Exponential Periods for Integrals in Physics

TL;DR

This paper introduces a systematic framework for analyzing Feynman integrals as exponential periods, improving the understanding of their relations and master integrals through advanced cohomology techniques.

Contribution

It proposes a novel approach to identify and construct homology and cohomology for Feynman integrals, enabling better analysis of their structure and relations.

Findings

Framework captures wall crossing and Stokes phenomena.

Unifies perturbative and non-perturbative integrals in physics.

Enhances counting and reduction of master integrals.

Abstract

The study of Feynman integrals through the lens of intersection theory offers a unifying framework for their analysis, capturing both the linear and quadratic relations that arise among integrals. In doing so, it provides a powerful method for systematically reducing them to the so called master integrals, a necessary strategy for multiloop contributions, whose huge number make direct calculation unfeasible. The Twisted de Rham cohomology offers a powerful tool for describing integrals with multivalued integrands, arising in dimensional regularization. However, it fails whenever the underlying geometry shows richer structures, as singularities and intricate monodromies. In this thesis we propose a systematic approach to identify and construct the appropriate homology and cohomology that allows to interpret Feynman integrals in parameter representation as exponential periods. This reformulation, together with the analytic continuation of the dimensional regularizator, provides a perfect framework to properly analyze the wall crossing structure and to correctly take into account Stokes phenomena for a sharp counting of the number of Master integrals. This framework allows to embed within the same formalism not only perturbative integrals, coming both from quantum field theories and string theory, but also wide class of physically relevant integrals, from Fourier calculus to statistical mechanics partition functions, from quantum mechanics expectation values to conformal field theory correlators.