Duality for Knizhinik-Zamolodchikov and Dynamical Equations, and Hypergeometric Integrals

Abstract

We review results on the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the $(gl_k,gl_n)$ duality, and their implications for hypergeometric integrals. The KZ and dynamical equations naturally exchange under the duality, which provides two kinds of integral formulae for hypergeometric solutions of the equations. This fact yields identities for hypergeometric integrals of different dimensions, the dimension of the first integral being the parameter of the second one, and vice versa. Similar identites exist between hypergeometric and q-hypergeometric integrals of Mellin-Barnes type of different dimensions. Those identites are multidimensional analogues of the equality of two integral representations for the Gauss hypergeometric function.