| Description |
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).
|