Double BFV Quantisation of 3D Gravity.

Area of Science:

• Mathematical Physics
• Theoretical Physics
• Differential Geometry

Background:

• The Batalin-Fradkin-Vilkovisky (BFV) formalism provides a cohomological setting for coisotropic reduction using Hamiltonian dg manifolds.
• Nested coisotropic embeddings present a more complex geometric structure within symplectic manifolds.

Purpose of the Study:

• To extend the BFV cohomological setting to handle nested coisotropic embeddings.
• To develop a 'double BFV resolution' for these nested structures.
• To investigate the relationship between resolution and reduction, and to deduce quantization procedures.

Main Methods:

• Extension of the BFV formalism to nested coisotropic embeddings.
• Construction of a graded coisotropic embedding within the BFV dg manifold.
• Application of the BFV prescription for resolution and quantization.

Main Results:

• Demonstration that the double BFV resolution allows resolution to commute with reduction for nested coisotropic embeddings.
• Deduction of a quantization of the embedded manifold from the geometric quantization of the double BFV Hamiltonian dg manifold.
• Identification of a candidate quantum state space for three-dimensional Einstein-Hilbert theory.

Conclusions:

• The double BFV resolution provides a powerful tool for understanding complex geometric reductions and their quantization.
• The framework offers a pathway to quantizing gravitational theories, exemplified by 3D Einstein-Hilbert theory.
• This work advances the interplay between algebraic structures (dg manifolds) and geometric concepts (coisotropic embeddings) in theoretical physics.