Multiple elliptic gamma functions associated to cones

TL;DR

This paper introduces generalized multiple elliptic gamma and sine functions linked to rational cones, revealing their modular properties and infinite product representations, extending prior work on classical functions.

Contribution

It defines new functions associated with cones and proves their modular properties, generalizing previous results on elliptic gamma and sine functions.

Findings

Generalized functions associated to cones are well-defined.

These functions have infinite product representations.

They exhibit modular properties determined by the cone.

Abstract

We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of $SL_r(\mathbb{Z})$-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.