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Journal of High Energy Physics

, 2014:164 | Cite as

Partition functions of superconformal Chern-Simons theories from Fermi gas approach

  • Sanefumi Moriyama
  • Tomoki NosakaEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We study the partition function of three-dimensional \( \mathcal{N}=4 \) superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes. In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as Z(N) = e A C −1/3Ai[C −1/3(NB)] with coefficients C, B and A and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves. We find that many of the interesting properties in the ABJM theory are extended to the general superconformal Chern-Simons theories. Especially, we find an explicit expression of B for general \( \mathcal{N}=4 \) theories, a conjectural form of A for a special class of theories, and cancellation in the non-perturbative coefficients for the simplest theory next to the ABJM theory.

Keywords

Supersymmetric gauge theory Matrix Models Chern-Simons Theories M-Theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].ADSMathSciNetCrossRefGoogle 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. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSzbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Mariño and P. Putrov, Exact Results in ABJM Theory from Topological Strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    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
  11. [11]
    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
  12. [12]
    N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    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
  15. [15]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].Google Scholar
  16. [16]
    M. Hanada, M. Honda, Y. Honma, J. Nishimura, S. Shiba et al., Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, JHEP 05 (2012) 121 [arXiv:1202.5300] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A. Klemm, M. Mariño, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [INSPIRE].
  18. [18]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact Results on the ABJM Fermi Gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    P. Putrov and M. Yamazaki, Exact ABJM Partition Function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton Effects in ABJM Theory from Fermi Gas Approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton Bound States in ABJM Theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Grassi, J. Kallen and M. Mariño, The topological open string wavefunction, arXiv:1304.6097 [INSPIRE].
  24. [24]
    Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    Y. Hatsuda, M. Honda, S. Moriyama and K. Okuyama, ABJM Wilson Loops in Arbitrary Representations, JHEP 10 (2013) 168 [arXiv:1306.4297] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
  27. [27]
    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].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    A. Cagnazzo, D. Sorokin and L. Wulff, String instanton in AdS 4 × CP 3, JHEP 05 (2010) 009 [arXiv:0911.5228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    B.M. Zupnik and D.V. Khetselius, Three-dimensional extended supersymmetry in the harmonic superspace. (In Russian), Sov. J. Nucl. Phys. 47 (1988) 730 [Yad. Fiz. 47 (1988) 1147] [INSPIRE].
  30. [30]
    H.-C. Kao and K.-M. Lee, Selfdual Chern-Simons systems with an N = 3 extended supersymmetry, Phys. Rev. D 46 (1992) 4691 [hep-th/9205115] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    H.-C. Kao, K.-M. Lee and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories, Phys. Lett. B 373 (1996) 94 [hep-th/9506170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-Matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.L. Jafferis and A. Tomasiello, A Simple class of N = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The theta-Angle in N = 4 Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets, JHEP 07 (2008) 091 [arXiv:0805.3662] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    M. Benna, I. Klebanov, T. Klose and M. Smedback, Superconformal Chern-Simons Theories and AdS 4 /CFT 3 Correspondence, JHEP 09 (2008) 072 [arXiv:0806.1519] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    Y. Imamura and K. Kimura, On the moduli space of elliptic Maxwell-Chern-Simons theories, Prog. Theor. Phys. 120 (2008) 509 [arXiv:0806.3727] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  38. [38]
    S. Terashima and F. Yagi, Orbifolding the Membrane Action, JHEP 12 (2008) 041 [arXiv:0807.0368] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    Y. Imamura and K. Kimura, N = 4 Chern-Simons theories with auxiliary vector multiplets, JHEP 10 (2008) 040 [arXiv:0807.2144] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    R.C. Santamaria, M. Mariño and P. Putrov, Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories, JHEP 10 (2011) 139 [arXiv:1011.6281] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    D. Martelli and J. Sparks, The large-N limit of quiver matrix models and Sasaki-Einstein manifolds, Phys. Rev. D 84 (2011) 046008 [arXiv:1102.5289] [INSPIRE].ADSGoogle Scholar
  42. [42]
    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] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    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].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    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].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    T. Suyama, Eigenvalue Distributions in Matrix Models for Chern-Simons-matter Theories, Nucl. Phys. B 856 (2012) 497 [arXiv:1106.3147] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    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].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    D.R. Gulotta, C.P. Herzog and T. Nishioka, The ABCDEFs of Matrix Models for Supersymmetric Chern-Simons Theories, JHEP 04 (2012) 138 [arXiv:1201.6360] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Mariño and P. Putrov, Interacting fermions and N = 2 Chern-Simons-matter theories, JHEP 11 (2013) 199 [arXiv:1206.6346] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    T. Suyama, On Large-N Solution of N = 3 Chern-Simons-adjoint Theories, Nucl. Phys. B 867 (2013) 887 [arXiv:1208.2096] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    P.M. Crichigno, C.P. Herzog and D. Jain, Free Energy of D n Quiver Chern-Simons Theories, JHEP 03 (2013) 039 [arXiv:1211.1388] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    T. Suyama, A Systematic Study on Matrix Models for Chern-Simons-matter Theories, Nucl. Phys. B 874 (2013) 528 [arXiv:1304.7831] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    M. Mezei and S.S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Grassi and M. Mariño, M-theoretic matrix models, arXiv:1403.4276 [INSPIRE].
  54. [54]
    L. Anderson and K. Zarembo, Quantum Phase Transitions in Mass-Deformed ABJM Matrix Model, JHEP 09 (2014) 021 [arXiv:1406.3366] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    M. Honda and S. Moriyama, Instanton Effects in Orbifold ABJM Theory, JHEP 08 (2014) 091 [arXiv:1404.0676] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    S. Matsumoto and S. Moriyama, ABJ Fractional Brane from ABJM Wilson Loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N = 5, 6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds, JHEP 09 (2008) 002 [arXiv:0806.4977] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    H. Awata, S. Hirano and M. Shigemori, The Partition Function of ABJ Theory, Prog. Theor. Exp. Phys. (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
  61. [61]
    M. Honda, Direct derivation ofmirrorABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    M. Honda and K. Okuyama, Exact results on ABJ theory and the refined topological string, JHEP 08 (2014) 148 [arXiv:1405.3653] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    S. Hirano, K. Nii and M. Shigemori, ABJ Wilson loops and Seiberg Duality, arXiv:1406.4141 [INSPIRE].
  64. [64]
    J. Kallen, The spectral problem of the ABJ Fermi gas, arXiv:1407.0625 [INSPIRE].
  65. [65]
    M.-x. Huang and X.-f. Wang, Topological Strings and Quantum Spectral Problems, JHEP 09 (2014) 150 [arXiv:1406.6178] [INSPIRE].
  66. [66]
    Y. Imamura and S. Yokoyama, N=4 Chern-Simons theories and wrapped M-branes in their gravity duals, Prog. Theor. Phys. 121 (2009) 915 [arXiv:0812.1331] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  67. [67]
    S. Bhattacharyya, A. Grassi, M. Mariño and A. Sen, A One-Loop Test of Quantum Supergravity, Class. Quant. Grav. 31 (2014) 015012 [arXiv:1210.6057] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS 4 /CFT 3 holography, JHEP 1410 (2014) 90 [arXiv:1406.0505] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Kobayashi Maskawa Institute & Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

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