Mills Equations Research Articles

Higher dimensional generalizations of the chiral field equations

AbstractWe consider the self‐dual Yang–Mills equation and its reduction, the Manakov–Zakharov system. We discuss three‐ and four‐dimensional generalizations of the chiral field equations, and explain methods for constructing their exact solutions.

Explicit abelian instantons on S1-invariant Kähler Einstein 6-manifolds

We consider the dimensional reduction of the (deformed) Hermitian Yang–Mills condition on S1-invariant Kähler Einstein 6-manifolds. This allows us to reformulate the (deformed) Hermitian Yang–Mills equations in terms of data on the quotient Kähler 4-manifold. In particular, when the gauge group is U(1) we apply this construction to the canonical bundles of CP2 and CP1×CP1 endowed with the Calabi ansatz metric to find abelian instantons. We show that these are determined by a suitable subset of the spectrum of the zero section and are explicitly given in terms of certain hypergeometric functions. As a by-product of our investigation we find a coordinate expression for the holomorphic volume form on OCP2(−3) and use it to construct a new special Lagrangian foliation. We also find 1-parameter families of explicit deformed Hermitian Yang–Mills connections on certain non-compact S1-invariant Kähler Einstein 6-manifolds.

Exact gauge fields from anti-de Sitter space

In 1977 Lüscher found a class of SO(4)-symmetric SU(2) Yang–Mills solutions in Minkowski space, which have been rederived 40 years later by employing the isometry S3 ≅ SU(2) and conformally mapping SU(2)-equivariant solutions of the Yang–Mills equations on (two copies of) de Sitter space dS4≅R×S3. Here we present the noncompact analog of this construction via AdS3 ≅ SU(1, 1). On (two copies of) anti-de Sitter space AdS4≅R×AdS3 we write down SU(1,1)-equivariant Yang–Mills solutions and conformally map them to R1,3. This yields a two-parameter family of exact SU(1,1) Yang–Mills solutions on Minkowski space, whose field strengths are essentially rational functions of Cartesian coordinates. Gluing the two AdS copies happens on a dS3 hyperboloid in Minkowski space, and our Yang–Mills configurations are singular on a two-dimensional hyperboloid dS3∩R1,2. This renders their action and the energy infinite, although the field strengths fall off fast asymptotically except along the lightcone. We also construct Abelian solutions, which share these properties but are less symmetric and of zero action.

2D Discrete Yang–Mills Equations on the Torus

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form.

The Boundary Case for the Supercritical Deformed Hermitian–Yang–Mills Equation

The Boundary Case for the Supercritical Deformed Hermitian–Yang–Mills Equation

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

Global exterior stability of the Minkowski space: Coupled Einstein–Yang–Mills perturbations

Here we prove a global gauge-invariant radiation estimates for the perturbations of the 3 + 1 dimensional Minkowski spacetime in the presence of Yang–Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang–Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang–Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang–Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang–Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang–Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source.

On Recursion Operators for Full-Fledged Nonlocal Symmetries of the Reduced Quasi-classical Self-dual Yang–Mills Equation

We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}$$\\end{document}-algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows.

A Nakai–Moishezon Type criterion for supercritical deformed Hermitian–Yang–Mills equation

A Nakai–Moishezon Type criterion for supercritical deformed Hermitian–Yang–Mills equation

Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes

<abstract><p>Two numerical schemes are proposed and investigated for the Yang–Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang–Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $ L^2 $-norm of the potential and electrical fields in $ \mathcal O(h^{k+1}) $ (provided that the time step is of that order), where $ k $ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.</p></abstract>

Predicting angular-momentum waves based on Yang–Mills equations

As one of the most elegant theories in physics, Yang–Mills (YM) theory not only incorporates Maxwell’s equations unifying electromagnetism, but also underpins the standard model explaining the electroweak and strong interactions in a succinct way. Whereas the highly nonlinear terms in YM equations involving the interactions between potentials and fields retard the resolution for them. In the U(1) case, the solutions of Maxwell’s equations are the electromagnetic waves, which have been applied extensively in the modern communication networks all over the world. Likewise the operator solutions of the YM equations under the assumptions of weak-coupling and zero-coupling predict the SU(2) angular-momentum waves, which is the staple of this work. Such angular-momentum waves are hopefully realized in the experiments through the oscillation of spin angular momentum, such as the “spin Zitterbewegung” of Dirac’s electron.

J-equation on holomorphic vector bundles

We introduce the J-equation on holomorphic vector bundles over compact Kähler manifolds and investigate some fundamental properties as well as examples of solutions. In particular, we provide an algebraic condition called (asymptotic) J-stability in terms of subbundles on compact Kähler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian–Yang–Mills equation in the small volume regime.

