Vector Spaces of Generalized Euler Integrals

TL;DR

This paper explores the structure of vector spaces linked to generalized Euler integrals, connecting algebraic geometry and homological methods, with applications to Feynman integrals in physics.

Contribution

It introduces new relations between homological algebra and D-module approaches and offers novel algorithmic tools for analyzing these integrals.

Findings

Dimension of vector spaces equals the Euler characteristic of a very affine variety

Uncovered new relations between homological algebra and D-module theory

Developed new algorithms for analyzing generalized Euler integrals

Abstract

We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new algorithmic tools.