Teichmüller TQFT vs. Chern-Simons theory

  • Victor Mikhaylov
Open Access
Regular Article - Theoretical Physics


Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2, ℝ) Chern-Simons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal translates a known conjecture by Andersen and Kashaev into a conjecture about the Kapustin-Witten equations. We further explain that Teichmüller TQFT is dual to complex SL(2, ℂ) Chern-Simons theory at integer level k = 1, clarifying some puzzles previously encountered in the 3d-3d correspondence literature. We also present a new simple derivation of complex Chern-Simons theories from the 6d (2,0) theory on a lens space with a transversely-holomorphic foliation.


Chern-Simons Theories Supersymmetric Gauge Theory Topological Field Theories 


Open Access

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  1. [1]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  5. [5]
    T. Dimofte, Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017) 443009 [arXiv:1608.02961] [INSPIRE].
  6. [6]
    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  7. [7]
    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
  9. [9]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
  10. [10]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
  11. [11]
    M. Dedushenko and E. Witten, Some details on the Gopakumar-Vafa and Ooguri-Vafa formulas, Adv. Theor. Math. Phys. 20 (2016) 1 [arXiv:1411.7108] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T. Dimofte, 3d superconformal theories from three-manifolds, arXiv:1412.7129.
  15. [15]
    H.L. Verlinde and E.P. Verlinde, Conformal field theory and geometric quantization, IASSNS-HEP-89-58 (1989).
  16. [16]
    H.L. Verlinde, Conformal field theory, 2D quantum gravity and quantization of Teichmüller space, Nucl. Phys. B 337 (1990) 652 [INSPIRE].
  17. [17]
    R.M. Kashaev, Quantization of Teichmueller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    L. Chekhov and V.V. Fock, Quantum Teichmüller space, Theor. Math. Phys. 120 (1999) 1245 [Teor. Mat. Fiz. 120 (1999) 511] [math/9908165] [INSPIRE].
  19. [19]
    J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011) 471 [arXiv:1005.2846] [INSPIRE].
  20. [20]
    T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Teschner, An analog of a modular functor from quantized Teichmüller theory, math/0510174 [INSPIRE].
  22. [22]
    B. Bakalov and A. Kirillov Jr., Lectures on tensor categories and modular functors, AMS University Lecture Series volume 21, American Mathematical Society, U.S.A. (2001).Google Scholar
  23. [23]
    J. Ellegaard Andersen and R. Kashaev, A TQFT from quantum Teichmüller theory, Commun. Math. Phys. 330 (2014) 887 [arXiv:1109.6295] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    J. Ellegaard Andersen and R. Kashaev, A new formulation of the Teichmüller TQFT, arXiv:1305.4291 [INSPIRE].
  25. [25]
    T. Dimofte, Quantum Riemann surfaces in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 479 [arXiv:1102.4847] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. Dimofte, Complex Chern-Simons theory at level k via the 3d-3d correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
  29. [29]
    Y. Terashima and M. Yamazaki, ℝ Chern-Simons, Liouville and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
  30. [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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    J. Yagi, 3D TQFT from 6D SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    S. Lee and M. Yamazaki, 3D Chern-Simons theory from M 5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    C. Cordova and D.L. Jafferis, Complex Chern-Simons from M 5-branes on the squashed three-sphere, JHEP 11 (2017) 119 [arXiv:1305.2891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, From rigid supersymmetry to twisted holomorphic theories, Phys. Rev. D 90 (2014) 085006 [arXiv:1407.2598] [INSPIRE].
  38. [38]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    S. Gukov and E. Witten, Branes and quantization, Adv. Theor. Math. Phys. 13 (2009) 1445 [arXiv:0809.0305] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 Super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
  41. [41]
    E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
  42. [42]
    A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    D. Gaiotto and E. Witten, Knot invariants from four-dimensional gauge theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    D. Gang, N. Kim and S. Lee, Holography of 3d-3d correspondence at Large N, JHEP 04 (2015) 091 [arXiv:1409.6206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    A. Reznikov, Rationality of secondary classes, J. Diff. Geom. 43 (1996) 674.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    J.-B. Bae, D. Gang and J. Lee, 3d \( \mathcal{N}=2 \) minimal SCFTs from wrapped M 5-branes, JHEP 08 (2017) 118 [arXiv:1610.09259] [INSPIRE].
  47. [47]
    S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
  48. [48]
    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
  50. [50]
    D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].CrossRefzbMATHGoogle Scholar
  52. [52]
    J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, in the proceedings of the International Congress on Mathematical physics (ICMP 2003), July 28-August 2, Lisbon, Portugal (2003), hep-th/0308031 [INSPIRE].
  53. [53]
    L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995) 249 [hep-th/9504111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].
  55. [55]
    N.R. Constable, R.C. Myers and O. Tafjord, The noncommutative bion core, Phys. Rev. D 61 (2000) 106009 [hep-th/9911136] [INSPIRE].ADSMathSciNetGoogle Scholar
  56. [56]
    S. He and R. Mazzeo, The extended bogomolny equations, to appear.Google Scholar
  57. [57]
    R. Mazzeo and E. Witten, The Nahm pole boundary condition, arXiv:1311.3167 [INSPIRE].
  58. [58]
    K. Corlette, Flat G-bundles with canonical metrics, J. Diff. Geom. 28 (1988) 361.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes É tudes Sci. 103 (2006) 1.CrossRefzbMATHGoogle Scholar
  61. [61]
    M. Henningson, Boundary conditions for geometric-Langlands twisted N = 4 supersymmetric Yang-Mills theory, Phys. Rev. D 86 (2012) 085003 [arXiv:1106.3845] [INSPIRE].
  62. [62]
    D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the theta-angle in N = 4 Super Yang-Mills theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
  63. [63]
    D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    C.H. Taubes, Compactness theorems for SL(2; C) generalizations of the 4-dimensional anti-self dual equations, Part I, arXiv:1307.6447 [INSPIRE].
  65. [65]
    S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
  66. [66]
    V. Mikhaylov, Analytic torsion, 3D mirror symmetry and supergroup Chern-Simons theories, arXiv:1505.03130 [INSPIRE].
  67. [67]
    D. Gaiotto and M. Rapčák, Vertex algebras at the corner, arXiv:1703.00982 [INSPIRE].
  68. [68]
    S. Garoufalidis and R. Kashaev, From state integrals to q-series, Math. Res. Lett. 24 (2017) 781 [arXiv:1304.2705] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    S. Garoufalidis and R. Kashaev, Evaluation of state integrals at rational points, Commun. Number Theory Phys. 9 (2015) 549.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    T.T. Dumitrescu, An introduction to supersymmetric field theories in curved space, J. Phys. A 50 (2017) 443005 [arXiv:1608.02957] [INSPIRE].
  72. [72]
    Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
  73. [73]
    V. Pestun, Localization for \( \mathcal{N}=2 \) supersymmetric gauge theories in four dimensions, arXiv:1412.7134.
  74. [74]
    S. Kawai, The symplectic nature of the space of projective connections in Riemann surfaces, Math. Ann. 305 (1996) 161.MathSciNetCrossRefGoogle Scholar
  75. [75]
    A. Balasubramanian and J. Teschner, Supersymmetric field theories and geometric Langlands: The other side of the coin, talk given at String Math 2016, June 27-July 2, Paris, France (2016), arXiv:1702.06499 [INSPIRE].
  76. [76]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    S. Carlip, Conformal field theory, (2 + 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [INSPIRE].
  78. [78]
    D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann Surfaces and Related Topics, I. Kra and B. Maskit eds., Princeton University Press, Princeton, U.S.A. (1981).Google Scholar
  79. [79]
    C. McMullen, Riemann surfaces and the geometrization of 3-manifolds, Bull. Amer. Math. Soc. (N.S.) 27 (1992) 207.Google Scholar
  80. [80]
    D. Birmingham, I. Sachs and S. Sen, Exact results for the BTZ black hole, Int. J. Mod. Phys. D 10 (2001) 833 [hep-th/0102155] [INSPIRE].
  81. [81]
    L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94.Google Scholar
  82. [82]
    T. Dimofte, D. Gaiotto and R. van der Veen, RG domain walls and hybrid triangulations, Adv. Theor. Math. Phys. 19 (2015) 137 [arXiv:1304.6721] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
  84. [84]
    G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    J. Teschner, Quantization of moduli spaces of flat connections and Liouville theory, talk given at the nternational Congress of Mathematicians (ICM 2014), August 13-21, Seoul, Korea (2014), arXiv:1405.0359 [INSPIRE].
  86. [86]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3D indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Simons Center for Geometry and PhysicsState University of New YorkStony BrookU.S.A.

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