## Séminaire Motifs et intégrales de Feynman |

Javier Fresán (Ecole Polytechnique), Omid Amini (Ecole Polytechnique), Federico Zerbini (IPhT CEA-Saclay), Pierre Vanhove (IPhT CEA-Saclay)

salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

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 application, I will demonstrate that the coefficients of genus-zero closed string amplitudes are single-valued multiple zeta values. This talk will also serve as an introduction to Dupont's talk.

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 homology. 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 integration, 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].

Symanzik polynomials are defined on Feynman graphs and they are used in quantum field theory to compute Feynman amplitudes. I will present a generalization of these polynomials to the setting of higher dimensional 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.

salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

In this talk, scattering amplitudes in string theory are explored as a laboratory for modern number-theoretic concepts including multiple zeta values and their elliptic generalizations. String amplitudes are computed from moduli-space integrals over punctured Riemann surfaces. Since open strings give rise to surfaces with punctures ordered along their boundaries, one is naturally led to the iterated-integral representation of (elliptic) multiple zeta values. Closed strings in turn are associated with integrals over the sphere, the torus as well as higher-genus surfaces. In case of the sphere, one obtains the so-called ``single-valued'' subset of multiple zeta values. The modular invariant integrals over tori involve what is proposed to be an elliptic analogue of single-valued multiple zeta values.

salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

In this talk I will overview QFT and string theory and will show how the hybrid topology that relates Archimedean and non Archimedean analysis can be used as a tool to understand how string theory converges to QFT in low energies. Joint work with O. Amini, S. Bloch and J. Fresán based on an insight of P. Tourkine.

salle W - DMA - Ecole normale supérieur, 45 rue d'Ulm 75005 Paris

Je discuterai quelques problèmes mathématiques posés par l'idée de voir les amplitudes en théorie quantique des champs comme limite des amplitudes en théorie des cordes. Il s'agit entre autre de décrire la limite non-archimédienne de l'accouplement de hauteur, et de formaliser un énoncé de convergence de l'espace de modules des courbes algébriques vers l'espace de modules des graphes métriques. Travail en collaboration avec Spencer Bloch, José alle de conférences du Centre de mathématiques Laurent Schwartz, École poly techniqueurgos et Javier Fresan.

salle W - DMA - Ecole normale supérieur, 45 rue d'Ulm 75005 Paris

Based on a series of joint works with David, Guillarmou, Kupiainen, Rhodes.

salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

I will explain how amplitudes and Feynman integrals in general even-dimensional renormalisable quantum field theories can be expressed as periods of cohomology groups of algebraic varieties. A consequence is that the amplitudes can be upgraded to richer objects with a host of new structures. In particular, we deduce the action of a group on the upgraded periods, which is the group of the title of the talk. It can be viewed as a superselection procedure for amplitudes at all loop orders. It is currently only defined for generic kinematics, but recent evidence suggests that this group action also holds in QED, string perturbation theory, N=4 super Yang Mills, as well as phi^4 theory to all known orders.

References: F. Brown,

salle W - DMA - Ecole normale supérieur, 45 rue d'Ulm 75005 Paris

String theory naturally leads to associating real analytic modular functions to certain classes of graphs. In this talk I will generalize the construction of such modular graph functions from the case of genus one to the case of higher genus.

pdf de l'exposé

salle W - DMA - Ecole normale supérieur, 45 rue d'Ulm 75005 Paris

General arguments indicate that Feynman integrals can be thought as period integrals. This talk aims to make this relation concrete on a certain class of quantum field theory amplitudes at two- and three-loop orders evaluated in a two-dimensional space-time. We will show that these integrals are given by a period of a variation of a mixed Hodge structure, and can be expressed as Eisenstein series of regulator of a class in the motivic cohomology. This is based on work done in collaboration with Spencer Bloch and Matt Kerr : 1601.08181, 1406.2664, 1309.5865, 1401.6438

salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can be written as a series of integrals over compactified moduli spaces of Riemann surfaces with marked points, indexed by the genus. In genus zero it is known that the amplitude can be expressed in terms of periods of $M_{0,N}$, which are just multiple zeta values. In this talk I want to report on recent advances in the genus one amplitude, relying on the development of an elliptic generalization of multiple zeta values given by certain iterated integrals of modular forms.

Some References: Brown-Levin 1110.6917 , Enriquez 1301.3042, Broedel-Matthes-Schlotterer: 1507.02254, D'Hoker-Green-Gurdogan-Vanhove 1512.06779, Zerbini 1512.05689, Brown 1707.01230, 1708.03354, 1710.07912