Superspace formulation in a three-algebra approach toD=3,N=4, 5 superconformal Chern-Simons matter theories

Abstract

We present a superspace formulation of the D ¼ 3, N ¼ 4, 5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N ¼ 1 superconformal action and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N ¼ 1 to N ¼ 5. By decomposing the N ¼ 5 supermultiplets and the symplectic 3-algebra properly and proposing a new superpotential term, we construct the N ¼ 4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N ¼ 4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N ¼ 4, 5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be re-derived in our 3-algebra approach. All known N ¼ 4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.

OSp(4|4)superconformal currents in three-dimensionalN=4Chern-Simons quiver gauge t

“…In the literature, only the closure of the super Poincaré algebra of the N ¼ 4 Chern-Simons quiver gauge theory has been explicitly checked (in the framework of 3-algebra) [5]. In this section, we will verify the closure of the full OSpð4j4Þ superconformal superalgebra in the framework of Lie 2-algebra.…”

“…into the variation of the fermionic fields, we obtain 3 In our previous work [5], we have examined the closure of the N ¼ 4 super Poincaré algebra in the framework of 3-algebra.…”

“…ac mc b , where mc b are a set of representation matrices of the Lie algebra of gauge symmetry. The untwisted multiplets ðZ a A ; c a _ A Þ and twisted multiplets ðZ a 0 _ A ; c a 0 A Þ satisfy the reality conditions, 5 To avoid introducing too many indices, we still use the capital letters A; B; . .…”

“…(For a 3-algebra approach, see Ref. [5].) To fill this gap, in this paper we derive the law of the N ¼ 4 superconformal transformations and verify the action is invariant under these transformations.…”

“…To our best knowledge, so far only the closure of the N ¼ 4 super Poincaré algebra of the N ¼ 4 theory has been checked (in the framework of 3-algebra) in the literature [5]. Therefore, it is necessary to verify the closure of the full OSpð4j4Þ superconformal superalgebra.…”