Reducible connections and non-local symmetries of the self-dual Yang-Mills equations
James D. E. Grant
备注：
AMSLatex, 15 pages, no figures. Corrected in line with the referee's comments. In particular, restriction to simply-connected open sets now explicitly stated. Version to appear in Communications in Mathematical Physics
We construct the most general reducible connection that satisfies the self-dual Yang-Mills equations on a simply connected, open subset of flat $\mathbb{R}^4$. We show how all such connections lie in the orbit of the flat connection on $\mathbb{R}^4$ under the action of non-local symmetries of the self-dual Yang-Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang-Mills equations that are analogous to harmonic maps of finite type.
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 of such pairs.
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 energy.
A local monotonicity formula for the Yang-Mills-Higgs flow on $G$-bundles over $\mathbb{R}^{n}$ ($n>4$) 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.
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity; and we obtain a gap theorem for homothetically shrinking solitons.
In this note, we prove an ${L^{\frac{n}{2}}}$-energy gap result for Yang-Mills connections on a principal $G$-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).
The purpose of this work is to present a Non--Commutative Geometrical version of the Yang--Mills Theory and the Yang--Mills Scalar Matter Theory by constructing a concrete example using M. Durdevich's theory of quantum principal bundles.
We investigate Lie symmetries of general Yang-Mills equations. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on the Yang-Mills equations. Determining equations are then obtained, and solved completely. Provided that Yang-Mills equations are locally solvable, this allows for a complete classification of their Lie symmetries. Symmetries of Yang-Mills equations in the Lorentz gauge are also investigated.
The Yang-Mills field strength incorporating a non-Abelian feature is one of the cornerstones of the standard model. Although Yang-Mills gauge theories have been around for over fifty years, surprisingly the derivation of the Yang-Mills field strength using classical gauge theory does not appear anywhere in the literature. In their 1954 paper, Yang and Mills had to invent a non-Abelian field strength to satisfy certain criteria. In Section 5 we use Yang's gauge transformation in a heuristic derivation of the Yang-Mills field strength. The preceding sections cover material relating to the deriva…
In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We calculate their action and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions on Euclidean three-space.
Shown is a new duality for the moduli spaces of Yang-Mills connections over noncommutative vector bundles, using which one sees that total data of quantum field theory are preserved by dimension reduction.
In this short note we provide an expository account of the work of Leonard Gross and the author on the Yang-Mills heat equation over smooth three-manifolds with boundary.
A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mil…
We present a new class of covariantly constant solutions of the Yang-Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form.
The phenomena of emergent fuzzy geometry and noncommutative gauge theory from Yang-Mills matrix models is briefly reviewed. In particular, the eigenvalues distributions of Yang-Mills matrix models in lower dimensions in the commuting (matrix or Yang-Mills) phase of these models are discussed.
I articulate and discuss a geometrical interpretation of Yang-Mills theory. Analogies and disanalogies between Yang-Mills theory and general relativity are also considered.
In this work, we introduce a novel approach to the problem of gauge choice for the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, which uses the Yang-Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As the first demonstration of the `structure' offered by this new approach, we will give an alternative proof of the local well-posedness of the Yang-Mills equations for initial data in $(\dot{H}^{1}_{x} \cap L^{3}_{x}) \times L^{2}_{x}$, which is a c…
We construct a free-probability quantum Yang-Mills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that this theory is the $N=\infty$ limit of U(N) quantum Yang-Mills theory on the plane.
A new classical solution of euclidean Yang-Mills gauge theory, which is governed by $π_4 (SU(2))$, is given. Its relationship to knot theory and Hopfions is discussed.
In this letter we establish Yangian symmetry of planar N=4 super-Yang-Mills theory. We prove that the classical equations of motion of the model close onto themselves under the action of Yangian generators. Moreover we propose an off-shell extension of our statement which is equivalent to the invariance of the action and prove that it is exactly satisfied. We assert that our relationship serves as a criterion for integrability in planar gauge theories by explicitly checking that it applies to integrable ABJM theory but not to non-integrable N=1 super-Yang-Mills theory.
We determine all inhomogeneous Yang-Mills algebras and super Yang-Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this paper, the homogeneous parts are the Yang-Mills algebra and the super Yang-Mills algebra.
Yangian symmetry of amplitudes in $\mathcal{N}=4$ super Yang-Mills theory is formulated in terms of eigenvalue relations for monodromy matrix operators. The Quantum Inverse Scattering Method provides the appropriate tools to treat the extended symmetry and to recover as its consequences many known features like cyclic and inversion symmetry, BCFW recursion, Inverse Soft Limit construction, Grassmannian integral representation, $\mathrm{R}$-invariants and on-shell diagram approach.
The equations of a relative equilibrium in a pure Yang--Mills gauge theory with the Coulomb gauge fixing are obtained. They are derived as a direct consequence of the results of our previous work on Wong's equations in gauge theory.The obtained equations are similar to the equations of a relative equilibrium in reducible finite-dimensional dynamical systems with a symmetry. In all these equations, the description of the reduced motion is performed by making use of the dependent coordinates.
In the noncommutative geometry program of Connes there are two variations of the concept of Yang-Mills action functional. We show that for the quantum Heisenberg manifolds they agree.
We show that a class of previously defined maps, called self-dual and causal morphisms, form classical symmetries of Yang-Mills fields in four complex dimensions. These maps generalize conformal transformations, and admit a nonlocal pullback connection that preserves the equations of the theory. First it is shown that self-dual morphisms form symmetries of the anti-self-dual Yang-Mills equations under this pullback. Then a supersymmetric generalization of causal morphisms is defined, which preserves solutions of the field equations for N=3 supersymmetric Yang-Mills theory. As a special case, t…
Using methods of differential geometry, a discrete analog of the Yang-Mills equations in Minkowski space is constructed. The gauge transformation law in a discrete formulation is given and gauge invariance of discrete Yang-Mills equations is studied. Difference self-dual and anti-self-dual equations with respect to the Lorentz metric are presented.
The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra.
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a massive Yang-Mills field. The resolution is achieved by first solving the free eigenvalue problem for the gravitational field and then the constrained eigenvalue problem for the Yang-Mills field. In the latter case the mass of the Yang-Mills field assumes the role of the eigenvalue.
