Quantum modular group in (2+1)-dimensional gravity

The role of the modular group in the holonomy representation of $\mathrm{}(2+1)\mathrm{}$-dimensional quantum gravity is studied. This representation can be viewed as a “Heisenberg picture,” and for simple topologies, the transformation to the ADM “Schrödinger picture” may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal “fundamental regions” that are interchanged by modular transformations.