Witten Geometric Langlands Talk and Paper

I spent yesterday afternoon down in Princeton, and attended a talk by Witten at the Institute on his work relating gauge theory and the geometric Langlands program. He says that his paper with Kapustin is done, it’s about 220 pages long, and will appear on the arXiv in Monday’s listings. So Sunday night, this link to hep-th/0604151 should start working. He’s also working on a book on the subject, where he would be the sole author.

At the start of talk, Witten noticed that many of the Institute’s mathematicians were there, and warned them that they had come to the wrong talk, since it was one aimed at physicists. Pierre Deligne got up and left, but others, including Sarnak and Langlands himself, did stay for the whole thing, although I’m not sure how much they got out of it.

Witten began by giving an outline of the talk, emphasizing six main ideas that were crucial to what he wanted to explain. He also listed as number zero the idea of geometric Langlands itself, saying he would talk about it at the end if he had time (he didn’t). The six main ideas were:

1. From a certain twisting of N=4 supersymmetric Yang-Mills one can construct a family of 4d TQFTs parametrized by a sphere. The twisting is the same sort that occurs in his original TQFT for Donaldson theory, in that case coming from N=2 supersymmetric Yang-Mills. The TQFTs he considers have an S-duality, part of a larger SL(2,Z) symmetry.

2. Compactifying the theory on a Riemann surface leads to topological sigma models, based on maps from the Riemann surface into the Hitchin moduli space MH of stable Higgs bundles. The four dimensional S-duality corresponds here to a mirror symmetry of these topological sigma models.

3. Wilson and ‘t Hooft operators of the 4-d gauge theory act on the branes of the topological sigma models. Branes mapped in some sense to a multiple of themselves by these operators are called electric or magnetic “eigenbranes” respectively.

4. Electric eigenbranes correspond to representations of the fundamental group, this is one side of the geometric Langlands correspondence.

5. The ‘t Hooft operators of the gauge theory correspond to the Hecke operators of the geometric Langlands theory although these are now defined on the space of Higgs bundles, not G-bundles.

6. Using a certaing co-isotropic brane on MH, magnetic eigenbranes give D-modules on the moduli space of bundles. The electric-magnetic duality coming from S-duality in the gauge theory relates electric and magnetic eigenbranes, giving the geometric Langlands duality between representations of the fundamental group of the Riemann surface in the Langlands dual group, and “Hecke eigensheaves” on the moduli space of G-bundles.

By the time he got to the 6th of these ideas, he was running out of time and things got very sketchy.

Witten made clear that this work doesn’t directly give dramatic new physics or mathematics, but rather just explains some tantalizing relations between gauge theory and Langlands duality, ones that were first noticed in work of Goddard, Nuyts and Olive in 1976, and pointed out to him by Atiyah way back then. The geometric Langlands program is famous among mathematicians for its difficulty (I still have trouble getting my brain around the concept of a Hecke eigensheaf..), and for its tantalizing nature, bringing together a range of different mathematical ideas (many involving conformal field theory, although these seem to be different than what Witten is doing). The new relations between this subject and supersymmetric gauge theory and TQFTs that Witten has unearthed may very well lead to some very interesting new mathematical developments in the future. Undoubtedly it will take people a while to make their way through the new 220 page paper and absorb all that he and Kapustin have worked out since last summer.

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105 Responses to Witten Geometric Langlands Talk and Paper

  1. dan says:

    Has Witten made any reviews or comments or written any papers on background independent quantum gravity, esp LQG, spin foam, or CDT approaches? Has he shown any interest in these approaches?

  2. woit says:


    No, and those topics have nothing to do with this posting.

  3. bigbigbig says:

    dear experts and crackpots,

    about the 2d QFT or string theory argument, yeah, I think it’s only fair to call it string theory once you couple the topological sigma model to 2d gravity.

  4. Tony Smith says:

    Here are four questions that came to mind after initially looking at the Kapustin-Witten paper. I apologize in advance for their simple-mindedness.

    Kapustin and Witten say:
    “… N = 2 super Yang-Mills theory can be twisted to make a quantum field theory realization of Donaldson theory …
    Similarly, N = 4 super Yang-Mills theory can be twisted in three ways to make a topological field theory. Two of the twisted theories … are closely analogous to Donaldson theory … The third twist … turns out to be the twist relevant to the geometric Langlands program, and we will call it the GL twist. …
    A hyper-Kahler manifold has … a family of complex structures parameterized by CP1, obtained by taking linear combinations of I, J, and K …[with]… quaternion relations … “.

    Donaldson, in Bull. AMS 33 (January 1996) 45-70, said:
    “… in the latter part of 1994 … the work of Witten and Seiberg … used … N = 2 supersymmetric Yang-Mills theory … The essential ingredient … is … a duality between electricity and magnetism proposed in 1977 by Olive and Montonen. …”.

    Question 1:
    Is it fair to say that Seiberg-Witten Donaldson theory is based on complex (N=2 SYM) electricity-magnetism duality, and Kapustin-Witten Langlands theory is based on extending that duality to the quaternionic (N=4 SYM) case ?

    Kapustin and Witten also say:
    “… N = 4 super Yang-Mills is most easily constructed by dimensional reduction from ten dimensions. Ten dimensions is the maximum possible dimension for supersymmetric Yang-Mills theory … the metric of ten-dimensional Minkowski space R1,9 or Euclidean space R10 … have symmetry groups SO(1, 9) or SO(10) … To reduce to four dimensions, we simply take all fields to be independent of the coordinates x4, . . . , x9. The components AI , I = 0, . . . , 3 describe the four-dimensional gauge field … while the components AI , I at least 4, become four-dimensional scalar fields … the symmetry is really Spin(1, 9) reduced to Spin(1, 3) x Spin(6) (or Spin(10) reduced to Spin(4) x Spin(6)). The group Spin(6) is isomorphic to SU(4) and is known as the “R symmetry group” of the theory. We will call it SU(4)R. …”.

    Questions 2 and 3:
    Isn’t that SU(4)R the conformal group used by Jackiw and Rebbi in Phys. Rev. D 14 (1976) 517 entitled “Conformal properties of a Yang-Mills pseudoparticle” ?

    Might it not be useful to extend the quaternionic approach of Kapustin and Witten to octonionic structures based on the approach of Grossman, Kephart, and Stasheff in Commun. Math. Phys. 96 (1984) 431-437, where they said: “… In the present paper we will study the properties of the last fundamental Hopf map (i.e. the Hopf fibration of hte 15-sphere …) in relation to solutions of pure eight dimensional Euclidean Yang-Mills field equations with gauge group Spin(8). … The solution is invariant under the action of Spin(9) … which is a subgroup of the Euclidean conformal group O(9,1) in eight dimensions. … Let us proceed in analogy with Jackiw-Rebbi …” ?

    Question 4:
    As to octonions, in math.RA/0105155 John Baez said: “… the canonical octonionic line bundle over OP1 generates Bott periodicity …”.
    What might be the role of Bott periodicity in an octonionic generalization of the Kapustin and Witten paper ?

    Tony Smith

  5. Aaron Bergman says:

    Is it fair to say that Seiberg-Witten Donaldson theory is based on complex (N=2 SYM) electricity-magnetism duality, and Kapustin-Witten Langlands theory is based on extending that duality to the quaternionic (N=4 SYM) case ?

    Not really. The duality is most present in N=4 SYM. Seiberg-Witten theory is, in a sense, the residue of that duality in the N=2 case. When you do a topological twist of N=2 SYM, you get a theory that computes Donaldson invariants. Applying the Seiberg-Witten results on N=2 SYM to this twisted theory gives rise to Seiberg-Witten invariants of 4-manifolds. There are a number of different twists of N=4 SYM. The ones related to Donaldson invariants were investigated by Vafa and Witten. This is a different twist. Nonetheless, all of this is related to the S-duality of N=4 SYM.

    I don’t know what sense SU(4) is a conformal symmetry. The conformal symmetry in N=4 SYM in 3+1 D is PSU(2,2) ~= SO(4,2). The full symmetry supergroup is PSU(2,2|4). The SU(4) is the R symmetry, not the conformal symmetry.

    I’m not sure the relevance of quaternions, much less octonions, to this whole thing. You can’t add more supersymmetry to N=4 SYM, regardless, as that would entail particles with spin greater than one, and it wouldn’t be a gauge theory any more.

  6. urs says:

    all of this is related to the S-duality of N=4 SYM.

    Which again is thought to come from the modular transformations of a torus on which you have compactified some 6D theory to 4-dimensions.

    Do you know if the Langlands thing has a nice interpretation in these six dimensions?

  7. Aaron Bergman says:

    That’s certainly a question that has occurred to many people. I can’t remember if the relevant twist can be thought of as coming from six dimensions, however.

  8. Who says:

    In response to Dan’s question Has Witten made any reviews or comments or written any papers on background independent quantum gravity…?
    there is negative comment on the (Kodama) QG ground state employed by Lee Smolin and others. see this article:
    A Note On The Chern-Simons And Kodama Wavefunctions
    Edward Witten
    Exerpt from abstract: “Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction. … It is known to be unphysical … Similar properties can be expected for the analogous Kodama wavefunction of gravity.”

    Exerpt pages 1 and 2:
    “…This is what we call the Chern-Simons state. It is far from being normalizable,..
    The Chern-Simons wavefunction of Yang-Mills theory has an even more surprising gravitational analog, commonly called the Kodama state [2]. Some authors have proposed the Kodama wavefunction as a starting point for understanding the real universe; for a review and references, see [3]. Our discussion here will make it clear how the Kodama state should be interpreted. For example, in the Fock space that one can build (see [3]) in expanding around the Kodama state, gravitons of one helicity will have positive energy and those of the opposite helicity will have negative energy.


