Magnetic Monopoles E A Olszewski Outline I Duality

Magnetic Monopoles E. A. Olszewski

Outline I. Duality (Bosonization) II. The Maxwell Equations III. The Dirac Monopole (Wu-Yang) IV. Mathematics Primer V. The t’Hooft/Polyakov and BPS Monopoles a. Gauge groups SU(2) and SO(3) b. Gauge groups SU(N) and G 2

Outline (continued) VI. Montonen-Olive Conjecture (weak/strong duality) and SL(2, Z) VII. Montonen-Olive Duality and Type IIB Superstring Theory

Duality (Bosonization) n The sine-Gordon equation n The Thirring model Meson states → fermion-anti fermion bound states Soliton → fundamental fermion

The Maxwell Equations

The Maxwell Equations (continued)

The Maxwell Equations (continued)

The Maxwell Equations (continued) § Coupling electromagnetism to quantum mechanics

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(-iec) are elements of a U(1) group of gauge transformations. If electric charge is quantized, then c=0 and c=2 p/e 1 (where e 1 is the unit of charge) yield the same gauge transformation, i. e. the range of c 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 c is not compact the gauge group is called R. 3. Mathematically, we have constructed a non-trivial principal fiber bundle with base manifold S 2 and fiber U(1).

Mathematics Primer Magnetic monopole bundle

The t’Hooft/Polyakov and BPS Monopoles The Maxwell Equations (Minkowski space)

The t’Hooft/Polyakov and BPS Monopoles (continued) The Maxwell Equations (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(F) is chosen so that vacuum expectation value of F 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 G 2 t’Hooft/Polyakov magnetic monopole in SU(N) BPS dyon G 2 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 n I have reviewed the Dirac monopole and its natural extension to n spontaneously broken Yang. Mills gauge theories. I have explicitly constructed t’Hooft/polyakov magnetic monopole and BPS dyon solutions for SU(N). Suprisingly , the electric charge of the dyon is coupled strongly, as is the magnetic charge.