  A gauge PDE is a geometrical object underlying what physicists call a local
gauge field theory defined at the level of equations of motion (i.e. without
specifying Lagrangian) in terms of Batalin-Vilkovisky (BV) formalism. This
notion extends the BV formulation in terms of jet-bundles on the one hand and
the geometrical approach to PDEs on the other hand. In this work we concentrate
on gauge PDEs equipped with a compatible presymplectic structure and show that
under some regularity conditions this data defines a jet-bundle BV formulation.
More precisely, the BV jet-bundle arises as the symplectic quotient of the
super jet-bundle of the initial gauge PDE. In this sense, presymplectic gauge
PDEs give an invariant geometrical approach to Lagrangian gauge systems, which
is not limited to jet-bundles. Furthermore, the presymplectic gauge PDE
structure naturally descends to space-time submanifolds (in particular,
boundaries, if any) and, in this respect, is quite similar to AKSZ sigma models
which are long known to have this feature. We also introduce a notion of a weak
presymplectic gauge PDE, where the nilpotency of the differential is replaced
by a presymplectic analog of the BV master equation, and show that it still
defines a local BV system. This allows one to encode BV systems in terms of
finite-dimensional graded geometry, much like the AKSZ construction does in the
case of topological models.
