# A gerbe for the elliptic gamma function

@article{Felder2006AGF, title={A gerbe for the elliptic gamma function}, author={Giovanni Felder and Andr{\'e} Henriques and Carlo Antonio Rossi and Chenchang Zhu}, journal={arXiv: Quantum Algebra}, year={2006} }

The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three (it is a stack). Our identities can be summarized by saying…

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## References

SHOWING 1-10 OF 27 REFERENCES

### The Elliptic Gamma Function and SL(3, Z)⋉Z3

- Mathematics
- 2000

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…

### The modular properties and the integral representations of the multiple elliptic gamma functions

- Mathematics
- 2003

### Even powers of divisors and elliptic zeta values

- Mathematics
- 2002

Abstract We introduce and study elliptic zeta values, a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coeffcients of the…

### Gerbes on complex reductive Lie groups

- Mathematics
- 2000

We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition.…

### An elliptic analogue of the multiple gamma function

- Mathematics
- 2001

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a…

### Bundle gerbes

- Mathematics
- 1994

. Just as C × principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral…

### q-deformed KZB heat equation: completeness, modular properties and SL(3,Z)

- Mathematics
- 2002

Abstract We study the properties of one-dimensional hypergeometric integral solutions of the q-difference (“quantum”) analogue of the Knizhnik–Zamolodchikov–Bernard equations on tori. We show that…

### Loop Spaces, Characteristic Classes and Geometric Quantization

- Mathematics
- 1994

This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical…

### Differentiable stacks and gerbes

- Mathematics
- 2006

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the…