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

Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian spaces Gr(k+1,n+1)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)Gr(2,4) case which corresponds to Gauss' hypergeometric functions. The cases of Gr(2,n+1)Gr(2,n+1) in general lead to (n+1)(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)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 N=4N=4 super Yang–Mills theory.