Nekrasov, Pure Spinors and the Berkovits Superstring

There’s a new paper out tonight by Nikita Nekrasov entitled Lectures on curved beta-gamma system, pure spinors, and anomalies. Motivated by questions about the covariant superstring quantization method being studied in recent years by Berkovits, Nekrasov considers a sigma model with target space the space of “pure spinors”. For more about pure spinors I suggest consulting “Spin Geometry” by Lawson and Michelson, but in general they are a subspace of the full spinor space with remarkable properties. In R2n, a pure spinor determines a complex structure on R2n, one that doesn’t change when you multiply the spinor by a complex scalar. Furthermore, modding out by the action of the complex scalars, the space Q(2n) of projective pure spinors is a Kahler manifold, isomorphic to O(2n)/U(n). This is a projective algebraic variety, and geometric quantization of it gives back the space of spinors. There’s quite a lot of beautiful geometry in this story.

Unfortunately, in the Berkovits story the target space of the sigma model is not Q(2n), which is smooth and has every nice property one could ask for, but the space of pure spinors themselves which is a cone over Q(2n), and has a singularity at the origin. How to handle this singularity is the problem Nekrasov is addressing. This is a rather technical business, one about which I’m no expert (and I’m not sure there are many experts out there on this topic other than Berkovits and Nekrasov).

At the end of his paper Nekrasov makes what appear to be some remarkable comments. He describes two ways to deal with the singularity. The first is to just remove it and work with a non-compact target space. In his paper he shows that this removes certain potential anomalies, but he comments that doing this causes “some unclear issues with the definitions of string measure”. The second way to deal with the singularity is to blow it up, working with the total space of a complex line bundle over Q(2n). Nekrasov claims that if you do this the superstring “would cease to be consistent beyond tree and one-loop level, thereby killing at once the landscape [48] problem.” The reference is to Susskind’s anthropic landscape paper, although Nekrasov refers to Susskind as “Sussking”.

I’m assuming this is some sort of perverse joke, since if the superstring is inconsistent on flat ten-dimensional space, there’s every reason to believe it’s also going to be inconsistent on curved 10d spaces and what gets killed is not just the landscape, but the whole idea of unification based on the 10d superstring. Nekrasov goes on to end with the comment that “This is of course one of the unrealized, so far, hopes to solve some pressing predictive issues of string theory by capitalizing on its unusual, from the conventional quantum field theory point of view, perturbation theory”, referring to a 1987 paper of Greg Moore that I don’t have access to at the moment.

I’m curious to hear what people more expert in this subject think of all this. There are various relevant blog entries: Robert Helling and Urs Schreiber on Nekrasov’s talk a couple weeks ago about this in Hamburg, a recent posting by Jacques Distler, and a report on a talk by Berkovits at the KITP in August by Andrew Neitzke. For some relevant papers on the arxiv, see a paper by Berkovits and Nekrasov from earlier this year as well as quite a few papers by Berkovits and other collaborators written over the last few years.

Update: A commenter wrote in to point out that the Moore paper is available on-line as a scan of the preprint at KEK.

After my post appeared, there were later posts on this topic by Jacques Distler and Lubos Motl. Lubos seems to agree with me that Nekrasov’s comment about an inconsistency in the quantization of the superstring in flat 10d killing the landscape is rather bizarre, since such an inconsistency would probably then hold in all backgrounds.

Funny, but if you look at trackbacks for the Nekrasov paper, they’re there for Distler and Motl’s blog entries but not mine, even though mine appeared earlier. I guess whatever the moderation policy is for trackbacks these days, I’m in a separate category.

Update: After inquiring with the arXiv about what was going on about this trackback, I just heard that it has been posted. It’s still unclear to me what their moderation system is.

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21 Responses to Nekrasov, Pure Spinors and the Berkovits Superstring

  1. Anonymous says:

    Spires has a link to a KEK scanned version of the 1987 Greg Moore lecture.

  2. D R Lunsford says:

    That is fascinating.

    Here is some interesting work by Trautman, which is fascinating in its own right:

    They seem to be generalizations of twistors to other spaces. It sounds a little like line geometry (Pluecker).



  3. Wolfgang says:

    > Nekrasov claims that if you do this the superstring “would cease to be consistent beyond tree and one-loop level

    I thought it was explicitly shown that two-loop is consistent.
    So this would indicate that something is wrong with the PS approach ?

  4. Mafra says:

    Last month it was explicitly shown that the four point two-loop amplitude computation in the pure spinor formalism agrees with the RNS result (when all states are NS).
    So, there is no indication that the PS approach is wrong up to 2-loops.

  5. One needs to deal with the singularity of the pure spinor space. This is a technically issue of correctly working out the formalism.

    One way one might guess to deal with it is to ‘blow it up’.

