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.
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...
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.
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...
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.
A Feynman integral is called "primitive" if it is superficially divergent and does not contain subdivergences. The "period" of a primitive graph is the coefficient of logarithmic energy dependence, or equivalently the simple pole in minimal subtraction. In recent work [2305.13506, 2403.16217] together with Kimia Shaban, we numerically computed the periods of 2 million Feynman integrals in...
Polylogarithms on higher-genus Riemann surfaces are necessary for systematic calculations of certain Feynman integrals and loop amplitudes in string theory. Employing the Schottky uniformization of a Riemann surface we construct higher-genus generating functions of polylogarithmic integration kernels, coinciding with the set of meromorphic differentials defined by Enriquez. This allows for...
We consider the problem of computing all the linear relations between the integrals of the
functions lying in a given holonomic D-module. We present in this poster a new integration
algorithm designed for handling multiple integrals of holonomic functions. This novel algorithm can be regarded as both an extension of Lairez’s reduction-based algorithm, which is limited to integrals of...
We investigate the D-module structure of Feynman integrals and Euler-Mellin integrals by means of Griffiths theorem. We present first applications to special mathematical functions and one-and two-loop integrals, and discuss the generation of corresponding Pfaffian equation they obey, via Macaulay matrix method.
What happens when we let artificial intelligence tackle mathematical problems? This work explores how Transformer neural networks—initially designed for language processing—can learn and perform tasks in computational Algebraic Geometry. As a result, we introduce a neural network model that approximates psi-class intersection numbers on the moduli space of curves. Through our analysis, we...
Study of correlation functions in AdS/CFT and in-in correlators in de Sitter space often requires the computation of Witten diagrams. Due to the complexity of evaluating radial integrals for these correlators, several indirect approaches have been developed to simplify computations. However, in momentum space, these methods have been limited to fields with integer spin. Here, we formulate...
I will present a method to calculate the Landau singularities of a Feynman integrals using Whitney stratifications. Whitney stratifications themselves are microlocal in nature, meaning that they naturally do not only live in the space we are stratifying but in the cotangent bundle of this space. With this in mind I introduce a microlocal (or rather distributional) framework for Feynman...
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...
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...
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...
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...
