Spectral data for spin Higgs bundles

TL;DR

This paper characterizes spectral data for spin and twisted spin Higgs bundles in generic Hitchin fibers, linking spectral variations to Hecke transformations and confirming duality relations from the geometric Langlands program.

Contribution

It provides an explicit description of spectral data for spin Higgs bundles, connecting spectral variations with Hecke transformations and duality in Hitchin fibers.

Findings

Spectral data parametrizes generic Hitchin fibers for spin Higgs bundles.

Spectral variations relate to Hecke transformations for orthogonal bundles.

Fibers of the Hitchin map are dual abelian varieties in the geometric Langlands context.

Abstract

In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and "twisted" spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case).