Higgs bundles and SYZ geometry

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].