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Topical. I know there are good mathematical theories in which "Higgs" is used, in a geometrical sense. Would someone care to explain?

To clarify, I'd like to know about Higgs bundles on Riemann surfaces, rather than the standard model of high-energy physics.

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Dear Charles. I think that (1) it's kind of complicated, as reflected by the rather complicated form of the standard model's Lagrangian: (2) It's not fair to simply ask for someone to explain: you should first show that you have invested some effort in learning something about the Higgs mechanism, and then you may ask your question. – André Henriques Jul 4 '12 at 19:02
@Charles: If you'd listen to a previous advice and are ready to work to deserve an explanation ... there is a little book by D. Bleecker (Gauge Theory and Variational Principles) that does a good job of providing a dictionnary between a physicist's view of gauge theory and a mathematician's view of it. Section 10.3 is an introduction to the Higgs mechanism. – alvarezpaiva Jul 4 '12 at 20:02
I wasn't asking about the standard model. – Charles Matthews Jul 4 '12 at 21:26
@alvarezpaiva Actually this is uncalled-for. One of my undergraduate students went on to do Ph.D. work at Oxford on Higgs bundles. In those days I did work at understanding the mathematics, but it was quite some time ago. I said geometrical theories in that sense. – Charles Matthews Jul 4 '12 at 21:33
@Charles: I'm sorry, that's not what I meant with my comment. I find it ridiculous that one should have to "work to deserve" an explanation or that it is "unfair" to simply ask for an introduction to some topic. If that were the case, this site would make little sense, to me at least. The ellipsis after my comment was too feeble an indication of what I was thinking. Again, my apologies, and I hope the people that voted my comment up understood it's true meaning. – alvarezpaiva Jul 4 '12 at 23:31
up vote 17 down vote accepted

Since you are asking about Higgs bundles, I can say a few words here. These were introduced by Hitchin in the mid 1980's, although I'm not sure he used this term. One can look at the introduction to his paper "The self-duality equations on Riemann surfaces" for some of the motivation and background. In brief outline, he starts from a gauge theoretic perspective: the initial data is a pair consisting of a vector bundle with connection on a compact Riemann surface $X$ and additional vector valued $1$-form (the Higgs field) subject to the equations referred to above. One of his main results is that such a pair is equivalent to a pair $(E,\theta)$ -- now called a Higgs bundle -- consisting of a holomorphic vector bundle and holomorphic section $\theta\in \Gamma(\Omega_X^1\otimes End(E))$ satisfying a suitable stability condition. One consequence is:

Theorem. There is a correspondence between irreducible representations of $\pi_1(X)$ and stable Higgs bundles of degree $0$.

Simpson rightly christened (a generalization of) this the nonabelian Hodge theorem. It is indeed a beautiful and fundamental theorem. It generalizes an earlier theorem of Narasimhan and Seshadri that corresponds to the case where the Higgs field is zero. These objects also make their appearance in Geometric Langlands -- but I'm hardly the right person to discuss these aspects. I could go on, but I think I'll stop here.

Perhaps I may add that the underlying geometry is very rich. Both the representation and Higgs sides admit natural moduli problems. Under this correspondence, the two moduli spaces are homemorphic but are analytically distinct. In fact, the complex structures behave like $i$ and $j$ of the quaternions, and together with a natural symplectic structure, constitute a hyper Kähler manifold (away from the singularities). Such things are quite rare. On the Higgs side, there is some additional geometry: A $\mathbb{C}^*$-action obtained by dilating the Higgs field, and the Hitchin fibration obtained by sending a Higgs bundle $(E,\theta)$ to the characteristic polynomial of $\theta$. The latter turns out to form a completely integrable system, that is the smooth fibres are Langrangian with respect to the symplectic structure. Arguments from symplectic geometry show that these are tori, although in fact, these fibres turn out to be abelian varieties. What is even more remarkable is that part of this story carries over into positive characteristic and had played an important role in Ngo's work. I understand very little of this, so instead I'll direct interested readers to the recent Bulletin article by Nadler.

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