Saturday, October 08, 2005

Langlands duality

Someone has sent me an article by Edward Witten with his Stony Brook talk about the

I am using Witten's favorite word "duality" instead of "program" because it is a bit more concrete; it's puzzling why the mathematicians haven't realized that their terminology can be sharpened. I encourage everyone to respect that the official terminology has changed to a "duality" right now.

What does number theory, Wiles' proof of Fermat's last theorem, and the S-duality of N=4 Yang-Mills theory in d=4 for general groups have in common? Well, the answer is, once again, Langlands duality - a framework established in the 1960s. For a recent and brief discussion of it, see this article by Edward Frenkel.

The magic trick that transforms the Langlands duality into an umbrella that unifies the diverse insights in the list above is the fact that the duality may be formulated "above" different fields - finite fields similar to "Z_p" as well as continuous fields we know and like. The latter is the geometric Langlands duality.

Start with the N=4 theory in four dimensions and its SL(2,Z) S-duality group from which only the Z2 will be used. For U(1), the spectrum of electric charges is mapped, by S-duality, to the spectrum of magnetic charges which belong to the dual lattice, by Dirac's quantization logic. For non-Abelian groups, you expect the root lattice to be relevant. The dual lattice is typically the weight lattice, and one may hope that it is the root lattice of another group. U(N) is dual to U(N). SU(N) is dual to SU(N) (mod Z_N, to be more exact). USp(2N) is dual to O(2N+1), and you may try to figure out what's dual to O(2N), E6, E7, E8, F2, G4. Answers welcome.

Instead of working with some generally unstable monopoles, Kapustin recently used the S-duality between the 't Hooft loops and the Wilson loops. The Wilson loops are associated with any representation (although the adjoint is the most popular one among physicists since it's related to confinement), i.e. with the weight lattice, while the 't Hooft loops are associated with an embedding of U(1) into your group, i.e. with a root lattice. Again, the same duality emerges. The virtue of Kapustin's picture is that the instability of the monopoles does not matter once they're replaced by the 't Hooft loops.

Another useful step is to compactify the 4D gauge theory on a genus "g" Riemann surface to obtain a two-dimensional sigma model; we must twist some scalars to preserve 8 supercharges - like in B. J. S. Vafa; this step becomes vacuous for "g=1" like in H. M. Strominger. You end up with an N=(4,4) supersymmetric sigma model - which is really the physics side of the "geometric Langlands duality" - whose target space is the moduli space "M_H" of the Higgs bundles where "H" surprisingly stands for Hitchin who constructed them.

The Hitchin fibration allows one to identify a T-duality of this CFT which, combined with another classical symmetry, reconstructs the original S-duality. T-duality also maps the supersymmetric D-branes of this sigma model to each other but their spectrum is not too easy to analyze in this particular model.

If you found the demo above interesting, you may want to read Witten's paper including his discussion of the role of various topological twists of the N=4 Yang-Mills theory, A-branes, B-branes, and other things. Alternatively, you may find Peter Woit's comments (plus some additions) to be a good starting point.

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