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Closed RNS Light-Cone Mass Shell and Level Matching

Physical scope

Free flat ten-dimensional closed RNS string, in a nonzero-p^+ light-cone chart and the stated zeta-regularized normal ordering; GSO, BRST, interactions, and higher-genus sewing are excluded.

Statement

For the flat ten-dimensional closed RNS string in light-cone gauge, let $N_{L}, N_{R}$ be the left and right transverse oscillator numbers, and let $a_{N S} = 1 / 2$ , $a_{R} = 0$ . In any sector pair $(s_{L}, s_{R}) \in {N S, R}^{2}$ , the two zero-mode constraints give $\frac{α^{'} M^{2}}{4} = N_{L} - a_{s_{L}} = N_{R} - a_{s_{R}} .$ Their equality is level matching. This is the free-string formula sector by sector before the chosen GSO combination.

Statement references

Timo Weigand, Introduction to String Theory, Chapter 6.6

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.3 and 3.1.4

Facts & Assumptions

Given: Let $D \in Z_{> 0}$ and impose the local criticality condition in [F3], with a closed string, $p^{+} \neq 0$ , $p^{2} = - M^{2}$ , independent chiral modes, and the residual left and right light-cone zero-mode constraints in RNS Transverse Oscillator Fock System.

[F1]

RNS Flat Worldsheet Model: the target metric is Lorentzian and $α^{'}$ has dimension length squared.

[F2]

RNS Spin Structures and Sectors: each chirality has NS half-integer or R integer fermion modes.

[F3]

RNS Local Central-Charge Cancellation: the local flat RNS central term cancels exactly at $D = 10$ .

[F4]

RNS Transverse Oscillator Fock System: there are $D - 2$ transverse modes in each chirality, the closed zero modes are $α_{0}^{μ} = {\tilde{α}}_{0}^{μ} = \sqrt{α^{'} / 2} p^{μ}$ , and each chiral constraint is $\frac{1}{2} α_{0}^{2} + N + A_{B} + A_{F} = 0$ with the displayed zeta prescription.

[F5]

Hurwitz Zeta Values for RNS Normal Ordering: $ζ (- 1, 1) = - 1 / 12$ and $ζ (- 1, 1 / 2) = 1 / 24$ .

Proof

1.1givenF3F4F5algebra

By [F3], $D = 10$ , hence there are eight transverse bosonic modes per chirality by [F4]. Their vacuum term is $A_{B} = 4 ζ (- 1, 1) = - 1 / 3$ by [F5].

2.1givenF2F4F5step 1.1algebra

For an NS chirality, [F2] gives positive frequencies $n + 1 / 2$ , so $A_{F}^{N S} = - 4 ζ (- 1, 1 / 2) = - 1 / 6$ by [F4]–[F5]; thus $A_{N S} = - 1 / 2$ and $a_{N S} : = - A_{N S} = 1 / 2$ . For an R chirality, the positive frequencies are integers, so $A_{F}^{R} = - 4 ζ (- 1, 1) = + 1 / 3$ and $A_{R} = 0$ , hence $a_{R} = 0$ . R zero modes have frequency zero.

3.1givenF1F4step 2.1algebra

For either chirality, [F4] gives $\frac{1}{2} α_{0}^{2} = α^{'} p^{2} / 4 = - α^{'} M^{2} / 4$ . Its constraint becomes $- α^{'} M^{2} / 4 + N - a_{s} = 0$ , where $a_{s}$ is the value from step 2.1. Therefore $α^{'} M^{2} / 4 = N - a_{s}$ separately on the left and right.

4.1givenF2F4step 3.1algebra∎

The same momentum $p^{μ}$ occurs in both closed zero modes by [F4]. Equating the two expressions for $α^{'} M^{2} / 4$ from step 3.1 gives $N_{L} - a_{s_{L}} = N_{R} - a_{s_{R}}$ , exactly the level-matching condition. No open-string zero-mode normalization or post-GSO restriction has been used.

Remarks

For NS-NS, level matching reads $N_{L} - 1 / 2 = N_{R} - 1 / 2$ ; for R-R it reads $N_{L} = N_{R}$ . Mixed NS-R and R-NS sectors have the corresponding half-unit shift. The equality is a consequence of the two free closed-string constraints, not modular invariance or an interacting sewing theorem.

Closed RNS Light-Cone Mass Shell and Level Matching

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources