Unitarity of SL(2)-conformal blocks in genus zero

It has been conjectured for quite some time that a bundle of conformal blocks carries a unitary structure that is (projectively) flat for the Hitchin connection. This was recently established by T.R. Ramadas in the simplest nontrivial case, namely where the genus is zero and the group is SL(2). In this paper we present a shorter and more direct version of his proof. We also identify the conformal block space with the bidegree (N$N$, 0)-part of an eigenspace of a finite group acting on a Hodge structure of weight N$N$.