Quantum mechanics

Cellular algebra

Theoretical physics

World Wide Web

Linguistics

Engineering

Philosophy

Pure mathematics

Mathematical physics

Boundary (topology)

Operating system

Phase space

Economics

Computer science

Virasoro algebra

Slicing

Surface (topology)

Mechanical engineering

Physics

Degrees of freedom (physics and chemistry)

Central charge

Integrable system

Mathematics

Position (finance)

Homogeneous space

Algebra representation

Mathematical analysis

Geometry

Conformal map

Charge (physics)

Codimension

Finance

Algebra over a field

Space (punctuation)

Journal of High Energy Physics

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.

Symmetries at causal boundaries in 2D and 3D gravity