Neveu–Schwarz Fourier phase-space action ($S=\int (\dot{x}\cdot p+{\sum }_{k>0}i{\alpha }_{-k}{\dot{\alpha }}_k/k+i{\sum }_{r>0}{b}_{-r}{\dot{b}}_r-\lambda L-i\chi G)dt$)

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 $\{b_r^m,b_s^n\}={\eta }^{mn}{\delta }_{r+s,0}$.