of its zero section). A generalized Szego projector as-sociated to Σ (or Σ-Szego projector) is a self adjoint9 elliptic Fourier integralprojector S of degree 0 (S = S∗ = S2), whose complex canonical relation Cis 0, with real part the diagonal diag Σ (elliptic means that the principalsymbol of S does not vanish on Σ).
From [27, 21, 26], we recall:
1) A Σ-Szego projector S always exists. It is microlocally isomorphic (mod.some elliptic FIO transformation) to the model above.
We will denote H ⊂ C−∞(M) its range. Modulo C∞, it defines a sheafµH on Σ - a subsheaf supported by Σ of the sheaf of microfunctions onT •M .
2) Toeplitz operators defined by S are the operators on H of the form u ∈H 7→ TP (u) = SPS(u) with P a pseudodifferential operator on M . Moregenerally, if P is any FIO whose canonical relation is complex positive,with real part containing diag Σ, then SPS is a Toeplitz operator.
Modulo operators of degree−∞ (smoothing operators), Toeplitz operatorsform a sheaf AΣ of algebras on Σ, acting on µH; (AΣ, µH) is locallyisomorphic to the sheaf of pseudodifferential operators in p real variables(2p = dim Σ), acting on the sheaf of microfunctions. The principal symbol(principal part) of TP is σ(P )|Σ.
3) If S, S′ are two Σ-Szego projectors with range H,H′, S′ induces a quasiisomorphism H→ H′ (the restriction of SS′ to H is a positive (≥ 0) ellipticToeplitz operator).
More generally, if Σ ⊂ T •M,Σ′ ⊂ T •M ′ are two symplectic cones andf : Σ→ Σ′ a homogeneous symplectic isomorphism, there always exists aFourier integral operator F from M to M ′, inducing an “elliptic” Fredholmmap H → H′, e.g. there exists a complex canonical relation C 0 withreal part the graph of f , and we may take F = S′ F ′ where F ′ is anyelliptic FIO with canonical relation C (such elliptic FIO exist, they werecalled “adapted” in [27, 21]).
9the requirement that S be self adjoint is convenient but not essential
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Thus the pair (AΣ, µH) consisting of the sheaf of micro Toeplitz operators(i.e. mod smoothing operators), acting on µH is well defined, up to (non unique)isomorphism: it only depends on the symplectic cone Σ, not on the embedding.
9.1.3 Holomorphic case
A first example of Toeplitz structure is Σ = T •M (M a compact manifold),S = Id : the Toeplitz algebra is the algebra of pseudodifferential operatorsacting on the sheaf of microfunctions on M .
In general, as noted above, the basis X = Σ/R×+ of Σ is a contact manifold,and Σ can be canonically embedded in T •X as the set of positive multiples ofthe contact form. An important particular case is the holomorphic case: Xis the smooth, strictly pseudoconvex boundary of a Stein complex manifold;the contact form of X is the form induced by Im ∂φ where φ is any definingfunction (φ = 0, dφ 6= 0 on X, φ < 0 inside - e.g. if X is the unit spherebounding the unit ball of Cn, with defining function z · z − 1, the contact formis Im z · dz|X). Then the Szego projector S is the orthogonal projector on thespace of boundary values of holomorphic functions in L2(X) (the fact that it isFourier integral operator as above was proved in [15]).
The pseudodifferential algebra is a special case of holomorphic Toeplitz al-gebra: if M is a manifold, it has a real analytic compact manifold; if M c is acomplexification of M , small tubular neighborhoods of M in M c (for some her-mitian metric) are Stein manifold with strictly complex boundary X ∼ S∗M ,and the pseudodifferential algebra of M acting on microfunctions is isomorphicto the Toeplitz algebra of X acting on H. In fact there exists an adapted Fourierintegral operator from M to X which defines an isomorphism from C−∞(M)to H(X)10 and interchanges pseudodifferential operators on M and Toeplitzoperators on X.
Note: the Atiyah-Weinstein problem can be described as follows: If X is acompact contact manifold, and S, S′ two Szego projectors defined by two em-beddable CR structures giving the same contact structure, then the restrictionof S′ to H is a Fredholm operator H→ H′ (SS′ induces an elliptic Toeplitz op-erator on H). The Atiyah-Weinstein conjecture computes the index in terms oftopological data of the situation (topology of the holomorphic fillings of whichX is the boundary).
9.2 Equivariant trace and index
9.2.1 Equivariant Toeplitz algebra
Let G be a compact Lie group, dg its Haar measure (∫dg = 1), g its Lie algebra.
Let Σ be a G-symplectic cone (with compact basis), ω its (invariant) sym-plectic form, λ the Liouville form (ω = dλ). As mentioned above, the basis
10e.g. eiεA with A =√−∆ for some real analytic Riemannian metric on M , cf [16].
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X = Σ/R×+ is a G-compact oriented contact manifold; replacing it by its G-mean, we may choose an invariant form λX defining the contact structure, andΣ is canonically identified with the set of positive multiples of λX in T ∗X.
As was shown in [27, 21], the statements of §1 allow a compact group ac-tion: if M is a compact G-manifold and Σ is embedded as an invariant sym-plectic subcone of T •M , there exists a G-invariant generalized Szego projec-tor associated to Σ 11; if S′ is another one, it induces an equivariant Fred-holm map H → H′, and more generally if u is an equivariant isomorphismΣ ⊂ T •M → Σ′ ⊂ T •M ′, there exists an equivariant adapted FIO F inducingan equivariant elliptic Toeplitz FIO H→ H′.
If S is an equivariant generalized Szego projector, G acts on H and on theToeplitz algebra, so as on their microlocalization µH,AΣ. The infinitesimalgenerators of G (vector fields image of elements ξ ∈ g) define Toeplitz operatorsTξ of degree 1 on H. The elements of G act as unitary Fourier integral operators- or “Toeplitz-FIO’s”.
The Toeplitz space H (and its Sobolev counterparts) splits according tothe irreducible representations of G: H =
⊕Hα (the same will hold for the
equivariant “Toeplitz bundles” below).
9.2.2 Equivariant trace
The G-trace and G-index (relative index in [5]) were introduced by M.F. Atiyahin [5] for equivariant pseudo-differential operators on a G-manifold. The G-traceof P is a distribution on G, describing tr (g P ). Here we adapt this to Toeplitzoperators.
Below we will use the following extension: an equivariant Toeplitz bundleis the range of an equivariant Toeplitz projector P of degree 0 on some HN .The symbol of E is the range of the principal symbol of P ; it is an equivariantvector bundle on X; any equivariant vector bundle on X is the symbol of anequivariant Toeplitz bundle. We will denote by E(s) its space of Sobolev Hs
sections.If E,F are two equivariant Toeplitz bundles, there is an obvious notion of
Toeplitz (matrix) operator P : E → F, and of its principal symbol σd(P ) if itis of degree d, which is a homogeneous vector-bundle homomorphism E → Fon Σ. P is elliptic if its symbol is invertible; then it is a Fredholm operatorE(s) → F(s−d) and has an index which does not depend on s.
Definition 62 We denote char g (characteristic set of g) the closed subcone ofΣ where all symbols of infinitesimal operators Tξ, ξ ∈ g vanish.
char g contains the fixed point set ΣG, whose basis is the fixed point set XG
(because G is compact). The base Z of char g is the set of points of X where all
11e.g. the Szego projector of an invariant embeddable CR structure is invariant.
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Lie generators Lξ, ξ ∈ g are orthogonal to λX . Note that ΣG is always a smoothsymplectic cone and its base XG a smooth contact manifold; char g and Z maybe singular.
Let E be an equivariant Toeplitz bundle. If P : E→ E is a Toeplitz operatorof trace class (degP < −n), the trace function TrGP (g) = tr (gP ) is well defined;it is a continuous function on G. It is smooth if P is of degree −∞ (P ∼ 0). IfP is equivariant, its Fourier coefficient for the representation α is 1
dαtrP|Hα (dα
the dimension of α).
The following result is an immediate adaptation of the similar result of [5]for pseudo-differential operators.
Proposition 63 Let P : E→ E be a Toeplitz operator, with P ∼ 0 near char g.Then TrGP (g) = tr gP is well defined as a distribution on G. If P is equivariant,trP |Hα is well defined (finite), and we have, in distribution sense:
TrGP =∑ 1
dαtrP|Hα χα (48)
where α runs over the set of irreducible representation of G, with dimension dαand character χα.
We have seen above that this is true if P is of trace class. Let DG be abi-invariant elliptic operator of order m > 0 on G, e.g. the Casimir of a faithfulrepresentation (with m = 2); its image DX on X defines an invariant Toeplitzoperator E→ F, with characteristic set char g.
If P ∼ 0 near Σ, we can divide it repeatedly by DX (mod. smoothingoperators) and get for any N :
P = DNXQ+R with R ∼ 0
The degree of Q is degP −mN , so it is of trace class if N is large enough. Weset TrGP = DN
GTrGQ + TrGR: this is well defined as a distribution; the fact that itdoes not depend on the choice of DG, N,Q,R is immediate.
Formula 48 for equivariant operators, obviously follows. Note that the seriesconverges in distribution sense, i.e. the coefficients have at most polynomialgrowth (with respect to the eigenvalues of DG).
More generally assume that we have an equivariant Toeplitz complex of finitelength:
(E, d) : · · · → Ejd−→ Ej+1 → . . .
i.e. E is a finite sequence Ek of equivariant Toeplitz bundles, d = (dk : Ek →Ek+1) a sequence of Toeplitz operators such that d2 = 0. Then for a Toeplitzoperator P : E → E, P ∼ 0 near char g, its equivariant supertrace TrGP =∑
(−1)kTrGPk is well defined; it vanishes if P is a supercommutator.
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9.2.3 Equivariant index
Let E0,E1 be two equivariant Toeplitz bundles. We will say that an equivariantToeplitz operator P : E0 → E1 is G-elliptic (relatively elliptic in [5]) if it iselliptic on char g, i.e. the principal symbol σ(P ), which is a homogeneous equiv-ariant vector bundle homomorphism E0 → E1, is invertible on char g. Thenthere exists an equivariant Q : E0 → E1 such that QP ∼ 1E0 , PQ ∼ 1E1 nearchar g. The G-index IndIGP is then defined as the distribution TrG1−QP −TrG1−PQ.
More generally, an equivariant complex (E, d) as above is G-elliptic if theprincipal symbol σ(d) is exact on char g. Then there exists an equivariantToeplitz operator s = (sk : Ek → Ek−1) such that 1 − [d, s] ∼ 0 near char g([d, s] = ds + sd). The index (Euler characteristic) is the super trace IG(E,d) =supertr (1− [d, s]) =
∑(−1)jTrG(1−[d,s])j .
If P is G-elliptic, for any irreducible representation α, the restriction Pα :E0,α → E1,α is a Fredholm operator: its kernel, cokernel and index Iα arefinite dimensional (resp. more generally the cohomology H∗α of d|Eα is finitedimensional), and we have
Ind IGP =∑ Iα
dαχα (resp. Ind IG(E,d) =
∑(−1)j
dimHjα
dαχα ) (49)
9.2.4 Asymptotic index
The G-index IndGP is obviously invariant under compact perturbation and de-formation, so for fixed Ej it only depends on the homotopy class of the symbolσ(P ). However it does depend on the choice of Szego projectors: as mentioned,the Toeplitz bundles Ej are known in practice only through their symbols Ej ,and are only determined up to a space of finite dimension, so as the Toeplitzspaces H. However if E,E′ are two equivariant Toeplitz bundles with the samesymbol, there exists an equivariant elliptic Toeplitz operator U : E → E′ withquasi-inverse V (i.e. V U ∼ 1E, UV ∼ 1′E). This may be used to transportequivariant Toeplitz operators from E to E′: P 7→ Q = UPV . Then if P ∼ 0on X0, Q = UPV and V UP have the same G-trace, and since P ∼ V UP , wehave TGP − TGQ ∈ C∞(G). Thus the equivariant G-trace or index are ultimatelywell defined up to a smooth function on G.
Definition 64 We define the asymptotic G-trace AsTrGP as the singularity ofthe distribution TrGP (i.e. TrGP mod. C∞(G)).
If P ∼ 0, we have TrGP ∼ 0, i.e. the sequence of Fourier coefficients is ofrapid decrease, O(cα)−m for all m, where cα is the eigenvalue of DG in therepresentation α (where DG is as above a bi-invariant elliptic operator on G).
Definition 65 If P is elliptic on char g, the asymptotic G-index AsIndGP isdefined as the singularity of IndGP .
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It only depends on the homotopy class of the principal symbol σ(P ), and sinceit is obviously additive we get :
Theorem 66 The asymptotic index defines an additive map from KGX−Z(X)
to Sing(G) = C−∞/C∞(G)(Z ⊂ X denotes the basis of char g).
KGX−Z(X) denotes the equivariant K-theory of X with compact support in X−
Z, i.e. the group of stable classes of triples (E,F, u) where E,F are equivariantG-bundles on X, u an equivariant isomorphism E → F defined near Z, withthe usual equivalence relations ((E,F, a) ∼ 0 if a is stably homotopic near Z toan isomorphism on the whole of X). The asymptotic index is also defined forequivariant Toeplitz complexes, exact near Z.
Note the sequence of Fourier coefficients 1dα
trPα is at most of polynomialgrowth with respect to the eigenvalues of DG; if P ∼ 0 it is of rapid decrease.The Fourier coefficients of the asymptotic index are integers, so they are com-pletely determined, except for a finite number of them, by the asymptotic index:AsIndGP = 0 means that the Fourier series of IndGP has finite support.
Example : let Σ be a symplectic cone, with free positive elliptic action ofU(1), i.e. the Toeplitz generator A = 1
i ∂θ is elliptic with positive symbol (this isthe situation studied in [27]). Then the algebra of invariant Toeplitz operators(mod. C∞) is a deformation star algebra, setting as deformation “parameter”~ = A−1. char g is empty and the asymptotic trace or index is always defined.
The asymptotic trace of any element a is the series∑∞−∞ ake
kiθ, ak =tr a|Hk , mod. smooth functions of θ, i.e. the sequence (ak) is known mod.rapidly decreasing sequences. It is standard knowledge that the sequence (ak)has an asymptotic expansion:
ak ∼∑k≤k0
αjk−j . (50)
In this case the asymptotic trace is just as well defined by this asymptoticexpansion, which encodes essentially the same thing as the residual trace.
Remark. For a general the circle group action, with generator A = eiθ,all simple representations are powers of the identity representation, denoted T ,and all representations occurring as indices can be written as sums.∑
k∈ZnkT
k (mod. finite sums) (51)
In fact, using the sphere embedding below, it can be seen that the positive andnegative parts of the series have a weak periodicity property: they are of theform
P±(T, T−1)(1− T±k)k
for a suitable polynomial P± and some integer k; in other words they representrational functions whose poles are roots of 1, and whose Taylor series haveintegral coefficients.
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9.3 K-theory and embedding
It will be convenient (even though not technically indispensable) to reformulatesome constructions above in terms of sheaves of Toeplitz algebras and modules,in particular to follow the index in an embedding (§9.3.2).
9.3.1 A short digression on Toeplitz algebras and modules
As above we use the following notation: for distributions, f ∼ g means thatf − g is C∞; for operators, A ∼ B (or A = B mod. C∞) means that A − Bis of degree −∞, i.e. has a smooth Schwartz kernel. If M is a manifold, T •Mdenotes the cotangent bundle deprived of its zero section; it is a symplectic conewith base the cotangent sphere S∗M = T •M/R+.
As pointed out above, if Σ is a G-symplectic cone, the micro sheaf AΣ ofToeplitz operators acting on µH are well defined with the action of G, up to(non unique) isomorphism, independently of any embedding Σ → T •M . Theasymptotic trace AsTrGP resp. index AsIndGP are well defined for a section P ofAΣ vanishing (resp. invertible) near char g.
If M is a G-manifold and X = S∗M (Σ = T •M), AΣ identifies with thesheaf of pseudodifferential operators acting on the sheaf µH of microfunctionson X (note that even in that case the exact index problem does not make sense:a Toeplitz bundle E on X corresponds to a vector bundle on the cotangent Eon X, not necessarily the pull-back of a vector bundle on M , so E is in generalat best defined up to a space of finite dimension).
It will be convenient to use the language of E-modules. In the C∞ categoryE is not coherent and general E-module theory is not practical. We will juststick to two useful examples.12
IfM is an A-module, resp. a complex of A modules, it corresponds to a sys-tem of pseudodifferential (resp. Toeplitz) operators, whose sheaf of solutions isHomA(M, µH). E.g. a locally free complex of (E , d)-modules defines a Toeplitzcomplex (E, D) = Hom (L,H).
More generally we will say that a E-module M is “good” if it is finitelygenerated, equipped with a filtrationM =
⋃Mk (i.e. EpMq =Mp+q,
⋂Mk =
0) such that the symbol grM has a finite locally free resolution. We denoteσ(M) =M0/M−1, which is a sheaf of C∞ modules on the basis X; since thereexist global elliptic sections of E , grM is completely determined by the symbol,so as the resolution.
It is elementary that a resolution of σ(M) lifts to a “good resolution” ofM,i.e. a good finite locally free resolution of M13. It is also standard that tworesolutions of σ(M) are homotopic, and if σ(M) has locally finite locally free
12In proof of the Atiyah-Weinstein conjecture we need to patch together two smooth em-bedded manifolds near their boundaries: this cannot be done in the real analytic category,where things work slightly better
13the converse is not true: if d is a locally free resolution ofM its symbol is not necessarilya resolution of the symbol of M - if only because filtrations must be defined to define thesymbol and can be modified rather arbitrarily.
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resolutions it also has a global one (because we are working in the C∞ categoryon a compact manifold or cone with compact support, and dispose of partitionsof unity); this lifts to a global good resolution of M.
If M is “good”, it defines a K-theoretical element [M] ∈ KY (X) (Y =suppσ(M)), viz. the K-theoretical element defined by the symbol of any goodresolution (this does not depend on the resolution of σ(M) since any two suchare homotopic).
This works just as well in presence of a G-action (one must choose invariantfiltrations etc.).
The asymptotic trace and index extend in an obvious manner to endomor-phisms of good complexes or modules:
• if M = AN is free, EndA(M) identifies with the ring of N ×N matriceswith coefficients in the opposite ring Aop, and if A = (Aij) vanishes nearchar g we set AsTrG(A) =
∑AsTrG(Ajj).
• If M is isomorphic to the range PN of a projector P in a free module N(this does not depend on the choice of N ), or if A ∈ EndA(M) we setAsTrG(A) = AsTrG(PA).
• If (L, d) is a locally free complex and A is a A = (Ak) endomorphism,vanishing near char g, we set AsTrG(A) =
∑(−1)kAsTrG(Ak) (the Euler
characteristic or super trace; if A,B are endomorphisms of opposite de-grees m,−m, we have AsTrG[A,B] = 0, where [A,B] = AB − (−1)m
2BA
is the superbracket).
• If M is a good A-module, (L, d) a good locally free resolution of M,A ∈ EndA(M), we set AsTrG(A) = AsTrG(A), where A is any extensionof A to (L, d) (such an extension exists, and is unique up to homotopy i.e.up to a supercommutator).
• Finally if M is a locally free complex with symbol exact on char g, or agood A-module with support outside of char g, it defines a K-theoreticalelement [M] ∈ KG
Z (X), and its asymptotic index (the supertrace of theidentity), is the image of [M] by the index map of Theorem 66.
Remark. The equivariant trace or index are defined just as well for modulesadmitting a projective resolution (projective meaning direct summand of someAN , with a projector not necessarily of degree 0). What does not work for thesemore general objects is the relation to topological K-theory.
9.3.2 Embedding
Let Σ be a G-symplectic cone, embedded equivariantly in T •M with M a com-pact G-manifold, and S an equivariant Szego projector. As recalled in §1, the
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range µH of S is the sheaf of solutions of an ideal I ⊂ EM . The correspondingEM -module M = EM/I is good as one can see on the microlocal model.
We have End E(M) = [I : I], the set of ψDO a such that Ia ⊂ I, actingon the right. The map a 7→ TrGa (TrGa f(1) = fa(1)) is an isomorphism fromEnd E(M) to the algebra of Toeplitz operators mod. C∞. M is a E , E ′ bimodule.
If P is a (good) E ′-module, the transfered module is M ⊗E′ P, whichhas the same solution sheaf (Hom (M ⊗ P,H) = Hom (P,Hom (M,H)) andHom (M,H) = H′). Thus the transfer preserves traces and indices.
This extends obviously to the case where Σ is embedded equivariantly inanother symplectic cone Σ ⊂ Σ′: the small Toeplitz sheaf µH is realized asHomAΣ(M, µH′), with M = E/I and I ⊂ E is the annihilator of the Szegoprojector S of Σ.
Theorem 67 Let X ′, X be two compact contact G-manifolds and f : X →X ′ be an equivariant embedding. Then the K-theoretical push-forward (Botthomomorphism) KG
X−Z(X) → KGX′−Z′(X
′) commutes with the asymptotic Gindex.
Let F : AΣ → A′Σ be an equivariant embedding of the corresponding Toeplitzalgebras (above f), and let M be the A′Σ-module associated with the Szegoprojector SΣ. We have seen that transfer P 7→M⊗P preserves the asymptoticindex.
Lemma 68 The K-theoretical element (with support in Σ) [M] ∈ KGΣ (T •M)
is precisely the Bott element used to define the Bott isomorphism KG(X) →KGX(X ′).14
Proof: We have already noticed that M is good; it has, locally (and glob-ally), a good resolution. Its symbol is a locally free resolution of σ(M) =C∞(X)/σ(I). Let us identify a small equivariant tubular neighborhood of Σwith the normal tangent bundle N of Σ in Σ′; N is a symplectic bundle; theideal I endows it with a compatible positive complex structure N c, i.e. the firstorder jet of elements of σ(I) are holomorphic in the fibers of N c; if a, b are suchsymbols we have a, bN = 0; 1
i a, aN 0. In such a neighborhood a goodsymbol resolution is homotopic to the Koszul complex : the Koszul complexis the complex (E, d) with Ep =
∧−p(N c∗) (0 if p > 0), the differential d ata point with complex coordinates z of N is the interior product (contraction)dω = zyω. The K-theoretical element [(E, d)] ∈ GGΣ(Σ′) is precisely the Bottelement.
14if f : X → Y is a map between manifolds (or suitable spaces), the K-theoretical push-forward is the topological translation of the Grothendieck direct image in K-theory (for alge-braic or holomorphic spaces). Its definition requires a spinc structure on the virtual normalof f (cf [29], §1.3) and this always exists (canonically) if X,Y are almost symplectic or almostcomplex, or as here if f is an immersion whose normal tangent bundle is equipped with asymplectic or complex structure.
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E.g. if Σ′ = CN − 0, with Liouville form Im z.dz 15, with basis the unitsphere X = S2N−1, H the space of holomorphic functions on the sphere X ′ =S2N−1, X ⊂ X the diameter z1 = · · · = zk = 0, Σ′, H′ = the functionsindependent of z1, . . . , zk, I is the ideal spanned by the Toeplitz operators T∂k .The transfer moduleM is A/I with I =
∑k0 zjA, its resolution is the standard
Koszul complex.
Remark. It is always possible to embed a compact contact manifold in acanonical contact sphere with linear G-action:
Proposition 69 Let Σ be a G cone (with compact base), λ a G-invariant hor-izontal 1-form homogeneous of degree 1, i.e. Lρλ = λ, ρyλ = 0, where ρ isthe radial vector field, generating homotheties. Then there exists an equivarianthomogeneous embedding x 7→ Z(x) of Σ in a complex unitary representation V c
of G such that λ = Im Z.dZ
In this construction, Z must be homogeneous of degree 12 as above. This applies
of course if Σ is a symplectic cone, λ its Liouville form (the symplectic form isω = dλ and λ = ρyω). We first choose a smooth equivariant function Y = (Yj),homogeneous of degree 1
2 , realizing an equivariant embedding of Σ in V − 0,where V is a real unitary G-vector space (this always exists if the basis iscompact). Then there exists a smooth function X = (Xj) homogeneous ofdegree 1
2 such that λ = 2X.dY . We can suppose X equivariant, replacing it byits mean
∫g.X(g−1x) dg if need be. Since Y is of degree 1
2 we have 2ρydY = Yhence X.Y = ρy(2X.dY ) = 0. Finally we get
λ = Im Z.dZ with Z = X + iY
(the coordinates zj on V are homogeneous of degree 12 so that the canonical
form Im Z.dZ is of degree 1)
15the coordinates zj are homogeneous of degree 12
.
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10 Asymptotic equivariant index :Atiyah-Weinstein index formula.
This section introduces and illustrates the ”equivariant asymptotic index ofToeplitz operators”.16
Let Ω,Ω′ be two bounded Stein domains (or manifolds) with smooth strictlypseudoconvex boundaries X0, X
′0 (these are compact contact manifolds), and f0
a contact isomorphism X0 → X ′0. If H0,H′0 denote the spaces of CR functions(or distributions) on X0, X
′0 (boundary values of holomorphic functions), S, S′
the Szego projectors,17 the map E0 : u 7→ S′(u f−10 ) : H0 → H′0 is Fredholm
(it is an elliptic Toeplitz FIO). The index of E0 was introduced by C. Epstein[52, 53, 54, 55], who called it the relative index of the two CR structures. Aformula for the index was proposed in [114]. A special case was establishedin [56], and a proof of this index formula in the general case was given by C.Epstein in [54], based on an analysis of the situation using the “Heisenberg-pseudodifferential calculus”. (Another proof based on deformation quantizationshould also be possible, using the ideas in [23] and [24].) In this paper wepropose a simpler proof based on equivariant Toeplitz-operator calculus, whichgives a straightforward view. Our formula is described in section 10.3.4. It is isessentially equivalent to the formula proposed in [114], which was stimulated bya problem in the theory of Fourier Integral Operators, a subject in which HansDuistermaat was a pioneer [46].
It is awkward to keep track of the index in the setting of Toeplitz operatorson X0 and X ′0 alone, because we are dealing with several Szego projectors, andToeplitz operator calculus controls the range H of a generalized Szego projectorat best up to a vector space of finite rank18.
To make up for this, we use the ball Ω ⊂ C × Ω defined by tt < φ wheret is the coordinate on C, φ is a smooth defining function (φ = 0, dφ 6= 0 onX0 = ∂Ω, φ > 0 inside - note that this is the opposite sign from the usualone) chosen so that Log 1
φ is strictly plurisubharmonic, so that the boundary
X = ∂Ω is strictly pseudoconvex; such a defining function always exists, e.g.we can choose φ strictly pseudoconvex. X is then a compact contact manifoldwith (positive) action of the circle group U(1). We will identify X0 with thesubmanifold 0 ×X0 of X.
We perform the same construction for Ω′: we will see that there existsan equivariant germ near X0 of equivariant contact isomorphism f : X →X ′ extending f0 such that t′ f is a positive multiple of t, and an ellipticequivariant Toeplitz FIO E extending E0, associated 19 to the contact map f ;
16to appear in ’Geometric Aspects of Analysis and Mechanics’, a conference in honor ofHans Duistermaat, Progress in Math, Birkhaser 2010.
17 The definition of S requires choosing a smooth positive density on X0; nothing of whatfollows depends on this choice.
18 There is no index formula for a vector bundle elliptic Toeplitz operator, although thereis one for matrix Toeplitz operators - a straightforward generalisation of the Atiyah-Singerformula, cf. [17].
19 f is associated to E in the same manner as a canonical map is associated to a FIO.
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the holomorphic spaces H,H′ split in Fourier components Hk,H′k on which theindex is repeated infinitely many times. This construction has the advantage oftaking into account the geometry of the two fillings Ω,Ω′, which obviously mustcome into the picture.
The final result can be then expressed in terms of an asymptotic version ofthe relative index (G-index) of E, derived from the equivariant theory of M.F.Atiyah and I.M. Singer [5]: the asymptotic index, described in §10.3.4, ignoresfinite dimensional spaces and is well defined for Toeplitz operators or Toeplitzsystems; it is also preserved by suitable contact embeddings.
The asymptotic equivariant trace and index are described in §10.1,10.2. Therelative index formula is described and proved in §10.3 (Theorem 126 and 84).
10.1 Equivariant trace and index
10.1.1 Equivariant Toeplitz Operators.
Let G be a compact Lie group with Haar measure dg (∫dg = 1), g its Lie
algebra, and X a smooth compact co-oriented contact manifold with an actionof G: this means that X is equipped with a contact form λ (two forms definethe same oriented contact structure if they are positive multiples of each other);G acts smoothly on X and preserves the contact structure, i.e. for any g theimage g∗λ is a positive multiple of λ; replacing λ by the mean
∫g∗λdg, we may
suppose that it is invariant. The associated symplectic cone Σ is the set ofpositive multiples of λ in T ∗X, a principal R+ bundle over X, a half-line bundleover X.
We also choose an invariant measure dx with smooth positive density on X,so L2 norms are well defined. The results below will not depend on this choice.
It was shown in [27] that there always exists an invariant generalized Szegoprojector S which is a self adjoint Fourier-integral projector whose microsupportis Σ, mimicking the classical Szego projector. S extends or restricts to allSobolev spaces; for s ∈ R we will denote by H(s) the range of S in the Sobolevspace Hs(X), and by H the union.
A Toeplitz operator of degree m on H is an operator of the form f 7→TQf = SQf , where Q is a pseudodifferential operator of degree m. Here we usepseudodifferential operators in a strict sense, i.e. in any local set of coordinatesthe total symbol has an asymptotic expansion q(x, ξ) ∼
∑k≥0 qm−k(x, ξ) where
qm−k is homogeneous of degree m − k with respect to ξ, and the degree mand k ≥ 0 are integers 20. A Toeplitz operator of degree m is continuousH(s) → H(s−m) for all s. Recall that Toeplitz operators give rise to a symboliccalculus, microlocally isomorphic to the pseudodifferential calculus, that liveson Σ (cf. [27]).
In particular, the infinitesimal generators of G (vector fields determined byelements ξ ∈ g) define Toeplitz operators Tξ of degree 1 on H. An element P of
20 We will occasionally use as multipliers operators of degree m = 12
(or any other complexnumber), with k still an integer in the expansion.
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the universal enveloping algebra U(g) acts as a higher order Toeplitz operatorPX (equivariant if P is invariant), and the elements of G act as unitary Fourierintegral operators - or “Toeplitz-FIO”.
H (with its Sobolev counterparts) splits according to the irreducible repre-sentations of G: H =
⊕Hα.
Below we will use the following extended notions: an equivariant Toeplitzbundle E is the range of an equivariant Toeplitz projector P of degree 0 on adirect sum HN . The symbol of E is the range of the principal symbol of P ; it isan equivariant vector bundle on X. Any equivariant vector bundle on X is thesymbol of an equivariant Toeplitz bundle (this also follows from [27]).
10.1.2 G-trace
The G-trace and G-index (relative index in [5]) were introduced by M.F. Atiyahin his joint work with I.M. Singer [5] for equivariant pseudodifferential operatorson G-manifolds. The G-trace of such an operator A is a distribution on G,describing tr (g A). Here we adapt this to Toeplitz operators. Because theToeplitz spaces H or E are really only defined up to a finite dimensional space,their G-trace or index are ultimately only defined up to a smooth function,i.e. they are distribution singularities on G (distributions mod C∞); they aredescribed below, and renamed “asymptotic G-trace or index”.
If E,F are equivariant Toeplitz bundles, there is an obvious notion of Toeplitz(matrix) operator P : E→ F, and of its principal symbol σd(P ) (if it is of degreed), a homogeneous vector-bundle homomorphism E → F over Σ. P is ellipticif its symbol is invertible; it is then a Fredholm operator Es → Fs−d and has anindex which does not depend on s.
If E is an equivariant Toeplitz bundle and P : E→ E is a Toeplitz operatorof trace class21 (degP < −n), the trace function22 TrGP (g) = tr (g P ) is welldefined; it is a continuous function on G. It is smooth if P is of degree −∞(P ∼ 0). If P is equivariant, its Fourier coefficient for the representation α is1dα
trP |Eα (with dα the dimension of α, Eα the α-isotypic component of E).
Definition 70 We denote by char g ⊂ Σ the characteristic set of the G-action,i.e. the closed subcone where all symbols of infinitesimal operators Tξ, ξ ∈ gvanish (this contains the fixed point set ΣG). The base of char g is the set ofpoints of X where all Lie generators Lξ, ξ ∈ g, are orthogonal to the contactform λ; in the sequel we will usually denote it by Z ⊂ X.
21 dimX = 2n − 1. The Toeplitz algebra is microlocally isomorphic to the algebra ofpseudodifferential operators in n real variables, and operators of degree < −n are of traceclass.
22 We still denote by g the action of a group element g through a given representation; forexample if we are dealing with the standard representation on functions, gf = f g−1, alsodenoted by g∗f , g∗−1f , or g−1∗f .
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The fixed point set XG is the base of ΣG because G is compact (there is aninvariant section). Z contains the fixed point set XG. Note that ΣG is alwaysa smooth symplectic cone and its base XG is a smooth contact manifold; char gand Z may be singular.
The following result is an immediate adaptation of the similar result forpseudodifferential operators in [5].
Proposition 71 Let P : E→ E be a Toeplitz operator, with P ∼ 0 near char g(i.e. its total symbol vanishes near char g). Then TrGP = tr (gP ) is well definedas a distribution on G. If P is equivariant, we have, in distribution sense:
TrGP =∑ 1
dα(trP |Eα) χα (52)
where α runs over the set of irreducible representations, dα is the dimensionand χα the character.
We have seen above that this is true if P is of trace class. For the general case,let DG be a bi-invariant elliptic operator of order m > 0 on G (e.g. the Casimirof a faithful representation, with m = 2). Since DG is in the center of U(g),the Toeplitz operator DX : E→ E it defines is invariant, with characteristic setchar g.
If P ∼ 0 near char g, we can divide it repeatedly by DX (modulo smoothingoperators) and get for any N :
P = DNXQ+R with R ∼ 0.
The degree of Q is degP−Ndeg (DG), so it is of trace class if N is large enough.We set TrGP = DN
GTrGQ+ TrGR: this is well defined as a distribution; the fact thatthis does not depend on the choice of DG, N,Q,R is immediate.
Formula (52) for equivariant operators is obvious for trace class operators,and the general case follows by application of DN
X and DNG . Note that the series
in the formula converges in the distribution sense, i.e. the coefficients have atmost polynomial growth.
Slightly more generally, let (E, d)
· · · → Ejd−→ Ei+1 → · · ·
be an equivariant Toeplitz complex of finite length, i.e. E is a finite sequence Ekof equivariant Toeplitz bundles, d = (dk : Ek → Ek+1) a sequence of Toeplitzoperators such that d2 = 0. If the (degree zero) endomorphism P = Pk of thecomplex E is ∼ 0 near char g, its supertrace TrGP =
∑(−1)kTrGPk is well defined;
it vanishes if P = [P1, P2] is a supercommutator with one factor ∼ 0 on char g.
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10.1.3 G index
Let E0,E1 be two equivariant Toeplitz bundles. An equivariant Toeplitz opera-tor P : E0 → E1 is G- elliptic (relatively elliptic in [5]) if it is elliptic on char g,i.e. the principal symbol σ(P ), which is a homogeneous equivariant bundle ho-momorphism E0 → E1, is invertible on char g. Then there exists an equivariantQ : E1 → E0 such that QP ∼ 1E0 , PQ ∼ 1E1 near char g. The G-index IGP isdefined as the distribution TrG1−QP − TrG1−PQ.
More generally,23 an equivariant complex E as above is G-elliptic if theprincipal symbol σ(d) is exact on char g. Then there exists an equivariantToeplitz operator s = (sk : Ek → Ek−1) such that 1 − [d, s] ∼ 0 near char g([d, s] = ds + sd). The index (Euler characteristic) is the super trace IG(E,d) =str (1− [d, s]) =
∑(−1)jTrG(1−[d,s])j .
For any irreducible representation α, the restriction Pα : E0,α → E1,α is aFredholm operator with index Iα, (resp. the cohomology H∗α of d |Eα is finitedimensional), and we have
IGP =∑ 1
dαIαχα (resp. IG(E,D) =
∑j,α
(−1)dα
j
dimHjα χα).
The G-index IGA is obviously invariant under compact perturbation and defor-mation, so it only depends on the homotopy class of σ(P ) once Ej has beenchosen; it does depend on a choice of Ej (on the projector that defines it, or onthe Szego projector), because Ej is determined by its symbol bundle only up toa finite dimensional space; this inconvenience is removed with the asymptoticindex below.
It is sometimes convenient to note an index as an infinite representation(mod finite representations)
∑nαχα. For the circle group U(1), all simple
representations are powers of the tautological representation, denoted J , andall representations occurring as indices have a generating series∑
k∈ZnkJ
k (mod finite sums) (53)
In fact the positive and negative parts of the series have a weak periodicity prop-erty: they are of the form P±(J±1)/
∏i(1− (J±1)ki) for a suitable polynomial
P± and positive integers ki. 24
Here in our relative index problem, only very simple representations of theform m
∑∞0 Jk = m(1− J)−1 (for some integer m) will occur.
23 This reduces to the case of a single operator where the complex is concentrated in degrees0 and 1.
24 This notation denotes the series expansion in positive powers of J±1; it is obviouslyabusive but suggestive - especially if one thinks of the extension to a multidimensional torus;it also represents a rational function whose poles are roots of 1, and whose Taylor series hasintegral coefficients; of which the corresponding distribution on G is the boundary value fromone or the other side of the circle in the complex plane. Something similar occurs for anycompact group, cf. [5].
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10.2 K-theory and embedding
A crucial point in the proof of the Atiyah-Singer index theorem [3] consists inshowing how one can embed an elliptic system A in a simpler manifold where theindex theorem is easy to prove, preserving the index and keeping track of the K-theoretic element [A]. The new embedded system F+A is analogous to a deriveddirect image (as in algebraic geometry), and the K-theoretic element [F+A]is the image of [A] by the Bott homomorphism constructed out of R. Bott’speriodicity theorem (cf.[3]). Here we will do the same for Toeplitz operators.The direct image F+A is even somewhat more natural, as is its relation to theBott homomorphism (§10.2.4). The direct image for elliptic systems does notpreserve the exact index, since this is not defined (because the Toeplitz spaceH is at best only defined mod a space of finite rank); but it does preserve theasymptotic equivariant index.
10.2.1 A short digression on Toeplitz algebras
We use the following notation: for distributions, f ∼ g means that f − g isC∞; for operators, A ∼ B (or A = B mod C∞) means that A−B is of degree−∞, i.e. has a smooth Schwartz-kernel. If M is a manifold, T •M denotes thecotangent bundle deprived of its zero section; it is a symplectic cone with baseS∗M = T •M/R+, the cotangent sphere bundle.
As mentioned above, a compact contact G-manifold always possesses aninvariant generalized Szego projector. More generally, if M is a G manifold,Σ ⊂ T •M an invariant symplectic cone, there exists an associated equivariantSzego projector (cf [27]). If Σ ⊂ T •M,Σ′ ⊂ T •M ′ and f : Σ → Σ′ is anisomorphism of symplectic cones, there always exists an “adapted FIO” F whichdefines a Fredholm map u 7→ F u = S′(Fu) : H → H′ and an isomorphism ofthe corresponding Toeplitz algebras (A 7→ FAF−1, mod C∞).
One can choose F equivariant if f is. Indeed any adapted FIO can be definedusing a global phase function φ on T •(M ×M ′op) such that25
1) φ vanishes on the graph of f , and dφ coincides with the Liouville formξ · dx− η · dy there;
2) Imφ 0, i.e. Imφ > 0 outside of the graph of f , and the transversalhessian is 0; replacing φ by its mean gives an invariant phase; we may setFf(x) =
∫eiφaf(y)dy dηdξ where the density a(x, ξ, y, η)dy dηdξ is a symbol,
invariant and positive elliptic (F is of Sobolev degree deg (a dy dη dξ)− 34 (nx+ny)
(cf. L. Hormander [82]), so a is possibly of non integral degree if we want F ofdegree 0). The transfer map from H to H′ is S′FS.
If M is a manifold and X = S∗M , the cotangent sphere, X carries a canoni-cal Toeplitz algebra, viz. the sheaf ES∗M of pseudo-differential operators actingon the sheaf µ of microfunctions. In general, if X is a contact manifold, we willdenote by EX (or just E) the algebra of Toeplitz operators on X. It is a sheaf
25 op in M ′op refers to the change of sign in the symplectic form on T ∗M ′.
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of algebras on X acting on µH = H mod C∞, which is a sheaf of vector spaceson X; the pair (EX , µH) is locally isomorphic to the pair of sheaves of pseudo-differential operators acting on microfunctions. If X is a G- contact manifold,we can choose the Szego projector invariant, so G acts on EX and µX .
For a general contact manifold, EX is well defined up to isomorphism, in-dependently of any embedding - but no better than that. The correspondingSzego projector (not mod C∞) is defined only up to a compact operator (a littlebetter than that - see below).
10.2.2 Asymptotic trace and index
The symbol bundles Ej of the Toeplitz bundles Ej only determines these up toa space of finite dimension (because, as mentioned above, both the projectordefining them, and the Szego projector, are not uniquely determined by theirsymbols. However, if E,E′ are two equivariant Toeplitz bundles with the samesymbol, there exists an equivariant elliptic Toeplitz operator U : E → E′ withquasi-inverse V (i.e. V U ∼ 1E, UV ∼ 1E′). This may be used to transportequivariant Toeplitz operators from E to E′: P 7→ Q = UPV . Then if P ∼ 0 onZ, Q = UPV and V UP have the same G-trace, and since P ∼ V UP , we haveTrGP − TrGQ ∈ C∞(G).
Definition 72 We define the asymptotic G-trace of P as the singularity of TrGP(i.e. TrGP mod C∞(G)).
The asymptotic trace vanishes if and only if the sequence of Fourier coefficientsof TrGP is of rapid decrease, i.e. O(cα)−m for all m where cα is the eigenvalue ofDG in the representation α. This is the case if P is of degree −∞.
Definition 73 We will say that a system P of Toeplitz operators is G-elliptic(relatively elliptic in [5]) if it is elliptic on char g. When this is the case, theasymptotic G-index (or IGP ) is defined as the singularity of IGP . (We will stilldenote it by IGP if there is no risk of confusion.)
We denote by KG(X − Z) the equivariant K-theory with compact support.By the excision theorem KG(X −Z) is the same as KG
X−Z(X), the equivariantK-theory of X with compact support in X − Z, i.e. the group of stable classesof triples d(E,F, u) where E,F are equivariant G-bundles on X, u an equiv-ariant isomorphism E → F defined near the set Z (the equivalence relation is:d(E,F, a) ∼ 0 if a is stably homotopic (near Z) to an isomorphism on the wholeof X). The asymptotic index is also defined for equivariant Toeplitz complexes,exact near char g.
If u : E→ F is a G-elliptic Toeplitz system or complex, its principal symboldefines is a homogeneous linear map on Σ, invertible on char g. Its restrictionto any equivariant section of Σ defines a K-theoretic element [u] ∈ KG(X −Z) (in case of a complex, u defines the same K-theoretic element as u + u∗ :Eeven → Eodd). The asymptotic index depends only on the homotopy class ofthe principal symbol σ(P ), and since it is obviously additive we get:
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Theorem 74 The asymptotic index of u only depends on the K-theoretic el-ement [u]. It defines an additive map from KG(X − Z) to C−∞(G)/C∞(G),where Z is, as above, the base of char g.
Note that the sequence of Fourier coefficients tr Pαdα
is in any case of polyno-mial growth with respect to the eigenvalues of D or DX ; if P ∼ 0, it is of rapiddecrease. The coefficients Iα
dαof the asymptotic index are integers, so they are
completely determined, except for a finite number of them, by the asymptoticindex.
Remark: if V is a finite dimensional representation of G and V ⊗ P or V ⊗ dis defined in the obvious way, we have IGV⊗P = χV I
GP (i.e. Index (V ⊗ P )α =
(V ⊗ IndexP )α, except at a finite number of places).
E.g. Let G = SU2 acting on the sphere X of V = C2 in the usual manner,and E = SmV the m-th symmetric power . Then E × X is a G bundle withthe action g(v, x) = (gv, gx). The CR structure on the sphere gives rise toa first Szego projector S1(v · f) = v · S(f), where S is the canonical Szegoprojector on holomorphic functions. On the other hand since X is a free orbitof G, the bundle E × X is isomorphic to the trivial bundle E0 × X where E0
is some fiber (i.e. the vector space of homogeneous polynomials of degree m,with trivial action of G). This gives rise to a second Szego projector S0, notequal to the first, but giving the same asymptotic index; we recover the factthat SmV ⊗
∑SkV ∼ (m+ 1)
∑SkV (= in degree ≥ m).
10.2.3 E-modules
For the sequel, it will be convenient to use the language of E-modules. In the C∞
category, E is not coherent; general E-module theory is therefore not practicaland not usefully related to topological K-theory. We will just stick to the twouseful cases below (elliptic complexes or “good” modules).26. Note also that thenotion of ellipticity is slightly ambiguous; more precisely: a system of Toeplitzoperators (or pseudo- differential operators) is obviously invertible mod C∞ ifits principal symbol is, but the converse is not true. The notion of “good”system below partly compensates for this; it is in fact indispensable for a goodrelation between elliptic systems and K-theory.
If M is an E-module (resp. a complex of E modules), it corresponds to thesystem of pseudo-differential (resp. Toeplitz) operators whose sheaf of solutionsis Hom (M, µH); e.g. a locally free complex of (L, d) of E-modules defines theToeplitz complex (E, D) = Hom (L,H).
More generally we will say that an E-module M is “good” if it is finitelygenerated, equipped with a filtrationM =
⋃Mk (i.e. EpMq =Mp+q,
⋂Mk =
0) such that the symbol grM has a finite locally free resolution. We writeσ(M) =M0/M−1, which is a sheaf of C∞ modules on the basis X; since thereexist global elliptic sections of E , grM is completely determined by the symbol,as is the resolution.
26 Things work better in the analytic category.
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A resolution of σ(M) lifts to a “good resolution” of M, i.e. a finite locallyfree resolution 27 of M.
It is standard that two resolutions of σ(M) are homotopic, and if σ(M)has locally finite locally free resolutions it also has a global one (because weare working in the C∞ category on a compact manifold or cone with compactsupport, and dispose of partitions of unity); this lifts to a global good resolutionof M.
If M is “good”, it defines a K-theoretic element [M] ∈ KY (X) (where Y isthe support of σ(M)), viz. the K-theoretic element defined by the symbol ofany good resolution (this does not depend on the resolution since any two suchare homotopic).
All this works just as well in presence of a G-action (if the filtration etc. isinvariant).
As above (§10.1.2), the asymptotic G-trace TrGA [using subscripts as before]is well defined if A is an endomorphism of a good locally free complex of Toeplitzmodules. The same holds for a good module M: the asymptotic trace of A ∈End E(M) vanishing near char g is the asymptotic trace of any lifting of A to agood resolution ofM. (Such a lifting, vanishing near char g, exists and is uniqueup to homotopy, i.e. modulo supercommutators.) Likewise, the asymptotic G-index of a locally free complex exact on Z, or of a good E- module with supportoutside of Z, is defined: it is the asymptotic G-trace of the identity.
Definition 73 of the asymptotic index (or Euler characteristic) extends inan obvious manner to good complexes of locally free E-modules or to goodE-modules. The asymptotic G-index of such an object, when it is G-elliptic,depends only on the K-theoretic element which it defines on the base.
Let us note that the asymptotic trace and index are still well defined forlocally free complexes or modules with a locally free resolution, not necessarilygood; in that case, what no longer works is the relation to topological K-theoryon the base.
10.2.4 Embedding
If M is a manifold, Σ ⊂ T •M a symplectic subcone, the Toeplitz space H isthe space of solutions of a pseudodifferential system mimicking ∂b. If I ⊂ Eis the ideal generated by these operators (mod C∞), and M = E/I, we haveµH = Hom E(M, µ) (as a sheaf: f ∈ Hom (M, µ) 7→ f(1); here as above µdenotes the sheaf of microfunctions). E.g. in the holomorphic situation, I is theideal generated by the components of ∂b.
We have End E(M) = [I : I], the set of pseudo-differential operators a suchthat Ia ⊂ I, acting on the right: if a ∈ [I : I], the corresponding endomorphism
27 The converse is not true: if d is a locally free resolution ofM, its symbol is not necessarilya resolution of the symbol of M – if only because filtrations must be defined to define thesymbol and can be modified rather arbitrarily.
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of M takes f (mod I) to fa (mod I); this vanishes if and only if a ∈ I. Themap which takes a ∈ [I : I] to the endomorphism f 7→ af of H defines anisomorphism from End E(M) to the algebra of Toeplitz operators mod C∞. Mis thus an ET•M−EΣ bimodule (where EΣ ' EndM denotes the sheaf of Toeplitzoperators mod C∞).
This extends immediately to the case where T •M is replaced by an arbitrarysymplectic cone Σ′′ with base X”28. The small Toeplitz sheaf µH can be realizedas Hom E′′(M, µH′′), where M = E ′′/I and I ⊂ E ′′ is the annihilator of theSzego projector S of Σ (i.e. the null-sheaf of I in Hom E′′(M,H′′) = µH). IfP is a (good) E-module, the transferred module is M ⊗E P, which has thesame solution sheaf (Hom E”(M ⊗ P,H′′) = Hom E”(P,Hom E(M,H′′)) andHom E”(M,H′′) = H). Thus the transfer preserves traces and indices.
The moduleM = E ′′/I is generated by 1 (mod I) and has a natural filtration,which is a good filtration: in the holomorphic case, the good resolution is dualto the complex ∂b on (0, ∗) forms.
