Exact results for Wilson loops in superconformal Chern-Simons theories with matter

  • Anton Kapustin
  • Brian WillettEmail author
  • Itamar Yaakov


We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any \(\mathcal{N}= 2 \) theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.


Supersymmetric gauge theory Matrix Models AdS-CFT Correspondence Chern-Simons Theories 


  1. [1]
    D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the theta-angle in N = 4 super Yang-Mills theory, arXiv:0804.2907 [SPIRES].
  2. [2]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [SPIRES].MathSciNetADSGoogle Scholar
  4. [4]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
  5. [5]
    J.H. Schwarz, Superconformal Chern-Simons theories, JHEP 11 (2004) 078 [hep-th/0411077] [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    M. Mariño, Chern-Simons theory, matrix integrals and perturbative three-manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [SPIRES].CrossRefADSGoogle Scholar
  7. [7]
    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP 02 (2004) 010 [hep-th/0211098] [SPIRES].CrossRefADSGoogle Scholar
  8. [8]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351.zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. [9]
    S.-J. Rey, T. Suyama and S. Yamaguchi, Wilson loops in superconformal Chern-Simons theory and fundamental strings in anti-de Sitter supergravity dual, JHEP 03 (2009) 127 [arXiv:0809.3786] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    N. Drukker, J. Plefka and D. Young, Wilson loops in 3-dimensional N = 6 supersymmetric Chern-Simons theory and their string theory duals, JHEP 11 (2008) 019 [arXiv:0809.2787] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    B. Chen and J.-B. Wu, Supersymmetric Wilson loops in N = 6 super Chern-Simons-matter theory, Nucl. Phys. B 825 (2010) 38 [arXiv:0809.2863] [SPIRES].CrossRefADSGoogle Scholar
  12. [12]
    M. Mariño, Les Houches lectures on matrix models and topological strings, hep-th/0410165 [SPIRES].
  13. [13]
    D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    H. Lü, C.N. Pope and J. Rahmfeld, A construction of killing spinors on S n, J. Math. Phys. 40 (1999) 4518 [hep-th/9805151] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. [15]
    Y. Fujii and K. Yamagishi, Killing spinors on spheres and hyperbolic manifolds, J. Math. Phys. 27 (1986) 979 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. [16]
    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn/deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [SPIRES].zbMATHMathSciNetGoogle Scholar
  17. [17]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaU.S.A.

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