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.
Authors compute monodromy groups for SL(n) and GL(n) Hitchin fibrations.
problem Computing monodromy groups for Higgs bundles on Riemann surfaces.
method Using spectral curves and Picard-Lefschetz transformations, they construct and classify vanishing lattices.
result They determine the structure of monodromy groups for SL(n) and GL(n) Hitchin fibrations.
The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.
problem Solutions of the extended Bogomolny equations on $Σ imes \RP$ with specific singularities.
method Develops a Kobayashi-Hitchin type correspondence and partial correspondence for solutions with Nahm pole and knot singularities.
result Verifies a conjecture and proves existence and uniqueness of solutions with knot singularities.
We show that for every nonelementary representation of a surface group into SL(2,C) there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
Holomorphic Higgs bundles on Teichmüller space for surface groups.
problem Characterizing representations of surface groups admitting holomorphic Higgs data.
method Non-abelian Hodge correspondence, unitarity conditions, and existence proofs for higher ranks.
result Holomorphic dependency of Higgs data is equivalent to unitarity for SL(2,C) representations but fails for higher ranks. Study describes metrics on moduli space of Higgs bundles, proving exponential decay rate.
problem Analyzing the geometry of moduli space of Higgs bundles.
method Perturbing from approximate solutions to describe metrics, comparing to semi-flat metrics.
result Proves exponential decay rate of gL2−gsf=O(e−γt). Using Hitchin's parameterization of the Hitchin-Teichmüller component of the SL(n,R) representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the SL(n,R)-Hitchin component, we study the asymptotics of the Hermitian metric solvin…
The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.
problem Modeling hyperkähler 4-manifolds and their degenerations.
method Using parabolic SL(2,C)-Higgs bundles and Nakajima quiver varieties.
result ALG-D4 spaces degenerate to ALE-D4 spaces under a limit. Study Higgs bundles and their Langlands duals on surfaces.
problem Matching components of Higgs bundle moduli spaces under Langlands duality.
method Defined antiholomorphic involutions on Higgs bundle moduli spaces and analyzed fixed point loci.
result Components of fixed point loci match under Langlands duality for specific groups.
Paper studies complex Lagrangian surfaces and their relation to SL(3,C)-representations.
problem Minimal Lagrangian surfaces in bi-complex hyperbolic space and their representations.
method Introduces bi-complex Higgs bundles and parameterizes SL(3,C)-quasi-Fuchsian representations. result Parameterization of SL(3,C)-quasi-Fuchsian representations by an open set in Teichmüller space. The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the SL(r,C) charact…
The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.
problem Understanding Lagrangian structures in Hitchin moduli space.
method Analyzing semistable and polystable Higgs bundles, focusing on the intersection with Lagrangian sublocus.
result The conformal limit of stable Higgs bundles on a specific Lagrangian sublocus exists under certain conditions.
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
Geometric methods for surface group representations in higher rank SL(2m+1,R).
problem Representations of surface groups in higher rank SL(2m+1,R).
method Para-complex and pseudo-Riemannian geometric techniques.
result One-to-one correspondence between Higgs bundles and isotropic P-alternating surfaces.
Constructs approximate solutions for Higgs bundles, useful for moduli space geometry.
problem Understanding the asymptotic geometry of the Hitchin moduli space.
method Gluing methods to construct families of approximate solutions.
result Explicit approximate solutions differing from actual metrics by exponentially small error terms.
N. Hitchin recently introduced the notion of folded hyperKähler metrics, in relation with SL(\infty,R) Higgs bundles. We provide a construction of such metrics, and prove the local existence of the Hitchin component for SL(\infty,R).
A principal Higgs bundle (P,φ) over a singular curve X is a pair consisting of a principal bundle P and a morphism φ:X→AdP⊗ΩX1. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve X using the theory of decorated vector bundles. More precisely, given…
Paper generalizes Higgs bundle limits to parabolic setting.
problem Generalizing Higgs bundle limits to parabolic setting.
method Gauge theoretic construction of moduli space of parabolic Higgs bundles.
result Conformal limit always exists and defines holomorphic sections.
We develop a new geometric method of understanding principal G-Higgs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p), SU(p,p), and Sp(2p,2p). Further, we give some applications of our results, and d…
Constructs hyperkähler metrics on Higgs bundle moduli spaces using Gaiotto coordinates.
problem Constructing hyperkähler metrics on moduli spaces of Higgs bundles.
method Using Gaiotto coordinates and solving Riemann-Hilbert problems, constructing a twistorial hyperkähler metric.
result The difference between the constructed hyperkähler metric and a simpler semiflat metric is exponentially suppressed.
A gauge theoretic description of the Morgan-Shalen compactification of the $\SL$ character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map.
Defines and parametrizes sl(2)-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 sl(2)-type Hitchin fibres. This research connects Higgs bundles to projective structures via conformal limits.
problem Mapping Higgs bundles to projective structures.
method Using non-abelian Hodge correspondence and conformal limit.
result The family of connections in the conformal limit can be understood as complex projective structures.
