I would like to understand the quantization of the U(1) Chern-Simons theory in (space-time) dimension d=2n+1 > 3. The Chern-Simons action reads
Here $A$ is a compact 1-form gauge field and $F$ is the field strength $F=dA$.
For $n=1$, the theory is quadratic and its quantization has been very well-understood. For example, one can calculate the expectation values of Wilson loop operators which give the linking numbers of the worldlines. One can also do canonical quantization on a Riemann surface and explicitly obtain the dimension of the Hilbert space and the wavefunctions, from which other important quantities like the representation of mapping class group.can be derived.
I'm wondering anything similar is known for the higher dimensional generalization. A basic question is: what is the dimension of the Hilbert space on a $2n$-dimensional torus? Any references?