Magnetic Monopoles E.A. Olszewski Outline I. Duality (Bosonization) II. The Maxwell Equations III....
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Transcript of Magnetic Monopoles E.A. Olszewski Outline I. Duality (Bosonization) II. The Maxwell Equations III....
OutlineOutline
I. Duality (Bosonization)I. Duality (Bosonization)II. The Maxwell EquationsII. The Maxwell EquationsIII. The Dirac Monopole (Wu-Yang)III. The Dirac Monopole (Wu-Yang)IV. Mathematics PrimerIV. Mathematics PrimerV. The t’Hooft/Polyakov and BPS V. The t’Hooft/Polyakov and BPS
MonopolesMonopoles a. Gauge groups SU(2) and SO(3) a. Gauge groups SU(2) and SO(3) b. Gauge groups SU(N) and G2 b. Gauge groups SU(N) and G2
Outline (continued)Outline (continued)
VI. Montonen-Olive Conjecture VI. Montonen-Olive Conjecture (weak/strong duality) and SL(2,Z)(weak/strong duality) and SL(2,Z)
VII. Montonen-Olive Duality and Type IIB VII. Montonen-Olive Duality and Type IIB Superstring TheorySuperstring Theory
Duality (Bosonization)Duality (Bosonization) The sine-Gordon equationThe sine-Gordon equation
The Thirring modelThe Thirring model
Meson states → fermion-anti fermion bound states Soliton → fundamental fermion
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).
The Maxwell Equations (continued)The Maxwell Equations (continued)
The t’Hooft/Polyakov and BPS Monopoles (continued)
The t’Hooft/Polyakov and BPS Monopoles (continued)
The potential V(is chosen so that vacuum expectation value of is non-zero, e.g.
The t’Hooft/Polyakov and BPS Monopoles (continued)
The equations of motion can be obtained from the Lagrangian.
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)
Summary Summary
I have reviewed the Dirac monopole and its natural extension to I have reviewed the Dirac monopole and its natural extension to
spontaneously broken YangMills gauge theoriesspontaneously broken YangMills gauge theories.. I have explicitly constructed t’Hooft/polyakov magnetic monopole I have explicitly constructed t’Hooft/polyakov magnetic monopole
and BPS dyon solutions for SU(N) . Suprisingly , the electric and BPS dyon solutions for SU(N) . Suprisingly , the electric charge of the dyon is coupled strongly, as is the magnetic charge.charge of the dyon is coupled strongly, as is the magnetic charge.