[Paper Review] SYZ Mirror Symmetry of Hitchin's Moduli Spaces Near Singular Fibers I

Wenxuan Lu|arXiv (Cornell University)|Aug 18, 2012

Nonlinear Waves and Solitons16 references3 citations

TL;DR

This paper establishes that certain four-dimensional Hitchin moduli spaces degenerate to Ooguri-Vafa local models near singular fibers of the Hitchin fibration, and demonstrates that the wall-crossing behavior of hyperkahler metrics and hyperholomorphic connections in the Gaiotto-Moore-Neitzke construction matches that of these local models. The key contribution is a conjectural metrical mirror symmetry for the local geometry, linking instanton corrections and BPS state counting to SYZ duality near singular fibers.

ABSTRACT

We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near singular fibers of the Hitchin's fibration are understood.

Motivation & Objective

• To understand the behavior of hyperkahler metrics and hyperholomorphic connections near singular fibers of Hitchin’s moduli spaces.
• To identify local models—specifically Ooguri-Vafa spaces—that capture the asymptotic geometry of four-dimensional Hitchin moduli spaces near singular fibers.
• To match the wall-crossing behavior of the full geometry with that of the local model, particularly in the context of Gaiotto-Moore-Neitzke's 2d-4d wall crossing formula.
• To propose a conjectural metrical mirror symmetry for the local model, relating the hyperkahler quotient metric on the mirror to the Gaiotto-Moore-Neitzke construction.
• To lay the groundwork for studying SYZ mirror symmetry of branes and connections in the context of geometric Langlands duality.

Proposed method

• Uses the Gaiotto-Moore-Neitzke formalism to construct hyperkahler metrics and hyperholomorphic connections on SU(2) Hitchin moduli spaces via twistor techniques and wall-crossing formulas.
• Applies the Ooguri-Vafa metric as a local model for the hyperkahler structure near singular fibers, particularly at the point where the spectral curve degenerates to a nodal curve.
• Matches BPS state counts (via $Ω(\gamma)$) and central charges ($Z_\gamma$) between the full Hitchin moduli space and the Ooguri-Vafa space, showing consistency in wall-crossing behavior.
• Employs the Gibbons-Hawking ansatz and Picard-Fuchs equations to derive the asymptotic form of the central charge and identify the monodromy-induced $Δ = 4$ in the Ooguri-Vafa metric.
• Conjectures that the holomorphic symplectic form on the mirror moduli space $̅{\mathcal{M}}^0$ converges to that of the Ooguri-Vafa space as $R \to \infty$, with exponentially decaying corrections.
• Analyzes the SYZ mirror symmetry of the local model by considering the quotient $̅{\mathcal{M}}^0 = \mathcal{M}^0 / Q$, and proposes a conjectural isometry between the quotient metric and the induced hyperkahler metric.

Experimental results

Research questions

• RQ1 How does SYZ mirror symmetry act on hyperkahler metrics and hyperholomorphic connections in the Gaiotto-Moore-Neitzke construction?
• RQ2 What is the role of wall-crossing phenomena in the geometric Langlands program and how does it relate to instanton corrections?
• RQ3 Can Ooguri-Vafa spaces serve as local models for the degeneration of Hitchin’s moduli spaces near singular fibers?
• RQ4 How do the BPS state counts and central charges in the full geometry compare with those in the Ooguri-Vafa local model?
• RQ5 What is the metrical structure of the mirror geometry near singular fibers, and does it match the Gaiotto-Moore-Neitzke construction?

Key findings

• The Ooguri-Vafa space provides a valid local model for the hyperkahler metric of four-dimensional Hitchin moduli spaces near singular fibers, particularly at the nodal degeneration point.
• The wall-crossing behavior of the full geometry is matched by the local Ooguri-Vafa model, with the BPS index $\Omega(\gamma_a) = 1$ corresponding to a flip between simple zeros.
• The central charge $Z_{\gamma_m}$ in the Ooguri-Vafa space is derived from Picard-Fuchs equations, yielding a logarithmic divergence and $\Delta = 4$, consistent with the Gibbons-Hawking ansatz.
• The 4d wall-crossing formula is trivial in the Ooguri-Vafa model, indicating that the local model simplifies the full geometry while preserving essential features of the wall-crossing structure.
• Conjecture 6.2 proposes that the holomorphic symplectic form on the mirror space $\bar{\mathcal{M}}^0$ converges to that of the Ooguri-Vafa space as $R \to \infty$, with exponentially decaying corrections of order $O(c_1 R e^{-c_2 R})$.
• The value $\Omega_1 = 2$ is conjectured based on monodromy considerations, implying $\Delta = 2$ in the mirror, and consistent with the requirement that only $q=1$ contributes to the sum in the mirror fiber.