A New Three-Algebra Representation for the {\cal N}=6 su(N)Xsu(N) Superconformal Chern-Simons Theory

TL;DR

This paper introduces a new three-algebra framework for an { m N}=6 superconformal Chern-Simons theory with su(N)×su(N) symmetry, closely related to the ABJM model, providing insights into M2-brane dynamics.

Contribution

It develops u(N)-based extended three-algebras that reproduce known structures and constructs an explicit SO(8) invariant su(N)×su(N) Chern-Simons action for M2-branes.

Findings

Constructed an explicit SO(8) invariant action.

Reproduced the Bagger-Lambert algebra for N=2.

Connected the theory to the low-energy limit of N M2-branes.

Abstract

Based on the realization of three-algebras in terms of algebra of matrices and four-brackets [arXiv:0807.1570] we present the notion of u(N)-based extended three-algebras, which for N=2 reproduces the Bagger-Lambert three-algebra. Using these extended three-algebras we construct an su(N)\times su(N) Chern-Simons action with explicit SO(8) invariance. The dynamical fields of this theory are eight complex valued bosonic and fermionic fields in the bi-fundamental representation of the su(N)\times su(N). For generic N the fermionic transformations, however, close only on a subclass of the states of this theory onto the 3d, {\cal N}=6 superalgebra. In this sector we deal with four complex valued scalars and fermions, our theory is closely related to the ABJM model [arXiv:0806.1218], and hence it can be viewed as the (low energy effective) theory of N M2-branes. We discuss that our three-algebra structure suggests a picture of open M2-brane stretched between any two pairs of M2-branes. We also analyze the BPS configurations of our model.