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).
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).