On Complex Gamma-Function Integrals

Abstract

It was observed recently that relations between matrix elements of certain operators in the SL(2, ℝ) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, ℂ) symmetry group and L₂(ℂ) as a local Hilbert space give rise to a new type of Γ-function integrals. In this work, we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit, these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals.