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

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}

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 + ...

- 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

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 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

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.

## No comments:

## Post a Comment