The elliptic gamma function and SL(3,Z) x Z^3

TL;DR

This work develops a comprehensive analytic and cohomological theory of the elliptic gamma function Γ(z,τ,σ), establishing its defining properties, degenerations, and modular behavior. It derives new modular-type three-term relations under SL(3,Z) actions and interprets Γ as a degree-1 automorphic form tied to a nontrivial 2-cocycle in SL(3,Z) × Z^3 cohomology. The paper also analyzes parameter extensions, real-axis limits, and semiclassical regimes, connecting phase functions to theta-function limits and dilogarithm structures. Together, these results illuminate the elliptic gamma function as a bridge between special functions, integrable models, and higher-rank automorphic cohomology, with potential applications to elliptic qKZB equations and related quantum integrable systems.

Abstract

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3,Z) x Z^3 associated to a non-trivial class in H^3(G,Z).

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 3.1
• Proposition 3.2
• Theorem 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 4.1
• Proposition 5.1
• Theorem 5.2
• Corollary 5.3