Nekrasov on String Perturbation Theory

Nikita Nekrasov is giving a very interesting series of talks at the Jerusalem Winter School on the topic of “Introduction to modern covariant superstring theory.” The first of his talks was yesterday and is now on-line. In it he outlined the two main formalisms for superstring theory and discussed their advantages and drawbacks, while also giving a beautiful discussion of the quantization of the superparticle, and the use of twistor and pure-spinor methods in 10d super-Yang-Mills.

One of these two formalisms, the NSR formalism, uses supersymmetry on the world-sheet, with target space a usual (bosonic) space (i.e. 10d space-time). The advantage of this is that amplitudes are computed using a linear theory, supergravity on the worldsheet. One disadvantage of this is that spacetime supersymmetry is not manifest, only recovered after GSO projection. A very serious technical problem is that, while one ultimately wants to construct amplitudes by summing over spin structures and integrating over the moduli space, the formalism gives one for each spin structure an amplitude on the super-moduli space, not the moduli space (and these super-moduli spaces are different for different spin structures). In recent years D’Hoker and Phong have been able to deal with this problem for genus 2 (and they have some results for genus 3), but for higher genus how to consistently get amplitudes on moduli space remains an open problem. Note that the problem with these multi-loop amplitudes is not only that you aren’t sure they are finite, but you aren’t sure that they are even well-defined. Presumably this is purely a technical problem, not evidence of an inherent inconsistency problem with such amplitudes, but one can’t be sure of this until someone finds a way of resolving the problem.

The other formalism, the so-called Green-Schwarz formalism, uses a bosonic worldsheet, but takes the target space to be a supermanifold. This has the advantage of making space-time supersymmetry manifest, and avoiding the problem of integrating over super-moduli space, but it carries its own disadvantages. The world-sheet theory is now a highly non-linear, constrained theory, with both first-class and second-class constraints, constraints that Nekrasov describes as “hard to separate in a covariant way”. No one knows how to quantize this theory preserving super-Poincare invariance, so one typically uses a non-covariant gauge-fixing like light-cone gauge, something that runs into trouble at genus 2 or higher.

In recent years, Berkovits has been developing an improved version of the Green-Schwarz formalism, sometimes called the Berkovits formalism, and this is the main topic of Nekrasov’s lectures. Presumably Nekrasov will be discussing in his next two lectures how this works and some of the interesting problems with it, problems that he wrote a paper about a couple months ago, one which was discussed here. In his talk, Nekrasov seemed rather nervous that he would get into trouble because people might think he was raising the possibility of superstring perturbation theory being inconsistent. At one point he said that his “policy statement” was that he hoped that things could be made to work at any genus. He also seemed concerned in his talk yesterday about how his remarks might be reported today, saying:

There are really conc… well… I don’t want to call them conceptual problems because these days everything is recorded. If I say something is a conceptual problem, tomorrow there will be a blog on that, or a paper. So, there are some technical difficulties….

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34 Responses to Nekrasov on String Perturbation Theory

  1. MathPhys says:

    I think that this website has helped educate more people about the most recent developments in string theory (while keeping a sober view of its problems and shortcomings) more than any other resource on the internet, excluding Ginsparg’s archives.

    I think that this website is doing string theory as a discipline a great service. I doubt more than a fraction of the readers of this website read J Distler’s blog (for example).

  2. woit says:

    Thanks MathPhys,

    I’ve certainly noticed that a sizable fraction of the traffic here comes from machines at academic sites with “string” part of the machine name, and have also heard from several string theorists that, while they disagree with my point of view, they do find a lot of interesting material here.

    Recently there have been roughly 5000 people a day looking at this weblog, a number that really astounds me.

  3. Dear Peter and MathPhys,

    try to guess what error you have done that allows an intelligent reader, after going through all occurences of “MathPhys” on this blog, determine that you are the very same person. ūüėČ

    All the best

  4. MathPhys says:

    Dear Lubos,

    Would you be willing to put your money where your mouth is and bet anything above US$1,000 that Peter Woit and I are the same person?

    If not, would you be willing to apologize in public for insinuating that Peter Woit lacks the integrity that puts a stunt such as you proposed below him?

    I am willing to accept any method of verification by independent observers. I do not accept any bets below $1,000 because that makes them not worth my time.

    Thank you.

  5. Dear Peter or MathPhys,

    no, I don’t intend to make a bet with you because I don’t believe that you would pay me the money.

    All the best

  6. Peter says:


    You really are mad as a hatter, I can’t begin to guess what you have in mind here, do tell. I do know who MathPhys is, and no he’s not me. From his previous comment he has less of a sense of humor about your antics than I do.

  7. MathPhys says:

    Peter, Please! I can use the money!

