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Quivers, YBE and 3-manifolds

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Abstract

We study 4d superconformal indices for a large class of \( \mathcal{N} = 1 \) superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of “zig-zag paths” on a two-dimensional torus T 2. An exchange of loops, which we call a “double Yang-Baxter move”, gives the Seiberg duality of the gauge theory, and the invariance of the index under the duality is translated into the Yang-Baxter-type equation of a spin system defined on a “Z-invariant” lattice on T 2. When we compactify the gauge theory to 3d, Higgs the theory and then compactify further to 2d, the superconformal index reduces to an integral of quantum/classical dilogarithm functions. The saddle point of this integral unexpectedly reproduces the hyperbolic volume of a hyperbolic 3-manifold. The 3-manifold is obtained by gluing hyperbolic ideal polyhedra in \( {\mathbb{H}^3} \), each of which could be thought of as a 3d lift of the faces of the 2d bipartite graph. The same quantity is also related with the thermodynamic limit of the BPS partition function, or equivalently the genus 0 topological string partition function, on a toric Calabi-Yau manifold dual to quiver gauge theories. We also comment on brane realization of our theories. This paper is a companion to another paper summarizing the results [1].

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References

  1. Y. Terashima and M. Yamazaki, Emergent 3-manifold from 4d Superconformal Index, to appear.

  2. A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].

  4. S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. K.D. Kennaway, Brane Tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. M. Yamazaki, Brane Tilings and Their Applications, Fortsch. Phys. 56 (2008) 555 [arXiv:0803.4474] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  8. N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. V.V. Bazhanov and S.M. Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations, arXiv:1006.0651 [INSPIRE].

  12. V.V. Bazhanov and S.M. Sergeev, Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation, Nucl. Phys. B 856 (2012) 475 [arXiv:1106.5874] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. A. Volkov, Quantum Volterra model, Phys. Lett. A 167 (1992) 345.

    ADS  Google Scholar 

  14. L. Faddeev and A.Y. Volkov, Abelian current algebra and the Virasoro algebra on the lattice, Phys. Lett. B 315 (1993) 311 [hep-th/9307048] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  15. L.D. Faddeev, Currentlike variables in massive and massless integrable models, in Proc. Internat. School Phys. Enrico Fermi. Vol. 127: Varenna 1994, Quantum groups and their applications in physics, IOS press, Amsterdam The Netherlands (1996), pg. 117.

  16. A.I. Bobenko and B.A. Springborn, Variational principles for circle patterns and Koebe’s theorem, math/0203250.

  17. 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 (2007) 234 [hep-th/0703041] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748].

    Article  ADS  Google Scholar 

  19. T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, arXiv:1106.4550 [INSPIRE].

  20. Y. Terashima and M. Yamazaki, Semiclassical Analysis of the 3d/3d Relation, arXiv:1106.3066 [INSPIRE].

  21. T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, arXiv:1108.4389 [INSPIRE].

  22. S. Cecotti, C. Cordova and C. Vafa, Braids, Walls and Mirrors, arXiv:1110.2115 [INSPIRE].

  23. K. Nagao, Y. Terashima and M. Yamazaki, Hyperbolic 3-manifolds and Cluster Algebras, arXiv:1112.3106 [INSPIRE].

  24. T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, arXiv:1112.5179 [INSPIRE].

  25. T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, arXiv:1107.5588 [INSPIRE].

  27. S. Franco, Dimer Models, Integrable Systems and Quantum Teichmu¨ller Space, JHEP 09 (2011) 057 [arXiv:1105.1777] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. D. Xie, Network, Cluster coordinates and N = 2 theory I, arXiv:1203.4573 [INSPIRE].

  29. F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, arXiv:1109.0283 [INSPIRE].

  30. L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. H. Ooguri and M. Yamazaki, Emergent Calabi-Yau Geometry, Phys. Rev. Lett. 102 (2009) 161601 [arXiv:0902.3996] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. D.R. Gulotta, Properly ordered dimers, R-charges and an efficient inverse algorithm, JHEP 10 (2008) 014 [arXiv:0807.3012] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. K. Ueda and M. Yamazaki, A note on dimer models and McKay quivers, Commun. Math. Phys. 301 (2011) 723 [math/0605780] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].

