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Integrable Lattice Spin Models from Supersymmetric Dualities

  • Physics of Solid State and Condensed Matter
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Abstract

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang–Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories and make clear notes on the special functions used by several authors.

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References

  1. J. B. McGuire, “Study of exactly soluble one-dimensional n-body problems,” J. Math. Phys. 5, 622–636 (1964).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. C.-N. Yang, “Some exact results for the many body problems in one dimension with repulsive delta function interaction,” Phys. Rev. Lett. 19, 1312–1314 (1967).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. R. J. Baxter, “Partition function of the eight vertex lattice model,” Ann. Phys. 70, 193–228 (1972); Ann. Phys. 281, 187 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).

    MATH  Google Scholar 

  5. M. Jimbo, “Introduction to the Yang–Baxter equation,” Int. J. Mod. Phys. A 4, 3759–3777 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. P. P. Kulish and E. K. Sklyanin, “On the solution of the Yang-Baxter equation,” J. Sov. Math. 19, 1596–1620 (1982).

    Article  MATH  Google Scholar 

  7. L. D. Faddeev, E. K. Sklyanin, and L. A. Takhtajan, “The quantum inverse problem method. 1,” Theor. Math. Phys. 40, 688–706 (1980).

    MATH  Google Scholar 

  8. L. A. Takhtadzhan and L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg xyz model,” Russ. Math. Surv. 34 (5), 11 (1979). http://stacks.iop.org/0036-0279/34/i=5/a=R02.

    Article  Google Scholar 

  9. L. D. Faddeev, “How algebraic bethe ansatz works for integrable model,” in Proceedigns of the School of Physics, Les Houches, France, 1995, pp. 149–219; arXiv:hep-th/9605187 [hep-th].

    Google Scholar 

  10. E. Ising, “Contribution to the theory of ferromagnetism,” Z. Phys. 31, 253–258 (1925).

    Article  ADS  Google Scholar 

  11. L. Onsager, “Crystal statistics. 1. A two-dimensional model with an order disorder transition,” Phys. Rev. 65, 117–149 (1944).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. R. J. Baxter, “Star-triangle and star-star relations in statistical mechanics,” Int. J. Mod. Phys. B 11, 27–37 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. H. Au-Yang, J. H. Perk, et al., “Onsager’s star-triangle equation: master key to integrability,” in Integrable Systems in Quantum Field Theory and Statistical Mechanics (Elsevier, Amsterdam, 1989), pp. 57–94.

    Google Scholar 

  14. P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, “Yang-Baxter equation and representation theory. 1,” Lett. Math. Phys. 5, 393–403 (1981).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. V. A. Fateev and A. B. Zamolodchikov, “Selfdual solutions of the star triangle relations in Z(N) models,” Phys. Lett. A 92, 37–39 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  16. M. Kashiwara and T. Miwa, “A class of elliptic solutions to the star triangle relation,” Nucl. Phys. B 275, 121 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  17. K. Hasegawa and Y. Yamada, “Algebraic derivation of the broken Z(N) symmetric model,” Phys. Lett. A 146, 387–396 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  18. M. Gaudin, “La relation etoile-triangle d’un modele elliptique Z(N),” J. Phys. I 1, 351–361 (1991).

    MathSciNet  Google Scholar 

  19. G. von Gehlen and V. Rittenberg, “Z(N) symmetric quantum chains with an infinite set of conserved charges and Z(N) zero modes,” Nucl. Phys. B 257, 351 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  20. H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang, and M.-L. Yan, “Commuting transfer matrices in the chiral potts models: solutions of star triangle equations with genus >1,” Phys. Lett. A 123, 219–223 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  21. R. J. Baxter, J. H. H. Perk, and H. Au-Yang, “New solutions of the star triangle relations for the chiral potts model,” Phys. Lett. A 128, 138–142 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  22. L. D. Faddeev, “Current -like variables in massive and massless integrable models,” in Proceedings of the Conference on Quantum Groups and Their Applications in Physics, 1994, pp. 117–136; arXiv:hep-th/9408041 [hep-th].

    Google Scholar 

  23. A. Yu. Volkov, “Quantum Volterra model,” Phys. Lett. A 167, 345–355 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  24. V. V. Bazhanov and S. M. Sergeev, “A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations,” Adv. Theor. Math. Phys. 16, 65–95 (2012); arXiv:1006.0651 [math-ph].

