Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Open RNS Light-Cone Intercepts

Physical scope

Free flat ten-dimensional open RNS string with NN boundary conditions, in a nonzero-p^+ light-cone chart and the declared zeta-regularized normal-ordering prescription; no GSO, BRST, or no-ghost equivalence is inferred.

Statement

For the flat ten-dimensional open RNS string with NN boundary conditions in all target directions, in a light-cone chart with $p^{+} \neq 0$ and with the declared zeta-regularized normal-ordering prescription, the unprojected mass formulas are $α^{'} M^{2} = N - \frac{1}{2} in NS, α^{'} M^{2} = N in R,$ where $M^{2} = - p^{2}$ . Thus the unprojected NS vacuum is tachyonic with $M^{2} = - 1 / (2 α^{'})$ , and the R ground state is massless. The GSO projection and the identification of these states with covariant BRST cohomology are separate.

Statement references

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

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

Facts & Assumptions

Given: Let $D \in Z_{> 0}$ and impose the local criticality condition in [F3], as well as $α^{'} > 0$ , open NN boundary conditions in all target directions, $p^{+} \neq 0$ , $p^{2} = - M^{2}$ , and the residual light-cone zero-mode constraint and zeta prescription in RNS Transverse Oscillator Fock System. Keep the NS vacuum before GSO.

[F1]

RNS Flat Worldsheet Model: the target is flat $R^{1, D - 1}$ with Lorentzian metric and $α^{'}$ is the length-squared parameter in the action.

[F2]

RNS Spin Structures and Sectors: NS fermion modes are half-integral and R modes integral, with periodic R zero modes.

[F3]

RNS Local Central-Charge Cancellation: for this flat RNS model, local central-charge cancellation is equivalent to $D = 10$ .

[F4]

RNS Transverse Oscillator Fock System: there are $D - 2$ transverse oscillator directions; the open zero mode is $α_{0}^{μ} = \sqrt{2 α^{'}} p^{μ}$ , $L_{0}^{l c} = \frac{1}{2} α_{0}^{2} + N + A_{B} + A_{F} = 0$ , and the vacuum sums are $A_{B} = \frac{D - 2}{2} \sum_{n \geq 1} n$ and $A_{F} = - \frac{D - 2}{2} \sum_{r > 0} r$ under zeta regularization.

[F5]

Hurwitz Zeta Values for RNS Normal Ordering: $ζ (- 1, a) = - \frac{1}{2} (a^{2} - a + 1 / 6)$ , so $ζ (- 1, 1) = - 1 / 12$ and $ζ (- 1, 1 / 2) = 1 / 24$ .

Proof

1.1givenF3F4F5algebra

By [F3], $D = 10$ , so [F4] has eight transverse bosons. Their zero-point term is $A_{B} = \frac{8}{2} \sum_{n \geq 1} n = 4 ζ (- 1, 1) = - 1 / 3$ by [F5].

2.1givenF2F4F5step 1.1algebra

In NS the positive fermion frequencies are $r = n + 1 / 2$ , $n \geq 0$ , by [F2]. Therefore $A_{F}^{N S} = - (8 / 2) \sum_{n \geq 0} (n + 1 / 2) = - 4 ζ (- 1, 1 / 2) = - 1 / 6$ by [F4]–[F5]. The total vacuum term is $A_{N S} = - 1 / 3 - 1 / 6 = - 1 / 2$ , so its intercept, defined as $a_{N S} = - A_{N S}$ in $L_{0} = α^{'} p^{2} + N - a_{N S}$ , is $1 / 2$ .

2.2givenF2F4F5step 1.1algebra

In R the positive fermion frequencies are positive integers, again by [F2]. Hence $A_{F}^{R} = - 4 ζ (- 1, 1) = + 1 / 3$ by [F4]–[F5], and $A_{R} = A_{B} + A_{F}^{R} = 0$ , giving $a_{R} = 0$ . The R zero modes have frequency zero and make no vacuum-energy contribution.

3.1givenF1F4step 2.1step 2.2algebra∎

The open normalization in [F4] gives $\frac{1}{2} α_{0}^{2} = α^{'} p^{2} = - α^{'} M^{2}$ . The constraint $L_{0}^{l c} = 0$ is therefore $- α^{'} M^{2} + N - a = 0$ in each sector. Substituting $a_{N S} = 1 / 2$ from step 2.1 and $a_{R} = 0$ from step 2.2 gives the two stated mass formulas. At $N = 0$ they give respectively $M^{2} = - 1 / (2 α^{'})$ and $M^{2} = 0$ .

Remarks

The NS vacuum's negative mass squared is a pre-GSO result; RNS GSO Projections removes it from the chiral supersymmetric projection. This calculation is a light-cone oscillator deduction and proves neither BRST equivalence nor the no-ghost theorem.

Open RNS Light-Cone Intercepts

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources