Wall crossing structure from quantum phenomena to Feynman Integrals

TL;DR

This paper explores the geometric and topological structures underlying Feynman integrals, revealing a wall-crossing framework that clarifies their analytic continuation, decomposition into master integrals, and large-parameter expansions.

Contribution

It introduces a geometric wall-crossing perspective for Feynman integrals, connecting their analytic properties to twisted periods and thimble decompositions, advancing the understanding of their structure.

Findings

Explicit thimble decompositions for holomorphic exponents

Wall-crossing structure counts independent master integrals

Unification of perturbative expansions with geometric representation theory

Abstract

A growing body of evidence suggests that the complexity of Feynman integrals is best understood through geometry. Recent mathematical developments [Kontsevich and Soibelman, arXiv:2402.07343] have illuminated the role of exponential integrals as periods of twisted de Rham cocycles over Betti cycles, providing a structured approach to tackle this problem in many situations. In this paper, we apply these concepts to show how families of physically relevant integrals, ranging from exponentials to logarithmic multivalued functions, can be recast as twisted periods of differential forms over homology cycles. In the case of holomorphic exponents, we provide explicit decompositions as thimble expansions and reveal a geometric wall-crossing structure behind the analytic continuation in parameters. We then show that the generalization to multivalued functions provides the right framework to describe Feynman integrals in the Baikov representation, where the multivaluedness is governed by the logarithm of the Baikov polynomial. In this context, the thimble decomposition aligns with the decomposition into Master Integrals. We highlight how the wall-crossing structure allows for a sharp count of independent Master Integrals (or periods), circumventing complications arising from Stokes phenomena. Additionally, we study the large-parameter expansions of these integrals, whose coefficients correspond to periods of standard (co-)homology associated with families of algebraic varieties, and which reveal the dominant basis elements in different sectors of the wall crossing structure. This unifies perturbative expansions and geometric representation theory under a single cohomological framework.