RNS Transverse Oscillator Fock System

For $D\ge 3$, use the target metric $\eta =\mathrm{diag}(-,+,\dots ,+)$ and momentum convention of the bosonic matter oscillator definition: $p=(p^{\mu })\in {\mathbb{R}}^{1,D-1}$ is a contravariant target momentum with dimension $L^{-1}$. Work at fixed $p$ in the light-cone chart with $p^{+}=(p^0+p^{D-1})\mathrm{/}\sqrt{2}\mathrm{\ne }0$, and put $i=1,\dots ,D-2$. For open strings impose NN boundary conditions in every target direction; for closed strings use independent left and right mode systems. On the finite-particle algebraic Fock domain, transverse bosonic modes obey $[{\alpha }_m^i,{\alpha }_n^j]=m{\delta }^{ij}{\delta }_{m+n,0}I,$ and the transverse fermions obey $\{{\psi }_r^i,{\psi }_s^j\}={\delta }^{ij}{\delta }_{r+s,0}I,$ with $r\in \mathbb{Z}+\frac{1}{2}$ in NS and $r\in \mathbb{Z}$ in R. Closed strings have commuting barred copies. Positive modes annihilate the selected sector vacuum; R zero modes act in a chosen representation of the Euclidean Clifford algebra on $D-2$ transverse directions. No target-space spin representation is constructed here.

Use the open/closed bosonic zero-mode normalization ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\mathrm{\prime }}}{p}^{\mu }$ for open strings and ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{p}^{\mu }$ for closed strings. Define the nonnegative weighted oscillator level operator for the open system by $N=N_B+N_F,N_B={\sum }_{i=1}^{D-2}{\sum }_{n\ge 1}{\alpha }_{-n}^i{\alpha }_n^i,$ so the number operator ${\alpha }_{-n}^i{\alpha }_n^i\mathrm{/}n$ at mode $n$ contributes level $n$, and $N_F={\sum }_i{\sum }_{r>0}r{\psi }_{-r}^i{\psi }_r^i\mathrm{.}$ In R the zero modes do not contribute to $N_F$. The sums act finitely on each finite-particle vector. For closed strings define $N_L=N_{B,L}+N_{F,L}$ and $N_R=N_{B,R}+N_{F,R}$ by these same formulas applied separately to the left and right mode families. The normal-ordering constant is not included in any level operator and is specified separately below.

For the selected light-cone Hamiltonian prescription, the unregularized open-string vacuum terms are $A_B=\frac{D-2}{2}{\sum }_{n\ge 1}n$ and $A_F=-\frac{D-2}{2}{\sum }_{r>0}r$, where $r$ ranges over positive half-integers in NS and positive integers in R. For closed strings define $A_{B,L},A_{F,L}$ and $A_{B,R},A_{F,R}$ by the same sums for the corresponding left and right mode families, and set $A_L=A_{B,L}+A_{F,L}$ and $A_R=A_{B,R}+A_{F,R}$. The divergent sums are assigned zeta values as a stated regularization prescription; a separate lemma proves the needed continuation values. This prescription is part of the quantization convention, not an empirical measurement.

Impose the residual chiral zero-mode constraints in this gauge. For an open string $L_0^{\mathrm{l}\mathrm{c}}=\frac{1}{2}{\alpha }_0\cdot {\alpha }_0+N+A_B+A_F=0$; for a closed string impose both $L_0^{\mathrm{l}\mathrm{c}}=\frac{1}{2}{\alpha }_0^2+N_L+A_L=0$ and ${\tilde{L}}_0^{\mathrm{l}\mathrm{c}}=\frac{1}{2}{\tilde{\alpha }}_0^2+N_R+A_R=0$. These are the gauge-fixed free-string state constraints, not BRST cohomology claims.

RNS Flat Worldsheet Model

Flat RNS Action Euler–Lagrange Equations

Bosonic Matter Oscillator Modes

RNS Spin Structures and Sectors

Hurwitz Zeta Values for RNS Normal Ordering

Timo Weigand, Introduction to String Theory, Chapters 6.3–6.6

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.2 and 3.1.4

The commutator and open/closed zero-mode factors agree with Bosonic Matter Oscillator Modes; the fermion modes use the RNS sector convention in RNS Spin Structures and Sectors. The finite-particle Fock space is a chosen quantization convention. This definition does not assert the BRST/no-ghost identification of that space with a covariant physical-state space.