  35. I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar quad-graphs, Trans. Amer. Math. Soc. 357 (2005) 3443.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Ishii and K. Ueda, A note on consistency conditions on dimer models, in RIMS Kôkyûroku Bessatsu. Vol. B 24: Higher dimensional algebraic geometry, pg. 143 [arXiv:1012.5449].

  38. S. Mozgovoy and M. Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings, arXiv:0809.0117 [INSPIRE].

  39. Y. Imamura, Global symmetries and ’t Hooft anomalies in brane tilings, JHEP 12 (2006) 041 [hep-th/0609163] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. Y. Imamura, H. Isono, K. Kimura and M. Yamazaki, Exactly marginal deformations of quiver gauge theories as seen from brane tilings, Prog. Theor. Phys. 117 (2007) 923 [hep-th/0702049] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].

  42. K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].

  43. B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 3 [hep-th/0511287] [INSPIRE].

    MathSciNet  Google Scholar 

  44. A. Butti, Deformations of Toric Singularities and Fractional Branes, JHEP 10 (2006) 080 [hep-th/0603253] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].

  48. F. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. E.M. Rains, Transformations of elliptic hypergeometric integrals, Ann. Math. 171 (2010) 169.

    Article  MathSciNet  MATH  Google Scholar 

  50. V. Spiridonov and G. Vartanov, Elliptic Hypergeometry of Supersymmetric Dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. V. Spiridonov and G. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots and vortices, arXiv:1107.5788 [INSPIRE].

  52. V. Spiridonov, Elliptic beta integrals and solvable models of statistical mechanics, arXiv:1011.3798 [INSPIRE].

  53. R. Baxter, Solvable eight vertex model on an arbitrary planar lattice, Phil. Trans. Roy. Soc. Lond. 289 (1978) 315 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  54. R. Baxter, Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics, Proc. Roy. Soc. Lond. A 404 (1986) 1 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  55. A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  57. R. Baxter, J. Perk and H. Au-Yang, New solutions of the star triangle relations for the chiral Potts model, Phys. Lett. A 128 (1988) 138.

    MathSciNet  ADS  Google Scholar 

  58. H. Au-Yang, B.M. McCoy, J.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 (1987) 219 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  59. 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 (1985) 351 [INSPIRE].

    Article  ADS  Google Scholar 

  60. F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Physics Letters B 704 (2011) 234 [arXiv:1104.1787].

    MathSciNet  ADS  Google Scholar 

  61. A. Gadde and W. Yan, Reducing the 4d Index to the S3 Partition Function, arXiv:1104.2592 [INSPIRE].

  62. T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  63. N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716].

    Article  MathSciNet  ADS  Google Scholar 

  64. A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  65. D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, arXiv:1012.3210.

  66. N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512].

    Article  MathSciNet  ADS  Google Scholar 

  67. G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. L. Faddeev, R. Kashaev and A.Y. Volkov, Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality, Commun. Math. Phys. 219 (2001) 199 [hep-th/0006156] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  69. N.A. Nekrasov and S.L. Shatashvili, Supersymmetric Vacua and Bethe Ansatz, Nucl. Phys. Proc. Suppl. 192 (2009) 91 [arXiv:0901.4744].

    Article  MathSciNet  ADS  Google Scholar 

  70. N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, Proceedings of 16th International Congress on Mathematical Physics, Prague Czech Republic (2009), World Scientific, Singapore (2010) [arXiv:0908.4052] [INSPIRE].

  71. W.P. Thurston, The geometry and topology of three-manifolds, (1979).

  72. B.A. Springborn, Variational principles for circle patterns, math/0312363.

  73. S. Lee, Superconformal field theories from crystal lattices, Phys. Rev. D 75 (2007) 101901 [hep-th/0610204] [INSPIRE].

    ADS  Google Scholar 

  74. M. Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. 88 (2004) 204 [math/0012185]. With an appendix by Ian Agol and Dylan Thurston.

