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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>↪</mml:mo> <mml:msub><mml:mi>C</mml:mi> <mml:mo>∘</mml:mo></mml:msub> <mml:mo>↪</mml:mo> <mml:mi>F</mml:mi></mml:mrow> </mml:math> inside a symplectic manifold <i>F</i>. To this, we naturally assign <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munder><mml:mi>C</mml:mi> <mml:mo>_</mml:mo></mml:munder> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:munder><mml:msub><mml:mi>C</mml:mi> <mml:mo>∘</mml:mo></mml:msub> <mml:mo>_</mml:mo></mml:munder> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munder><mml:mi>C</mml:mi> <mml:mo>_</mml:mo></mml:munder> </mml:math> , whose reduction can further be resolved using the BFV prescription. We call this construction <i>double BFV resolution</i>, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munder><mml:mi>C</mml:mi> <mml:mo>_</mml:mo></mml:munder> </mml:math> , 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.