Journal of High Energy Physics

, 2012:132 | Cite as

Matrix models for supersymmetric Chern-Simons theories with an ADE classification

  • Daniel R. Gulotta
  • J. P. Ang
  • Christopher P. HerzogEmail author


We consider \( \mathcal{N} \) = 3 supersymmetric Chern-Simons (CS) theories that contain product U(N ) gauge groups and bifundamental matter fields. Using the matrix model of Kapustin, Willett and Yaakov, we examine the Euclidean partition function of these theories on an S 3 in the large N limit. We show that the only such CS theories for which the long range forces between the eigenvalues cancel have quivers which are in one-to-one correspondence with the simply laced affine Dynkin diagrams. As the A n series was studied in detail before, in this paper we compute the partition function for the D 4 quiver. The D 4 example gives further evidence for a conjecture that the saddle point eigenvalue distribution is determined by the distribution of gauge invariant chiral operators. We also see that the partition function is invariant under a generalized Seiberg duality for CS theories.


AdS-CFT Correspondence Chern-Simons Theories Duality in Gauge Field Theories 1/N Expansion 


  1. [1]
    S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [arXiv:0903.4172] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    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].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    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
  4. [4]
    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
  5. [5]
    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].ADSCrossRefGoogle Scholar
  6. [6]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, arXiv:1012.3210 [INSPIRE].
  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].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2 N (c)) and U(N (c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    B. Willett and I. Yaakov, N = 2 dualities and Z extremization in three dimensions, arXiv:1104.0487 [INSPIRE].
  11. [11]
    F. Benini, C. Closset and S. Cremonesi, Comments on 3 d Seiberg-like dualities, JHEP 10 (2011)075 [arXiv:1108.5373] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996)164 [hep-th/9604089] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    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
  14. [14]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, arXiv:1110.4066 [INSPIRE].
  15. [15]
    R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].MathSciNetADSGoogle Scholar
  16. [16]
    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].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, arXiv:0807.3720 [INSPIRE].
  18. [18]
    A. Amariti, C. Klare and M. Siani, The large-N limit of toric Chern-Simons matter theories and their duals, arXiv:1111.1723 [INSPIRE].
  19. [19]
    A. Kapustin, D(n) quivers from branes, JHEP 12 (1998) 015 [hep-th/9806238] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    T. Kitao, K. Ohta and N. Ohta, Three-dimensional gauge dynamics from brane configurations with (p, q)-five-brane, Nucl. Phys. B 539 (1999) 79 [hep-th/9808111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    O. Bergman, A. Hanany, A. Karch and B. Kol, Branes and supersymmetry breaking in three-dimensional gauge theories, JHEP 10 (1999) 036 [hep-th/9908075] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A geometric unification of dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].MathSciNetGoogle Scholar
  24. [24]
    N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
  25. [25]
    D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4/CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    D.L. Jafferis and A. Tomasiello, A simple class of N = 3 gauge/gravity duals, JHEP 10 (2008)101 [arXiv:0808.0864] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    C. Albertsson, B. Brinne, U. Lindström, M. Roček and R. von Unge, ADE quiver theories and mirror symmetry, Nucl. Phys. Proc. Suppl. 102 (2001) 3 [hep-th/0103084] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Daniel R. Gulotta
    • 1
  • J. P. Ang
    • 1
  • Christopher P. Herzog
    • 1
    Email author
  1. 1.Department of PhysicsPrinceton UniversityPrincetonU.S.A

Personalised recommendations