# On Complex Gamma-Function Integrals

## TL;DR

This paper introduces new complex gamma-function integrals arising from ${
m SL}(2, ext{C})$ spin chain models, analyzes their properties, and connects them to star-triangle and duality relations in conformal field theory.

## Contribution

It provides a direct calculation of novel gamma-function integrals related to ${
m SL}(2, ext{C})$ symmetry and explores their mathematical properties and physical implications.

## Key findings

- Derived new gamma-function integrals for ${
m SL}(2, ext{C})$ models
- Showed these integrals satisfy star-triangle relations
- Connected integral identities to Dotsenko-Fateev duality in the quasi-classical limit

## Abstract

It was observed recently that relations between matrix elements of certain operators in the ${\rm SL}(2,\mathbb R)$ spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with ${\rm SL}(2,\mathbb C)$ symmetry group and ${\rm L}_2(\mathbb C)$ as a local Hilbert space give rise to a new type of $\Gamma$-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.