The Torus Universe in the Polygon Approach to 2+1-Dimensional Gravity

In this paper we describe the matter-free toroidal spacetime in ’t Hooft’s polygon approach to 2+1-dimensional gravity (i.e. we consider the case without any particles present). Contrary to earlier results in the literature we find that it is not possible to describe the torus by just one polygon but we need at least two polygons. We also show that the constraint algebra of the polygons closes. E-mail: [email protected] Introduction As is well known nowadays, gravity in 2+1 dimensions is flat everywhere outside sources [1, 2]. This means that the gravitational field itself has no local degrees of freedom. One can make the theory non-trivial by adding sources (e.g. point-particles) or considering a non-trivial topology of a closed universe. For N point-particles that live on a genus-g surface for instance, the phase space is 12g−12+4N dimensional [4, 6, 8]. This formula is however wrong in the case of a torus in the absence of particle sources. This is due to the fact that the torus has some symmetries because of which the counting argument breaks down. The toroidal universe, with or without cosmological constant, has been extensively studied in the past. Its classical solutions [11, 12] as well as the quantum theory [8, 9, 10, 13, 16, 17] are well understood in both the ADM formalism and in the Chern-Simons formalism. From this work we know that the dimension of the space of ADM solutions is four (i.e. there are only four independent degrees of freedom) The torus is therefore a particularly simple model and a convenient starting point for a quantization program. As we are interested in the polygon description of 2+1-D gravity invented by ’t Hooft [3, 4] we decided to study the torus in this approach. Guadagnini and Franzosi had already worked on this problem [15]. But their counting of degrees of freedom was a bit puzzling to us. We found that they described a subset of all possible solutions for a torus universe. This is due to the fact that they use only one polygon for their slicing of spacetime. This is not enough to cover all possible tori. The simple solution to this problem is to add another polygon. This unfortunately implies that the description loses its simplicity due to the fact that polygon transitions may take place during evolution. This fact also considerably complicates the quantization. The temporary conclusion is that the polygon approach is not the most convenient description for the matter-free torus universe as compared with other approaches. In section 1 we recapitulate the way Carlip describes a toroidal spacetime and stress the fact that the phase space is 4 dimensional. Section 2 contains an introduction to the polygon approach. We compute the constraint algebra of the polygons and conclude that the algebra closes but is highly nonlinear. We propose to define a new constraint for which the constraint algebra closes linearly. In this section we also reproduce the one-polygon solution for the toroidal universe of Guadagnini and Franzosi and it is shown that it contains only part of phase space. In section 3 we propose a 2 polygon representation for the torus and show that The topology of the universe under consideration is: M = Σ(g)×R where Σ(g) is a genus-g spacelike surface and R is in the time direction.