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

Abstract

Freed (arXiv:hep-th/9712042) formulated special Kähler 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 Kähler structures. We show that near such a stratum the canonical special Kähler 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 Kähler potential extends continuously to these strata, and is $C^1$ on a portion of them.

This version: added references and an example, expanded proof details. Fixed typos, updated figures