Magnetic Monopoles E.A. Olszewski Outline I. Duality (Bosonization)

Magnetic Monopoles E.A. Olszewski Outline I. Duality (Bosonization)

Magnetic Monopoles E.A. Olszewski Outline I. Duality (Bosonization)

II. The Maxwell Equations III. The Dirac Monopole (Wu-Yang) IV. Mathematics Primer V. The tHooft/Polyakov and BPS Monopoles

a. Gauge groups SU(2) and SO(3) b. Gauge groups SU(N) and G2 Outline (continued) VI. Montonen-Olive Conjecture

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

The sine-Gordon equation

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(-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 Primer Magnetic monopole bundle The tHooft/Polyakov and BPS Monopoles

The Maxwell Equations (Minkowski space) The tHooft/Polyakov and BPS Monopoles (continued) The Maxwell Equations (continued)

The tHooft/Polyakov and BPS Monopoles (continued) Gauge groups SU(2) and SO(3) The tHooft/Polyakov and BPS Monopoles (continued)

Monopole construction The tHooft/Polyakov and BPS Monopoles (continued)

The potential V(is chosen so that vacuum expectation value of is non-zero, e.g. The tHooft/Polyakov and BPS Monopoles (continued) The equations of motion can be obtained from the Lagrangian.

The tHooft/Polyakov and BPS Monopoles (continued) The tHooft/Polyakov and BPS Monopoles (continued) The tHooft/Polyakov and BPS Monopoles (continued)

The tHooft/Polyakov and BPS Monopoles (continued) The tHooft/Polyakov and BPS Monopoles (continued)

The tHooft/Polyakov and BPS Monopoles (continued) BPS bound Gauge groups SU(N) and G2

tHooft/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 spontaneously broken YangMills gauge theories .

I have explicitly constructed tHooft/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.

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