Diffeomorphism invariant subspaces in Witten's 2+1 quantum gravity on $\mathbf{R}\times T^2$

We address the role of large diffeomorphisms in Witten’s 2+1 gravity on the manifold $\mathbf{R}\times T^2$. In a ``spacelike sector" quantum theory that treats the large diffeomorphisms as a symmetry, rather than as gauge, the Hilbert space is shown to contain no nontrivial finite dimensional subspaces that are invariant under the large diffeomorphisms. Infinite dimensional closed invariant subspaces are explicitly constructed, and the representation of the large diffeomorphisms is thus shown to be reducible. Comparison is made to Witten’s theory on $\mathbf{R}\times \mathrm{\Sigma }$, where $\mathrm{\Sigma }$ is a higher genus surface.