BV equivalence with boundary.

An extension of the notion of classical equivalence of equivalence in the Batalin-Vilkovisky (BV) and Batalin-Fradkin-Vilkovisky (BFV) frameworks for local Lagrangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of nonabelian Yang-Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV-BFV description, are shown to be pairwise equivalent as strict BV-BFV theories. This in particular implies that their BV complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV-BFV formulation. They are shown to be equivalent as lax BV-BFV theories and to have isomorphic BV cohomologies. This shows that strict BV-BFV equivalence is a strictly finer notion of equivalence of theories.

General Relativity and the AKSZ Construction.

In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein-Hilbert and of the Palatini-Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein-Hilbert theory, in the latter a BV theory for Palatini-Cartan theory with a partial implementation of the torsion-free condition already on the space of fields. All theories described here are BV versions of the same classical system on cylinders. The AKSZ implementations we present have the advantage of yielding a compatible BV-BFV description, which is the required starting point for a quantization in presence of a boundary.

Gravitational Constraints on a Lightlike Boundary.

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1-dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case.