Chern-Simons-Witten, Some History

Lubos Motl is promoting a revisionist history of topological quantum field theory according to which it was all inspired by string theory. Unlike him, I was working on the subject at the time it was developed, and remember the history quite clearly. I’ve recently checked my memories against the literature, learning some more details of what happened back then. Here’s an outline of the history of TQFT (or at least of one small part of it, the part leading to Witten’s Chern-Simons theory):

1982: Witten comes up with a beautiful reinterpretation of Morse theory in terms of supersymmetric quantum mechanics, writing an extremely influential paper on “Supersymmetry and Morse Theory”, which is published in a math journal, the Journal of Differential Geometry.

Spring 1987: Atiyah conjectures that Andreas Floer’s new homology groups (inspired by Witten’s supersymmetry and Morse theory paper) are the Hilbert space of a QFT. There are two cases where Floer theory works: 1+1 dimensions where the observables of the QFT would count curves (later to be known as Gromov-Witten invariants), and 3+1 dimensions where the observables count instantons (Donaldson invariants). Atiyah conjectures the existence of two corresponding QFTs, and also notes that the new knot polynomials of Vaughan Jones might correspond to a QFT in 2+1 d. He talks to Witten about this and gives an amazing lecture at a conference at Duke explaining these ideas. Witten tries to find a supersymmetric QFT that will do what Atiyah wants, but initially doesn’t succeed.

Late 1987: Atiyah visits Witten again at the IAS and keeps after him about the TQFT idea. Witten finally realizes that things work if he uses a “twisted” version of N=2 supersymmetry.

February 1988: Two papers by Witten appear, one “Topological Quantum Field Theory” about the 3+1 d case, one “Topological Sigma Models” about the 1+1 d case. The second paper contains some vague speculation at the beginning about the relation of these “topological strings” to physical string theory, perhaps in some kind of “unbroken phase”. At the end it also contains a sketch of an attempt to get Jones polynomials by using a 3+1d TQFT that would couple together his 3+1 topological gauge theory with a topological sigma model on the worldsheet swept out by a knot in 3 dimensions moving through time. This doesn’t actually work.

Summer 1988: At a conference in Swansea, talking to Atiyah and Segal about Segal’s ideas about conformal field theory and “modular functors”, Witten realizes that the right theory to get Jones polynomials is a 3d QFT whose Hilbert space is the finite dimensional space of conformal blocks of a 2d WZW theory. He also realizes that one can think of the Lagrangian of this theory as being the Chern-Simons functional. His paper “Quantum Field Theory and the Jones Polynomial” appears in September. There’s not a word about string theory anywhere in it and he has completely abandoned the idea of relating Jones polynomials to topological sigma models.

I was in Berkeley at MSRI for the academic year 1988-89. In January there was a workshop there involving Atiyah, Bott, Witten, and many other mathematicians and physicists. Initially many of the mathematicians were a bit skeptical, but by the end Witten had convinced the skeptics that what he had made complete sense, and they were very impressed. In the summer of 1990 he was awarded the Fields Medal for this work.

New ideas about relations between branes, topological strings, and Chern-Simons appeared about ten years later, and that’s an ongoing story, one which Lubos conflates with what was going on in 1988-9 that got Witten the Fields medal. These are two completely different stories.

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24 Responses to Chern-Simons-Witten, Some History

  1. Anonymous says:

    DRL,

    I assume you understand the context in which the instrument is being offerred for introspection?:)

    It is important that, while the argument presented in this paper can only be conclusive in a full treatment including gravity, it also indicates where one should look for a quantum mechanical theory that encompasses gravity. It makes clear that any approach to quantum gravity should treat matter and spacetime as two manifestations of the same thing. An approach that puts matter on a spacetime will encounter the cosmological constant problem.

    So here we might find Robert Hellings currents comments on his blog hepful?

    He offers this link for consideration and for me Witten summarizes for us as well, in regards to Glast determnations.

    I would like to understand Robert’s position very well.

    Leading to the undertsanding of kinetic potential values would say to me that the evolution of this instrument, is a continued effort of understanding Mercuries orbits? IN the quantum perceptin reductionistic physics asks us to consider something else, so while the quantum ranger suggests one position, we have to ask about the other?:)

  2. D R Lunsford says:

    Oh priceless! I love the harpsichord reference!! Well done! So string theory fails the Bach test..

    “Which came first, the string or the plectrum?”

