The Quantum Modular Group in (2+1)-Dimensional Gravity

TL;DR

This paper investigates the quantum modular group in (2+1)-dimensional gravity, revealing how it structures the holonomy representation space and connecting different quantum pictures for torus topologies.

Contribution

It constructs explicit operator representations of the modular group and demonstrates how it partitions the Hilbert space into equivalent regions.

Findings

Quantum modular group splits the Hilbert space into fundamental regions.

Explicit operator representations of the mapping class group are developed.

Transformation between holonomy and Schrödinger pictures is established.

Abstract

The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a "Heisenberg picture", and for simple topologies, the transformation to the ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal ``fundamental regions'' that are interchanged by modular transformations.