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Journal of High Energy Physics

, 2014:37 | Cite as

Three-sphere free energy for classical gauge groups

  • Márk MezeiEmail author
  • Silviu S. Pufu
Open Access
Article

Abstract

In this note, we calculate the S 3 free energy F of 3-d \( \mathcal{N} \) ≥ 4 supersymmetric gauge theories with U(N), O(N), and USp(2N) gauge groups and matter hypermultiplets in the fundamental and two-index tensor representations. Supersymmetric localization reduces the computation of F to a matrix model that we solve in the large N limit using two different methods. The first method is a saddle point approximation first introduced in [1], which we extend to next-to-leading order in 1/N. The second method generalizes the Fermi gas approach of [2] to theories with symplectic and orthogonal gauge groups, and yields an expression for F valid to all orders in 1/N . In developing the second method, we use a non-trivial generalization of the Cauchy determinant formula.

Keywords

Matrix Models Supersymmetric gauge theory AdS-CFT Correspondence Extended Supersymmetry 

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]
    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
  2. [2]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].Google Scholar
  3. [3]
    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) 1113] [hep-th/9711200] [INSPIRE].ADSzbMATHMathSciNetGoogle Scholar
  4. [4]
    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].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSzbMATHMathSciNetGoogle Scholar
  6. [6]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N} \) = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    D.L. Jafferis, Quantum corrections to \( \mathcal{N} \) = 2 Chern-Simons theories with flavor and their AdS 4 duals, JHEP 08 (2013) 046 [arXiv:0911.4324] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. Gaiotto and D.L. Jafferis, Notes on adding D6 branes wrapping RP 3 in AdS 4× CP 3, JHEP 11 (2012) 015 [arXiv:0903.2175] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    D.L. Jafferis and A. Tomasiello, A simple class of \( \mathcal{N} \) = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    S. Franco, I.R. Klebanov and D. Rodriguez-Gomez, M2-branes on Orbifolds of the Cone over Q 1,1,1, JHEP 08 (2009) 033 [arXiv:0903.3231] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    D. Martelli and J. Sparks, AdS 4 /CFT 3 duals from M2-branes at hypersurface singularities and their deformations, JHEP 12 (2009) 017 [arXiv:0909.2036] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    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
  15. [15]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    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
  17. [17]
    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
  18. [18]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: \( \mathcal{N} \) = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F -theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].ADSGoogle Scholar
  22. [22]
    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
  23. [23]
    O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CF T 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    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
  25. [25]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    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
  27. [27]
    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
  28. [28]
    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
  29. [29]
    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
  30. [30]
    D.R. Gulotta, C.P. Herzog and S.S. Pufu, Operator counting and eigenvalue distributions for 3D supersymmetric gauge theories, JHEP 11 (2011) 149 [arXiv:1106.5484] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    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
  32. [32]
    M. Mariño and P. Putrov, Interacting fermions and N = 2 Chern-Simons-matter theories, JHEP 11 (2013) 199 [arXiv:1206.6346] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    Y. Hyakutake, Y. Imamura and S. Sugimoto, Orientifold planes, type-I Wilson lines and nonBPS D-branes, JHEP 08 (2000) 043 [hep-th/0007012] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    J. de Boer et al., Triples, fluxes and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].Google Scholar
  35. [35]
    E.G. Gimon and J. Polchinski, Consistency conditions for orientifolds and d manifolds, Phys. Rev. D 54 (1996) 1667 [hep-th/9601038] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    A. Sen, Kaluza-Klein dyons in string theory, Phys. Rev. Lett. 79 (1997) 1619 [hep-th/9705212] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    O. Bergman, E.G. Gimon and S. Sugimoto, Orientifolds, RR torsion and k-theory, JHEP 05 (2001) 047 [hep-th/0103183] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    N. Seiberg, IR dynamics on branes and space-time geometry, Phys. Lett. B 384 (1996) 81 [hep-th/9606017] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  41. [41]
    A. Sen, A note on enhanced gauge symmetries in M and string theory, JHEP 09 (1997) 001 [hep-th/9707123] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    K. Landsteiner and E. Lopez, New curves from branes, Nucl. Phys. B 516 (1998) 273 [hep-th/9708118] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].ADSGoogle Scholar
  44. [44]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, math/0008184.
  46. [46]
    D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  47. [47]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Center for Theoretical Physics, Massachusetts Institute of TechnologyCambridgeU.S.A

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