In contrast to the reduced phase space quantization of the preceding Section 3.1, our understanding of the quantum Chern-Simons gravity depends strongly on the sign of the cosmological constant. For , the relevant gauge group is or its cover . This is the most poorly understood case; an explicit quantization of the algebra holonomies exists for genus one (see below) and genus two , but more general results do not yet exist.
For , the relevant gauge group is or its cover , a complex gauge group whose Chern-Simons theory is somewhat better understood [280, 155, 43] . As noted in Section 2.3, the Poisson brackets for this theory are related to the quantum double of the Lorentz group, and Buffenoir et al. have used this structure to write down an explicit quantization  . As far as I know, the relationship between this work, which is based on a Hamiltonian formalism and combinatorial quantization, and that of Witten and Hayashi [280, 155], which is based on geometric quantization, has not yet been explored.
For , the relevant gauge group is , the (2+1)-dimensional Poincaré group, or its universal cover. Here there is again a connection to the quantum double of the Lorentz group, which has been used in [37, 36, 201] to explore the quantum theory, although largely in the context of point particles. In this case, one has the nice feature that the phase space has a natural cotangent bundle structure, allowing us to immediately identify the holonomies of the spin connection as generalized positions, and their derivatives as generalized momenta. This provides a direct link to the loop variables of Ashtekar, Rovelli, and Smolin [26, 29], . Note that the generator may, in principle, be in any representation of , and that the trace in Equation (55) may depend on the choice of representation. I will return to the resulting quantum theory, loop quantization, in Section 3.5 .
As in reduced phase space quantization, matters simplify considerably for the torus universe . Let us again focus on the case . A complete - in fact, overcomplete - set of observables is given by the traces (47) of the holonomies, and our goal is to quantize the algebra (48). To do so, we proceed as follows:
The resulting algebra is defined by the relations . Classically, the observables , , and are not independent; in the quantum theory, the corresponding statement is that the quantities [87, 86] 
We must also implement the action of the modular group (50) on the operators . One can find an action preserving the algebraic relations (57), corresponding to a particular factor ordering of the classical modular group. The Nelson-Picken quantization (61) admits a similar modular group action.
For a full quantum theory, of course, one needs not only an abstract operator algebra, but a Hilbert space upon which the operators act. For the universe, Equation (60) suggests that a natural choice is to take wave functions to be square integrable functions of the . There is a potential difficulty here, however: The modular group does not act properly discontinuously on this configuration space. This means that the quotient of this space by the modular group is badly behaved; in fact, there are no nonconstant modular invariant functions of the [182, 143, 221] . We shall return to this problem in Section 3.4 .
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