Neveu–Schwarz sector

Antiperiodic worldsheet Majorana fermions have half-integer modes in this sector.

The lowest fermionic excitation has level $1/2$. Its old-covariant physical conditions are ${\alpha }^{\prime }{p}^2+1/2-a=0$ and $p\cdot A=0$. For nonzero real momentum, timelike $p$ gives a positive polarization space, null $p$ gives a nonnegative space with a longitudinal null direction, and spacelike $p$ permits a negative-norm time polarization. The limiting $a=1/2$ state is a massless gauge vector after quotienting null polarizations. This level-specific test does not prove positivity of every higher NS level.

The usual half-level NS claim that all physical vector polarizations have nonnegative norm for $a\le 1/2$ assumes nonzero particle momentum. At zero momentum and limiting intercept, all positive-mode constraints and transversality allow a timelike polarization with negative norm, and $A\sim A+\kappa p$ removes nothing. Excluding this exceptional momentum restores the ordinary null-vector argument; retaining it makes the unrestricted elementary criterion strict $a<1/2$.

The open NS action has integer bosonic modes and half-integer fermionic matter modes, together with even Virasoro and odd supercurrent multipliers. Opposite fermion signs at the two endpoints make the doubled fermion antiperiodic and remove a matter zero mode. The $b_r$ matter variables are distinct from the $bc$ reparameterization FP ghosts. Canonical quantization yields $[{\alpha }_k^m,{\alpha }_l^n]=k{\eta }^{mn}{\delta }_{k+l,0}$ and $\{b_r^m,b_s^n\}={\eta }^{mn}{\delta }_{r+s,0}$.

The unprojected level-one states form a transverse vector representation and an exterior square. Fermionic anticommutation makes the two-fermion states antisymmetric. Together they restrict from an antisymmetric rank-two tensor of $SO(D-1)$, not a massive spin-two field.

Before the GSO projection, $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k+{\sum }_{r>0}r{b}_{-r}\cdot b_r$ takes nonnegative half-integer values. The first vector at $N=1/2$ is massless for string intercept $a_{\mathrm{NS}}=1/2$.

Antiperiodic worldsheet Majorana fermions give transverse modes $b_r^I$, $r\in \mathbb{Z}+\frac{1}{2}$, with $\{b_r^I,b_s^J\}={\delta }^{IJ}{\delta }_{r+s,0}$. Positive modes annihilate the oscillator vacuum; negative modes are fermionic creation operators.