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

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.