Local Models for Special K\"ahler Metric Singularities Along the Discriminant Locus of the $\mathrm{SL}_2(\mathbb{C})$ Hitchin Base

TL;DR

This paper analyzes the singular behavior of special K"ahler metrics near the discriminant locus of the $ ext{SL}_2(C)$ Hitchin base, revealing logarithmic asymptotics and metric extensions on strata with nodal spectral curves.

Contribution

It provides a detailed description of the local structure and asymptotics of special K"ahler metrics near discriminant components with nodal spectral curves in the Hitchin base.

Findings

The metric exhibits logarithmic asymptotics near the discriminant strata.

Along complex lines, the metric restricts to a cone with angle $\pi$ at the origin.

The K"ahler potential extends continuously and is $C^1$ on parts of the strata.

Abstract

Freed (arXiv:hep-th/9712042) formulated special K\"ahler structures; in particular, the regular locus of the $\mathrm{SL}_2(\mathbb{C})$ Hitchin base $\mathcal{B}$ carries such a structure, while the associated metric $\omega_{\mathrm{SK}}$ is singular along the discriminant locus $\mathcal{D}$. Baraglia-Huang (arXiv:1707.04975) computed its Taylor expansion near points of $\mathcal{B}\setminus\mathcal{D}$. Hitchin (arXiv:1712.09928) then defined subsystems attached to those components of $\mathcal{D}$ whose spectral curves have only nodal singularities; these components form smooth strata with induced special K\"ahler structures. We show that near such a stratum the canonical special K\"ahler metric has logarithmic asymptotics in transversal directions, whereas its tangential part converges to a metric on the stratum agreeing with the one from Hitchin's subsystems. Along any complex line through the origin of $\mathcal{B}$ and a point of the stratum, the metric restricts to a cone flat metric with cone angle $\pi$ at the origin only. Finally, the special K\"ahler potential extends continuously to these strata, and is $C^1$ on a portion of them.