Solution space of 2+1 gravity on ${\bf R} \times T^2$ in Witten's connection formulation

Details Solution space of 2+1 gravity on ${\bf R} \times T^2$ in Witten's connection formulation Solution space of 2+1 gravity on ${\bf R} \times T^2$ in Witten's connection formulation

1993-08-19

arxiv.org/abs/gr-qc/9308018v2

Class.Quant.Grav.11:311-330,1994

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

23 pages, REVTeX v3.0, SU-GP-93/7-6, CGPG-93/8-3. (Discussion on the gauge equivalence of degenerate and nondegenerate metrics

Scientific paper

10.1088/0264-9381/11/2/005

We investigate the space ${\cal M}$ of classical solutions to Witten's formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is connected, unlike the spaces of classical solutions in the cases where $T^2$ is replaced by a higher genus surface. Although ${\cal M}$ is neither Hausdorff nor a manifold, removing from ${\cal M}$ a set of measure zero yields a manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space~${\cal B}$. We discuss the relation of the various parts of ${\cal M}$ to spacetime metrics, and various possibilities of quantizing~${\cal M}$. There exist quantizations in which the exponentials of certain momentum operators, when operating on states whose support is entirely on the part of ${\cal B}$ corresponding to conventional spacetime metrics, give states whose support is entirely outside this part of~${\cal B}$. Similar results hold when the gauge group ${\rm SO}_0(2,1)$ is replaced by ${\rm SU}(1,1)$.

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