Area law proven for 2D Yang-Mills in generalized axials.
problem Proving area law for 2D Yang-Mills theory in generalized axials.
method Homotopy invariance of iterated integrals, complex gauge-fixing, integration cycle deformation.
result Wilson loop expectation values obey area law up to second order.
This paper compares exact 2D Yang-Mills theory with perturbative approaches.
problem Investigating the relationship between exact and perturbative approaches in 2D Yang-Mills theory.
method Mathematical rigorous formulation of perturbative quantization and comparison of Wilson loop expectation values.
result Exact lattice Wilson loop expectations on S2 agree with perturbative expectations in holomorphic gauge for simple closed curves to all orders. We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…
We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of (n,m) torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the (m,n)↔(n,m) s…
Langmuir-Blodgett films (LB-films) consist from few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe structurization into LB-monolayers. Consequently, we develop on the 1-jet space J^1([0,\infty),R^2) the single-time Lagr…
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
New method defines Wilson surface observables on arbitrary 2D surfaces.
problem Defining Wilson surface observables on arbitrary surfaces.
method Using equivariant cohomology and Poisson σ-models.
result Wilson surface observables are nontrivial for non-simply connected groups.
Study on determinant properties of elliptic operators with counterexamples and positive results.
problem Does the determinant of a matrix solution to a second order elliptic equation satisfy the unique continuation property?
method Analyzes counterexamples and positive results for various operators, including reductions to special cases.
result Partial answers and counterexamples provided, with positive results for specific cases.
New 2D TQFT connects Chern-Simons theory on circle bundles to gauge theory.
problem Localization of Chern-Simons theory on circle bundles.
method Caloron correspondence and symplectic structure.
result Wilson loops on circle fibers described naturally.
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-ty…
We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example,…
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. 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 studies stability of F-Yang-Mills connections on complex projective spaces.
problem Stability of F-Yang-Mills connections on complex projective spaces.
method Inspired by Lawson-Simons, the paper proves stability conditions and structures for F-Yang-Mills connections.
result Conditions for weakly stable F-Yang-Mills connections on complex projective spaces.
Research explores vortices, Painlevé integrability, and projective geometry in Yang-Mills theory.
problem Exploring vortices and their properties in Yang-Mills theory.
method Analyzes Abelian vortices, proposes modified Abelian-Higgs model, applies Painlevé test, and investigates projective differential geometry.
result Identifies integrable cases of soliton equations and derives solutions in terms of third Painlevé transcendents.
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.
Proves local monotonicity for Yang-Mills-Higgs flow on bundles.
problem Ensuring monotonicity for Yang-Mills-Higgs flow solutions.
method Proved a local monotonicity formula for G-bundles over Rn (n>4). result Monotone quantity matches existing formulas for Yang-Mills and Yang-Mills-Higgs flows.
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.
New theorems in 2D and 4D for metrics with curvature or singularity.
problem Proving new versions of Huber theorem in dimensions 2 and 4.
method Using Coulomb frames and Bach tensor conditions to construct conformal metrics.
result Constructs conformal metrics with regularity across singularities in 4D.
We prove that the Yang-Mills α-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills α-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as α→1, a sequence of Yang-Mills α-connections converge…
New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.
problem Developing new mathematical tools for Yang-Mills theory.
method Introducing normalized exponential Yang-Mills energy functional, deriving monotonicity formula and vanishing theorem.
result Monotonicity and vanishing theorems for exponential Yang-Mills fields.
In this paper, we introduce some notions on the pair consisting of a Chern connection and a Higgs field closely related to the first and second variation of Yang-Mills- Higgs functional, such as strong Yang-Mills-Higgs pair, degenerate Yang-Mills-Higgs pair, stable Yang-Mills-Higgs pair. We investigate some properties …
Compact formulas for Yang-Mills conditions on conformal manifolds.
problem Finding variational analogs of Yang-Mills conditions.
method Provided compact formulas and proved their equivalence to Yang-Mills conditions under specific metrics.
result Conformal Yang-Mills condition is equivalent to vanishing of Fefferman-Graham obstruction tensor.
The paper defines and studies new types of submanifolds in a unit sphere.
problem Variational problems of curvature tensors for submanifolds.
method Euler-Lagrange equations for Normal-Yang-Mills and Tangent-Yang-Mills submanifolds.
result Infinitely many non-trivial examples of Normal-Yang-Mills and Tangent-Yang-Mills submanifolds are constructed.