    -drl

  3. Quantum_Ranger says:

    Lubos – “regarding the predictivity of string theory: be sure that string theory predicts a precise value for the mass of the electron”

    The electron rest mass,symbolized me,as measured when its speed is zero relative to an observer?

    What came first, the string or the Plectrum?

    The precise value is based on the two paramiters of OBSERVATION and ELECTRON, one see’s and one cannot be seen!

    Defining the accuracy is like stating that string theory, predicts and proofs the unpredictable, using observable quantities?, but only if one subscribes to the notion that this occurs as a process of ‘Seeing’ without looking!

    Its ok if you dont look, but just incase anyone happens to be ‘thinking’ of looking, string theorists like to hedge their bets on every conceivable notion of WE TOLD YOU SO!

    As if.

  4. DMS says:

    “I don’t have any morons comparable to DMS and similar stuff.

    There is absolutely no way how could I ever lose this argument, and I am sure that you know it very well – unlike DMS, who probably really believes that your position is defendable. Of course, if someone has excrements instead of brain inside his skull, like DMS, he will never understand that maths and physics did not end in 1989.”

    Seems like I have touched a nerve 🙂

    But if one can dish it, one should be be prepared to take it.

    Well, I really don’t care what Prof. Motl thinks of me. I actually learn from his posts about interesting work going on string theory and I (seriously) thank him for it; it is a valuable public service.

    But I know enough particle theory (and some mathematics) to know where factual statements end and heavily biased opinions (“String theory is the mother of everything”) begin. I simply read the arguments and drew my own conclusions.

    DMS

  5. sol says:

    DRL,

    Oops, I posted last my last comment in the wrong spot:)

  6. D R Lunsford says:

    Sol,

    In the same post you imagine the Mona Lisa on a trampoline, then let it drop in a sort of passive-aggressive way that Einstein was senile. The juxtaposition of these fantasies struck me as ironic.

    Forgive me for believing that Einstein will yet have his day. I do NOT forgive you for joining the ranks of Einstein and Dirac bashers.

    -drl

  7. Thomas Larsson says:

    I can think of no better way in which to steer promissing physics students AWAY from string faerie fantasies than to have them read the blog and usenet rantings of Lubos M-
    I often wonder if he is some kind of LQG activist using obvious reverse psychology on his readers to repulse them from strings- becasue that is what happens most of the time!

    /:set\AI:

    This idea has also struck my mind. However, in my experience things are usually precisely what they seem to be. In fact, this is one thing that makes me think that string theory is wrong. String theory seems to disagree with experiments, and thus I believe that it actually does disagree with experiments. LM seems to be a rabid string zealot, and thus I believe that he actually is a rabid string zealot. But I guess that only shows how naive I am.

  8. Aaron says:

    The citations for Witten’s Fields Medal are available online here for the curious.

  9. Anonymous says:

    DRL,

    Pardon me Sol, but it’s hardly a discourse when factual evidence from a direct participant is attributed to cerebral intra-aural doo-doo. LM doesn’t do discourse, he barks like a carvinal shill, and he’s usually wrong to boot. You can’t have a discourse with a living scowl.

    This is a sideline issue about Witten, but there are more important things that are mathematically being describe, and they seem to have a place in the hyperdimensional reality.

    A picture crossed my mind today with the Mona lisa on the trampoline. I do not think it so absurd that the definition of Ramanujan’s modular functions point on the stringy world sheet less relevant then other topological considerations in that abtract space(I’m still learning).

    It had to be consistant with what we know, and accepting Einstein’s position in regards to the beauty of GR(smolin’s positon?), and attempts to explain these ideas on a quantum mechanical scale have been a most troubling issue to contend with.

    Einstein did it for the last thirty years of his life, and we would have those from the quantum mechanical perspective who would of thought Einstein Senile? Yet his concept was complete, yet he decides to contend with the issue of a quantum gravity perspective, for the rest of his life?

    Why I needed to understand whether Peter supports Smolin.

  10. D R Lunsford says:

    Pardon me Sol, but it’s hardly a discourse when factual evidence from a direct participant is attributed to cerebral intra-aural doo-doo. LM doesn’t do discourse, he barks like a carvinal shill, and he’s usually wrong to boot. You can’t have a discourse with a living scowl.

