Study on existence of harmonic metrics for non-Hermitian Yang-Mills bundles.
problem Existence of harmonic metrics in non-Hermitian Yang-Mills bundles.
method Examined compact Kähler manifolds and equivalence to semisimplicity of NHYM bundles.
result Existence of harmonic metrics is equivalent to semisimplicity of NHYM bundles.
Study on hermitian Yang-Mills connections on blown-up manifolds.
problem Analyzing hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Investigates connections for pullback vector bundles under specific conditions.
result Provides numerical criterion for convergence of hermitian Yang-Mills connections.
We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from har…
Direct method finds Yang-Mills connections for SO(3) bundles.
problem Finding Yang-Mills connections for SO(3) bundles over closed 4-manifolds.
method Direct minimizing method with test connections and assumptions.
result Minimizing sequence converges to a minimizer or anti-selfdual/selfdual connection.
Study hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
problem Investigate hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
method Obtain a criterion for the existence of hermitian Yang-Mills connections on pullback bundles, using intersection numbers on the base.
result Determine conditions under which pullback bundles of stable or unstable bundles remain stable or unstable for adiabatic classes.
The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.
problem Stability and energy identity of Yang-Mills-Higgs pairs on vector bundles.
method Bubble-neck decomposition and analysis of weakly stable pairs.
result A sequence of Yang-Mills-Higgs pairs converges to a Yang-Mills-Higgs pair with uniformly bounded energy.
Introduces a new Yang-Mills functional for connections and scalars over circle bundles.
problem Yang-Mills functional over circle bundles with two-forms.
method Dimensional reduction and Euler-Lagrange equations.
result Special three-dimensional solutions satisfy a duality condition.
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
Develops a unified theory of Yang-Mills and GR using generalized principal bundles.
problem Combining Yang-Mills theories and General Relativity into a single framework.
method Using generalized principal bundle theory, the authors develop a new approach to field theories.
result Recover General Relativity within the framework of generalized principal connections.
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…
New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.
problem Approximating Kähler potentials in complex domains.
method New Wess-Zumino-Witten type equation and Berndtsson's theorem on direct image bundles.
result Approximation of Kähler potentials by Hermitian-Yang-Mills metrics.
Develops equivariant connections for Yang-Mills equations, simplifying interactions modeling.
problem Simplifying interactions modeling in Yang-Mills equations for different bundles.
method Introduces SO+(p,q)-equivariance to reduce Yang-Mills equations. result Models electroweak interaction and interactions with differential and wave equations.
Solves Dirac equation coupled to vector bundles.
problem Yang-Mills equations and vector bundles on Riemann surfaces.
method Analyzes coupled Dirac operators.
result Provides concrete solutions to the Dirac equation.
We study Yang-Mills connections on holomorphic bundles over complex Kähler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isom…
Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.
problem Constructing Hermitian-Einstein metrics on stable holomorphic vector bundles
method Dynamical construction
result Provided a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
Study extends Yang-Mills energy gap to Kähler surfaces.
problem Extend Yang-Mills energy gap to Kähler surfaces.
method Extend L2-energy gap to compact Kähler surfaces with generic Kähler metrics. result All ASD connections on principal bundles over Kähler surfaces are irreducible.
New system solves curvature for ample vector bundles, proving Griffiths conjecture.
problem Proving Griffiths conjecture on vector bundle positivity.
method Proposes Hermitian-Yang-Mills elliptic system for curvature.
result Solutions provide metrics with positive curvature in Griffiths sense.
Extends Yang-Mills theory to non-integrable Lie algebroids.
problem Developing a Yang-Mills theory for non-integrable Lie algebroids.
method Generalized Yang-Mills theory to Lie algebroids, introducing multiplicative Ehresmann connections.
result Derived self-dual solutions (instantons) in 4 and 5 dimensions.
In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle (E,H0) over a compact Kähler manifold (M,ω). We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorph…
Paper solves tensor problem for holomorphic vector bundles.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors.
method New comparison theorem for Hermitian-Yang-Mills tensors.
result Existence of unique smooth Hermitian metric for any positive-definite tensor.
The paper solves a problem related to Higgs bundles and Hermitian metrics.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors for Higgs bundles.
method Solving the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds.
result For any Hermitian positive definite tensor, there exists a unique smooth Hermitian metric on the Higgs bundle.
