February 15, 2006

Koszul-Tate

I’ve written before about Berkovits’s Pure Spinor formalism for the superstring and its relation to curved $\beta$-$\gamma$ systems. One of the issues one has to deal with is that the algebraic variety one is trying to study (the space of pure spinors) is singular. For $d=10$ Euclidean pure spinors, it’s a complex cone over $\tilde Q= SO(10)/SU(5)$. As I discussed, Nekrasov proposed to deal with this particular issue by omitting the tip of the cone.

Grassi and Policastro propose an alternate approach which, they claim, allows one to deal directly with the singular affine algebraic variety. Consider a (set of $n$) spin-1 $\beta$-$\gamma$ system(s) with action $S_{\beta\gamma} = \frac{1}{2\pi} \int \beta_i \overline{\partial} \gamma^i$ One would like to impose some polynomial constraints $\Phi^a(\gamma)=0$ Grassi and Policastro propose to implement this via the Koszul-Tate Theorem. Introduce a set of spin-1 anticommuting ghosts $S_{b c}= \frac{1}{2\pi} \int b_a \overline{\partial}c^a$ and the Koszul-Tate differential $\tilde\delta = \frac{1}{2\pi i} \oint b_a \Phi^a(\gamma)$ If there are relations among the constraints, $f_a(\gamma)\Phi^a(\gamma)=0$ then one may need to introduce ghosts-for-ghosts, and the procedure becomes a bit messy. This certainly happens in the pure spinor case, where the Fierz Identities introduce relations of this form.

We happen to be interested in the particular case where the $\Phi^a$ are homogeneous quadratic polynomials, so we can assign a grading $J^g(z)= J^{\beta\gamma}(z) + 2 J^{b c}(z)$ such that $\tilde\delta$ has grading equal to zero.

The claim is, now, that computing the cohomology of $\tilde\delta$, restricted to the subspace with non-negative values of the grading gives the correct spectrum of the constrained theory. Grassi and Policastro have checked this in a few simple examples.

In the general case, I’m not quite sure what “correct” means. In their examples, one can resolve the cone to a smooth variety, and the spectrum doesn’t change. That’s the sort of result one would expect from an application of Koszul-Tate. But, as we’ve already seen, resolving the conical singularity in the case of the cone on $\tilde Q$ give the wrong answer for the pure spinors. The resulting variety has nonzero $p_1$, and so the curved $\beta$-$\gamma$ system suffers from an anomaly.

Posted by distler at February 15, 2006 11:50 PM

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Re: Koszul-Tate

I don’t think we claim that in general our prescription is equivalent to the resolution of the singularity. It is so in the example we checked, but it may well not be in more complicated cases, and in particular what happens if there is more than one way to resolve, and they result in different spectra? I suppose our prescription somehow selects one possible way. That should be an interesting question. At any rate, by “correct” spectrum we mean all the operators that can be built out of $\beta_i$ and $\gamma^i$, that are gauge-invariant, modulo the constraints. To list these states is an algebraic problem, and whether the constraint defines a singular or a smooth variety does not seem to matter at this level. This is also how Berkovits originally defined the physical states in the pure spinor formalism.

I hope this clarifies a little what we wrote. In any case, we don’t have a complete understanding yet, so we are happy to receive comments and suggestions.

Posted by: policastro on February 17, 2006 1:25 AM | Permalink | Reply to this

Which resolution?

I don’t think we claim that in general our prescription is equivalent to the resolution of the singularity. It is so in the example we checked, but it may well not be in more complicated cases, and in particular what happens if there is more than one way to resolve, and they result in different spectra?

Indeed! That’s one of the questions that naturally presents itself.

All I was pointing out was that Nekrasov studied two possibilities for what to do in the specific case of the cone on $\tilde Q= SO(10)/SU(5)$. One was to resolve it to the total space of a line bundle $\hat Q \to \tilde Q$. The other was to simply remove the point at the tip of the cone.

Your algebraic prescription may be yet a third possibility. All I know for sure is that $\hat Q$ is wrong.

But, for understanding questions like how to construct the composite $B$ anti-ghost, one needs to settle this question first. So I’m interested to see how your prescription works out in the case of interest.

Posted by: Jacques Distler on February 17, 2006 1:43 AM | Permalink | PGP Sig | Reply to this

Re: Koszul-Tate

Dear Sir,

The Wikipedia project (en.wikipedia.org) has been in need of an article describing the “Koszul-Tate derivation” for two years now.

If you could donate your time to write a description of this, you would be doing us a great favor.