    If you’d blow up the singularity of the pure spinor space, then there’d be incosistencies above one loop.

    So it’s the wrong thing to do.

    Unless you argue like Nekrasov mentions one could argue. You could argue that you redefine your theory to be given by the beta/gamma model on the blown up space of pure spinors, by definition. This is now a different gadget than the original theory. Since it is inconsisztent on flat target space, it is apparently more constrained than the original theory. So maybe (that’s Nekrasov’s idea), maybe there are only very few target spaces which would make a pure spinor string with a blown-up space of pure spinors consistent.

  6. anonymous says:

    it appears that the “moderation policy is that you have to submit your own trackbacks.

  7. woit says:

    OK, I just tried submitting a manual trackback and will see what happens. The documentation does seem to claim that auto-discovery of trackbacks is supposed to be supported.

  8. Matt says:

    Peter, it’s not a secret that they won’t post your trackback on the arxiv. The arxiv is for scientific exchange only, to which you have not contributed at least since 1989.

  9. woit says:

    Hi “Matt” (aka “Michael”) from Brandeis,

    Whoever you are, you’re as much of a cowardly asshole as ever. What is it about string theory that causes its proponents to behave like this?

  10. D R Lunsford says:

    Peter, that’s a more interesting question than anything about string theory proper. I have my own theories about it, which I will keep to myself.


  11. Who says:

    I see where Steven Weinberg has bet the life of Andrei Linde plus one dog, apparently on the string theory Landscape.

    see the conclusions on page 13 of a paper by Weinberg posted today

    “As for me, I have just enough confidence about the multiverse to bet the lives of both Andrei Linde and Martin Rees’s dog. ”

    Do I hear two dogs?

  12. D R Lunsford says:

    That is really distressing. “Gravitation and Cosmology” is still the best gravity textbook.


  13. D R Lunsford says:

    It reminds me of Hegel’s “proof” that there were no more planets – just before Ceres was discovered.


  14. D R Lunsford says:

    OK I read the entire paper – what a mess. Apparently Weinberg has the now common attitude, “since I’m the smartest guy in town, and I can’t figure it out, we have to change the definition of science so that I can”. You’d think a little embarrassment would creep in.


  15. Chris Oakley says:

    Weinberg’s paper is depressing, and ultimately, irresponsible. As one of the most influential people in particle physics it is his duty to try to make the subject attractive to bright young mathematicians and physicists currently at school or university. The promise of nothing better than the application of anthropic reasoning to research problems certainly would not have drawn me into the subject, and I rather suspect that the same applies to the majority of young people today.

    Anthropic reasoning is in any case circular. We work out what we believe to be the physical processes necessary for life and then study the range of values of physical constants that allow this to work. But all we are doing is finding out what works for our particular form of life. What if a different set of constants led to varieties of life vastly different from our own? It is arrogant to assume that the set of nuclear and chemical reactions that led to us are the only ones possible.
    I do not agree with his analogy concerning Kepler’s unfulfilled wish to be able to predict planetary distances. At least Kepler and Newton had a full dynamical theory. With the string theory landscape everything is up in the air.

  16. Dissident says:

    That paper has a truly remarkable cringe factor.

  17. Arun says:

    Weinberg does say that it would help to know what string theory is (w.r.t. the work of classifying string vacua). So, we have to accept as a philosophical truth (as it is unverifiable scientifically) some conclusion from a theory which we cannot formulate completely?

  18. Juan R. says:

    Weinberg’s paper is fascinating.

    It is an open oportunity for any new ‘Weinberg’.

    Solve the problem that Weinberg cannot. This may be the message to the young generations.

    I, at least, am sure that ‘Weinbergs’ born each 50 years.

    Juan R.

    Center for CANONICAL |SCIENCE)

  19. Wolfgang says:


    I mentioned Weinberg’s latest opus on my blog (just click on my name).
    I would certainly not bet my dog on this …

  20. Chris Oakley says:

    I would happily bet my dog, car, house, life, your dog, your car, your life, anyone else’s life or anything else in a bet where it is never possible to say whether either party has won.
    What I would rather do, though, is to sell an option. Someone can pay me a million dollars now, and if it is established that a multiverse exists in some given time frame – twenty years, let’s say (that is, twenty years in this particular universe for an observer not moving at relativistic speeds relative to this planet), then I pay ten million dollars back.

  21. MathPhys says:

    Is it possible that a certain arXiv moderator just doesn’t like you, Peter? Seriously, I can imagine that certain string theorists would prefer not to divert too much traffic to this site.

    Incidentally, what’s the current job situation like, for string theorists, in the US, nowadays? In the mid 80’s, only string theorists could get postdoctoral positions, in the mid 90’s, no string theorists could get positions. What’s it like it in the mid 00’s?

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