    Article  MathSciNet  MATH  Google Scholar 

  25. V. P. Spiridonov, “Elliptic beta integrals and solvable models of statistical mechanics,” Contemp. Math. 563, 181–211 (2012); arXiv:1011.3798 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Yamazaki, “Quivers, YBE and 3-manifolds,” J. High Energy Phys. 05, 147 (2012); arXiv:1203.5784 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Yamazaki, “New integrable models from the gauge/YBE correspondence,” J. Stat. Phys. 154, 895 (2014); arXiv:1307.1128 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. J. Yagi, “Quiver gauge theories and integrable lattice models,” J. High Energy Phys. 10, 065 (2015); arXiv:1504.04055 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. M. Yamazaki and W. Yan, “Integrability from 2d N = (2. 2) dualities,” J. Phys. A 48, 394001 (2015); arXiv:1504.05540 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Gahramanov and V. P. Spiridonov, “The star-triangle relation and 3d superconformal indices,” J. High Energy Phys. 08, 040 (2015); arXiv:1505.00765 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. A. P. Kels, “New solutions of the star -triangle relation with discrete and continuous spin variables,” J. Phys. A 48, 435201 (2015); arXiv:1504.07074 [math-ph].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. K. Maruyoshi and J. Yagi, “Surface defects as transfer matrices,” PTEP 2016, 113B01 (2016); arXiv:1606.01041 [hep-th].

    Google Scholar 

  33. I. Gahramanov and A. P. Kels, “The star-triangle relation, lens partition function, and hypergeometric sum/integrals,” J. High Energy Phys. 02, 040 (2017); arXiv:1610.09229 [math-ph].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. M. Yamazaki, “Cluster-enriched Yang-Baxter equation from SUSY gauge theories,” arXiv:1611.07522 [hep-th].

  35. J. Yagi, “Branes and integrable lattice models,” Mod. Phys. Lett. A 32, 1730003 (2016); arXiv:1610.05584 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. J. Yagi, “Surface defects and elliptic quantum groups,” J. High Energy Phys. 06, 013 (2017); arXiv:1701.05562 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. A. P. Kels and M. Yamazaki, “Elliptic hypergeometric sum/integral transformations and supersymmetric lens index,” arXiv:1704.03159 [math-ph].

  38. A. P. Kels, “Exactly solved models on planar graphs with vertices in Z3,” J. Phys. A 50, 495202 (2017); arXiv:1705.06528 [math-ph].

    Article  MathSciNet  MATH  Google Scholar 

  39. N. A. Nekrasov and S. L. Shatashvili, “Quantization of integrable systems and four dimensional gauge theories,” in Proceedings of the 16th International Congress on Mathematical Physics ICMP09, 2009, pp. 265–289; arXiv:0908.4052 [hep-th].

    Google Scholar 

  40. N. A. Nekrasov and S. L. Shatashvili, “Supersymmetric vacua and Bethe ansatz,” Nucl. Phys. Proc. Suppl. 192–193, 91–112 (2009); arXiv:0901.4744 hep-th.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  41. V. V. Bazhanov, A. P. Kels, and S. M. Sergeev, “Quasiclassical expansion of the star-triangle relation and integrable systems on quad-graphs,” J. Phys. A 49 (2016, in press); arXiv:1602.07076.

    Google Scholar 

  42. J. H. Perk and H. Au-Yang, “Yang-Baxter equations,” in Encyclopedia of Mathematical Physics (Elsevier Science, Oxford, 2006), Vol. 5, pp. 465–473; mathph/0606053.

    MATH  Google Scholar 

  43. M. P. Bellon, J. M. Maillard, and C. Viallet, “On the symmetries of integrability,” Int. J. Mod. Phys. B 6, 1881–1904 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. T. Deguchi, “Introduction to solvable lattice models in statistical and mathematical physics,” in Classical and Quantum Nonlinear Integrable Systems: Theory and Application (CRC, Boca Raton, FL, 2003), p. 107.

    Google Scholar 

  45. H. Saleur and J. B. Zuber, “Integrable lattice models and quantum groups,” in Proceedings of the Spring School on String Theory and Quantum Gravity Trieste, Italy, Apr. 23–May 1, 1990, p. 0001–54.

    Google Scholar 

  46. S. Jafarzade and Z. Nazari, “A new integrable isingtype model from 2d N=(2,2) dualities (2017),” arXiv:1709.00070 [hep-th].

