Supergeometry of gauge PDE

We study (super)geometry of gauge PDE paying particular attention to globally well-defined definitions and equivalence of such objects. Gauge PDE is a notion that arises by abstracting what physicists call a local gauge field theory (not necessarily Lagrangian) defined in terms of BV-BRST differential. It gives a natural setup for studying global symmetries, conservation laws, deformations, and anomalies of gauge theories. We demonstrate that a natural geometrical language to work with gauge PDEs is that of $Q$-bundles (fiber bundles in the category of $Q$-manifolds) and associated super jet-bundles. In particular, we demonstrate that any gauge PDE can be embedded (at least locally) into a super-jet bundle of the $Q$-bundle. This gives a globally well-defined version of the so-called parent formulation, which in turn can be though of as a certain generalization of Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) sigma models.