I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks.


More precisely, on the one hand, we have the construction of Reshetikhin–Turaev as well as Blanchet–Habegger–Masbaum–Vogel of "quantum representations" of mapping class groups of Riemann surfaces (possibly with marked points): see e.g. Section 3 of [Masbaum - Quantum representations of mapping class groups](https://webusers.imj-prg.fr/%7Egregor.masbaum/Masbaum.pdf) for a nice survey. On the other hand, the book of Bezrukavnikov–Finkelberg–Schechtman (BFS) [Factorizable sheaves and quantum groups](https://arxiv.org/abs/q-alg/9712001) produces sheaves over certain stacks $\mathcal{M}\_{A, \delta}$, which are, roughly, (a line bundle over) moduli stacks of curves with marked points **and** non-zero tangent vectors at these marked points; see for example Section 1.5 on page 9.


Question: how are these two constructions related?


I believe that they should be closely related, and this is perhaps well known to experts. I am confused by the appearance of tangent vectors in the moduli stacks $\mathcal{M}\_{A, \delta}$ above, though I have perhaps misunderstood what are supposed to be the conformal blocks in BFS. Any comments or references would be greatly appreciated!



Relation between TQFT representations and factorizable sheaves
To sum up: these tangent vectors are always present. In the literature on WZW one usually choose local formal coordinates at the marked point, ie an identification of the neighborhood of those with a chosen formal punctured disc, but this amount to essentially the same thing. The reason is compatibility with gluing: the moduli space of algebraic curves of genus $g$ with $n$ marked points with tangent vectors, is equivalent to the moduli space of genus $g$ surfaces with $n$ boundary components **together with a parametrization of the boundary** which is what you need to glue.


Btw, I wouldn't say BFS is a rigorous construction of the conformal blocks of the WZW model (this can already be made rigorous). Rather, my understanding is that it is a geometric construction of the modular functor associated with the modular tensor category of a quantum group at root of unity (hence, in particular, a direct geometric construction of that category as well). I think the point is to have a construction that looks like the construction of conformal blocks in WZW but on the quantum group side.


Once you know the modular tensor categories you get as the value of the circle for either of those constructions are the same, you can use e.g. Andersen-Ueno result (<https://arxiv.org/abs/math/0611087>) that a modular functor (in a somewhat restrictive sense) is determined by its genus 0 part to conclude these are equivalent.


