Journal of High Energy Physics

, 2011:22 | Cite as

Non-perturbative topological strings and conformal blocks

  • Miranda C. N. Cheng
  • Robbert Dijkgraaf
  • Cumrun Vafa
Open Access


We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.


Topological Strings Matrix Models Duality in Gauge Field Theories 


  1. [1]
    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [SPIRES].CrossRefMATHADSGoogle Scholar
  2. [2]
    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP 02 (2004) 010 [hep-th/0211098] [SPIRES].CrossRefADSGoogle Scholar
  3. [3]
    cM. Aganagic and M. Yamazaki, Open BPS Wall Crossing and M-theory, Nucl. Phys. B 834 (2010) 258 [arXiv:0911.5342] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    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] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  6. [6]
    L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  7. [7]
    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].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d Correspondences, arXiv:1006.3435 [SPIRES].
  9. [9]
    F. David, Nonperturbative effects in matrix models and vacua of two-dimensional gravity, Phys. Lett. B 302 (1993) 403 [hep-th/9212106] [SPIRES].ADSGoogle Scholar
  10. [10]
    F. David, Phases of the large-N matrix model and nonperturbative effects in 2 − D gravity, Nucl. Phys. B 348 (1991) 507 [SPIRES].CrossRefADSGoogle Scholar
  11. [11]
    R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    R. Dijkgraaf, P. Sulkowski and C. Vafa, unpublished, (2009).Google Scholar
  13. [13]
    R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES].
  14. [14]
    R. Dijkgraaf and C. Vafa, A perturbative window into non-perturbative physics, hep-th/0208048 [SPIRES].
  15. [15]
    R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    R. Dijkgraaf and C. Vafa, On geometry and matrix models, Nucl. Phys. B 644 (2002) 21 [hep-th/0207106] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [SPIRES].
  18. [18]
    T. Dimofte, S. Gukov, J. Lenells and D. Zagier, Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group, Commun. Num. Theor. Phys. 3 (2009) 363 [arXiv:0903.2472] [SPIRES].MATHMathSciNetGoogle Scholar
  19. [19]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [SPIRES].CrossRefADSGoogle Scholar
  22. [22]
    T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP 07 (2010) 081 [arXiv:1006.0828] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    B. Eynard and M. Mariño, A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011) 1181 [arXiv:0810.4273] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  25. [25]
    G. Felder, BRST Approach to Minimal Methods, Nucl. Phys. B 317 (1989) 215 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    P.J. Forrester and S.O. Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008) 489 [arXiv:0710.3981].CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
  28. [28]
    S.B. Giddings and E.J. Martinec, Conformal Geometry and String Field Theory, Nucl. Phys. B 278 (1986) 91 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    S.B. Giddings, E.J. Martinec and E. Witten, Modular Invariance in String Field Theory, Phys. Lett. B 176 (1986) 362 [SPIRES].ADSMathSciNetGoogle Scholar
  30. [30]
    S.B. Giddings and S.A. Wolpert, A triangulation of moduli space from light-cone string theory, Commun. Math. Phys. 109 (1987) 177 [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  31. [31]
    C. Gomez and G. Sierra, A Brief history of hidden quantum symmetries in conformal field theories, hep-th/9211068 [SPIRES].
  32. [32]
    C. Gomez and G. Sierra, Quantum group meaning of the Coulomb gas, Phys. Lett. B 240 (1990) 149 [SPIRES].ADSMathSciNetGoogle Scholar
  33. [33]
    R. Gopakumar and C. Vafa, Topological gravity as large-N topological gauge theory, Adv. Theor. Math. Phys. 2 (1998) 413 [hep-th/9802016] [SPIRES].MATHMathSciNetGoogle Scholar
  34. [34]
    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [SPIRES].MATHMathSciNetGoogle Scholar
  35. [35]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett. B 685 (2010) 79 [arXiv:0911.4296] [SPIRES].ADSMathSciNetGoogle Scholar
  36. [36]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix Models, Geometric Engineering and Elliptic Genera, JHEP 03 (2008) 069 [hep-th/0310272] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [SPIRES].
  38. [38]
    S. Iguri and C.A. Núñez, Coulomb integrals for the SL (2, R) WZW model, Phys. Rev. D 77 (2008) 066015 [arXiv:0705.4461] [SPIRES].ADSGoogle Scholar
  39. [39]
    A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [SPIRES].MATHMathSciNetGoogle Scholar
  40. [40]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  41. [41]
    H. Itoyama and T. Oota, Method of Generating q-Expansion Coefficients for Conformal Block and N = 2 Nekrasov Function by beta-Deformed Matrix Model, Nucl. Phys. B 838 (2010) 298 [arXiv:1003.2929] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  42. [42]
    M. Jimbo and T. Miwa, QKZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A 29 (1996) 2923 [hep-th/9601135] [SPIRES].ADSMathSciNetGoogle Scholar
  43. [43]
