Witten on Gauge Symmetry Breaking for Mathematicians

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.

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17 Responses to Witten on Gauge Symmetry Breaking for Mathematicians

  1. Bee says:

    Hi Peter,

    don’t forget to have a toast on the guy who said

    “Das ist nicht nur nicht richtig, es ist nicht einmal falsch!”



  2. Jimbo says:

    Accd’g to the latest results (4/15/07), the new upper bound on the Higgs boson mass is ~144 Gev at the 95% CL:
    This and the lower bound U cite, put the mean at about 130 Gev.


  3. woit says:

    Thanks B.,

    I’ll definitely remember to toast Pauli on his birthday before the evening is out!

  4. DB says:

    Jimbo wrote:

    “Peter, Accd’g to the latest results (4/15/07), the new upper bound on the Higgs boson mass is ~144 Gev at the 95% CL:
    This and the lower bound U cite, put the mean at about 130 Gev.”

    At the end of the quoted article is the link to the next article:
    “Mathematician suggests extra dimensions are time-like”

    Oh well, the real physics was nice while it lasted.

  5. Hi Peter,

    Just to point out a couple of things:

    (a) One can find a more detailed historical account of the facts leading to the discovery of Spontaneous Symmetry Breaking in the following link: A Physics History of My part in the Theory of Spontaneous Symmetry Breaking and Gauge particles with a mix of modern ideas (PDF, 194Kb).

    (b) Thus, in all fairness, there should have been a link [in Witten’s article] to Global Conservation Laws and Massless Particles (GHK). As can be seen in Spontaneous Breakdown of Symmetry in Axiomatic Theory (page 515) and Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism, there are some “subtleties” treated in the GHK which had not been previously considered — and those were important for the future development of the ElectroWeak symmetry breaking and the Standard Model.

    Lastly, i would personally say that there have been some “new developments” in this area: Computational High Energy Physics Group at Brown. For instance, we have some new numerical methods (which we’re still improving and sharpening) that are able to deal with Symmetry Breaking and different phases in a given QFT (something that ordinary Monte Carlo simulations cannot do): We’ve been using some Stochastic Quantization techniques, together with some techniques from Wiener Polynomial Chaos, and have been able to tackle the issue of symmetry breaking in simulations of QFT quite well. Also, we’ve been working on the relation between Symmetry Breaking and Topology Change (there will be a new [sequel] paper coming out on this topic shortly); a topic that has potential implications to Quantum Gravity at large, once it deals with Topological Transitions, explaining them in terms Symmetry Breaking (and Morse Theory). So, i think the playfield is still active… 😉

    Cheers, Daniel.

  6. Jimbo says:

    Suggest U keep an open mind. That mathematician is Penrose’s former student, George Sparling. U might be impressed if U attempted to read the article. Itzhak Bars has also investigated Xtra time-like dimensions as well.

  7. Kea says:


    That’s fantastic! Thanks for the link. FINALLY, some decent physics makes the news.

  8. Just a quick note in order to stress points (a) and (b) of the comment above: Both, Weinberg’s (PDF, 76Kb) and Salam’s (PDF, 1.16Mb) Nobel Lectures credit GHK and, in fact, Salam goes as far as saying that he was tutored by Kibble [in the technique of Symmetry Breaking]. Furthermore, Glashow’s (PDF, 59Kb) [Nobel Lecture] also cites GHK and Symmetry Breaking, albeit in a “funny” remark (see paragraph 3 of page 498 — here i mean “funny” under the context given by Bob Marshak’s remarks on the 3rd Shelter Island conference).

    And, to drive those points home, note the acknowledgements in Spontaneous Symmetry Breakdown without Massless Bosons. 😉

    Cheers, Daniel.

  9. joseph smidt says:

    Thanks for posting this. I love these kind of papers.

  10. Jonathan says:


    We invite you to discuss science in our open forums. Subscription to the forum is anonymous.


  11. anon. says:

    Jonathan, is your invitation to the ‘reading men’ science forum open to women as well as to men? Or are women excluded? I hope women are included, for a variety of reasons.

  12. Kea says:

    Too late. We’re boycotting it.

  13. anon. says:

    Kea, ‘boycotting’ is a sexist term.

  14. Kea says:

    Oops, sorry! Personcotting.

  15. anon.2 says:

    I like how some blogs make it difficult to tell when someone is joking. Charles Boycott is the source of the eponym.

  16. Jonathan says:

    as I wrote registration is anonymous. We do not check, gender:))

  17. Jonathan says:

    We have changed our domain name. Now, It is

    and the forum is at


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