Graded Contact Geometry and the AKSZ Formalism

The AKSZ formalism is a construction of topological field theories where the target spaces are differential graded symplectic manifolds. In this paper, we describe an analogue of the AKSZ formalism where the target spaces are differential graded contact manifolds. We show that the space of fields inherits a weak contact structure, and we construct a solution to the analogue of the classical master equation, defined via the Jacobi bracket. In the $n=1$ case, we recover the Jacobi sigma model, and in the $n=2$ case, we obtain three-dimensional topological field theories associated to Courant-Jacobi algebroids.