Hyper-Kähler manifolds from Riemann-Hilbert problems I: Ooguri-Vafa-like model geometries

TL;DR

The paper develops a rigorous GMN-based framework to construct hyper-Kähler metrics near semi-flat limits by building local model geometries that generalize the Ooguri–Vafa construction. Starting from semi-flat data on a Coulomb-base $ obreak \\mathcal{B}$ and a lattice extension $0\to \Gamma_f\to \widehat{\Gamma}\to \Gamma\to 0$, it constructs a semi-flat model $ obreak \mathcal{M}'\to \mathcal{B}'$ with holomorphic Darboux coordinates and a $ obreak \mathbb{C}^ imes$-family of holomorphic symplectic forms $\varpi^{sf}(\zeta)$, then applies a GMN-style integral construction and twistor methods to produce model OV-like geometries. It introduces a detailed set of assumptions near the discriminant locus to guarantee convergence and the existence of smooth model geometries, and develops a two-step extension to a smooth manifold $\mathcal{M}\to \mathcal{B}$ via a generalized Gibbons–Hawking Ansatz, culminating in a model geometry that captures the higher-rank and non-unimodular lattice cases. Together, these developments lay the groundwork for proving global hyper-Kähler metrics via iterative GMN procedures in subsequent work and provide refined control on local degenerations and collapses to semi-flat limits.

Abstract

We construct model hyper-Kähler geometries that include and generalize the multi-Ooguri-Vafa model using the formalism of Gaitto, Moore, and Neitzke. This is the first paper in a series of papers making rigorous Gaiotto--Moore--Neitzke's formalism for constructing hyper-Kähler metrics near semi-flat limits. In that context, this paper describes the assumptions we will make on a sequence of lattices $0 \to Γ_{f} \to \widehatΓ \to Γ\to 0$ over a complex manifold $\mathcal{B}'=\mathcal{B} - \mathcal{B}''$ near the singular locus, $\mathcal{B}''$, in order to define a smooth manifold $\mathcal{M} \to \mathcal{B}$ and hyper-Kähler model geometries on neighborhoods of points of the singular locus. In follow-up papers, we will use a modified version of Gaiotto-Moore-Neitzke's iteration scheme starting at these model geometries to produce true global hyper-Kähler metrics on $\mathcal{M}$.

Figures (4)

• Figure 2: The map $\mathcal{X}^{\mathrm{sf}}(\zeta)$ is naturally a section of $\pi^* \mathcal{T}'_\zeta$. Lemmata \ref{['lem:interp1']}, \ref{['lem:interp2']}, \ref{['lem:interp3']} relate the holomorphic symplectic form $\varpi^\mathrm{sf}(\zeta)$ on $\mathcal{M}'$ to the holomorphic symplectic forms $\varpi_0$ on $\mathcal{T}'_u$ and $\varpi_\zeta$ on $\mathcal{T}'_\zeta$ (all shown in green) via various restrictions and trivializations (shown in gray) and pullbacks.
• Figure 6: The map $p: \widetilde{M'} \to Y$ described in Lemma \ref{['lem:gwOV']}\ref{['it:d']}-\ref{['it:bundle']}is shown for Ooguri-Vafa. More precisely, note that picture is of this is the $\mathbb{Z}^r$ universal cover of $\Theta$, with implicit $\mathbb{Z}^r$ action.
• Figure 7: (Left) Sectorial decomposition $\mathcal{V}=\{V_A\}_{A=1}^{2K}$ of $\mathbb{C}_\zeta^\times$ with rays $r_A$ labelled (Right)$V_A$ contains $r_{A-1}$ but not $r_A$
• Figure 8: $Z_{\gamma}(u) = Z^\parallel_{\gamma}(u) + Z^\perp_{\gamma}(u)$

Theorems & Definitions (110)

• Theorem 1.3: Theorem 3.16b of FZ:twistor
• Definition 1.5
• Proposition 1.6: Proposition 3.4 of FZ:twistor
• Corollary 1.8
• proof
• Lemma 2.1: \ref{['item:Z']} revisited
• Remark 2.5: Period matrix $\tau$
• proof
• Corollary 2.8: global consequence of \ref{['item:Z']}
• Lemma 2.9