Mills Field Research Articles

Inequalities of Simons type and gaps for Yang–Mills fields

In this paper, we establish an inequality of Simons type for Yang–Mills fields, and obtain a gap property, which generalize the results obtained in Bourguignon and Lawson (Comm Math Phys 79(2):189–230, 1981).

Role of division algebra in seven-dimensional gauge theory

The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields

We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\).

Charged black holes in colored Lifshitz spacetimes

We consider Einstein gravities coupled to a cosmological constant and SU(2) Yang–Mills fields in four and five dimensions. We find that the theories admit colored Lifshitz solutions with dynamic exponents z>1. We study the wave equations of the SU(2) scalar triplet in the bulk, and find that the vacuum color modifies the scaling dimensions of the dual operators. We also introduce a Maxwell field and construct exact solutions of electrically-charged black holes that approach the D=4, z=3 and D=5, z=4 colored Lifshitz spacetimes. We derive the thermodynamical first law for general colored and charged Lifshitz black holes.

Covariant extrinsic gravity and the geometric origin of dark energy

In this paper, we construct the covariant or model independent induced Einstein–Yang–Mills field equations on a four-dimensional brane embedded isometrically in an D-dimensional bulk space, assuming the matter fields are confined to the brane. Applying this formalism to cosmology, we derive the generalized Friedmann equations. We derive the density parameter of dark energy in terms of width of the brane, normal curvature radii and the number of extra large dimensions. We show that dark energy could actually be the manifestation of the local extrinsic shape of the brane. It is shown that the predictions of this model are in good agreement with observation if we consider an 11-dimensional bulk space.

Self-dual solutions to pseudo Yang–Mills equations

We study pseudo Yang–Mills fields on a compact 5-dimensional strictly pseudoconvex CR manifold M i.e. critical points to the functional Y Mb(D)=12∫M‖ΠHRD‖2θ∧(dθ)2 on the space C(E,h) of all connections D on a Hermitian vector bundle (E,h) over M, such that Dh=0. If A={D∈C(E,h):ξ⌋RD=0,Gθ∗(Tr(RD),dθ)=0} and D∈A is an absolute minimum to Y Mb:A→R then (i) ΔbTr(RD)=0 and (ii) D is self-dual or anti-self-dual according to the sign of c2(θ,D)=∫Mθ∧{P2(D)−m−12mP1(D)∧P1(D)} [where Pk(D) is the k-th Chern form of (E,D)] and provided c2(θ,D) is constant on A.

$$P$$ P – $$V$$ V criticality of AdS black hole in the Einstein–Maxwell–power-Yang–Mills gravity

We study the $$P$$ – $$V$$ critical behaivor of N-dimensional AdS black holes in Einstein–Maxwell–power-Yang–Mills gravity. Our results show the existence of the Van der Waals like small-large black hole phase transitions when taking some special values of charges of the Maxwell and Yang–Mills fields. Further to calculate the critical exponents of the black holes at the critical point, we find that they are the same as those in the Van der Waals liquid-gas system.

On composite fields approach to Gribov copies elimination in Yang-Mills theories

We suggest a method of introducing the Gribov--Zwanziger horizon functional, $H$, for Yang--Mills theories by using the composite fields technique: $\sigma (\phi )=H$. A different form of the same horizon functional in gauges $\chi $ and $\chi ^{\prime }$ is taken into account via (gauged) field-dependent BRST transformations connecting quantum Yang--Mills actions in these gauges. We introduce generating functionals of Green's functions with composite fields and derive the corresponding Ward identities. A study of gauge dependence shows that the effective action in Yang--Mills theories with the composite field $H$ does not depend on the gauge on the extremals determined by the Yang--Mills fields $\phi $ alone.

A semi-classical analogue of the relation between the chiral and the gluon QCD condensates

We present a semi-classical analogue of the relation between chiral and gluon QCD condensates, in which a condensation of a massless scalar field is provided by a classical field, instead of a quantum Yang–Mills field. For the classical field, we choose a gravitational field of a spherically symmetric object. The size of such an object becomes then a classical-physics analogue of the QCD vacuum correlation length, whereas the squared curvature of the gravitational field plays the role of the gluon condensate. Finally, by iterating the emerging contribution to the trace of the energy–momentum tensor, we prove that the energy density of the object remains of the same order of magnitude, i.e. no instability of the system occurs.

The instability spectrum of weakly-magnetized SU(2) Reissner–Nordström black holes

It is well known that the U(1) Reissner–Nordström black hole is stable within the framework of the Einstein–Maxwell theory. However, the SU(2) Reissner–Nordström black-hole solution of the coupled Einstein–Yang–Mills equations is known to be unstable. In fact, this magnetically charged black hole is characterized by an infinite set of unstable (growing in time) perturbation modes. In the present paper we study analytically the instability resonance spectrum of weakly-magnetized SU(2) Reissner–Nordström black holes. In particular, we obtain explicit analytical expressions for the infinite set {ωn}n=0n=∞ of imaginary eigenvalues that characterize the instability growth rates of the perturbation modes. We discuss the role played by these unstable eigenvalues as critical exponents in the gravitational collapse of the Yang–Mills field. Finally, it is shown that our analytical formulas for the characteristic black-hole instability spectrum agree with new numerical data that recently appeared in the literature.

