Various CFT models are known to produce representations of braid groups.
A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps on $S^2$ is
$$ B = \exp \left( \frac{4 \pi i}{k + 2} T^a \otimes T^a \right). $$
I only know this formula because WZW models are holographically dual to the 3-dimensional gauge theory with Chern-Simons action, which is known to produce knot polynomials related to quantum groups (the Jones polynomial). So I guessed the expression above by simply writing the $R$-matrix of $U_q(\mathfrak{su_2})$ for two fundamental irreps.
However, I would very much like to understand how the action of $B$ on conformal blocks of the WZW model can be explicitly calculated (at least for the simplest possible case of two fundamental irreps on $S^2$). The calculation should only use the first principles of CFT, not clever connections to Chern-Simons and quantum groups (as I already know how to obtain the answer from them).
I also admit to not fully understand the conformal blocks construction, so please add any information that you think is relevant.
As a follow-up question, which CFT models produce representations of the braid group? Is there a specific criterion?