Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

Batalin-Vilkovisky (BV) theories are classical field theories where the target space is Z-graded. They are particularly well-suited for gauge symmetry reduction. A BV theory comprises an action functional, a symplectic form, and a homological vector field: from the action functional one obtains the Euler-Lagrange (EL) field equations, the homological vector field encodes the gauge symmetries of the theory, and the symplectic form captures their interrelationship. Under physically reasonable assumptions, these data allows for a nice cohomological description of the tangent space to the so-called EL-moduli space (the space of solutions to the EL equations, modulo gauge symmetries), at a smooth point.In this two-parts seminar (based on the paper “Classical BV Theories on Manifolds with Boundary”, by A.S. Cattaneo et al., Commun. Math. Phys., 2014), I will review the main features of the BV formalism on closed manifolds. Then I will switch to manifolds with boundary, and show that a BV theory induces a so-called Batalin-Fradkin-Vilkovisky (BFV) theory on the boundary, in a compatible way: the result is a particular case of a BV-BFV theory. In the BV-BFV context, the symmetry reduction, and the corresponding cohomological “infinitesimal” description of the EL-moduli space at a smooth point, go along the same conceptual lines as in the BV case, but the actual procedure is more delicate, since it has to take into account the additional boundary structures. The final output will be the symplectic EL-moduli space, which fulfils the expected gluing properties, indispensable for quantisation.