    [2] H. Kodama, “Holomorphic Wave Function Of The Universe,” Phys. Rev. D42 (1990) 2548.
    [3] L. Smolin, “Quantum Gravity With A Positive Cosmological Constant,” hep-th/0209079.

    Lee Smolin just finished given a course to University of Waterloo students called Introduction to Quantum Gravity using the cited article as his main text. The course has 25 lectures, video is available online, and it goes thorough much of hep-th/0209079 section by section.

    For a modern view of the LQG ground state
    Graviton propagator in loop quantum gravity
    Rovelli et al.
    this cites Smolin on the Kodama ground state as reference [38] on page 17.
    It is still not clear how generally Witten’s apprehensions concerning the LQG ground state apply, if they do. For example see page 2 of the Rovelli et al paper:
    “…The physical correctness of these theories has been questioned … two “bad” terms: an exponential with opposite sign,… and a dominant term … We show here that only the “good” term contributes to the propagator. The others are suppressed by the rapidly oscillating phase in the vacuum state that peaks the state on its correct extrinsic geometry. Thus, the physical state selects the “forward” propagating [33] component of the transition amplitude. This phenomenon was anticipated in [34]…”

    so I would amplify what Peter said, in reply to Dan.
    YES Witten has published comment on LQG (in particular the notion that proposed LQG ground state might have forwards and backwards propagating gravitons—something addressed recently in a different context by Rovelli), however as Peter rightly points out this is NOT GERMANE TO THE POSTING.

  9. urs says:

    I can’t remember if the relevant twist can be thought of as coming from six dimensions, however.

    At least in 6D there are candidate strings which one could hope would have a topological twist.

  10. urs says:


    if you ever feel like saying more about that autofunctor on the category of branes and how it comes from line operators in a fashion considered in FRS formalism (as remarked by Witten/Kapustin on p. 100), please don’t hesitate. 🙂

    (I will be awfully busy the next days. After that I might try to have a look at this.)

  11. Aaron Bergman says:

    Dunno much about FRS. Sorry. Guess you’ll have to wait for the book.

  12. Tony Smith says:

    Aaron Bergman said that he doesn’t “… know what sense SU(4) is a conformal symmetry. The conformal symmetry in N=4 SYM in 3+1 D is PSU(2,2) ~= SO(4,2). … The SU(4) is the R symmetry, not the conformal symmetry. …”.

    SU(4) is the Euclidean version of conformal symmetry, since SU(4) = Spin(6) = Euclidean version of the conformal group.
    If you use nonEuclidean signature, the relevant conformal group is SU(2,2) = Spin(2,4).

    As to their use of the Euclidean SU(4) = Spin(6) version of the conformal group, Kapustin and Witten say:
    “… The twisted theory is a formal construction in the sense that twisting violates unitarity and only works in Euclidean signature. … To construct a twisted theory in Lorentz signature, we would have needed a suitable homomorphism Spin(1, 3) into Spin(6). Because of the compactness of Spin(6), a non-trivial homomorphism does not exist. If one replaces Spin(6) by Spin(1, 5) to work around this, the couplings to fermions will no longer be hermitian and the energy is no longer bounded from below. …
    however …
    we split off the time direction and take M = R x W, where R parametrizes the time, W is a three-manifold, and
    the metric on M is a product. Then time is not involved in the twisting, and the twisted theory makes sense with Lorentz or Euclidean signature and is unitary and physically sensible. …”.

    In other words, Kapustin and Witten transfer the signature complications from the conformal group to the manifold by continuing to use the Euclidean conformal group SU(4) = Spin(6) but changing the manifold from something like S4 (where time is on an equal footing) to something like S1 x S3 (where time is S1 and space is S3).

    Aaron Bergman also says that he is “… I’m not sure the relevance of quaternions, much less octonions …”.

    Perhaps octonionic structure might be relevant to Loop Operators. Kapustin and Witten say:
    “… What is really unusual about the four-dimensional TQFT’s … compared to other theories with a superficially similar origin, is that they admit operators that are associated to oriented one-manifolds …
    In gauge theory, the most elementary loop operator is the Wilson loop operator. ,,,
    In our problem, the dual of a Wilson operator is an ’t Hooft operator … “.

    Since the imaginary octonions generate the 7-sphere S7, and since S7, although not a Lie algebra, is a Malcev algebra (having no Jacobi identity, but instead a Moufang identity) with naturally occurring Moufang loops, the ideas in the paper by Loginov at hep-th/0109206 might be useful. Loginov says:
    “… the linear representations of analytic Moufang loops … are closely associated with the (anti-)self-dual Yang-Mills equations in R8. …”.

    Tony Smith

  13. Aaron Bergman says:

    The Euclidean version of the conformal group is SO(5,1). su(4) ~= su(6) is the compact form of the complexification of so(5,1). I’ll leave the rest for someone else.

