### CDO and Pure Spinors

One of the drawback of having other things to do besides blogging is the long list of half-completed posts sitting on my computer. One of them was adiscussion of the work of Witten and Kapustin on chiral differential operators. The basic idea is to recast the nontrivial aspects of 2D (supersymmetric) nonlinear $\sigma$-models. One works with an open covering of the target space. On each patch, one can replace the nontrivial $\sigma$-model action by a free theory, with a first-order action

On patch overlaps, one has nontrivial transition functions for the algebra of observables. All of the nontrivial aspects (the $\beta$-function, anomalies, *etc.*) of the $\sigma$-model are encoded in how the local observables patch together. The usual global observables (whatever they are) are obtained as global sections of the sheaf of local observables.

I have written some observations on Berkovits’s pure-spinor approach to the superstring. In that approach, the (bosonic) ghosts, $\lambda^\alpha$, (which contribute $c=+22$ to the total central charge) are a Majorana-Weyl spinor of $Spin(9,1)$ ($Spin(10)$, if you’re not too particular about reality conditions) satisfying the constraint
$\lambda^\alpha \gamma^m_{\alpha\beta}\lambda^\beta=0$
The ghosts, $\lambda^\alpha$, and the anti-ghosts, $w_\alpha$, have a free-field action. But, because of the constraints, rather than *really* being free fields, we have a nontrivial $\sigma$-model.

Berkovits and Nekrasov realize the space of (Euclidean) pure spinors as a cone over $SO(10)/U(5)$. Since the action is already of the “free”, first-order form, (1) one can adopt the same CDO technique to study it. Urs has posted a nice summary of a talk by Nikita on the subject.

Of course, even adopting this CDO approach, there’s a big difference with the examples studied by Kapustin and Witten. Here, the cone is not a smooth manifold. I’m not sure how that impacts the analysis. But, evidently, there’s some progress in understanding the composite operators which are the analogues of the $b$-antighosts, in the bosonic string, as Čech cohomology classes. This, as you’ll recall, was the point on which my understanding of the pure-spinor approach fell apart.

Posted by distler at October 26, 2005 4:30 PM
## Re: CDO and Pure Spinors

Actually, while listening to Nekrasov’s talk I was reminded of that old post of your’s. I don’t think that Nikita Nekrasov has presented in that talk the relevant details (on the other hand, maybe they escaped my attention), but it sure seemed to me that he was saying that the messy ghost issue which you discussed in detail becomes nice and natural once one adopts the Čech cohomology point of view.

Thanks a lot for the link to Kapustin’s work. I wasn’t even aware of that. Will have a look at it.

What I find exciting about this approach is the following:

Segal told us that a CFT is a certain ‘modular’ functor from a cobordism category into $\mathrm{Vect}$. Stolz and Teichner in their work point out that this description is a little too coarse. They say one has to replace cobordisms with surfaces whose in- and outgoing boundary may be open paths instead of closed circles, such that gluing them appropriately reproduces the cobordism structure. But such surfaces are (up to some additional structure they might be equipped with) essentially 2-path 2-morphisms as they appear in the study of parallel transport of surfaces in 2-bundles/gerbes.

Moreover, Stolz and Teichner say the target category $\mathrm{Vect}$ has to be replaced accordingly by some 2-category, which also matches the surface-transport constructions in 2-bundles.

So in conclusion, even though their work remains unfinished, it seems that the emerging picture is that a CFT ‘is’ a certain 2-functor from 2-paths (bigons) to a suitable target 2-category.

By analogy with what happens in 2-transport in 2-bundles, this alone suggests that when locally trivialized this 2-functor gives rise to precisely the Čech-cohomology structure that appears in the CFT literature now.

In a sense this is just a tautology. But that’s how it goes with the really good ideas. :-)