Laboratoire de Mathématiques Pures et Appliquées
Joseph Liouville

Rencontre du GDR Renormalisation

Equations de Dyson-Schwinger

Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
Université du Littoral Côte d’Opale, Calais

Du 30 septembre 2019 au 4 octobre 2019

Organisateurs : Loïc Foissy, Dominique Manchon, Frédéric Patras.

(Enghish version below).

La rencontre annuelle du GDR Renormalisation aura lieu cette année à Calais du 30 septembre au 4 octobre 2019 et s’articulera autour des équations de Dyson-Schwinger.
Quatre mini-cours de trois heures seront assurés par Pierre Clavier, Kurusch Ebrahimi-Fard, Frédéric Menous et Karen Yeats.

Le GDR dispose de fonds pour financer le déplacement ou le trajet d’étudiants ou de jeunes chercheurs.

English version

The annual meeting of the GDR Renormalisation will take place this year in Calais, from September, 30 to October, 4. It will be based on Dyson-Schwinger equations. Four series of mini-lectures of three hours will be given by Pierre Clavier, Kurusch Ebrahimi-Fard, Frédéric Menous and Karen Yeats.

We can provide some financial support for students and young researchers.

Exposés prévus à ce jour / Scheduled talks by now

Pierre CLAVIER (mini-cours)
Kurusch EBRAHIMI-FARD (mini-cours)
Frédéric MENOUS (mini-cours)
Karen YEATS (mini-cours)
Dominique MANCHON
Hoang Ngoc MINH
Julien QUEVA
Enrico RUSSO
Nikolas TAPIA

Autres participants / Other participants

Jean-David JACQUES
Claude ROGER
Yuanyuan ZHANG

Titres et résumés / Titles and abstracts

Mini-cours / Mini-lectures :

Pierre Clavier : Resurgence, well-behaved averages and Schwinger-Dyson equation

This series of three lectures will be devoted to the presentation of the theory of well-averages of Ecalle and Menous. This theory is part of the more general theory of resurgence and allow to resum a class of divergent series in a direction where the Borel transform has singularities. The thee parts of the lectures will be :

  1. Introduction to resurgence : the basic concepts of resurgence (Borel-Laplace resummation, alien derivations, resurgent functions) will be presented.
  2. The notion of well-behaved average. Averages will be introduced through their weights. We will then define the notion of well-behaved average and see their characterisation.
  3. Finally, we will see the application of these concepts on a Schwinger-Dyson equation. We will study the Schwinger-Dyson equation of the Wess-Zumino model together with its renormalisation group equation.

Kurusch Ebrahimi-Fard : Dyson-Schwinger Equations from a shuffle algebra viewpoint

We will provide a concise introduction to the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. The ultimate aim is to understand and study the notion of Dyson-Schwinger equations in Voiculescu’s theory of free probability, where it plays the role of an integration by parts rule for non-commutative laws. Based on joint work with F. Patras (CNRS).

Frédéric Menous : (Quasi)-shuffles, trees and dynamics

In these lectures we will establish the correspondence between Ecalle’s mould calculus and (infinitesimal) characters on combinatorial Hopf algebras. We will then give some elementary applications to vector fields and diffeomorphisms. The three lectures shall be as follows :

  1. We will explain what is mould calculus in terms of Hopf algebras and why it is so natural tu use it in the framework of formal vector fields and diffeomorphisms.
  2. We will apply the previous results to two elementary problems : conjugacy of formal vector fields and Birkhoff decomposition of formal diffeomorphisms.
  3. We will explain why computations with trees (arborescent moulds or, equivalently, characters on a Connes-Kreimer Hopf algebra) allows to get analytic properties in the previous problems.

Karen Yeats : Dyson-Schwinger equations from physics to combinatorics

We will discuss how to go from what a physicist would recognize as a Dyson-Schwinger equation to the more combinatorial equations that many of us work with. With this framework set, I will then proceed to discuss work of mine with various coauthors on chord diagram expansions solving Dyson-Schwinger equations.

Exposés / Talks :

Carlo Bellingeri : Quasi-geometric rough paths and rough change of variable formula (Slides)

Starting from the classical change of variable formula for smooth paths, in this talk we will explain how the theory of rough paths can allow us to extend this identity, when the underlying path has a generic Hölder regularity. In particular, we use the theory of quasi-shuffle algebra to simplify some general results obtained by David Kelly’s in his PhD thesis, that have been neglected in the last years.

Marc Bellon : Ward-Schwinger-Dyson equations

Usual Schwinger-Dyson equations have serious limitations in all but very specific theories. Overlapping divergences prevent them to be written in terms of renormalised Green functions and any truncation to finite order is incompatible with fundamental symmetries as expressed by Ward identities. Reviving an idea of Ward, we show how all these limitations can be nicely avoided.

