A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes

TL;DR

This paper reviews generalized hypergeometric functions on Grassmannian spaces, explores their integral representations and connections to KZ solutions, and applies these insights to improve the holonomy formalism for gluon scattering amplitudes.

Contribution

It clarifies the integral representations of hypergeometric functions on Grassmannians and links them to KZ solutions and gluon amplitude formalisms in super Yang-Mills theory.

Findings

Integral representations of hypergeometric functions are clarified.

KZ solutions relate to hypergeometric-type integrals.

Holonomy operators can be interpreted as hypergeometric integrals.

Abstract

Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian spaces $Gr(k+1,n+1)$ are reviewed. Particularly, their integral representations in terms of twisted homology and cohomology are clarified with an example of the $Gr(2,4)$ case which corresponds to Gauss' hypergeometric functions. The cases of $Gr(2, n+1)$ in general lead to $(n+1)$-point solutions of the Knizhnik-Zamolodchikov (KZ) equation. We further analyze the Schechtman-Varchenko integral representations of the KZ solutions in relation to the $Gr(k+1, n+1)$ cases. We show that holonomy operators of the so-called KZ connections can be interpreted as hypergeometric-type integrals. This result leads to an improved description of a recently proposed holonomy formalism for gluon amplitudes. We also present a (co)homology interpretation of Grassmannian formulations for scattering amplitudes in ${\cal N} = 4$ super Yang-Mills theory.