AKSZ constructions for topological membranes on $G_2$-manifolds

TL;DR

This paper develops two new AKSZ constructions for topological membranes on $G_2$-manifolds, explores their reductions to string models, and connects them through a unified threebrane theory.

Contribution

It introduces two inequivalent AKSZ models for topological membranes on $G_2$-manifolds and links them via a single threebrane theory, expanding the understanding of membrane and string topological models.

Findings

Two distinct AKSZ membrane models on $G_2$-manifolds.

Dimensional reduction yields new AKSZ formulations of the topological A-model.

Reduction of the threebrane theory produces the standard Courant sigma-model.

Abstract

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.