Symmetries at Causal Boundaries in 2D and 3D Gravity

TL;DR

This paper explores the structure of symmetries and boundary charges in 2D and 3D gravity theories with causal boundaries, revealing integrable slicings and algebraic structures like Heisenberg and Virasoro algebras.

Contribution

It constructs the solution phase space for gravity with causal boundaries and identifies boundary charges and their algebraic structures, including integrable slicings and symmetry algebras.

Findings

Existence of D+1 boundary charges in D dimensions.

Identification of integrable slicings where charges are well-defined.

Discovery of boundary charge algebra as Heisenberg plus Virasoro structures.

Abstract

We study 2d and 3d gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss Y and W freedoms and change of slicing in the solution space. For D dimensional case we find D+1 surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the 3d case there exists an integrable slicing where charge algebra takes the form of Heisenberg \oplus\ {\cal A}_3 where {\cal A}_3 is two copies of Virasoro at Brown-Henneaux central charge for AdS_3 gravity and BMS_3 for the 3d flat space gravity.