Survey on 2k-Hitchin equations and Higgs bundles from geometric perspective.
problem Understanding 2k-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 2k-Hitchin equations to a set of two equations for Higgs bundles. In this paper, we introduce an α-flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the α-flow with smooth initial value. We prove that the limit of solutions of the α-flow as α\to 1 is a weak solution to the Yan…
We consider the minimum Yang-Mills energy on the complete G2-manifolds and Calabi-Yau 3-folds,the connection A is a stability Yang-Mills connection on the G-bundle E.We prove that the connection must be a G2-instanton on G2-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $…
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Study vortices in Kähler-Yang-Mills equations on complex manifolds.
problem Solving coupled equations for Kähler metrics and connections.
method Dimensional reductions of Kähler-Yang-Mills equations to study vortices.
result Found solutions to Yang-Mills-Higgs equations related to vortices.
The paper examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. We investigate monotonicity properties of p-harmonic vector bundle-valued k-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formulæ for p-harmonic maps and Yang-Mills connections, proving a monotonicity formula for p-Yang-…
Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation…
A local monotonicity formula for the Yang-Mills-Higgs flow on G-bundles over Rn (n>4) is proved. It is shown that the monotone quantity coïncides on certain self-similar solutions with that appearing in existing non-local monotonicity formulæ for the Yang-Mills and Yang-Mills-Higgs flows.
Let X be a compact connected Kähler--Einstein manifold with c1(TX)≥0. If there is a semistable Higgs vector bundle (E,θ) on X with θ=0, then we show that c1(TX)=0, any X satisfying this condition is called a Calabi--Yau manifold, and it admits a Ricci--flat Kähler form \cite{Ya}. Let …
The paper presents new formulations of gauge and gravity theories using dynamical principal bundles.
problem Formulating gauge and gravity theories with a flexible principal bundle structure.
method Original variational formulations of Yang-Mills, Einstein's gravitation, and Kaluza-Klein theories with a dynamical principal bundle.
result The principal bundle structure and connection emerge from the dynamics, leading to solutions of Yang-Mills, Einstein-Cartan, or Yang-Mills-Einstein equations.
Extends weak continuity of Yang-Mills connections to a broader class.
problem Weak compactness of Ω-Yang-Mills connections. method Compensation compactness argument applied to Yang-Mills fields.
result Weak continuity result extended to Ω-Yang-Mills connections. Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our p…
Removes singularities for Yang-Mills-Higgs fields in higher dimensions.
problem Yang-Mills-Higgs fields with isolated singularities.
method Establishes decay estimates and conformally invariant energy bounds.
result Removable singularity theorem for Yang-Mills-Higgs fields.
Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.
problem Analyzing Ω-Yang-Mills connections on Riemannian manifolds. method Extending known results on Yang-Mills connections to Ω-Yang-Mills connections, proving weak compactness and removable singularity theorems. result Compactification of moduli spaces of smooth Hermitian-Yang-Mills connections on unitary bundles over balanced manifolds.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
Finite-time blow-up in Yang-Mills flow for small energy initial connections.
problem Finite-time blow-up of Yang-Mills flow solutions.
method Analyzing the Yang-Mills flow on Riemannian and Kähler manifolds.
result Finite-time blow-up occurs for small energy initial connections.
In this paper we introduce a set of equations on a principal bundle over a compact complex manifold coupling a connection on the principal bundle, a section of an associated bundle with Kähler fibre, and a Kähler structure on the base. These equations are a generalization of the Kähler-Yang-Mills equations introduced b…
The paper studies gauge fields on coherent sheaves and their Yang-Mills properties.
problem Analyzing gauge fields on coherent sheaves and their Yang-Mills properties.
method Defined necessary and sufficient conditions for Yang-Mills fields, introduced cohomology classes, and analyzed holomorphic and meromorphic gauge fields.
result Existence of curves of Yang-Mills fields connecting vacuum states on bundles over the torus T2. Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of Z-positive and Z-critical metrics leading to bundle stability. result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, a…
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. Formula found for energy slope in complex geometry.
problem Calculating the asymptotic slope of a K-energy.
method Established a formula for the asymptotic slope.
result Found a formula for the asymptotic slope of α-K-energy.
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable.