Elliptic hypergeometric sum/integral transformations and supersymmetric lens index

Abstract

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $B{C}_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between $S^1\times S^3/{\mathbb{Z}}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories related by Seiberg duality, with gauge groups $SU(n+1)$ and $Sp(2n)$. This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the $A_n$ integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.

29 pages, 4 figures; v2: published version

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• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Essays on the theory of elliptic hypergeometric functions
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Quivers, YBE and 3-manifolds
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• The Endless Beta Integrals
• Modularity of supersymmetric partition functions
• Integrable lattice spin models from supersymmetric dualities
• Elliptic hypergeometric functions associated with root systems
• Introduction to the theory of elliptic hypergeometric integrals
• The rarefied elliptic Bailey lemma and the Yang-Baxter equation
• Lens partition function, pentagon identity and star-triangle relation
• Orbifold Schur Index and IR formula
• Lens Generalisation of $\tau$-functions for the Elliptic Discrete Painlevé Equation
• Integrability As Duality: The Gauge/YBE Correspondence
• The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory