Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables

Abstract

Field Theories in Physics can be formulated giving a local Lagrangian density. Locality is imposed using the infinite jet bundle. That bundle is viewed as a pro-finite dimensional smooth manifold and that point of view has been compared to different topological and Frech\'et structures on it. A category of local (insular) manifolds has been constructed. Noether's second theorem is reviewed and the notion of Lie pseudogroups is explored using these concepts. The L-infinity algebra of local observables is defined depending only on the cohomology of the Lagrangian (using a result in multisymplectic manifold which has been extended to the local category). That local pre-multisymplectic form, called the Poincar\'e-Cartan can be thought of as a coordinate free, cohomological version of other similar structures in the field.