The Set of Destabilizing Curves for Deformed Hermitian Yang–Mills and Z-Critical Equations on Surfaces

Abstract We show that on any compact Kähler surface existence of solutions to the Z-critical equation can be characterized using a finite number of effective conditions, where the number of conditions is bounded above by the Picard number of the surface. This leads to a first PDE analogue of the locally finite wall-chamber decomposition in Bridgeland stability. As an application we characterize optimally destabilizing curves for Donaldson’s J-equation and the deformed Hermitian Yang–Mills equation, prove a non-existence result for optimally destabilizing test configurations for uniform J-stability, and remark on improvements to convergence results for certain geometric flows.

Yang–Mills–Stueckelberg theories, framing and local breaking of symmetries

We consider Yang–Mills theory with a compact structure group [Formula: see text] on a Lorentzian 4-manifold [Formula: see text] such that gauge transformations become identity on a submanifold [Formula: see text] of [Formula: see text] (framing over [Formula: see text]). The space [Formula: see text] is not necessarily a boundary of [Formula: see text] and can have dimension [Formula: see text]. Framing of gauge bundles over [Formula: see text] demands introduction of a [Formula: see text]-valued function [Formula: see text] with support on [Formula: see text] and modification of Yang–Mills equations along [Formula: see text]. The fields [Formula: see text] parametrize non-equivalent flat connections mapped into each other by a dynamical group [Formula: see text] changing gauge frames over [Formula: see text]. It is shown that the charged condensate [Formula: see text] is the Stueckelberg field generating an effective mass of gluons in the domain [Formula: see text] of space [Formula: see text] and keeping them massless outside [Formula: see text]. We argue that the local Stueckelberg field [Formula: see text] can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace.

The deformed Hermitian–Yang–Mills equation, the Positivstellensatz, and the solvability

Let (M,ω) be a compact connected Kähler manifold of complex dimension four and let [χ]∈H1,1(M;R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑iarctan⁡(λi)=θˆ, where λi are the eigenvalues of χ with respect to ω and θˆ is a topological constant. This conjecture was stated in [8], wherein they proved that the existence of a supercritical C-subsolution or the existence of a C-subsolution when θˆ∈[((n−2)+2/n)π/2,nπ/2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θˆ>(n−2)π/2, where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θˆ is close to the supercritical phase π from the right, then the existence of a C-subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation.

A Note on the Deformed Hermitian–Yang–Mills Equation with Gradient Terms on Compact Almost Hermitian Manifolds

A Note on the Deformed Hermitian–Yang–Mills Equation with Gradient Terms on Compact Almost Hermitian Manifolds

Classification of all constant solutions of SU(2) Yang–Mills equations with arbitrary current in pseudo-Euclidean space ℝp,q

We present a classification and an explicit form of all constant solutions of the Yang–Mills equations with [Formula: see text] gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space [Formula: see text] of arbitrary finite dimension [Formula: see text]. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang–Mills equations of [Formula: see text] cubic equations with [Formula: see text] unknowns and [Formula: see text] parameters in the general case. We present a new symmetry of this system of equations. All solutions in terms of the potential, strength and invariant of the Yang–Mills field are presented. Nonconstant solutions of the Yang–Mills equations can be considered in the form of series of perturbation theory using all constant solutions as a zeroth approximation.

Solutions to the SU([formula omitted]) self-dual Yang–Mills equation

In this paper we aim to derive solutions for the SU(N) self-dual Yang–Mills (SDYM) equation with arbitrary N. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev–Petviashvili hierarchy and the (matrix) Ablowitz–Kaup–Newell–Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

On the convexity of general inverse σk equations

We prove that if a level set of a degree n general inverse σk equation f(λ1,⋯,λn):=λ1⋯λn−∑k=0n−1ckσk(λ)=0 is contained in q+Γn for some q∈Rn, where ck are real numbers not necessary to be non-negative and Γn is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse σk type equations, for example, the Monge–Ampère equation, the Hessian equation, the J-equation, the deformed Hermitian–Yang–Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse σk equation is contained in q+Γn for some q∈Rn, which is a way to determine the convexity of this level set.

On the global well-posedness of the Einstein–Yang–Mills system

In this paper, we present a partial result on the global well-posedness of the Cauchy problem for the Einstein–Yang–Mills system in constant mean extrinsic curvature spatial harmonic and generalized Coulomb gauges as introduced in the work of Mondal [arXiv:2112.14273 (2021)]. We give a small-data global existence theorem for a family of n + 1 dimensional spacetimes with n ≥ 4, utilizing energy arguments presented in the work of Andersson and Moncrief [J. Differ. Geom. 89, 1–47 (2009)]. We observe that these energy arguments will fail for n = 3 due to the conformal invariance of 3 + 1 Yang–Mills equations and present a gauge-covariant formulation of the Einstein–Yang–Mills system in 3 + 1 dimensions to show that an energy argument cannot be used to prove the global well-posedness result, regardless of the choice of gauge.