Electromagnetic duality for children

In the Fall and Winter terms 1995-6, I gave a series of ten lectures on electromagnetic duality to members of the QMW String Theory group. The lectures started from very basic material and ended with Sen's test of the predicted dyonic spectrum at monopole number 2 for the gauge group SU(2).

The full set of lectures has not changed since late 1998. The tenth and last lecture has still to be typed. However there is a new section on the extension to arbitrary compact simple gauge groups.

The notes include about 150 exercises of varying difficulty, but the solutions have yet to be typed.


  1. Dirac monopole, dyons, Dirac-Schwinger-Zwanziger quantisation condition. The 't Hooft-Polyakov monopole.
  2. Magnetic charge as topology. Prasad-Sommerfield limit, Bogomol'nyi bound. The Montonen-Olive conjecture.
  3. The Witten effect and SL(2,ℤ) duality.
  4. Supersymmetry, central charges and the Bogomol'nyi bound revisited: short multiplets.
  5. N=2 supersymmetric Yang-Mills: BPS states revisited.
  6. Monopole moduli space, collective coordinates, low energy dynamics. Dyonic spectrum for 1-monopole sector.
  7. Moduli space of N=2 supersymmetric BPS monopoles, low energy dynamics. Bound states.
  8. "Forget it all for an instanton!"
    Details of the calculation of the number of collective coordinates: an example of an index theorem calculation for operators on open spaces.
  9. N=4 supersymmetric Yang-Mills from 10 dimensions -- yet another avatar of the BPS monopole.
  10. Effective action for N=4 supersymmetric Yang-Mills. The Atiyah-Hitchin metric on the 2-monopole moduli space. Sen's calculation.

Electromagnetic Duality

The original aim of the lectures was to study the Seiberg-Witten solution of N=2 supersymmetric gauge theories, and the working title was On the road to Seiberg-Witten. The road instead detoured towards the self-duality of N=4 supersymmetric Yang-Mills theory in four dimensions.