Séminaire Motifs et intégrales de Feynman

Javier Fresán (École Polytechnique)

Pierre Vanhove (Institut de Physique Théorique) Federico Zerbini (Institut de Physique Théorique)

Lundi 18 février 2019

Salle de conférences du CPHT – École polytechnique

10h00-11h00 – Frederico Zerbini (IPhT CEA-Saclay) "Single-valued multiple zeta values in genus-zero string amplitudes" Single-valued multiple zeta values are special values of single-valued solutions to the KZ equation on the punctured Riemann sphere. Such solutions, found by F. Brown for an arbitrary number of punctures, are called single-valued hyperlogarithms. In this talk, based on a joint work with P. Vanhove [arXiv:1812.03018], I will develop the theory of integration of single-valued hyperlogarithms. As an appli-cation, I will demonstrate that the coefficients of genus-zero closed string amplitudes are single-valued mul-tiple zeta values. This talk will also serve as an introduction to Dupont's talk. 11h30-12h30 – Clément Dupont (IMAG, Université de Montpellier) " Single-valued integration and superstring amplitudes" The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homol-ogy. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of inte-gration, that we call single-valued integration, pairs a differential form with a `dual differential form'. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown [arXiv:1810.07682].

14h00-15h00 – Matthieu Piquerez (CMLS, École polytechnique) "A multidimensional generalization of Symanzik polynomials" Symanzik polynomials are defined on Feynman graphs and they are used in quantum field theory to com-pute Feynman amplitudes. I will present a generalization of these polynomials to the setting of higher di-mensional simplicial complexes [arXiv:1901.09797]. This generalization has several interesting stability properties and is dual to the generalization of what we call the Kirchhoff polynomials. These properties could be partly explained by the link between the Symanzik polynomials, the Tutte polynomial and matroid theory.