Seminar talk, 17 November 2021

From Geometry of Differential Equations
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Speaker: Maxim Grigoriev

Title: Presymplectic gauge PDEs and Batalin-Vilkovisky formalism

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 Lagranian) in terms of BV-BRST formalism. Although gauge PDE can be defined as a PDE equipped with extra structures, the generalization is not entirely straightforward as, for instance, two gauge PDEs can be equivalent even if the underlying PDEs are not. As far as Lagrangian gauge systems are concerned the powerful framework is provided by the BV formalism on jet-bundles. However, just like in the case of usual PDEs it is difficult to encode the BV extension of the Lagrangian in terms of the intrinsic geometry of the equation manifold while working on jet-bundles is often very restrictive, especially in analyzing boundary behaviour, e.g., in the context of AdS/CFT correspondence. We show that BV Lagrangian (or its weaker analogs) can be encoded in the compatible graded presymplectic structure on the gauge PDE. In the case of genuine Lagrangian systems this presymplectic structure is related to a certain completion of the canonical BV symplectic structure. A presymplectic gauge PDE gives rise to a BV formulation of the underlying system through an appropriate generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) sigma-model construction followed by taking the symplectic quotient. The construction is illustrated on the standard examples of gauge theories with particular emphasis on the Einstein gravity, where this naturally leads to an elegant presymplectic AKSZ representation of the BV extension of the Cartan-Weyl formulation of gravity.