Double BFV quantisation of 3d Gravity

TL;DR

This work extends the BFV formalism to nested coisotropic embeddings by introducing a double BFV construction that resolves reductions in two steps and proves that resolution commutes with reduction for a broad class of embeddings. It then formulates a quantum version via geometric quantisation of the double BFV data, yielding a principled path to quantise reduced phase spaces, with a concrete application to three-dimensional gravity by relating Einstein–Hilbert, Palatini–Cartan, and BF theories. The authors provide both classical and quantum bridges between BF theory and EH gravity, presenting two quantisation routes (polarisation choices) and demonstrating that the resulting physical state spaces are well-defined candidates for quantum gravity in 3d. The framework clarifies how constraints and reductions can be handled in a cohomological setting, and it offers explicit constructions that connect canonical gravity formulations through BF theory. Overall, the paper advances a robust, mathematically precise route to quantise constrained gravity models via double BFV reduction and geometric quantisation.

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.

Theorems & Definitions (62)

• Remark 1
• Remark 2: Notation
• Definition 3
• Remark 4: Graded coisotropic
• Lemma 5
• proof
• Proposition 6
• proof
• Definition 7: Nested Coisotropic Embedding
• Remark 8