Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

SU(2) monopoles of charge 2

Using the methods of Hitchin, the moduli space of SU(2) monopoles of charge two is computed.     

Cylindrically and spherically symmetric monopoles in SU(3) gauge theory

We apply to the Atiyah-Ward ansätze a systematic procedure locating symmetric monopoles in SU(3) gauge theory broken to U(1) × U(1). In particular we recover the known spherically symmetric monopole as a limit of a cylindrically symmetric separated two monopole solution in SU(3). We also discuss the spherically symmetric monopole in SU(n). This latter is the only instance where we have properly shown the smoothness of the Higgs and gauge fields.     

Magnetic monopoles in SU(3) and SU(2) gauge theories without Higgs' scalars

We show that certain known singular solutions for pure SU(3) Yang-Mills theory carry magnetic charges with respect to both U(1) subgroups of SU(3). A topological characterisation in terms of monopoles is given to the SU(2) singular solutions of Wu and Yang.     

Magnetic monopoles in SU(N) gauge theories

The potential $\text{A}\text{(}\text{r}\text{) ≡}\text{M}\text{(}\text{r}\text{?}\text{×}\text{n}\text{?}\text{)(r?}\text{r}\text{·}\text{n}\text{?}\text{)}{}^{\mathrm{?1}}$ is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix $\text{M}$ in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of $\text{M}$ in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are $\text{±}\text{1}\text{2}$. In this paper, for the gauge groups G = SU(N), it is shown which sets of eigenvalues of $\text{M}$ are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case $\text{M}\text{=}\text{0}$. Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center $\text{Z}{}_{\mathrm{N}}\text{?Π}{}_1\text{(}\text{SU}\text{(N)/}\text{Z}{}_{\mathrm{N}}\text{)}$ of SU(N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude $\text{tr}\text{M}{}^2$. This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hooft's calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.     

SU(5) monopoles,magnetic symmetry and confinement

The monopoles of the unified SU(5) gauge theory broken down to H_E = SU(3)_c ? U(1)_EM [or to K_E = SU(3)_c ? SU(2) ? U(1)_Y], are classified. They belong to representations of a magnetic group H_M(K_M), which is found to be isomorphic to H_E(K_E). For SU(5) broken down to H_E, there exists a regular and stable monopole which is a colour magnetic triplet, and carries a non-zero abelian magnetic charge. It is suggested that composite operators made out of this monopole and its antiparticle fields develop a non-zero vacuum expectation value, and so lead to a squeezing of the colour electric flux. Finally, we comment on the cosmological production of SU(5) monopoles.     

A numerical study of SU(3) charge-two monopoles with minimal symmetry breaking

Charge-two SU(3) monopoles with minimal symmetry breaking can be generated via Nahm's equations. This paper investigates the detailed structure of such monopoles, through explicit calculation of the energy density, and the norm and discriminant of the Higgs field. Monopoles may be classified according to the maxima and minima of these quantities.     

Analytic approach to confinement and monopoles in lattice SU (2)

We extend the approach of Banks, Myerson, and Kogut for the calculation of the Wilson loop in lattice U (1) to the non-abelian SU (2) group. The original degrees of freedom of the theory are integrated out, new degrees of freedom are introduced in several steps. The centre group enters automatically through the appearance of a field strength tensor , which takes on the values 0 or 1 only. It obeys a linear field equation with the loop current as source. This equation implies that is non vanishing on a two-dimensional surface bounded by the loop, and possibly on closed surfaces. The two-dimensional surfaces have a natural interpretation as strings moving in euclidean time. In four dimensions we recover the dual Abrikosov string of a type II superconductor, i.e. an electric string encircled by a magnetic current. In contrast to other types of monopoles found in the literature, the monopoles and the associated magnetic currents are present in every configuration. With some plausible, though not generally conclusive, arguments we are directly led to the area law for large loops. Received: 27 August 1999 / Revised version: 29 October 1999 / Published online: 21 December 1999     

