arXiv research

Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.

Study pressure metrics for cusped Hitchin representations.

Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.

We consider Hitchin's hyperkähler metric $g_{L^2}$ on the $SU(n)$-Hitchin moduli space moduli space over a compact Riemann surface. We prove that the difference between the metric $g_{L^2}$ and a simpler "semiflat" hyperkähler metric $g_{\mathrm{sf}}$ is exponentially-decaying along generic rays in the Hitchin moduli s…

Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.

New metrics derived from Hölder distortion on Hitchin components.

Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.

Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.

Study Blaschke metrics to compactify Hitchin component.

We consider Hitchin's hyperkähler metric $g$ on the moduli space $\mathcal{M}$ of degree zero $\mathrm{SL}(2)$-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in $\mathcal{M}$, $g$ converges to an explicit "semiflat" metric $g^{\mathrm{sf}}$, wit…

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…

Numerical experiments support conjecture about opers and nonabelian Hodge.

Compactifies a component by studying metric degeneration.

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).

Paper describes a pseudo-Kähler structure on a specific Hitchin component.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

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…

The Kobayashi-Hitchin correspondence is proven for twisted vector bundles on Kähler manifolds.

Study local models for special Kähler metrics near discriminant locus components.

Study on the asymptotic geometry of Higgs bundles over projective line.

Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.

New insights into currents of Hitchin representations with combinatorial restrictions.

We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.

Study of Hitchin moduli spaces over Teichmüller space.

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the se…

The Legendre transform and its generalizations, originally found in supersymmetric sigma-models, are techniques that can be used to give constructions of hyperkahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli spa…

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 …

Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.

The dynamics of representations into PSL_d(R) are studied for surfaces of genus at least 3.

Constructs an asymptotic metric for moduli space of centred hyperbolic monopoles.

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…

New metrics for Anosov representations defined from Thurston's asymmetric metrics.

This is the first in a series of papers in which we develop a twistor-based method of constructing hyperkaehler metrics from holomorphic functions and elliptic curves. As an application, we revisit the Atiyah-Hitchin manifold and derive in an explicit holomorphic coordinate basis closed-form formulas for, among other t…

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is…

We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component $\mathcal{H}_{3}(S)$ of surface group representations into $PSL(3,\mathbb{R})$. The proof consists of the following elements: we compute first derivatives of the pressure metric…

Study proves correspondence for special bundles on complex surfaces.

A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,φ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $φ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and…

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of…

Extends Ooguri-Vafa symplectic form to framed Higgs bundles.

Study on Blaschke locus with covariance metric properties.

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such…

Study improves the exponential rate of metric difference in Higgs bundles.

The paper adapts metrics to anti-de Sitter structures, characterizing their degeneracies.

This paper proves Hitchin moduli spaces are ALG gravitational instantons.

Conditions for polyhedral Kähler metrics on CP^n with specific singularities.

Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.