  Given a simple, simply connected, complex algebraic group G, a flat
projective connection on the bundle of nonabelian theta functions on the moduli
space of semistable parabolic G-bundles over any family of smooth projective
curves with marked points was constructed by the authors in an earlier paper.
Here, it is shown that the identification between the bundle of nonabelian
theta functions and the bundle of WZNW conformal blocks is flat with respect to
this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an
application, we give a geometric construction of the Knizhnik-Zamolodchikov
connection on the trivial bundle over the configuration space of points in the
projective line whose typical fiber is the space of invariants of tensor
product of representations.

Algebraic Geometry;High Energy Physics - Theory;Differential Geometry;Quantum Algebra;Representation Theory