Registration

• Sorana Scholtes (Max-Planck-Institut für Physik )

Welcome

• Johannes Henn

Parametric cycles

• Ruth Britto (Trinity College Dublin)

I will discuss integration contours for integrals over Feynman parameters that reveal discontinuities. I explore their variation in phase space, with a view towards deriving cut relations and understanding sequential discontinuities.

Effective homology and periods of algebraic varieties

• Eric Pichon-Pharabod

The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of the period matrix allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. In this talk, we will present a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of periods of algebraic varieties, and showcase implementations for the case of hypersurfaces and elliptic surfaces.

Free Discussions

Positive geometries and canonical forms via mixed Hodge theory

• Clément Dupont

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.

Taming IBPs with Transverse Integration

• Gaia Fontana (University of Zürich)

Reduction of Feynman integrals to a basis of linearly independent master integrals is a crucial step in any perturbative calculation, but also one of its main bottlenecks. In this talk I will present an improvement over the traditional approach to IBP reduction, that exploits transverse integration identities. Given an integral family to be reduced, the key idea is to find sectors whose corner integrals correspond to either diagrams with fewer external legs or diagrams that can be factorized as products of lower-loop integrals. Then, using transverse integration identities, i.e. a tensor decomposition in the subspace that is transverse to the external momenta of the diagrams, we map integrals belonging to these sectors and their subsectors to (products of) integrals belonging to new and simpler integral families, characterized by either fewer generalized denominators, fewer external invariants, lower loops or combinations thereof. Integral reduction is thus drastically simpler for the newer families. I will include some applications to cutting-edge two-loop families which show significant improvements with respect to traditional methods.

Moduli spaces of graphs

• Karen Vogtmann (University of Warwick)

A natural setting for studying the n-loop contribution to a Feynman amplitude is the appropriate moduli space of graphs. I will describe these moduli spaces, then explain how methods from both geometric group theory and quantum field theory can be used to explore their structure.

Linear PDE with constant coefficients

• Bernd Sturmfels

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis-Palamodov from the 1960’s. Our audience will have a chance to gain hands-on experience with primary ideals and the schemes they represent.

Canonical differential equations for maximal cuts of hyperelliptic Feynman integrals

• Franziska Porkert (Universität Bonn)

Feynman integrals are the building blocks of multi-loop scattering amplitudes and beyond one loop, many Feynman integrals are related to interesting geometries. In this talk, I will focus on Feynman integrals related to hyperelliptic curves and discuss ongoing work on finding canonical differential equations for maximal cuts of such Feynman integrals. This includes new ideas about the connection between the intersection matrix of a twisted cohomology group related to the maximal cut and the form of its differential equation.

Relations in the differential equation for Feynman integrals

• Stefan Weinzierl (Universität Mainz)

The differential equation for a system of Feynman integrals is encoded in a connection matrix. In this talk I will discuss that a specific choice of master integrals can lead to relations among the entries of the connection matrix. I will discuss self-duality and Galois symmetries.

Advancing diversity in math and physics

• Diana López-Falcón
• Ruth Britto (Trinity College Dublin)
• Bernd Sturmfels

A hypergeometric view on Landau singularity

• Saiei Matsubara-Heo

Recent years have seen a resurgence of old topics in quantum field theory. One of them is Landau singularity, which should be defined as the singular locus of a Feynman integral. In this talk, we develop a way to understand and compute Landau singularity from the perspective of hypergeometric system, which is a special class of D-modules. This explains why Landau singularity is a natural generalization of principal A-determinant.

Euler Discriminant of Complements of Hyperplanes

• Claudia Fevola (Inria Saclay)

The Euler discriminant describes the locus of coefficients that cause a drop in the Euler characteristic of a very affine variety. In this talk, we focus on the case where the variety is the complement of hyperplanes. I will present formulas for two specific scenarios: when the coefficients are sparse and when they are restricted to a subspace of the parameter space. These formulas enable the computation of singularities in Euler integrals of linear forms, with applications in cosmology. This is joint work with Saiei Matsubara-Heo.

Free Discussions

IBP Reduction using Gröbner bases

• Tobias Huber (Siegen U)

Integral reduction based on integration-by-parts (IBP) identities are an indispensable tool for accomplishing higher-order calculations in perturbative quantum field theory. We approach this task from the point-of-view of algebraic geometry by solving the system of IBP equations symbolically. We formulate the problem as a non-commutative rational double-shift algebra Y and a left ideal that the IBP equations generate therein. For simple examples we compute the Gröbner basis of the left ideal in Y. For more complicated problems we compute so-called normal-form IBPs from a linear algebra ansatz, whose solution makes use of techniques from finite fields and rational reconstruction. Both, the Gröbner basis and the normal-form IBPs, allow for a fast and targeted reduction of integrals. We also report on open issues and ongoing developments, among which the extension of the algebra by including the dimensional shift operator. The presentation is based on arXiv2207.09275, arXiv2210.05347, and work in progress.

A Survey on (un)Twisted Logarithmic Comparison Theorems

• Daniel Bath (KU Leuven)

The comparison theorems of Grothendieck and Deligne tell us that to compute cohomology of the complement of a hypersurface with constant coefficients one must compute the cohomology of the meromorphic (or rational) de Rham complex. As this complex's objects lack finiteness, one wonders if a subcomplex of forms of order at most one along the hypersurface suffices. In verbiage: does the Logarithmic Comparison Theorem hold? We will survey the recent results on this problem, which, time permitting, may include: making the statement of the Logarithmic Comparison Theorem, and its twisted versions, explicit; verifying the (un)Twisted Logarithmic Comparison theorems for hyperplane arrangements, resolving a conjecture Terao from the 1970s; an explicit D-module theoretic formulation of the problem in the spirit of the Riemann Hilbert correspondence; applications to Bernstein—Sato polynomials; how to use the (un)Twisted Logarithmic Comparison theorem to turn computations into finite dimensional linear algebra. Based on solo work by the speaker (arXiv: 2202.01462) and joint work with Morihiko Saito (arXiv: 2203.11716).

Geometrizing Landau Analysis

• Qinglin Yang (Max Planck Institute for Physics)

In this talk, we will revisit Landau analysis for Feynman Integrals and their singularity study from a new geometrical viewpoint. The first part of the talk is about the foundation of our method. Through interpreting Landau loci by pinching of Schubert solutions, we will be able to uplift Landau singularities of an integral to its symbol letters automatically, and see our previous conjectural method for symbology study called Schubert analysis naturally arises. In the second part, we apply this method for studying alphabets of scattering amplitudes in N=4 Super Yang-Mills theory. Following amplituhedron picture and previous Landau diagrams study, we produce 2-loop MHV and NMHV amplitudes alphabet for all multiplicities nicely. As a byproduct conclusion, we also talk about cluster algebraic structures underlying the amplitude alphabets.

Triangulations of cosmological polytopes

• Martina Juhnke

A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. The goal of this talk to report on specific types of triangulations of these polytopes, obtained via algebraic techniques. More precisely, we show that the toric ideal of every cosmological polytope a Gröbner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including chains and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. Choosing the term order wisely, for the chain, we find a triangulation that respects the graph symmetry and whose facets seem to be in bijection with certain graph tubings which used have been used by Arkani-Hamed, Baumann, Hillman, Joyce, Lee and Pimentel to use certain wavefunction coefficients.

Completely monotone functions and applications

• Khazhgali Kozhasov (Université Côté d'Azur)

Complete monotonicity of a smooth function on a convex cone is a strong property given by infinitely many sign conditions on the directional derivatives of the function. I will discuss results and questions around this concept that are motivated by research in convex optimization (interior-point methods), algebraic statistics (exponential families) and real algebraic geometry (hyperbolic and nonnegative polynomials). The talk is based on joint works with M. Michalek, B. Sturmfels and J.-B. Lasserre.

