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.
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
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.
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.
Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
The paper defines and analyzes higher-order Yang-Mills-Higgs functionals and their gradient flows.
problem Analyzing the behavior of higher-order Yang-Mills-Higgs functionals and their gradient flows.
method Gauge fixing technique, L2-bound of the Higgs field, local L2-derivative estimates, energy estimates, blow-up analysis. result Solutions to the gradient flow do not hit finite time singularities under certain conditions.
The paper studies how adding a 'Gauge Mass' term breaks gauge symmetry in Yang-Mills-Higgs systems and analyzes the resulting behavior.
problem Breaking gauge symmetry in Yang-Mills-Higgs systems.
method Analyzing the asymptotic behavior of the system with a 'Gauge Mass' term added.
result The system's behavior is characterized by concentration phenomena and convergence to harmonic maps and minimal energies.
Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.
problem Quadratic one-forms on logarithmic Higgs bundles on pointed curves.
method Use elementary pole cancellation for invariant polynomials.
result Found a logarithmic quadratic one-form.
The paper analyzes flows related to Higgs energies on manifolds.
problem Analyzing flows related to Higgs energies on manifolds.
method Developing asymptotic analysis for gradient flow of self-dual U(1)-Higgs energies. result Solutions converge to codimension-two mean curvature flows.
The study finds critical points of Yang-Mills-Higgs energy on 3-manifolds.
problem Finding critical points of Yang-Mills-Higgs energy on 3-manifolds.
method 2-parameter min-max construction and energy gap analysis.
result Existence of non-trivial critical points on 3-manifolds with bounded geometry.
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.
Study investigates singularity formation in α-Yang-Mills-Higgs fields on spheres.
problem Singularity formation in α-Yang-Mills-Higgs fields on spheres. method Established α-energy identity, no-neck property through Hodge decomposition and new conservation law. result Unified and quantitative framework for singularity formation in variational gauge theories.
The paper studies decay near singularities of 3d Yang-Mills-Higgs fields.
problem Understanding isolated singularities of 3d Yang-Mills-Higgs fields.
method Derives decay estimates and applies removable singularity theorems.
result Generalizes removable singularity theorems for 3d Yang-Mills-Higgs fields.
Constructs Higgs bundle moduli spaces using Teichmüller space.
problem Holomorphic family of Higgs bundle moduli spaces over a curve.
method Uses a function f on the character variety to define flat Ehresmann connections.
result Reveals various aspects of moduli spaces and their metrics.
The paper studies entropy and free energy for harmonic metrics on cyclic Higgs bundles.
problem Quantifying the degree of mutual misalignment of metrics on Higgs bundles.
method Introduced entropy and free energy to quantify mutual misalignment; provided conditions for entropy and free energy to change.
result Extended work on boundedness of functions related to entropy and free energy on the unit disc.
We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scatterin…
In this paper we consider SU(2) monopoles on an asymptotically conical, oriented, Riemannian 3-manifold with one end. The connected components of the moduli space of monopoles in this setting are labeled by an integer called the charge. We analyse the limiting behavior of sequences of monopoles with fixed charg…
The study shows how energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces.
problem Understanding energy density and topological invariants in n-Fuchsian fibers of Higgs bundles. method Establishing an algebraic inequality generalizing a GIT theorem to prove energy density domination.
result Energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces. The Higgs field growth is studied on special geometric spaces, confirming a conjecture.
problem Growth of the Higgs field in special geometric spaces.
method Analyzing θ-Kapustin-Witten equations on ALX spaces. result Finite energy solutions on ALE and ALF instantons have vanishing commutator and flat connection.
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Boosted decision trees improved for particle identification in high-energy physics.
problem Overfitting in boosted decision trees hampers their performance in particle identification.
method Meta-learning techniques of boosting and bagging to mitigate overfitting.
result The proposed algorithm achieves performance close to that of deep neural networks on a benchmark data set.
We define a functional J(h) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that ${\c…
Energy functional on Teichmüller space is plurisubharmonic but not strictly so.
problem Characterizing points where energy functional fails to be strictly plurisubharmonic.
method Analyzing the kernel of the Levi form and relating it to Higgs bundles and Hitchin fibration.
result For generic choices, energy functional is strictly plurisubharmonic.
The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.
problem Proving uniqueness of blow-down for critical points of a Yang-Mills-Higgs functional.
method Using an Allard-type improvement of flatness to establish co-dimension-two analogue of Savin's theorem.
result Entire critical points have unique blow-down, two-dimensional in ambient dimensions 2-4 or any dimension assuming local minimizer.
Reformulates binary classification on manifolds using Yang-Mills-Higgs theory.
problem Binary classification on non-contractible spaces.
method Formulates binary classification as a Yang-Mills-Higgs variational problem, encoding data as a functor.
result Reveals a geometric interpretation of binary classification and solves XOR on the torus.
Minimal submanifolds are found as energy concentration sets in variational problems.
problem Understanding the structure of minimal submanifolds in codimension two.
method Purely variational approach, extending previous work on geodesics.
result Non-degenerate minimal submanifolds can be derived from critical maps of the Ginzburg-Landau functional.
We study Bogomolny equations on R2×S1. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using Nahm transform, we show that periodic monop…
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
The paper shows how Yang-Mills-Higgs energies converge to the (n−2)-area functional.
problem Understanding the convergence of Yang-Mills-Higgs energies to the (n−2)-area functional. method Analyzing the convergence of critical points of Yang-Mills-Higgs energies to minimal submanifolds and proving Γ-convergence. result Yang-Mills-Higgs energies converge to the (n−2)-area functional as εo0. We develop a new method for proving regularity for small energy stationary solutions of coupled gauge field equations. Our results duplicate those of Tian--Tao [7] for the pure Yang Mills equations, but our proof is simpler, and obtains bounded curvature without the use of Coulomb gauges. It relies instead on the Weitz…
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-…
The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.
problem Modeling hyperkähler 4-manifolds and their degenerations.
method Using parabolic SL(2,C)-Higgs bundles and Nakajima quiver varieties.
result ALG-D4 spaces degenerate to ALE-D4 spaces under a limit. Paper constructs moduli spaces of Higgs bundles and connects them to Teichmüller space structures.
problem Constructing moduli spaces of Higgs bundles on varying Riemann surfaces.
method Gauge theoretic construction, Teichmüller space, isomonodromic foliation, Atiyah-Bott-Goldman symplectic structure.
result Surprising relationships between Higgs bundles, isomonodromic foliation, and Teichmüller space structures.
Study examines Yang-Mills-Higgs energy convergence to codimension-three area functional.
problem Asymptotic behavior of Yang-Mills-Higgs energy in large mass limit.
method Investigates the asymptotic behavior of Yang-Mills-Higgs energy in the large mass limit, proving convergence to the codimension-three area functional.
result The (n−3)-currents dual to the Yang-Mills-Higgs energy converge to a relative integral (n−3)-cycle. A common problem in a high energy physics experiment is extracting a signal from a much larger background. Posed as a classification task, there is said to be an imbalance in the number of samples belonging to the signal class versus the number of samples from the background class. In this work we provide a brief overv…
Given a Hermitian line bundle L→M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as ε→0, of couples (uε,∇ε) critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dua…
In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the Riemann surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth Y…
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimi…
We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is give…
Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.
problem Understanding the asymptotic geometry of character varieties.
method Nonabelian Hodge correspondence and study of harmonic maps.
result Asymptotic correspondence between limiting configurations and geometric parameters.
Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.
problem Proving growth of spinors in GSW equations on R4 and R3. method Unified framework of GSW equations, averaged L2-norm, curvature decay assumption, Yang-Mills-Higgs energy. result Growth of spinors in GSW equations on R4 and R3 faster than a power of the radius under suitable curvature decay. Proposes a meta-algorithm for classification with overlapping classes in high-energy physics.
problem Challenges of class overlap in binary classification.
method Combines bagging and boosting techniques with a randomization trick.
result Improves statistical significance of Higgs discovery.
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.
Jets from boosted heavy particles have a typical angular scale which can be used to distinguish them from QCD jets. We introduce a machine learning strategy for jet substructure analysis using a spectral function on the angular scale. The angular spectrum allows us to scan energy deposits over the angle between a pair …
Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.
problem Analyzing convergence of Yang-Mills-Higgs flow for twisted Higgs pairs.
method Proves convergence to a reflexive twisted Higgs sheaf outside a closed subset.
result Limiting twisted Higgs sheaf is isomorphic to the double dual of graded twisted Higgs sheaves.
Classifies Higgs and co-Higgs bundles on symmetric spaces.
problem Classifying Higgs and co-Higgs bundles over Hermitian symmetric spaces.
method Defined homogeneous principal Higgs and co-Higgs bundles, provided a classification up to isomorphism.
result Defined and classified moduli spaces for each type of bundle.
The harmonic sections of the Kaluza-Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combin…
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…