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November 1, 2005

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 Q˜=SO(10)/U(5)\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 Q^Q˜\hat{Q}\to\tilde{Q}. That space has a nonvanishing c 1(Q^)c_1(\hat{Q}) and p 1(Q^)p_1(\hat{Q}), which means that the transition functions for the local β\beta-γ\gamma systems cannot be consistently-defined (don’t satisfy the requisite cocycle conditions, respectively over the worldsheet and over Q^\hat{Q}).

Deleting it, yields the total space of a *\mathbb{C}^* bundle, XQ˜X\to \tilde{Q}, with c 1(X)=p 1(X)=0c_1(X)=p_1(X)=0. That’s better, though, as Nikita notes, it’s a little strange to exclude the point λ=0\lambda=0 from the path integral.

Anyway, one can then go on to check that the SO(10)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, BB, where T={Q,B} T = \{Q, B\} does not. Rather, according to Nikita, the Čech coboundary of BB is QQ-exact. I’d like to see the details. But, in any case, it’s rather problematic. The genus-gg string measure involves insertions of BB. 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.

Posted by distler at November 1, 2005 11:50 PM

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Re: Nekrasov on Pure Spinors

Rather, according to Nikita, the Čech coboundary of BB is QQ-exact. I’d like to see the details.

Indeed. Nikita mentions this on p. 43 in passing, and refers to hep-th/0410079, hep-th/0503197 and hep-th/0509120.

I have trouble identifying in these references anything which would illuminate his statement. (But maybe I am missing it, I haven’t read every single line.)

The only formula that remotely and maybe superficially looks related is (3.13) in hep-th/0509120. There we are told that for QQ the BRST(-like) operator, λ α\lambda^\alpha the pure spinor bosonic ghosts and TT the stress tensor we have a ghost-like entity G αG^\alpha (with α\alpha a spinor index) such that

(1)λ αT={Q,G α} λ [αG β]=[Q,H [αβ]] λ [αH βγ]={Q,K [αβγ]} λ [αK βγδ]=[Q,L [αβγδ]] λ [αL βγδκ]=0 \array{ \lambda^\alpha T = \{Q,G^\alpha\} \\ \lambda^{[\alpha}G^{\beta]} = [Q,H^{[\alpha\beta]}] \\ \lambda^{[\alpha}H^{\beta\gamma]} = \{Q, K^{[\alpha\beta\gamma]}\} \\ \lambda^{[\alpha}K^{\beta\gamma\delta]} = [Q,L^{[\alpha\beta\gamma\delta]}] \\ \lambda^{[\alpha}L^{\beta\gamma\delta\kappa]} = 0 }

The first is the equation which you have discussed here before. Using some redefinitions as in equation (3.9) it should lead to the

(2)T=[Q,G] T = [Q,G]

from (3.7) in hep-th/0511008. With a little imagination the other equations in the list in this sense seem to follow a pattern vaguely similar to the Čech-pattern δG (i)=[Q,G (i+1)] \delta G^{(i)} = [Q, G^{(i+1)}] with

(3)H G (1) K G (2) L G (3) \array{ H &\to& G^{(1)} \\ K &\to& G^{(2)} \\ L &\to& G^{(3)} \\ }

and with

(4)δXλX \delta X \to \lambda X

But that’s a free association. I don’t know if it’s really related.

Posted by: Urs Schreiber on November 2, 2005 10:49 AM | Permalink | Reply to this
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