On Complex Gamma-Function Integrals

Abstract

It was observed recently that relations between matrix elements of certain operators in the $\mathrm{SL}(2,\mathbb{R})$ spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with $\mathrm{SL}(2,\mathbb{C})$ symmetry group and ${\mathrm{L}}_2(\mathbb{C})$ 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.

References in corpus (9)

• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Exact solution of the Faddeev-Volkov model
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• The importance of the Selberg integral
• New Compact Construction of Eigenstates for Supersymmetric Spin Chains
• Barnes-Ismagilov integrals and hypergeometric functions of the complex field
• $\mathrm{SL}(2,\mathbb{C})$ Gustafson Integrals
• On $b$-Whittaker functions
• On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models

Cited by in corpus (10)

• The Endless Beta Integrals
• Barnes-Ismagilov integrals and hypergeometric functions of the complex field
• Complex hypergeometric functions and integrable many-body problems
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group
• Fishnet four-point integrals: integrable representations and thermodynamic limits
• On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
• Complex and rational hypergeometric functions on root systems
• Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb{C})$ Spin Chains
• The Gelfand-Tsetlin basis for infinite-dimensional representations of $g{l}_n(\mathbb{C})$
• BC-type open $SL(2,\mathbb{C})$ spin chain