  47. S. E. Derkachov and A. N. Manashov, “General solution of the Yang-Baxter equation with symmetry group SL(N,C),” St. Petersburg Math. J. 21, 513–577 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  48. S. E. Derkachov and V. P. Spiridonov, “Yang–Baxter equation, parameter permutations, and the elliptic beta integral,” Russ. Math. Surv. 68, 1027–1072 (2013); arXiv:1205.3520 [math-ph].

    Article  MathSciNet  MATH  Google Scholar 

  49. D. Chicherin, S. E. Derkachov, and V. P. Spiridonov, “New elliptic solutions of the Yang-Baxter equation,” arXiv:1412.3383 [math-ph].

  50. R. J. Baxter, “Hard hexagons: exact solution,” J. Phys. A: Math. Gen 13, L61 (1980).

    Google Scholar 

  51. R. J. Baxter, “Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 1. Some fundamental eigenvectors,” Ann. Phys. 76, 1‒24 (1973).

    Google Scholar 

  52. R. J. Baxter, “Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 2. Equivalence to a generalized ice-type lattice model,” Ann. Phys. 76, 25–47 (1973).

    Article  ADS  MATH  Google Scholar 

  53. R. J. Baxter, “Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 1. Eigenvectors of the transfer matrix and Hamiltonian,” Ann. Phys. 76, 48–71 (1973).

    Article  ADS  MATH  Google Scholar 

  54. P. A. Pearce and K. A. Seaton, “A solvable hierarchy of cyclic solid-on-solid lattice models,” Phys. Rev. Lett. 60, 1347–1350 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  55. P. A. Pearce and K. A. Seaton, “Exact solution of cyclic solid-on-solid lattice models,” Ann. Phys. 193, 326–366 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. G. E. Andrews, R. J. Baxter, and P. J. Forrester, “Eight vertex SOS model and generalized Rogers-Ramanujan type identities,” J. Stat. Phys. 35, 193–266 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. E. H. Lieb, “Residual entropy of square ice,” Phys. Rev. 162, 162–172 (1967).

    Article  ADS  Google Scholar 

  58. B. Sutherland, “Exact solution of a two-dimensional model for hydrogen-bonded crystals,” Phys. Rev. Lett. 19, 103 (1967).

    Article  ADS  Google Scholar 

  59. A. G. Izergin and V. E. Korepin, “The inverse scattering method approach to the quantum Shabat-Mikhailov model,” Commun. Math. Phys. 79, 303 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  60. J. H. H. Perk and F. Y. Wu, “Graphical approach to the nonintersecting string model: star triangle equation, inversion relation and exact solution,” Phys. A (Amsterdam, Neth.) 138, 100–124 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  61. R. J. Baxter, “The inversion relation method for some two-dimensional exactly solved models in lattice statistics,” J. Stat. Phys. 28, 1–41 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  62. K. A. Intriligator and N. Seiberg, “Lectures on supersymmetric gauge theories and electric-magnetic duality,” Nucl. Phys. Proc. Suppl. 45BC, 1–28 (1996); arXiv:hep-th/9509066 [hep-th].

    Google Scholar 

  63. M. J. Strassler, “Erice lectures on confinement and duality,” Subnucl. Ser. 40, 154–193 (2003); ICTP Lect. Notes Ser. 7, 105 (2002).

    Google Scholar 

  64. M. J. Strassler, “The Duality cascade,” in Proceedings of the Summer School on Progress in string theory, TASI 2003, Boulder, USA, June 2–27, 2003 (2005). pp. 419–510; arXiv:hep-th/0505153 [hep-th].

    Google Scholar 

  65. J. Terning, Modern Supersymmetry: Dynamics and Duality (Oxford Univ. Press, Oxford, 2006).

    MATH  Google Scholar 

  66. N. Seiberg, “Electric -magnetic duality in supersymmetric non-Abelian gauge theories,” Nucl. Phys. B 435, 129–146 (1995); arXiv:hep-th/9411149 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. D. Berenstein and M. R. Douglas, “Seiberg duality for quiver gauge theories,” arXiv:hep-th/0207027 [hepth].

  68. M. Yamazaki, “Brane tilings and their applications,” Fortsch. Phys. 56, 555–686 (2008); arXiv:0803.4474 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. A. Hanany and D. Vegh, “Quivers, tilings, branes and rhombi,” J. High Energy Phys. 10, 029 (2007); arXiv:hep-th/0511063 [hep-th].

    Article  ADS  MathSciNet  Google Scholar 

  70. I. Gahramanov, “Mathematical structures behind supersymmetric dualities,” Archivum Math. 51, 273–286 (2015); arXiv:1505.05656 [math-ph].

    Article  MathSciNet  MATH  Google Scholar 

  71. I. B. Gahramanov and G. S. Vartanov, “Superconformal indices and partition functions for supersymmetric field theories,” in Proceedings of the 17th International Congress on Mathematical Physics, 2013, pp. 69–703; arXiv:1310.8507 [hep-th].

    Google Scholar 

  72. M. Yamazaki, “Four-dimensional superconformal index reloaded,” Theor. Math. Phys. 174, 154–166 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  73. J. Teschner and G. Vartanov, “6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories,” Lett. Math. Phys. 104, 527–551 (2014); arXiv:1202.4698 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. F. A. H. Dolan, V. P. Spiridonov, and G. S. Vartanov, “From 4d superconformal indices to 3d partition functions,” Phys. Lett. B 704, 234–241 (2011); arXiv:1104.1787 [hep-th].

    Article  ADS  MathSciNet  Google Scholar 

  75. V. P. Spiridonov, “Elliptic hypergeometric terms,” SMF Semin. Congr. 23, 385–405 (2011); arXiv:1003.4491 [math.CA]. http://inspirehep. net/record/849713/files/arXiv:1003.4491.pdf.

    MathSciNet  MATH  Google Scholar 

  76. A. P. Kels, “A new solution of the star-triangle relation,” J. Phys. A 47, 055203 (2014); arXiv:1302.3025 [math-ph].

    Article  ADS  MATH  Google Scholar 

  77. O. Aharony, S. S. Razamat, N. Seiberg, and B. Willett, “3d dualities from 4d dualities,” J. High Energy Phys. 07, 149 (2013); arXiv:1305.3924 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. O. Aharony, S. S. Razamat, and B. Willett, “From 3d duality to 2d duality,” J. High Energy Phys. 11, 090 (2017); arXiv:1710.00926 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Yu. G. Stroganov, “A new calculation method for partition functions in some lattice models,” Phys. Lett. A 74, 116–118 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  80. V. P. Spiridonov and G. S. Vartanov, “Vanishing superconformal indices and the chiral symmetry breaking,” J. High Energy Phys. 06, 062 (2014); arXiv:1402.2312 [hep-th].

    Article  ADS  Google Scholar 

  81. S. Elitzur, A. Giveon, and D. Kutasov, “Branes and N=1 duality in string theory,” Phys. Lett. B 400, 269–274 (1997); arXiv:hep-th/9702014 [hep-th].

    Article  ADS  MathSciNet  Google Scholar 

  82. S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici, and A. Schwimmer, “Brane dynamics and N=1 supersymmetric gauge theory,” Nucl. Phys. B 505, 202–250 (1997); arXiv:9704104 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys. 71, 983–1084 (1999); arXiv:hep-th/9802067 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  84. V. V. Bazhanov, A. P. Kels, and S. M. Sergeev, “Comment on star-star relations in statistical mechanics and elliptic gamma-function identities,” J. Phys. A 46, 152001 (2013); arXiv:1301.5775 [math-ph].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  85. V. V. Bazhanov and S. M. Sergeev, “Elliptic gammafunction and multi-spin solutions of the Yang-Baxter equation,” Nucl. Phys. B 856, 475–496 (2012); arXiv:1106.5874 [math-ph].

    Article  ADS  MATH  Google Scholar 

  86. I. Gahramanov and S. Jafarzade, “Comments on the multi-spin solution to the Yang-Baxter equation and basic hypergeometric sum/integral identity,” arXiv:1710.09106 [math-ph] (2017).

    Google Scholar 

  87. V. P. Spiridonov, “Theta hypergeometric integrals,” St. Petersburg Math. J. 15, 929–967 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  88. F. Benini, T. Nishioka, and M. Yamazaki, “4d index to 3d index and 2d TQFT,” Phys. Rev. D 86, 065015 (2012); arXiv:1109.0283 [hep-th].

