M2 to D2 revisited

TL;DR

This paper revisits the derivation of multiple D2-brane actions from M2-brane models, presenting two methods: one via Lie 3-algebra and another through M5-brane compactification, clarifying the algebraic and geometric origins.

Contribution

It provides two explicit derivations of the multiple D2-brane action from the Bagger-Lambert M2-brane model, connecting algebraic structures and M-theory compactification.

Findings

Derivation from Lie 3-algebra with indefinite metric.

Connection between M2, M5, and D2 via compactification.

Interpretation of the extra generator as M5 winding.

Abstract

We present two derivations of the multiple D2 action from the multiple M2-brane model proposed by Bagger-Lambert and Gustavsson. The first one is to start from Lie 3-algebra associated with given (arbitrary) Lie algebra. The Lie 3-algebra metric is not positive definite but the zero-norm generators merely correspond to Lagrange multipliers. Following the work of Mukhi and Papageorgakis, we derive D2-brane action from the model by giving a variable a vacuum expectation value. The second derivation is based on the correspondence between M2 and M5. We compactify one dimension and wind M5 brane along this direction. This leads to a noncommutative D4 action. Multiple D2 action is then obtained by suitably choosing the non-commutative parameter on the two-torus. It also implies a natural interpretation to the extra generator in Lie 3-algebra, namely the winding of M5 world volume around $S^1$ which defines the reduction of M theory to \IIA superstring.