    Lubos, It shouldn’t be too difficult to find someone that both of us accept who can administer the bet. I know at least one person right here that both of us trust. Or you can nominate anyone from your side that I know.

  8. Arun says:

    Looking forward to hearing about the technic-conceptu-al problems.

  9. WorldRen says:

    LuboŇ° Motl
    I visited your blog before using the link woit provided. But I don’t think I would visit your blog again.

  10. Peter says:

    Enough about Lubos, please! Unless he has an entertaining explanation of why he is sure I am MathPhys, I don’t want to hear any more about it. Further comments should be about higher order calculations in superstring perturbation theory.

  11. Dear “WorldRen”,

    indeed, I would appreciate if you’ll never come closer than 5 miles from my blog because I have roughly 10 times more of such WorldRens visiting the blog than what would be appropriate and their market value is heavily negative. Thanks! ūüėČ Incidentally, it is not hard to see that this is the first time that “WorldRen” appears in the physics blogosphere. What do you think is the conclusion?


    your comments about the stringy perturbative expansions are pathetic. I have explained you why it works in such a transparent way that a better than average high school student must have already understood the proof of the finiteness.

    Nikita is trying to get people’s attention and attract the people into the worldsheet because it is his world. Of course he will never say that there are conceptual problems with string perturbation theory because he knows that such a statement is unjustifiable, and if he said such a thing, everyone would be asking him what he means (and he would have nothing to answer) because – as we correctly observe – everything is recorded these days.


  12. David says:

    Peter, it creates a strange impression when on the one hand you bemoan the current dominance of string theory in particle physics (or at least the formal theory end of it) and on the other hand use your blog to publicise the work of string theorists. I can understand it when the purpose is to point out and discuss major problems with the program (landscape etc), but that doesn’t seem to be the case here. The admiring tone of your discussion of Nekraskov’s “very interesting series of talks” is hardly offset by the mention of the “conc-technical difficulties” bit at the end. Are you a closet string theory fan? (Perhaps analogous to a homophobe whose homophobia is really a reaction to doubts about his own sexuality…)

    In a previous post/comment you said that you thought LQG is advancing much more than string theory at present and that research efforts should be shifted from strings to LQG. Why is it then that you write admiring posts on string theory work but have never done the same for LQG work? (I know you mentioned the talks at Loops 05 but that isn’t the same.) Another area where interesting developments have occured in recent years, and which would have been natural for you to mention given you background in it, is lattice gauge theory. E.g. one of the developments in this area made the AIP’s `Top stories of 2005′ (; other developments include index theory on the lattice and have led to major progress on the very topic of your PhD work (see, e.g., hep-lat/0108009, hep-th/0407052). Is the reason why you don’t mention work in these areas that the people doing it are not high-powered or illustrious enough for your liking?

  13. Well I can think of three justifications for Lubos first comment here:
    1) A naive “social engineering” attemp to discover (for free) the identity of MathPhys

    2) A misread of an early intervention of MathPhys, where he appears as first commenter of a post titled “Hiding in the Mirror”. Lubos could had missed a previous apparition of this commenter some days before and thought that this one was the first one, then being a subtle play of Peter.

    3) I really thought a third reason, but I have forgotten it while I was writting this note.

  14. hobgoblin says:

    The issue is about finiteness, and the arguments on one side seem to consist of statements like ‘There’s no evidence that it isn’t finite and no reason to suppose that something magical will step in at higher orders and ruin everything,’ while the argument on the other side is ‘There’s no proof that we have finiteness at all genera and the assumption that we do have it without further evidence is wishful thinking.’

    A lot of the debate in theoretical physics seems to be like this lately. There’s an awful lot of politicization and taking sides going on, and that isn’t healthy for science. In fact, most of the side-taking seems to relate specifically to string theory (either ‘string theory is the best achievement of the human race since speech’ versus ‘string theory is a massive waste of resources that are better spent elsewhere’, or ‘the landscape is stupid’ versus ‘the landscape is the best achievement of the human race since string theory’).

    As for Lubos, he has been diagnosed as a narcissist.

  15. WorldRen says:

    In your posts, you try to publicize the most recent development in string theory and modestly criticize it at the same time. Well, the key problem with string theory is, of course, that there is no experimental evidence to prove it or disprove it yet. So string theory is special. I suggest everyone should be careful when making posts and comments in case not to mislead people. Maybe we need some humility in string theory.

    Dear LuboŇ° Motl,
    You seems to be confident. But if one becomes too confident, he/she may ignore others’ existence and may somewhat reluctantly trusts anything.

  16. Ben Compson says:

    Lubos, presumably you are going to apply for a faculty position at some point, then hope to be awarded tenure. Have you thought of the consequences of having written so many nasty comments and ad hominem attacks on this career path?

