${\rm SL}(2,\mathbb{C})$ Gustafson Integrals

TL;DR

This paper explores the ${ m SL}(2, ext{C})$ spin chain models, deriving new integrals that generalize Gustafson integrals, with implications for mathematical physics and integrable systems.

Contribution

It introduces new integrals related to the ${ m SL}(2, ext{C})$ spin chain models, extending previous work on ${ m SL}(2, ext{R})$ cases.

Findings

Derived several new integrals for ${ m SL}(2, ext{C})$ spin chains.

Connected integrals to boundary conditions in spin models.

Extended Gustafson integrals to complex symmetry groups.

Abstract

It was shown recently that many of the Gustafson integrals appear in studies of the ${\rm SL}(2,\mathbb{R})$ spin chain models. One can hope to obtain a generalization of the Gustafson integrals considering spin chain models with a different symmetry group. In this paper we analyse the spin magnet with the ${\rm SL}(2,\mathbb{C})$ symmetry group in case of open and periodic boundary conditions and derive several new integrals.