    Article  MathSciNet  MATH  Google Scholar 

  75. J.S. Purcell, An introduction to fully augmented links, in Contemp. Math. Vol. 541: Interactions between hyperbolic geometry, quantum topology and number theory, Amer. Math. Soc., Providence U.S.A. (2011), pg. 205.

  76. R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2002) 409.

    Article  MathSciNet  MATH  Google Scholar 

  77. B. de Tilière, Partition function of periodic isoradial dimer models, Probab. Theor. Rel. Fields 138 (2007) 451 [math/0605583].

    Article  MATH  Google Scholar 

  78. R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, Ann. Math. 163 (2006) 1019 [math-ph/0311005].

    Article  MathSciNet  MATH  Google Scholar 

  79. M. Yamazaki, Crystal Melting and Wall Crossing Phenomena, Int. J. Mod. Phys. A 26 (2011) 1097 [arXiv:1002.1709] [INSPIRE].

    ADS  Google Scholar 

  80. P. Sulkowski, BPS states, crystals and matrices, Adv. High Energy Phys. 2011 (2011) 357016 [arXiv:1106.4873] [INSPIRE].

    MathSciNet  Google Scholar 

  81. B. Szendroi, Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol. 12 (2008) 1171 [arXiv:0705.3419] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  82. H. Ooguri and M. Yamazaki, Crystal Melting and Toric Calabi-Yau Manifolds, Commun. Math. Phys. 292 (2009) 179 [arXiv:0811.2801].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  83. M. Aganagic, H. Ooguri, C. Vafa and M. Yamazaki, Wall Crossing and M-theory, Publ. Res. Inst. Math. Sci. Kyoto 47 (2011) 569 [arXiv:0908.1194] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  84. H. Ooguri, P. Sulkowski and M. Yamazaki, Wall Crossing As Seen By Matrix Models, Commun. Math. Phys. 307 (2011) 429 [arXiv:1005.1293] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  85. R.J. Szabo and M. Tierz, Matrix models and stochastic growth in Donaldson-Thomas theory, arXiv:1005.5643 [INSPIRE].

  86. W.-y. Chuang and D.L. Jafferis, Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions, Commun. Math. Phys. 292 (2009) 285 [arXiv:0810.5072] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  87. M. Aganagic and K. Schaeffer, Wall Crossing, Quivers and Crystals, arXiv:1006.2113.

  88. A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225].

    Article  MathSciNet  ADS  Google Scholar 

  89. A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].

    Article  ADS  Google Scholar 

  90. A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, arXiv:1110.3740.

  91. D. Gaiotto, N = 2 dualities, arXiv:0904.2715.

  92. N. Arkani-Hamed, Scattering Amplitudes and the Positive Grassmannian, lecture at Institute for Advanced Study, Princeton U.S.A. (2012).

  93. L. Faddeev and R. Kashaev, Quantum Dilogarithm, Mod. Phys. Lett. A 9 (1994) 427 [hep-th/9310070] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  94. L.D. Faddeev, Discrete Heisenberg-Weyl Group and modular group, Lett. Math. Phys. 34 (1995) 249 [hep-th/9504111].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  95. A.N. Kirillov, Dilogarithm Identities, Prog. Theor. Phys. Suppl. 118 (1995) 61 [hep-th/9408113].

    Article  MathSciNet  ADS  Google Scholar 

  96. R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Commun. Math. Phys. 222 (2001) 147.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  97. H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, J. Am. Math. Soc. 14 (2001) 297 [math/0008220].

    Article  MathSciNet  MATH  Google Scholar 

  98. T. Fujimori, M. Nitta, K. Ohta, N. Sakai and M. Yamazaki, Intersecting Solitons, Amoeba and Tropical Geometry, Phys. Rev. D 78 (2008) 105004 [arXiv:0805.1194] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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Yamazaki, M. Quivers, YBE and 3-manifolds. J. High Energ. Phys. 2012, 147 (2012). https://doi.org/10.1007/JHEP05(2012)147

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