Ashtekar Formulation of 2+1 Gravity on a Torus

TL;DR

This paper analyzes (2+1)-dimensional Einstein gravity on a torus using Ashtekar variables, revealing new observables, solving quantum constraints, and identifying a global time variable, thus advancing understanding of quantum gravity in lower dimensions.

Contribution

It introduces a set of globally defined observables forming a closed algebra and demonstrates that the Ashtekar formulation's reduced phase space exceeds that of the Witten formulation.

Findings

Found a closed algebra of global observables.

Solved quantum constraints explicitly.

Identified a globally defined extrinsic time variable.

Abstract

Pure (2+1)-dimensional Einstein gravity is analysed in the Ashtekar formulation, when the spatial manifold is a torus. We have found a set of globally defined observables, forming a closed algebra. This allowed us to solve the quantum constraints, and to show that the reduced phase space of the Ashtekar formulation is greater then the corresponding space of the Witten formulation. Furthermore, we have found a globally defined time variable which satisfies all the requiriments of an extrinsic time variable in quantum gravity.