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  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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