paper

Identities for hypergeometric integrals of different dimensions

arXiv:math/0305224

Abstract

Given complex numbers $m_{1}, l_{1}$ and positive integers $m_{2}, l_{2}$ , such that $m_{1} + m_{2} = l_{1} + l_{2}$ , we define $l_{2}$ -dimensional hypergeometric integrals $I_{a, b} (z; m_{1}, m_{2}, l_{1}, l_{2})$ , $a, b = 0, ..., min (m_{2}, l_{2})$ , depending on a complex parameter $z$ . We show that $I_{a, b} (z; m_{1}, m_{2}, l_{1}, l_{2}) = I_{a, b} (z; l_{1}, l_{2}, m_{1}, m_{2})$ , thus establishing an equality of $l_{2}$ and $m_{2}$ -dimensional integrals. This identity allows us to study asymptotics of the integrals with respect to their dimension in some examples. The identity is based on the $(g l_{k}, g l_{n})$ duality for the KZ and dynamical differential equations.

Preprint (2003), 9 pages, AmsLaTeX, misprints corrected