AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
Dmitry Roytenberg
TL;DR
The paper develops the AKSZ-BV framework within the language of differential graded manifolds and shows how Courant algebroid data yield a canonical three-dimensional topological membrane theory. By choosing a dg source $N=T[1]N_0$ and a dg symplectic target with a Hamiltonian $\Theta$ satisfying $\{\Theta,\Theta\}=0$, the master action $\mathbf{S}$ is built from a kinetic term and an interaction pulled back from $\Theta$, yielding a BV action with $(\mathbf{S},\mathbf{S})=0$. It unifies lower-dimensional models like the Poisson sigma-model and Chern–Simons within the AKSZ cycle and produces a concrete Courant algebroid–valued closed membrane action, including the comprehensive gauge structure with ghosts and antifields. The work clarifies how 3d TFTs can be constructed from Courant algebroids and provides explicit Master actions that generalize known topological membrane theories.
Abstract
We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz- Zaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model.