Parabolic Higgs Bundles Research Articles

On the motives of moduli of parabolic chains and parabolic Higgs bundles

Analogous to the Higgs bundles on a Riemann surface which played an important role in the study of representations of the fundamental group of the surface, the parabolic Higgs bundles play also their importance in the study of the fundamental group but of the punctured surface. In this paper, we give an algorithm to calculate the (virtual) motive (i.e. in a suitable Grothendieck group) of the moduli spaces of parabolic Higgs bundles of fixed rank and fixed parabolic structure, using localization with respect to the circle action.

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

On motives of parabolic Higgs bundles and parabolic connections

Let [Formula: see text] be a compact Riemann surface of genus [Formula: see text] and let [Formula: see text] be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over [Formula: see text] with the parabolic structure over [Formula: see text]. For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their [Formula: see text]-polynomials are the same. We also show that the Voevodsky and Chow motives of these two moduli spaces are also equal. We showed that the Grothendieck motivic classes and the [Formula: see text]-polynomials of parabolic Higgs moduli and of parabolic Hodge moduli are closely related. Finally, we considered the moduli spaces with fixed determinants and showed that the above results also hold for the fixed determinant case.

Mirror symmetry for parabolic Higgs bundles via p-adic integration

Applying the technique of p-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups SLn and PGLn, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the E-polynomial of the smooth moduli space of parabolic GLn-Higgs bundles is independent of the degree of the underlying vector bundles.

Pullback and Direct Image of Parabolic Connections and Parabolic Higgs Bundles

Abstract We provide an explicit algebraic construction—for the pullback and direct image of parabolic bundles, parabolic Higgs bundles, and parabolic connections—through nonconstant maps between compact connected Riemann surfaces. We show that these constructions preserve semistability and polystability. We also prove that these constructions are compatible with the nonabelian Hodge correspondence.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Hitchin fibration under ramified coverings

We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map π:Y→X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y, the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to describe this map.

Picard group of moduli of parabolic Higgs bundles

Let X be a compact Riemann surface of genus and let be the moduli space of semistable parabolic bundles over X. Let denote the moduli space of semistable parabolic Higgs bundles over X. In this article, we study the Picard group of and .

Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.

Parabolic Higgs bundles and cyclic monopole chains

We formulate a correspondence between SU(2) monopole chains and “spectral data”, consisting of curves in CP1×CP1 equipped with parabolic line bundles. This is the analogue for monopole chains of Donaldson's association of monopoles with rational maps. The construction is based on the Nahm transform, which relates monopole chains to Higgs bundles on the cylinder. As an application, we classify charge k monopole chains which are invariant under actions of Z2k. We present images of these symmetric monopole chains that were constructed using a numerical Nahm transform.

Parabolic Higgs bundles, $tt^\ast$ connections and opers

Parabolic Higgs bundles, $tt^\ast$ connections and opers

Generalizations of parabolic Higgs bundles, real structures, and integrability

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves with marked points. They are endowed with additional structures, which replace the parabolic structures at marked points in the parabolic Higgs bundles. The latter means that the coadjoint orbits are attached to the marked points. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case the coadjoint orbits are replaced by the cotangent bundles over some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of the parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of product of Cartan subgroups. Also, by changing the symmetric spaces we introduce quasi-compact and quasi-normal Higgs bundles. Then the fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic Calogero-Moser system and the rational ${\rm SL}(2)$ Gaudin system together with its real reductions.

Perverse filtrations, Hilbert schemes, and the $P=W$ Conjecture for parabolic Higgs bundles

We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$, $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.

Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

Let (X,D) and (X′,D′) be two compact Riemann surfaces of genus g≥4 with the set of marked points D⊂X and D′⊂X′. Fix a parabolic line bundle L with trivial parabolic structure. Let NSp(2m,α,L) and NSp′(2m,α,L) be the moduli spaces of stable symplectic parabolic Higgs bundles over X and X′ respectively, with rank 2m and fixed parabolic structure α, with the symplectic form taking values in L. We prove that if NSp(2m,α,L) is isomorphic to NSp′(2m,α,L), then there exist an isomorphism between X and X′ sending D to D′.

The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs 𝑉-bundles and components of parabolic 𝑆𝑝(2𝑛,ℝ)-Higgs moduli spaces

We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs V V -bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic S p ( 2 n , R ) \mathrm {Sp}\left ( 2n,\mathbb {R} \right ) -Higgs bundles with fixed parabolic structure.

Hitchin Fibration on Moduli of Symplectic and Orthogonal Parabolic Higgs Bundles

Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. Let $\mathcal{M}(r,d,\alpha)$ denote the moduli space of stable parabolic $G$-bundles (where $G$ is a complex orthogonal or symplectic group) of rank $r$, degree $d$ and weight type $\alpha$ over $X$. Hitchin discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. In this paper we study the Hitchin fibers for $\mathcal{M}(r,d,\alpha)$.

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. This generalizes Simpson's results for GL(n,C) to arbitrary complex and real reductive Lie groups. Three interesting features are the relation between the parabolic degree and the Tits geometry of the boundary at infinity of the symmetric space, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant–Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

Topological mirror symmetry for parabolic Higgs bundles

We prove the topological mirror symmetry conjecture of Hausel–Thaddeus (Hausel and Thaddeus, 2001, 2003) for the moduli space of strongly parabolic Higgs bundles of rank two and three, with full flags, for any generic weights. Although the main theorem is proved only for rank at most three, most of the results are proved for any prime rank.