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Visible Lagrangians for Hitchin Systems and Pillowcase Covers

Johannes Horn, Johannes Schwab

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.

Visible Lagrangians for Hitchin Systems and Pillowcase Covers

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.
Paper Structure (12 sections, 19 theorems, 29 equations, 4 figures)

This paper contains 12 sections, 19 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Symplectic geometry
  3. Completely integrable systems
  4. Visible Lagrangians
  5. Hitchin systems
  6. Preliminaries about the $\boldsymbol{\mathrm{GL}(n,\mathbb{C})}$ and $\boldsymbol{\mathrm{SL}(n,\mathbb{C})}$-Hitchin systems
  7. Langlands duality for Hitchin systems
  8. Visible Lagrangians in Hitchin systems
  9. Parallelogram-tiled surfaces and pillowcase covers
  10. Multifold pillowcase covers
  11. Visible Lagrangians over lines in the Hitchin base
  12. The Fourier--Mukai dual

Key Result

Theorem 1.1

Let $\mathcal{L} \subset \mathcal{M}_G$ be a visible Lagrangian, such that $\mathcal{B}'=\mathrm{Hit}(\mathcal{L}) \subsetneq \mathcal{B}_G$ and $\mathcal{B}' \cap \mathcal{B}^\mathrm{reg}_G \neq \varnothing$. Let $s\colon \mathcal{B}' \rightarrow \mathcal{L}$ be a section of $\mathrm{Hit}\: \,{}_{\

Figures (4)

  • Figure 1: Langlands duality between $\mathrm{SL}(n,\mathbb{C})$-Hitchin system and $\mathrm{PGL}(n,\mathbb{C})$-Hitchin system.
  • Figure 2: Pillowcase with canonical cover: Opposite sides identified, when not indicated otherwise. The involution on the cover acts as central symmetry in the two-torsion points.
  • Figure 3: A pillowcase.
  • Figure 4: Two non-isomorphic pillowcase covers on the same curve $X$.

Theorems & Definitions (32)

  • Theorem 1.1: Theorem \ref{['theo:FM']}
  • Theorem 1.2: Corollary \ref{['coro:visible_Lagr_over_line_sl2C']}
  • Theorem 1.3: Proposition \ref{['prop:non-isomorphic_pillow']}
  • Theorem 1.4: Corollary \ref{['coro:hyperholomorphic']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: visible Lagrangians
  • Theorem 2.4
  • Proposition 3.1: Baraglia
  • Proposition 3.2
  • ...and 22 more