Yang-Mills Equations Research Articles

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.

Evolution of eccentricities induced by geometrical and quantum fluctuations in proton-nucleus collisions

We compute the azimuthal eccentricities arising from initial stage fluctuations in high energy proton-nucleus collisions at proper times $\tau \geq 0^+$. We consider two sources of fluctuations, namely the geometrical structure of the proton and the fluctuation of color fields carried by both proton and nucleus. Describing these effects with Gaussian models allows us to analytically calculate the one- and two-point correlators of energy density, from which the eccentricities are obtained. We compute the proper time evolution of these quantities by approximating the Glasma dynamics in terms of linearized Yang-Mills equations, which we solve by assuming free field propagation and adopting the dilute-dense limit.

Spin polarization and correlation of quarks from the glasma

We investigate the interaction of strong color fields in the glasma stage of high-energy nuclear collisions with the spins of quarks and antiquarks. We employ the perturbative solution of the quantum kinetic theory for the spin transport of (massive) quarks in a background color field governed by the linearized Yang-Mills equation and derive expressions for the quark-spin polarization and quark-antiquark spin correlation at small momentum in terms of field correlators. For the Golec-Biernat W\"usthoff dipole distribution the quark-spin polarization vanishes, but the out-of-plane spin correlation of quarks and antiquarks is nonzero. Our order-of-magnitude estimate of the correlation far exceeds that caused by vorticity effects, but does not fully explain the data for vector meson alignment. We identify possible mechanisms that could further increase the predicted spin correlation.

The supercritical deformed Hermitian Yang–Mills equation on compact projective manifolds

In this paper, we extend a result of Chen regarding the solvability of the twisted deformed Hermitian Yang–Mills (dHYM) equations on compact Kähler manifolds to allow for the twisting function to be nonconstant and slightly negative in all dimensions. Using this result along with the methods of Datar and Pingali, we prove that the twisted dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied. As a corollary, we obtain a new proof in the projective case of a recent theorem of Chu, Lee, and Takahashi addressing a conjecture of Collins, Jacob, and Yau.

Solitons in openN= 2 string theory

AbstractThe open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW4) model. The equation of motion of the WZW4 model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW4 model in the split signature by calculating the action density of the WZW4 model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n-soliton solutions is a “non-linear superposition” of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory

We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0)-form Ω. If Ω is nowhere vanishing, it descends to a theory on 4d space-time with classical equations of motion equivalent to the anti-self-dual Yang-Mills equations. Examples include a 4d analogue of the Wess-Zumino-Witten model and a theory of a Lie algebra valued scalar with a cubic two derivative interaction. Under symmetry reduction, these yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces holomorphic Chern-Simons theory to the 4d Chern-Simons theory with disorder defects studied by Costello & Yamazaki. Finally we show that a similar reduction by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations.

Cosmological implications of photon-flux upper limits at ultrahigh energies in scenarios of Planckian-interacting massive particles for dark matter

Using the data of the Pierre Auger Observatory, we report on a search for signatures that would be suggestive of super-heavy particles decaying in the Galactic halo. From the lack of signal, we present upper limits for different energy thresholds above ≳108 GeV on the secondary by-product fluxes expected from the decay of the particles. Assuming that the energy density of these super-heavy particles matches that of dark matter observed today, we translate the upper bounds on the particle fluxes into tight constraints on the couplings governing the decay process as a function of the particle mass. Instantons, which are nonperturbative solutions to Yang-Mills equations, can give rise to decay channels otherwise forbidden and transform stable particles into metastable ones. Assuming such instanton-induced decay processes, we derive a bound on the reduced coupling constant of gauge interactions in the dark sector: αX≲0.09, for 109≲MX/GeV<1019. Conversely, we obtain that, for instance, a reduced coupling constant αX=0.09 excludes masses MX≳3×1013 GeV. In the context of dark matter production from gravitational interactions alone during the reheating epoch, we derive constraints on the parameter space that involves, in addition to MX and αX, the Hubble rate at the end of inflation, the reheating efficiency, and the nonminimal coupling of the Higgs with curvature.1 MoreReceived 9 August 2022Accepted 14 December 2022DOI:https://doi.org/10.1103/PhysRevD.107.042002© 2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCosmic rays & astroparticlesCosmologyDark matterParticle dark matterGravitation, Cosmology & Astrophysics

Limits to Gauge Coupling in the Dark Sector Set by the Nonobservation of Instanton-Induced Decay of Super-Heavy Dark Matter in the Pierre Auger Observatory Data.

Instantons, which are nonperturbative solutions to Yang-Mills equations, provide a signal for the occurrence of quantum tunneling between distinct classes of vacua. They can give rise to decays of particles otherwise forbidden. Using data collected at the Pierre Auger Observatory, we search for signatures of such instanton-induced processes that would be suggestive of super-heavy particles decaying in the Galactic halo. These particles could have been produced during the post-inflationary epoch and match the relic abundance of dark matter inferred today. The nonobservation of the signatures searched for allows us to derive a bound on the reduced coupling constant of gauge interactions in the dark sector: α_{X}≲0.09, for 10^{9}≲M_{X}/GeV<10^{19}. Conversely, we obtain that, for instance, a reduced coupling constant α_{X}=0.09 excludes masses M_{X}≳3×10^{13} GeV. In the context of dark matter production from gravitational interactions alone, we illustrate how these bounds are complementary to those obtained on the Hubble rate at the end of inflation from the nonobservation of tensor modes in the cosmological microwave background.

Stability and energy identity for Yang–Mills–Higgs pairs

In this paper, we study the properties of the critical points of Yang–Mills–Higgs functional, which are called Yang–Mills–Higgs pairs. We first consider the properties of weakly stable Yang–Mills–Higgs pairs on a vector bundle over Sn (n ≥ 4). When n ≥ 4, we prove that the norm of its Higgs field is 1 and the connection is actually Yang–Mills. More precisely, its curvature vanishes when n ≥ 5. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang–Mills–Higgs pairs over a four-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence that converges smoothly to a Yang–Mills–Higgs pair up to gauge modulo finitely many four-dimensional spheres with Yang–Mills connections.

A polyhedral discrete de Rham numerical scheme for the Yang–Mills equations

We present a discretisation of the 3+1 formulation of the Yang–Mills equations in the temporal gauge, using a Lie algebra-valued extension of the discrete de Rham (DDR) sequence, that preserves the non-linear constraint exactly. In contrast to Maxwell's equations, where the preservation of the analogous constraint only depends on reproducing some complex properties of the continuous de Rham sequence, the preservation of the non-linear constraint relies for the Yang–Mills equations on a constrained formulation, previously proposed in [10]. The fully discrete nature of the DDR method requires to devise appropriate constructions of the non-linear terms, adapted to the discrete spaces and to the need for replicating the crucial Ad-invariance property of the L2-product. We then prove some energy estimates, and provide results of 3D numerical simulations based on this scheme.

A Note on Coupled Dirac Operators

The article considers some concrete solutions to the Dirac equation coupled to a vector bundle with connection, arising in the study of Yang-Mills equations and vector bundles on Riemann surfaces.

A conformally invariant Yang–Mills type energy and equation on 6-manifolds

We define a conformally invariant action [Formula: see text] on gauge connections on a closed pseudo-Riemannian manifold [Formula: see text] of dimension 6. At leading order this is quadratic in the gauge connection. The Euler–Lagrange equations of [Formula: see text], with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang–Mills equations. For any gauge connection [Formula: see text] on [Formula: see text], we define [Formula: see text] by first defining a Lagrangian density associated to [Formula: see text]. This is not conformally invariant but has a conformal transformation analogous to a [Formula: see text]-curvature. Integrating this density provides the conformally invariant action. In the special case that we apply [Formula: see text] to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman–Graham obstruction tensor. So in this case, the Euler–Lagrange equations are exactly the “obstruction-flat” condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang–Mills equations of the Cartan-tractor connection.

Stability conditions for polarised varieties

Abstract We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.

A note on singular Hermitian Yang–Mills connections

A note on singular Hermitian Yang–Mills connections

Yang–Mills solutions on Minkowski space via non-compact coset spaces

We find a two-parameter family of solutions of the Yang–Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space H3≅SO(1,3)/SO(3), while the exterior of the lightcone employs de Sitter space dS≅3SO(1,3)/SO(1,2). The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang–Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang–Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.

Asymptotics of Finite Energy Monopoles on AC 3-Manifolds

We study the asymptotic behavior of finite energy \(\mathrm SU(2)\) monopoles, and general critical points of the \(\mathrm SU(2)\) Yang–Mills–Higgs energy, on asymptotically conical 3-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat 3-dimensional Euclidean space \({\mathbb {R}}^3\). Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of the Higgs field of a critical point of the energy decays quadratically along the end and that its transverse component with respect to the Higgs field, as well as the corresponding component of the curvature of the underlying connection, actually decay exponentially. Additionally, under the assumption of positive Gaussian curvature on the asymptotic link, we prove that the curvature of any critical point connection decays quadratically. Furthermore, we deduce that any irreducible critical point converges uniformly along the conical end to a limiting configuration at infinity consisting of a reducible Yang–Mills connection and a parallel Higgs field.

Hessian equations of Krylov type on compact Hermitian manifolds

Abstract In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Dynamical mechanisms for Kaluza–Klein theories

We present variational formulations of gauge theories and Einstein–Yang–Mills equations in the spirit of Kaluza–Klein theories. For gauge theories, only a topological fibration is assumed. For gravitation coupled with gauge fields, no fibration is assumed: Fields are defined on a ‘space-time’ \({\mathcal {Y}}\) of dimension \(4 + r\) without any structure a priori, where r is the dimension of the structure group. If the latter is compact and simply connected, classical solutions allow to construct a manifold \({\mathcal {X}}\) of dimension 4 to be the physical space-time, in such a way that \({\mathcal {Y}}\) acquires the structure of a principal bundle over \({\mathcal {X}}\) and leads to solutions of the Einstein–Yang–Mills systems. The special case of the Einstein–Maxwell system is also discussed: It suffices that at least one fiber closes in on a circle to deduce that the five-dimensional space-time has a fiber bundle structure.

Symplectic Flatness and Twisted Primitive Cohomology

We introduce the notion of symplectic flatness for connections and fibre bundles over symplectic manifolds. Given an A_infty -algebra, we present a flatness condition that enables the twisting of the differential complex associated with the A_infty -algebra. The symplectic flatness condition arises from twisting the A_infty -algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang–Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

The deformed Hermitian Yang–Mills equation on three-folds

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Amp\`ere equation with mixed sign coefficients.