The BV-BFV formalism is a combination of the BV approach to quantisation of Lagrangian field theories with local symmetries and the BFV approach to quantisation of constrained Hamiltonian systems. It aims to assign compatible bulk-boundary cohomological data to a Lagrangian field theory on a manifold with boundary (and higher codimension strata), in view of a perturbative quantisation scheme that is compatible with cutting and gluing.
General Relativity (GR), seen as a field theory, is a very important example to phrase within this setting, and one in which interesting new insight and complications emerge already at the classical level.
In this talk I will present a summary of investigations on GR within the BV-BFV formalism, as well as other diffeomorphism-invariant theories, which have given access to rich and nontrivial information about the boundary structure of gravitational models. However, I will argue that the featured examples present unexpected complications for the program of quantisation with boundary (and higher strata).
Indeed, I will show how the BV-BFV construction provides a filter to refine the notion of classical equivalence of field theories, which distinguishes theories in terms of their bulk-boundary behaviour, suggesting that some realisations — among the class of classically equivalent ones—may be more suitable for quantisation with boundary. This allows us to differentiate between, e.g., metric and coframe gravity as well as different string theory models and their 1d analogues.
This is a summary of joint works with G. Canepa and A.S. Cattaneo.