  11. sol says:

    Unfortunately I have to add my comment here not to slight any individual, but to encourage two indiviudals to continue the discourse they are having, about views of Witten, regardless of whether the fields medal was awarded for this or that.

    Those who intrude with comments, did not pay attention to the focus, on the landscape, instead focused, on the Mona Lisa’s mouth:)

    Maybe you like Salvador Dali’s’s version better?:)

    Previous, a discussion took place here in Peter’s Blog on Susskind and Smolin. I would like to know if Peter supports Smolin’s position?

    NATHAN MYHRVOLD

    I found the email debate between Smolin and Susskind to be quite interesting. Unfortunately, it mixes several issues. The Anthropic Principle (AP) gets mixed up with their other agendas. Smolin advocates his CNS, and less explicitly loop quantum gravity. Susskind is an advocate of eternal inflation and string theory. These biases are completely natural, but in the process the purported question of the value of the AP gets somewhat lost in the shuffle. I would have liked more discussion of the AP directly

    http://www.edge.org/discourse/anthropic.html#myhrvold

    The thing I like about the oppositon of minds who embrace the Solvay attitude, is that it forces another to bring forward a history that few of us would have seen. So outside of the comments of opposing views what kind of harmony could have been produced?

    So in following up, I thought it would have been a good opportunity to bring this debate forward again, in context, of opposing views?:) Why do you think Susskind wrong or Smolin wrong? Is there is a certain accpetance here, of what one chooses, that another view is recogized? A certain symmetry?:)Etc.

    SMOLIN VS. SUSSKIND: THE ANTHROPIC PRINCIPLE

    Leonnard Susskind and Lee Smolin

    While this is a conversation written by physicists for physicists, it should nonetheless be of interest for Edge readers as it’s in the context of previous Edge features with the authors, it’s instructive as to how science is done, and it’s a debate that clarifies, not detracts.

    http://www.edge.org/documents/archive/edge145.html

    Sorry for intruding, and all due respect to you both, Lubos and Peter.

  12. D R Lunsford says:

    Toutes-vous avez des excrements entre des oreilles! Vive les cordes! Vive les fonctionnes thetaines! Vive les blogs de savants-des-cordes! These things sound better in French! Voulez-vous des pommes frites avec cela?

    M. de Bois-Motile, Compte de Bourbaki

  13. Mubos Lotl says:

    Let me begin by reminding you all that you are all fools, morons, idiots, professors at MIT, monkeys, and have excrement between your ears. Right, with that out of the way, let me assure you that I am right about everything I have ever said, so resistance is futile, is not it? Finally, about all those taunts
    regarding the predictivity of string theory: be sure that string theory predicts a precise value for the mass of the electron, modulo a certain p-adic cohomology class which is of no interest because Andy Strominger already mentioned it in 1992 and so a computation of it would be entirely superfluous.
    Yours in eternal friendship,
    Mubos.

  14. Fabio says:

    “Of course, if someone has excrements instead of brain inside his skull, like DMS…”

    Years of debating usenet cranks have left poor Lubos with a wit as sharp as a plastic knife.

  15. /:set\AI says:

    oops- the last comment was by me- name got dropped from the post for some reason

  16. Anonymous says:

    I can think of no better way in which to steer promissing physics students AWAY from string faerie fantasies than to have them read the blog and usenet rantings of Lubos M-

    I often wonder if he is some kind of LQG activist using obvious reverse psychology on his readers to repulse them from strings- becasue that is what happens most of the time!

  17. Peter says:

    Hi Lubos,

    No, the Fields medal is not given for a single paper. Atiyah’s address to the 1990 ICM about the award to Witten explains what work of Witten’s mathematicians considered important at that time. String theory isn’t mentioned. I can assure you from my many conversations with mathematicians about this around that time that the Chern-Simons theory stuff was the deciding factor in awarding him the medal.

    Witten’s work on the the positive mass conjecture and Morse theory, both using supersymmetry, were also cited by Atiyah. But this work was done around 1980, long before Witten got interested in string theory. It really has nothing to do with string theory, unless you adopt the attitude that string theory encompasses everything.

  18. Lubos Motl says:

    I found a web page that explains, in detail, that he received the medal also for his proof of the positive mass theorem, using supersymmetry – a clearly superstring-inspired proof – and the relations between supersymmetry and Morse theory. In that work he used a supersymmetric quantum mechanical model – which is a typical stringy approach because all other (nonstringy) people who are interested in SUSY are focusing on d=4.

