I wish to understand the statement in this paper more precisely:
(1). Any 3d Topological quantum ﬁeld theories(TQFT) associates an innerproduct vector space $H_{\Sigma}$ to a Riemann surface $\Sigma$.

(2) In the case of abelian ChernSimons 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 $\det(K)^g$.

(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 ChernSimons 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) stepbystep for focusing on this abelian ChernSimons theory?
partial answer of (1)~(4) is fine.
I can understand the statements, but I cannot feel comfortable to derive them myself.
This post imported from StackExchange Physics at 20140625 21:02 (UCT), posted by SEuser mysteriousness