Local mirror symmetry and the sunset Feynman integral

TL;DR

This paper explores the sunset Feynman integral in two dimensions, revealing its connection to elliptic dilogarithms, motivic cohomology, and local mirror symmetry, and expressing it via Gromov-Witten prepotentials of a local Calabi-Yau 3-fold.

Contribution

It establishes a novel link between the sunset Feynman integral, elliptic dilogarithms, motivic cohomology, and local mirror symmetry, providing a new geometric interpretation.

Findings

The sunset integral can be expressed using elliptic dilogarithms evaluated at puncture divisors.

A local non-compact Calabi-Yau 3-fold is associated with the sunset elliptic curve.

The integral is related to the local Gromov-Witten prepotential via a strong form of local mirror symmetry.

Abstract

We study the sunset Feynman integral defined as the scalar two-point self-energy at two-loop order in a two dimensional space-time. We firstly compute the Feynman integral, for arbitrary internal masses, in terms of the regulator of a class in the motivic cohomology of a 1-parameter family of open elliptic curves. Using an Hodge theoretic (B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms evaluated at the divisors determined by the punctures. Secondly we associate to the sunset elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the Batyrev dual A-model, we arrive at an expression for the sunset Feynman integral in terms of the local Gromov-Witten prepotential of the del Pezzo surface of degree 6. This expression is obtained by proving a strong form of local mirror symmetry which identifies this prepotential with the second regulator period of the motivic cohomology class.