Elliptic gamma function

Abstract: We compute the supersymmetric partition function of an $\mathcal{N}=1$ chiral multiplet coupled to an external Abelian gauge field on complex manifolds with $T^2 \times S^2$ topology. The result is locally holomorphic in the complex structure moduli of $T^2\times S^2$. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions. We also devise a simple method to compute the chiral multiplet partition function on any four-manifold $\mathcal{M}_4$ preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of $\mathcal{M}_4=S^3\times S^1$, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the $S^3 \times S^1$ complex structure moduli.

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).

Abstract: A new solution to the star–triangle relation is given, for an Ising type model of interacting spins containing integer and real valued components. Boltzmann weights of the model are given in terms of the lens elliptic gamma function, and are based on Yamazaki's recently obtained solution of the star–star relation. The star–triangle relation given here, implies Seiberg duality for the 4−d index of the SU(2) quiver gauge theory, and the corresponding two component spin case of the star–star relation of Yamazaki. A proof of the star–triangle relation is given, resulting in a new elliptic hypergeometric summation/integration identity. The star–triangle relation in this paper contains the master solution of Bazhanov and Sergeev as a special case. Two other limiting cases are considered one of which gives a new star–triangle relation in terms of ratios of infinite q-products, while the other case gives a new way of deriving a star–triangle relation that was previously obtained by the author.

Abstract: We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite product. We also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions.