BRST and Dirac Cohomology

For the last couple years I’ve been working on the idea of using what mathematicians call “Dirac Cohomology” to replace the standard BRST formalism for handling gauge symmetries. So far this is just in a toy model: gauge theory in 0+1 dimensions, with a finite dimensional Hilbert space. Over the last few months I’ve finally got this to the point where I think I understand completely how this should work, at least for this toy model. I talked about this last week in St. Petersburg, and have a preliminary version of a paper on the subject, which is available here. Next weekend I’m leaving for a trip to Shanghai and Hong Kong (the plan is to be in Shanghai for the July 22 total solar eclipse, which will be visible there). After I get back at the beginning of August I’ll work on the paper a bit more, hoping to have a final version done by the beginning of September, when the academic year starts.

The paper uses quite a lot of mathematical technology, so I fear most people will find it hard to read. This fall I hope to get back to finishing the Notes on BRST I was writing up, the idea behind those was to give a more expository account of this subject. That project got bogged down when I realized there was something I was still confused about, and after getting unconfused it seemed like a good idea to get the basic ideas down on paper, since the expository project might take a while to complete.

First of all, what this is and what it isn’t. It’s a toy quantum mechanical model, with gauge symmetry treated using some new ideas from representation theory which are related to BRST, but different. It’s not a QFT, and not a treatment of gauge symmetry in the physical case of four space-time dimensions. I’ve been thinking about how to extend this to higher dimensions, but this requires some new ideas. Next on the agenda is to try and get something that works in 1+1 dimensions, where one can exploit a lot that is known about affine lie algebras and coset models. There also appear to be interesting possible connections to geometric Langlands in that case.

Given a quantum system with G-symmetry, the BRST method allows one to gauge a subgroup H, picking out the H-invariant subspace of the original Hilbert space using Lie algebra cohomology methods. The proposal here is to do something different, picking out a subgroup H of symmetries one wants to keep, and gauging the rest. In the special case where Lie G/Lie H is the sum of a Lie subalgebra and its conjugate, the method proposed here reduces to the standard BRST method, but it is more general.

An algebraic version of the Dirac operator plays a role here somewhat like that of the BRST operator in the standard formalism. One difference is that the square of this operator is not zero. However, it is in the center of the algebra of operators acting on the Hilbert space, so its action on operators squares to zero. This sort of thing has been studied a bit before in the physics literature, in the context of supersymmetric quantum mechanics models, but I do believe that the interpretation here as a method for handling gauge symmetry is new.

One thing I want to add to the paper is some comments about the relation to the physical Dirac operator. The point of view on the Dirac operator explained here that comes out of representation theory seems to me perhaps the most intriguing part of this story. Remarkably, this Dirac operator is in some sense a quantization of the Chern-Simons form. The full story of how to use this in higher dimensions remains obscure to me, but there is some hope it will bring together the physical Dirac operator, something like BRST, and something like supersymmetry in a new way.

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16 Responses to BRST and Dirac Cohomology

  1. Austin says:

    Peter, what are the testable predictions of the generalized physically unphysically topological Woit-Dirac-Kostant-BRST-supersymmetry non-nilpotent cohomology-non-cohomology and how can one falsify it, aside from pure mathematical proofs that it cannot work?

  2. Peter Woit says:


    The problem with string theory is not that it doesn’t now make testable predictions, but that 25 years of intensive effort have shown that it inherently can’t.

    Whether ideas related to Dirac Cohomology will ever lead to progress on doing better than the Standard Model, I don’t know. I think it’s reasonable to hope that they might lead to a better understanding of the Standard Model and its relation to fundamental mathematics. This seems to me worth working on and I’ll keep doing that. If 25 years from now this hasn’t led anywhere, but there are institutes of Dirac Cohomology and thousands of physicists working in the field, I’ll be the first to criticize this.

    Quantum gauge theories remain a largely unexplored rich mathematical territory. Exploring this may lead to new physics, or it may lead to new mathematics. What I’m writing about is just one idea about how to make progress on this. We’ll see if it gets anywhere.

    And, with that, from now on I’ll delete any more stupid anonymous comments to this posting. If you want to discuss the ideas I’m writing about here, I’m happy to do so. If you want to engage in juvenile argumentation, you’ll have to do it under your own name here, or find another posting.

  3. Per says:


    You must be a bit nervous now tho. All those string theory people has critized you over and over again for not publishing anything. Now it comes, so I am sure they’ll all jump on it.

    Best of lucks.

    // phd student.

  4. Christine says:


    Sounds interesting, but it will make a somewhat hard reading for me. I understand that it is just a toy model for the moment, but I missed the main motivation for it. Do you envision any methodological advantages over BRST or perhaps a possible connection to the Langlands programme?


  5. Tumbledried says:


    In my opinion, there in nothing wrong with pursuing a speculative line of enquiry – that is what research is supposed to be all about. “Search”, and “search again” ie Re- search. Results are never guaranteed. In fact the chance of a particular line of reasoning being successful, in absence of some very good heuristic/intuitive/formal/physical reason, is very, very small.

    And even if there are grounds to believe something to be true, that does not ensure success. String theory is an example of this. There originally were relatively good grounds and motivation to pursue it as a topic – 30 to 40 years ago. But it is pure foolishness in an area where results are not guaranteed to put all resources into one particular topic.

