Witten's 2+1 gravity on R x (Klein bottle)

TL;DR

This paper explores Witten's 2+1 gravity on a nonorientable manifold, analyzing classical solutions, recovering spacetime metrics, and discussing potential paths to quantization within a gauge-theoretic framework.

Contribution

It provides a detailed analysis of classical solutions of 2+1 gravity on a Klein bottle, including metric recovery and an initial discussion on quantization approaches.

Findings

Recovered nondegenerate spacetime metrics from multiple solution components

Identified conditions under which Klein bottles are spacelike in the solutions

Formulated an action principle for certain bundle configurations

Abstract

Witten's formulation of 2+1 gravity is investigated on the nonorientable three-manifold R x (Klein bottle). The gauge group is taken to consist of all four components of the 2+1 Poincare group. We analyze in detail several components of the classical solution space, and we show that from four of the components one can recover nondegenerate spacetime metrics. In particular, from one component we recover metrics for which the Klein bottles are spacelike. An action principle is formulated for bundles satisfying a certain orientation compatibility property, and the corresponding components of the classical solution space are promoted into a phase space. Avenues towards quantization are briefly discussed.