Invariant vector bundles and Hitchin systems

We obtain explicit formulae for the Donagi–Markman (Bryant–Griffiths, Yukawa) cubic for Hitchin systems of type A2, B2 and G2. This is achieved by evaluating the quadratic residues in the Balduzzi–Pantev formula, using a previous result of ours. For G2 we also recover earlier results of Hitchin.

Let X be a compact Riemann surface of genus g≥2, G be a complex semisimple Lie group, and MG(X) be the moduli space of stable principal G-bundles. This paper studies the fixed point set of involutions on MG(X) induced by an anti-holomorphic involution τ on X and a Cartan involution θ of G, producing an involution σ=θ∗∘τ∗. These fixed points are shown to correspond to stable GR-bundles over the real curve (Xτ,τ), where GR is the real form associated with θ. The fixed point set MG(X)σ consists of exactly 2r connected components, each a smooth complex manifold of dimension (g−1)dimG2, where r is the rank of the fundamental group of the compact form of G. A cohomological obstruction in H2(Xτ,π1(GR)) characterizes which bundles are fixed. A key result establishes a derived equivalence between coherent sheaves on MG(X)σ and on the fixed point set of the dual involution on the moduli space of G∨-local systems, where G∨ denotes the Langlands dual of G. This provides an extension of the Geometric Langlands Correspondence to settings with involutions. An application to the Chern–Simons theory on real curves interprets MG(X)σ as a (B,B,B)-brane, mirror to an (A,A,A)-brane in the Hitchin system, revealing new links between real structures, quantization, and mirror symmetry.

We establish an isomorphism of complex K -theory of the moduli space \widecheck{\mathcal{M}} of “ \mathrm{SL}_{n} ”-Higgs bundles of degree d and rank n (in the sense of Hausel–Thaddeus) and twisted complex K -theory of the orbifold \widehat{\mathcal{M}} of \mathrm{PGL}_{n} -Higgs bundles of degree e , where (n,d)=(n,e)=1 . Along the way, we prove the vanishing of torsion for H^{*}(\widecheck{\mathcal{M}}) and certain twisted complex K -theory groups of \widehat{\mathcal{M}} . We also extend Arinkin’s autoduality of compactified Jacobian to a derived equivalence between \mathrm{SL}_{n} - and \mathrm{PGL}_{n} -Hitchin systems over the elliptic locus. In the appendix, we develop a formalism of G -sheaves of spectra, generalising equivariant homotopy theory to a relative setting.

The workshop addressed a broad range of subjects in algebraic geometry. Recent results on moduli spaces of various types (of sheaves, complexes, varieties) played a prominent role in many of the talks, as well as derived categories of coherent sheaves. A number of talks were devoted to the geometry of special varieties (Calabi–Yau, hyperkähler) and to the geometry of the Hitchin system.

Mirror of orbifold singularities in the Hitchin fibration: The case (SL ,PGL )

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri-Vafa space with the Hitchin moduli spaces, we show that the Ooguri-Vafa space can be interpreted as a set of rank 2, framed wild harmonic bundles over $\mathbb{C}P^1$, with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperkähler geometry of the Ooguri-Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles.

ABSTRACT We give a complete, self-contained computation of the spectral data parametrizing Higgs bundles in the generic fibres of $\mathrm{SO}_{2n+1}$-Hitchin fibration where the Higgs fields are L-twisted endomorphisms. Although the spectral data are known in the literature, we develop a new approach, which takes advantage of Hecke modification. Furthermore, we present the computation for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n}$ cases while clarifying some aspects of the correspondence, which are not well explained in the pre-existing literature. We also compute the number of connected components of the generic fibres and demonstrate Langlands duality in the fibres via the canonical duality in the fibres.

Compactifying the rank 2 Hitchin system via spectral data on semistable curves

We study the summands of the decomposition theorem for the Hitchin system for \operatorname{GL}_{n} , in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.

Separation of Variables for Hitchin Systems with the Structure Group $$\mathrm{SO}(4)$$ on Genus $$2$$ Curves

For any almost-simple group [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero, we describe the automorphism group of the moduli space of semistable [Formula: see text]-bundles over a connected smooth projective curve [Formula: see text] of genus at least [Formula: see text]. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

Abstract We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.

Abstract For an irreducible representation $\rho :\pi _{1}(\Sigma _{g})\to \textrm{GL}(n,\mathbb{C})$, there is an energy functional $\textrm{E}_{\rho }: {{\mathcal{T}}}_{g}\to \mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\textrm{GL}(n,\mathbb{C})/\textrm{U}(n)$. It follows from a result of Toledo that $\textrm{E}_{\rho }$ is plurisubharmonic, that is, its Levi form is positive semi-definite. We describe the kernel of this Levi form, and relate it to the $\mathbb{C}^{*}$ action on the moduli space of Higgs bundles. We also show that the points in $ {{\mathcal{T}}}_{g}$ where strict plurisubharmonicity fails (i.e., this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,\rho )$, the map $\textrm{E}_{\rho }$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

We give a survey of some recent advances in parabolic Hitchin systems (parabolic Beauville-Narasimhan-Ramanan correspondence, mirror symmetry for parabolic Hitchin systems), and in exact methods of solving the non-parabolic Hitchin systems. Bibliography: 55 titles.

Abstract Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map $\overline{\Xi }$ between the algebraic compactification of the Dolbeault moduli space of ${\rm SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action and the analytic compactification of Hitchin’s moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that $\overline{\Xi }$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration.

Abstract Let $X$ be an $n$ -dimensional (smooth) intersection of two quadrics, and let ${T^{\rm{*}}}X$ be its cotangent bundle. We show that the algebra of symmetric tensors on $X$ is a polynomial algebra in $n$ variables. The corresponding map ${\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n}$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a $2$ -torsion subgroup. In dimension $3$ , ${\rm{\Phi }}$ is the Hitchin fibration of the moduli space of rank $2$ bundles with fixed determinant on a curve of genus $2$ .

In this paper, we talk about parahoric Hitchin systems over smooth projective curves with the structure group a semisimple simply connected group. We prove the equivalence of parahoric Hitchin systems over the curve with Hitchin systems over a corresponding root stack with a finite cyclic group action determined (up to conjugation) by the parahoric data. And we also show the compatibility of the equivalence with Hitchin maps. We work over an algebraically closed field with a mild assumption of the characteristic.

Perverse filtrations, Chern filtrations, and refined BPS invariants for local [formula omitted