How solvable is (2+1)-dimensional Einstein gravity?

Abstract

In this paper, the relationship between Witten’s approach to the (2+1)-dimensional, vacuum Einstein equations (for spatially compact space-times) and the conventional Annowitt, Deser, and Misner (ADM) Hamiltonian approach is discussed. It is argued (especially for the space-times with higher genus Cauchy surfaces) that neither approach is complete in itself; Witten’s because it does not provide a technique (even at the classical level) for recovering the space-time metric and the conventional approach because it provides no mechanism for solving a seemingly intractable set of Hamilton equations. It is also argued, however, that the two formulations are instead complementary in the sense that the Wilson loops, which play a key role in Witten’s approach, provide (at least in principle) a mechanism for solving the reduced Hamilton equations and thereby completing the picture at the classical level. An example of this synthesis for the (explicitly computable) case of genus-one hypersurfaces is provided. The more tenuous problem of whether this synthesis can be extended to the quantized Einstein equations will also be discussed. A principal open question is whether the Wilson loops, when expressed in terms of the ADM canonical variables, can be ordered in such a way as to preserve, quantum mechanically, their (classical) Poisson bracket algebra.