Orthogonal bundles, theta characteristics and the symplectic strange duality

TL;DR

This paper proves the symplectic strange duality conjecture for all curves of genus at least two by establishing projective flatness of a basis for generalized theta functions and leveraging Abe's proof for generic curves.

Contribution

It demonstrates the projective flatness of theta function bases over moduli spaces and confirms the symplectic strange duality conjecture for all relevant curves.

Findings

Basis for theta functions is projectively flat over moduli space.

Symplectic strange duality holds for all genus ≥ 2 curves.

Utilizes Abe's proof for generic curves to extend the result.

Abstract

A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the moduli space of curves (for Hitchin's connection). The symplectic strange duality conjecture, conjectured by Beauville is shown to hold for all curves of genus at least two, by using Abe's proof of the conjecture for generic curves, and the above monodromy result.