# Number of Physical States of a $U(1)$ Chern-SImons Theory on a Riemann Surface of Genus $g$

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 $$g$$.

• – d_b Dec 22 '18 at 11:06