Double BFV quantisation of 3d Gravity

TL;DR

This paper develops a double BFV quantization method for nested coisotropic embeddings in symplectic manifolds, enabling a new approach to quantizing 3D gravity models and ensuring resolution commutes with reduction.

Contribution

It introduces the concept of double BFV resolution for nested coisotropic embeddings and proves that resolution commutes with reduction in this context.

Findings

Established a natural graded coisotropic embedding inside BFV dg manifolds.

Proved that resolution commutes with reduction for nested coisotropic embeddings.

Provided a candidate space of quantum states for 3D Einstein--Hilbert gravity.

Abstract

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 $\underline{C}$ and $\underline{C_\circ}$, 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 $\underline{C}$, whose reduction can further be resolved using the BFV prescription. We call this construction \emph{double BFV resolution}, and we use it to prove that "resolution commutes with reduction" for a general class of nested coisotropic embeddings. We then deduce a quantisation of $\underline{C}$, 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.