SIGMA 22 (2026), 022, 24 pages      arXiv:2310.15716      https://doi.org/10.3842/SIGMA.2026.022

Visible Lagrangians for Hitchin Systems and Pillowcase Covers

Johannes Horn and Johannes Schwab
Goethe-Universität Frankfurt am Main, Institut für Mathematik, Robert-Mayer-Str. 6-8, 71732 Frankfurt, Germany

Received September 18, 2025, in final form February 18, 2026; Published online March 09, 2026

Abstract
We study complex Lagrangians in Hitchin systems that factor through a proper subvariety of the Hitchin base non-trivially intersecting the regular locus. This gives a general framework for several examples in the literature. We compute the fiber-wise Fourier-Mukai transform of flat line bundles on visible Lagrangians. This proposes a construction of mirror dual branes to visible Lagrangians. Finally, we study a new example of visible Lagrangians in detail. Such visible Lagrangian exists whenever the underlying Riemann surface is a pillowcase cover. The proposed mirror dual brane turns out to be closely related to Hausel's toy model.

Key words: Higgs bundles; flat surfaces; mirror symmetry.

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