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See [1] M. Bos and V.P. Nair. Coherent State Quantization of Chern-Simons Theory. International Journal of Modern Physics A, A5:959, 1990. and chapter 20 of [2] V.P. Nair. Quantum Field Theory: A modern perspective. Springer, 2005. These references have geometric quantization of abelian Chern-Simons theory for $\Sigma=S^1 \times S^1$ (the first ref. ...


I recommend reading this first: P. Cotta-Ramusino, E. Guadagnini, M. Martellini, and M. Mintchev. Quantum Field Theory and Link Invariants. Nuclear Physics B, B330:557, 1990. Also Guadagnini has a nice book("The Link Invariants of the Chern-Simons Field Theory") on the subject but it is hard to find.


Alright, let's go on a thrilling tour through the theory of representations. Notation in the following is the same as in the OP, except that we call the trivial representation $\boldsymbol{1}$, as is canon. We start from my expression in the question $$ I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \sum_\rho \sum_{\mu = ...

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