Advertisement

Journal of High Energy Physics

, 2011:141 | Cite as

Nonperturbative aspects of ABJM theory

  • Nadav Drukker
  • Marcos Mariño
  • Pavel Putrov
Article

Abstract

Using the matrix model which calculates the exact free energy of ABJM theory on \( {\mathbb{S}^3} \) we study non-perturbative effects in the large N expansion of this model, i.e., in the genus expansion of type IIA string theory on AdS4 × \( \mathbb{C}{\mathbb{P}^3} \). We propose a general prescription to extract spacetime instanton actions from general matrix models, in terms of period integrals of the spectral curve, and we use it to determine them explicitly in the ABJM matrix model, as exact functions of the ’t Hooft coupling. We confirm numerically that these instantons control the asymptotic growth of the genus expansion. Furthermore, we find that the dominant instanton action at strong coupling determined in this way exactly matches the action of an Euclidean D2-brane instanton wrapping \( \mathbb{R}{\mathbb{P}^3} \).

Keywords

Matrix Models AdS-CFT Correspondence Nonperturbative Effects Topological Strings 

References

  1. [1]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    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] [INSPIRE].CrossRefADSGoogle Scholar
  5. [5]
    M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].CrossRefADSGoogle Scholar
  6. [6]
    N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    N. Halmagyi and V. Yasnov, The spectral curve of the lens space matrix model, JHEP 11 (2009) 104 [hep-th/0311117] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    B. Haghighat, A. Klemm and M. Rauch, Integrability of the holomorphic anomaly equations, JHEP 10 (2008) 097 [arXiv:0809.1674] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].CrossRefMATHGoogle Scholar
  10. [10]
    S.H. Shenker, The strength of nonperturbative effects in string theory, in Random surfaces and quantum gravity, O. Álvarez, E. Marinari and P. Windey eds., Plenum, New York U.S.A. (1991) 191.Google Scholar
  11. [11]
    J. Polchinski, Combinatorics of boundaries in string theory, Phys. Rev. D 50 (1994) 6041 [hep-th/9407031] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    E.J. Martinec, The annular report on noncritical string theory, hep-th/0305148 [INSPIRE].
  13. [13]
    S.Y. Alexandrov, V.A. Kazakov and D. Kutasov, Nonperturbative effects in matrix models and D-branes, JHEP 09 (2003) 057 [hep-th/0306177] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    E. Brézin, J.-C. Le Guillou and J. Zinn-Justin, Perturbation theory at large order. 2. Role of the vacuum instability, Phys. Rev. D 15 (1977) 1558 [INSPIRE].ADSGoogle Scholar
  15. [15]
    M. Mariño, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, arXiv:0711.1954 [INSPIRE].
  16. [16]
    M. Mariño, R. Schiappa and M. Weiss, Multi-Instantons and multi-cuts, J. Math. Phys. 50 (2009) 052301 [arXiv:0809.2619] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    A. Klemm, M. Mariño and M. Rauch, Direct integration and non-perturbative effects in matrix models, JHEP 10 (2010) 004 [arXiv:1002.3846] [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    O. Aharony, A. Hashimoto, S. Hirano and P. Ouyang, D-brane charges in gravitational duals of 2+1 dimensional gauge theories and duality cascades, JHEP 01 (2010) 072 [arXiv:0906.2390] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    J.T. Liu and R. Minasian, Computing 1/N 2 corrections in AdS/CFT, arXiv:1010.6074 [INSPIRE].
  22. [22]
    W. Taylor, M(atrix) theory: matrix quantum mechanics as a fundamental theory, Rev. Mod. Phys. 73 (2001) 419 [hep-th/0101126] [INSPIRE].CrossRefMATHADSGoogle Scholar
  23. [23]
    F. David, Phases of the large-N matrix model and nonperturbative effects in 2-D gravity, Nucl. Phys. B 348 (1991) 507 [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    F. David, Nonperturbative effects in matrix models and vacua of two-dimensional gravity, Phys. Lett. B 302 (1993) 403 [hep-th/9212106] [INSPIRE].ADSGoogle Scholar
  25. [25]
    N. Seiberg and D. Shih, Branes, rings and matrix models in minimal (super)string theory, JHEP 02 (2004) 021 [hep-th/0312170] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    V.A. Kazakov and I.K. Kostov, Instantons in noncritical strings from the two matrix model, hep-th/0403152 [INSPIRE].
  27. [27]
    J.C. Le Guillou and J. Zinn-Justin, Large order behavior of perturbation theory, North Holland, Amsterdam The Netherlands (1990).Google Scholar
  28. [28]
    S. Pasquetti and R. Schiappa, Borel and Stokes nonperturbative phenomena in topological string theory and c = 1 matrix models, Annales Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [INSPIRE].CrossRefMATHADSMathSciNetGoogle Scholar
  29. [29]
    M. Aganagic, V. Bouchard and A. Klemm, Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [INSPIRE].CrossRefMATHADSMathSciNetGoogle Scholar
  30. [30]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485–486] [hep-th/9407087] [INSPIRE].
  31. [31]
    S. Garoufalidis, A. Its, A. Kapaev and M. Mariño, Asymptotics of the instantons of Painlevé I, arXiv:1002.3634 [INSPIRE].
  32. [32]
    S. Hikami and E. Brézin, Large order behavior of the 1/n expansion in zero-dimensions and one-dimensions, J. Phys. A 12 (1979) 759 [INSPIRE].ADSGoogle Scholar
  33. [33]
    B. Eynard and J. Zinn-Justin, Large order behavior of 2-D gravity coupled to D < 1 matter, Phys. Lett. B 302 (1993) 396 [hep-th/9301004] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].CrossRefADSGoogle Scholar
  35. [35]
    I.R. Klebanov, String theory in two-dimensions, hep-th/9108019 [INSPIRE].
  36. [36]
    D. Ghoshal and C. Vafa, C = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121 [hep-th/9506122] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    A. Polyakov, de Sitter space and eternity, Nucl. Phys. B 797 (2008) 199 [arXiv:0709.2899] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    A. Polyakov, Global warming of De Sitter space, talk presented at Strings, (2008).Google Scholar
  39. [39]
    S. Hohenegger and I. Kirsch, A note on the holography of Chern-Simons matter theories with flavour, JHEP 04 (2009) 129 [arXiv:0903.1730] [INSPIRE].CrossRefADSGoogle Scholar
  40. [40]
    Y. Hikida, W. Li and T. Takayanagi, ABJM with flavors and FQHE, JHEP 07 (2009) 065 [arXiv:0903.2194] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  41. [41]
    P. Koerber and L. Martucci, From ten to four and back again: how to generalize the geometry, JHEP 08 (2007) 059 [arXiv:0707.1038] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  42. [42]
    P. Koerber, Lectures on generalized complex geometry for physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  43. [43]
    N. Drukker, J. Plefka and D. Young, Wilson loops in 3-dimensional N = 6 supersymmetric Chern-Simons theory and their string theory duals, JHEP 11 (2008) 019 [arXiv:0809.2787] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    M. Cvetič, H. Lü and C. Pope, Consistent warped space Kaluza-Klein reductions, half maximal gauged supergravities and CP n constructions, Nucl. Phys. B 597 (2001) 172 [hep-th/0007109] [INSPIRE].CrossRefADSGoogle Scholar
  45. [45]
    T. Nishioka and T. Takayanagi, On type IIA Penrose limit and N = 6 Chern-Simons theories, JHEP 08 (2008) 001 [arXiv:0806.3391] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  46. [46]
    T. Nishioka and T. Takayanagi, Fuzzy ring from M2-brane giant torus, JHEP 10 (2008) 082 [arXiv:0808.2691] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  47. [47]
    M.C. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].CrossRefADSGoogle Scholar
  48. [48]
    M. Mariño, S. Pasquetti and P. Putrov, Large-N duality beyond the genus expansion, JHEP 07 (2010) 074 [arXiv:0911.4692] [INSPIRE].CrossRefADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.The Blackett Laboratory, Imperial College LondonLondonUK
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenèveSwitzerland
  3. 3.Section de Mathématiques, Université de GenèveGenèveSwitzerland

Personalised recommendations