Chern-Simons corner phase space in 4D gravity from BF-BB theory

TL;DR

This paper explores the phase space structure of 4D gravity on codimension 2 and 3 surfaces, revealing Chern-Simons and Kac-Moody algebraic structures and introducing a Maxwell algebra generalization of the gauge algebra.

Contribution

It introduces a BF-BB parametrisation that relaxes simplicity constraints, uncovering new algebraic structures in 4D gravity's corner phase space.

Findings

Gravity in 4D has Chern-Simons-like phase spaces on codimension 2 surfaces.

Corner Poisson brackets of the spin connection are off-shell commutative.

The corner metric exhibits noncommutativity.

Abstract

We investigate an approach to determine the correct Poisson brackets of fields restricted to codimension 2 and 3 surfaces in 4D gravity, which are of great potential use in holographic setups and discretisation. Employing a specific BF-BB type parametrisation of gravity which relaxes Plebanski's simplicity constraints, we find that gravity in 4 dimensions carries Chern-Simons like phase spaces in codimension 2 and Kac-Moody algebras in codimension 3. The necessary gauge algebra in this context shows that the appropriate generalisation of the double $\mathcal{D}\mathfrak{so}(1,2)$ of 3D gravity is the Maxwell algebra, $\mathfrak{g}=\mathfrak{so}(1,3)\ltimes(\mathbb{R}^{1,3}\tilde\oplus \mathfrak{so}(1,3)^\ast)$. This realises the corner Poisson bracket of the spin connection for the first time and shows it is off-shell commutative, while the corner metric is noncommutative.