Anomalous baryon number non-conservation in the presence of SU(5) monopoles

It is shown that the effects related to the Adler-Bell-Jackiw anomaly give rise to strong baryon number non-conservation in the presence of magnetic monopoles in the SU(5) grand unified theory. The main assumption made is that, as long as distances of order 1 GeV^?1 from the monopole center are considered, the masses of light fermions u, d, s, e, μ can be neglected, while heavy fermions decouple. Two types of the SU(5) monopoles are studied, namely: (i) the fundamental monopole with a non-vanishing chromomagnetic charge, (ii) the minimal purely electromagnetic monopole. In both cases the processes like

$\text{p}\text{+}\text{monopole}\text{→}\text{e}{}^{\mathrm{+}}\text{+}\text{momopole}\text{+}\text{everything}$

are allowed. It is argued that the cross sections of these processes are of order 1 GeV^?2β^?2, at least for sufficiently low monopole-proton relative velocities β.     

Thermal monopoles in the SU(2) gauge theory on a lattice

Color-magnetic thermal monopoles in SU(2) lattice gluodynamics with improved Simanzik action were studied. The density of the monopoles, the monopole chemical potential, the cluster susceptibility, and the cluster magnetization were studied. These results were compared with results that were reported elsewhere.     

SU (4) × SU (4) symmetry breaking and its implications for SU (3) × SU (3) breaking

We investigate models where the SU (4) × SU (4) symmetry breaking Hamiltonian, $\text{H}$_SB, belongs to the (15, 15) and $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ representations, and show how they are equivalent to models of SU (3) × SU (3) breaking where $\text{H}$_SB belongs to a mixture of different representations. The results for the ππ scattering lengths in the (15, 15) model are outside the experimental limits, but the $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ model yields solutions with a wide range of values for the scattering lengths within the experimental bounds.     

Remarks on the moduli space of SU(2) monopoles and Toda flow

Rational functions of degreek which parametrize the moduli space of SU(2)k-monopoles, can be regarded as transfer functions for certain linear dynamical systems. It is shown that a time shift for linear systems induces a finite nonperiodic complexk×k Toda flow on the parameter space of generic SU(2)k-monopoles. Thus, there exists an integrable flow of the Toda type over the moduli space of SU(2) monopoles.This research was supported in part by National Science Foundation under Grant ECS-8718026.     

Is the SU(3) pattern evidence for SU(3) symmetry?

The results obtained in this paper show that the SU(3) pattern in the spectra and transition rates of even-even nuclei appear qualitatively as a general feature of a large class of quadrupole-phonon models, independently of SU(3) symmetry. This pattern reflects the truncation of the quadrupole phonon space, incorporated via truncation operator in the fourth-order anharmonic quadrupole-phonon Hamiltonian. Three particular forms of the truncation operator are considered, corresponding to three models, namely the truncated quadrupole-phonon model (TQM), equivalent to the well-known interacting-boson model (IBM), the step-truncated quadrupole-phonon model (STQM), equivalent to a standard anharmonic quadrupole-phonon model and the linearly truncated quadrupole-phonon model (LTQM), introduced here. The results obtained provide a basis for a broader scope for the considerations in the SU(3) limit.     

Electromagnetic field in the gauge SU(3) × SU(3) chiral group

The massless electromagnetic Yang-Mills field is explicitly constructed as a linear combination of 16 gauge fields of the chiral SU(3) × SU(3) group within the framework of the plasmon generating mechanism [1]. The remaining 15 gauge fields acquire a mass through the non-zero vacuum expectation values of the auxiliary scalar multiplet which transforms according to the (8,8) representation of the gauge group. The tadpoles with non-zero hypercharge which are required for the existence of the only massless electromagnetic potential A_μ are due to the natural mixing of charged weak currents with ΔS = 0 and ΔS = 1. The relevance of this phenomenon to the Cabibbo angle is briefly discussed. Also presented is a theorem concerning an admissible form of the zero-order mass term of gauge fields when the canonical number is unknown.