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The Ooguri-Vafa Space as a Moduli Space of Framed Wild Harmonic Bundles

Iván Tulli

TL;DR

This work identifies the Ooguri--Vafa space $\mathcal{M}^{\rm ov}$ as a local hyperkähler model for Hitchin moduli near generic singular fibers by realizing $\mathcal{M}^{\rm ov}$ as a moduli space $\mathfrak{X}^{\rm fr}$ of rank-2 framed wild harmonic bundles on $\mathbb{C}P^1$ with one irregular singularity. The construction builds a twistor family of holomorphic Darboux coordinates from Stokes data of associated framed flat bundles, and shows $\mathcal{X}_e(\xi)=\mathcal{X}_e^{\rm ov}(\xi)$ and $\mathcal{X}_m(\xi)=\mathcal{X}_m^{\rm ov}(\xi)$ under an explicit parameter matching, thereby identifying $\mathcal{M}^{\rm ov}_*$ with $\mathfrak{X}^{\rm fr}_*(\mathcal{B})$. A careful analysis of holomorphic dependence, asymptotics, and non-vanishing of Stokes data establishes a robust, framed wild-Higgs-to-flat-bundle correspondence; the resulting hyperkähler structure on $\mathfrak{X}^{\rm fr}_*(\mathcal{B})$ extends to the OV central fiber, mirroring Gross--Wilson/Gross–Wilson-type degenerations. The main result is a precise, one-to-one correspondence between a framed wild-Higgs moduli and OV space, supporting GMN’s program linking Hitchin moduli metrics to local OV-like models and providing a blueprint for analogous constructions for other wild moduli problems.

Abstract

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri-Vafa space with the Hitchin moduli spaces, we show that the Ooguri-Vafa space can be interpreted as a set of rank 2, framed wild harmonic bundles over $\mathbb{C}P^1$, with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperkähler geometry of the Ooguri-Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles.

The Ooguri-Vafa Space as a Moduli Space of Framed Wild Harmonic Bundles

TL;DR

This work identifies the Ooguri--Vafa space as a local hyperkähler model for Hitchin moduli near generic singular fibers by realizing as a moduli space of rank-2 framed wild harmonic bundles on with one irregular singularity. The construction builds a twistor family of holomorphic Darboux coordinates from Stokes data of associated framed flat bundles, and shows and under an explicit parameter matching, thereby identifying with . A careful analysis of holomorphic dependence, asymptotics, and non-vanishing of Stokes data establishes a robust, framed wild-Higgs-to-flat-bundle correspondence; the resulting hyperkähler structure on extends to the OV central fiber, mirroring Gross--Wilson/Gross–Wilson-type degenerations. The main result is a precise, one-to-one correspondence between a framed wild-Higgs moduli and OV space, supporting GMN’s program linking Hitchin moduli metrics to local OV-like models and providing a blueprint for analogous constructions for other wild moduli problems.

Abstract

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri-Vafa space with the Hitchin moduli spaces, we show that the Ooguri-Vafa space can be interpreted as a set of rank 2, framed wild harmonic bundles over , with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperkähler geometry of the Ooguri-Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles.

Paper Structure

This paper contains 38 sections, 61 theorems, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary and strategy of the paper
  3. The Ooguri--Vafa space
  4. The Gibbons--Hawking ansatz
  5. Construction of the Ooguri--Vafa space
  6. Twistor coordinates
  7. Some properties of the magnetic twistor coordinate
  8. Framed wild harmonic bundles
  9. Filtered bundles on curves
  10. Wild harmonic bundles and their associated filtered objects
  11. Definition of our set of framed wild harmonic bundles
  12. Twistor coordinates Part 1: Preliminaries and definition
  13. Stokes data
  14. Associated compatibly framed flat bundles
  15. Stokes data of the associated compatibly framed flat bundles
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}^{\textnormal{ov}}$ be the Ooguri--Vafa space with cut-off $\Lambda=4{\rm i}$, and let $\mathcal{B}\subset \mathbb{C}$ be the base of the singular torus fibration $\mathcal{M}^{\textnormal{ov}}\to \mathcal{B}$. Fixing an affine coordinate $z\in \mathbb{C}\subset \mathbb{C}P^1$, let and let $\mathfrak{X}^{\textnormal{fr}}_*(\mathcal{B})\subset \mathfrak{X}^{\textnormal{fr}}(\mathca

Figures (3)

  • Figure 1: $\mathcal{M}^{\textnormal{ov}}$ as a singular torus fibration over $\mathcal{B}\subset \mathbb{C}$. The central fiber at $0\in \mathcal{B}$ degenerates to a torus with a node.
  • Figure 2: We illustrate the case where we put $q_1=-q_2=-\frac{1}{2w^2}$. The bold rays denote the anti-Stokes rays, while the dotted rays denote the Stokes rays.
  • Figure 5: Given $m \in \mathbb{C}^*$, the figure above shows the configuration of the half-planes $\mathbb{H}_{\pm m}$ and the rays $l_{\pm}(-2{\rm i}m)$.

Theorems & Definitions (92)

  • Theorem 1.1
  • Proposition 2.1: jumps of the twistor coordinate
  • Proposition 2.2: asymptotics of the twistor coordinate
  • Proposition 2.3: reality condition
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 82 more