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We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang--Mills wave equation over the extremal Reissner--Nordstr\"{o}m spacetime found in a recent work of Bizo\'{n} and Kahl, is the unique solution to an associated domain-wall type energy minimization problem. We also derive the accompanying interior Yang--Mills wave equation within the same formalism. We obtain a few more closed-form solutions, regular and singular, among other solutions of oscillatory behavior, and discuss several interesting features of the solutions based on some energy consideration., Comment: 16 pages, 2 figure

A known feature of electrically charged Reissner-Nordstr\"om-anti-de Sitter planar black holes is that they can become unstable when considered as solutions of Einstein--Yang-Mills theory. The mechanism for this is that the linearized Yang-Mills equations in the background of the Reissner-Nordstr\"om (RN) black holes possess a normalizable zero mode, resulting in non-Abelian (nA) magnetic clouds near the horizon. In this work we show that the same pattern may occur also for asymptotically flat RN black holes. Different from the anti-de Sitter case, in the Minkowskian background the prerequisite for the existence of the nA clouds is $i)$ a large enough gauge group and $ii)$ the presence of some extra interaction terms in the matter Lagrangian. To illustrate this mechanism we present two specific examples, one in four and the other in five dimensional asymptotically flat spacetime. In the first case, we augment the usual $SU(3)$ Yang-Mills Lagrangian with higher order (quartic) curvature term, while for the second one we add the Chern-Simons density to the $SO(6)$ Yang-Mills system. In both cases, an Abelian gauge symmetry is spontaneously broken near a RN black hole horizon with the appearance of a condensate of nA gauge fields. In addition to these two examples, we review the corresponding picture for anti-de Sitter black holes. All these solutions are studied both analytically and numerically, existence proofs being provided for nA clouds in the background of RN black holes. The proofs use shooting techniques which are suggested by and in turn offer insights for our numerical methods. They indicate that, for a black hole of given mass, appropriate electric charge values are required to ensure the existence of solutions interpolating desired boundary behavior at the horizons and spatial infinity., Comment: 19 pages, 5 figures

In this paper, we prove the existence of finite-energy electrically and magnetically charged vortex solutions in the full Chern–Simons–Higgs theory, for which both the Maxwell term and the Chern–Simons term are present in the Lagrangian density. We consider both Abelian and non-Abelian cases. The solutions are smooth and satisfy natural boundary conditions. Existence is established via a constrained minimization procedure applied on indefinite action functionals. This work settles a long-standing open problem concerning the existence of dually charged vortices in the classical gauge field Higgs model minimally extended to contain a Chern–Simons term.

Inspired by magnetic impurity considerations some broad classes of Abelian Higgs and Chern--Simons--Higgs BPS vortex equations are derived and analyzed., 19 pages

An existence theorem is established for the solutions to the non-Abelian relativistic Chern--Simons--Higgs vortex equations over a doubly periodic domain when the gauge group $G$ assumes the most general and important prototype form, $G=SU(N)$

Recent work of Harland shows that the $SO(3)$-symmetric, dimensionally-reduced, charge-$N$ self-dual Yang--Mills calorons on the hyperbolic space $ H^3\times S^1$ may be obtained through constructing $N$-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper we establish the existence of such minimal action charge-$N$ calorons by constructing arbitrarily prescribed $N$-vortex solutions of the Witten type equations., Comment: 19 pages, accepted for publication, Proceedings of Royal Society (London), Series A, 2015

We prove the existence and uniqueness of a static and radially symmetric BPS monopole of unit topological charge in an arbitrary (4p – 1)-dimensional space descended from the generalized Yang–Mills theory in 4p dimensions and formulated and presented in a recent study of Radu and Tchrakian. This monopole solution also gives rise to an electrically and magnetically charged soliton, called dyon, in the same spacetime setting through the well-known Julia–Zee correspondence. Our method is based on a dynamical shooting approach depending on two shooting parameters which provides an effective framework for constructing the BPS monopole and dyon solutions in general dimensions.

Existence and uniqueness of the solution are proved for the `master equation' derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with Nf charged matter hypermultiplets with eight supercharges. This proof finally establishes the fact that the moduli space of the BPS domain wall solution is CP^(N_f-1) for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(Nc) gauge theory with Nf hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(Nc) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences., Comment: 18 pages, a reference added, two typos corrected(v2), several typos corrected, explanation of total moduli space added(v3)

Two distinct Skyrme terms for the CP1 model are defined, only one of which preserves the equivalence with the corresponding O(3) model. A gauging prescription in which the composite connection of the CP1 model is replaced by the U(1) gauge connection is defined. A U(1) gauged CP1 model and two distinct U(1) gauged CP1 models with Skyrme terms are defined. The existence of topologically stable vortices in each of the three U(1) gauged models is proved analytically.

We prove the existence and uniqueness of a spherically symmetric monopole solution in the Abelian model of Lee and Weinberg in the BPS limit. The solution carries finite energy and unit charge and generalizes the classical SU (2) BPS monopole. We also prove an existence theorem for the model with a general Higgs field-dependent mass term and establish the equivalence of the second-order equations of motion and the first-order Bogomol'nyi equations within radial symmetry assumption and BPS limit. Our methods have wide applicability in other monopole problems.

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincare half plane.

We prove the existence of multisolitons in the gauged self-dual Maxwell and Chern–Simons sigma models with broken symmetry. In the context of topological solutions, the existence proofs rely on established results on earlier models. In the context of nontopological solutions with radial symmetry, the proof is by adapting a shooting argument.

We establish the existence and uniqueness of radially symmetric self-dual topological vortices in thep=2 members of the hierarchies of (generalised) Chern-Simons Higgs and Abelian Higgs models. We also obtain all possible symmetric nontopological vortices in the Chern-Simons model characterised by an additional parameter governing the decay rates of the fields.

This is a survey article concerning some recent mathematical results for the static selfdual cosmic-string solutions in the Abelian Higgs model and in the Weinberg-Salam standard model unifying electromagnetic and weak interactions, both coupled with gravity through the Einstein equations. For the Abelian Higgs strings there is a nearly complete picture. If the Riemann surface M on which the strings reside is compact, it can be shown that M must be S2 up to topological equivalence and there are only countably many values of the Higgs vacuum states for strings to exist. When M is noncompact and conformally a plane there are exact obstructions to the finiteness of energies and geodesic completeness of solutions. For the Weinberg-Salam strings, much is to be achieved. It can be shown in this case that self-dual strings generated from W and Higgs condensation lead to an explicit formula for a positive cosmological constant and the gravitational metric is always noncomplete. This feature leads to the properties that the metric decays sufficiently rapidly at infinity and there exist non-Abelian electroweak strings of finite energies. It is established that for any integer N there are always suitable ranges of the electroweak parameters to allow the existence of W- and Higgs-condensed N-vortex solutions of finite energies.

We construct multi-string solutions of the coupled Einstein and Abelian Higgs equations so that the spacetime is uniform along the time axis and a vertical direction and nontrivial geometry is coded on a Riemann surfaceM. We concentrate on the critical Bogomol'nyi phase. WhenM is compact, the Abelian Higgs model is defined by a complex line bundleL overM. We prove that, due to the coupling of the Einstein equations, the Euler characteristic ofM and the first Chern number of the line bundleL identified as the total string number impose an exact obstruction to the existence of a string solution. Such an obstruction leads to some interesting implications. We then study the existence of multi-string solutions which can realize a prescribed string distribution. We show that there are such solutions when the local string winding numbers do not exceed half of the total string number. WhenM is noncompact and globally conformal to a plane, we show that the energy scale of symmetry breaking plays a crucial role and there are finite-energy radially symmetric string solutions realizing a given string number if and only if the symmetry breaking scale is sufficiently small but nonvanishing. Finally, we obtain finite-energy multistring solutions with an arbitrary string distribution and associated local winding numbers. These solutions are not radially symmetric and are regular everywhere and topologically nontrivial so that both the energy of the matter-gauge sector and the energy of the gravitational sector viewed as the total Gauss curvature ofM are quantized.

It is shown that there is a critical value of the Chern-Simons coupling parameter so that, below the value, there exists self-dual doubly periodic vortex solutions, and, above the value, the vortices are absent. Solutions of such a nature indicate the existence of dyon condensates carrying quantized electric and magnetic charges.

A series of sharp existence and uniqueness theorems are established for the multiple vortex solutions in the supersymmetric Chern--Simons--Higgs theory formalism of Aharony, Bergman, Jaferis, and Maldacena, for which the Higgs bosons and Dirac fermions lie in the bifundamental representation of the general gauge symmetry group $U(N)\times U(N)$. The governing equations are of the BPS type and derived by Kim, Kim, Kwon, and Nakajima in the mass-deformed framework labeled by a continuous parameter.

We explicitly construct all kink solitons arising in the recent study of Auzzi, Bolognesi, and Shifman of a monopole confinement problem in $\mathcal{N}=2$ supersymmetric QCD. In particular, we show that all finite-energy kink solitons must be Bogomol'nyi-Prasad-Sommerfield.

There are obstructions in the Bogomol'nyi critical phase to the existence of a cylindrically symmetric cosmic string solution of the Einstein equations coupled with the Abelian Higgs model. In the compact case, the obstruction is exact, which forbids the presence of gravitational condensates and leads to the quantization of the energy scale of symmetry breaking. In the noncompact case, the obstruction is an inequality which says that, at a given level of broken symmetry, the topological charge or winding number of a string cannot be too large in order to maintain finite energy.

String-like static solutions of the Einstein matter-gauge equations have interesting implications in cosmology. It has been shown recently that, at a critical coupling phase, this system of equations allows a reduction into a coupled Einstein-Bogomol'nyi system. In this Letter, we prove that, in the important case where the underlying two-dimensional Riemannian manifold is either compact or asymptotically Euclidean, the two systems are actually equivalent. Moreover, we show that the standard assumption that the strings reside in a conformally Euclidean surface will give us a metric which fails to be asymptotically Euclidean. In particular, in the radially symmetric case, we establish under the finite energy condition the boundary behavior of the metric. These results may indicate that a string solution will inevitably lead to nonflatness of the space at infinity even on the cross-section.

A necessary and sufficient condition is obtained for the existence of multivortex solutions of the Bogomol’nyi system arising in the abelian Higgs theory defined on a rectangular domain and subject to a ’t Hooft type periodic boundary condition. In particular, the number of vortices of a solution is confined by the size of the domain. Such solutions realize the magnetic periodic cell structure in a superconductor predicted by Abrikosov. If the periodic boundary condition is removed, the Bogomol’nyi equations on a bounded domain possess solutions with an arbitrary number of vortices, and these solutions may be used to approximate the unique finite energy solution over the full plane. Moreover, it is shown that, for any given vortex distribution in the plane, the Bogomol’nyi system has a continuous family of gauge-distinct solutions with infinite energy.

In this paper we consider the bosonic sector of the electroweak theory. It has been shown in the work of Ambjorn and Olesen that when the Higgs mass equals to the mass of theZ boson, the model in two dimensions subject to the 't Hooft periodic boundary condition may be reduced to a Bogomol'nyi system and that the solutions of the system are vortices in a “dual superconductor”. We shall prove using a constrained variational reformulation of the problem the existence of such vortices. Our conditions for the existence of solutions are necessary and sufficient when the vortex numberN=1,2.

SynopsisRecent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.

In the recently discovered Chern-Simons model, the reduction to a Bogomol'nyi bound or self-duality depends crucially on the specific form of the Higgs potential energy function, which is characterized by a φ6 type self-interaction. The purpose of this paper is to show that a much wider class of Higgs self-interaction may be allowed to achieve self-duality provided that the kinetic energy term of the Higgs scalar is suitably modified. The existence of topological multivortex solutions is also established. Furthermore, it is remarked that the Meissner effect may occur in the model.

In the massive SO(3) gauge field theory, a critical choice of the configuration ansatz leads to a system of the Bogomol’nyi type equations governing the unstable modes of the model. A ’t Hooft periodic condition implies the quantization of the flux lines confined in a cell domain if a certain class of solutions exist. The present paper establishes an existence theorem for such solutions via a constrained variational principle. These solutions exhibit the analogous properties of the vortex‐like solutions predicted in the theory of type II superconductivity by Abrikosov.

The main result of this note is a proof of a conjecture of Burzlaff, O'Se and Tchrakian [1] concerning the existence of a certain kind of finite action solutions of the seven-dimensional generalized Yang-Mills-Higgs model derived from an eight-dimensional SO(8) Yang-Mills theory.

