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Summing up all genus free energy of ABJM matrix model

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Abstract

The localization technique allows us to compute the free energy of the U(N) k  × U(N)−k Chern-Simons-matter theory dual to type IIA strings on AdS 4 × CP 3 from weak to strong ’t Hooft coupling λ = N/k at finite N, as demonstrated by Drukker, Mariño, and Putrov. In this note we study further the free energy at large ’t Hooft coupling with the aim of testing AdS/CFT at the quantum gravity level and, in particular, sum up allthe1/N corrections, apart from the worldsheet instanton contributions. The all genus partition function takes a remarkably simple form — the Airy function, \( {\text{Ai}}\left( {{{\left( {{{{\pi {k^2}}} \left/ {{\sqrt {2} }} \right.}} \right)}^{{{2} \left/ {3} \right.}}}{\lambda_{\text{ren}}}} \right) \), with the renormalized ’t Hooft coupling λren.

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References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Beisert et al., Review of AdS/CFT Integrability: An Overview, arXiv:1012.3982 [SPIRES].

  3. N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  4. C. Kristjansen, M. Orselli and K. Zoubos, Non-planar ABJM Theory and Integrability, JHEP 03 (2009) 037 [arXiv:0811.2150] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  5. P. Caputa, C. Kristjansen and K. Zoubos, Non-planar ABJ Theory and Parity, Phys. Lett. B 677 (2009) 197 [arXiv:0903.3354] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  6. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].

  7. J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  8. N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  10. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  11. N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, arXiv:1007.3837 [SPIRES].

  12. M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, arXiv:1104.0783 [SPIRES].

  13. N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, arXiv:1103.4844 [SPIRES].

  14. 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] [SPIRES].

    ADS  Google Scholar 

  15. D. Martelli and J. Sparks, The large-N limit of quiver matrix models and Sasaki-Einstein manifolds, arXiv:1102.5289 [SPIRES].

  16. S. Cheon, H. Kim and N. Kim, Calculating the partition function of N = 2 Gauge theories on S 3 and AdS/CFT correspondence, JHEP 05 (2011) 134 [arXiv:1102.5565] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  17. D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, arXiv:1012.3210 [SPIRES].

  18. 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] [SPIRES].

    Article  ADS  Google Scholar 

  19. A. Amariti and M. Siani, Z-extremization and F-theorem in Chern-Simons matter theories, arXiv:1105.0933 [SPIRES].

  20. A. Amariti and M. Siani, F-maximization along the RG flows: a proposal, arXiv:1105.3979 [SPIRES].

  21. Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  22. Y. Imamura, D. Yokoyama and S. Yokoyama, Superconformal index for large-N quiver Chern-Simons theories, arXiv:1102.0621 [SPIRES].

  23. C. Krattenthaler, V.P. Spiridonov and G.S. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [SPIRES].

    Article  ADS  Google Scholar 

  24. A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  25. A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like Duality in Three Dimensions, arXiv:1012.4021 [SPIRES].

  26. M.-x. Huang and A. Klemm, Holomorphicity and Modularity in Seiberg-Witten Theories with Matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  27. O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  28. O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  29. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  30. N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  31. N. Halmagyi and V. Yasnov, The spectral curve of the lens space matrix model, JHEP 11 (2009) 104 [hep-th/0311117] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  32. M. Mariño and P. Putrov, Exact Results in ABJM Theory from Topological Strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [SPIRES].

    Article  ADS  Google Scholar 

  33. M. Mariño, Chern-Simons theory, matrix integrals and perturbative three-manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [SPIRES].

    Article  ADS  Google Scholar 

  34. 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].

    Article  ADS  Google Scholar 

  35. 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] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  37. B. Eynard, M. Mariño and N. Orantin, Holomorphic anomaly and matrix models, JHEP 06 (2007) 058 [hep-th/0702110] [SPIRES].

    Article  ADS  Google Scholar 

  38. B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].

  39. 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] [SPIRES].

    Article  ADS  Google Scholar 

  40. G. Akemann, Higher genus correlators for the Hermitian matrix model with multiple cuts, Nucl. Phys. B 482 (1996) 403 [hep-th/9606004] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  41. M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string/string duality: A one-loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  42. M.B. Green and P. Vanhove, D-instantons, strings and M-theory, Phys. Lett. B 408 (1997) 122 [hep-th/9704145] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  43. M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  44. E. Cremmer and S. Ferrara, Formulation of Eleven-Dimensional Supergravity in Superspace, Phys. Lett. B 91 (1980) 61 [SPIRES].

    ADS  MathSciNet  Google Scholar 

  45. L. Brink and P.S. Howe, Eleven-Dimensional Supergravity on the Mass-Shell in Superspace, Phys. Lett. B 91 (1980) 384 [SPIRES].

    ADS  Google Scholar 

  46. R. Kallosh and A. Rajaraman, Vacua of M-theory and string theory, Phys. Rev. D 58 (1998) 125003 [hep-th/9805041] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  47. A.A. Tseytlin, R 4 terms in 11 dimensions and conformal anomaly of (2, 0) theory, Nucl. Phys. B 584 (2000) 233 [hep-th/0005072] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  48. B. Allen and S. Davis, Vacuum Energy in Gauged Extended Supergravity, Phys. Lett. B 124 (1983) 353 [SPIRES].

    ADS  Google Scholar 

  49. N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  50. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  51. O. Aharony and E. Witten, Anti-de Sitter space and the center of the gauge group, JHEP 11 (1998) 018 [hep-th/9807205] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  52. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. H. Ooguri, C. Vafa and E.P. Verlinde, Hartle-Hawking wave-function for flux compactifications, Lett. Math. Phys. 74 (2005) 311 [hep-th/0502211] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. M. Mariño, Exact Results and Stringy Effects in ABJM Theory, talk at the conference Strings 2011, Uppsala Sweden, June2011.

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Authors and Affiliations

  1. Department of Physics, Nagoya University, Nagoya, 464-8602, Japan

    Hiroyuki Fuji & Shinji Hirano

  2. Kobayashi Maskawa Institute, Nagoya University, Nagoya, 464-8602, Japan

    Sanefumi Moriyama

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  1. Hiroyuki Fuji
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  2. Shinji Hirano
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  3. Sanefumi Moriyama
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Correspondence to Sanefumi Moriyama.

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ArXiv ePrint: 1106.4631

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Fuji, H., Hirano, S. & Moriyama, S. Summing up all genus free energy of ABJM matrix model. J. High Energ. Phys. 2011, 1 (2011). https://doi.org/10.1007/JHEP08(2011)001

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