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Pure Spinor Flat BRST Nilpotence

Physical scope

Formal flat-space holomorphic operator-algebra deduction for the minimal pure-spinor BRST charge. It does not prove BRST cohomology, curved-background consistency, or existence of a global measure.

Statement

In the flat ten-dimensional pure-spinor model with $Q = \int d z λ^{α} d_{α}$ and OPE $d_{α} (z) d_{β} (w) \sim - (γ^{m})_{α β} Π_{m} (w) / (z - w)$ , one has $Q^{2} = 0$ because the only simple-pole obstruction is proportional to $λ γ^{m} λ$ , which vanishes by the pure-spinor constraint. This is a flat-background operator-algebra statement and does not establish the full BRST cohomology or quantum consistency in an arbitrary curved background.

Statement references

Nathan Berkovits and Carlos R. Mafra, Pure spinor formulation of the superstring and its applications, arXiv:2210.10510v2

Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring, arXiv:hep-th/0209059v1

Facts & Assumptions

Given: The minimal flat pure-spinor model and its holomorphic OPE convention.

[F1]

Pure Spinor Superstring Model fixes $Q = \oint \frac{d z}{2 π i} λ^{α} d_{α}$ , the regular $λ$ OPE, the simple-pole OPE $d_{α} (z) d_{β} (w) \sim - (γ^{m})_{α β} Π_{m} (w) / (z - w)$ , and the constraint $λ γ^{m} λ = 0$ .

Proof

1.1givenF1algebra

Since $λ$ is bosonic and has regular OPE with itself and with $d$ , the singular part of the BRST-current product $j_{B} (z) j_{B} (w)$ , where $j_{B} = λ^{α} d_{α}$ , is only the contraction of the two $d$ factors. Substitution of [F1] gives $j_{B} (z) j_{B} (w) \sim - \frac{(λ γ^{m} λ) Π_{m} (w)}{z - w} .$ The displayed pole coefficient vanishes identically by the pure-spinor constraint in [F1].

2.1F1step 1.1algebra∎

The square of an odd contour charge is one half of its graded anticommutator; the contour residue of the simple pole in step 1.1 is therefore the only contribution to $Q^{2}$ . Since that residue is zero as an operator, $Q^{2} = 0$ . The proof uses only the flat chiral OPE and does not assert any cohomology or curved-background result.

Depends on

Pure Spinor Superstring Model