From positive geometries to a coaction on hypergeometric functions

TL;DR

This paper develops a coaction framework for hypergeometric functions related to positive geometries, extending concepts from Feynman integrals and ensuring consistency with known polylogarithm coactions.

Contribution

It introduces a coaction on hypergeometric functions using intersection theory, connecting positive geometries with hypergeometric integral representations.

Findings

Coaction on hypergeometric functions is consistent with multiple polylogarithms.

Explicit coaction formulas are derived for various hypergeometric and Appell functions.

The framework links positive geometries to hypergeometric functions through intersection theory.

Abstract

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter $\epsilon$. We show that the coaction defined on this class of integral is consistent, upon expansion in $\epsilon$, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric ${}_{p+1}F_p$ and Appell functions.