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

Yasuhiro Abe

doi:10.1016/j.nuclphysb.2016.03.032

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 N=4 super Yang–Mills theory.

Highlights

We show that (n + 1)-point KZ solutions in general can be represented by generalized hypergeometric functions on Gr(2, n+1)
The (n + 1)-point KZ solutions can be represented by the hypergeometric-type integrals on Gr(k + 1, n+ 1) but we find that there exist ambiguities in the construction of such integrals for k ≥ 2
In what follows we show that RJ (t, z)dt1 ∧ · · · ∧ dtk can be interpreted as an element of the k-th cohomology group Hk(X, Lg) for k < n and ji ∈ {0, 1} (i = 1, 2, · · ·, n)

Summary

Aomoto’s generalized hypergeometric functions

The generalized hypergeometric function (2.7) is determined by the following bilinear form. To avoid redundancy in configuration of hyperplanes, we assume the set of hyperplanes are nondegenerate, that is, we consider the hyperplanes in general position This can be realized by demanding that any (k + 1)-dimensional minor determinants of the (k + 1) × (n + 1) matrix Z are nonzero. The integral F (Z) in (2.7) satisfies the defining equations (2.2)-(2.4) of the generalized hypergeometric functions on Gr(k + 1, n + 1). Z is more relaxed since it allows some (k + 1)-dimensional minor determinants vanish, that is, Z ⊆ Z In this sense F (Z) is conventionally called the generalized hypergeometric functions on Gr(k + 1, n + 1) and we follow this convention in the present note.

Reduction to Gauss’ hypergeometric function

Integral representations of the KZ solutions

Holonomy operators of KZ connections

Holonomy formalism for gluon amplitudes

Grassmannian formulations of gluon amplitudes

Conclusion