Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index

Symmetry Integrability and Geometry-Methods and Applications

article

Andrew P. Kels¹  Masahito Yamazaki² 

[1] Institute of Physics, University of Tokyo;Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo

关键词: elliptic hypergeometric;    elliptic gamma;    supersymmetric;    Seiberg duality;    integrable;    exactly solvable;    Yang–Baxter;    star-star;   

DOI  :  10.3842/SIGMA.2018.013

来源: National Academy of Science of Ukraine

【 摘 要 】

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/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.

【 授权许可】

Unknown