2024/10/30 by Giovanni Canepa, G. Canepa, Michele Schiavina +2
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Black Holes and Theoretical Physics
paper · pdf · doi:10.1007/s00220-025-05507-y
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_∘ \hookrightarrow F <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 F . To this, we naturally assign \underlineC <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 \underlineC_∘ <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 \underlineC <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 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 \underlineC <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.