[Paper Review] Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli Spaces

Wenxuan Lu|arXiv (Cornell University)|Oct 17, 2010

Nonlinear Waves and Solitons50 references11 citations

TL;DR

This paper establishes a deep equivalence between two instanton correction problems in Hitchin's moduli spaces: corrections to the hyperkähler metric (via Fock-Goncharov coordinates and quadratic differential foliations) and corrections to the complex structure (via Gross-Siebert log Gromov-Witten theory). The key result is that the wall-crossing formula of Kontsevich and Soibelman unifies both problems, providing a nontrivial mirror symmetry statement for Hitchin moduli spaces beyond geometric Langlands.

ABSTRACT

We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense via the identification of the wall crossing formulas in the metric problem with consistency conditions in the complex structure problem. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined.

Motivation & Objective

• To understand instanton corrections to the hyperkähler metric of Hitchin's moduli spaces using special coordinates derived from quadratic differential foliations.
• To analyze instanton corrections to the complex structure of Calabi-Yau varieties via Gross-Siebert's log Gromov-Witten theory.
• To establish a precise mathematical equivalence between the two instanton correction problems in Hitchin's moduli spaces.
• To demonstrate that the wall-crossing formula of Kontsevich and Soibelman serves as a unifying consistency condition and computational tool across both problems.
• To provide explicit examples where instanton corrections for both metric and complex structure are fully determined, offering new Calabi-Yau examples in mirror symmetry.

Proposed method

• Constructing Fock-Goncharov coordinates on Hitchin's moduli space using foliations from quadratic differentials on a Riemann surface.
• Applying the wall-crossing formula of Kontsevich and Soibelman as a consistency condition for instanton corrections in the hyperkähler metric problem.
• Using Gross-Siebert's framework to determine instanton corrections to the complex structure of Calabi-Yau varieties from singular affine structures with log data.
• Identifying the wall-crossing formula in the metric problem with the consistency conditions in the complex structure problem, establishing equivalence.
• Employing cluster algebra structures and scattering diagrams to compute wall-crossing transformations via ordered products of automorphisms.
• Verifying equivalence through explicit examples, including the derivation of defining equations for the Calabi-Yau variety via variable transformations and cluster mutations.

Experimental results

Research questions

• RQ1 How are instanton corrections to the hyperkähler metric of Hitchin's moduli space related to those of the complex structure?
• RQ2 Can the wall-crossing formula of Kontsevich and Soibelman serve as a unifying mechanism between metric and complex structure corrections?
• RQ3 What is the precise mathematical equivalence between the Gross-Siebert construction of complex structure corrections and the Fock-Goncharov construction of metric corrections?
• RQ4 How do cluster algebra mutations and scattering diagrams encode the wall-crossing behavior in both problems?
• RQ5 What explicit Calabi-Yau varieties arise from this equivalence, and how are their defining equations derived?

Key findings

• The wall-crossing formula of Kontsevich and Soibelman provides a consistency condition that links instanton corrections in the hyperkähler metric and complex structure problems.
• The two instanton correction problems—metric via Fock-Goncharov coordinates and complex structure via Gross-Siebert log Gromov-Witten theory—are mathematically equivalent under the identification of wall-crossing structures.
• Explicit defining equations for a Calabi-Yau variety are derived as the intersection of two quadric hypersurfaces: $XY = (1+W)^2$, $ZW = (1+Y)^2$.
• The cluster transformation $y_2 = x_1^{-1}, x_2 = y_1(1+x_1)^2$ generates the same defining equations under variable reparameterization, confirming consistency.
• The wall-crossing formula is derived without truncations by following continuous evolution along a loop crossing stability walls in both directions, yielding the transformation $K_{ar{ heta}_1 + ar{ heta}_2}^{-2}$.
• The system of degenerations over $\mathrm{Spec}\, \mathbf{C}[t]/(t^{k+1})$ for increasing $k$ realizes a consistent, finite-order approximation of the full instanton correction.