arXiv research

Paper constructs counterexamples of higher solutions to self-duality equations.

New estimates for Hitchin's equations at high energy.

New hyper-Kähler manifolds found from Riemann surfaces.

Paper fills in technical details for Hitchin's self-duality equations proof.

We review recent work on the compactification of the moduli space of Hitchin's self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top bound…

Paper studies compactifications of Higgs bundles and self-duality equations.

We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.

Defines a functional for Riemann surfaces, proving a unique solution.

New duality found between harmonic maps and self-dual solutions.

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces $Σ_R$ converging for $R\searrow0$ to a surface $Σ_0$ with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration gives rise to a graph-continuous…

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…

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin-Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on the 4-sphere for a family of met…

Study explores Nahm-Schmid equations and their connection to hypersymplectic geometry.

In this paper, we write down Seiberg-Witten equations on contact metric manifolds of dimension 5. Any contact metric manifold has a spin^c structure. For Dirac equation we use Dirac type operators associated to the generalized Tanaka-Webster connection on spin^c spinor bundle of a contact metric manifold. For curvature…

We consider supersymmetric gauge theories with impurities in various dimensions. These systems arise in the study of intersecting branes. Unlike conventional gauge theories, the Higgs branch of an impurity theory can have compact directions. For models with eight supercharges, the Higgs branch is a hyperKahler manifold…

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

Survey on $2k$-Hitchin equations and Higgs bundles from geometric perspective.

The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.

We consider the octonionic self-duality equations on eight-dimensional manifolds of the form $M_8=M_4\times \R^4$, where $M_4$ is a hyper-Kähler four-manifold. We construct explicit solutions to these equations and their symmetry reductions to the non-abelian Seiberg-Witten equations on $M_4$ in the case when the gauge…

Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.

The paper constructs solutions for Higgs fields on a 4-punctured sphere.

The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.

In the context of D-dimensional Euclidean gravity, we define the natural generalisation to D-dimensions of the self-dual Yang-Mills equations, as duality conditions on the curvature 2-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), G_2 and Spin(7) holo…

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

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

Maps self-duality in little disks operad to framed manifolds.

The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.

Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.

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

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…

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…

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

We consider nonlinear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations -- the so-called vortex equations. We find the moduli space and energy spectrum of the solutions of these equations when the gauge group is a torus T^n, the…

The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.

Solves generalized Kazdan-Warner equations on foliated manifolds.

New equations reveal moduli space rigidity in geometric deformations.

Study of mixed equation combining gauge theory and symplectic geometry.

We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal…

Symplectic vortex equations link Sasakian manifolds to Kahler cones.

Study of limiting configurations for SU(1,2) Hitchin equation solutions.

We present a 1-parameter family of finite action solutions to the $S0(2,1)$ Hitchin's equations and explore some of its basic properties. For a fixed value of the parameter, the solution is smooth. We conclude by showing a multi-particle generalization of our basic solutions.

We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192,…

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…

The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.

Study of Hitchin moduli spaces over Teichmüller space.

Study extended Bogomolny equations on curved space with special boundary conditions.

We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The …

Study of twisted Kapustin-Witten equations on Riemann surfaces.