For simplicity, let us initially restrict our attention to the case . The group of Section 2.2, or, equivalently, the gauge group in the Chern-Simons formalism of Section 2.3, is then . The fundamental group has two generators, and , satisfying a single relation similar to Equation (12):

The holonomy group (11) is therefore generated by two commuting matrices, unique up to overall conjugation.It is a bit more convenient to describe the holonomies as elements of the covering group [211] . Let denote the two holonomies corresponding to the curve . An matrix is called hyperbolic, elliptic, or parabolic according to whether is greater than, equal to, or less than 2, and the space of holonomies correspondingly splits into nine sectors. It may be shown that only the hyperbolic-hyperbolic sector corresponds to a spacetime in which the slices are spacelike [117, 119, 182, 209] . By suitable overall conjugation, the two generators of the holonomy group in this sector can then be taken to be

where the are four arbitrary parameters. Note that this gives the right counting: The Riemann moduli space of the torus is two dimensional, so from Section 2.4 we expect a four-dimensional space of solutions.To obtain the corresponding geometry, we can use the quotient space construction of Section 2.2 . Note first that three-dimensional anti-de Sitter space can be represented as the submanifold of flat (with coordinates and metric ) defined by the condition that

This gives an isometry between and the group manifold of . The quotient of by the holonomy group (38) may now be obtained by allowing the to act on by left multiplication and the to act by right multiplication.It is straightforward to show that the resulting induced metric is

where and are coordinates with period . An easy calculation confirms that this is a space of constant negative curvature. The triad may be read off directly from Equation (40), and it is easy to solve Equation (15) for the spin connection . The resulting Chern-Simons connections of Equation (23) are flat, and their holonomies reproduce the holonomies (38) of the geometric structure we began with.To relate these expressions to the ADM formalism of Section 2.4, we must first find the slices of constant extrinsic curvature . For the metric (40), the extrinsic curvature of a slice of constant is , which is independent of and . A constant slice is thus also a slice of constant York time. The standard flat metric on , the genus one version of the standard metric (31), is

where is the modulus. Comparing (40), we see that a slice of constant has a modulus The conjugate momentum can be similarly computed from Equation (33), while the ADM Hamiltonian of Equation (35) becomes In the limit of vanishing , these relations go over to those of [66] .To quantize this system, we will need the classical Poisson brackets, which can be obtained from Equation (26):

These, in turn, determine the brackets among the moduli and momenta and , a result consistent with Equation (36). It may be shown that the version of Hamilton’s equations of motion coming from these brackets reproduces the time dependence (42) of the moduli; see [87, 135] for details. The Poisson brackets among the traces of the holonomies (38) are also easy to compute. If we let it is not hard to check that reproducing the Poisson algebra of Nelson, Regge, and Zertuche [211] .Finally, let us consider the action of the torus mapping class group. This group is generated by two Dehn twists, which act on by

These transformations act on the parameters and the ADM moduli and momenta as These transformations are consistent with the relationships between the ADM and holonomy variables, and that they preserve all Poisson brackets.For a torus universe with zero or positive cosmological constant, similar constructions are possible. I refer the reader to [81] for details.

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