Bagger-Lambert Theory for General Lie Algebras

TL;DR

This paper constructs a 3-algebra structure for a superconformal 3d theory describing multiple M2 branes, starting from arbitrary semi-simple Lie algebras, and discusses its properties and issues.

Contribution

It introduces a method to derive totally antisymmetric structure constants for 3-algebras from semi-simple Lie algebras, enabling new superconformal theories for M2 branes.

Findings

Constructed 3-algebra structure constants from semi-simple Lie algebras.

Formulated a maximally superconformal 3d theory with M2 brane degrees of freedom.

Discussed unitarity issues and the relation to 3d Yang-Mills theory.

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.