    Article  ADS  Google Scholar 

  89. J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An index for 4 dimensional super conformal theories,” Commun. Math. Phys. 275, 209–254 (2007); arXiv:hep-th/0510251 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  90. C. Romelsberger, “Counting chiral primaries in N = 1, d=4 superconformal field theories,” Nucl. Phys. B 747, 329–353 (2006); arXiv:hep-th/0510060 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. F. A. Dolan and H. Osborn, “Applications of the superconformal index for protected operators and qhypergeometric identities to N=1 dual theories,” Nucl. Phys. B 818, 137–178 (2009); arXiv:0801.4947 [hep-th].

    Article  ADS  MATH  Google Scholar 

  92. V. P. Spiridonov, “On the elliptic beta function,” Russ. Math. Surv. 56, 185 (2001). http://stacks.iop.org/0036-0279/56/i=1/a=L21.

    Article  MathSciNet  MATH  Google Scholar 

  93. V. P. Spiridonov, “A Bailey tree for integrals,” Theor. Math. Phys. 139, 536–541 (2004); math/0312502.

    Article  MathSciNet  MATH  Google Scholar 

  94. S. Kim, “The complete superconformal index for N=6 Chern–Simons theory,” Nucl. Phys. B 821, 241–284 (2009); arXiv:0903.4172 [hep-th]; Nucl. Phys. B 864, 884(E) (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  95. J. Bhattacharya and S. Minwalla, “Superconformal indices for N = 6 Chern–Simons theories,” J. High Energy Phys. 01, 014 (2009); arXiv:0806.3251 [hepth].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  96. A. Kapustin, B. Willett, and I. Yaakov, “Exact results for Wilson loops in superconformal Chern–Simons theories with matter,” J. High Energy Phys. 03, 089 (2010); arXiv:0909.4559 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. I. Gahramanov and G. Vartanov, “Extended global symmetries for 4D N = 1 SQCD theories,” J. Phys. A 46, 285403 (2013); arXiv:1303.1443 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  98. I. Gahramanov and H. Rosengren, “A new pentagon identity for the tetrahedron index,” J. High Energy Phys. 11, 128 (2013); arXiv:1309.2195 [hep-th].

    Article  ADS  Google Scholar 

  99. I. Gahramanov and H. Rosengren, “Integral pentagon relations for 3d superconformal indices,” in Proceedings of the String-Math 2014, Alberta, Canada, June 9–13, 2014, arXiv:1412.2926 [hep-th].

    Google Scholar 

  100. I. Gahramanov and H. Rosengren, “Basic hypergeometry of supersymmetric dualities,” Nucl. Phys. B 913, 747–768 (2016); arXiv:1606.08185 [hep-th].

    Article  ADS  MATH  Google Scholar 

  101. H. Rosengren, “Rahman’s biorthogonal functions and superconformal indices,” arXiv:1612.05051 [math.CA].

  102. Y. Imamura and D. Yokoyama, “S3=Zn partition function and dualities,” J. High Energy Phys. 11, 122 (2012); arXiv:1208.1404 [hep-th].

    Article  ADS  MATH  Google Scholar 

  103. F. Nieri and S. Pasquetti, “Factorisation and holomorphic blocks in 4d,” J. High Energy Phys. 11, 155 (2015); arXiv:1507.00261 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  104. Y. Imamura and D. Yokoyama, “N = 2 supersymmetric theories on squashed three-sphere,” Int. J. Mod. Phys. Conf. Ser. 21, 171–172 (2013).

    Article  Google Scholar 

  105. K. Hosomichi, “A review on SUSY gauge theories on S3,” in New Dualities of Supersymmetric Gauge Theories (Springer Int., Switzerland, 2016), pp. 307–338; arXiv:1412.7128 [hep-th].

    Chapter  MATH  Google Scholar 

  106. J. Nian, “Localization of supersymmetric Chern–Simons-matter theory on a squashed S3 with SU(1.2) U(1.1) isometry,” J. High Energy Phys. 07, 126 (2014); arXiv:1309.3266 [hep-th].

    Article  ADS  MATH  Google Scholar 

  107. N. Hama, K. Hosomichi, and S. Lee, “SUSY gauge theories on squashed three-spheres,” J. High Energy Phys. 05, 014 (2011); arXiv:1102.4716 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  108. Y. Imamura and D. Yokoyama, “N=2 supersymmetric theories on squashed three-sphere,” Phys. Rev. D 85, 025015 (2012); arXiv:1109.4734 [hep-th].

    Article  ADS  Google Scholar 

  109. J. V. Stokman, “Hyperbolic beta integrals,” Adv. Math. 190, 119–160 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  110. A. Y. Volkov and L. D. Faddeev, “Yang-baxterization of the quantum dilogarithm,” J. Math. Sci. 88, 202–207 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  111. V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, “Faddeev–Volkov solution of the Yang–Baxter equation and discrete conformal symmetry,” Nucl. Phys. B 784, 234–258 (2007); arXiv:hep-th/0703041 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  112. V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, “Exact solution of the Faddeev–Volkov model,” Phys. Lett. A 372, 1547–1550 (2008); arXiv:0706.3077 [cond-mat.stat-mech].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  113. F. J. van de Bult, “Hyperbolic hypergeometric functions,” PhD Thesis (Univ. Amsterdam, 2007).

    Google Scholar 

  114. V. P. Spiridonov and G. S. Vartanov, “Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices,” Commun. Math. Phys. 325, 421–486 (2014); arXiv:1107.5788 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  115. I. Gahramanov, S. Jafarzde, and G. Mogol, (2018, to appear).

  116. F. Benini and S. Cremonesi, “Partition functions of N = (2. 2) gauge theories on S2 and vortices,” Commun. Math. Phys. 334, 1483–1527 (2015); arXiv:1206.2356 [hep-th].

    Article  ADS  MATH  Google Scholar 

  117. N. Doroud, J. Gomis, B. le Floch, and S. Lee, “Exact results in D = 2 supersymmetric gauge theories,” J. High Energy Phys. 05, 093 (2013); arXiv:1206.2606 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  118. A. Gadde and S. Gukov, “2D Index and Surface operators,” J. High Energy Phys. 03, 080 (2014); arXiv:1305.0266 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  119. F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of 2d N = 2 gauge theories,” Commun. Math. Phys. 333, 1241–1286 (2015); arXiv:1308.4896 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  120. F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups,” Lett. Math. Phys. 104, 465–493 (2014); arXiv:1305.0533 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  121. A. Kuniba, M. Okado, and S. Sergeev, “Tetrahedron equation and generalized quantum groups,” J. Phys. A 48, 304001 (2015); arXiv:1503.08536 [math.QA].

    Article  MathSciNet  MATH  Google Scholar 

  122. V. V. Bazhanov and S. M. Sergeev, “Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups,” J. Phys. A 39, 3295–3310 (2006); arXiv:hepth/0509181 [hep-th].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  123. Yu. G. Stroganov, “Tetrahedron equation and spin integrable models on a cubic lattice,” Theor. Math. Phys. 110, 141–167 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  124. V. V. Bazhanov and Yu. G. Stroganov, “On commutativity conditions for transfer matrices on multidimensional lattice,” Theor. Math. Phys. 52, 685–691 (1982).

    Article  MathSciNet  Google Scholar 

  125. I. Frenkel and G. Moore, “Simplex equations and their solutions,” Commun. Math. Phys. 138, 259–271 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  126. A. Zamolodchikov, “Tetrahedra equations and integrable systems in three-dimensional space,” Sov. Phys. JETP 52, 325 (1980).

    ADS  Google Scholar 

  127. S. Fomin and A. Zelevinsky, “Cluster algebras I: foundations,” J. Am. Math. Soc. 15, 497–529 (2002); math/0104151.

    Article  MathSciNet  MATH  Google Scholar 

  128. S. N. M. Ruijsenaars, “First order analytic difference equations and integrable quantum systems,” J. Math. Phys. 38, 1069–1146 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  129. G. Felder and A. Varchenko, “The elliptic gamma function and SL(3.Z) Z3,” Adv. Math. 156, 44–76 (2000); arXiv:math/9907061.

    Article  MathSciNet  MATH  Google Scholar 

  130. E. Friedman and S. Ruijsenaars, “Shintani-barnes zeta and gamma functions,” Adv. Math. 187, 362–395 (2004).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Mimar Sinan Fine Arts University, Istanbul, Bomonti, 34380, Turkey

    Ilmar Gahramanov & Shahriyar Jafarzade

  2. Department of Mathematics, Khazar University, Baku, AZ1096, Azerbaijan

    Ilmar Gahramanov & Shahriyar Jafarzade

  3. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, D-14476, Germany

    Ilmar Gahramanov

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  1. Ilmar Gahramanov
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Gahramanov, I., Jafarzade, S. Integrable Lattice Spin Models from Supersymmetric Dualities. Phys. Part. Nuclei Lett. 15, 650–667 (2018). https://doi.org/10.1134/S1547477118060079

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