From positive geometries to a coaction on hypergeometric 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 $\epsilon$. 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.
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https://hal.archives-ouvertes.fr/hal-02350159
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Submitted on : Tuesday, November 5, 2019 - 11:45:44 PM
Last modification on : Wednesday, November 13, 2019 - 2:14:56 PM

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Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi, James Matthew. From positive geometries to a coaction on hypergeometric functions. 2019. ⟨hal-02350159⟩

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