Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 003, 20 pages      arXiv:1908.01530      https://doi.org/10.3842/SIGMA.2020.003
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

On Complex Gamma-Function Integrals

Sergey É. Derkachov a and Alexander N. Manashov ba
a) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
b) Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany

Received October 15, 2019, in final form January 14, 2020; Published online January 18, 2020

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.

Key words: Mellin-Barnes integrals; star-triangle relation.

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