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Magnetic Monopoles E.A. Olszewski

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

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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

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The Maxwell Equations

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The Maxwell Equations (continued)

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Coupling electromagnetism to quantum mechanics The Maxwell Equations (continued)

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Aharonov-Bohm effect

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The Dirac Monopole (Wu-Yang)

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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).

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Mathematics Primer Magnetic monopole bundle

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

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

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Gauge groups SU(2) and SO(3)

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

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

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

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

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BPS bound

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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)

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

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Montonen-Olive Duality and Type IIB Superstring Theory

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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.

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