Wick Products and Combinatorial Hopf Algebras

• Kurusch EBRAHIMI-FARD (NTNU Trondheim)

Room: Marilyn and James Simons Conference Center 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.

Geometrical Splitting and Reduction of N-point Feynman Diagrams

• Andrei Davydychev (Moscow State University)

Room: Marilyn and James Simons Conference Center 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.

Renormalization Hopf Algebras and Gauge Theories: an Overview

• Walter van SUIJLEKOM (Radboud Universiteit Nijmegen)

Room: Marilyn and James Simons Conference Center 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.

Random Loops and T-algebras

• Martin HAIRER (Imperial College London)

Room: Marilyn and James Simons Conference Center 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.

Bogoliubov Type Recursions for Renormalisation in Regularity Structures

• Yvain BRUNED (University of Edinburgh)

Room: Marilyn and James Simons Conference Center 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.

From Complementations on Lattices to Locality

• Sylvie PAYCHA (Institut für Mathematik Potsdam)

Room: Marilyn and James Simons Conference Center 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

Hopf-algebraic Renormalization of Multiple Zeta Values and their q-analogues

• Dominique MANCHON (CNRS & Université Clermont-Auvergne)

Room: Marilyn and James Simons Conference Center 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.

Cointeracting Bialgebras

• Loïc FOISSY (Université du Littoral Côte d'Opale)

Room: Marilyn and James Simons Conference Center 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.

On the enumerative structures in QFT

• Ali Assem Mahmoud (University of Waterloo)

Room: Marilyn and James Simons Conference Center 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.

Toric Hall Algebras

• Matt SZCZESNY (Boston University)

Room: Marilyn and James Simons Conference Center 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.

Categorical Interactions in Algebra, Geometry and Physics: Cubical Structures and Truncations.

• Ralph KAUFMANN (Purdue University)

Room: Marilyn and James Simons Conference Center 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.

Spaces of Graphs, Tori and Other Flat Gamma-complexes

• Karen VOGTMANN (University of Warwick)

Room: Marilyn and James Simons Conference Center 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.

Renormalization and Galois Theory

• Alain CONNES (IHES)

Room: Marilyn and James Simons Conference Center

Cohomology of Graph Complexes, Invariant Differential Forms and Feynman Periods

• Francis BROWN (University of Oxford)

Room: Marilyn and James Simons Conference Center 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.

Gauge/Gravity Double Copy from a Lie Bracket on the Shuffle Algebra

• Hadleigh Frost (University of Oxford)

Room: Marilyn and James Simons Conference Center 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.

On a Theorem of Kreimer

• Marko BERGHOFF (University of Oxford)

Room: Marilyn and James Simons Conference Center 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.

Elliptic Curves Associated to Two-loop Graphs

• Spencer BLOCH (University of Chicago)

Room: Marilyn and James Simons Conference Center 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.

Generalized Gross-Neveu Universality Class with Non-abelian Symmetry

• John GRACEY (University of Liverpool)

Room: Marilyn and James Simons Conference Center 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.

Solvable Dyson-Schwinger Equations

• Raimar WULKENHAAR (Westfälische Wilhelms-Universität Münster)

Room: Marilyn and James Simons Conference Center 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.

Connes-Kreimer Hopf Algebras : from Renormalisation to Tensor Models and Topological Recursion

• Thomas KRAJEWSKI (CPT Aix-Marseille)

Room: Marilyn and James Simons Conference Center 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.

Solution of $\phi^4_4$ on the Moyal Space

• Alexander Hock (Westfälische Wilhelms-Universität Münster)

Room: Marilyn and James Simons Conference Center 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.

Resurgent Trans-series Analysis of Hopf Algebraic Renormalization

• Gérald DUNNE (University of Connecticut)

Room: Marilyn and James Simons Conference Center 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.

New Techniques for Worldline Integration

• Christian SCHUBERT (Universidad Michoacana de San Nicolas de Hidalgo)

Room: Marilyn and James Simons Conference Center 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.

2010-2020: a Decade of Quantum Computing

• Oliver SCHNETZ (FAU Erlangen-Nürnberg)

Room: Marilyn and James Simons Conference Center 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.

The Euler Characteristic of Out(F_n) and the Hopf Algebra of Graphs

• Michael BORINSKY (Nikhef)

Room: Marilyn and James Simons Conference Center 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).

Dirk Kreimer through my Looking Glass

• Marc BELLON (LPTHE (Sorbonne Université))

Room: Marilyn and James Simons Conference Center 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.

Tree-like Equations from the Connes-Kreimer Hopf Algebra and the Combinatorics of Chord Diagrams

• Lukas Nabergall (University of Waterloo)

Room: Marilyn and James Simons Conference Center 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.

Classical Gravity at High Precision

• Johannes BLÜMLEIN (DESY Zeuthen)

Room: Marilyn and James Simons Conference Center We report on recent progress in post-Newtonian precision calculations for the motion of conservative binary gravitating systems.

Tasmanian Adventures

• David BROADHURST (The Open University)

Room: Marilyn and James Simons Conference Center I report on two adventures with Dirk Kreimer in Tasmania, 25 years ago. One of these, concerning knots, is not even wrong. The other, concerning a conjectural 4-term relation, is either wrong or right. I suggest that younger colleagues have powerful tools that might be brought to bear on this 4-term conjecture.