Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet

TL;DR

This paper extends Gustafson's integrals, originally related to $SL(2,R)$ spin chains, to the complex $SL(2, ext{C})$ case, revealing new multidimensional Mellin--Barnes integrals relevant for noncompact spin magnets.

Contribution

It introduces generalized Gustafson integrals for $SL(2, ext{C})$ spin magnets, expanding the mathematical framework for noncompact spin chain models.

Findings

Derived new multidimensional Mellin--Barnes integrals for $SL(2, ext{C})$

Connected Gustafson's integrals to complex spin chain models

Extended the applicability of these integrals to noncompact cases

Abstract

It was observed recently that the multidimensional Mellin--Barnes integrals (Gustafson's integrals) arise naturally in studies of the $SL(2,R)$ spin chain models. We extend this analysis to the noncompact $SL(2,\mathbb{C})$ spin magnets and obtain integrals which generalize Gustafson's integrals to the complex case.