Advertisement

Orientifolding of the ABJ Fermi gas

  • Kazumi OkuyamaEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

The grand partition functions of ABJ theory can be factorized into even and odd parts under the reflection of fermion coordinate in the Fermi gas approach. In some cases, the even/odd part of ABJ grand partition function is equal to that of \( \mathcal{N}=5\;\mathrm{O}(n)\times \mathrm{U}\mathrm{S}\mathrm{p}\left({n}^{\prime}\right) \) theory, hence it is natural to think of the even/odd projection of grand partition function as an orientifolding of ABJ Fermi gas system. By a systematic WKB analysis, we determine the coefficients in the perturbative part of grand potential of such orientifold ABJ theory. We also find the exact form of the first few “half-instanton” corrections coming from the twisted sector of the reflection of fermion coordinate. For the Chern-Simons level k = 2,4,8 we find closed form expressions of the grand partition functions of orientifold ABJ theory, and for k = 2, 4 we prove the functional relations among the grand partition functions conjectured in arXiv:1410.7658.

Keywords

Nonperturbative Effects M-Theory Matrix Models 

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]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
  2. [2]
    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].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact instanton expansion of the ABJM partition function, PTEP 2015 (2015) 11B104 [arXiv:1507.01678] [INSPIRE].
  4. [4]
    S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N}=8 \) Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Grassi, Y. Hatsuda and M. Mariño, Quantization conditions and functional equations in ABJ(M) theories, J. Phys. A 49 (2016) 115401 [arXiv:1410.7658] [INSPIRE].ADSGoogle Scholar
  6. [6]
    P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations, J. Phys. A 32 (1999) L419 [hep-th/9812211] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    A. Voros, Zeta-regularisation for exact-WKB resolution of a general 1D Schrödinger equation, J. Phys. A 45 (2012) 4007 [arXiv:1202.3100] [INSPIRE].MathSciNetGoogle Scholar
  8. [8]
    M. Mezei and S.S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S. Moriyama and T. Nosaka, Superconformal Chern-Simons Partition Functions of Affine D-type Quiver from Fermi Gas, JHEP 09 (2015) 054 [arXiv:1504.07710] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. Assel, N. Drukker and J. Felix, Partition functions of 3d D-quivers and their mirror duals from 1d free fermions, JHEP 08 (2015) 071 [arXiv:1504.07636] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Moriyama and T. Suyama, Instanton Effects in Orientifold ABJM Theory, arXiv:1511.01660 [INSPIRE].
  12. [12]
    K. Okuyama, Probing non-perturbative effects in M-theory on orientifolds, JHEP 01 (2016) 054 [arXiv:1511.02635] [INSPIRE].
  13. [13]
    M. Honda, Exact relations between M2-brane theories with and without Orientifolds, arXiv:1512.04335 [INSPIRE].
  14. [14]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    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
  16. [16]
    H. Awata, S. Hirano and M. Shigemori, The Partition Function of ABJ Theory, PTEP 2013 (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
  17. [17]
    M. Honda, Direct derivation of “mirror” ABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    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
  19. [19]
    S. Matsumoto and S. Moriyama, ABJ Fractional Brane from ABJM Wilson Loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum Geometry of Refined Topological Strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Hanada, M. Honda, Y. Honma, J. Nishimura, S. Shiba and Y. Yoshida, Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, JHEP 05 (2012) 121 [arXiv:1202.5300] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    Y. Hatsuda and K. Okuyama, Resummations and Non-Perturbative Corrections, JHEP 09 (2015) 051 [arXiv:1505.07460] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  24. [24]
    O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 02 (2016) 093 [arXiv:1512.00161] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    C.A. Tracy and H. Widom, Proofs of two conjectures related to the thermodynamic Bethe ansatz, Commun. Math. Phys. 179 (1996) 667 [solv-int/9509003] [INSPIRE].
  28. [28]
    P. Putrov and M. Yamazaki, Exact ABJM Partition Function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S. Sinha and C. Vafa, SO and Sp Chern-Simons at large-N , hep-th/0012136 [INSPIRE].
  30. [30]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Moriyama and T. Suyama, to appear.Google Scholar
  32. [32]
    Y. Hatsuda, Spectral zeta function and non-perturbative effects in ABJM Fermi-gas, JHEP 11 (2015) 086 [arXiv:1503.07883] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    R. Kashaev, M. Mariño and S. Zakany, Matrix models from operators and topological strings, 2, arXiv:1505.02243 [INSPIRE].
  34. [34]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    M.-x. Huang and A. Klemm, Direct integration for general Ω backgrounds, Adv. Theor. Math. Phys. 16 (2012) 805 [arXiv:1009.1126] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Aganagic, V. Bouchard and A. Klemm, Topological Strings and (Almost) Modular Forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    H. Nakajima and K. Yoshioka, Lectures on instanton counting, math/0311058 [INSPIRE].
  38. [38]
    B. Eynard and M. Mariño, A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011) 1181 [arXiv:0810.4273] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    J. Walcher, Evidence for Tadpole Cancellation in the Topological String, Commun. Num. Theor. Phys. 3 (2009) 111 [arXiv:0712.2775] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    D. Krefl, S. Pasquetti and J. Walcher, The Real Topological Vertex at Work, Nucl. Phys. B 833 (2010) 153 [arXiv:0909.1324] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    I. Kostov and N. Orantin, CFT and topological recursion, JHEP 11 (2010) 056 [arXiv:1006.2028] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of PhysicsShinshu UniversityMatsumotoJapan

Personalised recommendations