Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

Abstract

This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel–Whitney class and the associated space of twisted spin bundles. In particular, we prove a Verlinde type formula and a dimension equality that was conjectured by Oxbury–Wilson. Modifying Hitchin’s argument, we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection. We furthermore address the issue of strange duality for odd orthogonal bundles, and we demonstrate that the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. Finally, we answer a question of Nakanishi–Tsuchiya about rank-level duality for conformal blocks on the pointed projective line with spin weights.