2. Magnetic Monopoles E.A. Olszewski

3. 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 G2

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

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

6. The Maxwell Equations

7. The Maxwell Equations (continued)

9.  Coupling electromagnetism to quantum mechanics The Maxwell Equations (continued)

10.  Aharonov-Bohm effect

11. The Dirac Monopole (Wu-Yang)

12. Dirac Monopole (continued) 1.The existence of a single magnetic charge requires that electric charge is quantized. 2.The quantities exp(-ie  are elements of a U(1) group of gauge transformations. If electric charge is quantized, then  and  e 1 (where e 1 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 S 2 and fiber U(1).

13. Mathematics Primer Magnetic monopole bundle

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

15. The Maxwell Equations (continued) The t’Hooft/Polyakov and BPS Monopoles (continued)

16. Gauge groups SU(2) and SO(3)

17. The t’Hooft/Polyakov and BPS Monopoles (continued) Monopole construction

18. The t’Hooft/Polyakov and BPS Monopoles (continued) The potential V(  is chosen so that vacuum expectation value of  is non-zero, e.g.

19. The t’Hooft/Polyakov and BPS Monopoles (continued) The equations of motion can be obtained from the Lagrangian.

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

25. BPS bound

26. 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)

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

28. Montonen-Olive Duality and Type IIB Superstring Theory

29. Summary I have reviewed the Dirac monopole and its natural extension to spontaneously broken YangMills gauge theories. I have reviewed the Dirac monopole and its natural extension to spontaneously broken YangMills 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. 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.