Journal of High Energy Physics

, 2012:19 | Cite as

Quantum geometry of refined topological strings

  • Mina Aganagic
  • Miranda C. N. Cheng
  • Robbert Dijkgraaf
  • Daniel KreflEmail author
  • Cumrun Vafa


We consider branes in refined topological strings. We argue that their wavefunctions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schrödinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and \( \mathcal{N}=2 \) gauge systems in four dimensions observed by Nekrasov and Shatashvili.


Topological Strings Matrix Models 


  1. [1]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].MathSciNetGoogle Scholar
  2. [2]
    R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
  3. [3]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    D. Krefl and J. Walcher, Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. 95 (2011) 67 [arXiv:1007.0263] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. [5]
    D. Krefl and J. Walcher, Shift versus extension in refined partition functions, arXiv:1010.2635 [INSPIRE].
  6. [6]
    M.-x. Huang and A. Klemm, Direct integration for general Ω backgrounds, arXiv:1009.1126 [INSPIRE].
  7. [7]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
  8. [8]
    A. Braverman, Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors, math/0401409 [INSPIRE].
  9. [9]
    A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential, math/0409441 [INSPIRE].
  10. [10]
    A. Negut, Laumon spaces and the Calogero-Sutherland integrable system, Inventiones Mathematicae 178 (2009) 299 [arXiv:0811.4454].MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    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] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  12. [12]
    A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of β-ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. [13]
    M. Taki, Surface operator, bubbling Calabi-Yau and AGT relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    N. Dorey, S. Lee and T.J. Hollowood, Quantization of integrable systems and a 2d/4d duality, JHEP 10 (2011) 077 [arXiv:1103.5726] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].
  17. [17]
    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].MathSciNetGoogle Scholar
  18. [18]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
  19. [19]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  21. [21]
    R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [INSPIRE].
  22. [22]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
  23. [23]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    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].MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. [27]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
  28. [28]
    N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [INSPIRE].
  31. [31]
    A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2004) 457 [hep-th/0212279] [INSPIRE].MathSciNetGoogle Scholar
  32. [32]
    A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  33. [33]
    D. Gaiotto, Surface operators in N = 2 4d gauge theories, arXiv:0911.1316 [INSPIRE].
  34. [34]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. [36]
    C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    F. Cachazo, K.A. Intriligator and C. Vafa, A large-N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, arXiv:math-ph/0702045.
  40. [40]
    L. Chekhov and B. Eynard, Hermitean matrix model free energy: Feynman graph technique for all genera, JHEP 03 (2006) 014 [hep-th/0504116] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    L. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, JHEP 12 (2006) 026 [math-ph/0604014] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    A. Brini, M. Mariño and S. Stevan, The uses of the refined matrix model recursion, J. Math. Phys. 52 (2011) 052305 [arXiv:1010.1210] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. [44]
    E. Witten, Quantum background independence in string theory, arXiv:hep-th/9306122.
  45. [45]
    B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and non-commutative algebraic geometry, JHEP 03 (2009) 094 [arXiv:0809.3367] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    L. Chekhov, B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and quantum algebraic geometry, arXiv:0911.1664.
  47. [47]
    L. Chekhov, B. Eynard and O. Marchal, Topological expansion of β-ensemble model and quantum algebraic geometry in the sectorwise approach, Theor. Math. Phys. 166 (2011) 141 [arXiv:1009.6007] [INSPIRE].CrossRefGoogle Scholar
  48. [48]
    A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    I.K. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [INSPIRE].
  50. [50]
    L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  51. [51]
    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].MathSciNetGoogle Scholar
  52. [52]
    N. Nekrasov and S. Shatashvili, Bethe ansatz and supersymmetric vacua, AIP Conf. Proc. 1134 (2009) 154 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192193 (2009) 91 [arXiv:0901.4744] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  54. [54]
    N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  55. [55]
    A.A. Gerasimov and S.L. Shatashvili, Higgs bundles, gauge theories and quantum groups, Commun. Math. Phys. 277 (2008) 323 [hep-th/0609024] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. [56]
    A.A. Gerasimov and S.L. Shatashvili, Two-dimensional gauge theories and quantum integrable systems, arXiv:0711.1472 [INSPIRE].
  57. [57]
    H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A new 2d/4d duality via integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    E. Sklyanin, Separation of variables - New trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    O. Babelon and M. Talon, Riemann surfaces, separation of variables and classical and quantum integrability, Phys. Lett. A 312 (2003) 71 [hep-th/0209071] [INSPIRE].MathSciNetADSGoogle Scholar
  60. [60]
    A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE].MathSciNetADSGoogle Scholar
  62. [62]
    D.J. Gross and I.R. Klebanov, One-dimensional string theory on a circle, Nucl. Phys. B 344 (1990) 475 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. I.H.P. A 39 (1983) 211.MathSciNetGoogle Scholar
  64. [64]
    M. Mariño and P. Putrov, Multi-instantons in large-N matrix quantum mechanics, arXiv:0911.3076 [INSPIRE].
  65. [65]
    A. Mironov, A. Morozov, A. Popolitov and S. Shakirov, Resolvents and Seiberg-Witten representation for gaussian β-ensemble, Theor. Math. Phys. 171 (2012) 505 [arXiv:1103.5470] [INSPIRE].CrossRefGoogle Scholar
  66. [66]
    M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, arXiv:hep-th/0105045.
  67. [67]
    A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP 08 (2010) 066 [arXiv:1004.2917] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [INSPIRE].MathSciNetADSGoogle Scholar
  70. [70]
    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  71. [71]
    N.E. Nørlund, Vorlesungen über Differenzenrechnung, Springer, Berlin Germany (1924).Google Scholar
  72. [72]
    D. Krefl and J. Walcher, unpublished (2010).Google Scholar
  73. [73]
    M.C. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  74. [74]
    R. Schiappa and N. Wyllard, An A r threesome: matrix models, 2D CFTs and 4d N = 2 gauge theories, J. Math. Phys. 51 (2010) 082304 [arXiv:0911.5337] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Mina Aganagic
    • 1
    • 2
  • Miranda C. N. Cheng
    • 3
    • 4
    • 5
  • Robbert Dijkgraaf
    • 6
  • Daniel Krefl
    • 2
    Email author
  • Cumrun Vafa
    • 4
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Center for Theoretical PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  3. 3.Department of MathematicsHarvard UniversityCambridgeU.S.A.
  4. 4.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  5. 5.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  6. 6.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

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