  19. D R Lunsford says:

    Bon-Motl has once again invoked “argumentum ad cellulosum”. My understanding is, if this happens 3 times in the same month, Einstein’s ghost will tear the universe a new nether region and we’ll all be sucked through it into the missing 11 dimensions, in a process called “Calabi YEOW”. I mention in passing that Halloween is near, and toying with the unseen world at this dicey time is just plain uncivilized.

    So, for God’s sake, let’s act before it’s too late. Millions of lives are at stake.

  20. Lubos Motl says:

    One more comment, Peter:

    This “Fields medal only” is a completely ridiculous game. By the way, can you show that the medal was given him for a specific paper? No one around is aware of it. I personally don’t think that it is important for anything, but it seems that even these statements about the nature of Fields medal – is something that you invented.

  21. Lubos Motl says:

    Dear Peter,

    I must admit that even though some of the contributors to my blog (comments) may be disappoitining, I don’t have any morons comparable to DMS and similar stuff.

    There is absolutely no way how could I ever lose this argument, and I am sure that you know it very well – unlike DMS, who probably really believes that your position is defendable. Of course, if someone has excrements instead of brain inside his skull, like DMS, he will never understand that maths and physics did not end in 1989.

    I don’t feel that it is quite appropriate for us to waste so much time with these idiots, but I must state it clearly, in a language that attracts enough attention, that this whole Peter Woit’s way of describing history of maths and physics is simply a result of the fact that Peter Woit, as a talented mathematician and physicist, died back in 1989, and unless he will be able to wake up, he will never be able to say anything reasonable about physics and maths after 1989.

    And be sure that physics of 1990s has brought many key new discoveries, much like physics of the 1980s.

    http://www-spires.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+WOIT%2CP+and+date+before+2000

    Best
    Lubos

  22. Peter says:

    Hi Lubos,

    As I said in my initial posting, the whole topic of the relation of string theory to mathematics is a huge and complicated one and I don’t have the time or energy to do it justice now. However, as I also said in my initial posting, the work that mathematicians in 1990 thought was worth a Fields medal was not work that came out of string theory. You claim that I am wrong about this, I’ve presented a factual history to back up what I said. If you have any facts to demonstrate that the work for which Witten was given the Fields medal came out of string theory, let’s see those facts. Show me a pre-1990 paper explaining Chern-Simons theory in terms of topological strings.

    Peter

  23. DMS says:

    When losing an argument,

    get personal

    “1989: Peter Woit loses any contact with theoretical physics for at least 15 years. This prevents him from seeing all new important connections between these subjects that were found between 1989 and 2004.”

    or move the goalposts,

    “Moreover, it’s just incredibly silly if someone judges the importance of ideas according to some fictitious links between a specific paper and some medal.”

    or, better still, change the subject

    “Moreover, this CS story is just one of tens of important examples how string theory – not just “field theory” – influences math. Another class of these influences is associated with mirror symmetry.”

    but never accept defeat.

  24. Lubos Motl says:

    I apologize that I must write it so explicitly, but it is sort of necessary.

    Your history is missing one important entry.

    1989: Peter Woit loses any contact with theoretical physics for at least 15 years. This prevents him from seeing all new important connections between these subjects that were found between 1989 and 2004. Some of these connections are described in my “revisionist” history. This lost contact with reality in science also implies that even in 2004, Peter Woit writes entries on his blog that are very far from being up-to-date. These entries may have been fair in 1989, but they don’t express the correct relation between the ideas as understood in 2004.

    Moreover, it’s just incredibly silly if someone judges the importance of ideas according to some fictitious links between a specific paper and some medal. If you care about these irrelevant medals so much, there are also different stories.

    In the spring 2004 or so, for example, Witten received the national science medal from the US president, and be sure that it was not mainly because of the only papers you want to see. No doubt, he is credited for many other discoveries that you don’t want to see, and most of us think that these other discoveries are more important.

    Today, it is also very important that the 3D CS-theory is the worldvolume theory living on the D-branes of the A-model, and it allowed completely new methods to calculate all these invariants – correlators of the Wilson lines etc.

    Moreover, this CS story is just one of tens of important examples how string theory – not just “field theory” – influences math. Another class of these influences is associated with mirror symmetry. I hope that you would not argue that even *mirror symmetry*, which has been converted to many rigorous statements in mathematics, has nothing to do with string theory.

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