Explains Calabi-Yau integrable and Hitchin systems connections.
problem None explicitly stated, focuses on explaining relationships.
method Review of existing work by Diaconescu-Donagi-Pantev and the author.
result Highlights relationships between Calabi-Yau integrable and Hitchin systems.
The parahoric Hitchin system is completely integrable.
problem Integrability of the parahoric Hitchin system on curves of genus > 1.
method Proof of Hitchin map properties and integrability using isotropy and Poisson structure.
result The Hitchin map is a completely integrable system.
The paper quantizes the Hitchin system using geometric quantization.
problem Quantizing the Hitchin system using geometric methods.
method Geometric quantization of the Hitchin moduli space by modifying the Quillen metric.
result The modified Kahler form is integral and descends as a prequantum line bundle.
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.
Study special Kähler geometry of Hitchin system using spectral curves and topological recursion.
problem Investigate special Kähler geometry of Hitchin system.
method Use spectral curves and topological recursion.
result Compute the symmetric quartic of second derivatives of the period matrix.
Study rational sections of Picard scheme, leading to Hitchin fibration group description.
problem Understanding rational sections of Picard schemes on smooth projective varieties.
method Analyzing rational sections of relative Picard scheme, proving them from line bundles.
result All rational sections of relative Picard scheme come from line bundles under certain conditions.
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.
Introduces generalized hyperpolygons and their geometric and algebraic properties.
problem Understanding moduli spaces of generalized hyperpolygons.
method Representation of a comet-shaped quiver, associated meromorphic Higgs bundles, Hitchin systems, and integrable Hamiltonian systems.
result Generalized hyperpolygons admit the structure of a completely integrable Hamiltonian system.
Integrable systems and Kähler metrics linked via spectral curves.
problem Connecting integrable systems to Kähler geometry.
method Describing Special Kähler structure on spectral curve base.
result Simple formula for Kähler potential derived.
Paper describes integrable structure of Hitchin moduli spaces.
problem Integrable structure of Hitchin moduli spaces.
method Explicit parameterizations and Separation of Variables method.
result Clear analogy with Drinfeld's geometric Langlands correspondence.
Oper families and solutions of Hitchin's equations converge in a scaling limit.
problem Understanding the relationship between oper families and solutions of Hitchin's equations.
method Scaling limit analysis of families of flat connections over a compact Riemann surface.
result Oper families converge to solutions of Hitchin's equations in the scaling limit.
Study on GL(1∣1) Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of GL(1∣1) Higgs bundles. method Explicit description of moduli space, study of Narasimhan-Seshadri theorem, nonabelian Hodge correspondence, Hitchin equations.
result Derive an explicit description of the moduli space and study its properties.
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. 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.
Study of conformal limits for special opers in Lie groups.
problem Analyzing conformal limits in specific Lie groups.
method Investigates Gaiotto's conformal limit for GR-Hitchin equations. result Identifies Θ-positive opers as a new class of solutions. We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow…
Visible Lagrangians in Hitchin systems are studied for pillowcase covers.
problem Visible Lagrangians in Hitchin systems intersecting non-trivially.
method Computation of Fourier-Mukai transforms and study of mirror dual branes.
result Mirror dual branes are closely related to Hausel's toy model.
New flows defined on Hitchin components for PGL(V).
problem Understanding dynamics on Hitchin components.
method Define and analyze new flows on Hitchin components.
result Construct a global coordinate system on the Hitchin component.
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.
The Goldman symplectic form is trivialized for a class of surfaces.
problem Trivializing the Goldman symplectic form on specific surfaces.
method Using ideal triangulations and compatible bridge systems, the authors prove the existence of a symplectic trivialization.
result A global Darboux coordinate system is established for the PGL(V)-Hitchin component.
The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems…
The paper studies how to transform a sequence of cmc planes into a minimal surface.
problem Transforming a sequence of constant mean curvature planes into a minimal surface.
method Using algebraic-geometric correspondence and solving the Gauss-Codazzi equations.
result A sequence of solutions to the sinh-Gordon system converges to a solution of Liouville's equation, which is related to the Korteweg-de Vries system.
Study of Hitchin map on specific Higgs bundles.
problem Understanding the Hitchin map on even very stable Higgs bundles.
method Defined even very stable Higgs bundles and studied the Hitchin map on their upward flows.
result Classification of type (1,...,1) examples and their relation to root systems.
Geodesic coordinates derived for a specific metric in surface group representations.
problem Computing geodesic coordinates for a specific metric in surface group representations.
method Using thermodynamic formalism and gauge-theoretic formulas, computing first and second derivatives of the pressure metric.
result First derivatives of the pressure metric vanish at the Fuchsian locus.
Explicitly constructs moduli spaces of stable parabolic bundles.
problem Understanding moduli spaces of stable parabolic bundles over the Riemann sphere.
method Explicit construction and quotient of stable parabolic structures by bundle automorphisms.
result Explicit models of moduli spaces as smooth, compact complex manifolds.
