A Calabi-Yau-to-Curve Correspondence for Feynman Integrals

TL;DR

This paper establishes a novel correspondence between Calabi-Yau threefolds and genus-two curves, providing new insights into the periods related to Feynman integrals, specifically applied to the four-loop banana integral.

Contribution

It introduces a general Calabi-Yau-to-curve correspondence that relates Calabi-Yau threefold periods to genus-two curve periods, with detailed application to a four-loop Feynman integral.

Findings

Calabi-Yau-to-curve correspondence is established.

The four-loop banana integral's period can be viewed via genus-two curves.

Framework suggests broader applicability to similar integrals.

Abstract

It has long been known that the maximal cut of the equal-mass four-loop banana integral is a period of a family of Calabi-Yau threefolds that depends on the kinematic variable $z=m^2/p^2$. We show that it can also be interpreted as a period of a family of genus-two curves. We do this by introducing a general Calabi-Yau-to-curve correspondence, which in this case locally relates the original period of the family of Calabi-Yau threefolds to a period of a family of genus-two curves that varies holomorphically with the kinematic variable $z$. In addition to working out the concrete details of this correspondence for the equal-mass four-loop banana integral, we outline when we expect a correspondence of this type to hold.