A Chern–Simons–Yang–Mills–Higgs system in 3+1 dimensions

We study spherically symmetric solutions of an SO(5) Chern–Simons–Yang–Mills–Higgs system in 3+1 dimensions. The Chern–Simons densities are defined in terms of both Yang–Mills fields and a five-component isomultiplet Higgs. The SO(3) × SO(2) solutions are analyzed in a systematic way, by employing numerical methods. These finite energy configurations possess both electric and magnetic global charges, differing radically, however, from Julia–Zee dyons. When two or more of these Chern–Simons densities are present in the Lagrangian, solutions with vanishing electric charge but nonvanishing electrostatic potential may exist.

Volume independence for Yang–Mills fields on the twisted torus

We review some recent results related to the notion of volume independence in SU (N) Yang–Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l), given by lN2/d, for d even. We discuss the well-known relation to noncommutative field theories and address certain threats to volume independence associated to the occurrence of tachyonic instabilities at one-loop order. We end by presenting some numerical results in 2+1 dimensions that extend these ideas to the nonperturbative domain.

The nonexistence theorems for F-harmonic maps and F-Yang–Mills fields

Let M be an n(n≥3)-dimensional complete Riemannian manifold with radial curvature K:−a2≤K≤−b2<0 with a≥b>0. In this paper, we consider the F-harmonic maps from M and F-Yang–Mills fields on M. By the monotonicity formulae, we can prove that (1) If (n−2)b≥(2dF−1)a, then every F-harmonic map from M to any Riemannian manifold with finite F-energy is constant, which improves the Dong and Wei's result in [3]; (2) If (n−2)b≥(4dF−1)a, then every F-Yang–Mills fields R∇ with finite F-Yang–Mills energy vanishes on M, which improves the Dong and Wei's result in [3].

Physical theories in hypercomplex geometric description

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

Abelian-Higgs model from Cho–Faddeev–Niemi decomposition

Applying Cho–Faddeev–Niemi decomposition for SU(2) Yang–Mills field, we obtain the Abelian–Higgs Lagrangian by some approximation. Abelian–Higgs Lagrangian with a spontaneous symmetry breaking potential has vortex solutions known as Nielsen–Olesen solutions. We conclude that vortices as well as magnetic monopoles can exist in Cho–Faddeev–Niemi decomposition of SU(2) Yang–Mills field.

Initial state, parton saturation and thermalization

Abstract The early time behavior of heavy ion collisions involves new forms of matter such as the Color Glass Condensate and the Glasma. An area of much recent work involves attempts to understand the development of the Glasma into a thermalized Quark Gluon Plasma. The evolution of the Glasma involves turbulent Yang–Mills fields, and might also have condensates and non-trivial topological excitations. The proton–nucleus collision experiments at the LHC accelerator may provide unique insight into these forms of matter and the role they play at various times in hadronic collisions. I introduce and review aspects of these forms of matter as related to experiments at the LHC and RHIC accelerators.

On a path integral representation of the Nekrasov instanton partition function and its Nekrasov–Shatashvili limit

In this work we study the Nekrasov–Shatashvili limit of the Nekrasov instanton partition function of Yang–Mills field theories with 𝒩 = 2 supersymmetry and gauge group SU(N). The theories are coupled with fundamental matter. A path integral expression of the full instanton partition function is derived. It is checked that in the Nekrasov–Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.

Spinfoam fermions

We give the definition of a minimal coupling of fermions and Yang–Mills fields to the loop quantum gravity covariant dynamics. The coupling takes a surprisingly simple form. Here we only define the dynamics; physical implications are considered in a subsequent paper.

GRAVIWEAK UNIFICATION, INVISIBLE UNIVERSE AND DARK ENERGY

We consider a graviweak unification model with the assumption of the existence of a hidden (invisible) sector of our Universe, parallel to the visible world. This Hidden World (HW) is assumed to be a Mirror World (MW) with broken mirror parity. We start with a diffeomorphism invariant theory of a gauge field valued in a Lie algebra [Formula: see text], which is broken spontaneously to the direct sum of the space–time Lorentz algebra and the Yang–Mills algebra: [Formula: see text] — in the ordinary world, and [Formula: see text] — in the hidden world. Using an extension of the Plebanski action for general relativity, we recover the actions for gravity, SU(2) Yang–Mills and Higgs fields in both (visible and invisible) sectors of the Universe, and also the total action. After symmetry breaking, all physical constants, including the Newton's constants, cosmological constants, Yang–Mills couplings, and other parameters, are determined by a single parameter g present in the initial action, and by the Higgs VEVs. The dark energy problem of this model predicts a too large supersymmetric breaking scale (M SUSY ~1010 GeV ), which is not within the reach of the LHC experiments.

RELATIVISTIC DIFFUSION OF QUARKS IN RANDOM GLUON FIELDS

We consider Wong's equations for a particle with a continuous mass spectrum in a random Yang–Mills field approximating the quantum field at finite temperature. We show that particle time evolution can be approximated by a relativistic diffusion. Kubo's generator of the relativistic diffusion is defined as an expectation value of the square of the Liouville operator.