  14. Michael says:

    you do know that SUSY gauge theories are intimately related, in many cases exactly equivalent, to certain string theory backgrounds, don’t you? SUSY gauge theories “know” automatically about extra dimensions and many other things. If you can draw a reasonably clear line between SUSY gauge theories and string theory, do that and publish it. If not, you better stop claiming that the “new work of Witten has little to do with string theory”.

  15. woit says:


    Sure, the same theory Witten is twisting to get his TQFT (N=4 supersymmetric YM), is related to string theory on AdS space. But that plays zero role in the Kapustin/Witten paper. This may be the only paper in recent years involving N=4 supersymmetric YM that doesn’t reference Maldacena’s AdS/CFT paper.

    Feel free to point out where in Witten’s new work string theory plays an important role. Otherwise, stop making a fool of yourself.

  16. “The Euclidean version of the conformal group is SO(5,1). su(4) ~= su(6) is the compact form of the complexification of so(5,1).”

    My guess is that Tony is talking the Lie Algebra for the group and you are talking just the group and since Kapustin and Witten seem to be talking Lie Algebra, I’ll vote for Tony’s rendition even if you would have preferred Tony to be more precise. (I am of course biased and have plenty of trouble getting the general idea without worrying about things like signatures and the group vs. the algebra).

  17. Michael says:


    string theory provides a geometric picture that is *equivalent* to the quantum dynamics of the SUSY gauge theory. Whether you talk about one or the other is a question of terminology.

    Since you asked, let me give you two examples, which you missed because you either didn’t read or didn’t understand the paper. The first shows the importance of string theory to the present work, the second the importance of the present work to string theory.

    (1) The massless open-string states of the canonical co-isotopic brane furnish the brane’s endomorphism algebra. In perturbation theory, open strings form a sheaf of noncommutative algebras. These statements are at the heart of understanding the mathematical significance of eigenbranes. Rest assured that these facts would have remained obscure were it not for the authors’ extensive experience with “physical” string theories.

    (2) The Blau-Thomson construction of Wilson operators of TQFTs and their magnetic duals plays an important role in recent string theory research. References are given at the beginning of Section 6. The present work deepens our understanding of the construction.

  18. Aaron Bergman says:

    I’ll vote for Tony’s rendition

    You’re welcome to think whatever you want.

  19. woit says:


    I don’t doubt that the author’s experience with string theory taught them many things that were useful in this work, and I also don’t doubt that lots of facts about these kinds of TQFT are useful to to string theorists. But, neither of these makes string theory an important part of this work. By your logic, string theory is an important part of virtually anything anyone does these days in pure QFT, since it is likely they learned some facts about QFT by thinking about string theory, and it is likely that most anything about QFT will have some application to string theory somehow somewhere.

    If you want to feel better about the failure of string theory by going on about how gauge theory and 2d TQFT is really string theory, no one’s going to stop you, but it is kind of silly.

  20. anonymous says:

    By your logic, string theory is an important part of virtually anything anyone does these days in pure QFT

    Peter sees the light! 😉

  21. Tony Smith says:

    It seems necessary for me to clarify terminology and notation.

    Lorentz group = Spin(1,3) of Minkowski space

    anti de Sitter group = Spin(2,3) = Sp(2) used in supergravity

    Conformal group = Spin(2,4) = SU(2,2)
    It is the group Spin(p+1,q+1) of which the Lorentz group Spin(p,q) = Spin(1,3) of Minkowski space is a subgroup.

    I used the term Euclidean Conformal group in the sense that the 6-dim space over which the Spin group acts has Euclidean signature (0,6), so that in my terminology Euclidean Conformal group = Spin(6) = SU(4).

    I think that in Aaron’s terminology Euclidean Conformal group is taken to mean Spin(1,5) in the sense that it is the group Spin(p+1,q+1) of which the group Spin(p,q) = Spin(0,4) of Euclidean 4-space is a subgroup.

    I hope that this clarifies the terminology I was using. Even if you prefer Aaron’s terminology, and I think that it is fair for you to do so, the meaning of my comments should now be clear.

    Tony Smith

  22. Kea says:

    Peter sees the light!

    I must say, I agree with anonymous. Come on, Peter! Why not start calling yourself a String theorist? It’s just a name. Then we could stop having these silly arguments.

  23. Chris Oakley says:

    I feel that I ought to now apply for a grant from the Templeton Foundation. Using what title, I wonder? Let me see: “0-dimensional, non-supersymmetric string theory in 3+1 dimensions”.

    No, too boring. They’ll fall asleep as soon as they open the envelope. How about “Searching for God using 0-dimensional, non-supersymmetric strings in 3+1 dimensions?”

    Better, but I still don’t think that they’ll go for it. I know! “Searching for remnants of the Primordial Sneezing by studying 0-dimensional, non-supersymmetric string theory in 3+1 dimensions”. That might do it! Of course, I’ll mention God if I have to.

  24. urs says:

    Why not start calling yourself a String theorist?

    I predict that in the future everybody will be a string theorist for 15 minutes.

