From current algebras for p-branes to topological M-theory

In this note we generalize a result by Alekseev and Strobl for the case of $p$-branes. We show that there is a relation between anomalous free current algebras and “isotropic” involutive subbundles of $T\oplus {\wedge }^p{T}^{\ast }$ with the Vinogradov bracket, that is a generalization of the Courant bracket. As an application of this construction we go through some interesting examples: topological strings on symplectic manifolds, topological membrane on $G_2$-manifolds and topological 3-brane on $Spin(7)$ manifolds. We show that these peculiar topological theories are related to the physical (i.e., Nambu-Goto) brane theories in a specific way. These topological brane theories are proposed as microscopic description of topological M/F-theories.