Kac-Moody Extensions of 3-Algebras and M2-branes

TL;DR

This paper explores extensions of 3-algebras relevant to M2-brane theories, introducing infinite dimensional structures and analyzing their properties, including central charges and metric signatures, to better understand gauge symmetries in M2-branes.

Contribution

It constructs new infinite dimensional 3-algebras with various metrics and analyzes finite dimensional 3-algebras with different signatures, revealing novel algebraic structures and their physical implications.

Findings

Central charge appears naturally in the extended algebra.

New algebra with zero eigenvalue metric leads to a decoupled abelian gauge theory.

No negative kinetic terms arise from the new algebra structures.

Abstract

We study the 3-algebraic structure involved in the recently shown M2-branes worldvolume gauge theories. We first extend an arbitrary finite dimensional 3-algebra into an infinite dimensional 3-algebra by adding a mode number to each generator. A unique central charge in the algebra of gauge transformations appears naturally in this extension. We present an infinite dimensional extended 3-algebra with a general metric and also a different extension with a Lorentzian metric. We then study ordinary finite dimensional 3-algebras with different signatures of the metric, focusing on the cases with a negative eigenvalue and the cases with a zero eigenvalue. In the latter cases we present a new algebra, whose corresponding theory is a decoupled abelian gauge theory together with a free theory with global gauge symmetry, and there is no negative kinetic term from this algebra.