    More seriously, it seems to be true that string theory is like a large dictionary of field theories. Many subtle things about field theories find their explanation when regarding their embedding into string theory. In the case under discussion here, it is not only AdS/CFT. There is an embedding of 4D SYM into a 6D theory which is believed to explain the S-duality of the SYM. That 6D theory is some theory of strings on 5-branes. (I guess that’s in the end the more relevant embedding for the Langlands program. But what do I know.)

    Fields and strings are tightly interwoven. Many feel that this suggests that strings play a role in nature. This is often refered to as the “qualitative” agreement of string theory with observation. And it’s true.

    Quantitative agreement is a different issue, as readers of this blog might have heard somebody say before.

    While it is only fair to admit the phenomenological problems of the idea of string unification, it is also only reasonable to acknowledge the usefulness of strings for understanding issues in field theory.

  25. Zelah says:

    Here is my piece regarding what Witten et al has wrought!

    Whatever you think of Mr Witten’s acheivements in regards to phyiscial predictions, as a leader of physics he is our Grothendieck.

    Like Grothendieck, he did not actually reach the promised land, but revolutionised the way EVERYONE THOUGHT ABOUT MATH/PHYSICS.

    Grothendieck showed mathematicians how to be CREATIVE yet rigourous! Witten is now showing physicists how to UNDERSTAND the world, without experimental data!

    I think that Mr woit has constantly underestimated this problem of lack of expermental data for the high energy community since the 1980’s. Mr Witten’s brilliance was to create a physics program enlightened enough that some the worlds best minds like Maldecena and Brian Greene became physicists and not say molecular biologists!

    Reading Mr Witten latest efforts are truely awe inspiring! Even thought I understand maybe 1% of what was going on, The way Witten illuminates areas of Mathematics I was always fascinated by in with such ease is magical to observe.

    Finally, it is obvious to me that String Theory is changing everyone way of thinking regarding QFT! The reason for this is SUSY, which is natural from a Stringy point of view, but completely unmotivated from a QFT point of view! SUSY/Strings is allowing theorists to examine models which are intractable without these insights! Seiberg_Witten is unthinkable without these insights!

    This is the point. How does one think about SUSY concepts starting from QFT? If you Mr Woit can generate INTERESTING theoretical physics models from pure QFT, please feel free to publish.

    An amateur mathematician.

  26. Thomas Larsson says:

    This is the point. How does one think about SUSY concepts starting from QFT?

    As something which for all practical purposes has already been disproven by experiments. We just wait for the LHC to be the last nail in the coffin 🙂

  27. Chris Oakley says:

    The position of supersymmetry as a physical phenomenon is summarised by the plight of the Black Knight here.

  28. Michael says:

    Not much remains of your original statement. You admit that tools from string theory were important. You also admit that the results can be important in the further development of string theory. You know, and didn’t debate, that there is an exact formal equivalence between the field theories at hand and certain string backgrounds.

    Clearly, then, the “new work of Witten has little to do with string theory”, right? Look, you are being pathetic once again.

    “[…] string theory is an important part of virtually anything anyone does these days in pure QFT, since it is likely they learned some facts about QFT by thinking about string theory, and it is likely that most anything about QFT will have some application to string theory somehow somewhere”

    You said that like a real prodigy. Of course, that is most likely exactly the point. Let me be clear about this. Absent SUSY, it is not clear what we can learn by realizing the field theory as a string background. The problem is the limit which we must eventually take to decouple the stringy modes and isolate the gauge dynamics. SUSY is the only thing that guarantees no phase transitions can occur in taking this limit. There are three ways out: either we learn that SUSY exists in nature, or we learn how to master the difficulty I mentioned, or both. Giving up? Only Peter Woit could suggest that…

  29. woit says:


    Lots of tools developed by people doing string theory will get used in other non-string theory contexts. This doesn’t turn non-string theory into string theory. Lots of tools developed by people doing non-string theory will get used in string theory. This doesn’t turn non-string theory into string theory.

    The fact that N=4 SYM has a string dual is of course interesting, but this doesn’t explain the relation to geometric Langlands that Witten is investigating. If you know otherwise, write a paper about it and it will make your career.

  30. Kea says:


    It doesn’t matter what you say! If it’s interesting and has anything at all to do with graded algebras, well of course it must be String theory. You wouldn’t have to be a real String theorist. You could be a “String theorist”.

  31. Anon says:

    Yes Peter, don’t you understand! If the paper has any Greek letters (translation for Michael to understand: “squiggly symbols”) it has to be string theory. See, just like the argument Michael pointed out earlier, this makes your entire blog useless. Way to go Michael!

  32. Benni says:

    sometimes ago woit wrote commenting Douglas talk at solvay where he wrote about the nichtmare of stringhteory:

    “under this circumstances, it is standard scientific practice, to acknoledge that [string theory] is a failed project and go on something else”

    It will be interesting if Witten will continue to publish String papers or if he leaves the field.

