M2 to D2 revisited

Abstract

We present two derivations of the multiple D2 action from the multiple M2brane 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 II A superstring.

Supersymmetric Yang-Mills theory from lorentzian three-algebras

“…As already mentioned in [13] [14] [15], the BF membrane model has ghosts in the classical theory, arising from the timelike direction in the "three-algebra". Despite the fact that ghosts cannot propagate in loops due to the special nature of the interactions of the BF membrane model, it remains an important issue to understand whether the theory has negative norm states.…”

“…Our starting point is the action derived in [13][14] [15]. These authors constructed a threealgebra with Lorentzian metric corresponding to an arbitrary Lie algebra G. The action for the corresponding Bagger-Lambert-Gustavsson theory is given by…”

“…5 A proposal for the worldvolume theory of coincident M2-branes in flat space has been made in [13] [14] [15], henceforth called the BF membrane model. This theory is based on a Lorentzian "three-algebra," 6 and its gauge structure is that of a three dimensional BF theory.…”

“…We show that the combined action is symmetric under a BRST 5 The details of the orbifold depend on a discrete parameter that is present in the theory, the level associated with a Chern-Simons term. 6 The fact that the algebras in [13][14] [15] are the unique undecomposable three-algebras has been proven in [17].…”

“…In section 2 we review the main features of the BF membrane model constructed in [13][14] [15]. In section 3 we gauge the shift symmetry in [13][14] [15] and gauge fix all the associated gauge symmetries and show that the gauge fixed action is described by the Lagrangian in [13][14] [15] together with a supersymmetric ghost action. By analyzing the BRST transformations of the gauge fixed action we show that the theory has no negative norm states.…”