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Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum\slash integrals associated to the $A_n$ and $BC_n$ 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 ${\rm SU}(n+1)$ and ${\rm Sp}(2n)$. 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.
Keywords: elliptic hypergeometric; elliptic gamma; supersymmetric; Seiberg duality; integrable; exactly solvable; Yang–Baxter; star-star. 
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     author = {Andrew P. Kels and Masahito Yamazaki},
     title = {Elliptic {Hypergeometric} {Sum/Integral} {Transformations} and {Supersymmetric} {Lens} {Index}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a12/}
}
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Andrew P. Kels; Masahito Yamazaki. Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a12/

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