Higgs bundles and SYZ geometry

TL;DR

This paper constructs numerous hyperbolic affine spheres using non-Abelian Hodge theory, leading to new semi-flat Calabi-Yau metrics on certain torus bundles, thus addressing a question in SYZ geometry.

Contribution

It introduces a novel method to generate hyperbolic affine spheres with specific monodromy, expanding the understanding of SYZ mirror symmetry and Calabi-Yau metrics.

Findings

Constructed infinitely many non-congruent hyperbolic affine spheres.

Produced non-isometric semi-flat Calabi-Yau metrics on torus bundles.

Answered a previously open question in SYZ geometry.

Abstract

Using non-Abelian Hodge theory for parabolic Higgs bundles, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathrm{SL}_3(\mathbb{Z})$. These give rise to non-isometric semi-flat Calabi-Yau metrics on special Lagrangian torus bundles over an open ball in $\mathbb{R}^{3}$ minus a Y-vertex, thereby answering a question raised by Loftin, Yau, and Zaslow in [LYZ], [LYZerr].