In general it always has a good locally free resolution, well defined up tohomotopy equivalence. In a small tubular neighborhood of Σ one can choosethis so that its symbol is the Koszul complex on
∧N ′, where N ′ is the dual of the
normal tangent bundle of Σ equipped with a positive complex structure (as inthe holomorphic case). The corresponding K-theoretic element [M] ∈ KG
X(X ′′)is precisely the element used to define the Bott isomorphism (with supportY ⊂ Σ) KG
Y (Σ) → KGY (Σ′′). (Here, Y is some set containing the support
of σ(M) and the map is the product map: [E] 7→ [M][E], where the virtualbundle [E] on Σ is extended arbitrarily to some neighborhood of Σ in Σ′′. 29)
For example if Σ′′ is CN \ 0, with Liouville form Im z · dz and base theunit sphere X ′′ = S2N−1, H′′ is the space of boundary values of holomorphicfunctions, Σ ⊂ Σ′′ consists of the nonzero vectors in the subspace z1 = · · · =zk = 0, and X ⊂ X ′′ is the corresponding subsphere, then H consists of thefunctions independent of z1, . . . , zk, and I is the ideal spanned by the Toeplitzoperators T∂1 , . . . T∂k . In this example the ideal I is generated by z1, . . . , zk, orby Tzj , j = 1 . . . k (On the sphere we have T∂j = (A+N)Tzj with A = TPN
1 zj∂j).
The E-moduleM itself has a global resolution with symbol the Koszul complexconstructed on z1, . . . , zk.
What precedes works exactly as well in the presence of a compact groupaction. If P is a good module with support outside of Z (or a complex withsymbol exact on Z), the transferred module has the same property (Z ⊂ Z ′′),and it has the same G-index (the G-index of the complex Hom E(M,H) 'Hom E′′(M′′,H′′)).
28 We use a double prime here because, eventually, we will be embedding two cones in athird one.
29Toeplitz operators (mod C∞) live on Σ and their principal symbols are homogeneousfunctions on Σ. However the K-theoretic element [u] ∈ KG(X − Z) of a G-elliptic elementlives on the base X, so as the support of “good” E-modules or complexes - in contrast withwhat happens for pseudodifferential operators.
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If X,X ′′ are (compact) contact G-manifolds, f : X → X ′′ an equivariantembedding, P a good G − E-module with support outside of Z (the base ofchar g in Σ), or a Toeplitz complex, exact on Z, the transferred module on X isf+P =M⊗f∗E′ f∗P ′. This is exact outside of f(Σ) and has the same G- indexas P; its K-theoretic invariant [P] is the image of [P] by the equivariant Botthomomorphism. The K- theoretic element [f+P] ∈ KG
X−Z(X) is the image of[P] by the Bott homomorphism (it is well defined since f(Z) ⊂ Z ′′). Thus
Theorem 75 Let f : X → X ′′ be an equivariant embedding. The Bott homo-morphism KG
X−Z(X)→ KGX′′−Z′′(X
′′) commutes with the asymptotic G index.30
It is always possible to embed a compact contact manifold in a canonicalcontact sphere with linear G-action. In fact, it is easier to work with the corre-sponding cones, as follows:
Proposition 76 Let Σ be a G-cone (with compact base), λ a horizontal 1-form, homogeneous of degree 1, i.e. ρyλ = 0 and Lρλ = λ, where ρ is theradial vector field, generating homotheties. Then there exists a homogeneousembedding x 7→ z(x) of Σ in a unitary representation space V c of G such thatλ = Im z · dz.
In the proposition, z(x) must be homogeneous of degree 12 . This applies of
course if Σ is a symplectic cone, λ its Liouville form. (The symplectic form isω = dλ and λ = ρyω).
We first choose a smooth equivariant function y = (yj), homogeneous ofdegree 1
2 , realizing an equivariant embedding of Σ in V −0, where V is a realunitary G-vector space (this always exists if the base is compact; (the coordi-nates zj on V are homogeneous of degree 1
2 so that the canonical form Im z · dzis of degree 1)). Then there exists a smooth function x = (xj) homogeneousof degree 1
2 such that λ = 2x · dy. We can suppose x equivariant, replacingit by its G-mean if need be. Since y is of degree 1
2 we have 2ρydy = y hencex · y = ρyλ = 0. Finally we get
λ = Im z · dz with z = x+ iy.
10.3 Relative index
As indicated in the introduction, we are considering the index of the Fredholmmap E0 : u 7→ S′(uf−1
0 ) from H0 to H′0, where X0, X′0 are the boundaries of two
smooth strictly pseudoconvex Stein manifolds Ω,Ω′, H,H′ the spaces CR dis-tributions (ker ∂b, equal to space of boundary values of holomorphic functions),S, S′ the Szego projectors, and f0 a contact isomorphism X0 → X ′0.
30 As mentioned above the interplay between the Bott isomorphism and embeddings ofsystems of differential or pseudodifferential operators lies at the root of Atiyah-Singer’s proofof the index theorem; it is described in M.F. Atiyah’s works [2, 3, 4, 5], cf also [29] in thecontext of holomorphic D-modules, close to the Toeplitz context.
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As announced we modify the problem and move to the larger boundariesX,X ′ of “balls” |t|2 < φ, |t′|2 < φ′ in C × Ω,C × Ω′, on which the circle groupacts (t 7→ eiλt) (§10.3.1). We will see (§10.3.2) that the Toeplitz FIO E0 de-fines almost canonically an equivariant extension F which is U(1)- elliptic, andIndex (F |Hk) = Index (E0) for all k (Hk ⊂ H(X) is the subspace of functionsf = tkg(x)), so that our relative index Index (E0) appears as an asymptoticequivariant index, easier to handle in the framework of Toeplitz operators.
In §10.3.3 we will show that the whole situation can be embedded in a largesphere, with action of U(1) as in the examples above. In the final result (section10.3.4) the relative index appears as the asymptotic index of an equivariantU(1)-elliptic Toeplitz complex on this large sphere. In general the equivariantindex (asymptotic or not) is rather complicated to compute, but in our case theU(1)-action is quite simple 31, it reduces naturally to the standard Atiyah-SingerK-theoretic index formula on a symplectic ball. The result is better stated interms of K-theory anyway, but it can be translated via the Chern character interms of cohomology or integrals. We give here a (rather clumsy) cohomological-integral translation, essentially equivalent to the result conjectured in [114].
We will also see below (§10.3.2) that f0 has an almost canonical extension fnear the boundary, well defined up to isotopy, not holomorphic but symplectic.We can then define a space Y by gluing together Y+, Y− by means of f . Y is nota Hausdorff manifold, but it is symplectic and both Y+, Y− carry orientationswhich agree on their intersection (as do the symplectic structures). We canfurther choose differential forms ν± representatives of the Todd classes of Y± sothat they are equal near the boundary X0 (the symplectic structures agree, notthe complex structures, but they define the same Todd classes).
Theorem 77 The relative index (index of E0) is the integral∫Y
(ν+ − ν−),where ν± are representatives of Todd(Y±) as above, so that the difference hascompact support in Y −X0.
This will be explained in more detail below (§10.3.4). This formula is relatedto the Atiyah-Singer index formula on the glued space Y , but is not quite thesame since Y is not a symplectic manifold.
To prove the index theorem we will give an equivalent equivariant descriptionof the situation, where the index of E0 is repeated infinitely many times, andembed everything in a large sphere where the index is given by the K- theoreticindex character (§10.3.4).
10.3.1 Holomorphic setting
Let Ω be a strictly pseudoconvex domain (or Stein manifold), with smoothboundary X0 (Ω = Ω∪X0 is assumed to be compact); we write Ω ⊂ C× Ω theball |t|2 < φ, where φ is a defining function (φ = 0, dφ 6= 0 on X0, φ > 0 inside),
31 it is free on the support of the K-theoretic symbol of our complex.
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chosen so that the boundary X = ∂Ω is strictly pseudoconvex, i.e. Log 1φ is
strictly plurisubharmonic (i.e. Im ∂∂ 1φ 0).
The circle group U(1) acts on X by (t, x) 7→ (eiλt, x). We choose as volumeelement on X the density dθ dv where dv is a smooth positive density on Ω(t = eiθ|t|): this is a smooth positive density on X; it is invariant by the actionof U(1), so as the Szego projector S and its range H, the space of boundaryvalues of holomorphic functions.
The infinitesimal generator of the action of U(1) is ∂θ, and we denote by Dthe restriction to H of 1
i ∂θ, which is a self-adjoint, ≥ 0, Toeplitz operator. D isthe restriction of TtT∂t .
The model case is the sphere S2N+1 ⊂ CN+1 with the action
(t = z0, z = (z1, . . . , zN )) 7→ (eiθt, z).
The Fourier decomposition of H
H = ⊕k≥0 Hk (Hk = ker (D − k) )
corresponds to the Taylor expansion of holomorphic functions: the k-th compo-nent of f =
∑fk(x)tk ∈ H is fktk.
H0 identifies with the set of holomorphic functions on X0 (it is the set ofboundary values of holomorphic functions on Ω with moderate growth at theboundary, i.e. |f | ≤ cst d(·, X0)−N for some N , where d(·, X0) is the distanceto the boundary).
Remark: If f = tkg(x) with g continuous, in particular if f ∈ Hk, its L2(X)norm is
‖f‖L2(X) =π
k + 1
∫Ω
φk+1|g(x)|2dv
where as above dv is the chosen smooth volume element on Ω. The restrictionof the Szego projector to functions of the form tkg(x) is thus identified withthe orthogonal projector on holomorphic functions in L2(Ω, φk+1dv). Such se-quences of projectors were considered by F.A. Berezin [10] and further exploitedby M. Englis [49, 50, 51], whose presentation is closely related to the one usedhere.
For the sequel, it will be convenient to modify the factorisation D = t∂t. Webegin with the easy following result.
Lemma 78 Let D = PQ be any factorisation where P,Q are Toeplitz operatorsand [D,P ] = P . Then there exists a (unique) invariant invertible Toeplitzoperator U such that P = tU,Q = U−1∂t.
Indeed it is immediate that any homogeneous function a on σ such that1i ∂θa = ±a is a multiple mt of t (resp. of t), with m invariant. For the
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same reason (or by successive approximations) a Toeplitz operator A such that[D,A] = ±A is a multiple of TtM (or M ′Tt) Tt with M or M ′ invariant (resp.of T∂t , on the right or on the left) . Thus in the lemma above we have P =TtU,Q = U ′T∂t , where U,U ′ are Toeplitz operators which necessarily commutewith D, and are elliptic and inverse of each other.
Note that D = PQ, [D,P ] = P is equivalent to D = PQ, [Q,P ] = 1.
In particular, since D = D∗ = T ∗∂tT∗t , there exists a Toeplitz operator A
such that T∂t = AT ∗t . A is elliptic of degree 1 (in fact invertible), positive sinceD = TtAT
∗t is self-adjoint ≥ 0; it is also invariant: [D,A] = 0.
Definition 79 We will set T = TtA12 ; its symbol is denoted by σ(T ) = τ .
Note that τ is homogeneous of degree 12 , and T is of degree 1
2 , so it is not aToeplitz operator in our strict sense, but for multiplications and automorphismsP 7→ UPU−1 it is just as good. We have
T ∗ = A12T ∗t , [D, T ] = T . D = T T ∗ (54)
(for any other such factorisation D = BB∗ with [D,B] = B, B is of degree 12 ,
and we have B = T U with U invariant and unitary. T is the unique Toeplitzoperator giving such a factorisation and such that T = TtA
′ with A′ a Toeplitzoperator of degree 1
2 , A′ 0).
In what precedes, all = signs can be replaced by ∼ (= mod C∞); we thenget local statements.
The symbol τ = σ(T ) is the unique homogeneous function of degree 12 such
that σ(D) = |τ |2, ∂θτ = iτ, τt > 0.
We also have the following (easy) local result:
Lemma 80 Given any Toeplitz operator K (mod C∞) on H such that D ∼KK∗, [D,K] = K near the boundary, there exists a unique unitary equivariantToeplitz FIO F such that F |H0 ∼ Id , FT ∼ KF .
The geometric counterpart is: given any function k on Σ homogeneous of degree12 such that σ(D) = kk there exists a unique germ of homogeneous symplecticisomorphism f such that f |Σ0 = Id , k f = τ . This is immediate because thetwo hamiltonian pairs Hτ , Hτ , Hk, Hk define real 2-dimensional foliations, andan isomorphism Σ ∼ Σ0 × C near Σ0. Note that this would not work if wereplaced k, k by two functions a, b such that σ(D) = ab, ∂θa = ia but not b = a,because then the ’foliation’ defined by the Hamiltonian vector fields Ha, Hb,although it is formally integrable, is not real.
The operator statement follows, e.g. by successive approximations. Notethat F is completely determined by its restriction F0 if it commutes with T .(In fact in EΣ, the commutator sheaf of T and T ∗ identifies with the inverseimage of EΣ0 - at least as far as the leaves of the Hamiltonian fields HT , HT ∗define a fibration over Σ0: EΣ is the (completed) tensor product of the Toeplitzalgebra Toepl(T , T ∗) generated by T and T ∗ and this commutator: EΣ ∼ EΣ0⊗
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Toepl(T , T ∗) (in a neighborhood of Σ0). In this statement, (T , T ∗) cannot bereplaced by (Tt, T∂t) whose commutator sheaf is only defined in the algebra ofjets of infinite order along Σ0, because the Hamiltonian leaves are complex, nolonger real.)
Note that, in our case, the base of char g is the boundary X0 (the set of fixedpoints), outside of which D is elliptic.
10.3.2 Collar isomorphisms
Let now Ω′ be another strictly pseudoconvex domain (or Stein manifold) withsmooth boundary X ′. We do the similar constructions Ω′, H′, and D′, . . . as inthe previous subsection. Let f0 : X0 → X ′0 be a contact isomorphism.
We define the Fourier Toeplitz operator E0 : u 7→ S′(u f−10 ) : H → H′,
which is a Fredholm operator. It will be convenient to replace E0 by F0 =(E0E
∗0 )−
12E0, which has the same index and is ∼ unitary (E0E
∗0 is an elliptic
≥ 0 Toeplitz operator on H′); (E0E∗0 )−
12 is defined to be 0 on kerE∗0 (mod C∞
would be quite enough). As for Ω, we construct a Toeplitz operator T ′ suchthat D′ = T ′T ′∗, [D′T ′] = T ′, T−1
t T ′ 0.Exactly as in Lemma 10.3.2, there exists a unique (unitary) Toeplitz FIO
F , defined near the boundary X0 and mod C∞, elliptic, such that F |H0 = F0,and FT ∼ T ′F near the boundary (mod C∞).
The geometric counterpart is: there exists a unique equivariant germ ofcontact isomorphism f : X → X ′ (defined and invertible near the boundary)such that f |X0 = f0, τ
′ = τ f .We may extend F , using an invariant cut off Toeplitz operator, so that
it vanishes (mod C∞) away from the boundary. There is an invariant FIOparametrix F ′, i.e. F ′F ∼ 1H, FF
′ ∼ 1H′ , near the boundary.
Proposition 81 For any k, Fk = F |Hk has an index, equal to Index F0.
Proof: both F ′F and FF ′ are invertible on the boundary, so have a G-index;the index of Fk = F |Hk is tr (1 − F ′F )k − tr (1 − FF ′)k. Now T , resp. T ′ isan isomorphism Hk → Hk+1, resp. H′k → H′k+1, and we have Index (Fk+1A) =Index (A′Fk), so IndexFk+1 = IndexFk, i.e. the index does not depend on kand is equal to IndexE0.32
The asymptotic index is stable by embedding; here the index is constant,and the asymptotic index of F (which is essentially a Toeplitz invariant) givesthe index of F0 itself.
10.3.3 Embedding
Theorem 82 Let f : X → X ′ be a collar isomorphism defined in some invari-ant neighborhood of X0 in X. Then for large N there exists equivariant contact
32 For a more general situation where P is a Toeplitz operator elliptic on X0, or where thecanonical Szego projector is replaced by some other general equivariant one, we would onlyget that the index Index (Pk) is constant for k 0. Here the fact that IndexPk = IndexP0
is obvious but important.
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embeddings U : X → S2N+1, U ′ : X ′ → S2N+1 such that U = U ′ f near theboundary, and tX , t′X′ map to positive multiples of tS2N+1 (as above the contactsphere S2N+1 is equipped with the U(1)-action (t, z) 7→ (eiθt, z)).
As usual, it will be more comfortable to work with the symplectic cones. Thesymplectic cone of X is Σ = R+ × X, where we choose the radial coordinateinvariant.
The symbol of D is τ τ with τ/t > 0 as in Definition 79. The Liouville formis Im (τ dτ) + λ0 where λ0 is a horizontal form, i.e. the pull-back of a form onΣb = U(1)\Σ ' R+ × Ω (equivalently: ∂θyλ0 = L∂θλ0 = 0).
Lemma 129 provides an embedding x 7→ zb(x) of Σb in CN ′ − 0 (withthe trivial action of U(1)). We now choose real functions ψ1, ψ2 invariant,homogeneous of degree 0, such that ψ2
1 +ψ22 = 1, with suppψ1 contained in the
domain of definition of f and ψ2 vanishing near the boundary, and we constructa new embedding z in 3 pieces: z = (z1, z2, z3) with z1 = ψ1z0, z2 = ψ2z0, z3 = 0in CN ′′ , N ′′ to be defined below.
Since Im zjzjψjdψj = 0 (zjzjψjdψj is real) we still have Im (z1 · dz1 + z2 ·dz2) = (ψ2
1 + ψ22)Im z0 · dz0 = Im z0 · dz0 inducing λ0. The first embedding
U = (τ, v) : Σ→ C1+N (N = 2N ′ +N ′′).Similarly there exists an embedding x′ 7→ z′0(x′) of Σ′b in CN ′′ − 0 (with
the trivial action of U(1)).
We replace this by z′ = (z′1, z′2, z′3) with z′1 = ψ′1z1 f−1, z′2 = 0, z′3 = ψ′3z
′0
where ψ′1, ψ′3 again are invariant, homogeneous of degree 0, ψ′21 + ψ′
23 = 1, and
suppψ′1 is contained in the domain of definition of f−1, ψ′3 vanishes near theboundary. This also defines an embedding U ′ = (a′, z′) : Σ′ → CN+1; we haveU = U ′ f near the boundary since ψ2, ψ
′3 vanish there.
10.3.4 Index
We are now reduced to the case where both U(1)-manifolds X,X ′ sit in a largesphere S = S2N+1 and coincide near the set of fixed points S0.
As in the preceding section, we can embed the U(1) sheaves µHX , µHX′ assheaves of solutions of two good equivariant ES- modules MX ,MX′ , and theidentification F gives an equivariant Toeplitz isomorphism F near X0 (we canmake the construction so that MX =MX′ , F = Id near X0).
The asymptotic index then only depends on the difference element
d([MX ], [MX′ ], σ(F )) ∈ KU(1)(S− S0).
Now U(1) acts freely on S− S0, with quotient space U(1)\(S− S0) the openunit ball B2N ⊂ CN . We have
Proposition 83 The pull back map is an isomorphism K(B)→ KU(1)(S−S0).We have K(B) ∼ Z, with generator the symbol of the Koszul complex kx at
the origin (or any point of the interior), whose index is 1.
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Its pull-back is the generator of KU(1)S−S0
(S): the symbol is the same, but nowacting on H(S). Its index is
∑∞0 Jk, where (as in (53)) J is the tautological
character of U(1): J(eiλ) = eiλ.
The first assertion is immediate (cf. [5]): if G is a compact group acting freelyon a space Y , the pull back defines an equivalence from the category of vectorbundles on G\Y to that of G-vector bundles on Y (an inverse equivalence isgiven by E 7→ G\E), and this induces a bijection on K-theory (with supports).
The fact that kx defines the generator of K(B)(= K0(B)) is just a re-statement Bott’s periodicity theorem. Its pullback is then the generator ofKU(1)(S − S0): the corresponding complex of Toeplitz operators is then thestandard Koszul complex, acting on holomorphic functions, whose index is thespace of holomorphic functions of z0 = t alone.
Thus if [u] ∈ KU(1)(S − S0), its asymptotic index is m∑∞k=0 J
k, where theinteger m is the value of the K-theoretic character K(B) on the element [uB]whose pull-back is [u].
Let us now come back to our index problem: we have constructed the dif-ference bundle d([MX ], [MX′ ], σ(F )). We may replaceMX ,MX′ by good res-olutions in small equivariant tubular neighborhoods of X, resp. X ′, whoseK-theoretic symbol is the Bott element - the Koszul complex for a positive com-plex structure on the normal symplectic bundle of X, resp. X ′. F lifts to theresolutions (uniquely up to homotopy), and the symbol of the lifting u is anisomorphism near X0 (we can make the construction so that u = Id near X0),so our K-theoretic element is [u] = d(βX , βX′ , u) (the equivariant K-theoreticelement attached to the double complex defined by u).
Theorem 84 Let m be the index of E0 we are investigating. Then, notationsand embeddings being as above,
1) the asymptotic index of our equivariant extension F is the asymptoticindex of the difference element [u] = d(βX , β′X , u) ∈ KU(1)(S − S0), where u isthe symbol of F (i.e. the identity map near S0, where X and X ′ coincide).
2) the index m itself is the value of the index character of K(B) on theelement [uB] = d(βΩ, βΩ′ , u).
The first part has just been proved. The asymptotic index is ∼ m(1− J)−1 forsome integer m.
To prove the second we go down to B2N . The bases of X,X ′ are the em-beddings Y+, Y− of Ω,Ω′ in B, which coincide near the boundary, and as abovethe pullback is an isomorphism KY±(B) → KG
X±(S − S0). The Bott complexes
βX± descend as Bott elements βY± on B, realized as Koszul complexes of pos-itive complex structure of the normal symplectic bundle 33; u descends as anisomorphism near the boundary.
The index m we are looking for is the K-theoretic index character of thedifference element d(βY+ , βY− , u). This can be as usual translated in terms of
33 note that Y± are symplectic submanifolds, not complex; but all positive complex struc-tures are homotopic.
96
cohomology, or as an integral:
m =∫
Bω
where ω is a differential form with compact support, representative of the Cherncharacter of our difference element d(βY+ , βY− , u).
We can push this down further. The construction can be made so thatu = Id near the boundary, choose differential forms ω± with support in smalltubular neighborhoods of Y± so that they coincide near the boundary (so as thetubular neighborhoods), so that ω is the difference ω+ − ω−.
The integral ν± of ω± over the fibers of the respective tubular neighborhoodsis then a representative of the Todd class of Y±; ν+ and ν− coincide near theboundary, so that the difference ν+− ν− has compact support in Y = Y+ ∪ Y−.
Finally the index m is the integral∫Y
(ν+ − ν−) as announced in Theorem126.
The integral can also be thought of as the constant limit∫Y+,ε
ν+−∫Y−,ε
ν−,where the subscript ε means that we have deleted the neighborhood φ < ε inY+ and the corresponding image in Y−.
10.4 Appendix
In this section we show how various symplectic extensions of f0 are related. Itis a little intriguing that, although in our proof, the extension f must be chosenrather carefully so that the asymptotic index of the corresponding Toeplitz FIOE is (asymptotically) the index of E0, the final result, expressed as an integralon the bases glued together by means of f near their boundaries, depends onlyon the isotopy class of f , which is unique.
10.4.1 Contact isomorphisms and base symplectomorphisms
Let X be as above, with X0 the fixed point set of codimension 2. Near theboundary, X is identified with X = X0×C and the base U(1)\X ∼ Ω identifieswith X0×R+; we have φ = tt and the C-coordinate is t =
√φ eiθ (it is smooth on
X). The contact form is λX = Im (tdt− ∂φ) = φdθ + λΩ, where λΩ = −Im ∂φis a smooth basic form. The connection form is γ = dθ − λΩ
φ , and the baseΩ = X0 × R+ is equipped with the (basic) symplectic curvature form
µ = dγ (with γ =λΩ
φ, λΩ = −Im ∂φ) .
We will still use the symplectic cone of X: this is Σ = char g ' R+×X, withLiouville form aλX and symplectic form its derivative, with the R+ coordinatea defined below: with the notation of Lemma 79, we have a = σ(A), i.e. σ(D) =aφ = τ τ , τ = t
√a (as above D = 1
i T∂θ denotes the infinitesimal generator ofrotations). We will also write in polar coordinates τ = ρ eiθ (ρ =
√φa).
Let F be a homogeneous equivariant symplectic transformation of Σ: then Fpreserves σ(D) = τ τ , so we have necessarily F∗τ = u τ , with u invariant, |u| = 1.
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F is then completely determined by its restriction to the boundary, since itcommutes with the two real commuting hamiltonian vector fields ReHτ , ImHτ ,which are linearly independent and transversal to Σ0.
Thus there is a one to one correspondence between unitary functions on thebase Ω and germs near Σ0 = char g of equivariant symplectomorphisms inducingId on char g - or equivalently of contact automorphisms of X inducing Id onX0.
If F is such a contact automorphism, the base map FΩ is obviously a dif-feomorphism of Ω which induces Id on the boundary X0 and preserves thesymplectic form µ.
The converse is not true. If FΩ is a smooth symplectomorphism of Ω inducingthe identity on X0, we have F ∗Ω(λΩ
φ ) = λΩφ + α with α a closed form. It is
elementary that α = cdφφ + β where c is a constant and β is smooth on theboundary. Locally on X0, FΩ lifts to X or Σ: the lifting is F : (x, τ) 7→ (x′, τ ′ =τeiψ) (θ′ = θ + ψ) where ψ is a primitive of α (this is not smooth at theboundary, only continuous). It is immediate that conversely any α of the formabove gives rise to such a contact isomorphism with smooth base map. (on Σthe horizontal (invariant) coordinates satisfy Hτeiψf = 0; the horizontal partof the Hamiltonian Hτeiψ is −iτeiψ(∂ρ − H0
ψ) (with H0ψ = ψξj∂xj − ψxj∂ξj );
finally ∂ρ − H0ψ is smooth so the horizontal coordinates (x′, ξ) are determined
by smooth differential equations.) Summing up:
Theorem 85 The map which to a germ of contact isomorphism F (near X0)assigns the invariant unitary smooth function u such that F ∗τ = τu is one toone (and continuous). In particular the homotopy class of F is determined bythat of u (an element of H1(X,Z)).
The map which to a smooth germ of symplectomorphism FΩ (near X0) as-signs the closed one-form α = c dφφ + smooth is one to one, the group of suchsymplectomorphisms is contractible. The contact lifting (which exists locally,and globally if α is exact) is smooth on X0 if and only if c = 0.
The fact that this group is contractible (connected) simplifies the final result,namely: in the proof of Theorem 84 it was essential that the base map FΩ havea smooth symplectic extension preserving τ > 0; for Theorem 126 however anysymplectic FΩ will do since these are all isotopic.
10.4.2 Example
(A smooth symplectic automorphism of the base does not lift to a smoothequivariant contact automorphism of the sphere.)
Let S be the unit sphere in CN+1, with coordinates x0 = t, x1, . . . , xN .U(1) acts by t 7→ eiθt. The base is B = S/U(1), the unit ball of CN .The contact form is Im tdt+λ = φdθ+λ with λ =
∑xjdxj , φ = tt = 1− xx.
The connection form is γ = dθ + λφ , its curvature is the symplectic form
µ = dλφ (on the interior of B).
98
Let FB be the diffeomorphism of B defined by x 7→ x′ = FB(x) = eciφx, c aconstant; this preserves φ and the inverse is x = e−ciφx′. We have
F ∗Bλ = Im (x(dx+ cix dφ)) = λ+ c(1− φ)dφ
Since d(1− φ)dφφ = 0, FB is symplectic (F ∗Bµ = µ).But FB does not lift to a smooth equivariant contact automorphism of S:
such a lifting F must preserve the connection form, so it is of the form
t 7→ e−iαt (θ 7→ θ − α) with α = cLogφ− φ+ cst
(dα = c(1− φ)dφφ ), and this is not smooth at the boundary t = 0 if c 6= 0.
Of course the reverse works: if F is a smooth equivariant contact automor-phism of the sphere S (or a germ of such near the fixed diameter S0), the basemap FB is a smooth symplectomorphism of the ball B (up to the boundary).
10.4.3 Final remarks
1) The preceding construction applies in particular to the following situation:let V,W be two compact manifolds, and f0 a contact isomorphism S∗V → S∗W .
We may suppose V real analytic; then S∗V is contact isomorphic to theboundary of small tubular neighborhoods of V in its complexification. Forexample if V is equipped with an analytic Riemannian metric, and (x, v) 7→ex(v) denotes the geodesic exponential map, the map (x, v) 7→ ex(iv) is welldefined for small v and for small ε it realizes a contact isomorphism of thetangent (or cotangent) sphere of radius ε to the boundary of the complex tubularneighborhood of radius ε (cf. [16]).
The corresponding FIO’s can be described as follows: as above there existsa complex phase (as in [102, 101] function φ on T ∗W × T ∗V 0 such that 1) φvanishes on the graph of f0 and dφ = ξ.dx − η.dy there, 2) Imφ 0 i.e. it ispositive outside of the graph and the transversal hessian is 0. φ is then aglobal phase function for FIO associated to f0 (φ is not unique, but obviouslythe set of such functions is convex, hence contractible).
The elliptic FIO’s we are interested in are those that can be defined by apositive symbol (or a symbol isotopic to 1):
f 7→ g(x) =∫eiφa(x, ξ, y, η)f(y)dydηdξ with a > 0 on the graph .
The degree of such operators depends on the degree of a, but they all have thesame index, given by the formula above.
2) The formula above extends also to vector bundle cases: if E,E′ are holo-morphic vector bundles (or complexes of such) on Ω,Ω′, f0 a contact isomor-phism (∂Ω → ∂Ω′) as above, and A a smooth (not holomorphic) isomorphismf0∗E → E′ on the boundaries, the Toeplitz operator a 7→ S′(Af0∗a) is Fredholm
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and its index is given by the same construction as above. For this constructionf0 only needs to be defined where the complexes are not exact.
In particular let Ω,Ω′ have singularities (isolated singularities, since we stillwant smooth boundaries): we can embed Ω,Ω′ in smooth strictly pseudoconvexdomains Ω, Ω′ of the same (higher) dimension; the contact isomorphism extendsat least in a small neighborhood of ∂Ω in ∂Ω. The coherent sheaves OΩ,OΩ′
have global locally free holomorphic resolutions on Ω, Ω′; near the boundarythese are homotopy equivalent to a Koszul complex, hence equivalent.
The theorem above shows that the relative index is the K-theoretical charac-ter of the difference virtual bundle d([OΩ], [OΩ′ ]) (vanishing near the boundary).Note however that the virtual bundles [OΩ], [OΩ′ ] lie in the K-theory of Ω withsupport in Ω. This can be readily described in terms of cohomology classeson Ω etc. with support in Ω, not on Ω itself (the relation between coherentholomorphic modules and topological K-theory, or K-theory and cohomology, isnot good enough when there are singularities).
100
11 Complex Star Algebras.
In this chapter we describe a classification of star algebras on the cotangentbundle of a complex manifold, locally isomorphic to the algebra of pseudo-differential operators ; this requires a slight extension of the usual definition ofstar algebras. We show that in dimension ≥ 3 these are essentially trivial andcome from algebras of differential operators on X ; in dimension 1 and 2 thereare many more, which we describe. 34
11.1 Introduction
Let us first recall what a star-product is (detailed definitions are given in section2) : let X be a manifold and let O denote the algebra of formal series
f =∑k≥k0
fkhk
where the fk are smooth functions on X and h is a “small” formal parameter.A star product on X is a unitary algebra law on O for which the unit is 1 andthe product is local, i.e. given by a formula :
f, g → B(f, g) = fg +∑k≥k0
hkBk(f, g)
where the Bk are bidifferential operators on X : in local coordinates Bk(f, g) =∑aαβ∂
αf∂βg with smooth coefficients aαβ (it is further required that the unitis 1, i.e. B0(f, g) = fg and Bk(1, f) = Bk(f, 1) = 0 for any k > 0 and anyf ; the addition law is the usual addition of O. A star product can be thoughtof as a non-commutative deformation of the usual product. The leading termof commutators f, g = hB1(f, g)− hB1(g, f) defines a Poisson bracket on X(star products are also called “deformation quantization of Poisson manifolds”).
In this paper I will use a slightly extended definition, where star products liveon cones. A cone Σ with basis BΣ = X is the complement of the zero sectionin a line bundle L → X (a complex line bundle if X is a complex manifold,and preferably a half-line bundle if X is real) ; in the semi-classical case aboveΣ = X × R×+ and h = 1
r if r denotes the fiber variable (the small “Planckconstant” plays the role of the inverse of a large frequency). In this contextO is the set of formal series f =
∑k≤k0
fk where for each k, fk is a functionhomogeneous of degree k on Σ and, locally, a star product is defined as aboveas a bidifferential product law on O
f, g → B(f, g) = fg +∑k≤k0
Bk(f, g)
where Bk is now a bidifferential operator on Σ, homogeneous of degree k → −∞with respect to fiber homotheties. The Bk may involve derivations in any direc-tion, so there is no longer a distinguished “Planck constant” commuting with
34Mathematical Physics, Analysis and Geometry 00: 1-27, 1999.
101
the rest 35. The associated Poisson bracket now lives on Σ and is homogeneousof degree −1. This definition includes the algebras of pseudo-differential oper-ators or Toeplitz operators, which are after all among the most important andbelong to the same formalism.
Complex star algebras arrive naturally and unavoidably in many problemsconcerning differential operators, whose symbols are polynomials and alwayslive on a complex manifold. So it is important to study them, and to studytheir relations with “polynomial” objects associated to differential operators.
In his paper [91] M. Kontsevitch has shown that any homogeneous Poissonbracket on a real manifold comes from a star product. His proofs extend withoutchanging a word to star -products on a cone. Kontsevitch’s formula giving astar product from a Poisson bracket on an affine space also works without anymodification in the complex case (i.e. Σ = Cn×C×). But the argument used togo from local to global does not work for complex manifolds, because it uses in anunavoidable manner partitions of unity and tubular neighborhoods. In generalI do not know if a global star product exists for a given Poisson bracket, even inthe symplectic case, nor do I know what the classification of such algebras lookslike (see however [88], where it is shown that even if such an algebra E may notexist, the category of sheaves of E-modules can be defined up to equivalence).
In this paper I investigate those star algebras which live on a complex cotan-gent cone T ∗X − 0 deprived of its zero section, equipped with its canonicalsymplectic Poisson bracket. All star algebras associated to this Poisson bracketare locally isomorphic, and there exists a global such algebra, viz. the algebraof pseudo-differential operators ; so there is at least a starting point for theclassification. This will turn out to be essentially trivial in dimension n ≥ 3(Theorem100), but instructively not in dimension 2. More precisely algebrasover a manifold X of dimension 2 or ≥ 2 are described in section 4, and com-pared to D-algebras, i.e. sheaves of algebras over X locally isomorphic to E , thealgebra of pseudo-differential operators coming from differential operators on X.It turns out that if dimX ≥ 3 we get nothing new : the functor which takes a D-algebra to the associated star-algebra is an equivalence. If dimX = 2 the samefunctor is fully faithful, i.e. two D-algebras are isomorphic if and only if theassociated star-algebras are isomorphic, and an isomorphism between such star-algebras comes from a unique isomorphism between the original D-algebras ;however there are in general many more “exotic” star-algebras which do notcome from a D-algebra.
If X is of dimension 1 the classification depends on whether X is open, ofgenus ≥ 2, of genus 1 or of genus 0.
An inner automorphism of the algebra E of pseudo-differential operators onX (U : P → APA−1) has a symbol σ(U) = dLog σ(A), which is a section ofthe sheaf ω (on the “basis” BΣ = Σ/C× of closed forms homogeneous of degree0 on Σ, and an exponent which is the degree of A ; we will see in section 2 thatany automorphism U of E has likewise a symbol and an exponent ∈ C. Similarly
35 There is absolutely no reason that the Planck constant should commute with the rest,especially when it is a parameter without physical significance
102
a star algebra has a symbol σ(A) ∈ H1(BΣ, ω) and an exponent ∈ H1(BΣ,C).We will see in section 3 that if X is an open curve or a curve of genus ≥ 1, staralgebras on Σ are completely determined by their exponent. The classificationis more subtle when X is closed of genus 1 or 0.
The techniques used in this paper are a mixture of non-commutative coho-mology, holomorphic cohomology, and the relation between the cohomology ofa sheaf with a filtration and the cohomology of the associated graded sheaf.This contains nothing really new or difficult, but the mixture can be somewhatmuddling.
As far as I know the questions studied here have not been investigated beforeand the results are new.
In sections 2 and 3 we recall the definition of star algebras, and some clas-sification principles.
In section 4 we describe the classification when dimX ≥ 2.
In section 5 we describe the case where X is a curve (dimX = 1) : resultsare substantially different if X is open, X = P1, X is of genus 1, or X is ofgenus g ≥ 2.
11.2 Star Algebras
11.2.1 Cones
Definition 86 A real (resp. complex) cone is a C∞ (resp. holomorphic) prin-cipal bundle Σ with group R×+ (resp. C×). The basis is BΣ = Σ/R×+ (resp.Σ/C×).
A real cone is isomorphic to a product cone BΣ × R×+. 36 A complex cone isisomorphic to L−0 (L deprived of its zero section) where L is a complex linebundle over BΣ. L will usually not be a trivial bundle.
Definition 87 (i) We denote O(m) the sheaf on BΣ of homogeneous functionsof degree m of Σ (holomorphic in the complex case).
(ii) We denote O the sheaf on BΣ of formal symbols (“asymptotic expansions”for ξ →∞ in Σ) :
f ∈ O if f =∑m≤m0
fm with fm ∈ O(m) (55)
(m an integer, m→ −∞).
36 at least if we are dealing with paracompact manifolds, which will always be the case inthis article.
103
Definition 88 For an integer k ≥ 1 we denote Dk the sheaf (on BΣ) of formalk-differential operators : P (f1, . . . , fk) =
∑m≤m0
Pm(f1, . . . , fk) with Pm a k-linear differential operator homogeneous of degree m with respect to homotheties(m an integer, m→ −∞).
If k = 1 we will just write D.
Locally Σ is a product cone and we may choose homogeneous coordinates(real or complex) xj of degree 0 on the basis, and r of degree 1 on the fiber.Then Pm(f1, . . . , fk) is a sum of monomials
ϕ(x) rm ∂α1x (r∂r)m1(f1) . . . ∂αkx (r∂r)mk(fk).
There is no restriction on the order of Pm.
The presence of two “degrees” is confusing so in what follows degree willalways refer to the degree with respect to homotheties, and order refers to thedegree as a differential operator; thus if P ∈ Dk each term Pm of degree m is offinite order, although the resulting infinite sum P may be of infinite order.
We will denote D× ⊂ D the sheaf of invertible formal differential operators :P =
∑Pk ∈ D× is invertible iff its leading term σ(P ) = Pm0 is invertible, i.e.
Pm0 is of order 0, the multiplication by a nonvanishing function homogeneousof degree m0. We denote by D×− the subsheaf of those invertible P such thatP (1) = 1, i.e. P is of degree 0, its leading term is P0 = 1 and terms of lowerdegree have no constant term : Pm(1) = 0 if m < 0.
Remark 1 Sheaves are of course useless in the real case but must be used inthe complex case where global sections do not necessarily exist.
Remark 2 For analytic cones there is also a notion of convergent symbol(introduced by the author in [13] to define analytic pseudodifferential operators).These are in fact the more important and for many questions it is essential touse convergent rather than formal symbols.37 However for the classificationresults below, there is no significant qualitative difference between formal andconvergent symbols, so we will stick to formal symbols and avoid convergencetechnicalities.
11.2.2 Star Products on a Real or Complex Cone.
Definition 89 A star product on Σ is a sheaf A on the basis BΣ, locally iso-morphic to O as a sheaf of vector spaces (the structural sheaf of groups is de-scribed below), equipped with an associative unitary algebra law whose product(star product) f ∗ g = B(f, g) is locally a formal bidifferential operator.
37 e.g. convergent rather than formal symbols are essential in the finiteness theorems ofT. Kawai and M. Kashiwara [91], or for going from E-modules to D-modules in the thesisof D. Meyer [103], and probably in most problems involving a comparison between E andD-modules.
104
Locally f ∗ g =∑Bm(f, g) with Bm a bidifferential operator homogeneous
of degree m→ −∞, B0 = 1. The first idea is that the structural sheaf of groupsused to patch together local frames of A is the sheaf D× (on BΣ) of invertibleformal differential operators, but there is a unit that we can choose equal to 1in all local frames so this obviously reduces to D×−.
Note that homotheties (hence degrees) are not respected by D×−. Howeverif P ∈ D×−, f and Pf have the same leading term ; so P respects the filtrationdefined by degrees (f ∈ Om if f =
∑j≤m fj) and grP is the identity on gr O =⊕
O(m).
In the semi-classical definition, Σ is a product cone Σ = BΣ × L (L = R×+or C×), the star product law is defined on O and does not involve verticalderivatives, so the “Planck constant” h = r−1 plays the role of a constant.The definition above includes the “semi-classical” case and also the algebras ofpseudodifferential or Toeplitz operators. This conic framework for star productswas described in [20].
In the real case, using partitions of unity, it is immediate to see that Ais always isomorphic to O as a sheaf (“there exists a global total symboliccalculus”). This is no longer true in the complex case, and in particular it isnot true in the most simple and natural examples as we will see below, so thesheaf theoretic presentation cannot be avoided.
11.2.3 Associated Poisson bracket
If A is a star algebra on Σ it has a canonical filtration coming from the filtrationof O by homogeneity degrees, and there is a canonical isomorphism :
grA ' gr O
because the structural sheaf of groups D×− induces the identity on gr O). Thecommutator law then defines a Poisson structure on grA = gr O i.e. the leadingterm of the commutator law
f, g = B1(f, g)−B1(g, f)
is a Poisson bracket on Σ, homogeneous of degree −1. This means that it is askew-symmetric bivector field
f, g = −g, f, f, gh = f, gh+ gf, h
satisfying the Jacobi identity (i.e. it is a Lie bracket) :
fg, h = f, gh+ gf, h
and it is homogeneous of degree −1 with respect to homotheties
deg f, g = deg f + deg g − 1 if f, g are homogeneous.
105
Existence of a global star-algebra on a real symplectic cone Σ was provedby V. Guillemin and myself in [27] (see also [17]), and by M. De Wilde andP. Lecomte ([38],[39]) in the semiclassical symplectic case (cf. also the nicedeformation proof of B.V. Fedosov [60]).
In [91] M. Kontsevitch proved that any Poisson bracket comes from a star-product in the real semiclassical case. More precisely he proves that there isa one to one correspondence between isomorphic classes of star-products andisomorphic classes of formal families of Poisson brackets depending on the “smallparameter” h. His result extends, without changing a word, to star-productson a real cone with the definition above ; families of Poisson brackets should bereplaced by formal Poisson brackets on Σ :
c =∑k≤−1
cm. (56)
Kontsevitch’s formula giving a star product from a Poisson bracket on an affinespace also works without any modification in the complex case (i.e. Σ = Cn ×C×). But as mentioned above the argument used to go from local to global doesnot work for complex manifolds, and in general I do not know if a global starproduct exists for a given Poisson bracket, even in the symplectic case, nor do Iknow what the classification of such algebras looks like (see however [88], whereit is shown that even if E may not exist, the category of sheaves of E-modulesis defined up to equivalence).
In the rest of the paper we investigate a special class of star algebras, i.e.those which live on a cotangent bundle Σ = T ∗X −0, X a complex manifold,equipped with its canonical Poisson bracket. In this case there is a canonicalglobal star-algebra, viz. the algebra E of pseudo-differential operators, whichis the “microlocalization” of the sheaf DX of differential operators on X. Itis known and easy (cf. below) that any two star algebras with the same sym-plectic Poisson bracket are locally isomorphic, so our algebras are classified byH1(BΣ,Aut E). It is also interesting to compare these with algebras of differen-tial operators, locally isomorphic to DX on X hence classified by H1(X,AutD) :this is done in the next three sections.
11.3 Pseudo-differential Algebras
11.3.1 E-algebras
Let Σ = T ∗X − 0 be the cotangent bundle (minus the zero section) of acomplex manifold X, equipped with its canonical symplectic structure. Thebasis is BΣ = Σ/C× = PX, the projective cotangent bundle. There is acanonical star algebra on Σ, viz. the algebra of pseudo-differential operators,microlocalization of the algebra of differential operators on X, whose Poissonbracket is the standard Poisson bracket of T ∗X. If we choose local coordinatesx = (x1, . . . , xn) on X and the dual cotangent coordinates ξ = (ξ1, . . . , ξn) on
106
the fibers, the pseudodifferential product is given by the Leibniz rule for symbolsf, g ∈ O :
f ∗ g =∑ 1
α!∂αξ f ∂
αx g. (57)
The patching cocycle is the cocycle defined by changes of coordinates : this isa cocycle because it does patch together total symbols of differential operators(locally : polynomials in ξ), to give the global sheaf DX of differential operators.
We are interested in star algebras on Σ associated to the canonical Poissonbracket : we will call E-algebra such an algebra.
Proposition 90 Any E-algebras is locally isomorphic to E through an operatorP ∈ D×−.
This result is well known and we just give an indication of the proof : locallythe pseudo-differential algebra E has (topological) generators xi, ξi satisfyingthe canonical relations
[xi, xj ] = [ξi, ξj ] = [ξi, xj ]− δij = 0.
If A is a star algebra with the same Poisson bracket, one can construct bysuccessive approximations symbols Xi,Ξi with the same principal part as xi, ξiand satisfying the same canonical relations
[Xi, Xj ]A = [Ξi,Ξj ]A = [Ξi, Xj ]A − δij = 0.
Now there is a unique isomorphism U : E → A which takes xi to Xi and ξi toΞi and this is always a differential operator U ∈ D×−.
Remark 3 The construction also works globally over any open subcone U ⊂T ∗Cn which is Stein and contractible (e.g. the set ξi 6= 0 ⊂ T ∗B, B a ball inCn, or a Stein contractible set). Over such a set, any E-algebra A is isomorphicto E , and any section α of O(m) is the symbol of a section of Am.
Thus one obtains all E-algebras by gluing together models of E over a cover-ing of Σ by open conic subsets Σi, using automorphisms of E on the intersections.The following proposition sums up what was said above :
Proposition 91 Star algebras on Σ = T ∗X −0 are locally isomorphic to thepseudo-differential algebra E. The set Alg E of isomorphy classes is canonicallyisomorphic to H1(PX,Aut E).
Aut E denotes the sheaf of automorphisms of E ; the noncommutative coho-mology H1(PX,Aut E) is described below in section 3.4.
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11.3.2 Differential Operators and D-algebras
If X is a complex manifold, the sheaf DX of differential operators on X iswell defined. If U is an automorphism of DX preserving symbols, it fixes thesubalgebra OX ⊂ DX of operators of order 0, (because it fixes symbols andpreserves invertible operators, which are necessarily of order 0). It follows thatU is locally an inner automorphism of the form Int ef (f holomorphic). Wehave Int ef = Id iff f is (locally) constant, so the automorphism sheaf is
Aut DX ' O×X/C× ' OX/C. (58)
We will call D-algebra a sheaf of algebras on X locally isomorphic to DX (suchalgebras appear in [7] where they are called “twisted algebras of differentialoperators”). The set AlgD of isomorphic classes of these algebras is canonicallyisomorphic to H1(X,OX/C).
A D-algebra obviously also defines a star-algebra on PX, and it is naturalto compare the two sets AlgD and Alg E .
11.3.3 Automorphisms and Symbols of Automorphisms
To understand how local E-algebras can be patched together to make globalobjects, we have to know what automorphisms of E look like.
Let U ∈ D×− be an automorphism of E : U preserves symbols and the unit 1,so U−1 is of degree ≤ −1 and the logarithm D = LogU = −
∑n≥1−
1n (1−U)n
is well defined ; it is a derivation of degree ≤ −1 of E .Now if D is a derivation of degree ≤ k its symbol δ = σk(D) is a homogeneous
derivation of degree k of the Poisson algebra O, i.e. a symplectic vector field onΣ, homogeneous of degree k. This corresponds, via the symplectic structure ofΣ, to a closed differential form α, homogeneous of degree k + 1.
Let ρ denote the radial vector field, infinitesimal generator of the action ofC× (ρ =
∑ξj∂ξj in local coordinates on X,T ∗X as above) : the associated Lie
derivation is Lρ = iρd+ diρ (iρ denotes the interior product) so
diρ α = (k + 1)α.
Hence α is exact (the differential of a homogeneous function) if k + 1 6= 0. Ifk + 1 = 0, s = iρ α is locally constant, and α is locally the differential of ahomogeneous function of degree 0 iff s = 0.
By successive approximations, it follows that locally any derivation D of Eis of the form s ad(LogP1) + adQ with P1 elliptic of degree 1, Q ∈ E , and anyautomorphism of E is locally of the form
U = (IntP1)sIntQ0 (59)
with P1 elliptic of degree 1, Q0 elliptic of degree 0. 38 IntP denotes the innerautomorphism a→ P aP−1.
38 as usual in the context of pseudodifferential operators, elliptic = invertible.
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If U is an automorphism of E , we define its symbol σ(U) as the closed 1-formon Σ homogeneous of degree 0 corresponding to the leading term of LogU .
We have σ(U) = dLogσ(P ) if U = IntP .global section of ω (this is a closed1-form on Σ). If σ(U) = 0 (LogU of degree ≤ 2) there exists a unique P ∈ E×of degree 0 and symbol 1 such that U = IntP . Summing up we have proved :
Proposition 92 There is an exact sequence of sheaves of groups on PX:
0→ E×− → Aut E → ω → 0 (60)
where E×− denotes the multiplicative sheaf of groups on BΣ of sections of E ofsymbol 1, and ω is the sheaf on PX of closed 1-forms homogeneous of degree 0on Σ.
If A ∈ Alg E ' H1(PX,Aut E) its symbol σ(A) ∈ H1(PX,ω) is defined asthe image cocycle.
Remark 4 If U is an automorphism of A, it defines a one parameter groupUs = exp sLogU, s ∈ C. This is polynomial in s mod.An for any n < 0.
11.3.4 Non Commutative Cohomology Classes
In this section we recall the elementary resuls of noncommutative cohomologythat we will use (for more information see [69]). Let Y be a space and G a sheafof groups on Y . We denote H0(Y,G) = Γ(Y,G) the set of global sections of Gover Y : this is a group.
We denote H1(Y,G) the set of equivalence classes of cocycles
uij ∈ Γ(Yij = Yi ∩ Yj ,G) such that uijujk = uik
associated to open coverings Y =⋃Yi ; two cocycles are equivalent if, after
a suitable refinement of the covering, we have uij = uiu′iju−1j for some family
ui ∈ Γ(Yi,G).H1(Y,G) classifies the set of isomorphy classes of G principal homogeneous
right G sheaves, i.e. sheaves α on Y , equipped with a right action of G, locallyisomorphic to G considered as a right G-sheaf.
Proposition 93 Let0→ A→ B → C → 0 (61)
be an exact sequence of sheaves of groups on Y , with A normal in B. Thenthere is a long cohomology sequence ;
0→ H0(Y,A)→ H0(Y,B)→ H0(Y,C)→ (62)
→ H1(Y,A)→ H1(Y,B)→ H1(Y,C).
This is “exact” in the sense that
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i) it is exact at the first three places (the H0 are groups, the H1 are pointedsets).
ii) The group H0(Y,C) acts on the set H1(Y,A), and its orbits are the fibersof the map H1(Y,A) → H1(Y,B) (the action is given by c · (aij) = (biaijb−1
j )if c is a global section of B, and bi ∈ Γ(Yi, B) a lifting of c to B over a fineenough covering Yi).
iii) If β ∈ H1(Y,B) it defines twisted sheaves of groups Aβ ⊂ Bβ (where Bβis the sheaf of B-automorphisms of the principal B-sheaf β), and the fiber ofthe map H1(Y,B)→ H1(Y,C) is the image of H1(Y,Aβ) in H1(Y,C).
More explicitly if β, β′ are two principal B-sheaves, then γ = HomB(β, β′)is a principal Bβ-sheaf. If β, β′ have the same image in H1(Y,C) then γ/Aβ =HomC(β/A, β′/A) has a global section, i.e. is trivial, so γ is the image of asheaf α ∈ H1(Y,Aβ). Finally β′ ∼ α×Aβ β is in the image of H1(Y,Aβ).
In this paper the noncommutative cohomology sequence stops there, and wewill not use higher cohomology Hj , j ≥ 2 whose definition is more elaborate(the substitutes are more complicated objects sometimes described by means of“stacks”). Exact sequences concerning torsors as above were introduced by J.Frenkel [65, 66]. Of course if A,B,C are commutative, the higher cohomologygroups Hj , j ≥ 0 are well defined commutative groups, and we will occasionallyuse the long cohomology exact sequence in that case up to order j = 2.
11.3.5 Symbols
If A ∈ Alg E ' H1(PX,Aut E) we have defined its symbol as the image of itsdefining cocycle in H1(PX,ω). To compute H0 and H1 for automorphisms, itwill be useful to compute them first for symbols.
The following exact sequences of sheaves are also useful to handle ω :
0→ OPX/C→ ω → C→ 0 (63)
0→ C→ OPX → OPX/C→ 0 (64)
These give rise to long exact cohomology sequences. We will call “exponentmap” the cohomology maps coming from the map ω → C in (63).
With slight abuse we will call “Chern maps” 39 the maps :
ch : Hj(Y,O/C)→ Hj+1(Y,C). (65)
in the long exact cohomology sequence derived from (64).
The sheaf O/C (Y = X or PX) identifies with the sheaf of closed holomor-phic 1-forms on Y . If Y is a Stein manifold we have Hj(Y,O) = 0 for j ≥ 1 sothe Chern map Hj(Y,O/C)→ Hj+1(Y,C) is an isomorphism for j ≥ 1.
39 The standard Chern map : H1(Y,O×)→ H2(Y,C) factors through H1(Y,O/C).
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If Y is a compact Kahler manifold, the long exact cohomology sequence from(64) splits into a sequence of short split exact sequences :
0→ Hj−1(Y,O/C)→ Hj(Y,C)→ Hj(Y,O)→ 0 (j ≥ 0)
and for j ≥ 0 we have an isomorphism
Hj(Y,O/C) =∑
p+q=j+1,p>0
Hpq (66)
where (here, and whenever possible) Hpq denotes the space of harmonic formsof type p, q on Y .
Proposition 94 (i) If n = dimX ≥ 2, or if X is a closed curve of genus 6= 1,the map H0(X,O/C)→ H0(PX,ω) is an isomorphism.
(ii) If X is an open curve or a closed curve of genus 1, then ω is split and
H0(PX,ω) ' H0(X,O/C)⊕H0(X,C).
Proof : A global section of ω is a closed 1-form on T ∗X − 0, homogeneousof degree 0. Locally on X such a form α reads
α =∑
αkdxk + βkdξk (67)
where the coefficients αk resp. βk are of degree 0 resp. −1. If n ≥ 2 this impliesβk = 0 so the αk only depend on x. Hence (i) for n ≥ 2.
If X is a closed curve of genus 6= 1 (n = 1 so PX = X), then the Chern mapH0(X,C) ' C → H1(X,O/C) = C is injective : it maps s ∈ C to s chO(1)(where as above O(1) denotes the sheaf of homogeneous functions of degree 1 onT ∗X) and chO(1) 6= 0 if g 6= 1.40 So the exponent map H0(X,ω)→ H0(X,C)vanishes, and the map H0(X,O/C) → H0(X,ω) is an isomorphism, hence (i)in this case.
If n = 1 and X is open or of genus 1, there exists a global nonvanishingvector field, so ω is split : ω = O/C⊕ C hence (ii).
Proposition 95 (i) If n = dimX ≥ 2 the map H1(X,O/C) → H1(PX,ω) isan isomorphism.
(ii) If n = dimX = 1 (PX = X) and X is open or closed of genus 1 (ωsplit), then H1(X,ω) = H1(X,O/C)⊕H1(X,C).
(iii) If X is a closed curved of genus g 6= 1 the exponent map H1(X,ω) →H1(X,C) ' C2g is an isomorphism.
40 The corresponding cocycle is dLog ( ξiξj
) if ξi is the symbol of a nonvanishing vector field
on a covering Xi of X , whose image in H1(X,ω) is dξiξi− dξj
ξj, obviously a coboundary.
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This should be complemented as follows in case (ii) : if X is an open curve,H1(X,O/C) = 0 so H1(X,ω) ' H1(X,C).
If X is closed of genus 1, then H20 = 0 so H1(X,O/C) ' H20 + H11 'H11 ' C, and H1(X,ω) ' H11 +H1(X,C) ' C3.
Lemma 96 If X is a ball (or more generally Stein contractible space), we haveH1(PX,ω) = 0.
Proof : We have PX ' X×Pn−1, so H1(PX,C) = 0 (PX is simply connected)and the map H1(PX,O/C)→ H1(PX,ω) is onto (if n = 1 we are finished).
Next we wave Hj(PX,O) = 0 for any j > 0 (O has no cohomology onPn−1) so the Chern map H1(PX,O/C) → H2(PX,C) ' C is one to one.Now, as above for curves of genus 6= 1, H2(PX,C) ' C is generated by theChern class of O(1), corresponding to the cocycle dLog ( ξiξj ) for ξi an ellipticsymbol of degree 1 over a covering Ui of PX. This is also precisely the imageof 1 ∈ H0(PX,C) ' C by the exponent map H0(PX,C) → H1(PX,O/C), sothe exponent map is onto and the map H1(PX,O)→ H1(PX,ω) vanishes.Thisproves the lemma.
Proof of Proposition 95. (i) Let α be a principal ω-sheaf on PX corre-sponding to a cocycle in H1(PX,ω), and let X =
⋃Xi be a covering of X
by complex balls (or Stein contractible open sets). Then αi = α|Xi is triv-ial. The patching isomorphism uij : αj → αi is the translation by a sectionuij ∈ H0(PXi ∩ PXj , ω) = H0(Xi ∩Xj ,O/C) ; thus α is defined by a cocycle(uij) ∈ H1(X,O/C). If n ≥ 2 and if (uij) = (αi − αj) ∼ 0 in H1(PX,ω) thenagain αi ∈ H0(Xi,O/C) by Proposition 94, so (uij) ∼ 0 in H1(X,O/C). Thisproves (i).
If n = 1 (PX = X) and X is open or of genus 1, ω is split so H1(X,ω) =H1(X,O/C)⊕H1(X,C).
If X is open then H1(X,O/C) = 0 because in the long exact cohomology se-quence from (64) we have H1(X,O) = H2(X,C) = 0, so H1(X,ω) ' H1(X,C).
If X is of genus g = 1, we have H1(X,O/C) = H20 + H11 = C andH1(X,ω) ' H11 +H1(X,C) ' C3.
If X is a closed curve of genus g 6= 1 we have seen that the map H0(X,C)→H1(X,O/C) is one to one, and H2(X,O/C) = H30 + H21 + H12 = 0 so fromthe long exact cohomology sequence from (63)
· · · → H0(X,C)→ H1(X,O/C)→ H1(X,ω)→ H1(X,C)→ . . .
we see that the map H1(X,ω)→ H1(X,C) is one to one.This proves Proposition 95 and its complement. Note that if X is a curve,
the only case where H1(X,ω) = 0 is when X is simply connected.
11.3.6 Filtrations
As mentioned above Aut E has a natural filtration (as well as E×− ⊂ Aut E) :any a ∈ Aut E is of degree ≤ 0 and a is of degree n < 0 if a = Ad (1 + b), b ∈ En.
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The corresponding graded sheaf is
gr Aut E =⊕k≤0
(Aut E)k/(Aut E)k−1 ' ω +⊕k<0
O(k). (68)
It is commutative, and this will help to extract more information. This alsoworks for any E-algebra A because the filtration above, and the leading terms,are by definition invariant by automorphisms so gr AutA ' gr Aut E .
Proposition 97 Let A be an E-algebra.
(i) The natural map grH0(PX,AutA) → H0(PX, gr Aut E) is injective. IfH1(PX, gr Aut E) = 0 it is one to one.
(ii) If H1(PX, gr Aut E) = 0 then H1(PX,A×) = 0.
(iii) If H2(PX, gr Aut E) = 0 the symbol map induces a surjective map
H1(PX, gr Aut E)→ grH1(PX,AutA). (69)
Proof : (i) The map grH0(PX,AutA) → H0(PX, gr Aut E) takes any U ofdegree m ≤ 0 to its symbol σm(U) which is a section of ω if m = 0 or of O(m)if m < 0. σm(U) = 0 means that U is really of degree ≤ m− 1 so the resultinggraded map is injective.
Conversely let Xi be a covering of PX, and ai ∈ H0(Xi,AutA) be suchthat aia−1
j is of degree m < 0 (this is true if σ(ai) = a, a global section ofgrm+1(PX,Aut E)). Then σm(aia−1
j ) is a 1-cocycle with coefficients in O(m).If H1(PX, E×− ) = 0 this is a coboundary i.e. of the form σm(bi) − σm(bj),bi ∈ Am so the Int (1 + bi)−1ai Int (1 + bj) are equal to ai mod. AutAm andpatch together mod. (AutA)m−1. Note that if the Xi and their intersections areStein it is not necessary to shrink the covering, so by successive approximationswe get a cocycle a ∈ H0(PX; AutA) equal to (ai) mod. (AutA)m.
(ii) If H1(PX, gr E×− ) = 0 and if (aij) is a cocycle of degree n < 0 withcoefficients in A×, then σn(aij) is a 1-cocycle with coefficients in O(n), hence acoboundary σn(bi)− σn(bj), bi ∈ An. So aij is equivalent to the cocycle
(1 + bi)−1aij(1 + bj)
which is of degree n− 1. Again we do not need to shrink the covering if it hasbeen chosen as above (Stein, contractible), so by successive approximations weget a ∼ 0.
(iii) If bij is a cocycle with coefficients in AutmA (m ≤ 0) its symbol σ(bij)is a cocycle with coefficients in grmAut E but in general this does not give riseto a map grH1(PX,AutA) → H1(PX, gr Aut E) nor the other way round, atbest an ill-defined “noncommutative spectral sequence”.
However if H2(PX, gr Aut E) = 0, the same argument as above shows thatif a cochain aij ∈ H0(Ui ∩ Uj ,AutA) is a cocycle mod. (AutA)m,m < 0,i.e. aijajkaki ∈ (AutA)m then σm(aijajkaki) is a coboundary
∑σm(bjk) with
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coefficients in O(m), and again by successive approximations there exists acocycle a′ij with coefficients in AutA equal to aij mod. Am.
In particular, by successive approximations, we see that any cocycle a withcoefficients in grmAut E (m ≤ 0) is the symbol of a cocycle b ∈ H1(PX,AutmA)which is well defined mod. H1(PX,Autm−1A) and vanishes if a is a cobound-ary. Thus our map is well defined and onto (if H0(PX,AutA) 6= 0 it may notbe injective because two cocycles of degree m with coefficients in AutA canthen be equivalent although their symbols are not).
11.4 E-Algebras on T ∗X, dim X ≥ 2
11.4.1 General Results.
We first point out the following results (which will also be useful in section 5) :
Lemma 98 (Global automorphisms of E) If A is an E-algebra A on X andn = dimX ≥ 2 the symbol map
H0(PX,AutA)→ H0(PX,ω) ' H0(X,OX/C).
is injective. It is bijective if A = E.
Proof : If n ≥ 2,A and AutA have no global section of degree< 0 so the symbolmap u→ σ(u) is injective. More generally if A,A′ are two E-algebras and u, vtwo isomorphisms A → A′ the difference symbol σ(u−1v) ∈ H0(PX,ω) is welldefined and completely determines v (given u) (note that we have σ(u−1v) =σ(vu−1)).
On the other hand if A = E (more generally if A comes from a D-algebra) thesymbol map is onto because, by Proposition 94, H0(PX,ω) ' H0(X,O/C) 'AutD, and this obviously lifts to Aut E .
If X is a ball of Cn or more generally a Stein contractible domain, we haveH1(PX,ω) = 0 (Lemma 96) so H1(PX, E×− )→ H1(PX,Aut E) is onto, i.e. anyE-algebra can be defined by a cocycle with coefficients inE×− .
Now let A,A′ be two algebras defined by cocycles a = (aij), a′ = (a′ij) ∈ E×−
and let u : A′ → A be an isomorphism, i.e. a family (ui) ∈ Aut E such thatuia′ij = aijuj . Then the symbols σ(ui) patch together since σ(aij) = σ(a′ij) = 0,
and the resulting symbol σaa′(u) is well defined. It only depends on the classesof a, a′ in H1(PX, E×− ) (however it does depend on a, a′ ∈ H1(PX, E×− ) andnot just on their images in H1(PX,Aut E) : any other representatives are ofthe form α · a, α′ · a′ with α, α′ ∈ H0(PX,ω) for the action of H0(PX,ω) ofProposition 93, and we get σα·a,α′·a′(u)) = σ(u) + α− α′.
If X ⊂ Cn, n ≥ 2, is a Stein contractible domain, the exponent of σaa′(u)vanishes : σaa′(u) ∈ H0(X,O/C), and again σaa′(u) completely determines u.
Let now A ∈ Alg E . There exists a covering X =⋃Xi where all finite
intersections are isomorphic to Stein contractible domains of Cn. Then Ai =
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A|Xi can be defined by a cocycle (ai) with coefficients in E×− ; this being sothe patching isomorphisms uij all have exponent 0 and are determined by theirsymbols σaiaj (uij) (for fixed Ai). In particular we have proved :
Proposition 99 If dimX ≥ 2 any E-algebra A has exponent 0 (the image ofσ(A) ∈ H1(PX,ω) in H1(PX,C) by the exponent map is zero), so A can bedefined by a cocycle with coefficients in Int E0 = E×0 /C
× (E0 is the sheaf ofpseudo-differential operators of degree ≤ 0).
11.4.2 The case dimX ≥ 3
If X is a ball and dimX ≥ 3 we have H1(PX,O(−k)) = 0 for all k > 0, i.e.H1(PX, gr E×− ) = 0 (this is also true if X is a Stein manifold).
It follows that we have H1(PX, E×− ) = 0, and more generally for any E-algebra A we have H1(PX,A×) = 0.
Hence if A is an E-algebra, it is built by patching together models of E overa covering Xi of X, where the patching cocycle belongs to H1(X,OX/C).
Moreover if A,B are two such algebras, any isomorphism B → A comesfrom a ϕ ∈ H0(X,OX/C) i.e. comes locally from an inner automorphism P →ϕP ϕ−1. Summing up we have proved :
Theorem 100 If dimX ≥ 3 the functor which to a D-algebra associates thecorresponding E-algebra is an equivalence.
This result is closely related to the result of [37] on microlocally free D-modulesin dimension ≥ 3.
11.4.3 The case dimX = 2
If dimX = 2 what was said above remains true, in particular any symbolα ∈ H1(PX,ω) is the symbol of an E-algebra (in fact of a D-algebra). Howeverthe picture changes considerably because H1(PX, E×− ) is usually very large. Thefollowing examples show what can happen, and also how, in global situationson compact manifolds, things can nevertheless at least partially cancel out.
Example 1. Let X be the unit ball of C2 (or more generally a Stein con-tractible manifold). 41
Then H1(PX,ω) = H1(X,O/C) = 0 so H1(PX,Aut E) is the quotient ofH1(PX, E×− ) by the action of H0(PX,Aut E) = H0(X,O/C).
Now PX is the union of the two Stein subcones Ui = ξi 6= 0 (i = 1, 2) soa cocycle is represented by just one section a12 ∈ E×− (U1 ∩U2). It is elementarythat any a ∈ H1(PX, E×− ) has a unique normalized representative of the form
a12 =∑p,q<0
apq(x)ξp1ξq2 (70)
41 what is used is H1(X,O/C) = 0 and the fact that T ∗X is a trivial holomorphic vectorbundle.
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i.e. with no holomorphic term in ξ1 or ξ2 (this is obvious for the additivecohomology H1(PX, gr E−1) and follows by successive approximation for E×− ).So H1(PX,Aut E) is the set of conjugate classes of normalized symbols a12 asabove, with a12 ∼ ϕ(x)a12ϕ(x)−1 for ϕ a nonvanishing function on X. This setis still very large ; on the other hand such algebras tend to have very few globalsections or automorphisms.
The analysis of these algebras is closely related to that of “microlocally” freeD-modules in dimension 2, made by M. Carette [32].
For global compact manifolds, some things may cancel out.
Example 2. Let X = P2(C) be the complex projective plane : thenPX is isomorphic to the incidence manifold x · ξ = 0 ⊂ X × X∗ (X∗
the dual projective space). T ∗X itself is the quotient of the incidence coneΓ = x.ξ = 0 ⊂ C− 0 × C by the group action (x, ξ) ∼ (λx, 1
λξ). The sheafOPX(n) of homogeneous functions of degree n on T ∗X identifies with the sheafof restrictions to Γ of functions f(x, ξ) such that f(λx, ξ) = f(x, λξ) = λnf(x, ξ)i.e. OPX(n) = OX(n) ⊗OX∗(n) (where exceptionally here OX(n) denotes thecanonical sheaf of the projective space). It follows easily that H1(PX, gr E×− ) =0 so H1(PX,A×) = 0 for any E-algebra A.
The symbol map H1(PX,Aut E) → H1(PX,ω) = H1(X,O/C) is one toone, and again, as in dimension ≥ 3, the correspondence D-algebras → E-algebras is an equivalence.
Note that in this case we have H1(PX,O/C) ' H20 +H11 ' H11 = C, andE-algebras ∼ D-algebras are parameterized by H11 = C.
Example 3 Let X be a holomorphic complex torus of dimension 2 (a torusC2/Γ with Γ ' Z4 acting by translations).
The group of automorphisms of E or D is
H0(PX,Aut E) = H0(X,O/C) = H10 = C2 (71)
and any automorphism comes from an inner automorphism of E and D on C2
of the form :
P = P (x, d)→ ea.xPe−a.x (x→ x, d→ d− a). (72)
Any E- or D-algebra on X lifts as the trivial algebra EC2 on the univer-sal cover C2, and is the quotient of EC2 by a group of isomorphisms over thetranslation group of periods Γ.
By Proposition 95 we have
H1(PX,ω) = H1(X,O/C) = H20 +H11 = C5.
More precisely an element α ∈ H20 +H11 is represented by a harmonic form
α = a dz1dz2 +∑
aijdzidzj . (73)
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There is a unique corresponding D-algebra, which is isomorphic to the quotientof DC2 by the lifting µ→ Uµ of the group Γ of periods (acting by translations) :
Uµ :x→ x+ µd→ d+ p(µ, µ) (74)
where p = (p1, p2) is a linear map C4 = C2 × C2 → C2 such that dp(z, z).dz =dp1dz1 + dp2dz2 = α, where z, resp. z denotes the variable in C2 resp. C2,and we use the notations of differential calculus. Such a map p splits intoholomorphic and antiholomorphic parts : p = p′(z) + p′′(z). They form an8-dimensional space, but it is classical that maps which differ by a symmetricholomorphic map (dp.dz = 0) define isomorphic algebras.
Remark 5 Cocycles coming from H11 are related to holomorphic line bundleson X : if L is a line bundle, DL the sheaf of differential operators on L, thecorresponding cocycle is the image in H1(X,O/C) of the multiplicative cocyclewith coefficients in O× defining L ; the corresponding harmonic form is anintegral form in H11, and such forms generate H11 if X is algebraic.
The cocycle associated to dz1dz2 ∈ H20 corresponds to the group of isomor-phisms
Uµ : z → z + µ, d1 → d1, d2 → d2 + µ1 (75)
This corresponds to the 1-form p(x) · dx = x1dx2 (which could be replaced byany holomorphic primitive of dx1dx2). It never appears in a context of linebundles.
We may now classify E-algebras. The map H1(PX,Aut E) → H1(PX,ω)is onto, and for α ∈ H1(PX,ω) ' H20 + H11 the fiber σ−1(α) is the image ofH1(PX,A×) for A the unique D-algebra as above with this symbol.
Let us examine H1(PX,A×) : by (72) two elements of H1(PX,A×) givethe same E-algebra iff there is a translation ξ → ξ + a which transforms one tothe other. An a ∈ H1(PX,A×) lifts to an element a ∈ H1(C2, E×− ) invariant bythe Uµ, so the normalized representative (70) is invariant:
a(x, ξ) = a(x+ µ, ξ + p′(µ) + p′′(µ)) (76)
where as above p′ : C2 → C2, resp. p′′ : C2 → C2 denote the holomorphic andantiholomorphic parts of p, which correspond to the H20, H11 components ofthe symbol α. Equivalently the symbol b(x, ξ) = a(x, ξ − p′(x)) satisfies
b(x, ξ) = b(x+ µ, ξ + p′′(µ)) =∑ 1
γ! ∂γξ b(x+ µ, ξ)(p′′(µ))γ . (77)
If p′′ = 0 this means that b does not depend on x (it is periodic hence constant).If p′′ 6= 0, the periodicity condition implies b = 0 : for if b is of degree n ≤ −1,its leading term is periodic in x hence independent of x : bn = bn(ξ) ; the nextterm satisfies
bn−1(x, ξ)− bn−1(x+ µ, ξ) = b′n(ξ) · p′′(µ) (78)
117
so it is linear in x : bn−1 = β(ξ) + γ(ξ) · x with p′′(µ) = −γ(ξ).µ. Since p′′ isantiholomorphic this implies b′n = 0 so bn = 0 since its degree is negative.
Summing up we have proved :
Proposition 101 If X is a torus (C2/Γ,Γ ' Z4), we have
H1(PX,ω) ' H1(X,O/C) ' H20(X) +H11(X) ' C5. (79)
Any symbol α ∈ H1(PX,ω) is the symbol of a unique D-algebra on X.
If the H11 component of α is 6= 0 there is no other E-algebra with this symbol.
If α ∈ H20 the E-algebras with symbol α can be defined by a mormalized cocycle
b(ξ) =∑p,q<0
bpqξp1ξq2 (bpq ∈ C) (80)
whose coefficients are translation invariant (independant of x). Two such cocy-cles b, b′ define the same E-algebra iff b′(ξ) ' b(ξ + a) for some constant vectora.
(this is an asymptotic relation between symbols : b(ξ + a) =∑
1α !∂αξ b a
α)
11.5 E-Algebras over Curves (dim X = 1)
We now describe E-algebras, and compare them to D-algebras, when X is acurve (dimX = 1). In this case PX = X. The general method is the same butas we will see the classification is strikingly different depending on whether Xis an open curve, or a closed curve of genus g = 0, 1 or ≥ 2.
