This is a very informal one-day workshop on topics related to the mathematical structure of Feynman integrals, for which tools from algebraic geometry and combinatorics, such as Hodge theory, moduli spaces, positive geometry, etc., are playing increasingly important roles. The goal is to foster discussion between local experts and current visitors with related interests.
All talks will be held in room 1104, on the 11th floor of Burnside Hall.
9:30-10:15: Giroux
10:20-11:05: Thomas
BREAK
11:30-12:15: Dupont
LUNCH (provided for speakers)
13:30-14:15 Ritland
14:20-15:05 Levinson
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15:30-16:15 Caron-Huot
16:20-17:05 Panzer
Title: Duals of Feynman integrals
Abstract: When dealing with Feynman integrals, one is often interested in the integrand modulo integration-by-parts relations. The finiteness of the associated cohomology groups is important for a number of applications, like finding differential equations that integrals satisfy, or to expand a given integral into a set basis of special functions. I will explain - at a physicist's level of rigor - why Feynman integrals are well-defined already in the original momentum space without requiring parametric representations, and introduce their cohomology dual. I'll discuss some conceptual advantages and technical complications of working in the dual space.
Title: Positive geometries and canonical forms via mixed Hodge theory
Abstract: Prompted by Arkani-Hamed and Trnka's discovery of the amplituhedra, the concept of positive geometry recently emerged as an important tool in the study of scattering amplitudes and related quantities in physics. Roughly speaking, a positive geometry is a semi-algebraic domain whose boundary structure matches the residue structure of a unique logarithmic form, called its canonical form. The goal of this talk is to recast these notions as natural byproducts of Deligne's mixed Hodge theory, a central organizing principle in complex algebraic geometry which is intimately linked to the study of logarithmic forms and their residues. This is joint work with Francis Brown.
Title: SOFIA: Singularities of Feynman Integrals Automatized
Abstract: We introduce SOFIA, a Mathematica package that automatizes the computation of singularities of Feynman integrals, based on new theoretical understanding of their analytic structure. Given a Feynman diagram, SOFIA generates a list of potential singularities along with a candidate symbol alphabet. The package also provides a comprehensive set of tools for analyzing the analytic properties of Feynman integrals and related objects, such as cosmological and energy correlators. We showcase its capabilities by reproducing known results and predicting singularities and symbol alphabets of Feynman integrals at and beyond the high-precision frontier.
Title: Fundamental groups of real genus zero Hassett spaces
Abstract: The ordinary and Sn-equivariant fundamental groups of the moduli space of real (n+1)-marked stable curves of genus 0 are known as cactus groups and arise in geometry, combinatorics and representation theory. I will discuss recent joint work with Haggai Liu, in which we compute the ordinary and Sn-equivariant fundamental groups of the real locus of Hassett's moduli space of weighted stable curves (with Sn-equivariant-symmetric weight vector), which we call weighted cactus groups. Our proof is by decomposing the moduli space as a union of cells. These cells are in fact polytopes, in fact products of permutahedra.
Title: Combinatorial Feynman integrals
Abstract: A graph is a simple combinatorial object, and graph theory studies many invariants of graphs (e.g. the Tutte polynomial) which essentially count the number of various structures (e.g. colourings, flows, spanning trees). In particle physics, however, a very different flavour of invariant is assigned to a graph: The Feynman integral. This integral is very hard to compute and typically evaluates to a transcendental number (in fact, a period of an algebraic variety). I will explain a bridge between these seemingly unrelated kinds of invariants: The Martin sequence. This is an integer sequence of combinatorial flavour (counting circuit or spanning tree partitions) that fully determines the Feynman integral. One way to extract the integral from this sequence is via a limiting procedure (called Apery limit) similar to that used by Apery to prove irrationality of zeta(3). This relationship generalizes beyond Feynman integrals to a large class of periods of hypersurfaces (and their diagonal sequences). This talk covers past work with Karen Yeats (arXiv:2304.05299) and ongoing work with Francis Brown.
Title: Deformation quantization generates all multiple zeta values
Abstract: It was proven by Banks-Panzer-Pym that the Feynman type integrals appearing in Kontsevich's deformation quantization formula are always an integer linear combination of multiple zeta values (MZVs). We outline a proof of a converse: all MZVs are generated by these integrals. This result gives further evidence for a conjectured close connection between the algebra of MZVs and classes of quantizations up to gauge equivalence. The talk will give an overview of these Feynman integals, MZVs, and polylogarithms. We will then introduce a set of these Feynman integrals which generate all MZVs, and introduce a remarkably simple algorithm to compute their value by using a new kind of polylogarithm which we call nautical polylogarithms.
Title: u-equations from finite-dimensional algebras
Abstract: In this talk, I will explain how to write down and solve a system of u-equations associated to any finite dimensional algebra with finitely many indecomposable representations. These vastly generalize the system of equations written down by Koba and Nielsen in 1969, which from our point of view are associated to the representation theory of a Dynkin type A quiver. This talk reports on joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori, specifically on a paper which (I strongly hope!) will be on the arxiv by the 18th.
The workshop is supported, in part, by the FRQ-CRM-CNRS exchange program, which is funded by FRQNT.