Mirror symmetry and Hitchin systems on DM curves: SYZ Duality

TL;DR

This paper extends Higgs bundle theory to Deligne-Mumford curves and demonstrates SYZ duality for their Higgs moduli spaces, revealing new geometric dualities in the context of stacky curves.

Contribution

It generalizes classical Higgs bundle theory to DM curves and proves SYZ duality for their Higgs moduli spaces under mild conditions.

Findings

Established a generalized theory of Higgs bundles on DM curves.

Proved SYZ duality for moduli spaces of Higgs bundles on hyperbolic stacky curves.

Connected mirror symmetry concepts with stacky curve geometry.

Abstract

We generalize the classical theory of Higgs bundles, spectral curves and Norm maps to Deligne-Mumford curves. As an application, under some mild conditions, we prove the Strominger-Yau-Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.