Remarks on the Reduced Phase Space of (2+1)-Dimensional Gravity on a Torus in the Ashtekar Formulation

TL;DR

This paper analyzes the reduced phase space of (2+1)-dimensional gravity on a torus using the Ashtekar formulation, revealing a finite-dimensional space of gauge-inequivalent solutions due to residual gauge invariance.

Contribution

It demonstrates that the reduced phase space is finite-dimensional, clarifying the structure of physical degrees of freedom in (2+1)-dimensional Ashtekar gravity on a torus.

Findings

Reduced phase space is finite-dimensional

Residual gauge invariance reduces solution space

Consistent with general arguments on degrees of freedom

Abstract

We examine the reduced phase space of the Barbero-Varadarajan solutions of the Ashtekar formulation of (2+1)-dimensional general relativity on a torus. We show that it is a finite-dimensional space due to existence of an infinite dimensional residual gauge invariance which reduces the infinite-dimensional space of solutions to a finite-dimensional space of gauge-inequivalent solutions. This is in agreement with general arguments which imply that the number of physical degrees of freedom for (2+1)-dimensional Ashtekar gravity on a torus is finite.