Self-dual magnetic monopoles

The present status of self-dual monopoles is reviewed with a particular attention to a duality conjecture.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–21, 1981.     

Self-dual monopoles in a seven-dimensional gauge theory

The existence of self-dual or anti-self-dual monopoles of a seven-dimensional generalized Yang-Mills-Higgs theory is proved using the second-order equations of motion. The behavior of solutions can be used to recognize self- or anti-self-duality. Moreover, it is shown that, in the class of the field configurations under discussion, the solutions are, in fact, unique.     

Interacting monopoles

The Bogomolny equation for interacting monopoles of like charge in the case of vanishing Higgs mass is fulfilled by an asymptotic expansion in powers of the inverse monopole distance. Consequently, the monopoles exert no long range forces on each other. However, the expansion is not summable. The degrees of freedom of the multi-monopole system are discussed.     

Magnetic monopoles

These are notes of the first part of the lectures given at the JINR-ISU Baikal Summer School on Physics of Elementary Particles and Astrophysics (July 2010). I review classical monopole solutions of the SU(2) Yang-Mills-Higgs theory providing a pedagogical introduction into to the theory of non-Abelian monopoles both in the BPS limit and beyond of it. I briefly discuss monopole dynamics, the idea of the moduli space and some of the basic properties which are connected with the field theoretical aspects of these classical solutions.     

Magnetic monopoles

Summary Theoretical foundations for the existence of magnetic monopoles are given. Consequences extracted from cosmological and astrophysical considerations on the abundance of magnetic monopoles today are reviewed. The latest experimental results in the search for classical monopoles as well as for grand unified monopoles are summarized.

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Riassunto Si danno i fondamenti teorici per l'esistenza di monopoli magnétici. Si esaminano le conseguenze tratte da considerazioni cosmologiche e astrofisiche sull'abbondanza di monopoli magnetici attuale. Si riassumono i piú recenti risultati sperimentali sulla ricerca di monopoli classici e di quelli della grande unificazione.

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Резюме Приводятся теоретические обоснования сушствования магнитных монополей. Анализируются следствия из космологических и астрофизических рассмотрений, касающиеся распространенноси магнитных мононолей в настоящее время. Приводятся последние экспериментальные результаты по поискы классических монополей и монополей Великого Объединения.

     

Quaternionic monopoles

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated withSpin ^h (4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kähler surface the quaternionic monopole equations decouple and lead to the projective vortex equation for holomorphic pairs. This vortex equation comes from a moment map and gives rise to a new complex-geometric stability concept. The moduli spaces of quaternionic monopoles on Kähler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along a Donaldson instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories.Partially supported by: AGE-Algebraic Geometry in Europe, contract No ERBCHRXCT940557 (BBW 93.0187), and by SNF, nr. 21-36111.92     

Tetrahedral and cubic monopoles

Using a numerical implementation of the ADHMN construction, we compute the fields and energy densities of a charge three monopole with tetrahedral symmetry and a charge four monopole with octahedral symmetry. We then construct a one parameter family of spectral curves and Nahm data which represent charge four monopoles with tetrahedral symmetry, which includes the monopole with octahedral symmetry as a special case. In the moduli space approximation, this family describes a novel kind of four monopole scattering and we use our numerical scheme to construct the energy density at various times during the motion.Address from September 1995, Institute of Mathematics, University of Kent at Canterbury, Canterbury CT2 7NZ. Email: p.m.sutcliffe@ukc.ac.uk     

Frenkel-Kontorova model with toda interactions

We have studied the Frenkel-Kontorova (FK) model with Toda interactions. The phase diagram is found to be asymmetric due to the exponential form of the Toda interaction. The reflection symmetry observed in the standard FK model is broken here. The singularity spectrum and the generalized dimension are calculated and their dependence on the nonlinearity parameter is discussed. The critical exponents of the gap in the phonon spectrum, the correlation length, and the Peierls-Nabarro barrier are found to be the same as those in the standard FK model and are independent of the nonlinearity parameter.     

The quantum mechanical toda lattice

The aim of this paper is to establish the exact quantization conditions for the three-body Toda lattice. The Hamiltonian consists of the kinetic energy for three particles in one dimension, and of the potential energy which couples each particle to its two companions through an exponential spring. After eliminating the center of mass motion, one is left with a system of two degrees of freedom and two constants of motion, the total energy E and a third integral A which commute. Nevertheless, no transformation has been found to separate the classical equations of motion or Schrödinger's equation. The wave function is written as a double Laurent series. Its coefficients have to satisfy two sets of recursion relations on a triangular grid where each set insures that we have a simultaneous eigenfunction of E and A. The condition for the convergence of this series can be expressed as the vanishing of a tridiagonal infinite determinant with 1 in the diagonal and the inverse of a third-order polynomial in the first off-diagonals. The coefficients in this polynomial are E and A, and the variable corresponds to a component of the wave vector associated with the wave function. This determinant can be treated exactly as Hill's, and yields the 3 components. The condition for the square integrability of the wave function requires the phase angle of the principal minors to be equal to 0, $\text{π}\text{3}$, or $\text{2π}\text{3}$ according as the representation of the cyclic groups, for each component of the wave vector. But the third condition follows from the two others. The analogy with the corresponding two-body problem is pointed out.     

Colored magnetic monopoles

We extend Dirac's theory of magnetic monopoles to the case of non-Abelian color gauge groups. Exact classical solution is obtained by making use of the gauge-independent method of Yang-Mills field. The case of broken gauge symmetry is also briefly discussed.     

Chaos in a damped and driven toda system

We describe our numerical studies of chaos for a damped and driven single Toda particle and in a Toda lattice. Comparisons are made with chaos in other classes of nonlinear potentials.     

Gravitationally bound monopoles

We construct monopole solutions in SU(2) Einstein-Yang-Mills-Higgs theory carrying magnetic charge n. For vanishing and small Higgs self-coupling, these multimonopole solutions are gravitationally bound. Their mass per unit charge is lower than the mass of the n = 1 monopole. For large Higgs self-coupling only a repulsive phase exists.     

Cosmological Kaluza-Klein monopoles

A solution of a pair of monopoles in 5D Kaluza-Klein theory without matter fields is given. The scale of the internal space is the same as that of the outer space. Its euclidean version is useful for calculating the wave function of the universe with monopoles.     

Non-Abelian magnetic monopoles

It is shown that a general, irreducible, SU(n), Sp(n), SO(2n) monopole with maximal symmetry breaking is determined by its spectral data. For SU(n) with minimal symmetry breaking the spectral data is defined and also shown to determine the monopole.Research supported in part by NSF Grant 8120790