The paper proves a conjecture linking Higgs bundles and Lie algebra actions.
problem Linking Higgs bundles and Lie algebra actions for a conjecture.
method Using Hecke correspondences and cup-product by tautological classes, the paper constructs an action of H2 on cohomology. result The perverse filtration on Higgs bundle cohomology matches the sl2 filtration, proving the P=W conjecture. Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of SL2(C)-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.
We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combin…
New parameterization of Teichmüller spaces for complex Lie groups.
problem Classifying special components of moduli spaces of Higgs bundles.
method Introducing magical sl2-triples and using their decomposition data. result Constructs previously unknown components of higher Teichmüller spaces.
Computes intersection cohomology of moduli space of Higgs bundles on a genus 2 curve.
problem Computing the intersection cohomology of the moduli space of Higgs bundles.
method Constructs a semismall desingularization and uses the decomposition theorem to compute the cohomology.
result Proves the mixed Hodge structure on the intersection cohomology is pure.
Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.
problem Investigating the Hitchin metric on moduli spaces of Higgs bundles.
method Using Hitchin hyperkähler metric and parabolic Deligne-Hitchin moduli space.
result Rescaled Hitchin metric converges to hyperpolygon space's hyperkähler metric in the semiclassical limit.
Study shows Hitchin's metric converges to a semiflat metric on the Hitchin section.
problem Understanding the asymptotic behavior of Hitchin's metric on the Hitchin section.
method Analyzing the hyperkähler metric on moduli spaces of Higgs bundles.
result The Hitchin metric converges exponentially to a semiflat metric on the Hitchin section.
The paper studies the mapping class group's action on Lie group representations and Higgs bundles.
problem Understanding the action of mapping class groups on character varieties and Higgs bundles.
method Identifying fixed points of the action in terms of G-Higgs bundles with specific equivariant structures.
result Generalization of pseudoreal Higgs bundles to parabolic Higgs bundles.
Study moduli spaces of solutions to Bogomolny equations on surfaces.
problem Understanding solutions to Bogomolny equations on surfaces with specific boundary conditions.
method Refined Kobayashi-Hitchin correspondence and identification with Higgs bundles.
result Identified moduli spaces as holomorphic lagrangian sub-manifolds.
Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the SL(2,C)-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.
Study Higgs bundles on CP1 with specific singularities, linking fixed points to W-algebra representations.
problem Analyzing Higgs bundles with irregular singularities and their fixed points.
method Introduced a C×-action on MK,N and classified its fixed points. result Found a 1-1 correspondence between fixed points and certain W-algebra representations.
Construct opers with apparent singularities from λ-connections on Riemann surfaces.
problem Constructing opers with apparent singularities from λ-connections.
method Using Riemann surfaces and λ-connections to define rational maps and Poisson structures.
result Defines a rational map capturing important data of λ-connections and apparent singularities.
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.
Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of d…
Develops correspondence for Bogomolny equations with specific boundary conditions.
problem Solving the extended Bogomolny equations with generalized Nahm pole boundary conditions.
method Kobayashi-Hitchin correspondence for stable Higgs pairs and holomorphic line bundles.
result Corroborates Gaiotto and Witten's prediction and extends previous work to higher dimensions.
Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.
problem Understanding the asymptotic geometry of character varieties.
method Nonabelian Hodge correspondence and study of harmonic maps.
result Asymptotic correspondence between limiting configurations and geometric parameters.
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 introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G=GL(2,C), SL(2,C) and PSL(2,C). We also determine the twisted Chern class of the regula…
Classifies Higgs and co-Higgs bundles on symmetric spaces.
problem Classifying Higgs and co-Higgs bundles over Hermitian symmetric spaces.
method Defined homogeneous principal Higgs and co-Higgs bundles, provided a classification up to isomorphism.
result Defined and classified moduli spaces for each type of bundle.
The paper constructs solutions for Higgs fields on a 4-punctured sphere.
problem Constructing self-duality solutions for Higgs fields on a 4-punctured sphere.
method Complex analytic methods, twistor approach, λ-connections interpretation.
result Identifies the rescaled limit hyper-Kähler moduli space at t=0.
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
Higgs bundles used in new applications.
problem None explicitly stated in the abstract.
method Overview of recent applications.
result Applications of Higgs bundles.
Researchers extend Higgs bundle theory to parabolic bundles, counting components.
problem Counting components in moduli spaces of Higgs bundles with parabolic structures.
method Generalized Beauville-Narasimhan-Ramanan correspondence, Bott-Morse theory.
result Exact component count for maximal parabolic Sp(2n,ℝ)-Higgs bundles.
Identifies images of determinant morphism for specific co-Higgs bundles.
problem Determining images of determinant morphism for co-Higgs bundles.
method Identifying images of the determinant morphism of trace-free co-Higgs bundles modeled on rank 2 Schwarzenberger bundles.
result Identified images of the determinant morphism for specific co-Higgs bundles.
The paper examines critical points in Higgs bundle moduli spaces.
problem Understanding critical points in Higgs bundle moduli spaces.
method Analyzes the Higgs field vanishing on divisors and related integrable systems.
result Topological and differential-geometric properties of critical loci are addressed.