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

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Abstract
Witten's formulation of 2 + 1 gravity is investigated on the non-orientable 3-manifold . The gauge group is taken to consist of all four components of the 2 + 1 Poincaré group. We analyse in detail several components of the classical solution space, and we show that from four of the components one can recover non-degenerate 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.
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