Double BFV Quantisation of 3D Gravity

We extend the cohomological setting developed by Batalin, Fradkin and Vilkovisky (BFV), which produces a resolution of coisotropic reduction in terms of hamiltonian dg manifolds, to the case of nested coisotropic embeddings $C\hookrightarrow C_{\circ }\hookrightarrow F$ inside a symplectic manifold F. To this, we naturally assign $\underset{\_}{C}$ and $\underset{\_}{C_{\circ }}$ , as well as the respective BFV dg manifolds. We show that the data of a nested coisotropic embedding defines a natural graded coisotropic embedding inside the BFV dg manifold assigned to $\underset{\_}{C}$ , whose reduction can further be resolved using the BFV prescription. We call this construction double BFV resolution, and we use it to prove that "resolution commutes with reduction" for a large class of nested coisotropic embeddings. We then deduce a quantisation of $\underset{\_}{C}$ , from the (graded) geometric quantisation of the double BFV Hamiltonian dg manifold (when it exists), following the quantum BFV prescription. As an application, we provide a well defined candidate space of (physical) quantum states of three-dimensional Einstein-Hilbert theory, which is thought of as a partial reduction of the Palatini-Cartan model for gravity.