Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory

TL;DR

This paper constructs Lorentzian Lie 3-algebras with infinite dimensions, demonstrating their role in supersymmetric models, and relates massive modes to Kaluza-Klein modes in toroidal compactification of M/string theory.

Contribution

It introduces a new class of Lorentzian Lie 3-algebras, including infinite-dimensional examples, and connects these to toroidal compactification and brane dynamics in M/string theory.

Findings

Constructed Lorentzian Lie 3-algebras with arbitrary generator pairs.

Demonstrated that these models describe supersymmetric massive vector multiplets.

Identified massive fields with Kaluza-Klein modes in compactification.

Abstract

We construct a class of Lie 3-algebras with an arbitrary number of pairs of generators with Lorentzian signature metric. Some examples are given and corresponding BLG models are studied. We show that such a system in general describes a supersymmetric massive vector multiplets after the ghost fields are Higgsed. Simple systems with nontrivial interaction are realized by infinite dimensional Lie 3-algebras associated with the loop algebras. The massive fields are then naturally identified with the Kaluza-Klein modes by the toroidal compactification triggered by the ghost fields. For example, Dp-brane with an (infinite dimensional) affine Lie algebra symmetry $\hat g$ can be identified with D(p+1)-brane with gauge symmetry $g$.