Wednesday, November 02, 2005

Nikita on pure spinor cone

Nikita Nekrasov has analyzed Berkovits' pure spinor formalism or, using his more general words, curved beta-gamma systems.

As we discussed previously, the pure spinor formalism makes the whole superPoincaré symmetry manifest. The price we pay is a curved system of ghosts - bosonic objects "lambda" that live in the space of "pure spinors". It means that they transform as a spinor in ten dimensions constrained by vanishing of most of its bilinears except for the self-dual five-form

  • lambda . Gamma_{ijklm} . lambda

Consequently, the space of pure spinors is equivalent to the space of these five-forms that can be obtained from spinors. Every SO(10) rotation gives a new solution, but a U(5) subgroup keeps the five-form invariant, and therefore the space of the projective pure spinors is the quotient SO(10)/U(5).

But Nathan Berkovits has to work with a cone whose base is this manifold - the normalization of lambda is not determined - and several people including Nikita seem to worry about the conical singularity of this cone.

This set of people includes not only one-dimensional Peter Woit who applies his projection operator on all of physics - projection along his axis whose negative direction is "string theory works well" and whose positive direction is "possible doubts about string theory"; he only picks the "positive" eigenvalues. It also includes Jacques Distler (who calls Nathan "Berkovit") and, more importantly in this case, Nikita himself (who calls Lenny "Sussking").

In the light cone gauge, we obtain the integrals over the Riemann surface moduli from the integrals over the moments of the interactions and over the relative splitting of the longitudinal momentum in between the intermediate particles.

In the RNS formalism (incidentally, NSR meant West Germany in Czech), we obtain it directly from the integrals over the zero modes of the "b" (and "beta") antighost - in a procedure that may be interpreted as a canonical BRST or Faddějev-Popov treatment.

However, the pure spinor formalism has no explicit "b" ghosts and the loop integrals are more subtle. In some sense, "b" is a composite field. Nathan has offered his own formula for the loop amplitudes and demonstrated that it has many expected properties - some of which are more difficult to prove in the other formalisms. At this moment, I don't have any continuing specific counter-arguments that would imply that Nathan's formula is wrong.

More generally, I find the worries about the conical singularity in the ghost space highly exaggerated. Ladies and Gentlemen, these guys are just ghosts! They're not a physical target space and we are not supposed to probe all of it. The place where Nathan's ghosts together with the other degrees of freedom are equivalent to the older formulations of the superstring is a generic point away from the tip of the cone. The stringy perturbative expansion involves various operations with the ghosts, but as far as I see, all of them may be viewed as small excitations around a non-singular point of the space.

Note that normally we just want to set the ghosts and antighosts to zero, at least their positive frequency modes. The ghosts may give you some determinants or measure from their zero modes, but all of these things are small deviations from a point in the ghost target manifold - and in Nathan's case, this point around which we expand is simply not the tip of the cone. I completely share Nathan's intuition that the effects of the tip of the cone should be "brutally removed", to use Nikita's words, and moreover it seems that Nathan has more reasons to do so than just this intuition and precise formulae showing what it exactly means to remove them.

Non-perturbatively in "g_{string}", one may imagine that the tip of the cone starts to influence the amplitudes, but generic non-perturbative physics indeed does not have any useful conformal symmetry (non-perturbative string theory is not just a theory of strings) so it should not be surprising if the ghost structure appropriate for the perturbative expansions breaks down non-perturbatively.

I can't guarantee that Nathan's loop formula is correct but it is pretty obvious that if one does the "field redefinition" from RNS to Berkovits properly, she will obtain some form of the amplitudes that may be more or less useful. This strictly speaking holds for the NS-NS states' amplitudes where the RNS is well-defined, but because Nathan makes spacetime SUSY manifest, the translated formula will also give good amplitudes for the other sectors.

Nikita's comment about "killing of the landscape" sounds bizarre to me. It would be great if someone killed it but Nikita's prescription - some hypothetical loop inconsistencies he may imagine in the pure spinor formalism - just don't seem to do the job. At best, they kill a whole formalism because such problems would already occur in 10 dimensions; I think it is unlikely. In Nikita's viewpoint, it seems that such hypothetical bugs would kill the formalism for all backgrounds, not just a majority of the landscape.

Or does Nikita want to say, as some of our friends, that string theory as such has some problems beyond one-loop? (Note that for two-loop, that would directly contradict many existing calculations.) The content of such a statement is very different from some detailed struggles with an unphysical singularity in a particular infrequently used formalism, and I would probably think the same thing about such a hypothetical statement what I think if this statement is presented by the non-stringy friends of ours.

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