Witten's description of WZW conformal blocks

Question

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a simply-laced and simply-connected Lie Group $G$ at level 1. He claims that on a Riemann Surface $W$, this vector space is the Heisenberg representation of the Heisenberg group associated to $H^1 (W, \mathcal{Z})$ where $\mathcal{Z} = \pi_1 (G)$. Later he shows that the dimension of this vector space is $(\#\mathcal{Z})^g$, which I have checked indeed coincides with the dimension computed through e.g. the Verlinde formula.

My questions are as follows - (1) What is the reason behind the appearance of $H^1 (W, \mathcal{Z})$ in this context?

(2) Does the vector space of conformal blocks have an interpretation in terms of a Heisenberg group associated to a cohomology group when the level is not equal to 1?

Answer 1

I am no expert on this, but it sounds as though the blocks are labeled by the Wilson lines around the 1-cycles. This is what happens on a torus.