Graded geometry of local gauge theories

Abstract: Gauge PDEs generalize AKSZ sigma models to the case of general local gauge theories. Despite being very flexible and invariant these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion of a weak gauge PDE where the nilpotency of the BRST differential is relaxed in a controllable way. In this approach a nontopological local gauge theory can be described in terms of a finite-dimensional geometrical object. Moreover, among such objects one can find a minimal one which is unique in a certain sense. In the case of a Lagrangian system, the respective weak gauge PDE naturally arises from the presymplectic structure. We prove that any weak gauge PDE determines the standard jet-bundle BV formulation of the underlying gauge theory, giving an unambiguous field-theoretical interpretation of these objects. The relation to the covariant phase space and the multisymplectic approaches is also discussed. The formalism is illustrated by a variety of models including (super) gravity, (chiral) Yang-Mills, and a non-Lagrangian self-dual Yang-Mills theory.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( slides | video )

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.