Remarks on the reduced phase space of (2+1)-dimensional gravity on a torus in the Ashtekar formulation

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 the 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.