The Hitchin fibration under degenerations to noded Riemann surfaces

TL;DR

This paper investigates the behavior of the Hitchin fibration and associated linearized equations as Riemann surfaces degenerate to nodal surfaces, revealing stability of the Fredholm index and comparing with large Higgs field limits.

Contribution

It demonstrates the graph-continuity and stability of the Fredholm index of the linearized Hitchin fibration during degenerations to nodal surfaces, and compares these properties with large Higgs field limits.

Findings

The linearized Hitchin fibration forms a graph-continuous Fredholm family during degeneration.

The Fredholm index remains stable in the degeneration process.

Similarities and differences are identified between degenerations and large Higgs field limits.

Abstract

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces $\Sigma_R$ converging for $R\searrow0$ to a surface $\Sigma_0$ with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields.