Wednesday, March 29, 2006

Warren Siegel and Kiyoung Lee: ghost pyramids

Tonight, I recommend you the preprint
by Warren Siegel and Kiyoung Lee. It is actually a followup of a 2005 preprint that most of us missed
In these papers, the authors propose a formalism that is arguably (and certainly according to the authors) better than the four existing frameworks to compute perturbative type II superstring amplitudes, namely than
  • Ramond-Neveu-Schwarz Lorentz-covariant approach
  • light-cone-gauge Green-Schwarz approach
  • the hybrid RNS-GS approach
  • Berkovits' pure spinor approach
In the picture of Siegel et al., the super-Poincaré invariance is manifest and no exotic picture-changing insertions are needed. The price you pay - and I guess that Nathan Berkovits would argue that it is a high price - is the infinite number of fields.

What you actually need is to extend the spacetime spinor variables "theta_a" where "a" is a spinor index in the 32-dimensional representation (the same variables that occur in the pure spinor or other covariant Green-Schwarz frameworks) into a ghost pyramid
  • theta^{mn}_{a}
where "m,n" are non-negative integers which is why you span a pyramid (or a quadrant, if you draw "m,n" as vertical and horizonal axes). The original ordinary group of observables "theta_a" becomes "theta^{00}_{a}". The number "m+n" measures the "ghost level" (where 0,1,2 means humans, ghosts, and ghosts for ghosts, and so on) while the difference "m-n" counts the ghost number. Note that the statistics of theta's is flipped for odd values of "m+n".

If you keep "m+n" fixed, there are "m+n+1" sibbling fields "theta". Their central charge should be counted as "(-1)^{m+n}" times the central charge "C00" of the fields "theta^{00}". This means that the total central charge of the whole ghost pyramid is
  • Cpyramid / C00 = 1 - 2 + 3 - 4 + 5 - 6 + ...
The contributions 1,-2,+3,-4 come from the theta variables with m+n=0,1,2,3 and so on while the alternating signs arise from the alternating statistics of the ghosts for the previous ghosts. This sum plays an important philosophical role in the new formalism, and it is useful to evaluate it. You can use the zeta-function regularization with the usual allowed tricks. A simple reasoning shows that
  • 1 - 2 + 3 - 4 + 5... =
  • 1 + 2 + 3 + 4 + 5... - 2 ( 2 + 4 + 6 + ...) =
  • 1 + 2 + 3 + 4 + 5... - 4 (1 + 2 + 3 + ...) =
  • (1 + 2 + 3 + 4 + 5...) x (1-4) =
  • (-1/12) x (-3) = +1/4
That's one quarter. Note that this heuristic calculation is inevitably confirmed by the rigorous theta functions when you check the one-loop modular invariance. One quarter is significant because it tells you that the 32-component spinorial ghost pyramid has the same "number of degrees of freedom" as a single 8-component spinor, such as the spinor found in the light-cone gauge description.

Your worldsheet field content then only has the usual fields "X,b,c" much like in the bosonic string plus the "theta^{mn}_{a}" ghost pyramid. No "beta,gamma" systems occur. Pure spinors "lambda" are absent, too. The total central charge cancels because of the identity
  • 10 (X) - 26 (bc) + 16 (theta pyramid) = 0
The BRST charge is constructed as some annoying conjugation by an operator "U" (to make the non-invertibility of the following operator harmless) of the operator defined as follows:

The operator in the core of the BRST operator is essentially the usual bosonic BRST term, i.e. the integral of "cT(sigma)", plus a fermionic term that is equal to the integral of
  • (1/4) Pi . Gammapyramid . Pi
Here, "Pi" are the conjugate momenta for the fields "theta" in the pyramid and "Gammapyramid" is a generalization of the Dirac matrices that acts on the "Pi" and "theta" components of the pyramid in a particular way. The operator "U" needed for the conjugation is defined as a particular exponential.

You can then define vertex operators for physical states and if you read and understand the paper that appeared one hour ago, you can also compute some particular scattering amplitudes.

And because I don't want to write their whole paper again, anyone who is interested in details should try to read the original papers. It should be possible to prove the equivalence with the Berkovits picture, and it is even conceivable that the proof is known.

An expert from the Western hemisphere confirms that Siegel's and Kiyoung's approach is analogous to working with the picture 0 operators in the RNS superstring while the picture -1 may be more natural on the worldsheet. These picture 0 operators are enough for tree-level and one-loop graphs but it seems obvious that for genus 2 diagrams and higher, one needs an extension of the formalism and the ghost pyramid approach could face problems at this level.

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