Presymplectic minimal models of local gauge theories

We describe how the BV-AKSZ construction (or, more generally, finite dimensional symplectic gauge PDE) can be extended to generic local gauge field theories including non-topological and non-diffeomorphism-invariant ones. The minimal formulation of this sort has a finite-dimensional target space which is a pre Q-manifold equipped with a compatible presymplectic structure. The nilpotency condition for the homological vector field is replaced with a presymplectic version of the classical BV master equation. Given such a presymplectic BV-AKSZ formulation, it defines a standard jet-bundle BV formulation by taking a symplectic quotient of the respective super jet-bundle. In other words all the information about the underlying PDE, its Lagrangian, and the corresponding BV formulation turns out to be encoded in the finite dimensional graded geometrical object. Standard examples include Yang-Mills, Einstein gravity, conformal gravity etc.