[Paper Review] Fully extended BV-BFV description of General Relativity in three dimensions

G. Canepa, Michele Schiavina|arXiv (Cornell University)|May 22, 2019

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes a fully extended BV-BFV formalism for three-dimensional General Relativity in triad variables, demonstrating that the theory extends consistently to all codimension strata—bulk, boundaries, corners, and vertices—via symplectic reductions. It proves strong equivalence between 3D GR and nondegenerate BF theory at every level, enabling a consistent quantization framework compatible with cutting and gluing, and paving the way for holographic and observable studies.

ABSTRACT

We compute the extension of the BV theory for three-dimensional General Relativity to all higher-codimension strata - boundaries, corners and vertices - in the BV-BFV framework. Moreover, we show that such extension is strongly equivalent to (nondegenerate) BF theory at all codimensions.

Motivation & Objective

• To develop a fully extended BV-BFV formulation of 3D General Relativity that includes all higher-codimension strata: bulk, boundaries, corners, and vertices.
• To demonstrate that this extension is strongly equivalent to nondegenerate BF theory across all codimensions, preserving symplectic structure and gauge symmetries.
• To provide explicit cohomological data at each stratum, enabling the computation of constraints, reduced phase space, and boundary insertion representations.
• To extend the BV-BFV framework to 3D GR in triad variables, where previous attempts in 4D failed due to obstructions in the induction procedure.
• To enable future applications in quantum gravity, such as edge modes, asymptotic symmetries, and Witten descent equations, via a consistent, gluing-compatible quantization scheme.

Proposed method

• The authors use the BV-BFV formalism to systematically extend 3D GR to all codimension strata, starting from the bulk and recursively constructing data at boundaries, corners, and vertices.
• They employ a cohomological approach to define the BV-BFV data at each stratum, ensuring compatibility with gauge symmetries and the failure of gauge invariance at boundaries.
• The key technical tool is the construction of symplectomorphisms between GR and BF theory data at each codimension, proving strong equivalence via explicit change of coordinates.
• The method relies on the triadic formulation of 3D GR, where the tetrad and spin connection are treated as independent variables, allowing for a clean BV-BFV extension.
• The authors verify equivalence by checking that the actions and symplectic structures are preserved under the proposed maps at all codimensions, including the cosmological term extension.
• They extend the known strong equivalence between BF theory and GR in the bulk [CSS18] to higher codimensions, showing that the equivalence maps remain valid throughout the stratified structure.

Experimental results

Research questions

• RQ1 Can the BV-BFV formalism be consistently extended to all codimension strata—bulk, boundaries, corners, and vertices—for 3D General Relativity in the triadic formulation?
• RQ2 Is the fully extended BV-BFV data of 3D GR strongly equivalent to that of nondegenerate BF theory at every codimension?
• RQ3 How does the inclusion of a cosmological constant affect the fully extended BV-BFV structure of 3D GR and its equivalence to BF theory?
• RQ4 What cohomological data can be extracted from the BV-BFV structure at each stratum, and how do they relate to physical observables and reduced phase spaces?
• RQ5 Can the extended BV-BFV framework support a consistent quantization of 3D GR that respects cutting and gluing, and how does it relate to edge modes and holography?

Key findings

• The fully extended BV-BFV structure of 3D General Relativity is constructed explicitly up to codimension 3, including all strata: bulk, boundaries, corners, and vertices.
• The theory exhibits nontrivial symplectic reductions at every codimension, providing a cohomological presentation of the reduced phase space at codimension 1.
• The BV-BFV data of 3D GR are strongly equivalent to those of nondegenerate BF theory at all codimensions, with the equivalence realized via explicit symplectomorphisms that preserve actions and symmetries.
• The cosmological term can be consistently added to the BV-BFV structure without breaking the strong equivalence, and the actions at higher codimensions are explicitly computed.
• The equivalence maps remain valid under the inclusion of the cosmological term, as verified by direct computation of the pullbacks of actions.
• The framework enables the direct computation of observables via Witten descent equations and provides a foundation for studying edge modes and asymptotic symmetries in 3D quantum gravity.