    Or do you think yourself so smart that that all your online activities will be overlooked?

  17. secret milkshake says:

    hobgoblin: I have a problem with what you wrote:
    1) such a diagnose requires some experienced psychiatrist
    2) who needs to be at least as smart as the patient.
    3) and who does carefull examination of the case history + interviews + tests with the patient.

    The results are not likely to be posted on a blog. To the point 1) and 2): I am not sure you can find somebody qualified, Dr. Lecter has been travelling.
    You better re-phrase your statement to something like this: “Lubos has been widely conjectured to be a piece of work”.

    Also, check out the most recent Motl, Vafa paper on arXiv; they are trying to drasticaly narrow the permisible lansdscape by introducing a new condition (by hand). If you ask me, the new hand-drawn line between the landscape and swampland on Fig.1 looks a lot like map of Czech Rep…

  18. woit says:

    Hi David,

    You’re probably right that I take too great an interest in following exactly what string theorists are up to. One justification I can give is that if I’m going to criticize them I should at least understand what they are doing, but sure, that’s not sufficient.

    I should try and make more clear what this blog is and what it isn’t. It’s my own take on what I for one reason or another take an interest in, it’s not in any way an attempt to make uniform judgements about what is or isn’t worth pursuing in theoretical physics. The fact that I don’t take enough of an interest in something to write much about it doesn’t mean it’s not interesting, it just means that for one personal reason or another it’s not something I’ve spent much time thinking about, virtually always simply due to a lack of time. There are many, many, many topics I wish I had more time to look into and think about.

    About LQG: I have written some about it, but one of the main reasons I haven’t written more is that it is an active, healthy subject that already has some really excellent expositors (such as Smolin and Rovelli). I encourage people to read what they have to say, and if quantum gravity is your main interest, you should be paying close attention to what is going on in LQG. But quantum gravity is not my main interest, particle physics is, so I’m mostly spending my time thinking about other things than LQG. While I think what the LQGers are doing is great and should be encouraged, I also want to draw attention to ideas closer to my own interests that are getting virtually none.

    About lattice gauge theory: Again, this is an important, healthy subject, with a lot of good work going on. But it’s not something I actively think much about these days. Some of my current interests grew out of first thinking a lot about spinor geometry in an attempt to find better ways to put spinors on the lattice. This is a problem I keep hoping to find time to come back to. If I ever do, I’ll probably be writing more about lattice theories here.

    As for my interest in string theory. First of all, while I do bemoan endlessly its current dominance, the formal theory end of it is actually the part that I think is worth paying some attention to. What seems to me really a complete waste of time are some of the phenomenological ends of the subject. Unlike “string phenomenology”, the more formal, mathematical ends may very well lead somewhere, at least in terms of producing spin-offs of good new mathematics or new ideas about quantum field theory. The Nekrasov talks are a good example. If you look at his second talk, which I just watched, it’s all about an interesting 2d qft, one with target space one of great geometrical significance, the space of pure spinors (=space of orthogonal complex structures). This is a nice QFT story, involving some spinor geometry that deserves to be much better known. Ignoring the string perturbation theory context, on its own it’s a nice piece of work on quantum field theory. Some of the other topics Nekrasov discussed in his first lecture involve again some beautiful ideas that are independent of string theory, especially the use of twistor methods to study Yang-Mills.

    One of my reasons for taking an interest in the multi-loop issue is purely idiosyncratic: I have friends and colleagues who have worked on this, and have enjoyed discussing their work with them and follwing this story over the years. Finally, I think it is also an excellent example of what’s wrong with how string theory is often being pursued. Instead of honestly evaluating what is known and what isn’t, and trying to push forward and learn more (as Nekrasov, D’Hoker, and Phong are doing), string theory partisans loudly make untrue claims and personally attack anyone who disagrees with them. Lubos is just the most extreme case, others like Distler aren’t much better.

  19. anon says:

    Lubos is symptomatic of the bigotry of many theoretical physicists.

    In a way you could say it is refreshing honest to see a person behaving so outrageously instead of quietly suppressing opposition.

    Lubos loudly proclaims without a fig leaf of evidence that string theory must be right, and that alternatives are a waste of time.

    This in my opinion is more honest than the usual “conspiracy of silence” which theoretical physicists use. Without people like Lubos, there wouldn’t be much evidence of the misery caused!

  20. Michael says:

    “[…] if I‚Äôm going to criticize them I should at least understand what they are doing […]”

    Peter, do you believe you do understand what string theorists are doing? May I quote you as saying that you believe you understand recent developments in string theory?


  21. secret milkshake says:

    since the Bogdanov affair, it is clear that that some PhD comitees and article editors do not have much clue about string theory. (At least at one minor French university and several less-known journals, they were accepting/publishing 100% balooney stringy thesis + several articles. This was done on the face value of the summary of this bogus work – because the comitees/referees were unable to understand the gibberish.)

