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Magnetic Monopoles

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 G2.

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Magnetic Monopoles

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  1. Magnetic Monopoles E.A. Olszewski

  2. 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

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

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

  5. TheMaxwellEquations

  6. The Maxwell Equations (continued)

  7. The Maxwell Equations (continued)

  8. The Maxwell Equations (continued) • Coupling electromagnetism to quantum mechanics

  9. The Maxwell Equations (continued) • Aharonov-Bohm effect

  10. The Dirac Monopole (Wu-Yang)

  11. Dirac Monopole (continued) • The existence of a single magnetic charge requires that electric charge is quantized. • The quantities exp(-iec) are elements of a U(1) group of gauge transformations. If electric charge is quantized, then c=0 and c=2p/e1 (where e1 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 S2 and fiber U(1).

  12. Mathematics Primer Magnetic monopole bundle

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

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

  15. The t’Hooft/Polyakov and BPS Monopoles (continued) Gauge groups SU(2) and SO(3)

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

  17. 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.

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

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

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

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

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

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

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

  25. 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)

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

  27. Montonen-Olive Duality and Type IIB Superstring Theory

  28. Summary • 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.

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