We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow li…
We prove the first mathematical result relating the Yang-Mills measure on a compact surface and the Yang-Mills energy. We show that, at the small volume limit, the Yang-Mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the Yang-Mills energ…
Explains a 1978 construction for Yang-Mills instantons.
problem None explicitly stated; focuses on explaining a construction.
method Atiyah-Drinfeld-Hitchin-Manin construction
result Explains the ADHM construction of Yang-Mills instantons.
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. Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.
problem Obstructing higher conformal Yang-Mills equations on conformally compact manifolds.
method Formal asymptotics, Dirichlet-to-Neumann maps, higher transverse derivative boundary operators.
result Obstructing current is the variation of a conformally invariant coefficient in the interior Yang-Mills energy expansion.
Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
problem Resolving conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
method Study compact Kähler manifolds and resolves conjectures of Collins-Yau.
result Resolves two conjectures of Collins-Yau.
Study the deformation theory of Einstein-Yang-Mills system on compact manifolds.
problem Deformation theory of Einstein-Yang-Mills system on compact manifolds.
method Slice theorem, linearization analysis, essential deformation characterization.
result Realize moduli space of Einstein-Yang-Mills pairs as an analytic set in a finite-dimensional tame Fréchet manifold.
Sharp gap theorem found for Yang-Mills connections in 4D.
problem Yang-Mills connections in 4D.
method Exploiting an associated Yamabe-type problem.
result Sharp conformally invariant gap theorem proved.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Proves symmetries of extremal horizons in spacetimes.
problem Proving symmetries of extremal horizons in arbitrary dimensions.
method Analyzes Killing vector fields and near-horizon geometry.
result Enhanced isometry groups and shifted Aretakis instability.
Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.
problem Proving uniqueness of Yang-Mills field tangent cones.
method Log-epiperimetric inequality, Luckhaus type lemma, and curvature concentration exclusion.
result Uniqueness of tangent cones for Yang-Mills fields in arbitrary dimensions.
Study Yang-Mills connections on conformally compact manifolds, proving existence of extensions.
problem Existence of Yang-Mills connections on conformally compact manifolds.
method Proving existence of Yang-Mills connections for small deformations.
result Confirming the existence of Yang-Mills connections in the interior of the manifold.
The paper constructs examples of coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
problem Constructing coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
method Constructing spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with SU(2) structure group.
result The construction yields coupled solutions, including on S^1(r_1) x S^2(r_2) for certain radii.
Study on a deformed Hermitian-Yang-Mills equation on compact Kähler manifolds.
problem Existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
method Introduce coerciveness and properness of the J-functional on almost calibrated (1,1)-forms.
result Equivalence of coerciveness and properness to the existence of solutions.
The paper applies gauge fixing to Yang-Mills flow on manifolds.
problem Existence and uniqueness of weak solutions for Yang-Mills flow.
method Parabolic gauge fixing theorems, key lemma improvements.
result Existence and uniqueness of weak solutions for Yang-Mills flow on 4D manifolds.
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.
Paper proves no stable Yang-Mills fields on spheres.
problem Existence of stable Yang-Mills fields on spheres.
method Analyzes C2 neighborhoods of Euclidean sphere metrics and warped product manifolds. result No nontrivial weakly stable Yang-Mills connections in specified neighborhoods.
Gravitational symmetries linked to Yang-Mills squared.
problem Understanding gravitational symmetries through Yang-Mills theories.
method Introducing a dictionary linking gravitational fields to the product of two Yang-Mills theories.
result Symmetries of each Yang-Mills theory generate symmetries of the corresponding gravity theory.
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.
Proves existence of Yang-Mills fields for specific curvature conditions.
problem Yang-Mills theory with specific curvature constraints.
method Proof of existence for non-degenerate traceless Ricci curvature.
result Locally exists SO(3) Yang-Mills field matching curvature. The paper finds asymmetric Type-I blowup solutions for Yang-Mills flow.
problem Existence of asymmetric Type-I blowup solutions for Yang-Mills flow.
method Constructing an infinite-dimensional family of solutions for the Yang-Mills flow on RnimesSO(n) for 5≤n≤9. result Existence of asymmetric Type-I blowup solutions for the Yang-Mills flow.
Study on a new Yang-Mills type functional and its critical points.
problem Deriving a new functional from Yang-Mills and Born-Infeld theories.
method First and second variation formulas established.
result Existence of critical points proved.
This paper surveys geometric invariant theory for Yang-Mills equations on Riemann surfaces.
problem Analyzing Yang-Mills equations on Riemann surfaces using geometric invariant theory.
method Exposition of Atiyah-Bott picture, focusing on semistable and unstable orbits.
result New proof of moment-weight inequality and convergence of Yang-Mills flow.