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Open NS Vacuum Before the GSO Projection

Physical scope

A worked open-string oscillator/GSO calculation in the flat ten-dimensional NN RNS model, in a chart with nonzero k^+ and with the declared zeta-regularized normal-ordering prescription and residual zero-mode constraint; it checks the stated light-cone mass and sector projection only, not BRST cohomology, spacetime supersymmetry, or observed physics.

Example

In the flat ten-dimensional open NN RNS model, work in the light-cone chart $k^{+} = (k^{0} + k^{9}) / \sqrt{2} \neq 0$ with the declared zeta-regularized normal-ordering prescription and residual zero-mode constraint. Choose the mass-shell momentum separately at each oscillator level. The unprojected NS vacuum has $N = 0$ , $F = 0$ , $G_{N S} = - 1$ , and $α^{'} M^{2} = - 1 / 2$ , so the chiral supersymmetric GSO projection removes it. For any transverse index $i \in {1, \dots, 8}$ , the first excitation $ψ_{- 1 / 2}^{i} ∣ 0; k ⟩$ has $N = 1 / 2$ , $F = 1$ , $G_{N S} = + 1$ , and $M^{2} = 0$ , so it passes this chiral projection. These are oscillator-level checks only; they do not verify all BRST or super-Virasoro constraints or establish a spacetime supersymmetry theorem.

Example references

Timo Weigand, Introduction to String Theory, §§6.4.3–6.5

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.3.5 and 3.1.4

Facts & Assumptions

Given: Use the ten-dimensional open NN model, one chiral NS sector, the unprojected vacuum and the mode/fermion-number conventions in RNS GSO Projections. For each tested level choose momentum $k \in R^{1, 9}$ of dimension $L^{- 1}$ , with $k^{+} = (k^{0} + k^{9}) / \sqrt{2} \neq 0$ and $M^{2} = - η_{μ ν} k^{μ} k^{ν}$ , where $η = diag (-, +, \dots, +)$ . Adopt the zeta-regularized vacuum terms and impose $L_{0}^{l c} = 0$ in RNS Transverse Oscillator Fock System, as required by Open RNS Light-Cone Intercepts. The momenta at the vacuum and first-excited levels lie on their respective mass shells and need not agree.

[F1]

RNS GSO Projections: $F$ counts nonzero fermion excitations modulo two, the NS vacuum has $F = 0$ , $G_{N S} = (- 1)^{F + 1}$ , and the chiral supersymmetric projection retains the $G = + 1$ eigenspace.

[F2]

Open RNS Light-Cone Intercepts: for the flat ten-dimensional open NN string in the nonzero- $k^{+}$ chart with the declared zeta prescription and residual zero-mode constraint, $α^{'} M^{2} = N - 1 / 2$ in NS before GSO.

[F3]

RNS Transverse Oscillator Fock System: NS modes are half-integral, there are eight transverse directions at $D = 10$ , and each fermion creation mode $ψ_{- r}^{i}$ contributes oscillator level $r$ .

[F4]

RNS Flat Worldsheet Model: $α^{'} > 0$ , so dividing by $α^{'}$ preserves the sign of $M^{2}$ .

Verification

1.1givenF1algebra

On the vacuum, $N = 0$ by definition and $F = 0$ by the vacuum convention in [F1]. Thus $G_{N S} = (- 1)^{0 + 1} = - 1$ , which fails the retained eigenvalue $+ 1$ .

2.1givenF1F3step 1.1algebra

Applying one creation mode $ψ_{- 1 / 2}^{i}$ adds level $1 / 2$ and one fermion excitation by [F3], so $N = 1 / 2$ and $F = 1$ . Hence $G_{N S} = (- 1)^{1 + 1} = + 1$ , which satisfies [F1]'s projection.

3.1givenF2F4step 1.1step 2.1algebra∎

Substitution of $N = 0$ and $N = 1 / 2$ into [F2] gives respectively $α^{'} M^{2} = - 1 / 2$ and $α^{'} M^{2} = 0$ . Since $α^{'} > 0$ , the first has negative mass squared and the second has $M^{2} = 0$ . The calculation checks only the level and GSO criteria stated in the example.

Remarks

The first NS excitation passes the displayed sector projection, but this short calculation does not separately check BRST closure or all residual super-Virasoro constraints. It is not a complete physical-state classification.

Open NS Vacuum Before the GSO Projection

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources