# Algebraic Structures in Perturbative Quantum Field Theory

Europe/Paris
Marilyn and James Simons Conference Center (IHES)

### Marilyn and James Simons Conference Center

#### IHES

35 route de Chartres, F-91440 Bures-sur-Yvette, France
Description

----------------------------------------- IMPORTANT INFORMATION ------------------------------------------

Due to the evolution of the health situation related to the Coronavirus epidemic, the conference will finally be totally on line. The Zoom link will be sent in the confirmation mail.

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Algebraic Structures in Perturbative Quantum Field Theory
A conference in honour of Dirk Kreimer's 60th birthday

On the occasion of Dirk Kreimer's birthday, there will be a special issue of SIGMA on "Algebraic Structures in Perturbative Quantum Field Theory".

Perturbative quantum field theory is essential for precision calculations of observables measured in experiments like the LHC, and therefore it is crucial for our understanding of the physics of the universe. At the same time, it is an extremely rich source of connections to a wide range of active research areas in mathematics. For example, Feynman integrals give rise to interesting motives and periods in algebraic geometry, their renormalization rests on combinatorial Hopf algebras underlying Feynman graphs, and further relations to noncommutative geometry and the moduli space of tropical curves and outer space have also been discovered.

This growing program keeps expanding in both breadth and depth, and exciting young researchers are entering the field. Now is an opportune time to bring together scientists working on all related aspects, to review old and new connections and to advance the state of the art. Lectures by established scientists will be accompanied by talks from young researchers, including a session dedicated to present and discuss open problems.

Close collaborations between mathematicians and physicists have been absolutely key for this kind of research, and many were initiated by Dirk Kreimer. Throughout his career, he made substantial contributions across these topics and led students and collaborators to the profound mathematical structures in perturbative quantum field theory that we are aware of today. Dirk Kreimer spent a particularly productive time at the IHES, and it is an honour that this workshop takes place in its inspiring and interdisciplinary environment.

Organisers: Erik PANZER (University of Oxford) & Karen YEATS (University of Waterloo)

Invited speakers include:

• Ali Assem Mahmoud, University of Waterloo
• Marc Bellon, LPTHE (Sorbonne Université)
• Marko Berghoff, Humboldt-Universität
• Spencer Bloch, University of Chicago
• Johannes Blümlein, DESY Zeuthen
• Michael Borinsky, Nikhef
• David Broadhurst, The Open University
• Francis Brown, University of Oxford
• Yvain Bruned, University of Edinburgh
• Alain Connes, IHES & Collège de France
• Andrei Davydychev, Moscow State University
• Gérald Dunne, University of Connecticut
• Kurusch Ebrahimi-Fard, NTNU Trondheim
• Loïc Foissy, Université du Littoral Côte d'Opale
• Hadleigh Frost, University of Oxford
• John Gracey, University of Liverpool
• Martin Hairer, Imperial College London
• Ralph Kaufmann, Purdue University
• Thomas Krajewski, CPT Aix-Marseille
• Dominique Manchon, CNRS & Université Clermont-Auvergne
• Lukas NABERGALL, University of Waterloo
• Sylvie Paycha, Institut für Mathematik Potsdam
• Oliver Schnetz, FAU Erlangen-Nürnberg
• Christian Schubert, Universidad Michoacana de San Nicolas de Hidalgo
• Matt Szczesny, Boston University
• Walter van Suijlekom, Radboud Universiteit Nijmegen
• Karen Vogtmann, University of Warwick
• Raimar Wulkenhaar, Westfälische Wilhelms-Universität Münster

Organized with the support of:

Contact: Elisabeth Jasserand
• Monday, 16 November
• 13:10 14:00
Wick Products and Combinatorial Hopf Algebras 50m

Wick products play a central role in both quantum field theory and stochastic calculus. They originated in Wick’s work from 1950. In this talk we will describe Wick products using combinatorial Hopf algebra. Based on joint work with F. Patras, N. Tapia, L. Zambotti.

Speaker: Prof. Kurusch EBRAHIMI-FARD (NTNU Trondheim)
• 14:00 14:45
Geometrical Splitting and Reduction of N-point Feynman Diagrams 45m

A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown how the geometrical splitting of N-point diagrams can be used to simplify the parametric integrals and reduce the number of variables in the occurring functions. Moreover, such a splitting yields useful connections between Feynman integrals with different momenta and masses. Calculation of the one-loop two-, three-and four-point functions in general kinematics is presented. The work on this approach was started in the 1990s in Tasmania, within a project where Bob Delbourgo and Dirk Kreimer were involved.

Speaker: Prof. Andrei Davydychev (Moscow State University)
• 14:45 15:00
Break 15m

Introduction to gather

• 15:00 15:50
Renormalization Hopf Algebras and Gauge Theories: an Overview 50m

We give an overview of the Hopf algebraic approach to renormalization, with a focus on gauge theories. We illustrate this with Kreimer's gauge theory theorem from 2006 and sketch a proof. It relates Hopf ideals generated by Slavnov-Taylor identities to the Hochschild cocycles that are given by grafting operators.

In the second part of the talk I will briefly present Kreimer's unexpected influence on noncommutative geometry via my more recent research. In joint work with Teun van Nuland we uncover a rich structure of the spectral action functional. We express its Taylor expansion in an inner perturbation in terms of Yang-Mills and Chern-Simons forms integrated against even Hochschild and odd cyclic cocycles, respectively.

Speaker: Prof. Walter van SUIJLEKOM (Radboud Universiteit Nijmegen)
• 15:50 16:30
Break 40m
• 16:30 17:20
Random Loops and T-algebras 50m

The stochastic quantization of the 1d non-linear sigma model (i.e. the natural Langevin dynamic on loop space) naturally leads to the study of an algebraic structure we call a T-algebra. We will discuss how they arise, a few of their properties, as well as a concrete example of their application.

Speaker: Prof. Martin HAIRER (Imperial College London)
• 17:20 18:10
Bogoliubov Type Recursions for Renormalisation in Regularity Structures 50m

Hairer's regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. We will revisit the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation. This is a joint work with Kurusch Ebrahimi-Fard.

Speaker: Prof. Yvain BRUNED (University of Edinburgh)
• Tuesday, 17 November
• 11:30 12:20
From Complementations on Lattices to Locality 50m

A complementation proves useful to separate divergent terms from convergent terms. Hence the relevance of complementation in the context of renormalisation. The very notion of separation is furthermore related to that of locality. We extend the correspondence between Euclidean structures on vector spaces and orthogonal complementation to a one to one correspondence between a class of locality structures and orthocomplementations on bounded lattices. This is joint work with P. Clavier, Li Guo and Bin Zhang

Speaker: Prof. Sylvie PAYCHA (Institut für Mathematik Potsdam)
• 12:20 13:30
Lunch break 1h 10m
• 13:30 14:20
Hopf-algebraic Renormalization of Multiple Zeta Values and their q-analogues 50m

Multiple zeta values are real numbers which appeared in depth one and two in the work of L. Euler in the Eighteenth century. They first appear as a whole in the work of J. Ecalle in 1981, as infinite nested sums. A systematic study starts one decade later with M. Hoffman, D. Zagier and M. Kontsevich, with multiple polylogarithms and iterated integral representation as a main tool. After a brief historical account, I'll explain how a quasi-shuffle compatible definition (by no means unique) can be given through Connes-Kreimer's Hopf-algebraic renormalization when the nested sum diverges. I'll also give an account of the more delicate renormalization of shuffle relations. Finally, I'll introduce the Ohno-Okuda-Zudilin model of q-analogues for multiple zeta values, and describe the algebraic structure which governs it.

Speaker: Prof. Dominique MANCHON (CNRS & Université Clermont-Auvergne)
• 14:20 15:10
Cointeracting Bialgebras 50m

Pairs of cointeracting bialgebras recently appears in the literature of combinatorial Hopf algebras, with examples based on formal series, on trees (Calaque, Ebrahimi-Fard, Manchon), graphs (Manchon), posets... We will give several results obtained on pairs of cointeracting bialgebras: actions on the group of characters, antipode, morphisms to quasi-symmetric functions... and we will give applications to Ehrhart poylnomials and chromatic polynomials.

Speaker: Prof. Loïc FOISSY (Université du Littoral Côte d'Opale)
• 15:10 15:15
Break 5m
• 15:15 15:45
On the enumerative structures in QFT 30m

The aim of this talk is to display some enumerative results that are directly applied in quantum field theory. We shall see how the number of connected chord diagrams can be used to count one-particle-irreducible (1PI) diagrams in Yukawa theory. This translation of Feynman diagrams simplified the process of calculating the asymptotic behaviour of the corresponding Green functions.

Speaker: Dr Ali Assem Mahmoud (University of Waterloo)
• 15:45 16:30
Break 45m
• 16:30 17:20
Toric Hall Algebras 50m

The process of counting extensions in categories yields an associative (and sometimes Hopf) algebra called a Hall algebra. Applied to the category of Feynman graphs, this process recovers the Connes-Kreimer Hopf algebra. Other examples abound, yielding various combinatorial Hopf algebras. I will discuss joint work with J. Jun which attaches a Hopf algebra to a projective toric variety X. This Hopf algebra arises as the Hall algebra of a category of coherent sheaves on X locally modeled on n-dimensional skew partitions.

Speaker: Prof. Matt SZCZESNY (Boston University)
• 17:20 18:10
Categorical Interactions in Algebra, Geometry and Physics: Cubical Structures and Truncations. 50m

There are several interactions between algebra and geometry coming from polytopic complexes as for instance demonstrated by several versions of Deligne's conjecture. These are related through blow-ups or truncations. The polytopes and their truncations also appear naturally as regions of integration for products, which is an area of active study. Two fundamental polytopes are cubes and simplices. The importance of cubes as a basic appears naturally in various situations on which we will concentrate. In particular, we will discuss cubical Feynman categories, which afford a W-construction that is a cubical
complex.
These relate combinatorics to geometry. Furthermore using categorical notions of push-forwards, we show how to naturally construction Moduli Spaces of curves and several of their compactifications. The combinatorial ingredients are graphs and there is a universal way of decorating them to study different types. This makes the theory applicable to several different geometries appearing in Moduli Spaces and Outer space. With respect to physics, there is an additional relationship coming through Hopf algebras which in turn also are related to multiple zeta values.
We will discuss these constructions and relations on concrete examples.

Speaker: Prof. Ralph KAUFMANN (Purdue University)
• Wednesday, 18 November
• 11:30 12:20
Spaces of Graphs, Tori and Other Flat Gamma-complexes 50m

Spaces of finite graphs play a key role in perturbative quantum field theory, but also in many other areas of science and mathematics. Among these is geometric group theory, where they are used to model groups of automorphism of free groups. Graphs can be thought of as 1-dimensional flat metric spaces.In higher dimensions, spaces of flat n-dimensional tori model automorphism groups of free abelian groups.There are very interesting groups which interpolate between free groups and free abelian groups, called right-angled Artin groups. I will describe a space of “Gamma-complexes”, which are a hybrid of tori and graphs, and which model automorphism groups of right-angled Artin groups, by recent joint work with Bregman and Charney.

Speaker: Prof. Karen VOGTMANN (University of Warwick)
• 12:20 13:30
Lunch break 1h 10m
• 13:30 14:20
Renormalization and Galois Theory 50m
Speaker: Prof. Alain CONNES (IHES)
• 14:20 15:10
Cohomology of Graph Complexes, Invariant Differential Forms and Feynman Periods 50m

Kontsevich introduced the graph complex $GC_2$ in 1993 and raised the problem of determining its cohomology. This problem is of renewed importance following the recent work of Chan-Galatius-Payne, who related it to the cohomology of the moduli spaces $M_g$ of curves of genus $g$. It is known by Willwacher that the cohomology of $GC_2$ in degree zero is isomorphic to the Grothendieck-Teichmuller Lie algebra $grt$, but in higher degrees, there are infinitely many classes which are mysterious and have no such interpretation.

In this talk, I will define algebraic differential forms on a moduli space of graphs (outer space). Such a form is a map which assigns to every graph an algebraic differential form of fixed degree, satisfying some compatibilities. Using the tropical Torelli map, I will construct an infinite family of such differential forms, which can in turn be integrated over cells. Surprisingly, these integrals are always finite, and therefore one can assign numbers to homology classes in the graph complex. They turn out to be Feynman periods in phi^4 theory, and can be used to detect graph homology classes.

The upshot of all this is a new connection between graph cohomology, Feynman integrals and motivic Galois groups. I will conclude with a conjectural explanation for the higher degree classes in graph cohomology.

Speaker: Prof. Francis BROWN (University of Oxford)
• 15:10 15:15
Break 5m
• 15:15 15:45
Gauge/Gravity Double Copy from a Lie Bracket on the Shuffle Algebra 30m

The 'field theory KLT' or 'double copy' relations express gravity amplitudes in terms of gauge theory partial amplitudes. I present an elementary proof of these identities, using only the properties of Lie polynomials and the shuffle algebra. The work completes a project sketched by M Kapranov in 2012, and is joint with C Mafra.

Speaker: Hadleigh Frost (University of Oxford)
• 15:45 16:30
Break 45m
• 16:30 17:20
On a Theorem of Kreimer 50m

I will report on joint work with Dirk on his vision of exploring
quantum fields in outer space. Our expeditions have so far uncovered an exciting wonderland of algebraic, geometric, and topological relations in a magical galaxy, populated by Feynman graphs. Its inhabitants seem to play a mysterious game of hide-and-seek, hopfing around singularities of various kinds, all ruled by the mighty King Cutkosky.

Speaker: Dr Marko BERGHOFF (University of Oxford)
• 17:20 18:10
Elliptic Curves Associated to Two-loop Graphs 50m

Amplitudes of one-loop graphs are known to be dilogarithms. What can one say about two-loop graphs? In a surprising number of cases, the motive of the second Symanzik of a two-loop graph involves (indeed, the motive is actually built around) the motive of an elliptic curve, suggesting some relation between the amplitude and elliptic polylogarithms. I will discuss a number of examples. This is joint work with C. Doran, M. Kerr, and P. Vanhove.

Speaker: Prof. Spencer BLOCH (University of Chicago)
• Thursday, 19 November
• 11:30 12:20
Generalized Gross-Neveu Universality Class with Non-abelian Symmetry 50m

We use the large N expansion to compute d-dimensional critical exponents at O(1/N^3) for a generalization of the Gross-Neveu Yukawa universality class that includes a non-abelian symmetry. Specific groups correspond to certain phase transitions in condensed matter physics such as graphene. The effect of the non-abelian symmetry on the exponents is evidenced by the appearance of rank 4 Casimirs in the higher order 1/N analytic corrections. These in effect tag the light-by-light diagrams. The main benefit of the final expressions for the exponents is that one can recover previous results for a variety of universality classes in various limits.

Speaker: Prof. John GRACEY (University of Liverpool)
• 12:20 13:30
Lunch break 1h 10m
• 13:30 14:20
Solvable Dyson-Schwinger Equations 50m

Dyson-Schwinger equations provide one of the most powerful non-perturbative approaches to quantum field theories. The quartic analogue of the Kontsevich model is a toy model for QFT in which the tower of Dyson-Schwinger equations splits into one non-linear equation for the planar two-point function and an infinite hierarchy of affine equations for all other functions. The non-linear equation admits a purely algebraic solution, identified through insight from perturbation theory. The affine equations turn out to be affiliated with (and solved by) a universal structure in complex algebraic geometry: blobbed topological recursion. As such they connect to the geometry of the moduli space of complex curves.

Speaker: Prof. Raimar WULKENHAAR (Westfälische Wilhelms-Universität Münster)
• 14:20 15:10
Connes-Kreimer Hopf Algebras : from Renormalisation to Tensor Models and Topological Recursion 50m

At the turn of the millenium, Connes and Kreimer introduced Hopf algebras of trees and graphs in the context of renormalisation. We will show how the latter can be used to formulate the analogue of Virasoro constraints for random tensors, which are natural generalisations of random matrices. Motivated by this example, we will also sketch how these algebras appear in the formulation of topological recursion proposed by Kontsevich and Soibelman.

Speaker: Prof. Thomas KRAJEWSKI (CPT Aix-Marseille)
• 15:10 15:15
Break 5m
• 15:15 15:45
Solution of $\phi^4_4$ on the Moyal Space 30m

We show the exact solution of the self-dual $\phi^4$-model on the 4-dimensional Moyal space. Using the results explained in Raimar's talk, an implicitly defined function converges to a Fredholm integral, which is solved, for any coupling constant $\lambda>-\frac{1}{\pi}$, in terms of a hypergeometric function. We prove that the interacting model has spectral dimension $4-2\frac{\arcsin(\lambda\pi)}{\pi}$ for $|\lambda|<\frac{1}{\pi}$. It is this dimension drop which for $\lambda>0$ avoids the triviality problem of the $\phi^4_4$ model on the Moyal space.

Speaker: Dr Alexander Hock (Westfälische Wilhelms-Universität Münster)
• 15:45 16:30
Break 45m
• 16:30 17:20
Resurgent Trans-series Analysis of Hopf Algebraic Renormalization 50m

In the Kreimer-Connes Hopf algebraic approach to renormalization, for certain QFTs the Dyson-Schwinger equations can be reduced to nonlinear differential equations. I describe methods based on Ecalle's theory of resurgent trans-series to extract non-perturbative information from these Dyson-Schwinger equations. Even in the absence of exact results, there exist efficient methods to uncover non-perturbative information numerically from perturbative data.

Speaker: Prof. Gérald DUNNE (University of Connecticut)
• 17:20 18:10
New Techniques for Worldline Integration 50m

The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. I will summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and multiple zeta values.

Speaker: Prof. Christian SCHUBERT (Universidad Michoacana de San Nicolas de Hidalgo)
• Friday, 20 November
• 11:30 12:20
2010-2020: a Decade of Quantum Computing 50m

Supported by Dirk Kreimer, in 2010 I started analyzing and calculating high loop-order amplitudes in perturbative quantum field theory. The main tools were graphical functions, generalized single-valued hyperlogarithms (GSVHs), and the c_2-invariant. I will report on the progress that has been achieved in the past decade and give a brief account of what might be within reach (of these techniques) in the future.

Speaker: Dr Oliver SCHNETZ (FAU Erlangen-Nürnberg)
• 12:20 13:30
Lunch break 1h 10m
• 13:30 14:20
The Euler Characteristic of Out(F_n) and the Hopf Algebra of Graphs 50m

In their 1986 work, Harer and Zagier gave an expression for the Euler characteristic of the moduli space of curves, M_gn, or equivalently the mapping class group of a surface. Recently, in joint work with Karen Vogtmann, we performed a similar analysis for Out(Fn), the outer automorphism group of the free group, or equivalently the moduli space of graphs. This analysis settles a 1987 conjecture on the Euler characteristic and indicates the existence of large amounts of homology in odd dimensions for Out(Fn). I will illustrate these results and explain how the Hopf algebra of graphs, based on the works of Kreimer, played a key role to transform a simplified version of Harer and Zagier's argument, due to Kontsevich and Penner, from M_gn to Out(Fn). This combined technique can be interpreted as a renormalized topological field theory. I will also report on more recent results on the integer Euler characteristic of Out(Fn).

Speaker: Dr Michael BORINSKY (Nikhef)
• 14:20 15:10
Dirk Kreimer through my Looking Glass 50m

For the last 13 years, my research activities have been largely influenced by Dirk Kreimer’s research. I will the speak of the prehistory, the actuality and the future of this influence.

Speaker: Prof. Marc BELLON (LPTHE (Sorbonne Université))
• 15:10 15:15
Break 5m
• 15:15 15:45
Tree-like Equations from the Connes-Kreimer Hopf Algebra and the Combinatorics of Chord Diagrams 30m

We describe how certain analytic Dyson-Schwinger equations and related tree-like equations arise from the universal property of the Connes-Kreimer Hopf algebra applied to Hopf subalgebras obtained from combinatorial Dyson-Schwinger equations in the work of Foissy. We then show how these equations can be solved as weighted generating functions of certain classes of chord diagrams and obtain an explicit formula counting some of these combinatorial objects.

Speaker: Lukas Nabergall (University of Waterloo)
• 15:45 16:30
Break 45m
• 16:30 17:20
Classical Gravity at High Precision 50m

We report on recent progress in post-Newtonian precision calculations for the motion of conservative binary gravitating systems.

Speaker: Prof. Johannes BLÜMLEIN (DESY Zeuthen)
• 17:20 18:10