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

Abstract

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

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

References in corpus (9)

• Geometrodynamics vs. Connection Dynamics
• (2+1)-Dimensional Chern-Simons Gravity as a Dirac Square Root
• Loop Representations for 2+1 Gravity on a Torus
• The Modular Group, Operator Ordering, and Time in (2+1)-Dimensional Gravity
• Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
• Unitary Equivalence of the Metric and Holonomy Formulations of 2+1 Dimensional Quantum Gravity on the Torus
• Six ways to quantize (2+1)-dimensional gravity
• Ashtekar Formulation of 2+1 Gravity on a Torus
• Homogeneous 2+1 Dimensional Gravity in the Ashtekar Formulation

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