Musings

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.