Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0

by Prof. Prakash Belkale (University of North Carolina, USA)

Thursday 4 Aug 2011, 16:00 → 17:00 Asia/Kolkata

AG-69 (Colaba Campus)

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).