Strange duality and the Hitchin/WZW connection

Abstract

For a compact Riemann surface $X$ of positive genus, the space of sections of a certain theta bundle on moduli of bundles of rank $r$ and level $k$ admits a natural map to (the dual of) a similar space of sections of rank $k$ and level $r$ (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as $X$ varies in a family. We prove that this map is (projectively) flat.