# Séminaire Motifs et intégrales de Feynman

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

10h30
##### Omid Amini
Amplitudes de Feynman comme une limite des amplitudes de cordes

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 Burgos et Javier Fresan.

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

14h30
##### Vincent Vargas
An introduction to quantum Liouville theory and bosonic string theory In the 1981 seminal paper "Quantum geometry of Bosonic strings", Polyakov introduced a path integral formulation of (critical and non critical) string theory. Polyakov discovered that an esssential building block of the theory is a conformal field theory called Liouville theory. In this talk, I will explain our recent probabilistic construction of Liouville theory on Riemann surfaces. I will also explain how one can use Liouville theory to construct 2d string theory on Riemann surfaces; an essential feature of our approach is that it is non perturbative.
Based on a series of joint works with David, Guillarmou, Kupiainen, Rhodes.

## Lieu :salle de conférences du Centre de mathématiques Laurent Schwartz, École poly technique

10h30
##### Francis Brown
Périodes en physique des particules et groupe de Galois cosmique --- Periods in particle physics and cosmic Galois group

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, Feynman Amplitudes and Cosmic Galois group arXiv:1512.06409, Feynman amplitudes, coaction principle, and cosmic Galois group CNTP Volume 11 (2017) Number 3 pp 453-556

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

10h30
##### Eric D'Hoker
Higher genus modular graph functions

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

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

10h30
##### Pierre Vanhove
Motives and Feynman integrals

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

10h30

## Lieu : salle de conférences du Centre de mathématiques Laurent Schwartz, École polytechnique

page au CMLS

##### Federico Zerbini
Elliptic multiple zeta values and string theory

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