  33. Chris Oakley says:

    It will be interesting if Witten will continue to publish String papers or if he leaves the field.

    It looks as though EW has not been working on String theory in the proper sense in the last year or so anyway.

    BTW, If the Superstring locomotive hits the buffers I do not think that EW can be blamed. He just does what interests him, and however much he would like to be called a physicist the fact is (in my book, at least) that he is a mathematician. Physics may provide the raw material, but what seems to trip his trigger is playing around with the ideas in a highly abstract way. I do not sense, and never have sensed an urgency in bringing any of this hyper-theoretical stuff to market.

    No, the problem is the personality cult that has been built up around him, something he seems to be uncomfortable with himself, but has occurred because of the absence of any real individualism, originality or creative ability amongst the majority of theoretical high-energy physicists practising today.
    I think that the psychology is just this: if someone is demonstrably better at something that oneself, then putting them on a pedestal is a way of dealing with one’s inferiority.

  34. Moeen says:

    For anyone interested, it seems Edward Witten will be speaking about this topic again at the Conference in Honor of the 60th Birthday of John Morgan. An abstract for the talk is available here.

  35. woit says:

    Thanks Moeen,

    I should have mentioned that Witten will be speaking about this to mathematicians here next Thursday at the John Morgan conference. There will also be quite a few other interesting talks here as part of that conference, and I’ll try and write something about them here after the conference is over.

  36. Bert Schroer says:

    if you want to evaluate a person’s contribution to particle physics you must consider also older contributions. What you say about EW certainly does not take into account his pre-Atiyah contributions.
    When physics meets mathematics on a profound level, success is not always guarantied. A story with a very happy end developped when Born had only the diagonal part of the p-q relation and looked for a mathematically talented young collaborator to understand the rest. He first asked Pauli who arrogantly rejected the offer with the warning that Born’s mathematical inclination could spoil the immediate intuitive grasp of Heisenberg on the new mechanics (in other words: don’t meddle with Heisenberg). He then asked Jordan who hardly went to any physics course (because the tended to be in the early morning hours) but who was helping Courant with some chapters of the famous Courant-Hilbert mathematical physics book. This story had a very happy end indeed; not only did Jordan compute the off-diagonal part, but he also contributed the first avatar (yet still somewhat confused) glimpse into QFT and in the subsequent 3-man paper the three (Born, Jordan and Heisenberg) harmonized perfectly together. In this case you did not have to wait for history to tell.

  37. Chris Oakley says:


    Was it Dirac who said that that was a time when second-rate people could do first-rate work? None of the people you mention were second rate, but the same would not apply today. Nowadays, there are no obvious easy wins (not that I can see, anyway) and part of the problem is that some first-rate people have done second-rate work and been rewarded with Nobel prizes.
    Anyway … as regards Ed Witten … I have the highest respect for him and although I am no expert I am reliably informed that his early work was outstanding. The problem is with his fan club. Physics research, like most frontier areas, needs a small number of independent thinkers, not these hundreds or thousands of sheep.

  38. Bert Schroer says:

    you said it. Instead of the great traditional schools which cultivated not only disputes between them but also encouraged criticism within, you now have these globalized monocultures with a guru at the helm. It is not that people are less intelligent but rather that the Zeitgeist of globalized capitalism prevents them from reaching their true potential. A particularly impressive illustrative example is Lubos Motl whos crap he writes about anything non stringy serves a useful sociological purpose. String theorists love him, because reading that stuff spares them the toil to seriously look at other things. It works both ways, because in this way he becomes promoted and being close to the hegemon the danger that he will ever be confronted with his past crap is virtually zero; it is a win-win situation in which only physics looses.

  39. Tony Smith says:

    Bert Schroer said “… Instead of the great traditional schools which cultivated not only disputes between them but also encouraged criticism within, you now have these globalized monocultures with a guru at the helm. …”.

    That is clearly (to me) true. A question is Why are things different now?

    Take, for example, Sommerfeld, a student of Lindemann, who was a student of Felix Kelin.
    Lindemann’s students in addition to Sommerfeld included Hilbert and Minkowski.
    Sommerfeld’s students included Bethe, Heisenberg, Pauli, and Stueckelberg.

    (List from http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=31357 and Silvan Schweber’s book “QED and the Men Who Made It” (Princeton Un. Press 1994) which says at page 667: “… Stueckelberg … obtained his Ph.D. with Sommerfeld in 1927 …”. )

    Can you imagine a more independent group of brilliant minds ?

    Although all of them go back to Felix Klein, and I am sure that they all respected Klein and his work, at no stage in the family tree Klein – Lindemann – Sommerfeld – Bethe, Pauli etc did ANYONE set themselves up as a “guru” to a herd of sheep.
    Even though there was NO “guru”, space does not permit listing the accomplishments of those people.

    It is interesting to compare the math/genealogy of the current conventional superstring guru, Ed Witten:
    Ed Witten studied under David Gross (another student of Gross was Frank Wilczek);
    David Gross studied under Geoffrey Chew (Gross was Chew’s only PhD student);
    Geoffrey Chew studied under Enrico Fermi.