    The whole scientific community benefits if a variety of ideas/approaches are posed to attack fundamental questions. Contrariwise, the whole community suffers if all avenues of attack save a few – which furthermore have yet to be vindicated (if ever!) – are stifled.

    So the burden of proof is not on Peter in this case, since I do not think he is advocating/demanding that everybody drop everything and work on BRST.


  6. Peter Woit says:


    Thanks, but I’m not worried about that. The problem with working on something rather abstract and far from what other people are used to is that it’s likely to be ignored. If string theorists decide to learn about Dirac cohomology in order to criticize what I’m doing with it, so much the better…


    The BRST treatment of gauge symmetry has various problems, especially non-perturbatively. Maybe Dirac cohomology avoids these, too soon to tell. It also may be useful in a non-perturbative formulation of chiral gauge theory, something which doesn’t really exist at the moment. Another possible application would be to make sense of the vague idea that spinor fields are the analogs of ghost fields for local translation symmetry. All of this is rather speculative at the moment.

    One approach to geometric Langlands uses BRST, perhaps Dirac cohomology gives something of mathematical interest there, including a closer connection between geometric Langlands and 2d gauge theory. Again, all speculation at the moment…

  7. Michael Zeiler says:

    Off topic, but you mentioned that you’ll be in Shanghai for the total solar eclipse on July 22.

    I just posted some eclipse maps with lines of equal duration for totality and local circumstances at

    The map for Shanghai is TSE2009_China_120E_123E.pdf. If you need to dodge clouds and drive east, then you also want TSE2009_China_117E_120E.pdf

    For a larger view, try Regional_TSE2009_CentralChina.pdf and Regional_TSE2009_EastChinaSea.pdf

    Clear skies!

  8. Peter Woit says:

    Thanks Michael,

    The plan so far is to watch the eclipse from downtown Shanghai, we’ll see if that works out…

  9. Another eclipse! I am extremely envious! I still have to see one… I seem to be on the opposite side of the world when one occurs. Have fun!


  10. Tipster Dude says:

    Peter, if you add this WordPress shortcode in the HTML of a page, [archives], it will list links to all your posts on one page, which would be around 750 links.

    WordPress shortcodes.

    So you could create a page such as “Archives”, and then create a link to that page in one of your link categories, if inclined.

    Formalizing theory from a heavy mathematician’s perspective ==> magical mystery tour for all but a few select physicists. Serves them right.

  11. Thomas Larsson says:

    I cannot say that I understand what you are writing. However, it seems to me that you make the same crucial assumption in Dirac cohomology as in BRST: that the relevant algebra has representations. This becomes problematic if you want to study current groups in multi-dimensions. Classically, there are no problems; the representations typically act on fields over spacetime, i.e. on sections of some bundles. However, after quantization you run into various infinities, which can not be removed by normal ordering. This problem seems independent of whether you consider Dirac or BRST cohomology.

    One can construct representations acting on quantum fermions parametrized by a classical background gauge field. This idea, which you seem to favor, has been put forward by Jouko Mickelsson. Alas, I think it is a bad idea, at least if you want to study fundamental physics. It may work within a limited scope, but fundamentally everything must be quantized. The introduction of a classical gauge field is therefore manifestly unphysical.

    The reason why Jouko considered this type of representations is a no-go theorem by Pickrell. IMO, the moral of this theorem is that no interesting representations of current groups can be constructed within the framework of QFT, but one must consider a more general framework instead. Since I know that you don’t want me to mention my own work, let me link to an old paper by Rao and Moody, where they construct vertex representations of current algebras. The thing to note is that these representations do not depend on some background gauge potentional, so Pickrell’s no-go theorem is evaded. However, the Rao-Moody representations can not arise in QFT.

  12. Pingback: A semana nos arXivs… « Ars Physica

  13. Jonathan says:

    Were you in Shanghai for the eclipse? I made it West to Wuhan and saw a few fleeting glimpses, a few km South West had perfect skies, but such is life. Third eclipse, third cloudy eclipse – I’ll get my corona yet!

  14. Peter Woit says:

    Hi Jonathan,

    Yes, I was in Shanghai, which was cloudy, except for a few short glimpses of the partial phases (one just after totality). I can’t complain too much though: this was the fifth eclipse I’ve tried to see, the previous four had clear skies.

  15. Peter Woit says:


    I agree that the background gauge field needs to be quantized. But it appears to me that what you naturally are getting is not a representation, but a family of representations parametrized by the background field. Perhaps this can be quantized by a path integral sort of construction, basically integrating over the background fields.

    Interestingly, in geometric Langlands, Frenkel argues that you should think of the group as acting not on a representation, but on a category of representations.

  16. Thomas Larsson says:


    I strongly disagree, and much prefer the Rao-Moody class of representations. They cannot arise in QFT (their extension is proportional to the second Casimir rather than to the third), but we know that QFT must break down anyway since it is incompatible with gravity. The extra ingredient that must be added is the observer’s trajectory in (n-1)-dimensional space. This is secretely encoded in the n-1 null roots, defined in R-M’s Definition (3.3).

    Given that every experiment involves an observer, and none fundamentally involves classical fields, I believe that representations are more physical if they depend on the former rather than on the latter. But then again, my opinion might be colored by the fact that I have invested 15 years of my non-career into this class of representations.

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