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

TL;DR

This work provides precise local models for the special Kähler metric $\omega_{\mathrm{SK}}$ on the SL$_2(\mathbb{C})$ Hitchin base near discriminant-strata where spectral curves are nodal. Using a carefully constructed family of integration contours on the normalized spectral curves and their deformations to nearby fibers, the authors show that $\omega_{\mathrm{SK}}$ exhibits logarithmic asymptotics in transverse directions while the tangential component converges to the Hitchin-subsystem metric; along complex radial lines through the strata the metric reduces to a flat cone with cone angle $\pi$, and the Kähler potential extends continuously (even $C^1$ on parts) to the strata. The paper treats three main strata, $\mathcal{B}_d$ ($0\le d\le 2g-3$), $\mathcal{B}_{2g-2}$, and $\mathcal{B}_{\mathrm{ab}}$, deriving explicit local models, tangent-limit results, and potential extensions $\mathcal{K}_d$, $\mathcal{K}_{2g-2}$, and $\mathcal{K}_{\mathrm{ab}}$, respectively. These results connect the asymptotics of the Hitchin metric to Hitchin’s subintegrable systems and parallel lines of inquiry on metric collapse of hyperkähler manifolds, providing a robust framework for understanding discriminant singularities in rank-2 Hitchin systems.

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 $ω_{\mathrm{SK}}$ 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 $π$ at the origin only. Finally, the special Kähler potential extends continuously to these strata, and is $C^1$ on a portion of them.

Figures (5)

• Figure 1: Configuration of $\ell_{2k}|_{q_0}$ (orange) on $C$
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Theorems & Definitions (71)

• Definition 1.1
• Theorem 1.2: Theorem \ref{['main Bd']}, Theorem \ref{['main B2g-2']}
• Theorem 1.3: Theorem \ref{['main Bab']}
• Corollary 1.4: Corollaries \ref{['metric Bd']}, \ref{['metric B2g-2']}, and \ref{['metric Bab']}
• Corollary 1.5: Corollaries \ref{['extension Bd']}, \ref{['extension B2g-2']}, and \ref{['metric Bab']}
• Proposition 1.6: Corollaries \ref{['radial metric']}, \ref{['radial metric B2g-2']}, and \ref{['radial metric Bab']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof