Bagger-Lambert Theory for General Lie Algebras

TL;DR

We address the problem of obtaining a 3-d maximally supersymmetric conformal field theory describing multiple M2-branes by constructing a family of BL-type theories from arbitrary semi-simple Lie algebras. The method builds 3-algebras A_g with a Lorentzian metric, yielding a BL-like Lagrangian whose gauge sector reduces to a BF theory for an extended Lie algebra G, a semidirect sum of g with abelian generators. A key result is that giving a vacuum expectation value to a scalar field reduces the model to 3-dim maximally supersymmetric Yang-Mills with coupling g_YM = v, plus a decoupled ghost U(1) multiplet after dualizing a scalar; the theory thus connects to D2-brane physics while highlighting unitarity subtleties due to ghost modes. The work provides a broad class of maximally supersymmetric 3-d conformal theories with arbitrary g and reveals an explicit induced Lie algebra G (e.g., iso(3) for g = su(2)) that underlies the BF gauge structure. Further study is needed to fully assess unitarity and to understand the potential role of these theories as effective descriptions of M2-brane dynamics, including large-N regimes.

Abstract

We construct the totally antisymmetric structure constants f^{ABCD} of a 3-algebra with a Lorentzian bi-invariant metric starting from an arbitrary semi-simple Lie algebra. The structure constants f^{ABCD} can be used to write down a maximally superconformal 3d theory that incorporates the expected degrees of freedom of multiple M2 branes, including the "center-of-mass" mode described by free scalar and fermion fields. The gauge field sector reduces to a three dimensional BF term, which underlies the gauge symmetry of the theory. We comment on the issue of unitarity of the quantum theory, which is problematic, despite the fact that the specific form of the interactions prevent the ghost fields from running in the internal lines of any Feynman diagram. Giving an expectation value to one of the scalar fields leads to the maximally supersymmetric 3d Yang-Mills Lagrangian with the addition of two U(1) multiplets, one of them ghost-like, which is decoupled at large g_YM.