Generalized theta functions, strange duality, and odd orthogonal bundles on curves

TL;DR

This paper explores generalized theta functions for odd orthogonal bundles, proving a Verlinde formula, analyzing strange duality failures, and establishing flat projective connections on moduli spaces.

Contribution

It introduces a Verlinde type formula for twisted spin bundles, demonstrates the reducibility of certain representations, and addresses rank-level duality questions.

Findings

Proved a Verlinde type formula for odd orthogonal bundles.

Showed the naive strange duality conjecture fails in general.

Established flat projective connections on moduli spaces of twisted spin bundles.

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.