Equivalent Quantisations of (2+1)-Dimensional Gravity

TL;DR

This paper explores different quantisation methods for (2+1)-dimensional gravity on a torus topology, showing that they are equivalent up to small quantum corrections when the cosmological constant is small.

Contribution

It introduces a unique formulation of (2+1)-dimensional gravity using holonomies and compares two quantisation approaches, demonstrating their near equivalence.

Findings

Quantisation methods differ by negligible quantum corrections for small |Λ|

Holonomy parameters relate to moduli via a canonical transformation

Effective Hamiltonian describes the dynamics on constant mean curvature slices

Abstract

For spacetimes with the topology $\IR\!\times\!T^2$, the action of (2+1)-dimensional gravity with negative cosmological constant $\La$ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on $T^2$. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order $O(\hbar^3)$, negligible for small $|\La|$.