Higgs bundles used in new applications.
problem None explicitly stated in the abstract.
method Overview of recent applications.
result Applications of Higgs bundles.
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48 results for “Hitchin fibrations”
In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces converging for to a surface with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration gives rise to a graph-continuous…
Triality connects three polynomial bases in Lie algebra studies.
problem Understanding relationships between invariant polynomials.
method Triality of and Hitchin fibrations.
result Explicit automorphisms linking polynomial bases identified.
Study of Hitchin moduli spaces over Teichmüller space.
problem Metric aspects of Hitchin moduli spaces over varying complex structures.
method Gauge theoretical approach, Kähler fibrations, moment map interpretation, symplectic reduction.
result Establishes natural complex and pseudo-Kähler structures on universal Hitchin moduli spaces.
We compute the monodromy of the Hitchin fibration for the moduli space of -twisted and -Higgs bundles for any , on a compact Riemann surface of genus . We require the line bundle to either be the canonical bundle or satisfy . The monodromy group is gene…
We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near sing…
Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
problem Understanding the structure of holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds.
method Analyzing the properties of Lagrangian fibrations and special Kähler structures.
result Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Study Higgs bundles with non-regular fibers for SO(4,C) and SL(2,C)XSL(2,C).
problem Understanding non-regular fibers in Hitchin fibrations.
method Examined Higgs bundles for non-compact real forms of SO(4,C) and SL(2,C)XSL(2,C).
result Found interesting examples of non-abelian spectral data.
In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrict…
In this paper, we study the monodromy of the Hitchin fibration for rank 2 vector bundles over hyperelliptic curves. We reduce the problem to studying a surface braid group generalization of the classical Burau representation, and give a combinatorial method for computing this representation.
We establish global existence of smooth solutions to heat flow for Yang-Mills-Higgs functional on Kahler fibrations. As an application, we give a new proof of the key inequality for Mundet's Hitchin-Kobayashi correspondence theorem using the heat flow technique.
Generalizes G-opers for arbitrary parabolics, parameterizing by Hitchin base.
problem Parameterizing (G,P)-opers for arbitrary parabolic subgroups.
method Introduces a generalization of G-opers and parameterizes them by an object generalizing the Hitchin base.
result Describes families of opers associated to higher Teichmuller spaces.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
The Ooguri-Vafa space is linked to framed wild harmonic bundles via hyperkähler geometry.
problem Relating the Ooguri-Vafa space to Hitchin moduli spaces.
method Interpreting the Ooguri-Vafa space as framed wild harmonic bundles over CP1.
result The Ooguri-Vafa space is a moduli space of framed wild harmonic bundles.
Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the -Hitchin system.
method Using Hecke transformations and analysis of parameter spaces, the paper stratifies and compactifies the singular spaces.
result Large classes of singular fibres are shown to be fibre bundles over Prym varieties.
Explains Higgs bundles and their applications in gauge theory.
problem Understanding Higgs bundles and their mathematical properties.
method Introduction to Higgs bundles, Hitchin fibration, and study of subspaces of moduli spaces.
result Different subspaces of Higgs bundles correspond to flat connections and representations.
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of -Hitchin moduli space over singular fibers.
method Extension of exponential convergence results to locally fiducial Higgs bundles and subintegrable systems.
result Hyperkähler metric converges exponentially to semi-flat metric on subintegrable systems.
The paper studies the geometry of flat manifolds with controlled holonomy.
problem Investigating the geometry of asymptotically flat manifolds with specific properties.
method Analyzes torus fibrations and Hitchin-Thorpe inequalities for Ricci-flat 4-manifolds.
result Proves that certain flat metrics on 4-manifolds are isometric to Euclidean or Taub-NUT.
Paper studies compactifications of Higgs bundles and self-duality equations.
problem Compactification of Hitchin moduli space and Higgs bundles.
method Analyzes maps between algebraic and analytic compactifications.
result Map between compactifications fails to be continuous at boundary over discriminant locus.
We define and study spectral data associated to U(m,m)-Higgs bundles through the Hitchin fibration. We give a new interpretation of the topological invariants involved, as well as a geometric description of the moduli space.
The article describes canonical metrics on holomorphic fibre bundles.
problem Existence of canonical metrics on isotrivial Kähler fibrations.
method Induced from Hermite--Einstein connections on holomorphic principal bundles.
result Existence of optimal symplectic connections when principal bundles are polystable.
New equations reveal moduli space rigidity in geometric deformations.
problem Understanding moduli space rigidity in geometric deformations.
method Coupled Hitchin-He equations, Lax pair, nonlinear embedding.
result Moduli space is analytically isomorphic to the classical case for small deformations.
Mid-dimensional and -branes in the moduli space of flat -connections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.
We show that for every nonelementary representation of a surface group into there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
Develops complex harmonic maps for Teichmüller theory, proving new theorems.
problem Analyzing complex harmonic maps in Teichmüller theory.
method Complex harmonic maps and Higgs bundles.
result Proves a Bers-type theorem for rank 2 Hitchin components.
We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, a…
Energy functional on Teichmüller space is plurisubharmonic but not strictly so.
problem Characterizing points where energy functional fails to be strictly plurisubharmonic.
method Analyzing the kernel of the Levi form and relating it to Higgs bundles and Hitchin fibration.
result For generic choices, energy functional is strictly plurisubharmonic.
We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequ…
A principal pair consists of a holomorphic principal -bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobaya…
Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to…
The study shows how energy density of harmonic maps dominates in -Fuchsian fibers, leading to unique minimal surfaces.
problem Understanding energy density and topological invariants in -Fuchsian fibers of Higgs bundles.
method Establishing an algebraic inequality generalizing a GIT theorem to prove energy density domination.
result Energy density of harmonic maps dominates in -Fuchsian fibers, leading to unique minimal surfaces.
The paper studies the moduli space of Higgs pairs and their geometric properties.
problem The moduli space of Higgs pairs and its geometric properties.
method Introduced -stability of Higgs pairs and established the Kobayashi-Hitchin correspondence.
result Proved that the moduli space is a non-singular complex manifold for a suitable choice of .
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, -actions, Hecke transformations, Fourier-Mukai transforms.
result Precise formula for multiplicity of very stable components of global nilpotent cone.
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimi…
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspo…
In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…
Defines and parametrizes -type singular fibres in symplectic and odd orthogonal Hitchin systems.
problem Characterizing and understanding singular fibres in Hitchin systems.
method Stratification by semi-abelian spectral data, study of irreducible components, global description of degenerations.
result Extension of Langlands duality to -type Hitchin fibres.
Real projective surfaces with Hitchin holonomy can be related via grafting.
problem Real projective surfaces with Hitchin holonomy.
method Defining graftable curves and constructing them in the Hitchin case.
result Real projective structures with the same Hitchin holonomy are related via multi-graftings.
Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.
problem Analyzing the asymptotic behavior of the Hitchin metric on moduli spaces of Higgs bundles.
method Examined the decay rate of the difference between Hitchin and semi-flat metrics on smooth spectral curves.
result Exponential decay of the difference between Hitchin and semi-flat metrics as t approaches infinity.
New estimates for Hitchin's equations at high energy.
problem Solutions to Hitchin's self-duality equations at high energy.
method New estimates and asymptotic decoupling phenomenon.
result Generalization to arbitrary Higgs bundles.
Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.
problem Understanding moduli spaces of the basic Hitchin equation on Sasakian 3-folds.
method Construction of moduli space and calculation of its dimension.
result Moduli space admits a hyperKähler metric.
Study pressure metrics for cusped Hitchin representations.
problem Characterize cusped Hitchin representations of Fuchsian groups.
method Develop pressure metrics associated to fundamental weights and roots.
result New pressure metrics for Hilbert length when .
Study of energy functional on Deligne-Hitchin moduli space sections.
problem Understanding energy functionals on sections of Deligne-Hitchin moduli space.
method Generalizes energy of equivariant harmonic maps to holomorphic sections, links to meromorphic connections, and uses Willmore energy analogy.
result Shows functional is essentially Willmore energy for certain sections, distinguishes new components from twistor lines.
Survey on -Hitchin equations and Higgs bundles from geometric perspective.
problem Understanding -Hitchin equations through Higgs bundles and complex geometry.
method Review of Higgs bundles, holomorphic vector bundles, and Hermite-Yang-Mills equations; geometric tools applied to simplify equations.
result Simplified -Hitchin equations to a set of two equations for Higgs bundles.
The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.
problem Behavior of sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.
method Compactness result for connections and renormalized Higgs fields.
result Every harmonic 1-form can be deformed into a sequence of solutions.