Graded Contact Geometry and the AKSZ Formalism

TL;DR

This work develops a graded-contact analogue of the AKSZ formalism by targeting differential graded contact manifolds, showing that the field space inherits a graded weak contact structure and that a Jacobi-bracket framework yields a master-equation analogue. In low degrees, the construction recovers the Jacobi sigma model at $n=1$ and a 3D theory associated to Courant-Jacobi algebroids at $n=2$, with the classical action recovered by restricting to zero-degree fields. The approach unifies and explains Jacobi and Courant-Jacobi structures within a direct, geometrically natural AKSZ-type framework and clarifies the role of symplectization in relating to ordinary AKSZ theories. The results provide explicit actions and brackets, connecting graded geometry, transgression, and BV/MASTER equation formalisms to concrete topological theories.

Abstract

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.

Theorems & Definitions (26)

• Definition 2.1
• Proposition 2.2: Geiges
• Lemma 2.3: Lemma 3.49 in mcduff_salamon
• Proposition 2.4
• Proposition 2.5
• Example 3.1
• Example 3.2: Euler vector field
• Definition 3.3
• Example 3.4
• Example 3.5