The study of geometric objects over Riemann surfaces using Dynkin diagrams.
problem Classifying geometric objects over Riemann surfaces.
method Developing a global theory of Lie groups over Riemann surfaces and using Dynkin diagrams for classification.
result A theory of Dynkin diagrams for classifying geometric objects.
A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the system of Nahm equations is closely related to the Hitchin system, the curves appe…
The paper analyzes cyclic Higgs bundles using elliptic systems and immersion properties.
problem Analyzing cyclic Higgs bundles and their associated immersions.
method Derive a maximum principle for elliptic systems and apply it to the Hitchin equation.
result Obtain bounds on extrinsic curvature and complete the picture for specific representations.
Study on the Hitchin moduli space metric approximates it by a semiflat metric.
problem Understanding the asymptotic behavior of the Hitchin moduli space metric.
method Analyzing the L2 metric on the Hitchin moduli space for G=SU(2), proving polynomial convergence rates. result The Hitchin moduli space metric is well-approximated by the semiflat metric with polynomial decay.
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.
The Kobayashi-Hitchin correspondence is established for mini-holomorphic bundles on certain 3-folds.
problem Establishing a correspondence between mini-holomorphic bundles and HE-monopoles.
method Defining mini-holomorphic bundles, Dirac-type singularities, and admissible BHE-metrics.
result A special Hermitian metric (admissible BHE-metric) exists on slope stable mini-holomorphic bundles.
Integrability conditions for complex structures on product twistor spaces linked to Weyl geometry.
problem Integrability conditions for complex structures on product twistor spaces.
method Using Hitchin's generalized geometry and a metric connection with skew-symmetric torsion, the integrability conditions are derived in terms of Weyl geometry.
result Examples of structures satisfying the integrability conditions are supplied.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the PSL3(R)-Hitchin component with canonical coordinates. method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the PSL3(R)-Hitchin component. The paper studies properties of a Hitchin-Witten connection in Chern-Simons theory.
problem Exploring asymptotic properties of the Hitchin-Witten connection in SL(n,C)-Chern-Simons theory. method Defined a formal Hitchin-Witten connection, reduced to differential equations, identified cohomological obstructions, and found solutions.
result Identified and found solutions to a cohomological obstruction in the Hitchin-Witten connection.
Minimal surfaces in symmetric spaces linked to Hitchin representations studied.
problem Properties of minimal surfaces in symmetric spaces associated with Hitchin representations.
method Investigation of properties of immersed minimal surfaces in symmetric spaces associated with specific subloci of Hitchin component.
result Minimal surfaces have strong rigidity properties and are strictly negatively curved.
A correspondence between 1) rank 2 completely integrable systems of Jacobians of algebraic curves and 2) (holomorphically) symplectic surfaces was established in a previous paper by the first author. A more general abelian variety that occurs as a Liouville torus of integrable systems is a prym variety associated to a …
We present a sigma model field theoretic realization of Hitchin's generalized complex geometry, which recently has been shown to be relevant in compactifications of superstring theory with fluxes. Hitchin sigma model is closely related to the well known Poisson sigma model, of which it has the same field content. The c…
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.
We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in …
We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a cl…
The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.
problem Classifying quasi-Einstein structures and understanding their properties.
method Analyzing quasi-Einstein equations and exploring their connections to Hitchin's equations.
result A class of quasi-Einstein structures on closed manifolds must admit a Killing vector field.
New system modifies constant scalar curvature Kähler condition with a 'Higgs field'.
problem Extending constant scalar curvature Kähler condition to higher-dimensional manifolds.
method Explicit construction of hyperkähler metrics, Hitchin's equations for harmonic bundles, and Hermitian Yang-Mills equation.
result Existence of solutions to the modified system on specific cases (Riemann surfaces, ruled surfaces, abelian and toric surfaces).
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.
Identifies spectral curves for SU(3) coadjoint orbits.
problem Understanding the geometry of coadjoint orbits in SU(3).
method Using Hitchin pairs and spectral curves, identifies a Hamiltonian circle action and finds Darboux coordinates.
result Identifies a differential equation for the Hamiltonian.
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.
Defines loop Hodge structures and connects them to harmonic bundles.
problem Understanding harmonic bundles using Hodge theory.
method Introduces loop Hodge structures and proves their equivalence to harmonic bundles.
result Establishes equivalence between loop Hodge structures and harmonic bundles, enabling classical Hodge theory tools.
Study of holonomy behavior along differential rays on Riemann surfaces.
problem Analyzing the asymptotic behavior of holonomy along differential rays on Riemann surfaces.
method Investigates the asymptotic formulas for holonomy eigenvalues and growth rates using local fourth roots of the differential.
result Explicit asymptotic formulas for holonomy eigenvalues and their growth rates are derived.
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.