We present several different classes of selfdual Yang-Mills instantons in all even d backgrounds with Euclidean signature. In d=4p+2 the only solutions we found are on constant curvature dS and AdS backgrounds, and are evaluated in closed form. In d=4p an interesting class of instantons are given on black hole backgrounds. One class of solutions are (Euclidean) time-independent and spherically symmetric in d-1 dimensions, and the other class are spherically symmetric in all d dimensions. Some of the solutions in the former class are evaluated numerically, all the rest being given in closed form. Analytic proofs of existence covering all numerically evaluated solutions are given. All instantons studied have finite action and vanishing energy momentum tensor and do not disturb the geometry., 41 pages, 3 figures

In this chapter we present a study of an important planar Abelian gauge field theory arising in condensed matter physics in which electromagnetism is governed by a Chern—Simons dynamics. In §5.1 we consider the gauged Schrodinger equation, which is nonrelativistic, and we present its explicit solution. In §5.2 we introduce the relativistic Chern—Simons model and state our main results concerning its topological solutions. In §5.3 we make a systematic analysis of the problem and prove the existence theorem stated in §5.2. In §5.4 we obtain all possible symmetric non-topological solutions. In §5.5 and §5.6 we construct spatially periodic solutions modelling a condensed lattice structure.

The Hodge theorem states that, on a compact oriented manifold, each de Rham cohomology class can be represented by a harmonic form. Such a result has an important parallel in the Yang—Mills theory: each second Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. The purpose of this chapter is to establish the general theorem that, for each m = 1, 2,..., a similarly defined 2m-th cohomology class on S 4m generalizing the Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. In §3.1 we review some basic facts in 4 dimensions. In §3.2 we solve the Liouville equation. In §3.3 we present the explicit solutions of Witten in 4 dimensions based in the solution of the Liouville equation. In §3.4 we discuss the problem in all 4m dimensions and state our general representation theorem. In §3.5–§3.7 we prove the theorem.

In this chapter we present static multisoliton solutions of the non-Abelian Chern—Simons equations. In §6.1 we review some basic facts about a complex semi-simple Lie algebra such as the Cartan—Weyl bases and Cartan matrices to be used in the development to follow. In §6.2 we consider the solution of the non-Abelian gauged Schrodinger equations coupled with a Chern—Simons dynamics via the Toda equations, which is a non-relativistic Chern—Simons theory. In §6.3 we introduce the relativistic Chern—Simons equations and state a general existence theorem. In §6.4 we reduce the governing equations into a nonlinear elliptic system and formulate a variational principle. In §6.5 we present an analysis of the elliptic system and prove all the results stated. In §6.6 we apply our existence theory to some concrete examples.

In this chapter we present a series of results concerning the existence of cosmic strings. Such solutions arise from the Einstein equations coupled with a suitable matter and gauge field system and lead to local concentration of energy and gravitational curvature, a structure relevant in the theory of galaxy formation in the early-time cosmology [321, 322]. In §10.1 we discuss some basic notions in the study of cosmic strings. In particular, we present the elegant solution of Comtet and Gibbons [86] of the harmonic map string equations. In §10.2–§10.6 we present a study of the existence of multiple strings arising from the self-dual Abelian Higgs theory. In §10.7 we consider non-Abelian strings arising from the Weinberg—Salam electroweak theory.

It is remarked that, if the gauge group is not compact, finite−time blowing−up Yang–Mills fields over the Minkowski spaces may exist. The example taken is G=SU(2,C). It is shown that blow−up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen.

"String-like static solutions of the Einstein-matter-gauge equations have interesting implications in cosmology. It has been shown recently that, at a critical coupling phase, this system of equations allows a reduction into a coupled Einstein-Bogomol'nyi system. In this paper, we prove that, in the important case where the underlying two-dimensional Riemannian manifold is either compact or asymptotically Euclidean, the two systems are actually equivalent. Moreover, we show that the standard assumption that the strings reside in a conformally Euclidean surface will give us a metric which fails to be asymptotically Euclidean. In particular, in the radially symmetric case, we establish under the finite energy condition the boundary behavior of the metric. These results may indicate that a string solution will inevitably lead to nonflatness of the space at infinity even on the cross section."

"This paper considers the Einstein equations coupled with the nonabelian gauge and Higgs fields. It is shown that, when cosmic string solutions are sought in the Einstein-Georgi-Glashow system and the Einstein-Weinberg-Salam system governing the gravitational-electromagnetic weak interaction forces, the self duality conditions lead to positive values of the cosmological constant which can be expressed by some fundamental parameters in particle physics."

An important question concerning the classical solutions of the equations of motion arising in quantum field theories at the BPS critical coupling is whether all finite-energy solutions are necessarily BPS. In this paper we present a study of this basic question in the context of the domain wall equations whose potential is induced from a superpotential so that the ground states are the critical points of the superpotential. We prove that the definiteness of the Hessian of the superpotential suffices to ensure that all finite-energy domain-wall solutions are BPS. We give several examples to show that such a BPS property may fail such that non-BPS solutions exist when the Hessian of the superpotential is indefinite., 25 pages