  Shape Dynamics is a formulation of General Relativity where refoliation
invariance is traded for local spatial conformal invariance. In this paper we
explicitly construct Shape Dynamics for a torus universe in 2+1 dimensions
through a linking gauge theory that ensures dynamical equivalence with General
Relativity. The Hamiltonian we obtain is formally a reduced phase space
Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus
surfaces is not explicitly possible, but we give an explicit expansion of the
Shape Dynamics Hamiltonian for large CMC volume. The fact that all local
constraints are linear in momenta allows us to quantize these explicitly, and
the quantization problem for Shape Dynamics turns out to be equivalent to
reduced phase space quantization. We consider the large CMC-volume asymptotics
of conformal transformations of the wave function. We then use the similarity
of Shape Dynamics on the 2-torus with the explicitly constructible strong
gravity (BKL) Shape Dynamics Hamiltonian in higher dimensions to suggest a
quantization strategy for Shape Dynamics.
