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

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 N = (1, 1) sigma model on the boundary, together with the Wess-Zumino term, paralleling the approach of Ševera in the bosonic case.