Weak gauge PDEs

Gauge PDEs are flexible graded geometrical objects that generalise AKSZ sigma models to the case of local gauge theories. However, aside from specific cases - such as PDEs of finite type or topological field theories - gauge PDEs are inherently infinite-dimensional. It turns out that these objects can be replaced by finite dimensional objects called weak gauge PDEs. Weak gauge PDEs are equipped with a vertical involutive distribution satisfying certain properties, and the nilpotency condition for the homological vector field is relaxed so that it holds modulo this distribution. Moreover, given a weak gauge PDE, it induces a standard jet-bundle BV formulation at the level of equations of motion. In other words, all the information about PDE and its corresponding BV formulation turns out to be encoded in the finite-dimensional graded geometrical object. Examples include scalar field theory and self-dual Yang-Mills theory.