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.
Strict plurisubharmonicity proven for Teichmüller energy on Hitchin representations.
problem Proving strict plurisubharmonicity of Teichmüller energy for Hitchin representations.
method Analyzing energy functional E on Teichmüller space associated to Hitchin representations. result Strict plurisubharmonicity of energy functional E proven. In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…
The abstract discusses rational approximations for Hitchin representations on surfaces.
problem Density of Hitchin representations in the Hitchin component for surfaces of genus g≥2. method Dynamical proof for g≥3; extension to other Q-groups. result Density of Hitchin representations for various Q-groups. The dynamics of representations into PSL_d(R) are studied for surfaces of genus at least 3.
problem Dynamics of representations into PSL_d(R) for surfaces of genus at least 3.
method Showed quasi-convex subsets of infinite diameter for the Weil--Petersson metric have finite diameter for the path metric of the pressure metric through controlled bounded length of biinfinite paths of bending deformations.
result Biinfinite paths of bending deformations have controlled bounded length.
Goldman parametrizes the PSL3(R)-Hitchin component of a closed oriented hyperbolic surface of genus g by 16g−16 parameters. Among them, 10g−10 coordinates are canonical. We prove that the PSL3(R)-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admi…
Sequences of Hitchin representations on surfaces are studied to describe their limits on trees.
problem Understanding the limits of sequences of Hitchin representations on surfaces.
method Using Fock-Goncharov coordinates on moduli spaces of flags.
result Non-trivial sufficient conditions for describing the limit of a sequence of Hitchin representations as an action on a tree.
Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let BunG denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on T∗BunG. We first show that the parahori…
Generic Hitchin representations avoid hyperplanes in Lie algebras.
problem Properties of Hitchin representations in Lie algebras.
method Defined J(ρ) and used hyperplanes in Lie algebras to show J(ρ)∩H=∅. result Generic G-Hitchin representations avoid hyperplanes in the Lie algebra of G. The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of λ-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli spa…
The paper constructs positive scalar curvature metrics via immersions.
problem Existence of positive scalar curvature metrics on manifolds.
method Extrinsic Gromov-Lawson surgery procedure.
result Positive scalar curvature metrics can be realized by immersions.
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 …
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.
Paper fills in technical details for Hitchin's self-duality equations proof.
problem Existence of solutions to Hitchin's self-duality equations.
method Reduction to minimizing energy functional, Coulomb gauge construction.
result Smooth solution constructed using unitary gauge transformation.
Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisa…
Smooth HP^2 bundle over S^4 with nontrivial A-genus found.
problem Existence of Riemannian metrics with positive sectional curvature.
method Explained existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
result Existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense 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.
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…
Geometric structures defined for G2′-Hitchin component on surfaces.
problem Understanding the geometric structures of G2′-Hitchin component on surfaces. method Explicit geometric structures interpretation and moduli space construction.
result Proves Hit(S,G2′) is homeomorphic to a moduli space of (G,X)-structures. We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin…
We compute the monodromy of the Hitchin fibration for the moduli space of L-twisted SL(n,C) and GL(n,C)-Higgs bundles for any n, on a compact Riemann surface of genus g>1. We require the line bundle L to either be the canonical bundle or satisfy deg(L)>2g−2. The monodromy group is gene…
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
We explore extensions to SL(n,C)-Chern-Simons theory of some results obtained for SU(n)-Chern-Simons theory via the asymptotic properties of the Hitchin connection and its relation to Toeplitz operators developed previously by the first named author. We define a formal Hitchin…
Canonical maps connect complex structures to Hitchin components.
problem Connecting higher complex structures to Hitchin components.
method Equivariant vector bundle construction and canonical isomorphisms.
result Canonical diffeomorphisms between spaces of higher complex structures and Hitchin components.
New minimal surface theory disproves a conjecture in symmetric spaces.
problem Proving the existence of unstable minimal maps in symmetric spaces.
method Using Hitchin representations and equivariant maps, with a new index bound.
result Disproves the Labourie conjecture for PSL(n,R) with n≥4. Holomorphic map connects Hitchin components to character varieties.
problem Complex affine spheres and their properties.
method Mapping class group equivariant holomorphic map from Hitchin components to character varieties.
result Holomorphic map includes holonomies of SL(3,C)-opers.
The paper defines and calculates Reidemeister torsion for a specific class of representations.
problem Defining and calculating Reidemeister torsion for G-Anosov representations.
method Symplectic chain complex method to establish a novel formula for R-torsion.
result Reidemeister torsion is well-defined and calculated for G-Anosov representations.
Minimal and CMC surfaces in S3 can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construc…
Uniformizes branched surfaces into Higgs bundles.
problem Uniformizing branched surfaces into cone metrics.
method Describes Higgs bundles corresponding to uniformization of conical metrics.
result Family of Higgs bundles parametrized by open subset of cohomology space.
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 provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller TQFT, and we prove it in detail in the case of the knot complements of 41 and 52. The conjecture states that the level-N Andersen-Kashaev invariant, JM,K(b,N), is annihilated by the non-homogeneous $\hat{…
Study of Einstein structures for surface group representations in specific Lie groups.
problem Understanding the geometric structures of representations in SO0(p,p+1). method Explicitly constructing fiber bundles and determining their diffeomorphism types.
result Many fiber bundles are trivial, revealing unique homotopy types.
New Einstein-Weyl spaces derived from hyperelliptic curves.
problem Constructing new Lorentzian Einstein-Weyl spaces.
method Using hyperelliptic curves and minitwistors.
result Spaces are diffeomorphic to 3D deSitter space with closed geodesics.
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. 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.
Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …
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 study the character variety of representations of the fundamental group of a closed surface of genus g≥2 into the Lie group SO(n,n+1) using Higgs bundles. For each integer 0<d≤n(2g−2), we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain…
We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space o…
For an oriented 2-dimensional manifold Σ of genus g with n boundary components the space Cπ1(Σ)/[Cπ1(Σ),Cπ1(Σ)] carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtratio…
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 the topology of spaces of pleated surfaces.
problem Topology of spaces of pleated surfaces.
method Real-analytic diffeomorphism and connected components analysis.
result Space of conjugacy classes of d-pleated surfaces is diffeomorphic to a specific manifold.
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 d=3.