Bagger-Lambert theory on an orbifold and its relation to Chern-Simons-matter theories

TL;DR

This work shows how an orbifold projection of the Bagger-Lambert theory, formulated with a 3-algebra realized by the Nambu bracket on a 3-torus, followed by a second truncation to $T^2$, yields a large-$N$ limit that reproduces Chern-Simons-matter theories. In particular, orbifolding by $\mathbb{C}^4/\mathbb{Z}_n$ and matrix regularization lead to a discretized ABJM model with gauge group $U(N)\times U(N)$ and Chern-Simons level $k$, with a matching vacuum moduli space $\text{Sym}_N(\mathbb{C}^4/\mathbb{Z}_k)$. The construction provides a bridge from BL-type 3-algebra dynamics to ABJM-like descriptions of multiple M2-branes, clarifying how higher-algebraic structures can give rise to familiar CS-matter theories in the appropriate large-$N$ limit. The approach also highlights the role of orbifold data and truncations in shaping the effective degrees of freedom and supersymmetry of the resulting theory.

Abstract

We consider how to take an orbifold reduction for the multiple M2-brane theory recently proposed by Bagger and Lambert, and discuss its relation to Chern-Simons theories. Starting from the infinite dimensional 3-algebra realized as the Nambu bracket on a 3-torus, we first suggest an orbifolding prescription for various fields. Then we introduce a second truncation, which effectively reduces the internal space to a 2-torus. Eventually one obtains a large-N limit of Chern-Simons gauge theories coupled to matter fields. We consider an abelian orbifold C^4/Z_n, and illustrate how one can arrive at the N=6 supersymmetric theories with gauge groups U(N) x U(N) and Chern-Simons levels (k,-k), as recently constructed by Aharony, Bergman, Jafferis and Maldacena.