M2 to D2 revisited

TL;DR

This work addresses deriving the multiple D2-brane action from the BLG M2-brane model by exploiting Lie 3-algebra structures with nonpositive-definite metrics in two complementary ways. The first route introduces a Lie 3-algebra extension of a Lie algebra, and after fixing a constant background $X_0^I=v^I$ (with $v$ often aligned along $X^{10}$), ghosts are removed and the standard D2 action emerges without higher-order terms, maintaining key symmetries. The second route builds on the M5–BLG connection via a Nambu–Poisson structure on a 3-manifold, winds one direction of the M5 along $S^1$, and yields a nonlinear D4 action with infinitesimal noncommutativity; finite-$N$ D2 is obtained by quantizing the Nambu–Poisson bracket on a torus, with the winding mode interpreted as an extra 3-algebra generator. Together, these derivations illuminate how M-theory degrees of freedom reorganize into Type IIA gauge dynamics, provide ghost-free realizations, and suggest a geometric origin for the additional algebraic structures involved.

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.