Feynman integrals, toric geometry and mirror symmetry

TL;DR

This paper explores how toric geometry and mirror symmetry can be used to evaluate Feynman integrals, revealing their connection to hypergeometric series, Calabi-Yau hypersurfaces, and elliptic functions.

Contribution

It introduces a novel approach linking Feynman integrals to toric geometry and mirror symmetry, providing explicit methods for their evaluation using Calabi-Yau hypersurfaces.

Findings

Maximal cuts of Feynman integrals are GKZ hypergeometric series.

Sunset integrals relate to elliptic and trilogarithms.

Mirror symmetry techniques evaluate complex Feynman integrals.

Abstract

This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of 1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and 2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface.