Spinors, Twistors and Classical Geometry

The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results.

Topological mirror symmetry for rank two character varieties of surface groups

The moduli spaces of flat $${\text{SL}}_2$$ SL 2 - and $${\text{PGL}}_2$$ PGL 2 -connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.

Moduli Spaces of Generalized Hyperpolygons

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System

We describe spectral data for singular fibres of the $\textsf{SL}(2,{\mathbb{C}})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations, we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations, this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications, we study irreducible components of singular Hitchin fibres and give a description of $\textsf{SL}(2,{\mathbb{R}})$-Higgs bundles in terms of these semi-abelian spectral data.

Spectral Curves for the Triple Reduced Product of Coadjoint Orbits for SU(3)

This chapter gives an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus zero curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, the chapter identifies the moment map for a Hamiltonian circle action on the reduced product. Finally, the chapter makes use of results from Adams, Harnad and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

This chapter proposes a natural quantization condition for the Hitchin system and relate this to the generating function for the variety of opers within the Hitchin space of local systems. Links with the geometric Langlands programme are investigated.

Complete Integrability of the Parahoric Hitchin System

Let $\mathcal {G}$ be a parahoric group scheme over a complex projective curve X of genus greater than one. Let $\mathrm {Bun}_{\mathcal {G}}$ denote the moduli stack of $\mathcal {G}$-torsors on X. We prove several results concerning the Hitchin map on $T^{\ast }\!\mathrm {Bun}_{\mathcal {G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm {Bun}_{\mathcal {G}}$ is “very good” in the sense of Beilinson–Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.

Geometric quantization of the Hitchin system

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.