Musings

Nekrasov on Pure Spinors

Nikita’s paper, based on the talk discussed by Urs, and mentioned here previously, has appeared.

I’ll refer to my previous posts for the background on Berkovits’s pure spinor approach. The space of (Euclidean) pure spinors is a complex cone over $\tilde{Q}=SO(10)/U(5)$. As I guessed, one of the difficulties Nikita faced is what to do with singularity at the tip of the cone.

Smoothing it, turns the space of pure spinors into the total space of a certain line bundle $\hat{Q}\to\tilde{Q}$. That space has a nonvanishing $c_1(\hat{Q})$ and $p_1(\hat{Q})$, which means that the transition functions for the local $\beta$-$\gamma$ systems cannot be consistently-defined (don’t satisfy the requisite cocycle conditions, respectively over the worldsheet and over $\hat{Q}$).

Deleting it, yields the total space of a $\mathbb{C}^*$ bundle, $X\to \tilde{Q}$, with $c_1(X)=p_1(X)=0$. That’s better, though, as Nikita notes, it’s a little strange to exclude the point $\lambda=0$ from the path integral.

Anyway, one can then go on to check that the $SO(10)$ generators, say, are global sections of the sheaf of local operators.

The stress tensor, too, exists globally. But, as discussed by Urs, the composite operator, $B$, where $$T = \{Q, B\}$$ does not. Rather, according to Nikita, the Čech coboundary of $B$ is $Q$-exact. I’d like to see the details. But, in any case, it’s rather problematic. The genus-$g$ string measure involves insertions of $B$. Since the latter is not globally-defined in field space, the string measure then depends on some choice of “partition-of-unity” in field space. The difference between two such choices is, at least formally, a total derivative on moduli space.

If those words give you nightmares about “integration ambiguities,” they should.