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Theoretical and Mathematical Physics

, Volume 198, Issue 2, pp 189–196 | Cite as

Pentagon Identities Arising in Supersymmetric Gauge Theory Computations

  • D. N. Bozkurt
  • I. B. GahramanovEmail author
Article
  • 8 Downloads

Abstract

The partition functions of three-dimensional N=2 supersymmetric gauge theories on different manifolds can be expressed as q-hypergeometric integrals. Comparing the partition functions of three-dimensional mirror dual theories, we derive complicated integral identities. In some cases, these identities can be written in the form of pentagon relations. Such identities are often interpreted as the Pachner 3–2 move for triangulated manifolds using the so-called 3d–3d correspondence. From the physics perspective, another important application of pentagon identities is that they can be used to construct new solutions of the quantum Yang–Baxter equation.

Keywords

pentagon identity exact results in supersymmetric gauge theories hypergeometric integral 

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References

  1. 1.
    J. Allman and R. Rimányi, “Quantum dilogarithm identities for the square product of A–type Dynkin quivers,” arXiv:1702.04766v3 [math.RT] (2017).zbMATHGoogle Scholar
  2. 2.
    A. Dimakis and F. Müller–Hoissen, “Simplex and polygon equations,” SIGMA, 11, 042 (2015); arXiv: 1409.7855v2 [math–ph] (2014).MathSciNetzbMATHGoogle Scholar
  3. 3.
    I. Gahramanov and H. Rosengren, “Integral pentagon relations for 3d superconformal indices,” in: String–Math 2014 (Proc. Symp. Pure Math., Vol. 93, V. Bouchard, C. Doran, S. Méndez–Diez, and C. Quigley, eds.), Amer. Math. Soc., Providence, R. I. (2016), pp. 165–173; arXiv:1412.2926v2 [hep–th] (2014).Google Scholar
  4. 4.
    V. Pestun, “Localization of gauge theory on a four–sphere and supersymmetric Wilson loops,” Commun. Math. Phys., 313, 71–129 (2012); arXiv:0712.2824v3 [hep–th] (2007).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. Kashaev, F. Luo, and G. Vartanov, “A TQFT of Turaev–Viro type on shaped triangulations,” Ann. Henri Poincaré, 17, 1109–1143 (2016); arXiv:1210.8393v1 [math.QA] (2012).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. Dimofte, D. Gaiotto, and S. Gukov, “Gauge theories labelled by three–manifolds,” Commun. Math. Phys., 325, 367–419 (2014); arXiv:1108.4389v1 [hep–th] (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T. Dimofte, D. Gaiotto, and S. Gukov, “3–Manifolds and 3d indices,” Adv. Theor. Math. Phys., 17, 975–1076 (2013); arXiv:1112.5179v1 [hep–th] (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I. Gahramanov and H. Rosengren, “A new pentagon identity for the tetrahedron index,” JHEP, 1311, 128 (2013); arXiv:1309.2195v3 [hep–th] (2013).ADSCrossRefGoogle Scholar
  9. 9.
    I. Gahramanov and H. Rosengren, “Basic hypergeometry of supersymmetric dualities,” Nucl. Phys. B, 913, 747–768 (2016); arXiv:1606.08185v2 [hep–th] (2016).ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Y. Imamura and D. Yokoyama, “S3/Zn partition function and dualities,” JHEP, 1211, 122 (2012); arXiv: 1208.1404v2 [hep–th] (2012).ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    R. M. Kashaev, “Beta pentagon relations,” Theor. Math. Phys., 181, 1194–1205 (2014); arXiv:1403.1298v2 [math–ph] (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    U. von Pachner, “Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten,” Abh. Math. Sem. Univ. Hamburg, 57, 69–86 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    U. Pachner, “P. L. homeomorphic manifolds are equivalent by elementary shellings,” Eur. J. Combin., 12, 129–145 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. V. Galakhov, A. D. Mironov, A. Yu. Morozov, and A. V. Smirnov, “Three–dimensional extensions of the Alday–Gaiotto–Tachikawa relation,” Theor. Math. Phys., 172, 939–962 (2012); arXiv:1104.2589v3 [hep–th] (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    T. Dimofte, “3d superconformal theories from three–manifolds,” in: New Dualities of Supersymmetric Gauge Theories (J. Teschner, ed.), Springer, Cham (2016), pp. 339–373; arXiv:1412.7129v1 [hep–th] (2014).Google Scholar
