Higgs bundles and SYZ geometry

Abstract

Using non-Abelian Hodge correspondence for parabolic Higgs bundles and surface group representations, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathbf{SL}_3 (\mathbb{Z})$. These give rise to non-isometric semi-flat Calabi–Yau metrics on special Lagrangian torus bundles over an open ball in $\mathbb{R}^3$ with a $\mathrm{Y}$-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow in [$\href{https://dx.doi.org/10.4310/jdg/1143644314}{32}$, $\href{https://doi.org/10.48550/arXiv.math/0405061}{33}$].