11.5.1 Open curves
If X is an open curve, the exponent map H1(X,ω) → H1(X,C) is an isomor-phism (Proposition 95). Also X is Stein, so Hj(X,O(n)) = 0 for j > 0 and forall n, j ≥ 1, so Hj(X, gr E×− ) = 0 for j = 1, 2, and H1(X, E×− ) = 0 (Proposition97). Finally we have
H1(X; Aut E) ' H1(X,ω) ' H1(X,C). (81)
Typically if (sij) is a cocycle with coefficients in C, the corresponding algebrais defined by a cocycle with symbol (Int ξ)sij , ξ a global nonvanishing vectorfield.
These algebras have many sections because we have H1(X, E×− ) = 0 so byProposition 97 the map H0(X, gr E) ' O(X)[ξ, ξ−1] → grH0(X,A) is oneto one. They also have many automorphisms, because the sequence 0 →H0(X,A×)→ H0(X,AutA)→ H0(X,ω)→ 0 is exact.D-algebras are classified by H1(X,O/C) = 0 and all give isomorphic E-
algebras. All non trivial E-algebras come from the exponent map. 42
42 the fact that such “exotic” algebras exist is related to the fact that coherent D-modulesdo not always possess global good filtrations.
118
11.5.2 Curves of genus g ≥ 2
Note that in any case OPX(1) identifies with the sheaf of sections of TX (vectorfields) and the dual OPX(−1) identifies with the sheaf of sections of T ∗X. If Xis of genus ≥ 2, we have H1(X,O(−k)) = 0 if k > 1, but H1(X,O(−1)) = C.43
Proposition 102 If X is a closed curve of genus g > 1, E-algebras on X areclassified by H1(X,Aut E) = C⊕C2g; D-algebras are classified by H1(X,O/C) =H0(X,C) = C and give isomorphic E-algebras.
We have a split exact sequence of sheaves of groups
0→ E−1 → Int E0 → O×X/C× → 0
It follows that we have H1(X, Int E) = C ⊕ C : the second factor comes fromH1(X,O×X/C×) ∼ H1(X,O(−1)) = C; it classifies D-algebras.
The first factor is the image of H1(X, E−1) ∼ H1(X,O(−1)) = C (E−kdenotes the multiplicative group of elements (1 + a),deg a ≤ −k ; the gradedsheaf associated to E−1 is
⊕k≤−1O(k), and O(k) is cohomologically trivial for
k < −1).
Consider the cohomology exact sequence (where the four last terms are com-mutative groups, even though Int E0 and Aut E are not):
0→ H0(X, Int E0)→ H0(X,Aut E)→ H0(X,C)→
→ H1(X, Int E0)→ H1(X,Aut E)→ H1(X,C)→ 0.
The second factor C2g in prop.10 liftsH1(X,C) ; it classifies“exotic” algebrasas for open curves. Such an algebra can be defined by a cocycle Ad ξsij where ξis a nonvanishing vector field over X minus one point, subordinate to a coveringX =
⋃Xj where all Xj except one avoid the point.
In the long exact sequence, the unit 1 ∈ H0(X,C) maps to the cocycle(ξiξj−1) ∈ H1(X,O×X/C×) ⊂ H1(X, Int E0), where (ξi) is a family on non-vanishing vector fields over a covering (Xi) of X. This is is not zero since theChern class of TX is not zero ; it is obviously killed by the map H1(X, Int E0)→H1(X,Aut E) (the E-algebra defined by a D-algebra is always trivial). In prop.10 the first factor, range of the map H1(X, Int E)→ H1(X,Aut E), is isomorphicto H1(X, E−1) ∼ C.
Here again E-algebras on X have many sections of negative degree and manyautomorphisms.
11.5.3 Curves of genus 1
This is the most complicated of the cases examined here. Let X be a closedcurve of genus 1 : X = C/Γ where the group of periods is Γ ' Z2 acts bytranslations.
43the original manuscript contained an error, corrected by P.Polesello.
119
We denote ξ the symbol of the constant vector field ∂/∂x on C.
Since TX is trivial, ω is split : ω = O/C + C. Also, for all n, we haveH0(X,O(n)) = H00 ' H1(X,O(n)) = H01 ' C, H2(X,O(n)) = 0.
We denoteG , resp. G− ⊂ G (82)
the group of automorphisms of E of the form Ad ξsAd (1 + a(ξ−1)), resp. thesub-group s = 0 : this is the commutant of ad ξ, it is a constant subsheaf ofAut E .
For any a ∈ C we set ξa = eaxξ. This is only defined up to a multiplicativeconstant eaµ, µ ∈ Γ, but the inner automorphism
Ad (eaxξ)
is well defined, as well as the corresponding commutator sheaf
Ga− ⊂ Ga (83)
which is a locally constant subsheaf of Aut E .
Proposition 103 We have
H0(X,ω) = H0(X,O/C) +H0(X,C) = H10 +H00 ' C2
H1(X,ω) = H1(X,O/C) +H1(X,C) = H11 + (H10 +H01) = C3.
For the commutative locally constant sheaf Ga− we have 44
Hj(X,Ga−) = Hj(X,C)⊗ G− if a = 0, 0 if a 6= 0.
We have gr E− =⊕
n<0O ξn and with an obvious notation
H0(X, gr E×− ) = grG ' ξ−1C[ξ−1]
H1(X, gr E×− ) ' H10 ⊗ ξ−1C[ξ−1]
H2(X, gr E×− ) = 0.
Theorem 104 If X is of genus 1, the symbol map Alg E → H1(X,ω) is onto.We will denote σ(A) = α = (α11, α10, α01) ∈ H11×H10×H01 the symbol of anE-algebra A. Then
(i) Algebras such that α11 = 0 are characterized by the fact that they possessa global section of degree 6= 0, or an automorphism of symbol dξξ = σ(Int ξ). Forsuch an algebra the set of global sections is C((ξ−1)) and except for E the groupof automorphisms is G.
44 the cohomology of the locally constant sheaf generated by eaxξ or enaxξn vanishes ifna 6= 0, because eanµ cannot be identically 1 for µ ∈ Γ, so H∗(X, grGa− ) = 0.
120
E is distinguished by the fact that its symbol map H0(X,Aut E)→ H0(X,ω)is onto.
(ii) If α11 6= 0, A has no section of degree 6= 0 (H0(X,A) = C), and A iscompletely determined by its symbol, in other words the image of H1(X,A×) inH1(X,AutA) is reduced to a single point.
For such an algebra the group of automorphisms is a one parameter groupwith symbol C(a dx + bdξξ ) for some (a, b) 6= 0, except in the case α01 = 0, α11
and α10 6= 0, where there is no automorphism other than Id.
(iii) Among these, E-algebras associated to a nontrivial D-algebra are thosefor which σ(A) = α11 ∈ H11 (α10 = α01 = 0). They are characterized by thefact that their group of automorphisms is a one parameter group with symbolσ(Int etx), (t ∈ C).
Thus for a torus X of genus 1, D-algebras which give isomorphic E-algebrasare already isomorphic as D-algebras, and E-automorphisms are the same asD-automorphisms, except for the canonical algebra E .
Let A be an E-algebra. The symbol map Alg E → H1(X,ω) is onto becauseH2(X, gr E×− ) = 0 so (Proposition 97) any cocycle with coefficients in ω is thesymbol of an E-algebra.
Next note that any star algebra on X lifts as the trivial E-algebra EC on Cwith an action of the group Γ over the translation group :
µ→ Tµ = TµUµ (84)
where Tµ is the translation (x→ x+ µ, ξ → ξ) and the Uµ are automorphismsof EC, subjected to the cocycle condition expressing that µ→ TµUµ is a grouphomomorphism.
Here are typical examples (models) :
Example 4 Let µ ∈ Γ → Uµ = α(ξ) ∈ G be an additive map. This definessuch a cocycle, because the Uµ commute with translations, hence an E-algebra,obviously of the first type. since ξ is invariant. Typically the period group
µ→ Uµ = (Int ξ)α10µ+α01µ
defines such an E-algebra with symbol α10 + α01 (α11 = 0).
Example 5 Let α(µ) = α10µ + α01µ be an additive map Γ → C and a ∈ C.Then the automorphisms Int (exp α(µ)(ax+Log ξ)) commute and also commutewith translations (because the commutator [ξ, ax + Log ξ] = a is a constant soexp s(ax+ Log ξ) commutes with translations, mod. constant factors which givetrivial inner automorphisms). So the group homomorphism
µ→ Uµ = exp α(µ)(ax+ Log ξ)
121
defines an E-algebra, whose symbol is aα01, α10, α01 ∈ H11 ×H10 ×H01. If weidentify the Hpq with spaces of differential forms, the symbol of A writes
σ(A) = (a dα(x, x)dx , dα(x, x)).
Such an algebra admits the automorphisms Int exp s(ax+ Log ξ), s ∈ C.
The only symbols we have missed are those for which
a = α11 6= 0, b = α10 6= 0, α01 = 0
As model for this case we can take the algebra defined by
Uµ = (Int ξ)bµ Int (eaµx)
with x = x(1 + bξ )−1 so that σ(x) = x, [ξ + bLog ξ , x] = 1 : with this choice
the Tµ (Int ξ)bµ (symbolically expµ(ξ + bLog ξ)) commute with the Int eaµx, soagain the Uµ define an E-algebra with symbol α11 = a , α10 = b , α01 = 0.
We now prove Theorem 104.
(i) First suppose that A has a nonzero section s of degree 6= 0. Then σ(s) = cξk
for some constant c 6= 0 and integer k 6= 0; c−1s has a unique k-th root withsymbol ξ (this is true locally because it works for pseudo-differential calculus;the roots with symbol ξ are unique and patch together into a global section).Similarly if a is an automorphism with symbol σ(Int ξ), locally there exists aunique section s with symbol ξ such that a = Int s (a = Int b determines blocally up to a constant, and σ(b) = ξ fixes the constant so again these patchinto a global section with symbol ξ).
If A has a section s with symbol ξ, then clearly all global sections of A are ofthe form
∑k≤k0
cksk, and H0(X,AutA) contains the group G (formula (82)).
Furthermore, again by elementary pseudo-differential calculus, any two sec-tions, resp. automorphisms of E of symbol ξ are locally conjugate, so (A, a) islocally isomorphic to (E , Int eξ), and A can be defined by a cocycle with coef-ficients in G, the commutator of Int ξ (formula (82)). Hence α = σ(A) belongsto the image of H1(X,G) in H1(X,ω) i.e. α ∈ H1(X,C) and α11 = 0.
Conversely let α ∈ H1(X,C) (α11 = 0). Example 4 gives an algebra A withsymbol α which has a section of symbol ξ.
Now any other algebra A′ with symbol α is defined by a cocycle (aij) ∈H1(X,A×). We know that there is a surjective map from H1(X, grA×) togrH1(X,A×) and also that the map H1(X,G) → H1(X, E×− ) = H1(X,A×)is surjective. It follows that the embedding G− → A× gives a surjective mapH1(X,G) → H1(X,A×) (the symbol map (gr) is onto, and surjectivity followsby successive approximations). Thus any E-algebra with symbol α can be de-fined from A by a cocycle with coefficients in G−, or from E with coefficients inG; in particular it has a section with symbol ξ.
122
Lemma 105 If such an algebra A (α11 = 0) is not trivial, it has no other globalautomorphism than those of G.
Proof : If A has two automorphisms a, b with independent symbols, we maysuppose that these symbols are σ(Int ξ), σ(Int ex). So A has a section α withsymbol ξ such that a = Intα. The section Log b = β is locally well defined up toan additive constant, so the section γ = [α, β] is globally defined and commuteswith α (as any global section of degree < 0).
Now the symbol of γ is 1, so γ is invertible, and replacing β by βγ−1,we see that we can suppose [α, β] = 1 (or equivalently b−1ξb = ξ + 1). Itfollows again, by successive approximations, that A equipped with two suchautomorphisms is locally isomorphic to E equipped with Int ξ and Int ex; butthe only automorphisms which commute with both are obviously trivial (theleading term is constant because it commutes with x and ξ), so A is isomorphicto E .
(ii) Suppose now α11 6= 0. Then any any section is a constant (of degree 0)and there is no global section of degree 6= 0. Let us choose an algebra A withsymbol α (one of the models above). Here again since H2(X, gr E×− ) = 0 thegraded map H1(X, grA×)→ grH1(X,A×) is surjective, and any cochain withcoefficients in A× which is a cocycle mod. An(n < 0) is equivalent mod. An toa cocycle.
Lemma 106 We have H1(X,A×) = C.
Proof : Let A be defined by a cocycle Uij relative to some Stein coveringX =
⋃Xi. We can choose Uij = Int eαijxInt ξsij mod. lower order terms, with
constant αij , sij .Then α11 = σ(A)11 ∈ H11 corresponds to the (01) part of the cocycle αij ,
and does not vanish. Let a = (ai) be the “constant” 1-cochain ai = ξn (n ≤ 0)
on Xi. Its coboundary with coefficients in A× (computed in A(Xi)) is of degreen− 1; more precisely on Xi we have
ai − αij(aj) = ξn − aij(ξn) = ξn − (ξ − αij)n + · · · = nαijξn−1 + . . . (85)
where the . . . are lower order terms, because Int ξ commutes with ξ and the restonly contributes to terms of degree ≤ n− 2.
It follows by successive approximations that any cocycle of degree < −1is equivalent to 0, and the same computation shows that two cocycles withthe same leading term are equivalent. Since the symbol map H1(X,A×) →H1(X,O(−1)) = H01 is onto, we have H1(X,A×) = C : any cocycle βijξ−1 isthe symbol of a unique element of H1(X,A×).
Lemma 107 The map H1(X,A×)→ H1(X,AutA) is constant.
Proof : The fibers of this map are the orbits of the action of H0(X,ω) (Propo-sition 93). We will prove that this is transitive.
123
For this action σ(Intξ) acts by u = (uij) → us = usij with Intusij =Int ξsi Intuij Int ξsj (mod. coboundary equivalence), where ξi ∈ A(Xi) has sym-bol ξ, and multiplication is the multiplication of A. Now in the local frame onXi we have ξj = Uij(ξ) = x− αij + . . . so for leading terms we get
σ(usij) = σ(uij)ξ
ξ − αij= σ(u) (1 + αijξ
−1) (86)
or with additive notation σ(us) = σ(u) + sα11.
This proves Lemma 107, and the other assertions of Theorem 104 are imme-diate consequences.
11.5.4 The projective line
Let X be the projective line (X = P1(C)). It is the union of the two open setsX0 = z 6=∞, X∞ = z 6= 0, and since these are Stein, contractible (' C), Eor D-algebras are classified by cocycles reduced to one function on X0 ∩X∞.
D-algebras are classified by H1(X,O/C) = H2(X,C) = C. The D-algebraDs (s ∈ C = H1(X,O/C)) is defined by the cocycle (Int z)s.
Let us introduce homogeneous coordinates x, y (z = xy ). We make use of
the sheaf of homogeneous differential operators Dhom on C2, i.e. differentialoperators of x and y which commute with the generator of homotheties ρ =x∂x + y∂y ; this algebra is generated by ρ and the operators
e = x∂y, h = x∂x − y∂y, f = y∂x (87)
which satisfy the relations
[h, e] = 2e, [h; f ] = −2f, [e, f ] = h, h2 + 2(ef + fe) = ρ(ρ+ 2). (88)
Ds is isomorphic to the quotient sheaf Dhom/(ρ+s) and can be thought of asthe sheaf of differential operators on the virtual sheaf O(s) on X of homogeneousfunctions of degree s of x, y (which really only exists s when s is an integer).
We now turn to E-algebras.
Lemma 108 If X = P1 is the projective line then
(i) H0(X, gr Aut E) = 0 hence H0(X,AutA) = 0 for any E-algebra A.
(ii) H1(X,ω) = 0 and H1(X, E×− )→ H1(X,Aut E) is one to one.
Proof : In homogeneous coordinates as above, O(n) is the sheaf of homoge-neous functions of degree 2n of x, y and has no global section if n < 0 , henceH0(X, gr E×− ) = H0(X, E×− ) = 0.
124
We have proved above (Proposition 94, 95) that for the projective line wehave H0(X,ω) = H1(X,ω) = 0, hence the lemma.
Note that any q ∈ H1(X,Aut E) has a unique “normalized” representative :
q0∞ =∑
0>p>2q
apqzpζq ∈ O(X0 ∩X∞). (89)
This is because the two vector fields ∂0 = ∂z, ∂∞ = ∂1/z are globally holomorphicand elliptic on X0, resp. X∞ and their symbols are ζ,−z2ζ, so any cocycle canuniquely be reduced to the form above, as for the additive cohomology groupH1(X, gr E×− ).
To compare D-algebras and E-algebras it is convenient to use the follow-ing intermediate exact sequence ; let Int E0 ' E×0 /C
× be the group of innerautomorphisms of E0 ; we have an exact sequence :
0→ Int E0 → Aut E → C→ 0
hence a surjectionH1(X, Int E0)→ H1(X,Aut E) (90)
whose fibers are the orbits of the action of C = H0(X,C) on H1(X, Int E)(q0∞ → (Int ∂0)s q0∞ (Int ∂∞)−s, cf. Proposition 93).
Lemma 109 We have the following relation :
(Int z)−s−2 = (Int ∂0)s+1(Int z)s(Int ∂∞)−s−1. (91)
Proof : If s = k is a positive integer we have
z−k−2(z2∂)k+1 = ∂k+1zk.
Indeed both are ordinary differential operators of order k+1, with leading termzs∂s+1, which kill all monomials zj , 0 ≥ j ≥ −k.
Identity (91) for arbitrary s follows, because it is polynomial in s mod.(Aut E)m, for any m < 0.
It follows that Ds resp. D−s−2 give isomorphic E-algebras, although theyare not isomorphic D-algebras. This is the only case where two D-algebras onX = P1 give isomorphic E-algebras: the algebra of global sections is obviouslyan invariant of an E-algebra, and in this the global sections e, h, f (with thenotations above) are well defined (up to an additive constant by their symbols,and the commutation relations fix these constants). It follows that s(s + 2) =h2 + 2(ef + fe) is an invariant of the E-algebra coming from Ds.
Note that D-algebras form a one-parameter family, so there are many E-algebras which do not come from an D-algebra.
As last remark we turn to the following problem: does there exist a globalsymbolic calculus, i.e. is the underlying sheaf of a given E-algebra isomorphic to
125
O? This is always true for real E-algebras, where one can patch global symbolsusing a partition of the unity.
Let us examine what happens on X = P1(C) . There is a canonical 2-covering of T ∗X − 0 by C2 − 0 : (u, v) → (z = u/v, ζ = 1
2v2). If A is a
E-algebra on Σ = T ∗X its pull-back on Σ′ = C2 − 0 is a star-algebra for thecanonical Poisson bracket (v, u = 1), equipped with an involution above thesymmetry (u, v) → (−u,−v) (note that on Σ′, u and v are of degree 1
2 ). If Ahas a global symbolic calculus, its pull-back defines a star-product on O(Σ′).
Now on O(Σ′) there is (up to isomorphism) only one star-algebra law for thecanonical Poisson bracket, generated by u, v with the relation [v, u] = 1. Up toisomorphism this is given by the representation
∑apqu
pvq →∑apqu
p∂qu. Forthis law there are many global sections (i.e. all polynomials of u, v) : the globalsections e, h, f are necessarily
e = −12u2, h = 2u ∗ v +
12, f =
12v2 (92)
because their respective symbols are
σ(e) = −z2ζ ∼ −12u2 σ(h) = 2zζ ∼ u v, σ(f) = ζ ∼ 1
2v2
these determine e, h, f up to additive constants, and the commutation relations(88) determine the constants as above.
For these constants we get
h2 + 2(ef + fe) = −34
so s = −12. or s = −3
2(93)
We have proved :
Proposition 110 The only D-algebras on P1 for which there is a global totalsymbolic calculus are D−1/2 and D−3/2. In particular there is no global totalsymbolic calculus for D.
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12 Related symplectic star algebras.
12.1 Geometric quantization
In [92] B. Kostant introduced pre-quantization or geometric quantization, inorder to describe some remarkable representations of semi-simple groups. Wedescribe this here in terms of Toeplitz operators, as in [27].
For this the data is- a complex holomorphic cone Γ of complex dimension n in a numeric space
CN , smooth outside of the origin. Xe set Γ• = Γ − 0; the complex baseY = Γ•/C× is a smooth projective complex manifold Y ⊂ PN (C), and Γ•
identifies with L′− the zero section, where L is an ample line bundle over Y .- a smooth strictly pseudo-convex hermitian metric r on Γ, i.e. r is homoge-
neous of degree 1 (r(λz) = |λ|r(z)), > 0 and smooth outside of the origin, andthe matrix of second derivatives ∂zp∂zq is hermitian 0. E.g. r2 =
∑zjzj ; r
is given on Γ, but it can always be extended to CN ).Note that r is strictly pseudo-convex⇔ rs is for any s > 0⇔ ∂∂Log r defines
a Kahler metric on X.
Let X be the sphere r = 1 in Γ. The Szego projector and Toeplitz operatorsare well defined on X. The corresponding contact form is i∂r|X , the symplecticcone, set of positive multiples of λ, can be canonically identified with Γ.
The circle group U(1) acts on the whole situation; the Szego projector S isinvariant (provided we choose an invariant volume element to define it - in factthere is a canonical one : dλ(λ)n−1).
We denote θ the infinitesimal generator; it is elementary that it is the hamil-tonian field θ = Hr.
Let A be the Toeplitz operator iTθ : f 7→ iSθf = iθf =∑zj∂zjf . Then the
eigenspace Hk = kerH− k is the space of holomorphic functions homogeneousof degree k, and O0(X) is the Hilbert sum of the Hk.
Recall that Toeplitz operators are those of the form TP : f 7→ SP (f) with Pa pseudo-differential operator on X. Modulo smoothing operators they definea symplectic star algebra A on Σ.
A Toeplitz operator TP is invariant iff [A, TP ] = 0; it is always equal to a TQwith Q invariant (replace P by its mean over U(1)); such an operator preserveseach Hk. Mod. smoothing invariant operators form a sub algebra B ⊂ A whichis a deformation algebra, iff we set the deformation parameter ~ = A−1 (Ais invertible mod. smoothing operators - in fact its kernel H0 is the space ofconstants).
Remark 12 Any Toeplitz operator os degree 0 is of the form Tf mod. operatorsof degree −1 (its symbol is homogeneous of degree 0 so it is a function f on X).By successive approximations we see that any Toeplitz operator ha a uniqueformal expansion
∑m≤m0
TfkAk. For an in variant Toeplitz operator, the fk
are invariant i.e. they are smooth (not holomorphic) functions on Y ; this canbe rewritten P ∼
∑k ≥ k0Tfk~k.
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12.2 Homomorphisms between Star Algebras
If A,A′ are two star-algebras over cones Σ,Σ′ we have described in section4.1homomorphisms U : A → A′ (preserving the filtrations); the symbol map isf 7→ u∗f = f u where u is a smooth homogeneous map Σ′c → Σc preservingthe Poisson brackets, i.e. u∗f, g = u∗f, u∗g.
Below we will only consider algebras whose Poisson bracket is real (or pureimaginary), and (positive) homomorphisms whose symbol map maps Σ′ to Σ
For instance the algebra of semi-classical pseudo-differential P (x, h∂, h) ona manifold V is isomorphic to the algebra of pseudo-differential operators onon the sub-cone τ = σ(∂t) > 0 of T •(X × R) which do not depend on t) (i.e.commute with ∂t) : the map takes h to ∂−1
t . It is easy to see that any 1-codimensional embedding of a symplectic deformation algebra to a symplecticstar algebra is locally isomorphic to the embedding above.
Using this example it is easy to embed a symplectic deformation algebra AX over
a symplectic manifold X in a symplectic star algebra AΣ where Σ is a disjoint union
of pieces as above. However without more information on the projection BΣ → X
this only gives very poor information: to reconstruct AX from AΣ one would need
to know how various components patch together, which requires further non trivial
information.
Definition 111 We will say that a symplectic deformation algebra B on X is“related” to A if there exists an injective homomorphism B → A, where thecorresponding projection Y → X is a principal circle bundle (this definition willbe slightly refined below).
A typical example comes from the “geometric prequantization” above. Wewill determine in the section which pairs A,B of symplectic star algebras anddeformation algebras can be related in that manner. We need first to investigatehow the circle group or more generally a compact group acts on a symplecticalgebra.
12.3 Action of a Compact Group
Let A be a symplectic star algebra on Σ (resp. a deformation algebra on X) asabove.
Theorem 112 Let G be a compact group acting on Σ by symplectic homoge-neous isomorphisms. We suppose g∗A ∼ A for all g ∈ G, i.e. g∗RA − RAis exact, with RA the Fedosov curvature of A (this is always true if G is con-nected).Then
1. the action of G lifts to A.2. Any two liftings are conjugate through an automorphism of A.
The same result holds for symplectic deformation algebras.
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The theorem follows from the fact that the group of global automorphismsAutA is a complete filtered group, and grA is a G-vector space, so since G iscompact any continuous cocycle is a coboundary. Note that if G is connected,it fixes all cohomology classes (because it fixes integral cohomology), so theinvariance condition is automatic.
The first assertion is also seen using Fedosov connections:
1. If G acts on Σ (resp. X), its action extends functorially to W resp. Wh.By hypothesis the cohomology class of A is invariant, so it has an invariantrepresentative R since G is compact. Then Fedosov’s construction yields aninvariant connection ∇ (we first choose ∇s invariant so the starting point ∇s−τis invariant), and G acts on ker∇ ∼ A.
2. If U0(g), denoted below g (f 7→ gf ∈ A), and U1(g) = Ug are two liftings,we set
ug = Ug g−1 ∈ AutA so that we have ugh = ug guhg
−1
This means that σ(ug) is a 1-G-cocycle with coefficients in gr AutA (i.e. wehave σ(g · uh)− σ(ugh) + σ(ug) = 0), so it is of the form σ(ug) = g.σ(v)− σ(v)(the cohomology vanishes in positive degree since G is compact). By successiveapproximations we get v ∈ AutA such that ug = v−−1 gvg−1 i.e. Ug = v−1gV .
12.4 Circle Action
We suppose now that G is the circle group G = U(1). The action of G on Σhas an infinitesimal generator θ, which is a symplectic vector field homogeneousof degree 0, and a generating function a which is a homogeneous function ofdegree 1 :
θ = Ha, with a = Iθ(λΣ) (94)
where Iθ =∑θjL ∂
∂xj
denotes the interior product by θ, λΣ the Liouville form
of Σ.
We will also denote
Lθ the vector field with coefficients in W lifting θ (95)
Let A be a symplectic star algebra on Σ. As noted above (theorem 112) thecircle group action lifts to A. The infinitesimal generator D of this action is welldefined; it is unique up to conjugation by an automorphism of A,45 and there
45Here is an alternate proof of this: we first choose a derivation D0 of degree 0 in A suchthat σ(D0) = ∂θ, for instance D0 = adA0 where A0 ∈ A is any element with symbol a. Thenfor t ∈ R, exp tD0 is a well defined group of isomorphisms above exp tθ. In particular e2πD0
is an automorphism of A (over exp 2πθ = Id ): it is of the form exp 2πδ with δ a derivation ofdegree −1 which commutes with D0. Then D = D0 − δ is the infinitesimal generator of anaction of U(1), lifting the action on Σ (Ut = exp tD).
If D1 = D + δ1 with δ1 a derivation of degree −1), we see by successive approximations
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exists an equivariant embedding u : A → W , e.g. we choose an invariant repre-sentative of the curvature R: the corresponding canonical Fedosov connection∇ is then invariant:
[Lθ,∇] = 0 (96)
We denote A ⊂ W the image of A, and set
∇θ = [Iθ,∇], Rθ = [Iθ, R] (97)
here and everywhere else [, ] is the superbracket: [f, g] = (−1)fg(f ∗ g − g ∗ f),where f, g denotes the degree of f resp. g as differential forms. ∇θ is a vectorfield with coefficients in W , and we have
Lθ = ∇θ + α (α ∈ W ) (98)
We have α(x, 0) = 0 if ∇ “vanishes” on the zero section, in particular if ∇ isthe canonical connection associated to R.
We have Rθ = [Iθ,∇2] = [[Iθ,∇],∇]. Since [Iθ,∇] = ∇θ = Lθ − b and ∇ isinvariant, i.e. [Lθ,∇] = 0, we have
Rθ = [Lθ − α,∇] = ∇(α) (99)
Lemma 113 Notations being as above, the infinitesimal generator D of theaction of U(1) on A is an inner derivation if and only if Rθ is exact (as a formon Σ with coefficients in O).
Proof : adLθ coincides with adα on A. In other words the O-derivation ofW corresponding to D is adα. If D is an inner derivation there exists b ∈ A(∇b = 0) such that α− b is central, so Rθ = ∇α = d(α− b) is exact.
12.5 Elliptic Circle Action
Definition 114 An action on Σ of the circle group U(1) is elliptic if its gen-erating function a is > 0 (there is an obviously symmetric case a < 0).
From now on we suppose that Σ is equipped with a free elliptic action ofU(1), with generating function a > 0 (θ = Ha, a ∈ OΣ(1)). We will use a asprivileged radial coordinate. The basis of Σ is identified with the“unit sphere”
Y = a = 1 ⊂ Σ ∼ Σ/R×+ (100)
Y is a principal U(1)-bundle, with basis X and connection form λY :
X = Y/U(1) , λY = λΣ|Y (101)
that D1 is conjugate to D+ δ2 where δ2 commutes with D (if U = 1+u with u of degree −N ,U−1D1U = [D,u] + v with v of degree −N − 1, and if the Fourier series of u is u =
Puk, we
have [D,u] =Pikuk). Then if exp 2πD1 = Id we have exp 2πδ2 = Id i.e. δ/2 = 0 since δ2
is of degree < 0.
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where as above λΣ denotes the Liouville form λΣ of Σ; we will also denoteλY = λΣ
a the pull-back to Σ.
The basis X = Y/U(1) is a symplectic manifold with symplectic form
ωX such that p∗(ωX) = dλY (102)
The U(1) action lifts canonically to WΣ, and as above Lθ denotes the vectorfield with coefficients in WΣ defined by the infinitesimal generator θ.
Let Γ = Σ/U(1). We have TΓ = TΣ/TU(1), where the tangent generator is(obviously) the
∑θj
∂∂ξj
; it corresponds to a vector field τθ with coefficients in
W :
τθ = Iθ · τ (103)
where Iθ =∑θjI( ∂
∂xj) is the extension to Ω⊗ W of the interior product by θ,
and τ is the canonical 1-form of W (def. 38).
Lemma 115 1) WΓ is identified with the sub-algebra of WΣ of sections invari-ant by the tangent group TU(1), i.e. killed both by Lθ and ad τθ.2) The Weyl algebra Wh(X) is identified with the quotient WΓ / (τθ = 0)
(if U(1) acts by translations (x1, . . . , xn) 7→ (x1 +u, x2, . . . , xn), which is alwaystrue in a suitable set of local coordinates, then TU(1) acts by (x, ξ) 7→ (x1 +u, . . . , xn, ξ1 + v, . . . , ξn)).
Let now A be a symplectic star algebra on Σ, equipped with an extension ofthe action of U(1) (we have seen this exists, and is unique up to conjugation).
Up to equivalence, A is determined by the cohomology class of its Fedosovcurvature R. We may choose R invariant, so as the Fedosov connection (if westart with an invariant torsionless symplectic connection, the canonical con-struction of §4.6 produces an invariant connection). There is a correspondingequivariant embedding, which we will adjust further suitably below.
The curvature is R = ωΣ + r, with r closed, of weight w(r) ≥ 0. As anyclosed 2-form on Σ, r is cohomologous to an invariant form homogeneous ofdegree 0, so we may suppose:
r = µX + λY νX + (γX + cλY )da
a(104)
where µX , νX , γX are the pull-backs of 2 and 1-forms on X.Since dr = 0, c is a constant, and dγX+cωX = 0, dνX = 0, dµX+ωXνX = 0.We necessarily have c = 0 if ωΣ is not exact (on any component of X); this
is always the case if X is compact.
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Since IθωΣ = −da we have, with the notations above:
Rθ = IθR = −da+ νX + cda
a(105)
We have set Lθ = ∇θ + α. Since ∇θ = [Iθ,∇ and [Lθ,∇] = 0 we getRθ = [Iθ,∇2] = [∇θ,∇] = −[α,∇, i.e.
Rθ = ∇(α) (106)
Le us further note that the leading term of α is τθ = Iθ(τ).
Lemma 116 There exists an invertible section U of W such that UαU−1 = τθ.
This is immediate by successive approximations as above: if w(αn) ≥ n2 there
exists u such that w(u) ≥ n+12 , [τθ, u] =
∑θj
∂∂ξj
= αn (there is a uniquesolution such that u(, ξ) = 0 if ξ belongs to the hyperplane orthogonal to theLiouville form λ, which is transversal to ad τθ). Then (1 + u)−1τθ(1 + u) =τθ + αn + (w ≥ n+1
2 ).
Let B = AU(1) ⊂ A be the invariant sub-algebra. The image B ⊂ A is
B = kerLθ ∩ A = ker∇∩ ker adα (107)
It is also contained in WΓ if α = τθ.
Lemma 117 B possesses a non-trivial central element iff the component νX ofthe Fedosov curvature of A vanishes. Then B possesses a structure of relatedsymplectic deformation algebra.
Proof : Let B 6= 0 be a central element of B, of degree k 6= 0. The symbol of Bis c ak for some constant c 6= 0, and replacing B by (Bc )
1k ∈ B we may suppose
σ(B) = a = h−1. Then the infinitesimal generator is necessarily of the form
D = ad (B′), with B′ = B + cLogB +∞∑1
ckB−k
With the notations above, α′ = α − B′ is central, and Tθ = ∇α = dα′. Theonly term of B′ not in B is cLogB with differential cdaa +exact, so Rθ − cdaa isexact, i.e. νX ∼ 0 and c is the coefficient of da
a which appears in (104),(105).
12.6 End of description
We may now finish the analysis of related symplectic algebras. We will supposethe basis X compact (or more generally that ωX is not exact, on each componentof X). We leave as an exercise the case where ωX is exact so the constant cabove can be 6= 0.
We refine definition 111 as follows:
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Definition 118 We will say that a relating homomorphism B → A is good(or that we have a good relation) if the operator A corresponding to h−1 is aninfinitesimal generator of U(1)
Theorem 119 We suppose X compact.1) Let A be a symplectic star algebra over Σ with elliptic action of U(1) asabove, and Fedosov curvature RA, and let B be the invariant sub-algebra. ThenB possesses a structure of symplectic deformation algebra if and only if νX =IθR ∼ 0. In this case the infinitesimal generator of the action of U(1) on A is aninner derivation adA and there exists a unique semiclassical structure on B suchthat h = f(A) = A−1 + . . . for any formal series f(T ) = T−1 +
∑k>1 fkT
−k.It is well-related to A iff h−1 = A+ c, c ∈ C a constant.
2) Let B be a semiclassical algebra over X = Y/U(1) as above. Then B has arelated symplectic star algebra if and only if its Fedosov curvature RB is constantmod. ωX , i.e. of the form RB = h−1ωX + constant+ϕ(h)ωX . It is well-relatedA iff RB = h−1ωX + µ, µ a constant 2-form on X (independent of h).
The Fedosov curvature of A is then RA = p∗RB + daa p∗γX for some γX ∈
H1(X). In particular there is a unique (up to isomorphism) “non exotic” suchA (γX = 0).
Proof : Let A be a symplectic star algebra on Σ, and choose R, ∇ as above, soLθ = ∇θ + α, α = τθ. Since X is compact we have c = 0 anyway. If A containsa symplectic deformation algebra, we have νX ∼ 0 so we may as well supposeνX = 0, R = ωΣ + µX + γX
daa .
This being so we have Rθ = ∇α = −da so ∇(a+ α) = 0. The infinitesimalgenerator is adA where A is the element with total symbol a (A = a + α).Moreover with this choice A is a polynomial of degree 1 with respect to ξ soA ∗ f = Af + 1
2A, f. It follows in particular that A ∗ f = Af (or a ∗ f = af inA for the star product and total symbol defined by this choice of embedding) ifA commutes with f , and we get a well related semiclassical algebra by settingh = A−1.
If V is a vector field on X, we denote V the unique vector field on Σ whichprojects on V , such that IθV = IρV = 0, with ρ the generator of homotheties(i.e. V is orthogonal to the conic rays and fiber circles and projects on V - inparticular it is homogeneous of degree 0 and rotation invariant). With our choiceof embedding (D = adα, α = τθ), ∇V obviously preserves ΩΓ = kerLθ∩ker adα,and kills a and α so goes down to Ωh. The Fedosov connection ∇B of B is thenobtained by restriction:
∇BV = the derivation of Ωh defined by ∇V |Ωh (108)
The curvature RB is the induced by R: RB = µX . Note that it doesnot depend on γX , and changing γX , which is equivalent to twisting A by acocycle Asij , (sij) a cocycle on X with coefficients in C, does not change B, thecommutator of A. Among the symplectic star algebras related to B the onlyone which is not “exotic” has curvature µX .
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Any other related symplectic deformation structure is obtained by replacingh by ϕ(h) for some formal series ϕ(h) = h +
∑∞2 ϕkh
k; as seen in Prop.42,the corresponding symplectic deformation algebras are pairwise non-isomorphic.Among them, are well-related to A those for which ϕ(h) = h
1+γh (ϕ(h)−1 =h−1 + γ).
Remark 13 If νX is not cohomologous to 0, the invariant sub-algebra B is nota semiclassical algebra, because its center is not big enough. However its liftingto the universal cover X is; B itself is obtained by gluing together local modelsof symplectic deformation algebras with isomorphisms which preserve symbols,but do not fix h (h 7→ U(h) = h+
∑k≥2 ukh
k).
Example 4 In the canonical model e have SS = Cn − 0, a = |z|2. The sym-plectic form is ωΣ = i∂∂a = i
∑dzjdzj (twice the usual one - the coordinates
zj , zj are homogeneous of degree 12 so a is of degree 1); we have zp, zq = iδpq
(the Poisson bracket is c = i∑∂zj ∧ ∂zj . The infinitesimal generator of homo-
theties is ρ = 12
∑zj
∂∂zj
+ zj∂∂zj
. Rotations are the usual ones: z 7→ eitz, ithinfinitesimal generator θ = i
∑zj
∂∂zj− zj ∂
∂zj= Ha.
A is the Weyl algebra, with product f ∗g = exp 12c(∂x, ∂y) f(x)g(y)|y=x. The
action of U(1) lifts to A, with generator ad a. The canonical embedding A → Wis f 7→ f(x+ξ), the corresponding connection is ∇ = d−τ , τ = i(dzjζj−dzjζj).
The invariant subalgebra B is a semiclassical algebra over X = Pn−1(C), ithh = a−1.
Note that in this case, with the notations above, we have a+α = |z+ζ|2 6= τθ(we still have to modify ∇ to get the connection used above).
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13 Toeplitz operators and asymptoticequivariant index.
This is an account of a lecture given at the 7th Issac congress (july 12-18 2009),where I described a joint work with E. Leichtnam, X. Tang and A. Weinsteingiving a proof of the Atiyah-Weinstein index formula. This concerns the indexof an operator closely related to Toeplitz operators, for which analogues of theAtiyah-Singer index formula does not make sense. Instead we used an equiv-ariant asymptotic index formula, which does; it is an outgrowth of Atiyah andSingers theory of equivariant index for transversally elliptic pseudodifferentialoperators. 46
13.1 Szego projectors, Toeplitz operators
We first describe generalized Szego projectors and Toeplitz operators, whichgeneralize pseudo-differential operators on arbitrary contact manifolds. An im-portant case arises from complex (CR) analysis.
Let M be a compact manifold, and Σ ⊂ T •M a symplectic subcone 47.
Definition 120 A generalized Szego projector associated to Σ (or Σ-Szego pro-jector) is a self adjoint elliptic Fourier integral projector S of degree 0 (S =S∗ = S2), whose complex canonical relation C is 0, with real part the diag-onal diag Σ (elliptic means that the principal symbol of S does not vanish onΣ).
Specially useful examples are1) Σ is the full cotangent bundle T •M , S is the identity operator.2) M is the boundary of a strictly pseudoconvex bounded complex domain,
S is the Szego projector (see below). More generally, M is a compact orientedcontact manifold, Σ ⊂ T •M is the set of positive multiples of the contact form(a generalized Szego projector always exists, see below).
13.1.1 Example 1: Microlocal model
The following example was described in [6]. It is universal in the sense thatany generalized Szego projector is micolocally isomorphic to it, via some ellipticFourier integral transformation (with dim Σ = 2p,dimM = p+ q).
Let (x, y) = (x1, . . . , xp, y1, . . . , yq) denote the variable in Rp+q. Set D =(Dj), with
Dj = ∂yj + |Dx|yj (j = 1, . . . , q)
The Dj commute; the complex involutive variety charD is defined by thecomplex equations ηj − i|ξ|yj = 0; it is 0, in the sense of [20, 101]. Its realpart is the symplectic manifold Σ : ηj = yj = 0.
46MSC2010: 19L47, 32A25, 53D10, 58J40.47T • denotes the cotangent bundle deprived of its zero section.
135
The kernel of D in L2 is the range of the Hermite operator H (in the senseof [6]) defined by its partial Fourier transform:
f ∈ L2(Rp) 7→ Hf with FxHf(ξ, y) =( |ξ|π
) q4 e−
12 |ξ|y
2f(ξ)
The orthogonal projector on kerD is S = HH∗:
f 7→ (2π)−p∫
R2p+qei(<x−x
′,ξ>+i|ξ|2 (y2+y′2)(
|ξ|π
) q2 f(x′, y′)dx′dy′dξ
As H, it is a Fourier integral operator, whose complex canonical relation is 0,with real part the graph of Id Σ.48
13.2 Example 2 : holomorphic model
Let X be the boundary of a strictly pseudoconvex Stein complex manifold (withsmooth boundary); the contact form of X is the form induced by Im ∂φ whereφ is any defining function (φ = 0, dφ 6= 0 on X, φ < 0 inside).
e.g. if X is the unit sphere bounding the unit ball of Cn, with definingfunction z · z − 1, the contact form is Im z · dz|X .
The Szego projector S is the orthogonal projector on the holomorphicsubspace H = ker ∂b boundary values of holomorphic functions - the fact thatS is Fourier integral operator as above was proved in [15]). The system of(pseudo)differential operators playing the role of D is the tangential CauchyRiemann system ∂b.49
Remark: A basic example of Toeplitz structure is Σ = T •M (M a compactmanifold), S = Id : the Toeplitz algebra is the algebra of pseudodifferential op-erators acting on the sheaf of microfunctions on M . This is in fact a specialcase of the holomorphic case - example 2.
13.3 Main properties
Cf. [27, 9, 26]
1) A Σ-Szego projector S always exists. All such projectors have a uniquemicrolocally model (via some elliptic FIO transformation) depending onlyon dimΣ, dimM .
2) Toeplitz operators defined by S are the operators on H of the form u ∈H 7→ TP (u) = SPS(u) with P a pseudodifferential operator on M . They
48Fourier integral operators are described in [82]. Fourier integral operators with complexcanonical relation are described in [20, 101]
49at least if the dimension n is > 1 - if n = 1, S is the Hilbert projectorP∞−∞ fkz
k 7→P∞0 fkz
k , it is a pseudodifferential projector
136
form an algebra EX (or EΣ or E 50. Mod smoothing operators, they forma sheaf acting on µH, locally isomorphic to the sheaf of pseudodifferentialoperators acting on the sheaf of microfunctions (in p variables if dim Σ =2p).
3) If S, S′ are two Σ-Szego projectors with range H,H′, S′ induces a quasiisomorphism H→ H′ (the restriction of SS′ to H is a positive (≥ 0) ellipticToeplitz operator).
More generally, if Σ ⊂ T •M,Σ′ ⊂ T •M ′ are two symplectic cones andf : Σ→ Σ′ a homogeneous symplectic isomorphism, there always exists aFourier integral operator F from M to M ′, inducing an “elliptic” Fredholmmap H → H′ (such elliptic FIO exist, they were called “adapted” in [27,9]).
The pair EΣ, µH consisting of the sheaf of micro Toeplitz operators (i.e.smoothing operators), acting on µH is well defined, up to (non unique) isomor-phism: it only depends on the symplectic cone Σ, not on the embedding.
4) H is the set of solutions of a system (an ideal) of pseudo-differential equa-tions described by a pseudo-differential complex DΣ mimicking the ∂b inthe holomorphic case (see below).
The K-theoretic element [DΣ] ∈ KX(§∗M) it defines is precisely the Bottelement, defining the Bott periodicity isomorphism K(X)→ KX(S∗M).
5) All these constructions allow a compact group action.
We also use a vector bundle extension: an equivariant G-bundle is an in-variant direct factor E of a trivial bundle G vector-bundle X × V , defined byan invariant projector p (V a finite representation of G). The correspondingToeplitz space (or Toeplitz bundle) HE , with symbol E, is the range of anequivariant Toeplitz projector P of degree 0 in H ⊗ V , with symbol p. Hereagain HE is only defined up to a Fredholm map. Equivalently, H is defined by a’good’ projective E moduleM, i.e. the range of Toeplitz projector P ′ of degree0 in some free left-module EN : E = HomE(M,H).
If E,F are two equivariant Toeplitz bundles, there is an obvious notion ofToeplitz operator P : E→ F, and of its principal symbol σd(P ) if it is of degreed, which is a homogeneous vector-bundle homomorphism E → F on Σ.
P is elliptic of degree d if its symbol is invertible; then it is a Fredholmoperator E(s) → F(s−d) and has an index (which does not depend on s)51.
50if M is a manifold one writes EM for ES∗M51E(s) its space of Sobolev Hs sections of E.
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13.4 Miscellaneous
Toeplitz-Fourier integral operatorsThe analogue of Fourier integral transformations is the following: let X,X ′
be two contact manifolds, S, S′ generalized Szego projectors, anf f : X → X ′ acontact isomorphism. The pushforward map u 7→ u f−1 does not send H toH′: we correct it as for Toeplitz operators Tf (u) = S′(u u−1); this behaves asan elliptic Fourier operator attached to the contact map f . Other analogues ofF.I.O attached to f are of the form u 7→ A′Tfu, A′ a Toeplitz operator on X ′.
Atiyah-Weinstein problem:The Atiyah-Weinstein problem can be described as follows: If X is a compact
contact manifold, and S, S′ to Szego projectors defined by two embeddable CRstructures giving the same contact structure, then the restriction of S′ to His a Fredholm operator H → H′ (SS′ induces an elliptic Toeplitz operator onH). In this case the spaces H,H′ and the index are well defined. The Atiyah-Weinstein conjecture computes the index in terms of topological data of thesituation (topology of the holomorphic fillings of which X is the boundary).
13.5 Equivariant Toeplitz algebra
In the sequel we use the following notations:G: a compact Lie group, with Haar measure dg (
∫dg = 1), Lie algebra g.
Σ: a G- symplectic cone, basis X (a compact oriented contact G-manifold).ω its symplectic form, λ the Liouville form (ω = dλ) (G- invariant).Σ is canonically identified with the set of positive multiples of λX in T ∗X.)
S a G-invariant generalized Szego projector, with range H =⊕
Hα
(where α runs over the set of irreducible representations, and Hα is the corre-sponding isotypic component of H).
13.6 Equivariant trace
The G-trace and G-index were introduced by M.F. Atiyah in [5] for equivariantpseudo-differential operators on a G-manifold. The G-trace of P is a distributionon G, describing tr (g P ). We adapt this to Toeplitz operators.
Any v ∈ g defines a vector field Lv on X and a Toeplitz operator Tv on H(or any Toeplitz bundle E).
Definition 121 char g (characteristic set of g) denotes the closed subcone of Σwhere all symbols of infinitesimal operators Tv, ξ ∈ g vanish.
The base Z of char g is the set of points of X where all Lie generatorsLv, v ∈ g are orthogonal to the Liouville form λX . char g contains the fixedpoint set ΣG, whose basis is the fixed point set XG because G is compact. Notethat ΣG is always a smooth symplectic cone and its base XG a smooth contactmanifold; char g and Z may be singular.
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Let E be an equivariant Toeplitz bundle as above, E =⊕
Eα its the de-composition in isotipic components. If P : E → E is a Toeplitz operator oftrace class (degP < −n), the trace function TrGP (g) = tr (g P ) is a continuousfunction on G (it is smooth if P is of degree −∞), and we have
TrGP (g) =∑α
1dα
trP |Hα χα (109)
where χα is the character of α, dα the dimension (the Fourier coefficient is1dα
trP |Eα).
The following result is an immediate adaptation of the similar result of [5]for pseudo-differential operators.
Theorem 122 Let P : E → E be a Toeplitz operator, with P ∼ 0 near char g.Then TrGP (g) = tr g P is defined as a distribution on G; P |Eα is of trace classfor each α and formula (109) holds.
We have TrGPQ(g) = TrGQP (g) if one of the two operators is equivariant andone ∼ 0 near char g; so TrG defines a trace map on the algebra of equivariantToeplitz operators.
Proof: this is true if P is of trace class. For the general case, we choose abi-invariant elliptic operator D of order m > 0 on G, e.g. the Casimir of afaithful representation, with m = 2; it defines an invariant Toeplitz operatorDX : E → E, elliptic outside of char g. If P ∼ 0 near Σ, we can divide itrepeatedly by DX (mod. smoothing operators) and get for any N :
P = DNXQ+R (with R ∼ 0)
Then TrGP = DNTrGQ + TrGR: this is well defined as a distribution since Q is oftrace class if N is large, and it does not depend on the choice of D,N,Q,R.
The series is convergent in distribution sense, i.e. the coefficients have atmost polynomial growth with respect to the eigenvalues of D.
