# geometric quantization of the moduli space of abelian Chern-Simons theory

I wish to understand the statement in this paper more precisely:

(1). Any 3d Topological quantum ﬁeld theories(TQFT) associates an inner-product vector space $H_{\Sigma}$ to a Riemann surface $\Sigma$.

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(2) In the case of abelian Chern-Simons theory $H_{\Sigma}$ is obtained by geometric quantization of the moduli space of ﬂat $T_{\Lambda}$-connections on ${\Sigma}$. The latter space is a torus with a symplectic form

$$ω =\frac{1}{4π} \int_{\Sigma} K_{IJ} \delta A_I \wedge d \delta A_J.$$

(3) Its quantization is the space of holomorphic sections of a line bundle $L$ whose curvature is $\omega$. For a genus g Riemann surface $\Sigma_g$, it has dimension $|\det(K)|^g$.

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(4) The mapping class group of $\Sigma$ (i.e. the quotient of the group of diffeomorphisms of $\Sigma$ by its identity component) acts projectively on $H_{\Sigma}$. The action of the mapping class group of $\Sigma_g$ on $H_\Sigma$ factors through the group $Sp(2g, \mathbb{Z})$.

We are talking about this abelian Chern-Simons theory: $$S_{CS}=\frac{1}{4π} \int_{\Sigma} K_{IJ} A_I \wedge d A_J.$$

Can some experts walk through this (1) (2) (3) (4) step-by-step for focusing on this abelian Chern-Simons theory?

partial answer of (1)~(4) is fine.

I can understand the statements, but I cannot feel comfortable to derive them myself.

• You may have interests to read this related paper: arxiv.org/abs/1212.4863; which is from a more physical perspective, at least related to low energy physics of topological orders. – wonderich Jun 22 '14 at 20:24

These references have geometric quantization of abelian Chern-Simons theory for $\Sigma=S^1 \times S^1$