$e^{2\pi ic_{-}/8}$ and $e^{2\pi ic_{-}/24}$ in unitary modular category (UMC)

Question

Background

1. Unitary modular categories (UMC) do not capture the central charge $c_{-}$ of the topological quantum field theory (TQFT). However, there is a relation that fixes, $c_{-}\bmod 8$: \begin{equation} \Theta = D^{-1}\sum_{a}d_{a}^{2}\exp(i 2 \pi s_a)\,=\, D^{-1}\sum_{a}d_{a}^{2}\theta_{a}\,=\, e^{2\pi ic_{-}/8}, \quad \tag{1} \end{equation} where $D = \sqrt{\textstyle\sum_{a}d_{a}^{2}}$, $d_{a}$ is the {quantum dimension} of the superselection sector $a$ (of some line operators of TQFT) and $s_a$ is the topological spin for spin statistics (or exchange statistics). Roughly, $\Theta=S^{-1}TSTST$, where $S$ is a so-called topological modular $S$-matrix and modular $T$-matrix is the diagonal matrix with entries $\theta_{a}$. The $\Theta$ is a root of unity, hence implies that $c_{-}$ is rational.

2. However, the modular properties of theory seems not to be captured by $$ c_{-} =0 \mod 8 $$ but by $$ c_{-} =0 \mod 24. $$ (Correct me if I was mistaken.) So eq.(1) is not ideal to pin down all $c_{-} \mod 24$.

3. There is another way to obtain $c_{-} \mod 24$, which is the transformation of the vacuum character under the Dehn twist: $$ \chi_{1}(w+1)\,=\,e^{-2\pi ic_{-}/24}\chi_{1}(w), \tag{2} $$ where $w$ is the modulus of the torus and the subscript $1$ refers to the vacuum sector.

Questions

1). What are the original literature obtaining the formulas of eqs.(1) and (2)?

I believe that it is summarized in reviews or textbooks like:

• J. Fr ̈ohlich, F. Gabbiani, “Braid statistics in local quantum field theory”, Rev. Math. Phys. 2 no. 3, 251–353 (1990).

• K.H.Rehren, “Braid group statistics and their superselection rules” in The algebraic theory of superselection sectors, D. Kastler (ed.) Proceedings Palermo 1989, pp. 333–355, World Scientific Publishing (1990).

• P. Di Francesco, P. Mathieu, D. S ́en ́echal, “Conformal field theory”, Springer (1996).

But I am hoping to know the history and the original literatures.

2). What are some ideas behind to obtaining/deriving the formulas of eqs.(1) and (2)?

Like eq. (2), I know the Dehn twist, but how to prove the phase $e^{-2\pi ic_{-}/24}$.