    Look at their records. Fermi was brilliant.
    Chew, with his S-matrix bootstrap program, became a guru shepherd with many sheep, only to see the program fail, being eclipsed by quarks and the Standard Model.
    Gross and Wilczek published the details of asymptotic freedom more or less simultaneously with Politzer, but all of them had been anticipated by ‘t Hooft who had already announced at a meeting (but did not publish in detail) the key idea.
    Witten (like Gross and Wilczek) has done some very interesting work, particularly math-oriented, but (like Chew) has become a guru with a herd of sheep, being the guru of superstring theory which (according to Distler et al hepph0604255) “… is constructed to produce an S-matrix with precisely these properties. … analyticity … unitarity … Lorentz invariance …”, which is very reminiscent of the Chew bootstrap.

    The difference between those groups that stands out to me is that
    Klein – Lindemann – Sommerfeld – Bethe, Pauli etc were anchored in European culture,
    Fermi – Chew – Gross – Witten were, since Fermi came to the USA, anchored in the USA.

    You might extend the Klein – Lindemann – Sommerfeld – Bethe line to the USA with Bethe, and then you might claim Feynman (not a PhD student of Bethe, but certainly heavily influenced by contact with him at Cornell) for that line. However, I don’t think that anyone would see Feynman as typical of USA physicists.

    If you try to extend the Fermi – Chew – Gross – Witten line, you see that the most prominent protege of Witten is Harvard Professor Lubos Motl.

    Is there really a cultural difference (European v. USA) that accounts for the guru-sheep phenomenon in today’s high-energy theoretical physics ?

    If not, then why the difference between the
    Klein – Lindemann – Sommerfeld – Bethe, Pauli etc line
    and the
    Fermi – Chew – Gross – Witten line ?

    Tony Smith

  40. Bert Schroer says:

    as a small footnote one should add the role of Kurt Symanzik in preparing the soil of asymptotic freedom discussion. He started the renaissance of perturbative short distance investigations in terms of parametric differential equations featuring coupling-dependent functions beta and gamma (Kallen-Symanzik equation). That signs in those functions are hugely important he illustrated in the pedagogical but unphysical model of a quadrilinear scalar selfcoupling with changed sign of the coupling term and it was immediately clear to Sid Coleman that all the physical sign of all standard couplings did not permit Symanzik’s searched “holy grail” of short-distance weakening of interactions. Sid, who (in the best tradition of Pauli) was a brilliant art critic before his health failed, had a deep admiration for Symanzik and he popularized his program in the US. Parisi in Rome got attracted to it and after ‘t Hooft’s thesis on renormalization of Yang-Mills couplings, Symanzik prodded ‘t Hooft to look also into that sign question. At a conference in Marseille ‘t Hooft finally told Symanzik that the model seemingly provided his looked for exception, but he did not announce this publically (probably because he wanted to cross check the nontrivial calculation).
    It is interesting that Symanzik in later years on several occasions pointed out that since the asymptotic freedom calculation is done in the wrong phase (i.e. not with a parametric differential equation in terms of a physical mass parameter in the confined phase), the asymptotic freedom statement has the status of a consistency check and not of a structural theorem. In this repect the situation in certain two dimensional models (e.g. Gross-Neveu) is better.
    With all appologies to Peter for having somewhat strolled away from his given theme.

  41. Bert Schroer says:

    Sorry for misspelling Callen’s name

  42. Tony Smith says:

    Two additional details:

    1 – Curt Callan (of the Callan-Symanzik equations) was Peter’s advisor at Princeton; and

    2 – Since Symanzik studied under Heisenberg at Goettingen, he might be considered part of the line
    Klein – Lindemann – Sommerfeld – Bethe, Pauli, Heisenberg, etc

    Tony Smith

  43. Aaron Bergman says:

    I know I shouldn’t respond to such silliness, but can you tell me any sense that Lubos is “the most prominent protege of Witten”? They’ve never even been at the same university. Witten also has had a number of students who, oustide this bizarre little internet bubble we all inhabit from time to time, are much more prominent than Lubos.

  44. Tony Smith says:

    Aaron Bergman says “… Witten … has had a number of students who … are much more prominent than Lubos. …”.

    According to http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=31293 Witten has two PhD descendants:

    1 – Dror Bar-Natan ( web page at http://www.math.toronto.edu/~drorbn/LOP.html ) who has written many nice papers on knots, links, tangles, cobordism, etc., as well as some very interesting papers on Torah codes and equidistant letter sequences in War and Peace. I like his work very much, but to me it seems that his most important work is math and not physics.

    2 – Scott Axelrod, who was at one time an Assistant Professor in the math department of MIT (resigned around 1998).Searching the arXiv in math and physics, I did not find any papers by him since 1995. More recently, he seems to be at IBM doing interesting work on speech recognition, natural language generation, etc. He has in the past done interesting physics and math work, but his current work, while also very interesting, does not seem to me to be physics.