More generally if we have let an equivariant Toeplitz complex of finite length:
(E, d) : · · · → Ejd−→ Ei+1 → · · ·
i.e. E is a finite sequence Ek of equivariant Toeplitz bundles, d = (dk : Ek →Ek+1) a sequence of Toeplitz operators such that d2 = 0. Then for a Toeplitzoperator P : E → E, P ∼ 0 near char g, its equivariant supertrace TrGP =∑
(−1)kTrGPk is well defined; it vanishes if P is a supercommutator [A,B] whereA,B are equivariant, and one of them vanishes near char g.
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13.7 Equivariant index
Let E0,E1 be two equivariant Toeplitz bundles.
Definition 123 We will say that an equivariant Toeplitz operator P : E0 → E1
is G-elliptic (transversally elliptic in [5]) if it is elliptic on char g, i.e. theprincipal symbol σ(P ), which is a homogeneous equivariant vector bundle ho-momorphism E0 → E1, is invertible on char g.
If P is G-elliptic it has a G-parametrix Q, i.e. Q : F → E is equivariant, andQP ∼ 1E, PQ ∼ 1F near char g.
The G-index Ind GP is then defined as the distribution
IndGP = TrG1−QP − TrG1−PQ. (110)
More generally, an equivariant complex (E, d) as above is G-elliptic if theprincipal symbol σ(d) is exact on char g. Then there exists an equivariantToeplitz operator s = (sk : Ek → Ek−1) such that 1 − [d, s] ∼ 0 near char g([d, s] = ds+ sd). The index (Euler characteristic) is the super trace
IG(E,d) = supertr (1− [d, s]) =∑
(−1)jTrG(1−[d,s])j
If P is G-elliptic, the restriction Pα : E0,α → E1,α is a Fredholm operator for
any irreducible representation α. Its index Iα is finite (resp. more generally thecohomology H∗α of d|Eα is finite dimensional), and we have
IndGP =∑ 1
dαIα χα (or Ind G
(E,d) =∑ (−1)
dα
j
dim Hjα χα) (111)
13.8 Asymptotic index
The G-index IndGP is obviously invariant under compact perturbation and de-formation, so for fixed Ej it only depends on the homotopy class of the symbolσ(P ). But it does depend on the choice of Szego projectors: the Toeplitz bun-dles Ej are known in practice only through their symbols Ej , and are onlydetermined up to a space of finite dimension, just as the Toeplitz spaces H.
However if E,E′ are two equivariant Toeplitz bundles with the same symbol,there exists an equivariant elliptic Toeplitz operator U : E → E′ with quasi-inverse V (i.e. V U ∼ 1E, UV ∼ 1′E). This may be used to transport equivariantToeplitz operators from E to E′: P 7→ Q = UPV . Then if P ∼ 0 on X0,Q = UPV and V UP have the same G-trace, and since P ∼ V UP , we haveTP − TQ ∈ C∞(G). Thus the equivariant G-trace or index are ultimately welldefined up to a smooth function on G.
Definition 124 We define the asymptotic G-trace AsTrGP as the singularity ofthe distribution TrGP (i.e. TrGP mod. C∞(G)).
140
If P ∼ 0, we have TrGP ∼ 0, i.e. the sequence of Fourier coefficients is ofrapid decrease, O(cα)−m for all m, where cα is the eigenvalue of DG in therepresentation α.
Definition 125 If P is elliptic on char g, the asymptotic G-index AsIndGP isdefined as the singularity of of IndGP .
It can also be viewed as a virtual trace-class representation or character∑nαχα
of G, mod finite representations.
It only depends on the homotopy class of the principal symbol σ(A), andsince it is obviously additive we get :
Theorem 126 (Main theorem) 1) The asymptotic index defines an additivemap from AsInd : KG
X−Z(X) to Sing(G) = C−∞/C∞(G) (Z ⊂ X denotes thebasis of char g).
2) If u : X → X ′ is a contact map, the the asymptotic index map AsIndcommutes with the Bott periodicity map K(X − Z)→ K(X ′ − u(Z))
The Bott periodicity map is described below.
KGX−Z(X) denotes the equivariant K-theory of X with compact support
in X − Z, i.e.the group of stable classes of triples (E,F, u) where E,F areequivariant G-bundles on X, u an equivariant isomorphism E → F defined nearZ, with the usual equivalence relations ((E,F, a) ∼ 0 if a is stably homotopicnear Z to an isomorphism on the whole of X).
The asymptotic index is as well defined for equivariant Toeplitz complexes,exact on Z.
Example : let Σ be a symplectic cone, with free positive elliptic action of U(1),i.e. the Toeplitz generator A = 1
i ∂θ is elliptic with positive symbol (this is thesituation studied in [27]). Then the algebra of invariant Toeplitz operators (mod.C∞) is a deformation star algebra, setting as “deformation parameter” ~ =A−1. char g is empty and the asymptotic trace or index is always defined. Theasymptotic trace of any element A is the series
∑∞−∞ ake
kiθ, ak = trA|Hk , modsmooth functions of θ, i.e. the sequence (ak) is known mod rapidly decreasingsequences. It is standard knowledge that the sequence (ak) has an asymptoticexpansion in (negative) powers of k:
ak ∼∑j≤j0
αjkj . (112)
In this case the asymptotic trace is as well defined by this asymptotic expan-sion; it encodes the same thing as the residual trace, viewed as a power seriesof ~ = k−1.
Remark. For a general the circle group action, with generator A = eiθ, allsimple representations are powers of the identity representation, denoted T , and
141
all representations occurring as indices can be written as formal power serieswith integral coefficients:∑
k∈ZnkT
k (mod. finite sums)
In fact, using the sphere embedding below, it can be seen that the positive andnegative parts of the series are “weakly periodic”, of the form
P±(T, T−1)(1− T±k)k
for suitable polynomials P± and some integer k, i.e. both the positive andnegative parts are the Taylor series of rational functions whose poles are rootsof 1; the asymptotic index corresponds to the polar parts.
13.9 K-theory and embedding
It is convenient (even though not technically indispensable), in particular tofollow the index in an embedding (§129), to reformulate some constructionsabove in terms of sheaves of Toeplitz algebras and modules. in the C∞ categoryE is not coherent and general E-module theory is not practical. We will juststick to two useful examples 52
As above we use the following notation: for distributions, f ∼ g means thatf − g is C∞; for operators, A ∼ B (or A = B mod. C∞) means that A − Bis of degree −∞, i.e. has a smooth Schwartz kernel (if M is a manifold, T •Mdenotes the cotangent bundle deprived of its zero section; it is a symplectic conewith base the cotangent sphere S∗M = T •M/R+). As mentionned earlier, if Σis a G-symplectic cone, the sheaf EΣ of Toeplitz operators (mod C∞) acting onµH is well defined, with the action of G, up to isomorphism, independently ofany embedding Σ → T •M . The asymptotic trace AsTrGP resp. index AsIndGPare well defined for a section P of EΣ vanishing (resp. invertible) near char g.(If M is a G-manifold and X = S∗M (Σ = T •M), EΣ identifies with the sheafof pseudodifferential operators acting on the sheaf µH of microfunctions on X;even in that case the exact index problem does not make sense: a Toeplitzbundle E corresponds to a vector bundle E on the cotangent sphere X = S∗M ,not necessarily the pullback of a vector bundle on M , and E is in general atbest defined up to a space of finite dimension).
An E-moduleM, corresponds to a system of Toeplitz operators, whose sheafof micro-solutions is Hom E(M, µH); likewise a locally free complex (L, d) of E-modules defines a Toeplitz complex (E, D) = Hom (L,H).
We will say that the E-module M is “good” if it is finitely generated,equipped with a filtration M =
⋃Mk (i.e. EpMq = Mp+q,
⋂Mk = 0)
52In the proof of the Atiyah-Weinstein conjecture we need to patch together two smoothembedded manifolds near their boundaries: this cannot be done in the real analytic category,even if things work slightly better there.
142
such that the symbol σ(M) =M0/M−1 has a finite locally free resolution (asa C∞(X)-modul 53). A locally free resolution of σ(M) lifts to a “good resolu-tion” ofM (i.e. locally free and whose symbol is a resolution of σ(M)).54 Tworesolutions of of σ(M) are homotopic, and if σ(M) has locally finite locally freeresolutions it also has a global one (because on comact X (or on the cone Σwith compact basis) we dispose of smooth (homogeneous) partitions of unity);this lifts to a global good resolution of M.
Similarily we will say that aG-elliptic complex (E, d) is “good” if its symbol isexact on char g. Note that “good” is not indispensable to define the asymptoticindex, but it is to define the K-theoretic element [(E, d)] ∈ KG(X − Z).
All this works just as well in presence of a G-action (one must choose invari-ant filtrations etc.).
The asymptotic trace and index extend in an obvious manner to endomor-phisms of good complexes or modules:
• if M = EN is free, End E(M) identifies with the ring of N ×N matriceswith coefficients in the opposite ring Eop, and if A = (Aij vanishes nearchar g we set AsTrG(A) =
∑AsTrG(Ajj).
• If M is isomorphic to the range PN of a projector P in a free moduleN (this does not depend on the choice of N our if A ∈ End E(M) we setAsTrG(A) = AsTrG(PA).
• If (L, d) is a locally free complex and A is a A = (Ak) endomorphism,vanishing near char g, we set AsTrG(A) =
∑(−1)kAsTrG(Ak) (the Euler
characteristic or super trace; if A,B are endomorphisms of opposite de-grees m,−m, we have AsTrG[A,B] = 0, where [A,B] = AB − (−1)m
2BA
is the superbracket).
• If M is a good E-module, (L, d) a good locally free resolution of M,A ∈ End E(M), we set AsTrG(A) = AsTrG(A, where A is any extensionof A to (L, d) (such an extension exists, and is unique up to homotopy i.e.up to a supercommutator).
• Finally if M is a locally free complex with symbol exact on char g, or agood E-module with support outside of char g, it defines a K-theoreticalelement [M] ∈ KG
Z (X), and its asymptotic index (the supertrace of theidentity), is the image by the index map of theorem 126 of [M].
Remark. The equivariant trace or index are defined just as well for modulesadmitting a projective resolution (projective meaning direct summand of some
53The symbol map identifies E0/E−1 with C∞(X); since there exist global elliptic sectionsof E, grM is completely determined by the symbol, same for the resolution.
54the converse is not true: if d is a locally free resolution ofM its symbol is not necessarilya resolution of the symbol of M - if only because filtrations must be defined to define thesymbol and can be modified rather arbitrarily.
143
EN , with a projector not necessarily of degree 0). What does not work for thesemore general objects is the relation to topological K-theory.
13.10 Embedding and transfer
Let Σ be a G-symplectic cone, embedded equivariantly in T •M with M a com-pact G-manifold, and S an equivariant Szego projector. As recalled in §1, therange µH of S is the sheaf of solutions of an ideal I ⊂ EM . The correspondingEM -module is M = EM/I; it is “good”, as is obvious on the microlocal modelor the holomorphic model (for which a good resolution near Σ is ∂b).
Endomorphisms of M are induced by right multiplications m 7→ ma whereaI ⊂ I (a ∈ [I : I], so E ′ = EndMop ' [I : I]/I. The map which to a ∈ [I : I]associates the Toeplitz operator Ta gives an isomorphism from End E(M)op tothe Toeplitz algebra (mod C∞). (this is is easily seen by successive approxi-mations since the symbol of Ta is σ(a)|Σ, or because, as indicated in [27], anyToeplitz operator is also of the form TP where P commutes with the Szegoprojector).
If P is a Toeplitz module, i.e. a left E ′-module supported by Σ, the trans-ferred module isM⊗E′P (also supported by Σ); it has the same solution sheaf asP, since we have HomM⊗P,H) = Hom (P,Hom (M,H)) and Hom (M,H) =H′. In this equality we can replace P by its global good resolution (i.e. replaceHom by Rhom 0, because this resolution is locally isomorphic to ∂b which hasno cohomology mod C∞ near Σ in degree > 0. Thus the transfer preservesasymptotic traces and indices.
This extends obviously to the case where Σ is embedded equivariantly inanother symplectic cone Σ ⊂ Σ′: the Toeplitz sheaf µH is Hom EΣ(M, µH′),with M = E/I and I ⊂ E is the annihilator of the Szego projector S of Σ.
Theorem 127 Let X ′, X be two compact contact G-manifolds and f : X →X ′ be an equivariant embedding. Then the K-theoretical push-forward (Botthomomorphism) KG
X−Z(X) → KGX′−Z′(X
′) commutes with the asymptotic Gindex of G-elliptic equivariant Toeplitz operators.
Let F : EΣ → EΣ′ be an equivariant embedding of the corresponding Toeplitzalgebras (over f), and let M be the E ′Σ-module associated with the Szego pro-jector SΣ (transfer module). We have seen that transfer P 7→M⊗P preservesthe asymptotic index.
Lemma 128 Notations being as above, the K-theoretical element (with supportin Σ) [M] ∈ KG
Σ (T •M) is precisely the Bott element used to define the Bottisomorphism KG(X)→ KG
X(X ′); [M⊗P] is the Bott image of [P].55
55if f : X → Y is a map between manifolds (or suitable spaces), the K-theoretical push-forward is the topological translation of the Grothendieck direct image in K-theory (for al-gebraic or holomorphic coherent modules). Its definition requires a spinc structure on thevirtual normal bundle of f (cf [27], §1.3) and this always exists canonically if X,Y are almostsymplectic or almost complex, or as here if f is an immersion whose normal tangent bundleis equipped with a symplectic or complex structure.
144
Proof: The tranfer module M is good: it has, locally (and globally), a goodresolution. Its symbol is a locally free resolution of σ(M) = C∞(X)/σ(I). Wemay identify a small equivariant tubular neighborhood of Σ with the normaltangent bundle N of Σ in Σ′; N is a symplectic bundle; the ideal I endows itwith a compatible positive complex structure N c (for which the first order jet ofelements of σ(I) are holomorphic in the fibers of N c). In such a neighborhood agood symbol resolution is homotopic to the Koszul complex of N (or the symbolof ∂b in the holomorphic case): the K-theoretical element it defines is preciselythe Bott element.
Example : Let X = S2N−1 be the unit sphere of CN , H the space of holo-morphic functions (the symplectic cone Σ can be identified with CN ). SimilarlyX ′ = C2k−1 and H′. We can embed X ′ as a subsphere of X (equivariantly if weare given suitable unitary group actions).
We can identify H′ with the subset of functions independent of zk+1, · · · , zN .The corresponding operators are the ∂zj , k < j ≤ N and the correspondingcomplex of Toeplitz operators is the partial De Rham complex.
Another way of relating the two is to identify H′ to H/∑Nk+1 zjH, identifying
H′ with the cohomology of the Koszul complex.
Note that we have ∂zm = (N +∑N
1 zj∂j)Tzm so up to a positive factor, theDe Rham complex is the adjoint of the Koszul complex, and both define thesame K-theoretical (equivariant) element.
Remark. It is always possible to embed equivariantly a compact contact mani-fold in a canonical contact sphere with linear G-action (this reduces the problemof computing asymptotic indices to the case where the base space is a sphere -but if G 6= 1 this is still complicated):
Lemma 129 Let Σ be a G cone (with compact base), λ a horizontal 1-form,homogeneous of degree 1 (Lρλ = λ, ρyλ = 0, where ρ is the radial vector field,generating homotheties). Then there exists a homogeneous embedding x 7→ Z(x)of Σ in a complex representation V c of G such that λ = Im Z.dZ
In this construction, Z is homogeneous of degree 12 as above. This applies
of course is Σ is a symplectic cone, λ its Liouville form (the symplectic form isω = dλ and λ = ρyω. We first choose a smooth equivariant function Y = (Yj),homogeneous of degree 1
2 , realizing an equivariant embedding of Σ in V − 0,where V is a real unitary G-vector space (this always exists if the basis iscompact).
Then there exists a smooth function X = (Xj) homogeneous of degree 12
such that λ = 2X.dY . We can suppose X equivariant, replacing it by itsmean
∫g.X(g−1x) dg if need be. We have 2ρydY = Y (Y is of degree 1
2 ) soX.Y = ρyX.dY = 0. Finally we get λ = Im Z.dZ with Z = X + iY (thecoordinates zj on V are homogeneous of degree 1
2 so that the canonical formIm Z.dZ is of degree 1)
145
13.11 Relative index
Let Ω,Ω′ be two strictly pseudo convex Stein domains with smooth boundariesX,X ′. Let f be a smooth contact isomorphism X → X ′. Then the holomorphicpush-forward
W : u ∈ H 7→ S′(u f−1) ∈ H′ (113)
is well defined, and is a (Toeplitz FIO) Fredholm map. The Atiyah-Weinsteinformula computes its index in terms of the geometrical data.
The original Atiyah question was : if M,M ′ are two smooth manifolds,f : S∗M → S∗M ′ a contact isomorphism, F an elliptic FIO associated to f ,then F has an index, which should be given by a similar formula.
This reduces to the former problem since ΨDO on M are the same thing asToeplitz operators on the boundary of a small tubular neighborhood of M in acomplexification M c (cf. [16]) 56
The main difficulty in this problem is that, with a fixed contact structure,we are changing the CR structure, hence the Szego projectors, and there is noformula, using only the contact boundary data, telling how the index behaves.
To overcome this, we enlarge the spaces of holomorphic boundary valuesin such a manner that the index is repeated infinitely many times and can beinterpreted as an asymptotic index, which can be handled geometrically.
13.12 Enlargement
Let Ω be as above, with defining function −φ (φ > 0 , I have changed the sign).We denote the boundary by X0 rather than X).
Ω ⊂ C×Ω denotes the ball |t|2 < φ. Its boundary X is strictly pseudoconvex,provided that Log 1
φ is strictly psh. (e.g. −φ strictly psh. on Ω 57). We stilldenote by Σ ⊃ Σ0 the symplectic cones.
The circle group U(1) acts on X : (t, x) 7→ (eiθt, x).
The volume element on X is dθ dv (smooth, positive, invariant) with dv asmooth positive density on Ω; S denotes the Szego projector, H its range (spaceof boundary values of holomorphic functions of moderate growth near X).
D denotes the Toeplitz operator defined by 1i ∂θ on H. It is self-adjoint, ≥ 0,
equal to TtT∂t .
56except one should also take into account the homotopy class (“winding number”) of theprincipal symbol.
57φ can always be chosen so.
146
The expansion of a function in the Fourier decomposition
H =∑k≥0
Hk (Hk = ker (D− k) )
is equivalent to its Taylor expansion:
f =∑
fk(x)tk.
H0 identifies with the set of holomorphic distributions onX0 (set of boundaryvalues of holomorphic functions on Ω with moderate growth at ∂Ω).
Note that the L2 norm of a holomorphic function tkf(x) on X is∫X
|tkf |2 = 2π∫
Ω
φk|f2|dv
(because |t|2 = φ on X and the measure on X is dθ dv)
If we decompose S in its equivariant components S =∑Sk, we get a se-
quence closely related to that of Berezin (see [5, 49])It will be convenient to replace the Toeplitz FIO operator W by a unitary
multipleE0 = (WW ∗)−
12W : H0 → H′0 (114)
with the convention that (WW ∗)−12 vanishes on the kernel of W ∗; E0 is in
any case unitary mod smoothing operators and has obviously the same indexas W . We are using the norm of H0 i.e. the L2 nor of X (or of Ω), which is notthe L2 norm of X0 (it is rather related to the Sobolev H−
12 norm) - but for the
index this makes no difference)As mentioned above the Toeplitz operator corresponding to rotations is
D = t∂t (=1i∂θ).
we have D = D∗ = T ∗∂tT∗t ; it follows that
∂∗t = tC, (t∗ = C−1∂t).
for an invariant Toeplitz operator C > 0 (unique) 58
We set τ = tC12 (115)
This is a Toeplitz operator of degree 12 , not an integer, but for the commu-
tation constructions below this does not matter
D = ττ∗, [D, τ ] = τ, [τ∗, τ ] = 1 (116)
58if we have a factorization D = PQ with [D,P ] = P , there exists a (unique) invariantinvertible Toeplitz operator U such that P = tU,Q = U−1∂t. Here we have D = tCt∗, soC = C∗ > 0 since D is ≥ 0 and Tt injective.
147
τ is globally defined, a positive multiple of t , τHk = Hk+1
τ is uniquely defined by by these conditions.59
There is a similar construction for Ω′.
Theorem 130 (embedding) There exists an equivariant Toeplitz FIO:E : X → X ′ (with microsupport close to X0) such that (mod C∞)
1) E is unitary elliptic (mod C∞) near X0
2) E induces E0 on H0 (mod smoothing operators)3) Eτ = τ ′EThen the Ek = Hk → H′k all have the same index IndexE0
If 2) holds, E is elliptic on X0 hence G-elliptic (because here G = U(1) actsfreely, with a positive action, on the “interior” X − X0). The last assertionfollows: we have E− τ ′Ek = Ek+1τ and since τ is a bijection Ek → Ek+1 (samefor τ ′), Ek, Ek+1 have same index.
The theorem replaces the relative index Index (E0) by the G-asymptoticindex AsInd (E).
13.13 Collar isomorphism
The geometric counterpart is: there is a (unique) equivariant homogeneoussymplectic isomorphism f of some equivariant neighborhood of Σ0 in Σ to (samefor Σ′) such that f |Σ0 = Id , and σ(τ) f = σ(τ ′), i.e. f commutes with thehamiltonians of the real and imaginary parts of τ, τ ′. This works because thehamiltonians of Re τ, Im τ commute.60
The operator statement follows from the geometric one in the usual manner.Notice that E is at first only defined mod smoothing operators near X0. Weextend it globally using any Toeplitz cut-off.
13.14 Embedding
We have mentioned that any G-contact manifold (compact) can be embeddedin a standard contact sphere with linear unitary action of G. Here we chooseembeddings more precisely.
Let X = S2N+1 ⊂ CN+1 be a large sphere, with variables (T,Z).The circle group G = U(1) acts by
(T,Z) 7→ (eiθT,Z)
The base of char g is the diameter Z(T = 0); it is equal to the fixed point set.59in fact we need a little less than that: τ should be globally defined over Ω, and τk :
Hk → Hk+1 should have index zero; The the hamiltonians of the real and imaginary parts ofτ should commute.
60This would not work if we replaced τ by t because the hamitonians of Re t, Im t do notcommute in general
148
Theorem 131 There exist equivariant contact embeddings F, F ′ of X,X ′ inthe sphere S2N+1 (with U(1)-action as above) such that F = F ′ f near theboundary X0.
We are now reduced to the case where X,X ′ sit in a large sphere S andcoincide near the fixed points. The trivial bundle of X defines, via the transferhomomorphism, a complex A of Toeplitz operators on the large sphereX, whoseK-theoretical element in KG
X(S) is the equivariant Bott image. Same for X ′.
The Toeplitz FIO E of theorem 130 provides a Toeplitz isomorphism A→ A′
near the boundary X0, thus defining a G-elliptic complex on X, whose asymp-totic G-index is precisely what we want to compute.
13.15 Index
Now U(1) acts freely on S−S0 and U(1)\(S−S0) is the open unit ball B ⊂ CN ,so the pull back is an isomorphism K0(CN ) = Z→ KG
S−S0(S) (the generator is
the symbol of the partial De Rham complex ∂X , or of the Koszul complex).
We may now go back to the original situation: Ω and Ω′ are complex mani-folds, glued together by the symplectic map f0 the result Y is not a manifold,but the K-theoretical index is well defined: χ : Kcomp(Y )→ Z:
Theorem 132 The relative index is χ(1Ω − 1Ω′); χ is the K-theoretical char-acter defined by the Bott periodicity theorem; the two trivial bundles 1Ω − 1Ω′
are glued together along the boundary to give an element of compact support.
The K-theoretical element defined by the complex above is the differenceelement between the K-theoretical (spinc) images of Ω and Ω′ defined by F0 onthe boundary (or its extension near the boundary defined by F ; any symplecticdiffeomorphism near the boundary would do as well since these are all isotopic.
This can be readily translated in terms of cohomology, using the Chern char-acters and Todd class, as done in [28]; the Todd class appears when comparingthe Chern class of the Bott element with the Euler class used for integrationalong fibers.
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Universite Pierre et Marie Curie - Paris 6Analyse Algebrique, Institut de Mathematiques de Jussieue-mail: [email protected]
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Index
A× 29A×0 33D 9Dk 9D0 10H∗hom 33O(m) 8O 8Ω 8τ 52W 49automorphisms 29, 32, 33, 35
morphisms 28involutions 30symbol, exponant 34inner -29, 49
Cartan formula 12cone 8
complexified cone 8radial vector 8
connection 52curvature 53embedding 94functional calculus 16group star product 21involution 30
automorphism preserving - 35homomorphisms 28Hochschild cohomology 37isomorphism 40Moyal star product 19, 20non commutative cohomology 38starproduct, star algebra 10Poisson bracket 12
symplectic manifold 13cotangent bundle 14, 14
pseudo-differential operators 23formal – 22Leibniz’ rule 10oscillatory asymptotics 22semi-classical – 25
soft (sheaf) 39
star algebra 10subprincipal symbol 31
automorphism preserving - 35Toeplitz operators 26valuation 48
160
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