Free Discussions

Classical radiation at one loop

• Ingrid Vazquez-Holm

We study classical radiation fields at next-to-leading order using the methods of scattering amplitudes. The fields of interest to us are sourced when two massive, point-like objects scatter inelastically, and can be computed from one- loop amplitudes. The real and imaginary parts of the amplitudes play important but physically distinct roles in the radiation field. We argue that the imaginary part captures the effects of radiation reaction. This aspect of radiation reaction is directly linked to cuts of one-loop amplitudes which expose Compton trees.

Symbol Alphabets from the Landau Singular Locus

• Christoph Dlapa (DESY)

I present work which provides evidence through two loops that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal A-determinant. Focusing on one loop, I further discuss how all square-root letters may also be obtained, by re-factorizing the principal A-determinant with the help of Jacobi identities. The letters are verified by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta.

Symbolic summation and integration techniques to simplify Feynman integrals

• Carsten Schneider

We present tools from symbolic summation and integration that are tailored for Feynman integrals. In particular, we will present algorithms that enable one to produce linear differential or difference equations that contain the input expression (e.g., in form of definite hypergeometric multisums, hyperexpontial multiintegrals or coupled systems of linear differential equations) as solution. In a nutshell, we show constructively that the given input expressions of Feynman integrals are holonomic. Given such equations one can then look for closed form solutions, e.g., in terms of iterative integrals and sums (which are again holonomic). This leads usually to rather compact expressions that can be used for asymptotic expansions or analytic continuation. We will report on these different tools that we have applied within our long-term cooperation between RISC and DESY (Deutsches Elektronen-Synchrotron) and demonstrate their flexibility.

The arithmetic of resurgent topological strings

• Claudia Rella (Institut des Hautes Études Scientifiques, Université Paris-Saclay)

Factorially divergent power series naturally arise as perturbative expansions in quantum theories but do not uniquely determine the original functions due to hidden non-analytic terms. In favourable circumstances, these terms can be systematically understood within the framework of resurgence. Growing evidence indicates that this is the case for topological string theory. In this talk, I will discuss the resurgence of the strong and weak coupling limits of the spectral traces of a toric Calabi-Yau threefold, which are captured by the free energy of the refined topological string on the same background. In the case of the spectral trace of local P^2, a remarkable arithmetic structure unfolds, revealing an exact strong-weak resurgent symmetry exchanging the perturbative/non-perturbative sectors in the dual regimes. Guided by this example, I will propose a new perspective on the resurgence of particular formal power series, which are conjectured to possess specific summability and quantum modularity properties, leading us to introduce the general paradigm of modular resurgence. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.

Single-valued Integration on the Elliptic Curve: Monodromy Relations and Twisted (Co)homology

• Stephan Stieberger

Free Discussions

Non-perturbative Methods for Planar N=4 SYM at Conformal Point

• Zahra Zahraee

In this pedagogical talk, we discuss non-perturbative methods to study planar N=4 SYM theory. Focusing on the four-point correlation function of the stress-energy tensor at the conformal point, we show how sum rules based on dispersion relations can be used to numerically bootstrap various objects in the theory, such as OPE coefficients, the four-point correlation function, and the energy-energy correlator. We show, for the first time, rigorous non-perturbative results for the planar OPE coefficients of single-trace operators, as well as the correlation function at various points in cross-ratio space. Additionally, focusing on the energy-energy correlator (EEC), we present rigorous bounds for its spin-2 and spin-3 Legendre coefficients, as well as the full EEC function at various angles. These results were obtained for a wide range of 't Hooft couplings, highlighting the power of the bootstrap in probing the non-perturbative aspects of planar N=4 SYM theory.

Closing Remarks

• Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences, Leipzig)