**Prakash Belkale**

University of North Carolina

August 4, 2011

**Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0**:
Associated to a (finite dimensional, simple) Lie algebra, and
a finite set of irreducible representations (and a level), there are vector bundles of conformal blocks on suitable moduli spaces of curves with marked points. These conformal block bundles carry flat projective connections (KZ/Hitchin).
We prove that conformal block bundles in genus zero (for arbitrary simple Lie algebras) carry geometrically defined unitary metrics (of
Hodge-theoretic origin, as conjectured by Gawedzki) which are preserved by the KZ/Hitchin connection. Our proof builds upon the work of Ramadas who proved this unitarity statement in the case of the Lie algebra sl(2) (and genus zero).