  We consider AKSZ constructions of BV actions for closed topological
membranes, and their dimensional reductions to topological string sigma-models.
Two inequivalent AKSZ constructions for topological membranes on
$G_2$-manifolds are proposed, in each of which the two existing topological
membrane theories appear as different gauge fixed versions. Their dimensional
reductions give new AKSZ constructions for the topological A-model, which on
further dimensional reduction gives an AKSZ formulation of supersymmetric
quantum mechanics. We show that the two AKSZ membrane models originate through
worldvolume dimensional reduction of a single AKSZ threebrane theory, which
gives the standard 2-Courant bracket as the underlying derived bracket. Double
dimensional reduction of the twisted topological threebrane theory on a circle
yields the standard Courant sigma-model for string theory with NS-NS flux.
