Relating U(N) × U(N) to SU(N) × SU(N) Chern-Simons membrane theories

Article

Abstract

By integrating out the U(1)B gauge field, we show that the U(n) × U(n) ABJM theory at level k is equivalent to a \( {\mathbb{Z}_k} \) identification of the \( {{\left( {{\text{SU}}(n) \times {\text{SU}}(n)} \right)} \mathord{\left/{\vphantom {{\left( {{\text{SU}}(n) \times {\text{SU}}(n)} \right)} {{\mathbb{Z}_n}}}} \right.} {{\mathbb{Z}_n}}} \) Chern-Simons theory, but only when n and k are coprime. As a consequence, the k = 1 ABJM model for two M2-branes in \( {\mathbb{R}^8} \) can be identified with the \( \mathcal{N} = {{8\left( {{\text{SU}}(2) \times {\text{SU}}(2)} \right)} \mathord{\left/{\vphantom {{8\left( {{\text{SU}}(2) \times {\text{SU}}(2)} \right)} {{\mathbb{Z}_2}}}} \right.} {{\mathbb{Z}_2}}} \) theory. We also conjecture that the U(2) × U(2) ABJM model at k = 2 is equivalent to the \( \mathcal{N} = 8\;{\text{SU}}(2) \times {\text{SU}}(2) \)-theory.

Keywords

M-Theory D-branes 

References

  1. [1]
    J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [SPIRES].MathSciNetADSGoogle Scholar
  2. [2]
    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
  3. [3]
    A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    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
  6. [6]
    N. Lambert and D. Tong, Membranes on an orbifold, Phys. Rev. Lett. 101 (2008) 041602 [arXiv:0804.1114] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    J. Distler, S. Mukhi, C. Papageorgakis and M. Van Raamsdonk, M2-branes on M-folds, JHEP 05 (2008) 038 [arXiv:0804.1256] [SPIRES].CrossRefADSGoogle Scholar
  8. [8]
    J. Bagger and N. Lambert, Three-algebras and N = 6 Chern-Simons gauge theories, Phys. Rev. D 79 (2009) 025002 [arXiv:0807.0163] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    M. Schnabl and Y. Tachikawa, Classification of N = 6 superconformal theories of ABJM type, arXiv:0807.1102 [SPIRES].
  11. [11]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    G.V. Dunne, Aspects of Chern-Simons theory, hep-th/9902115 [SPIRES].
  13. [13]
    S. Terashima, On M5-branes in N = 6 membrane action, JHEP 08 (2008) 080 [arXiv:0807.0197] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    M.A. Bandres, A.E. Lipstein and J.H. Schwarz, Studies of the ABJM theory in a formulation with manifest SU(4) R-symmetry, JHEP 09 (2008) 027 [arXiv:0807.0880] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    A. Gustavsson and S.-J. Rey, Enhanced N = 8 supersymmetry of ABJM theory on R(8) and R(8)/Z(2), arXiv:0906.3568 [SPIRES].
  16. [16]
    O.-K. Kwon, P. Oh and J. Sohn, Notes on supersymmetry enhancement of ABJM theory, JHEP 08 (2009) 093 [arXiv:0906.4333] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    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] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    D. Berenstein and J. Park, The BPS spectrum of monopole operators in ABJM: towards a field theory description of the giant torus, arXiv:0906.3817 [SPIRES].

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonU.K.

Personalised recommendations