RNS GSO Projections

Take the ten-dimensional flat target $M={\mathbb{R}}^{1,9}$ with $\eta =\mathrm{diag}(-,+,\dots ,+)$ and orientation $d{x}^0\wedge \cdots \wedge d{x}^9$. Fix Clifford generators ${\mathrm{\Gamma }}^{\mu }$ on a complex target Dirac module with $\{{\mathrm{\Gamma }}^{\mu },{\mathrm{\Gamma }}^{\nu }\}=2{\eta }^{\mu \nu }I$, and define the chirality involution by ${\mathrm{\Gamma }}_{11}={\mathrm{\Gamma }}^0{\mathrm{\Gamma }}^1\cdots {\mathrm{\Gamma }}^9$; it squares to $I$, and reversing the chosen target orientation reverses its sign. In a single chiral sector let $F$ count nonzero fermionic oscillator excitations modulo two, with the NS vacuum assigned $F=0$; Ramond zero modes are not counted in $F$ and act through the target Clifford module. Set $G_{\mathrm{N}\mathrm{S}}=(-1{)}^{F+1}\mathrm{.}$ For the Ramond sector choose an independent sign ${ϵ}_{\chi }\in \{+1,-1\}$ for each closed chirality $\chi \in \{L,R\}$; for an open string use a single label $\chi =O$ for its doubled chiral sector and choose one sign ${ϵ}_O$. Set $G_{\mathrm{R}}^{(\chi )}={ϵ}_{\chi }{\mathrm{\Gamma }}_{11}(-1{)}^{F_{\chi }}\mathrm{.}$ The chiral supersymmetric projection retains the $+1$ eigenspace of $G_{\mathrm{N}\mathrm{S}}$ or the selected $G_{\mathrm{R}}^{(\chi )}$. It removes the NS vacuum. On a Ramond ground state, $F_{\chi }=0$, so the retained target chirality is the ${\mathrm{\Gamma }}_{11}$ eigenspace with eigenvalue ${ϵ}_{\chi }$; the projector equation gives the opposite ground-state chirality for odd $F_{\chi }$. For closed Type IIB choose ${ϵ}_L={ϵ}_R=+1$, so the retained Ramond chiralities agree; for Type IIA choose ${ϵ}_L=+1$ and ${ϵ}_R=-1$, so they are opposite. Reversing both signs gives the primed presentation of the same relative Type II choice. For an open string, use the endpoint conditions and doubled single chiral sector defined in RNS Spin Structures and Sectors, and apply one such projection.

Write $N{S}^{\pm }$ for the $G_{\mathrm{N}\mathrm{S}}=\pm 1$ eigenspaces and $R^{\pm }$ for Ramond sectors whose ground state has ${\mathrm{\Gamma }}_{11}=\pm 1$. The Type 0A closed sector set is $(N{S}^{+},N{S}^{+}),(N{S}^{-},N{S}^{-}),(R^{+},R^{-}),(R^{-},R^{+})$; Type 0B is $(N{S}^{+},N{S}^{+}),(N{S}^{-},N{S}^{-}),(R^{+},R^{+}),(R^{-},R^{-})$. Thus both retain the NS-NS vacuum, remove the mixed NS-R and R-NS sectors, and have no spacetime fermions. These are sector-space definitions; the torus spin-structure sum, modular invariance and locality of projected operator products are separate checks, not consequences of the notation $G$.

RNS Spin Structures and Sectors

Free two-dimensional fermion CFT with spin sectors

Timo Weigand, Introduction to String Theory, Chapters 6.5–6.7

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §3.1.7

The signs use the NS/R mode convention in RNS Spin Structures and Sectors. The common-spin sum in Free two-dimensional fermion CFT with spin sectors belongs to a different nonchiral theory and is not the Type II/0 projection defined here. No modular-invariance or operator-locality theorem is claimed.