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.
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48 results for “Hitchin theory”
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
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 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.
New Teichmüller spaces found for orbifolds.
problem Understanding geometric structures on orbifolds.
method Extending Hitchin components to orbifold groups and proving properties.
result Hitchin components are homeomorphic to open balls and have explicit dimensions.
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.
Study on projective structures linked to Hitchin representations.
problem Understanding the topology and geometric properties of projective structures.
method Using gauge theory, flat bundles, Higgs bundles, and geometric constructions.
result Some projective structures are fibered in a standard way.
The paper studies properties of a Hitchin-Witten connection in Chern-Simons theory.
problem Exploring asymptotic properties of the Hitchin-Witten connection in -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.
This paper shows connections between two complex mathematical theories are equivalent.
problem Establishing equivalence between two complex mathematical theories.
method Using geometric quantisation and conformal field theory, the paper establishes equivalence between the Hitchin connection and the Knizhnik-Zamolodchikov connection.
result The Hitchin and Knizhnik-Zamolodchikov connections are projectively equivalent in genus zero.
By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1 dimensional vacuum general relativity theory on S x R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary S, a compact, oriented Riemann surface…
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.
Constructs a new geometric structure on surfaces to generalize Teichmüller theory.
problem Exploring new geometric structures in Teichmüller theory.
method Uses the punctual Hilbert scheme of the plane to construct a higher complex structure and explores its properties.
result Establishes a canonical diffeomorphism between the moduli space of higher complex structures and Hitchin's component.
Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.
problem Stability analysis of nearly Kähler and parallel G2 manifolds.
method Gradient flow of Hitchin functionals, spectral decomposition of Hessians, Hitchin index.
result Hitchin index provides a lower bound for the Einstein co-index.
Proves a simplified version of Hitchin's theorem.
problem Constructing hyper-Kähler structures.
method Concrete variant of Hitchin's theorem.
result Applies to real manifolds without constructing hyper-Kähler structures.
Topological field theories of 2- and 3-forms in 6D, showing no propagating degrees of freedom.
problem Exploring field theories of 2- and 3-forms in six dimensions.
method Analyzing field theories with kinetic term and varying potential terms.
result The theories remain topological even with added potential terms, and their reductions yield 3D gravity.
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.
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler struct…
We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit toward…
Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groupsmath.DG
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
problem Analytic and geometric properties of harmonic maps.
method Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
result Proves Dai-Li's conjecture on the monotonicity of the energy density and negative curvature conjecture for Coxeter cyclic G-Higgs bundles.
In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …
We prove an extension of Basmajian's identity to -Hitchin representations of compact bordered surfaces. For , we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the L…
Paper constructs counterexamples of higher solutions to self-duality equations.
problem Whether all real sections give rise to global solutions of self-duality equations.
method Using Willmore surfaces and generalized Whitham flow.
result Higher solutions exist on an open dense subset of the Riemann surface, not globally.
Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.
problem Defining and linking novel Floer homologies for different manifold dimensions.
method Physics of a topologically-twisted 5d N=2 gauge theory, Vafa-Witten, Hitchin, and BF configurations.
result Derived novel gauge-theoretic and symplectic Floer homologies, and Atiyah-Floer correspondences.
Study of monopoles and q-difference modules correspondence.
problem Finding a correspondence between algebraic and differential geometric objects.
method Analogue of non-abelian Hodge theory for -difference modules.
result Doubly periodic monopoles and parabolic -difference modules correspond.
We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets …
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 Hitchin-Kobayashi correspondence is extended to foliated Riemannian manifolds.
problem Proving a Hitchin-Kobayashi correspondence for foliated manifolds.
method Adapting Uhlenbeck and Yau's proof to the foliated setting, defining stability for foliated bundles, and relating to higher-dimensional instantons.
result Foliated Hermitian-Einstein connections correspond to polystability, with implications for higher-dimensional instantons.
The real spectrum compactification of character varieties: characterizations and applicationsmath.GT
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Study of non-Archimedean Hitchin map for SL2(F) characters.
problem Characterizing representations of SL2(F) characters.
method Equivariant harmonic maps into R-trees, Jenkins-Strebel differentials.
result The non-Archimedean Hitchin map is continuous and its image is contained in Jenkins-Strebel differentials.
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.
Develops twistorial approach to define monopoles on R^5.
problem Defining monopoles on R^5.
method Twistorial approach, spectral curves, and generalized Nahm's equations.
result Found a new system of equations for monopoles on R^5.
Study non-orientable surfaces in gauge theories, matching fluxes with homotopy classes.
problem Understanding non-orientable surfaces in gauge theory and their electric-magnetic duality.
method Reduction of twisted N=4 gauge theory on non-orientable manifolds, matching fluxes with homotopy classes.
result Verification of mirror symmetry of branes as predicted by S-duality.
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.
On Higgs bundlesmath.AG
Study on Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of 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.
Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.
problem Structure of domains of discontinuity in PSL(4,R)-Hitchin representations.
method Detailed study and proof of foliations by projective line segments and convex domains.
result Foliated component has exactly two group-invariant foliations.
Surveying recent developments in Hitchin moduli space geometry.
problem Understanding the asymptotic geometry of Hitchin moduli space.
method Introduction to Hitchin moduli space and hyperkähler geometry.
result Recent developments in asymptotic geometry of Hitchin moduli space.
Hitchin's generalized complex geometry has been shown to be relevant in compactifications of superstring theory with fluxes and is expected to lead to a deeper understanding of mirror symmetry. Gualtieri's notion of generalized complex submanifold seems to be a natural candidate for the description of branes in this co…
In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particular, we recall the differential geometric construction of Hitchin's pro…
An introduction to the theory of bundle gerbes and their relationship to Hitchin-Chatterjee gerbes is presented. Topics covered are connective structures, triviality and stable isomorphism as well as examples and applications.
Hitchin representations are identified by curve spectral radii.
problem Classifying isometries of Hitchin components.
method Establishing transversality for positive quadruples of flags.
result Hitchin representations are uniquely determined by spectral radii of curves.
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.
Study topology and geometry of hyperkähler quotients using Morse theory.
problem Topology and geometry of hyperkähler quotients and related non-compact Kähler quotients.
method Employ Morse theory with moment maps, Lojasiewicz inequality, Kirwan surjectivity, and Jeffrey-Kirwan residue formula.
result Obtained new proofs and explicit inductive procedures for Betti numbers of fixed-point sets.
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.
In this paper we provide a review of asymptotic results of Toeplitz operators and their applications in TQFT. To do this we review the differential geometric construction of the Hitchin connection on a prequantizable compact symplectic manifold. We use asymptotic results relating the Hitchin connec- tion and Toeplitz o…
New findings on Frobenius structures on Kodaira manifolds.
problem Understanding Frobenius structures on Kodaira manifolds.
method Extended deformation theory and Frobenius structures.
result Frobenius structure on Kodaira manifolds is trivial on degree-2 component.