  16. 16.
    Y. Terashima and M. Yamazaki, “Semiclassical analysis of the 3d/3d relation,” Phys. Rev. D, 88, 026011 (2013); arXiv:1106.3066v3 [hep–th] (2011).ADSCrossRefGoogle Scholar
  17. 17.
    L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville correlation functions from four–dimensional gauge theories,” Lett. Math. Phys., 91, 167–197 (2010); arXiv:0906.3219v2 [hep–th] (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    L. D. Faddeev and R. M. Kashaev, “Quantum dilogarithm,” Modern Phys. Lett. A, 9, 427–434 (1994); arXiv: hep–th/9310070v1 (1993).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. D. Faddeev, “Volkov’s pentagon for the modular quantum dilogarithm,” Funct. Anal. Appl., 45, 291–296 (2011); arXiv:1201.6464v1 [math.QA] (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Yu. Volkov, “Beyond the ‘pentagon identity’,” Lett. Math. Phys., 39, 393–397 (1997); arXiv:q–alg/9603003v1 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Yu. Volkov, “Pentagon identity revisited I,” Int. Math. Res. Notices, 2012, No. 20, 4619–4624 (2012); arXiv: 1104.2267v1 [math.QA] (201).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    R. M. Kashaev and S. M. Sergeev, “On pentagon, ten term, and tetrahedron relations,” Commun. Math. Phys., 195, 309–319 (1998); arXiv:q–alg/9607032v1 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    R. M. Kashaev, “On the spectrum of Dehn twists in quantumTeichmüller theory,” in: Physics and Combinatorics (A. N. Kirillov and N. Liskova, eds.), World Scientific, Singapore (2001), pp. 63–81; arXiv:math/0008148v1 (2000).Google Scholar
  24. 24.
    N. A. Nekrasov, “Seiberg–Witten prepotential from instanton counting,” Adv. Theor. Math. Phys., 7, 831–864 (2003); arXiv:hep–th/0206161v1 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    K. Hosomichi, “The localization principle in SUSY gauge theories,” Prog. Theor. Exp. Phys., 2015, 11B101 (2015); arXiv:1502.04543v1 [hep–th] (2015).Google Scholar
  26. 26.
    B. Willett, “Localization on three–dimensional manifolds,” J. Phys. A: Math. Theor., 50, 443006 (2017); arXiv: 1608.02958v3 [hep–th] (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Cremonesi, “Localization and supersymmetry on curved space,” PoS(Modave2013), 201, 002 (2013).Google Scholar
  28. 28.
    K. A. Intriligator and N. Seiberg, “Mirror symmetry in three–dimensional gauge theories,” Phys. Lett. B, 387, 513–519 (1996); arXiv:hep–th/9607207v1 (1996).ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. Strassler, “Aspects of N=2 supersymmetric gauge theories in three–dimensions,” Nucl. Phys. B, 499, 67–99 (1997); arXiv:hep–th/9703110v1 (1997).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Kapustin and B. Willett, “Generalized superconformal index for three dimensional field theories,” arXiv: 1106.2484v1 [hep–th] (2011).Google Scholar
  31. 31.
    A. Kapustin, B. Willett, and I. Yaakov, “Exact results for Wilson loops in superconformal Chern–Simons theories with matter,” JHEP, 1003, 089 (2010); arXiv:0909.4559v4 [hep–th] (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    N. Hama, K. Hosomichi, and S. Lee, “Notes on SUSY gauge theories on three–sphere,” JHEP, 1103, 127 (2011); arXiv:1012.3512v3 [hep–th] (2010).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    D. L. Jafferis, “The exact superconformal R–symmetry extremizes Z,” JHEP, 1205, 159 (2012); arXiv: 1012.3210v2 [hep–th] (2010).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    N. Hama, K. Hosomichi, and S. Lee, “SUSY gauge theories on squashed three–spheres,” JHEP, 1105, 014 (2011); arXiv:1102.4716v1 [hep–th] (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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:1107.5788v4 [hep–th] (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    I. Gahramanov and A. P. Kels, “The star–triangle relation, lens partition function, and hypergeometric sum/integrals,” JHEP, 1702, 040 (2017); arXiv:1610.09229v1 [math–ph] (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    I. Gahramanov and S. Jafarzade, “Integrable lattice spin models from supersymmetric dualities,” arXiv: 1712.09651v1 [math–ph] (2017).Google Scholar
  38. 38.
    C. Krattenthaler, V. Spiridonov, and G. Vartanov, “Superconformal indices of three–dimensional theories related by mirror symmetry,” JHEP, 1106, 008 (2011); arXiv:1103.4075v2 [hep–th] (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    A. Tanaka, H. Mori, and T. Morita, “Superconformal index on RP2 × S1 and mirror symmetry,” Phys. Rev. D, 91, 105023 (2015); arXiv:1408.3371v3 [hep–th] (2014).ADSCrossRefGoogle Scholar
  40. 40.
    A. Tanaka, H. Mori, and T. Morita, “Abelian 3d mirror symmetry on RP2 × S1 with Nf = 1,” JHEP, 1509, 154 (2015); arXiv:1505.07539v2 [hep–th] (2015).ADSCrossRefGoogle Scholar
  41. 41.
    H. Mori and A. Tanaka, “Varieties of Abelian mirror symmetry on RP2 ×S1,” JHEP, 1602, 088 (2016); arXiv: 1512.02835v3 [hep–th] (2015).ADSCrossRefGoogle Scholar
  42. 42.
    V. Spiridonov, “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, P. B. Acosta–Humánez, F. Finkel, N. Kamran, and P. J. Olver), Amer.Math. Soc., Providence, R. I. (2012), pp. 181–211; arXiv:1011.3798v2 [hep–th] (2010).Google Scholar
  43. 43.
    S. Benvenuti and S. Pasquetti, “3d N=2 mirror symmetry, pq–webs, and monopole superpotentials,” JHEP, 1608, 136 (2016); arXiv:1605.02675v2 [hep–th] (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    K. Hikami, “Generalized volume conjecture and the A–polynomials: The Neumann–Zagier potential function as a classical limit of quantum invariant,” J. Geom. Phys., 57, 1895–1940 (2007); arXiv:math/0604094v1 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    R. M. Kashaev, “The hyperbolic volume of knots from quantum dilogarithm,” Lett. Math. Phys., 39, 269–275 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    D. Gang, N. Kim, and S. Lee, “Holography of wrapped M5–branes and Chern–Simons theory,” Phys. Lett. B, 733, 316–319 (2014); arXiv:1401.3595v3 [hep–th] (2014).ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    V. V. Bazhanov, A. P. Kels, and S. M. Sergeev, “Quasi–classical expansion of the star–triangle relation and integrable systems on quad–graphs,” J. Phys. A, 49, 464001 (2016); arXiv:1602.07076v4 [math–ph] (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    S. Jafarzade and Z. Nazari, “A new integrable Ising–type model from 2d N=(2, 2) dualities,” arXiv:1709.00070v2 [hep–th] (2017).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Koç UniversityIstanbulTurkey
  2. 2.Mimar Sinan Fine Arts UniversityKhazar UniversityTurkeyAzerbaijan
  3. 3.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)BerlinGermany

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