Yvain Bruned : BPHZ renormalisation and vanishing subcriticality limit of the fractional $\Phi^3_d$ model

In this talk, we consider the fractional $\Phi^3_d$ model which is a stochastic PDEs on the d-dimensional torus with fractional Laplacian and quadratic nonlinearity driven by space-time white noise. We obtain precise asymptotics on the renormalisation counterterms as the mollification parameter becomes small and the parameter of the fractional Laplacian approaches its critical value.
This is a joint work with Nils Berglund.

Hoang Ngoc Minh : Families of eulerian functions involved in regularizations of divergent polyzetas (Slides)

Cécile Mammez : Weak stuffle products

Study the multiple zeta function in algebraic terms leads to the definition of the stuffle product. It is recursively defined by

$x_i$ $u$ $\star$ $x_j$ $v$=$x_i$ ($u$ $\star$ $x_j$ $v$)+$x_j$ ($x_i$ $u$ $\star$ $v$)+$x_i$ $_+$ $_j$ ($u$ $\star$ $v$)

where $x_i$, $x_j$ are letters and $u$, $v$ are words.
In his thesis, Singer gives the algebraic translation of multiple zeta values with the Schlesinger-Zudilin model (S-Z model) and the Bradley-Zhao model (B-Z model). This leads to the definition of two products where the recursive relation depends on prefix of each words. For instance, we can find letters $y$ and $p$ such as, for any words $u$ and $v$ :

$yu\diamond pv=pv\diamond yu=y(u\diamond pv).$

In this talk, we give a generalization of the stuffle product, called weak stuffle product, including the case of the S-Z model and the B-Z model. Then we study the structure of the underlying algebra.

Dominique Manchon : L’algèbre pré-Lie-Rinehart universelle des forêts aromatiques

Les algèbres de Lie-Rinehart sont la traduction algébrique de la notion d’algébroïde de Lie. Parmi celles-ci figurent les algèbres pré-Lie-Rinehart, qui correspondent à des fibrés vectoriels munis d’une connexion plate et sans torsion. Nous montrerons que l’algèbre pré-Lie-Rinehart libre engendrée par un ensemble X est donnée par les forêts aromatiques décorées par X, c’est-à-dire des forêts dont certains arbres sont munis d’une arête supplémentaire créant ainsi un cycle. Travail en cours avec Gunnar Fløystad et Hans Z. Munthe-Kaas.

Romain Pascalie : Schwinger-Dyson equations in Tensor Field Theory (Slides)

We will present the derivation of Schwinger-Dyson equations in Tensor Field Theory, which are obtained using Ward-Takahashi identities, focusing on the 2-point function. After taking the large N limit, we will find the 2-point function of a particular model in term of Lambert’s W-function.

Raul Penaguiao : Pattern Hopf algebras in combinatorial presheaves (Slides)

The study of substructures in combinatorics is seminal in mathematics : a striking example is the characterization of planar graphs through its description of minors due to Kuratowski. Finding motivation in a new construction of Hopf algebras from patterns in permutations due to Vargas, we introduce the notion of pattern Hopf algebra in a combinatorial presheaf. We start a journey to encode substructures like permutations and graphs as combinatorial presheaves : there, we conjecture that the resulting pattern Hopf algebra is always free. Some particular cases will be dealt with and the general strategy will be described.

Julien Queva : Resurgence of the $\lambda$ $\Phi$ $^2$ $^k$ model in zero dim. QFT

Lucia Rotheray : Dyson-Schwinger equations for (2) incidence bialgebras

In their 2006 paper, Bergbauer and Kreimer used an operad-based argument to prove that the solutions to combinatorial Dyson-Schwinger equations generate sub-Hopf algebras in the Connes-Kreimer Hopf algebra of rooted trees. Incidence bialgebras are defined using the monoidal product (as multiplication) and the composition (for comultiplication) of a monoidal category. In particular, there is an incidence bialgebra of rooted trees which admits the Hopf algebra of rooted trees as a quotient. We will see how Bergbauer & Kreimer’s operadic proof fits nicely into this picture and how this could lead us to us define cDSEs for other combinatorial Hopf algebras, in particular the Hopf algebra of skew shapes.

Enrico Russo : TBA

Nikolas Tapia : Non-commutative Wick polynomials (Slides)

Solutions to the Dyson-Schwinger equation with Gaussian weights can be obtained by inverting the normal ordering operator. This operator is related to Wick polynomials of annihilation and creation operators.
In this talk I will focus on Wick polynomials for non-commuting random variables. I will present a shuffle algebra approach to the combinatorics of such polynomials in the free, boolean and monotone settings, building on previous work by K. Ebrahimi-Fard, F. Patras and a joint work with K. Ebrahimi-Fard, F. Patras and L. Zambotti.

Yannic Vargas : Universal Hopf algebras from Hopf monoids (Slides)