I recently asked this question on phys.SE and it was suggested to me to ask it here.
One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces.
Can one perform a similar construction for affine Lie algebras? In particular, I am interested in two features:
Can the central extension be understood in terms of the geometry of the loop space $\mathcal{L}G$?
Can the representation theory of these algebras be understood in terms of similar geometry?
More broadly, central extensions of infinite-dimensional algebras feature often in conformal field theories. Usually, these are understood as a consequence of the fact that symmetry groups need only be represented projectively in quantum mechanics. Is there a more geometric point of view of all this?
As a final note, in Witten's paper on non-abelian bosonization, he mentions in footnote 6 that the loop space carries a complex structure and the symplectic structure of the WZW model represents the first Chern class of a certain holomorphic line bundle (essentially what I described in the first part of the question). However, he does not go into much more detail on this point of view (at least in this paper) and mainly sticks to discussing the classical (Poisson bracket) version of the symmetry algebra of the WZW model. Wouldn't it be much more straightforward (and less handwaving) to start from this view and geometrically quantize the holomorphic structure to obtain the full quantum commutators?