  22. woit says:

    I have no idea who you are, but from your behavior I’m quite sure that I understand a lot more about recent developments in string theory than you do. And yes, you can quote me on that.

  23. shantanu says:

    Peter , what about neutrino physics. It is one field in particle physics in which
    there have been lots of exciting developments in the past 7 years both in
    experiments (neutrino mass discovered, solar neutrino problem solved,
    direct evidence for tau neutrino, etc.) IS it something which can impact rest of particle physics and what do you think?

  24. MathPhys says:


    If Peter Woit had no clue what he’s talking about and/or what’s going on in string theory, people like you (and most notably Motl) wouldn’t even bother to log onto his web site, let alone take the trouble to write comments on it.

    I just don’t see what your problem is with someone who says that the formal/technical aspects of string theory are great, but that the theory has made no contact with physics.

    It seems like a very reasonable point of view (also made by people like ‘tHooft, and in different ways by Witten) and one that any of us may wish to disagree with but can also live with.

    Logging onto this site simply to write insulting comments anonymously is something that doesn’t become a scientist.

  25. woit says:


    There certainly has been a lot of progress in neutrino physics in recent years, and it looks like there will be more in years to come (MiniBoone and Numi-Minos should soon be reporting results). It’s one piece of particle physics where the limitations on the energy to which we can accelerate particles is not very relevant.

    But the problem is that we still have no idea where the entries in fermion mass matrices come from, neither the ones we already knew about, nor the new ones we’re learning about in the neutrino sector. So what is being discovered about neutrinos is just adding a new puzzle to solve, pretty similar to one we’ve already failed at solving for quite a while now. Maybe there will be some hint hidden in the structure of these new numbers that gives an important clue…

  26. dan says:

    Does theory predict neutrinos travel at a constant speed (presumably less than c due to mass) at all times, or can they vary, and can gravity slow down (or accelerate) their speed?

    i wonder if dark matter is nothing else but slow moving neutrinos?

    does string theory have anything to say about neutrinos?

  27. woit says:


    I don’t want to get into a discussion of this here, it’s off topic. But very quickly: yes, neutrinos travel at variable speed less than c, string theory says nothing about neutrinos, and dark matter doesn’t seem to be just neutrinos (their mass is too small).

  28. hobgoblin says:

    Dr. Milkshake,

    You’re right about needing a qualified psychiatrist to make a diagnosis, but you don’t need a psychiatrist to see that something is abnormal. Psychiatrists have crazy people brought to them by non-psychiatrists – they don’t have to prowl around the population looking for candidate crazies who others just aren’t qualified to spot.

    I’ll have a look at the landscape paper today. Best of luck to everybody for 2006 from the goblin.

  29. David says:

    Hi Peter,
    Thanks very much for your detailed, balanced reply to my somewhat knee-jerk comment. I understand your points. The problem I have with the original post, while sympathising with the reasons you gave in your comment, is that it adds to the impression that string theory is where the action is and that if someone wants to make an impact in particle physics then that is what they should be working on. I know this wasn’t really your intention, but still, it’s an impression that someone reading it could easily get.

    More generally, when attempting to counter the “sociological” problem of string dominance I think it is crucial to emphasise and discuss breakthroughs and interesting developments taking place in other areas. I do understand though that there is only so much that you as an individual can do in finite time. And for what it’s worth I think you have generally been doing an excellent job of pointing a critical spotlight on string theory. There was a real need for someone to step in and counter the string hype, and you have been performing this role admirably imho. It’s too bad there aren’t other people to help you out with this. (Sure there are other string critics but as far as I can tell no one else has been prepared to immerse themselves in the details in the way that you have done.)

    Best wishes,

  30. Pingback: The String Coffee Table

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  32. Arun says:

    I had technical difficulties in getting to Nekrasov Lecture 2, and I wouldn’t understand most of it anyway. Was there anything that throws new light on the superstring perturbation expansion?
    Thanks in advance!

  33. woit says:


    Nekrasov’s second and third lectures were rather technical, mainly about the sigma model discussed in his recent paper. Unfortunately the video cut off at the end of his third lecture while he was about to answer questions from the audience. He describes his approach as a more geometrical version of that of Berkovits, I don’t think he has actually used it to do multi-loop calculations himself. He did say that Berkovits claimed to be able to use his formalism to reproduce some of the 2-loop results of D’Hoker-Phong and even come up with some new ones. But nothing in his talk that I noticed dealt explicitly with the issue of what happens at higher order than 2-loops.

  34. mathjunkie says:

    Peter appeared as MathPhys to attack the stringy guys?

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