Edward Witten has a new expository article, aimed at mathematicians, to appear at some point in the Bulletin of the AMS, but now available here. It’s based on colloquium-style lectures to mathematicians he has given over the last few years (including here at Columbia) and is entitled “From Superconductors and Four-Manifold to Weak Interactions”. The paper is organized around describing various aspects of gauge-symmetry breaking, but pretty much sticks to aspects of the problem that don’t involve the full quantum theory, just analysis of classical Lagrangians.
He begins with a description of the Landau-Ginzburg model of superconductivity, and various physical phenomena that it describes including the Meissner effect, Abrikosov-Gorkov flux lines, and Type I and II superconductors. Solutions for a special case are described using complex-analytic techniques. Exploiting an analogy to the Landau-Ginzburg case, he next takes up the Seiberg-Witten equations and their use by Taubes to get invariants for symplectic 4-manifolds and existence theorems for pseudo-holomorphic curves in them.
Witten’s final topic is electroweak gauge symmetry breaking and the Higgs mechanism in the Standard Model. He ends by remarking that in the superconducting case the analog of the Higgs field is just an effective field for a different underlying physics, and mentioning technicolor as an implementation of something similar in the electroweak case, while noting that precision electroweak data shows no signs of anything other than an elementary Higgs field. He comments “But it is always possible that the right alternative has not yet been proposed” and explains how the LHC should definitively see a Higgs particle if the SM is correct since current bounds place its mass between 115 and 200 GeV.
The paper is purely expository, and aimed at mathematicians. It’s interesting to see that, even though there aren’t any really new developments in the area of gauge symmetry breaking, Witten clearly sees it as a fundamental problem every bit as deserving of being explained to non-physicists as string theory.