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Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant

  • Masanori Hanada
  • Masazumi HondaEmail author
  • Yoshinori Honma
  • Jun Nishimura
  • Shotaro Shiba
  • Yutaka Yoshida
Article
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Abstract

We show that the ABJM theory, which is an \( \mathcal{N} = {6} \) superconformal U(N) × U(N) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS4/CFT3 duality at the quantum level. Here we calculate the free energy, and confirm the N 3/2 scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models.

Keywords

Gauge-gravity correspondence AdS-CFT Correspondence M-Theory 
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References

  1. [1]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    J.H. Schwarz, Superconformal Chern-Simons theories, JHEP 11 (2004) 078 [hep-th/0411077] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    O. Aharony, O. Bergman, D.L. Jafferis and J.M. Maldacena, \( \mathcal{N} = {6} \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  7. [7]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  8. [8]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  9. [9]
    N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, Supergravity and the large-N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004 [hep-th/9802042] [INSPIRE].MathSciNetADSGoogle Scholar
  10. [10]
    T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    W. Bietenholz and J. Nishimura, Ginsparg-Wilson fermions in odd dimensions, JHEP 07 (2001) 015 [hep-lat/0012020] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    W. Bietenholz and P. Sodano, A Ginsparg-Wilson approach to lattice Chern-Simons theory, hep-lat/0305006 [INSPIRE].
  13. [13]
    J. Giedt, Progress in four-dimensional lattice supersymmetry, Int. J. Mod. Phys. A 24 (2009) 4045 [arXiv:0903.2443] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    M. Hanada, L. Mannelli and Y. Matsuo, Large-N reduced models of supersymmetric quiver, Chern-Simons gauge theories and ABJM, JHEP 11 (2009) 087 [arXiv:0907.4937] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, N = 4 super Yang-Mills from the plane wave matrix model, Phys. Rev. D 78 (2008) 106001 [arXiv:0807.2352] [INSPIRE].MathSciNetADSGoogle Scholar
  16. [16]
    H. Kawai, S. Shimasaki and A. Tsuchiya, Large-N reduction on group manifolds, Int. J. Mod. Phys. A 25 (2010) 3389 [arXiv:0912.1456] [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    M. Honda, G. Ishiki, J. Nishimura and A. Tsuchiya, Testing the AdS/CFT correspondence by Monte Carlo calculation of BPS and non-BPS Wilson loops in 4d N = 4 super-Yang-Mills theory, PoS(Lattice 2011)244 [arXiv:1112.4274] [INSPIRE].
  18. [18]
    J. Nishimura, Non-lattice simulation of supersymmetric gauge theories as a probe to quantum black holes and strings, PoS(LAT2009)016 [arXiv:0912.0327] [INSPIRE].
  19. [19]
    M. Honda, G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Supersymmetry non-renormalization theorem from a computer and the AdS/CFT correspondence, PoS(Lattice 2010)253 [arXiv:1011.3904] [INSPIRE].
  20. [20]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [INSPIRE].
  21. [21]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    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] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. [23]
    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].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, arXiv:1012.3210 [INSPIRE].
  25. [25]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [arXiv:0903.4172] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    Y. Imamura, D. Yokoyama and S. Yokoyama, Superconformal index for large-N quiver Chern-Simons theories, JHEP 08 (2011) 011 [arXiv:1102.0621] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].ADSGoogle Scholar
  30. [30]
    N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    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].ADSzbMATHCrossRefGoogle Scholar
  33. [33]
    C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].ADSGoogle Scholar
  34. [34]
    N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].ADSGoogle Scholar
  36. [36]
    H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    K. Okuyama, A note on the partition function of ABJM theory on S 3, Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  38. [38]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. (2012) P03001 [arXiv:1110.4066] [INSPIRE].
  39. [39]
    C. Kristjansen, M. Orselli and K. Zoubos, Non-planar ABJM theory and integrability, JHEP 03 (2009) 037 [arXiv:0811.2150] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    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
  41. [41]
    C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, math.AG/9810173.