    J. Kaneko, Selberg Integrals and Hypergeometric Functions Associated with Jack Polynomials, Siam. J. Math. Anal. 24 (1993) 1086. CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    I. Kostov, Matrix models as conformal field theories: genus expansion, Nucl. Phys. B 837 (2010) 221 [arXiv:0912.2137] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  45. [45]
    I.K. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [SPIRES].
  46. [46]
    C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  47. [47]
    M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [SPIRES].CrossRefADSGoogle Scholar
  48. [48]
    M. Mariño, Les Houches lectures on matrix models and topological strings, hep-th/0410165 [SPIRES].
  49. [49]
    M. Mariño, R. Schiappa and M. Weiss, Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings, arXiv:0711.1954 [SPIRES].
  50. [50]
    A. Marshakov, A. Mironov and A. Morozov, Combinatorial Expansions of Conformal Blocks, Theor. Math. Phys. 164 (2010) 3 [arXiv:0907.3946] [SPIRES].CrossRefGoogle Scholar
  51. [51]
    A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: Discrete case, Phys. Lett. B 265 (1991) 99 [SPIRES].ADSMathSciNetGoogle Scholar
  52. [52]
    K. Maruyoshi and F. Yagi, Seiberg-Witten curve via generalized matrix model, JHEP 01 (2011) 042 [arXiv:1009.5553] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  53. [53]
    A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys. B 843 (2011) 534 [arXiv:1003.5752] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  54. [54]
    A. Mironov, A. Morozov and S. Shakirov, Conformal blocks as Dotsenko-Fateev Integral Discriminants, Int. J. Mod. Phys. A 25 (2010) 3173 [arXiv:1001.0563] [SPIRES].ADSMathSciNetGoogle Scholar
  55. [55]
    A. Mironov, A. Morozov and Sh. Shakirov, On Dotsenko-Fateev representation of the toric conformal blocks, J. Phys. A 44 (2011) 085401 [arXiv:1010.1734] [SPIRES].ADSMathSciNetGoogle Scholar
  56. [56]
    A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  57. [57]
    A. Morozov, Matrix models as integrable systems, hep-th/9502091 [SPIRES].
  58. [58]
    A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP 08 (2010) 066 [arXiv:1004.2917] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  59. [59]
    S. Mukhi, Topological matrix models, Liouville matrix model and c = 1 string theory, hep-th/0310287 [SPIRES].
  60. [60]
    Y. Nakayama, Liouville field theory: A decade after the revolution, Int. J. Mod. Phys. A 19 (2004) 2771 [hep-th/0402009] [SPIRES].ADSMathSciNetGoogle Scholar
  61. [61]
    N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  62. [62]
    N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  63. [63]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
  64. [64]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  65. [65]
    A. Pakman, Liouville theory without an action, Phys. Lett. B 642 (2006) 263 [hep-th/0601197] [SPIRES].ADSMathSciNetGoogle Scholar
  66. [66]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
  67. [67]
    B. Ponsot, Recent progresses on Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 311 [hep-th/0301193] [SPIRES].
  68. [68]
    B. Ponsot, Liouville theory on the pseudosphere: Bulk-boundary structure constant, Phys. Lett. B 588 (2004) 105 [hep-th/0309211] [SPIRES].ADSMathSciNetGoogle Scholar
  69. [69]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [SPIRES].
  70. [70]
    B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U q(sl(2,R)), Commun. Math. Phys. 224 (2001) 613 [math/0007097].CrossRefADSMathSciNetGoogle Scholar
  71. [71]
    R. Schiappa and N. Wyllard, An A r threesome: Matrix models, 2d CFTs and 4d N = 2 gauge theories, arXiv:0911.5337 [SPIRES].
  72. [72]
    J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, arXiv:1005.2846 [SPIRES].
  73. [73]
    J. Teschner, A lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [SPIRES].ADSMathSciNetGoogle Scholar
  74. [74]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  75. [75]
    J. Teschner, On structure constants and fusion rules in the SL(2,C)/SU(2) WZNW model, Nucl. Phys. B 546 (1999) 390 [hep-th/9712256] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  76. [76]
    J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [SPIRES].ADSGoogle Scholar
  77. [77]
    N.P. Warner, Supersymmetry in boundary integrable models, Nucl. Phys. B 450 (1995) 663 [hep-th/9506064] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  78. [78]
    E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [SPIRES].
  79. [79]
    E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [SPIRES].
  80. [80]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  81. [81]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [SPIRES].
  82. [82]
    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].CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2011

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

  • Miranda C. N. Cheng
    • 1
    • 2
  • Robbert Dijkgraaf
    • 3
  • Cumrun Vafa
    • 4
  1. 1.Department of MathematicsHarvard UniversityCambridgeU.S.A.
  2. 2.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  3. 3.Institute for Theoretical Physics & KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  4. 4.Center for Theoretical PhysicsMassachusetts Institute of TechologyCambridgeU.S.A.

Personalised recommendations