Duality and strings

In class, I compared a dual theory to a two-faced coat that is still a fully functional garment when turned inside-out. For a better insight into duality, consider Maxwell's theory of electromagnetism. In this theory there is an almost perfect correspondence between electric and magnetic phenomena. The only difference is that there are isolated electrical charges, but nobody has ever isolated a magnetic charge: a magnet always has a positive effect on one side (north pole) and a negative effect of equal strength on the opposite side (south pole). We can however postulate that isolated magnetic charges exist and call them magnetic monopoles. With the introduction of magnetic monopoles, Maxwell's equations exhibit a precise duality between electric and magnetic phenomena: we could exchange all magnetic quantities with the corresponding electric quantities and the "laws of nature" (the Maxwell equations) would remain essentially the same. The question is then: why has nature abandoned this perfect symmetry of duality and chosen to favor electricity by banning magnetic monopoles? This may seem an idle question, but there is more to it when the laws of electromagnetism are combined with the principles of quantum mechanics. First, we must accept that electric charge is quantized: it comes attached to elementary particles that all have the same charge. To simplify the discussion, consider only electrons. Dirac (the same man who introduced antimatter) showed that, if there are magnetic monopoles corresponding to electrons, their magnetic charge m is related to the electron's charge e by the equation

em = hc/4

where h is Planck's constant and c is the speed of light. This is an amazing prediction, and it is useful in guiding a search for monopoles: we now know the strength of their magnetic charge, which turns out to be large.

Now, it is found that some string theories "naturally" exhibit a duality property of the type we just discussed: this is called S-duality. One string theory can be the "dual" of another string theory, or it can be "self-dual", like Maxwell's theory with monopoles added. S-duality ties together apparently unrelated things, like the charge of the electron and the charge of the monopole. One can use S-duality to predict the behavior of strongly interacting particles, like the monopoles, from the known behavior of weakly interacting particles, like the electrons.

There is another kind of duality in string theory, called T-duality. I will try to explain it in class. Much more can be found at:

http://theory.caltech.edu/people/jhs/strings/intro.html
http://www.physics.ucsb.edu/~jpierre/strings/intro.htm