  42. [42]
    M. Mariño, S. Pasquetti and P. Putrov, Large-N duality beyond the genus expansion, JHEP 07 (2010) 074 [arXiv:0911.4692] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    R. Dijkgraaf and C. Vafa, N = 1 supersymmetry, deconstruction and bosonic gauge theories, hep-th/0302011 [INSPIRE].
  46. [46]
    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
  47. [47]
    R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].MathSciNetADSGoogle Scholar
  48. [48]
    I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    A. Cagnazzo, D. Sorokin and L. Wulff, String instanton in AdS 4 × C P 3, JHEP 05 (2010) 009 [arXiv:0911.5228] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    H. Ooguri, C. Vafa and E.P. Verlinde, Hartle-Hawking wave-function for flux compactifications, Lett. Math. Phys. 74 (2005) 311 [hep-th/0502211] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  51. [51]
    J. Ambjørn, L. Chekhov, C.F. Kristjansen and Y. Makeenko, Matrix model calculations beyond the spherical limit, Nucl. Phys. B 404 (1993) 127 [Erratum ibid. B 449 (1995) 681] [hep-th/9302014] [INSPIRE].
  52. [52]
    G. Akemann, Higher genus correlators for the Hermitian matrix model with multiple cuts, Nucl. Phys. B 482 (1996) 403 [hep-th/9606004] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    N. Kawahara, J. Nishimura and A. Yamaguchi, Monte Carlo approach to nonperturbative strings — demonstration in noncritical string theory, JHEP 06 (2007) 076 [hep-th/0703209] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    W. Krauth, H. Nicolai and M. Staudacher, Monte Carlo approach to M-theory, Phys. Lett. B 431 (1998) 31 [hep-th/9803117] [INSPIRE].MathSciNetADSGoogle Scholar
  57. [57]
    I. Kanamori, A method for measuring the Witten index using lattice simulation, Nucl. Phys. B 841 (2010) 426 [arXiv:1006.2468] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    N. Halmagyi and V. Yasnov, The spectral curve of the lens space matrix model, JHEP 11 (2009) 104 [hep-th/0311117] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    I.S. Gradshteyn and M. Ryzhik, Table of integrals, series, and products, Academic Press, New York U.S.A. (1980).zbMATHGoogle Scholar
  60. [60]
    M.-X. Huang and A. Klemm, Holomorphic anomaly in gauge theories and matrix models, JHEP 09 (2007) 054 [hep-th/0605195] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    H.J. Rothe, Lattice gauge theories: an introduction, World Sci. Lect. Notes Phys. 74 (2005) 1 [INSPIRE].MathSciNetGoogle Scholar
  63. [63]
    M.A. Clark and A.D. Kennedy, The RHMC algorithm for two flavors of dynamical staggered fermions, Nucl. Phys. Proc. Suppl. 129 (2004) 850 [hep-lat/0309084] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 191602 [arXiv:0811.3102] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Monte Carlo studies of matrix theory correlation functions, Phys. Rev. Lett. 104 (2010) 151601 [arXiv:0911.1623] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Direct test of the gauge-gravity correspondence for Matrix theory correlation functions, JHEP 12 (2011) 020 [arXiv:1108.5153] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    D.R. Gulotta, C.P. Herzog and S.S. Pufu, From necklace quivers to the F-theorem, operator counting and T (U(N)), JHEP 12 (2011) 077 [arXiv:1105.2817] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    D.R. Gulotta, J.P. Ang and C.P. Herzog, Matrix models for supersymmetric Chern-Simons theories with an ADE classification, JHEP 01 (2012) 132 [arXiv:1111.1744] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    D.R. Gulotta, C.P. Herzog and T. Nishioka, The ABCDEF’s of matrix models for supersymmetric Chern-Simons theories, JHEP 04 (2012) 138 [arXiv:1201.6360] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    M. Hanada, C. Hoyos and H. Shimada, On a new type of orbifold equivalence and M-theoretic AdS 4 /CFT 3 duality, Phys. Lett. B 707 (2012) 394 [arXiv:1109.6127] [INSPIRE].MathSciNetADSGoogle Scholar
  71. [71]
    M. Hanada, C. Hoyos and A. Karch, Generating new dualities through the orbifold equivalence: a demonstration in ABJM and four-dimensional quivers, JHEP 01 (2012) 068 [arXiv:1110.3803] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Masanori Hanada
    • 1
    • 2
  • Masazumi Honda
    • 2
    • 3
    Email author
  • Yoshinori Honma
    • 3
  • Jun Nishimura
    • 1
    • 2
    • 3
  • Shotaro Shiba
    • 1
    • 4
  • Yutaka Yoshida
    • 1
  1. 1.High Energy Accelerator Research Organization (KEK)TsukubaJapan
  2. 2.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  3. 3.Department of Particle and Nuclear PhysicsGraduate University for Advanced Studies (SOKENDAI)TsukubaJapan
  4. 4.Blackett LaboratoryImperial College LondonLondonU.K.

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