Yang-Mills Equations Research Articles

Covariantly Constant Solutions of the Yang–Mills Equations

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.

A new construction of rational electromagnetic knots

We set up a correspondence between solutions of the Yang–Mills equations on R×S3 and in Minkowski spacetime via de Sitter space. Some known Abelian and non-Abelian exact solutions are rederived. For the Maxwell case we present a straightforward algorithm to generate an infinite number of explicit solutions, with fields and potentials in Minkowski coordinates given by rational functions of increasing complexity. We illustrate our method with a nontrivial example.

Evolution of pressures and correlations in the glasma produced in high energy nuclear collisions

We consider the SU(2) Glasma with gaussian fluctuations and study its evolution by means of classical Yang-Mills equations solved numerically on a lattice. Neglecting in this first study the longitudinal expansion we follow the evolution of the pressures of the system and compute the effect of the fluctuations in the early stage up to $t\approx 2$ fm/c, that is the time range in which the Glasma is relevant for high energy collisions. We measure the ratio of the longitudinal over the transverse pressure, $P_L/P_T$, and we find that unless the fluctuations carry a substantial amount of the energy density at the initial time, they do not change significantly the evolution of $P_L/P_T$ in the early stage, and that the system remains quite anisotropic. We also measure the longitudinal fields correlators both in the transverse plane and along the longitudinal direction: while at initial time fields appear to be anticorrelated in the transverse plane, this anticorrelation disappears in the very early stage and the correlation length in the transverse plane increases; on the other hand, we find that the longitudinal correlator decreases for a small longitudinal separation while being approximately constant for larger separation, which we interpret as a partial loss of longitudinal correlation induced by the dynamics.

A note on the deformed Hermitian Yang-Mills PDE

ABSTRACTWe prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob–Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.

BRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fock space

We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extension of the free theory. It is verified that the evolution equations implied by the resulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations. The modifications and requirements for the inclusion of matter are discussed in detail.

Exact Cosmological Models with Yang–Mills Fields on Lyra Manifold

The present study deals with the Friedmann-Robertson-Walker cosmological models of the Yang-Mills (YM) fields in Lyra geometry. The energy-momentum tensor of the YM fields for our models is obtained with the help of exact solution for the corresponding YM equations subject to their minimal coupling to gravity. Two specific exact solutions of the model are obtained regarding the effective equation of state and the exponential law of expansion. Some physical and geometrical behavior of the model are also discussed.

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

A survey of the GIT picture for the Yang–Mills equation over Riemann surfaces

The purpose of this paper is to give a self-contained exposition of the Atiyah–Bott picture [AB83] for the Yang–Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This includes the analogue of the Ness uniqueness theorem for Yang–Mills connections, the Kempf–Ness theorem, the Hilbert–Mumford criterion and a new proof of the moment-weight inequality following an approach outlined by Donaldson [Don4]. A central ingredient in our discussion is the Yang–Mills ow for which we assume longtime existence and convergence (see [Rad]).

Special Lagrangian and deformed Hermitian Yang–Mills on tropical manifold

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mari\~no, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.

Critical phenomena in the general spherically symmetric Einstein-Yang-Mills system

We study critical behavior in gravitational collapse of a general spherically symmetric Yang-Mills field coupled to the Einstein equations. Unlike the magnetic ansatz used in previous numerical work, the general Yang-Mills connection has two degrees of freedom in spherical symmetry. This fact changes the phenomenology of critical collapse dramatically. The magnetic sector features both type I and type II critical collapse, with universal critical solutions. In contrast, in the general system type I disappears and the critical behavior at the threshold between dispersal and black hole formation is always type II. We obtain values of the mass scaling and echoing exponents close to those observed in the magnetic sector, however we find some indications that the critical solution differs from the purely magnetic discretely self-similar attractor and exact self-similarity and universality might be lost. The additional "type III" critical phenomenon in the magnetic sector, where black holes form on both sides of the threshold but the Yang-Mills potential is in different vacuum states and there is a mass gap, also disappears in the general system. We support our dynamical numerical simulations with calculations in linear perturbation theory; for instance, we compute quasi-normal modes of the unstable attractor (the Bartnik-McKinnon soliton) in type I collapse in the magnetic sector.

A Conformally Invariant Gap Theorem in Yang–Mills Theory

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Reexamining the gluon spectrum in the boost-invariant glasma from a semianalytic approach

In high energy heavy-ion collisions, the degrees of freedom at the very early stage can be effectively represented by strong classical gluonic fields within the Color Glass Condensate framework. As the system expands, the strong gluonic fields eventually become weak such that an equivalent description using the gluonic particle degrees of freedom starts to become valid. We revisit the spectrum of these gluonic particles by solving the classical Yang-Mills equations semi-analytically with the solutions having the form of power series expansions in the proper time. We propose a different formula for the gluon spectrum which is consistent with energy density during the whole time evolution. We find that the chromo-electric fields have larger contributions to the gluon spectrum than the chromo-magnetic fields do. Furthermore, the large momentum modes take less time to reach the weak-field regime while smaller momentum modes take more time. The resulting functional form of the gluon spectrum is exponential in nature and the spectrum is close to a thermal distrubtion with effective temperatures around $0.6$ to $0.9\, Q_s$ late in the Glasma evolution. The sensitiveness of the gluon spectrum to the infrared and the ultraviolet cut-offs are discussed.

Chiral higher spin theories and self-duality

We study recently proposed chiral higher spin theories — cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidden symmetries, which ensures their integrability. Secondly, we show that off-shell amplitudes in chiral higher spin theories satisfy the generalised BCJ relations with the usual colour structure constants replaced by the structure constants of higher spin gauge algebras. We also propose generalised double copy procedures featuring higher spin theory amplitudes. Finally, using the light-cone deformation procedure we prove that the structure of the Lagrangian that leads to all these properties is universal and follows from Lorentz invariance.

Direct test of the integral Yang-Mills equations through SU(2) monopoles

We use the SU(2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory.

Finite-action solutions of Yang-Mills equations on de Sitter dS4 and anti-de Sitter AdS4 spaces

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS4 and anti-de Sitter AdS4 spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in {mathbb{R}}^3 under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and action. We propose a lower bound for the action on both backgrounds. Employing another metric on AdS4, the SU(2) Yang-Mills equations are reduced to an analytic continuation of the above particle mechanics from {mathbb{R}}^3 to {mathbb{R}}^{2,1} . We discuss analytical solutions to these equations, which produce infinite-action configurations. After a Euclidean continuation of dS4 and AdS4 we also present self-dual (instanton-type) Yang-Mills solutions on these backgrounds.

The decay of the SU(2) Yang–Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data

Abstract We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang–Mills fields valued in the Lie algebra associated to the Lie group S U ( 2 ) . We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there do not exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang–Mills fields within this setting, using the Yang–Mills equations directly. We then adapt the proof constructed in previous work by the first author to show local energy decay and uniform decay of the L ∞ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon.

Twistor theory at fifty: from contour integrals to twistor strings.

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics-anti-self-duality equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.

Instantons on Calabi-Yau and hyper-K\xe4hler cones

The instanton equations on vector bundles over Calabi-Yau and hyper-Kähler cones can be reduced to matrix equations resembling Nahm’s equations. We complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements, by a new set of (singular) boundary conditions which have a known instanton solution in one direction. This approach extends the classic results of Kronheimer by probing a relation between generalised Nahm’s equations and nilpotent pairs/tuples. Moreover, we consider quaternionic instantons on hyper-Kähler cones over generic 3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back to the intersection of three Hermitian Yang-Mills conditions.

Wilson lines in the MHV action

The MHV action is the Yang-Mills action quantized on the light-front, where the two explicit physical gluonic degrees of freedom have been canonically transformed to a new set of fields. This transformation leads to the action with vertices being off-shell continuations of the MHV amplitudes. We show that the solution to the field transformation expressing one of the new fields in terms of the Yang-Mills field is a certain type of the Wilson line. More precisely, it is a straight infinite gauge link with a slope extending to the light-cone minus and the transverse direction. One of the consequences of that fact is that certain MHV vertices reduced partially on-shell are gauge invariant — a fact discovered before using conventional light-front perturbation theory. We also analyze the diagrammatic content of the field transformations leading to the MHV action. We found that the diagrams for the solution to the transformation (given by the Wilson line) and its inverse differ only by light-front energy denominators. Further, we investigate the coordinate space version of the inverse solution to the one given by the Wilson line. We find an explicit expression given by a power series in fields. We also give a geometric interpretation to it by means of a specially defined vector field. Finally, we discuss the fact that the Wilson line solution to the transformation is directly related to the all-like helicity gluon wave function, while the inverse functional is a generating functional for solutions of self-dual Yang-Mills equations.

Manton’s five vortex equations from self-duality

We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang–Mills equations in four dimensions. In particular the Jackiw–Pi vortex and the Ambjørn–Olesen vortex correspond to the gauge group , and respectively the Euclidean or the symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang–Mills theory in 4 + 1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.