Supersymmetric Chern-Simons theory in the presence of a boundary

Abstract

In this paper we analyze super-Chern-Simons theory in N ¼ 1 superspace formalism, in the presence of a boundary. We modify the Lagrangian for the Chern-Simons theory in such a way that it is supersymmetric even in the presence of a boundary. Also, even though the Chern-Simons theory is not gauge invariant in the presence of a boundary, if it is suitably coupled to a gauged Wess-Zumino-Witten model, then the resultant theory can be made gauge invariant. Thus, by suitably adding extra boundary degrees of freedom, the gauge and supersymmetry variations of the boundary theory exactly cancel the boundary terms generated by the variations of the bulk Chern-Simons theory. We also discuss how this can be applied to the Aharony-Bergman-Jafferis-Maldacena model in N ¼ 1 superspace, and we then describe the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries of the resultant gauge invariant supersymmetric theory.

Polyakov Loops for the ABJ Theory

“…A system of M2-branes ending on other objects in M-theory has been studied by analysing the ABJM theory and the BLG theory on a manifold with boundaries [51,52,53]. Boundary conditions for the M2-branes ending on M5-branes, M9-branes and gravitational waves have been studied [54].…”

Nonanticommutativity in the presence of a boundary

“…Thus, we would need to add another boundary term (or choose some suitable boundary conditions) such that the combined gauge variation of the original theory along with this boundary piece will be gauge invariant. This has been done in the bosonic case [43] and in N = 1 superspace [35,36,37] where it was shown that, not unexpectedly, the Lagrangian on the boundary is (a gauged version of) a WZW model. It is expected that a similar result will hold in N = 2 superspace but the non-Abelian Chern-Simons action is more complicated in this case, and the detailed construction of the boundary action has not yet been performed.…”

“…It may be noted that if the gauge part included Chern-Simons terms then this theory would not be gauge invariant, but gauge invariance could be restored by the addition of further boundary terms which would cancel the boundary piece generated by the gauge transformation. This has been considered in the context of the ABJM model in [35,43,36] in component form, in N = 1 superspace, and for the Abelian case in N = 2 superspace. However, the full boundary action for the non-Abelian N = 2 case has not yet been constructed.…”

“…This has been done for three dimensional theories in N = 1 superspace [33,34]. Such boundary effects for M2-branes have also been analysed in N = 1 superspace [35,36,37]. In this paper we will first generalize these results to a three dimensional theory in N = 2 superspace and then analyse the non-anticommutative deformation of this theory.…”

Near horizon geometry of strings ending on intersecting D8/D4-branes

“…In the dual field theory, this corresponds to putting ABJM theory on a half-space with supersymmetric boundary conditions, possibly coupled to 2d degrees of freedom localized on the boundary. Such supersymmetric boundary conditions were studied in [33][34][35] and dual supergravity solutions were constructed in [31], although a precise holographic dictionary has yet to be constructed. Since the M2-branes end on the M5-branes, the supersymmetry does not necessarily need to be a subgroup of OSp(8|4, R), as there is no way to remove the M5-branes and recover the full M2-brane theory.…”