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In A Duality Web in 2+1 Dimensions and Condensed Matter Physics, the authors claimed in Appendix B that for a $U(1)_{k}$ Chern-Simons theory defined on a Riemann surface $\Sigma$ of genus $g$, the number of physical states is $k^{g}$.

Can anybody tell me how to calculate the number of physical states of an Abelian Chern-Simons theory on a Riemann surface? Is there any reference that I can follow to understand the above statement?


New Eddition: As far as I see it, since a Chern-Simons theory is a topological field theory, it must be diffeomorphism invariant. As a result, its Hilbers space has only vacum states.

The dimension of its vacua is, therefore, the dimension of its Hilbert space. This can be computed from its partition function.

$$\mathcal{Z}=\frac{1}{\mathrm{vol\mathcal{G}}}\int\mathcal{D}Ae^{i\mathrm{CS}[A]}=\mathrm{Tr}_{\mathcal{H}}(1)=\dim\mathcal{H}.$$

The $U(1)$ Chern-Simons path integral is known to be

$$\mathcal{Z}=e^{i\left(\frac{\pi\eta(0)}{4}-\mathrm{CS}_{g}\right)}\sqrt{T},$$

where $T$ is the Ray-Singer analytic torsion, which is topological invariant, and $\eta(0)$ is the APS eta-invariant of the Dirac type operator $d\ast+\ast d$, and $\mathrm{CS}_{g}$ is a gravitational Chern-Simons counter term contributing the framing anomaly.

Now the problem reduces to how $\eta(0)$ is related with the level $k$ and genus $g$.

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