Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 013, 29 pages      arXiv:1704.03159      https://doi.org/10.3842/SIGMA.2018.013
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index

Andrew P. Kels a and Masahito Yamazaki b
a) Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b) Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Chiba 277-8583, Japan

Received April 24, 2017, in final form February 02, 2018; Published online February 16, 2018

Abstract
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.

Key words: elliptic hypergeometric; elliptic gamma; supersymmetric; Seiberg duality; integrable; exactly solvable; Yang-Baxter; star-star.

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References

  1. Au-Yang H., Perk J.H.H., McCoy B.M., Tang S., Yan M.L., Commuting transfer matrices in the chiral Potts models: solutions of star-triangle equations with genus $>1$, Phys. Lett. A 123 (1987), 219-223.
  2. Baxter R.J., Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228.
  3. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  4. Baxter R.J., Star-triangle and star-star relations in statistical mechanics, Internat. J. Modern Phys. B 11 (1997), 27-37.
  5. Baxter R.J., Perk J.H.H., Au-Yang H., New solutions of the star-triangle relations for the chiral Potts model, Phys. Lett. A 128 (1988), 138-142.
  6. Bazhanov V.V., Kels A.P., Sergeev S.M., Comment on star-star relations in statistical mechanics and elliptic gamma-function identities, J. Phys. A: Math. Theor. 46 (2013), 152001, 7 pages, arXiv:1301.5775.
  7. Bazhanov V.V., Kels A.P., Sergeev S.M., Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs, J. Phys. A: Math. Theor. 49 (2016), 464001, 44 pages, arXiv:1602.07076.
  8. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry, Nuclear Phys. B 784 (2007), 234-258, hep-th/0703041.
  9. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Exact solution of the Faddeev-Volkov model, Phys. Lett. A 372 (2008), 1547-1550, arXiv:0706.3077.
  10. Bazhanov V.V., Sergeev S.M., Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation, Nuclear Phys. B 856 (2012), 475-496, arXiv:1106.5874.
  11. Bazhanov V.V., Sergeev S.M., A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations, Adv. Theor. Math. Phys. 16 (2012), 65-95, arXiv:1006.0651.
  12. Benini F., Nishioka T., Yamazaki M., 4d index to 3d index and 2d TQFT, Phys. Rev. D 86 (2012), 065015, 10 pages, arXiv:1109.0283.
  13. Date E., Jimbo M., Kuniba A., Miwa T., Okado M., Exactly solvable SOS models. II. Proof of the star-triangle relation and combinatorial identities, in Conformal Field Theory and Solvable Lattice Models (Kyoto, 1986), Adv. Stud. Pure Math., Vol. 16, Academic Press, Boston, MA, 1988, 17-122.
  14. Dolan F.A., Osborn H., Applications of the superconformal index for protected operators and $q$-hypergeometric identities to ${\mathcal N}=1$ dual theories, Nuclear Phys. B 818 (2009), 137-178, arXiv:0801.4947.
  15. Faddeev L., Volkov A.Yu., Abelian current algebra and the Virasoro algebra on the lattice, Phys. Lett. B 315 (1993), 311-318, hep-th/9307048.
  16. Fateev V.A., Zamolodchikov A.B., Self-dual solutions of the star-triangle relations in $Z_{N}$-models, Phys. Lett. A 92 (1982), 37-39.
  17. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in The Arnold-Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171-204.
  18. Gahramanov I., Kels A.P., The star-triangle relation, lens partition function, and hypergeometric sum/integrals, J. High Energy Phys. 2017 (2017), no. 2, 040, 41 pages, arXiv:1610.09229.
  19. Gahramanov I., Spiridonov V.P., The star-triangle relation and $3d$ superconformal indices, J. High Energy Phys. 2015 (2015), no. 8, 040, 23 pages, arXiv:1505.00765.
  20. Kashiwara M., Miwa T., A class of elliptic solutions to the star-triangle relation, Nuclear Phys. B 275 (1986), 121-134.
  21. Kels A.P., A new solution of the star-triangle relation, J. Phys. A: Math. Theor. 47 (2014), 055203, 6 pages, arXiv:1302.3025.
  22. Kels A.P., New solutions of the star-triangle relation with discrete and continuous spin variables, J. Phys. A: Math. Theor. 48 (2015), 435201, 19 pages, arXiv:1504.07074.
  23. Kinney J., Maldacena J., Minwalla S., Raju S., An index for 4 dimensional super conformal theories, Comm. Math. Phys. 275 (2007), 209-254, hep-th/0510251.
  24. Narukawa A., The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004), 247-267, math.QA/0306164.
  25. Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169-243, math.QA/0309252.
  26. Razamat S.S., Willett B., Global properties of supersymmetric theories and the lens space, Comm. Math. Phys. 334 (2015), 661-696, arXiv:1307.4381.
  27. Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
  28. Seiberg N., Electric-magnetic duality in supersymmetric non-abelian gauge theories, Nuclear Phys. B 435 (1995), 129-146, hep-th/9411149.
  29. Spiridonov V.P., On the elliptic beta function, Russian Math. Surveys 56 (2001), 185-186.
  30. Spiridonov V.P., Theta hypergeometric integrals, St. Petersburg Math. J. 15 (2004), 929-967, math.CA/0303205.
  31. Spiridonov V.P., Short proofs of the elliptic beta integrals, Ramanujan J. 13 (2007), 265-283, math.CA/0408369.
  32. Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Russian Math. Surveys 63 (2008), 405-472, arXiv:0805.3135.
  33. Spiridonov V.P., Elliptic beta integrals and solvable models of statistical mechanics, in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemp. Math., Vol. 563, Amer. Math. Soc., Providence, RI, 2012, 181-211, arXiv:1011.3798.
  34. Spiridonov V.P., Rarefied elliptic hypergeometric functions, arXiv:1609.00715.
  35. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometry of supersymmetric dualities, Comm. Math. Phys. 304 (2011), 797-874, arXiv:0910.5944.
  36. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices, Comm. Math. Phys. 325 (2014), 421-486, arXiv:1107.5788.
  37. Stokman J.V., Hyperbolic beta integrals, Adv. Math. 190 (2005), 119-160, math.QA/0303178.
  38. Terashima Y., Yamazaki M., Emergent 3-manifolds from 4d superconformal indices, Phys. Rev. Lett. 109 (2012), 091602, 4 pages, arXiv:1203.5792.
  39. Volkov A.Yu., Quantum Volterra model, Phys. Lett. A 167 (1992), 345-355.
  40. Yamazaki M., Quivers, YBE and 3-manifolds, J. High Energy Phys. 2012 (2012), no. 5, 147, 50 pages, arXiv:1203.5784.
  41. Yamazaki M., New integrable models from the gauge/YBE correspondence, J. Stat. Phys. 154 (2014), 895-911, arXiv:1307.1128.

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