Skip to main content
SpringerLink
Log in

Quantum geometry of refined topological strings

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].

    MathSciNet  Google Scholar 

  2. R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].

  3. A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. D. Krefl and J. Walcher, Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. 95 (2011) 67 [arXiv:1007.0263] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. D. Krefl and J. Walcher, Shift versus extension in refined partition functions, arXiv:1010.2635 [INSPIRE].

  6. M.-x. Huang and A. Klemm, Direct integration for general Ω backgrounds, arXiv:1009.1126 [INSPIRE].

  7. N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].

  8. A. Braverman, Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors, math/0401409 [INSPIRE].

  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. A. Negut, Laumon spaces and the Calogero-Sutherland integrable system, Inventiones Mathematicae 178 (2009) 299 [arXiv:0811.4454].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. M. Taki, Surface operator, bubbling Calabi-Yau and AGT relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  16. M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].

  17. E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].

    MathSciNet  Google Scholar 

  18. K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].

  19. H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  21. R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [INSPIRE].

  22. R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].

  23. T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].

  28. N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [INSPIRE].

  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].

    MathSciNet  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  33. D. Gaiotto, Surface operators in N = 2 4d gauge theories, arXiv:0911.1316 [INSPIRE].

  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].

    Article  MathSciNet  ADS  Google Scholar 

  35. T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  37. R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  39. B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, arXiv:math-ph/0702045.

  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].

    Article  MathSciNet  ADS  Google Scholar 

  41. L. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, JHEP 12 (2006) 026 [math-ph/0604014] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. E. Witten, Quantum background independence in string theory, arXiv:hep-th/9306122.

  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].

    Article  MathSciNet  ADS  Google Scholar 

  46. L. Chekhov, B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and quantum algebraic geometry, arXiv:0911.1664.

  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].

    Article  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  49. I.K. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [INSPIRE].

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    MathSciNet  Google Scholar 

  52. N. Nekrasov and S. Shatashvili, Bethe ansatz and supersymmetric vacua, AIP Conf. Proc. 1134 (2009) 154 [INSPIRE].

    Article  ADS  Google Scholar 

  53. N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192–193 (2009) 91 [arXiv:0901.4744] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  54. N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. A.A. Gerasimov and S.L. Shatashvili, Two-dimensional gauge theories and quantum integrable systems, arXiv:0711.1472 [INSPIRE].

  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].

    Article  MathSciNet  ADS  Google Scholar 

  58. E. Sklyanin, Separation of variables - New trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  60. A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  62. D.J. Gross and I.R. Klebanov, One-dimensional string theory on a circle, Nucl. Phys. B 344 (1990) 475 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  63. A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. I.H.P. A 39 (1983) 211.

    MathSciNet  Google Scholar 

  64. M. Mariño and P. Putrov, Multi-instantons in large-N matrix quantum mechanics, arXiv:0911.3076 [INSPIRE].

  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].

    Article  Google Scholar 

  66. M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, arXiv:hep-th/0105045.

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  70. M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  71. N.E. Nørlund, Vorlesungen über Differenzenrechnung, Springer, Berlin Germany (1924).

    Google Scholar 

  72. D. Krefl and J. Walcher, unpublished (2010).

  73. M.C. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, U.S.A.

    Mina Aganagic

  2. Center for Theoretical Physics, University of California, Berkeley, U.S.A.

    Mina Aganagic & Daniel Krefl

  3. Department of Mathematics, Harvard University, Cambridge, U.S.A.

    Miranda C. N. Cheng

  4. Department of Physics, Harvard University, Cambridge, U.S.A.

    Miranda C. N. Cheng & Cumrun Vafa

  5. School of Natural Sciences, Institute for Advanced Study, Princeton, U.S.A.

    Miranda C. N. Cheng

  6. Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands

    Robbert Dijkgraaf

Authors
  1. Mina Aganagic
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miranda C. N. Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robbert Dijkgraaf
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daniel Krefl
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Cumrun Vafa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Krefl.

Additional information

ArXiv ePrint: 1105.0630

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aganagic, M., Cheng, M.C.N., Dijkgraaf, R. et al. Quantum geometry of refined topological strings. J. High Energ. Phys. 2012, 19 (2012). https://doi.org/10.1007/JHEP11(2012)019

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2012)019

Keywords

Search

Navigation