Constants of motion and the conformal anti - de Sitter algebra in (2+1)-Dimensional Gravity

TL;DR

This paper explores the constants of motion in 2+1 dimensional gravity with negative cosmological constant, revealing their algebraic structure and quantization, and connecting them to the conformal algebra and modular group actions.

Contribution

It identifies the algebraic structure of conserved quantities in 2+1 gravity and extends the quantum algebra to the conformal algebra, including modular group actions.

Findings

Constants of motion form the anti-de Sitter algebra in ADM and holonomy variables.

Quantization is straightforward using holonomy parameters.

Inclusion of the Hamiltonian extends the algebra to the conformal algebra so(2,3).

Abstract

Constants of motion are calculated for 2+1 dimensional gravity with topology R x T^2 and negative cosmological constant. Certain linear combinations of them satisfy the anti - de Sitter algebra so(2,2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to that of the conformal algebra so(2,3). The modular group appears as a discrete subgroup of the conformal group. Its quantum action is generated by these conserved quantities.