Magnetic MonopolesE.A. OlszewskiE.A. Olszewski

Outline

I. Duality (Bosonization)I. Duality (Bosonization)II. The Maxwell EquationsII. The Maxwell EquationsIII. The Dirac Monopole (Wu-Yang)III. The Dirac Monopole (Wu-Yang)IV. Mathematics PrimerIV. Mathematics PrimerV. The t’Hooft/Polyakov and BPS V. The t’Hooft/Polyakov and BPS

MonopolesMonopoles a. Gauge groups SU(2) and SO(3) a. Gauge groups SU(2) and SO(3) b. Gauge groups SU(N) and G2 b. Gauge groups SU(N) and G2

Outline (continued)

VI. Montonen-Olive Conjecture VI. Montonen-Olive Conjecture (weak/strong duality) and SL(2,Z)(weak/strong duality) and SL(2,Z)

VII. Montonen-Olive Duality and Type IIB VII. Montonen-Olive Duality and Type IIB Superstring TheorySuperstring Theory

Duality (Bosonization) The sine-Gordon equationThe sine-Gordon equation

The Thirring modelThe Thirring model

Meson states → fermion-anti fermion bound states Soliton → fundamental fermion

The Maxwell Equations

The Maxwell Equations (continued)

The Maxwell Equations (continued)

Coupling electromagnetism to quantum mechanics

The Maxwell Equations (continued)

The Maxwell Equations (continued) Aharonov-Bohm effect

The Dirac Monopole (Wu-Yang)

Dirac Monopole (continued)

1. The existence of a single magnetic charge requires that electric charge is quantized.

2. The quantities exp(-ieare elements of a U(1) group of gauge transformations. If electric charge is quantized, then and e1 (where e1 is the unit of charge) yield the same gauge transformation, i.e. the range of is compact. In this case the gauge group is called U(1). In the alternative case when charge is not quantized and the range of is not compact the gauge group is called R.

3. Mathematically, we have constructed a non-trivial principal fiber bundle with base manifold S2 and fiber U(1).

Mathematics PrimerMagnetic monopole bundle

The t’Hooft/Polyakov and BPS Monopoles

The Maxwell Equations (Minkowski space)

The Maxwell Equations (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

Gauge groups SU(2) and SO(3)

The t’Hooft/Polyakov and BPS Monopoles (continued)

Monopole construction

The t’Hooft/Polyakov and BPS Monopoles (continued)

The potential V(is chosen so that vacuum expectation value of is non-zero, e.g.

The t’Hooft/Polyakov and BPS Monopoles (continued)

The equations of motion can be obtained from the Lagrangian.

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

The t’Hooft/Polyakov and BPS Monopoles (continued)

BPS bound

Gauge groups SU(N) and G2

t’Hooft/Polyakov magnetic monopole in SU(N)

BPS dyon

G2 monopoles and dyons consist of two copies of SU(3)

Montonen-Olive Conjecture (weak/strong duality) and SL(2,Z)

Montonen-Olive Duality and Type IIB Superstring Theory

Summary I have reviewed the Dirac monopole and its natural extension to I have reviewed the Dirac monopole and its natural extension to

spontaneously broken YangMills gauge theoriesspontaneously broken YangMills gauge theories.. I have explicitly constructed t’Hooft/polyakov magnetic monopole I have explicitly constructed t’Hooft/polyakov magnetic monopole

and BPS dyon solutions for SU(N) . Suprisingly , the electric and BPS dyon solutions for SU(N) . Suprisingly , the electric charge of the dyon is coupled strongly, as is the magnetic charge.charge of the dyon is coupled strongly, as is the magnetic charge.