Conditional RNS Physical-State BRST Cohomology

Fix the local criticality condition $c_{\mathrm{t}\mathrm{o}\mathrm{t}}=0$ in each chiral matter-plus-ghost system. The local central-charge calculation gives $c_{\mathrm{t}\mathrm{o}\mathrm{t}}=3D\mathrm{/}2-15$, so this condition is equivalent to $D=10$. This criticality assumption is separate from BRST nilpotence: even at $D=10$, this definition independently assumes that the displayed $Q_B$ is nilpotent and that the relative subspace is $Q_B$-stable; cancellation of the local central term alone is not asserted to prove either property.

At a fixed target momentum $p\in {\mathbb{R}}^{1,9}$ of dimension $L^{-1}$, take the chiral matter, $bc$, and bosonized $\beta \gamma$ algebraic Fock spaces of the flat RNS model. Let ${\mathcal{C}}_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{p}\mathrm{i}\mathrm{c}}(p)$ be their direct sum over the chosen NS or R scalar-exponent picture sectors, and let ${\mathcal{C}}_{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}}(p)=\ker {\eta }_0$ inside this sum. All operators below act on the supplied common invariant algebraic domain and preserve $p$; suppress $(p)$ in the chiral formulas. With the declared matter stress tensor $T_m$ and supercurrent $G_m$, define the charge on this total space, with $G_m$ twice the half-normalized source supercurrent, by $Q_B=Q_0+Q_1+Q_2,$ $Q_0=\oint \frac{dz}{2\pi i}[c(T_m+T_{\beta \gamma })+bc\mathrm{\partial }c],Q_1=-\frac{1}{2}\oint \frac{dz}{2\pi i}\gamma G_m,Q_2=-\frac{1}{4}\oint \frac{dz}{2\pi i}b{\gamma }^2\mathrm{.}$ Each term has ghost number $+1$; in the displayed bosonization they have picture shifts $0,1,2$, respectively. Closed strings use independent left and right copies. Fix the canonical representative pictures $-1$ in NS and $-1\mathrm{/}2$ in R when a fixed-picture representative is required.

Assume as an explicit input that this $Q_B$ is an odd differential on the chosen small total state space, $Q_B^2=0$, and that the relative subspace ${\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l}}=\{v\in {\mathcal{C}}_{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}}:b_{0}v=0,L_0^{\mathrm{t}\mathrm{o}\mathrm{t}}v=0\}$ is $Q_B$-stable. Here $L_0^{\mathrm{t}\mathrm{o}\mathrm{t}}=\oint \frac{dz}{2\pi i}z(T_m+T_{bc}+T_{\beta \gamma })$ is the covariant total stress-tensor zero mode in that chirality.

To select a GSO sector, additionally supply an even complex-linear involution $\widehat{G}$ on the total state space, preserving ghost number, the small space and ${\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l}}$. Require it to implement the matter sign action ${\psi }^{\mu }\mapsto -{\psi }^{\mu }$, to fix $X,b,c,\varphi$, and to send $\eta ,\xi \mapsto -\eta ,-\xi$, hence $\beta ,\gamma \mapsto -\beta ,-\gamma$. Fix its vacuum phases so that on canonical-picture matter states tensored with the chosen ghost vacuum it agrees with the NS or R involution of the GSO definition. Existence of an implementing involution with these domain and phase properties is additional conditional input, not a conclusion about arbitrary ghost modules. Each displayed $Q_j$ is invariant under this action: $Q_0$ contains invariant stresses, $Q_1$ contains the two sign-changing factors $\gamma ,G_m$, and $Q_2$ contains ${\gamma }^2$. Thus $[\widehat{G},Q_B]=0$. Using the matter involution alone would not give this commutation.

For a retained sign $g\in \{+1,-1\}$ put ${\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{g}}=\ker (\widehat{G}-gI)\cap {\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l}}$. It is a graded subcomplex by the preceding commutation. Define the open physical state space to be $H^1(Q_B,{\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{g}})=\frac{\ker (Q_B:{\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{g}}^1\to {\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{g}}^2)}{Q_B({\mathcal{C}}_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{g}}^0)},$ where superscripts denote ghost number, and $g=+1$ for the chiral supersymmetric projection. Nilpotence makes the denominator a subspace of the numerator.

For closed strings, take both chiral complexes at the same $p$, with the closed-string matter zero modes ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{p}^{\mu }$. Write $H_{\chi ,s,g}^1(p)$ for the above relative cohomology in chirality $\chi \in \{L,R\}$, sector $s\in \{NS,R\}$ and retained sign $g$. On the tensor product of relative complexes define the covariant level-difference operator $\mathrm{\Delta }=L_{0,L}^{\mathrm{t}\mathrm{o}\mathrm{t}}\otimes I-I\otimes L_{0,R}^{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{.}$ Each summand vanishes on its relative complex, so $\mathrm{\Delta }=0$ there and induces the zero operator on $H_{L,s_L,g_L}^1(p)\otimes H_{R,s_R,g_R}^1(p)$. Thus covariant level matching $\mathrm{\Delta }=0$ is already imposed by the two relative constraints; it requires no additional restriction by transverse oscillator operators. For a chosen Type II or Type 0 pairing, let $\mathcal{S}$ be its finite set of allowed tuples $(s_L,g_L;s_R,g_R)$, using the GSO conventions above, and define ${\mathcal{H}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}={⨁}_{p\in {\mathbb{R}}^{1,9}}{⨁}_{(s_L,g_L;s_R,g_R)\in \mathcal{S}}{H}_{L,s_L,g_L}^1(p){\otimes }_{\mathbb{C}}{H}_{R,s_R,g_R}^1(p)\mathrm{.}$ The momentum sum is algebraic: each vector has finite support in $p$; no momentum measure or Hilbert completion is asserted. This defines the selected state spaces only under the supplied differential, domain and ghost-inclusive involution hypotheses. The transverse equality $N_L-a_{s_L}=N_R-a_{s_R}$ belongs to the separately defined light-cone system; no map from its Fock spaces to these cohomologies is assumed or proved here.

This is a conditional cohomology definition. It does not assert that $Q_B$ is nilpotent from central-charge cancellation alone, establish picture-changing as an isomorphism at exceptional momentum, prove positivity/no-ghost, or define a global measure.

RNS Flat Worldsheet Model

RNS Spin Structures and Sectors

RNS GSO Projections

RNS Local Central-Charge Cancellation

Bosonic Bc Ghost System

RNS Beta-Gamma Superghost and Picture System

Local RNS Free-Field Operator Conventions

RNS Transverse Oscillator Fock System

Open RNS Light-Cone Intercepts

Closed RNS Light-Cone Mass Shell and Level Matching

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §3.1.2

Timo Weigand, Introduction to String Theory, Chapter 6.4

Schlotterer gives the charge decomposition with its half-normalized matter supercurrent. Since the local convention here is $G_m=2G_{\mathrm{S}\mathrm{c}\mathrm{h}}$, its $-\oint \gamma G_{\mathrm{S}\mathrm{c}\mathrm{h}}$ becomes $-\frac{1}{2}\oint \gamma G_m$; the $Q_2$ coefficient is unchanged. The source describes physical states as nilpotent cohomology, but the inspected section does not give a complete local proof of $Q_B^2=0$ or of stability of the stated small relative subspace. The bosonic BRST nilpotence theorem Bosonic BRST Nilpotence is a different system and is not used here. The exact missing RNS-specific proof is escalated in the batch notes; this definition remains explicitly conditional.