    As to the prominence of Lubos Motl, he is an Assistant Professor of Physics at Harvard University, and may be the best-known advocate of conventional superstring theory. Not only is his blog well-known, and allowed to do trackbacks with arXiv, but he is a founder and moderator of sci.physics.strings.
    As to why I consider him to be a protege of Witten, Motl’s web page at http://schwinger.harvard.edu/~motl/sf/arxiv-nytimes.html quotes a 2001 NY Times article as saying:
    “… In 1996, when Mr. Motl was a 22-year- old undergraduate … at Charles University in Prague … Dr. Witten, perhaps the premier figure in string theory, astonished Mr. Motl by writing to congratulate him on his 1996 posting. Mr. Motl still refers to Dr. Witten, only half-jocularly, as “the flying god knowing everything.” …”.
    It seems likely to me that Witten has supported Lubos Motl througout his career – moving to the USA – getting a PhD at Rutgers and now a Harvard Professorship.

    Of course, there is a certain amount of subjectivity in choosing “the most prominent protege” of anyone.
    I have stated my reasons for choosing Lubos Motl as the most promient protege of Ed Witten in the context of physics.
    If I have any facts wrong, I am willing to stand corrected and reconsider.

    Aaron is, of course, free to disagree and state his choice and give his reasons therefore.

    Tony Smith

  45. Bert Schroer says:

    Kurt Symanzik as well as Harry Lehmann and Wolfhart Zimmermann started their joint innovative work at Heisenberg’s institute, but scientifically they were kind of revolutionary “young Turks” in fierce opposition to H. who at that time was working at another of those ill-fated TOEs (Weltformel) in the veil of a nonlinear spinor theory. Hence to count them into the same genealogy is only true in a superficial geographical sense. There was also a strong subconcious rejection (generation gap) of the older generation because they were blamed for the mess of the Third Reich. The few post-war physics innovators in Germany were youngsters who were forced from school directly into Hitler’s army and spend a short time as POWs (Lehmann was a POW in North Afrika, Kurt Symanzik in Marseille and H.-J. Borchers in northern Germany); they were usually a bit older when they returned and started their studies and they were certainly very highly motivated. Rudolf Haag, whom I consider most original, was scientifically 100% selfmade. He went as a schoolboy to the UK to visit his older sister exactly on the eve of the war. During the war he was interned in a Canadian POW camp and that is where he started to become interested in physics. Although none of these people can be easily placed into a genealogy, they started a “school” in the sense of creating a common coherent stock of knowledge and scientific culture. Detlev Buchholz, Klaus Fredenhagen and several others (including myself) have been formed in that school and the very strong and original group of mathematical physicists at the University of Rome (all in the math. department, Sergio Doplicher, John Roberts, Danielle Guido and Roberto Longo) are either directly or indirectly also related to it.
    But since the “market value” of such usually more long-range scientific investments in the new globalized physics scene (which determines economical survival) is not very high, you may very well wittness already the end of this genealogical line.

  46. If you take this thread into genealogy, please let me remember that I moved my page to the wikipedia
    The condition to add branches in the main page is to keep at a reasonable distance of a prized (Nobel, Fields, etc) physicist, but other branches can be built in the talk page, as a provision for the future(?).

  47. Peter says:


    The students of Witten you mention are just his students who were mathematicians. He has had quite a few physics Ph.D. students. One of the first was my fellow student at Princeton Jon Bagger, who is now at Johns Hopkins, and I think could be characterized as more of a phenomenologist.

    I don’t think it’s at all fair to pin the blame for Lubos Motl on Witten. Undoubtedly Witten encouraged him when he was starting out, as he encouraged many people, but Lubos was not one of his students and never worked with him. I doubt that he approves of the kind of ranting that Lubos has made his trademark in recent years.

  48. Aaron Bergman says:

    Tony, that page is for mathematical “descendents” of Witten. His physicist students include Eva Silverstein, Shamit Kachru and Cumrun Vafa (IIRC). Lubos is also far from “the best-known advocate of conventional superstring theory.” That would probably be Brian Greene and (god help us) Michio Kaku. So, what you’re basing your statement on is a single e-mail. That’s quite a stretch.

  49. JC says:


    (slightly offtopic)

    Besides Heisenberg, Jordan, Stark, etc …, what other physicists went through “denazification” after the war?

  50. Bert Schroer says:

    probably most of those (Jordan, Stark,…) who were not interned in Farm Hall (for some reason von Laue was also at Farm all, although he was not a member of the uranium club and he was even recollected as anti-Nazi by Einstein). But I am not sure about the extent of de-nazification. The physicists who where in Russian occupied East Germany like Manfred von Ardenne, probably did not go through that process. I have heard that Euler, the collaborator of Heisenberg, who was a communist, got so fed up with what he saw going on at the University of Leipzig that he volunteered for the airforce and died (he was shot down) soon after.

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