Super AKSZ construction, integral forms, and the 2-dimensional $\mathcal N=(1,1)$ sigma model

TL;DR

This paper extends the AKSZ construction to supermanifolds with integral forms, deriving a super Chern-Simons theory and a 2D =(1,1) sigma model with Wess-Zumino term, linking supergeometry and topological field theories.

Contribution

It introduces a super AKSZ framework with integral forms and applies it to derive super Chern-Simons and =(1,1) sigma models from Courant algebroids.

Findings

Extended AKSZ construction to supermanifolds with integral forms.

Derived super Chern-Simons theory as a BV version.

Obtained 2D =(1,1) sigma model with Wess-Zumino term.

Abstract

We discuss a natural extension of the AKSZ construction to the case where the source is given by a supermanifold with a chosen integral form. We then focus on the special case with the target given by a Courant algebroid. In the simplest case this leads to the BV version of the super Chern-Simons theory, as developed by Grassi-Maccaferri and Cremonini-Grassi. In the case of exact Courant algebroids we derive the 2-dimensional $\mathcal N=(1,1)$ sigma model on the boundary, together with the Wess-Zumino term, paralleling the approach of \v{S}evera in the bosonic case.