October 26, 2005

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

(1)
$S = \int \eta_i \overline{\partial} X^i +\dots$

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

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Re: CDO and Pure Spinors

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.

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.

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. :-)

Posted by: Urs Schreiber on October 27, 2005 3:22 AM | Permalink | Reply to this

chiral de Rham complex

Here, the cone is not a smooth manifold. I’m not sure how that impacts the analysis.

It seems to me that this construction and the result that it is obstructed by the first two Chern classes is in fact well known for ‘target spaces’ being any smooth complex algebraic (or analytic) varieties.

That’s at least what it seems to say right on the first page of math.AG/9906117.

Posted by: Urs Schreiber on October 28, 2005 5:50 AM | Permalink | Reply to this

Re: chiral de Rham complex

The topological obstructions for a smooth target, $X$, are just the usual gravitational anomaly coefficients, $p_1(X)$ and $c_1(X)c_1(\Sigma)$, where $\Sigma$ is the worldsheet. Both are realized as obstructions to patching together the local operator algebras. You see the former when you try to patch together the operator algebras on triple overlaps, $U_i\cap U_j\cap U_k$ of the target space, $X$. The latter appears when you study the transition functions between patches on $\Sigma$.

My point was that Malikov et al only ever try to define these things for smooth varieties, $X$, which the cone certainly is not.

Posted by: Jacques Distler on October 28, 2005 8:05 AM | Permalink | PGP Sig | Reply to this
Read the post Sheaves of CDOs
Weblog: The String Coffee Table
Excerpt: Literature on Cech-methods in CFT.
Tracked: October 31, 2005 7:53 AM

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