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From positive geometries to a coaction on hypergeometric functions

Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi, James Matthew

TL;DR

This work defines a coaction Δ_{ epsilon} on broad classes of hypergeometric integrals whose integrands and contours are governed by positive geometries, using twisted (co)homology and intersection numbers to keep the structure symmetric between cycles and differential forms. The construction is designed to be consistent with the known coaction on multiple polylogarithms when the ε-expansion is taken, and is demonstrated across Gauss $_2F_1$, Appell functions, Lauricella $F_D^{(n)}$, and general $_{p+1}F_p$ by explicit canonical- and orthonormal-basis calculations of period and intersection matrices. The paper shows how endpoint singularities are regularized by ε, and provides concrete coaction formulas for a range of hypergeometric functions, including degenerate/limit cases that correspond to familiar polylogarithmic structures. The results offer a unifying framework linking hypergeometric-type integrals in Feynman diagram computations to the diagrammatic coaction on periods, with potential extensions to higher loops and broader function classes, and they connect to ongoing mathematical programs on motivic coactions for Lauricella and related functions.

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 $ε$. We show that the coaction defined on this class of integral is consistent, upon expansion in $ε$, 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.

From positive geometries to a coaction on hypergeometric functions

TL;DR

This work defines a coaction Δ_{ epsilon} on broad classes of hypergeometric integrals whose integrands and contours are governed by positive geometries, using twisted (co)homology and intersection numbers to keep the structure symmetric between cycles and differential forms. The construction is designed to be consistent with the known coaction on multiple polylogarithms when the ε-expansion is taken, and is demonstrated across Gauss , Appell functions, Lauricella , and general by explicit canonical- and orthonormal-basis calculations of period and intersection matrices. The paper shows how endpoint singularities are regularized by ε, and provides concrete coaction formulas for a range of hypergeometric functions, including degenerate/limit cases that correspond to familiar polylogarithmic structures. The results offer a unifying framework linking hypergeometric-type integrals in Feynman diagram computations to the diagrammatic coaction on periods, with potential extensions to higher loops and broader function classes, and they connect to ongoing mathematical programs on motivic coactions for Lauricella and related functions.

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 . We show that the coaction defined on this class of integral is consistent, upon expansion in , 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 and Appell functions.

Paper Structure

This paper contains 28 sections, 148 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The coaction on MPLs and resummation of the $\epsilon$ expansion
  3. The coaction on MPLs
  4. Resummation of the $\epsilon$ expansion
  5. Positive geometries and canonical forms
  6. A class of integrals
  7. Integrals and positive geometries
  8. Example: Euler's beta function.
  9. Pairings of cycles and forms
  10. Example: Euler's beta function.
  11. A coaction on integrals
  12. Example: Euler's beta function.
  13. Gauss' hypergeometric function $_{2}F_1$
  14. The coaction in the basis of canonical forms
  15. Coaction in an orthonormal basis
  16. ...and 13 more sections

Figures (3)

  • Figure 1: Geometry for the two-dimensional integral representation of the Appell $F_1$ integral of eq. \ref{['eq:f1AsDouble']}
  • Figure 2: Geometry for the Appell $F_3$ integral representation in eq. \ref{['eq:F3Def']}.
  • Figure 3: Geometry for the Appell $F_2$ integral representation of eq. \ref{['eq:f2Def']}