Advertisement

Instanton effects in orbifold ABJM theory

  • Masazumi HondaEmail author
  • Sanefumi Moriyama
Open Access
Article

Abstract

We study the partition function of the orbifold ABJM theory on S 3, which is the \( \mathcal{N} \) = 4 necklace quiver Chern-Simons-matter theory with alternating levels, in the Fermi gas formalism. We find that the grand potential of the orbifold ABJM theory is expressed explicitly in terms of that of the ABJM theory. As shown previously, the ABJM grand potential consists of the naive but primary non-oscillatory term and the subsidiary infinitely-replicated oscillatory terms. We find that the subsidiary oscillatory terms of the ABJM theory actually give a non-oscillatory primary term of the orbifold ABJM theory. Also, interestingly, the perturbative part in the ABJM theory results in a novel instanton contribution in the orbifold theory. We also present a physical interpretation for the non-perturbative instanton effects.

Keywords

Supersymmetric gauge theory Matrix Models AdS-CFT Correspondence 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]
    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].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    J.H. Schwarz, Superconformal Chern-Simons theories, JHEP 11 (2004) 078 [hep-th/0411077] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    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].
  4. [4]
    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
  5. [5]
    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].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the θ-angle in N =4 super Yang-Mills theory, JHEP 06 (2010) 097[arXiv:0804.2907] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    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].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    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].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    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].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Terashima and F. Yagi, Orbifolding the membrane action, JHEP 12 (2008) 041 [arXiv:0807.0368] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    Y. Imamura and K. Kimura, N = 4 Chern-Simons theories with auxiliary vector multiplets, JHEP 10 (2008) 040 [arXiv:0807.2144] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    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].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    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
  16. [16]
    N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    K. Okuyama, A note on the partition function of ABJM theory on S 3, Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 03 (2012) P03001 [arXiv:1110.4066] [INSPIRE].Google Scholar
  20. [20]
    M. Hanada 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
  21. [21]
    A. Klemm, M. Mariño, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [INSPIRE].
  22. [22]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    P. Putrov and M. Yamazaki, Exact ABJM partition function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Grassi, J. Kallen and M. Mariño, The topological open string wavefunction, arXiv:1304.6097 [INSPIRE].
  28. [28]
    Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, arXiv:1306.1734 [INSPIRE].
  29. [29]
    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
  30. [30]
    J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
  31. [31]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    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].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  36. [36]
    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
  37. [37]
    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].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    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].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    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].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    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].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    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].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    M. Mariño and P. Putrov, Interacting fermions and N = 2 Chern-Simons-matter theories, JHEP 11 (2013) 199 [arXiv:1206.6346] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M. Mezei and S.S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    A. Grassi and M. Mariño, M-theoretic matrix models, arXiv:1403.4276 [INSPIRE].
  45. [45]
    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
  46. [46]
    T. Suyama, Eigenvalue distributions in matrix models for Chern-Simons-matter theories, Nucl. Phys. B 856 (2012) 497 [arXiv:1106.3147] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  47. [47]
    T. Suyama, On large-N solution of N = 3 Chern-Simons-adjoint theories, Nucl. Phys. B 867 (2013) 887 [arXiv:1208.2096] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    T. Suyama, A systematic study on matrix models for Chern-Simons-matter theories, Nucl. Phys. B 874 (2013) 528 [arXiv:1304.7831] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    A. Cagnazzo, D. Sorokin and L. Wulff, String instanton in AdS 4 × CP 3, JHEP 05 (2010) 009 [arXiv:0911.5228] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  50. [50]
    I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    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
  52. [52]
    S. Matsumoto and S. Moriyama, ABJ fractional brane from ABJM Wilson loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    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
  54. [54]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    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].ADSCrossRefMathSciNetGoogle Scholar
  56. [56]
    H. Awata, S. Hirano and M. Shigemori, The partition function of ABJ theory, Prog. Theor. Exp. Phys. (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
  57. [57]
    M. Honda, Direct derivation ofmirrorABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    E.B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431 [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B 192 (1981) 125 [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    G.V. Dunne and M. Ünsal, Uniform WKB, multi-instantons and resurgent trans-series, arXiv:1401.5202 [INSPIRE].
  61. [61]
    J.G. Russo, A note on perturbation series in supersymmetric gauge theories, JHEP 06 (2012) 038 [arXiv:1203.5061] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteAllahabadIndia
  2. 2.High Energy Accelerator Research Organization (KEK)TsukubaJapan
  3. 3.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  4. 4.Kobayashi Maskawa InstituteNagoya UniversityNagoyaJapan
  5. 5.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

Personalised recommendations