Journal of High Energy Physics

, 2010:57 | Cite as

Gauge theory loop operators and Liouville theory

  • Nadav Drukker
  • Jaume Gomis
  • Takuya Okuda
  • Jörg Teschner
Open Access


We propose a correspondence between loop operators in a family of four dimensional \( \mathcal{N} \) = 2 gauge theories on S 4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these \( \mathcal{N} \) = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these \( \mathcal{N} \) = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.


Supersymmetric gauge theory Field Theories in Lower Dimensions Duality in Gauge Field Theories 


  1. [1]
    C. Montonen and D.I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977) 117 [SPIRES].ADSGoogle Scholar
  2. [2]
    E. Witten and D.I. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978) 97 [SPIRES].Google Scholar
  3. [3]
    H. Osborn, Topological charges for \( \mathcal{N} \) = 4 supersymmetric gauge theories and monopoles of spin 1, Phys. Lett. B 83 (1979) 321 [SPIRES].ADSGoogle Scholar
  4. [4]
    P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in \( \mathcal{N} \) = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    P.C. Argyres, S-duality and global symmetries in N = 2 supersymmetric field theory, Adv. Theor. Math. Phys. 2 (1998) 293 [hep-th/9706095] [SPIRES].MATHMathSciNetGoogle Scholar
  7. [7]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    D. Gaiotto, \( \mathcal{N} \) = 2 dualities, arXiv:0904.2715 [SPIRES].
  9. [9]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].
  10. [10]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
  11. [11]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  12. [12]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
  13. [13]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, arXiv:0906.3219 [SPIRES].
  14. [14]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  15. [15]
    J. Teschner, A lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [SPIRES].MathSciNetGoogle Scholar
  16. [16]
    J. Teschner, Nonrational conformal field theory, arXiv:0803.0919 [SPIRES].
  17. [17]
    J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  18. [18]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  19. [19]
    J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    N. Drukker and D.J. Gross, An exact prediction of \( \mathcal{N} \) = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  21. [21]
    A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, hep-th/0604151 [SPIRES].
  23. [23]
    J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in \( \mathcal{N} \) = 4 super Yang-Mills, arXiv:0904.4486 [SPIRES].
  24. [24]
    J. Gomis and T. Okuda, S-duality, ’t Hooft operators and the operator product expansion, JHEP 09 (2009) 072 [arXiv:0906.3011] [SPIRES].CrossRefGoogle Scholar
  25. [25]
    N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].CrossRefGoogle Scholar
  26. [26]
    A. Kapustin, Holomorphic reduction of \( \mathcal{N} \) = 2 gauge theories, Wilson-’t Hooft operators and S-duality, hep-th/0612119 [SPIRES].
  27. [27]
    J. Teschner, On the relation between quantum Liouville theory and the quantized Teichmueller spaces, Int. J. Mod. Phys. A 19S2 (2004) 459 [hep-th/0303149] [SPIRES].MathSciNetGoogle Scholar
  28. [28]
    J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, in Prospects in mathematical physics, Contemp. Math. 437 (2007) 231, American Mathematical Society, Providence U.S.A. (2007) [hep-th/0308031] [SPIRES].
  29. [29]
    J. Teschner, An analog of a modular functor from quantized Teichmüller theory, in Handbook of Teichmüller theory. Volume I, IRMA Lect. Math. Theor. Phys. 11 (2007) 685, European Mathematical Society, Zürich Switzerland (2007) [math.QA/0510174] [SPIRES].
  30. [30]
    H.L. Verlinde, Conformal field theory, 2D quantum gravity and quantization of Teichmüller space, Nucl. Phys. B 337 (1990) 652 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in \( \mathcal{N} \) = 2 gauge theory and Liouville modular geometry, arXiv:0909.0945 [SPIRES].
  32. [32]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  33. [33]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  35. [35]
    D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
  36. [36]
    M. Dehn, On curve systems on two-sided surfaces, with application to the mapping problem, in Papers on group theory and topology, translated from the German by J. Stillwell, Springer-Verlag, New York U.S.A. (1987), pg. 234.Google Scholar
  37. [37]
    W.P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc. 19 (1988) 417.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    A. Bytsko and J. Teschner, The integrable structure of nonrational conformal field theory, arXiv:0902.4825 [SPIRES].
  39. [39]
    E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  41. [41]
    G.W. Moore and N. Seiberg, Lectures on RCFT, presented at Trieste Spring School, Trieste Italy (1989) [SPIRES].
  42. [42]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [SPIRES].
  43. [43]
    R.C. Penner, The action of the mapping class group on curves in surfaces, Enseign. Math. 30 (1984) 39.MATHMathSciNetGoogle Scholar
  44. [44]
    P.G. Zograf and L.A. Takhtadzhyan, On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0, Mat. Sbornik 132(174) (1987) 147 [Math. USSR Sb. 60(1) (1988) 143].Google Scholar
  45. [45]
    P.G. Zograf and L.A. Takhtadzhyan, On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces, Mat. Sbornik 132(174) (1987) 304 [Math. USSR Sb. 60(2) (1988) 297].Google Scholar
  46. [46]
    R.C. Penner, The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys. 113 (1987) 299.MATHCrossRefMathSciNetADSGoogle Scholar
  47. [47]
    V.V. Fock, Dual Teichmüller spaces, dg-ga/9702018.
  48. [48]
    R.C. Penner, Weil-Petersson volumes, J. Diff. Geom. 35 (1992) 559.MATHMathSciNetGoogle Scholar
  49. [49]
    L.A. Takhtajan, Uniformization, local index theorem, and geometry of the moduli spaces of Riemann surfaces and vector bundles, in Theta functions — Bowdoin 1987, Proc. Symp. Pure Math. 49, U.S.A. (1989).Google Scholar
  50. [50]
    L. Takhtajan and P. Zograf, Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on \( \mathcal{N} \), Trans. Am. Math. Soc. 355 (2003) 1857 [math.CV/0112170] [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    L. Chekhov and V.V. Fock, A quantum Teichmüller space, Theor. Math. Phys. 120 (1999) 1245 [Teor. Mat. Fiz. 120 (1999) 511] [math.QA/9908165] [SPIRES].MATHCrossRefGoogle Scholar
  52. [52]
    R.M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105 [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    V.V. Fock and A.B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009) 223.MATHCrossRefMathSciNetADSGoogle Scholar
  54. [54]
    L.O. Chekhov and V.V. Fock, Observables in 3D gravity and geodesic algebras, Czech. J. Phys. 50 (2000) 1201 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  55. [55]
    G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1 [SPIRES].CrossRefADSGoogle Scholar
  56. [56]
    A. Kapustin and N. Saulina, The algebra of Wilson-’t Hooft operators, Nucl. Phys. B 814 (2009) 327 [arXiv:0710.2097] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    L. Chekhov, Lecture notes on quantum Teichmüller theory, arXiv:0710.2051.
  58. [58]
    L.O. Chekhov, Orbifold Riemann surfaces and geodesic algebras, J. Phys. A 42 (2009) 304007.MathSciNetGoogle Scholar
  59. [59]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, More supersymmetric Wilson loops, Phys. Rev. D 76 (2007) 107703 [arXiv:0704.2237] [SPIRES].MathSciNetADSGoogle Scholar
  60. [60]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  61. [61]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal \( \mathcal{N} \) = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].CrossRefGoogle Scholar
  62. [62]
    V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Inst. Hautes Études Sci. Publ. Math. 103 (2006) 1.MATHMathSciNetGoogle Scholar
  63. [63]
    N.J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449.MATHCrossRefMathSciNetGoogle Scholar
  64. [64]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  65. [65]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [SPIRES].CrossRefGoogle Scholar
  66. [66]
    E.W. Barnes, The theory of the double gamma function, Proc. Roy. Soc. Lond. 66 (1899) 265.CrossRefGoogle Scholar
  67. [67]
    T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Tokyo (Sect. 1A Math.) 24 (1977) 167.MATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Nadav Drukker
    • 1
  • Jaume Gomis
    • 2
  • Takuya Okuda
    • 2
  • Jörg Teschner
    • 3
  1. 1.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.DESY TheoryHamburgGermany

Personalised recommendations