RNS picture-changing insertion on a local superstring chart

Fix a closed RNS model in one local supermoduli chart, with the selected spin structure, GSO projection, beta-gamma bosonization, nilpotent BRST operator, and picture grading. For this local operation additionally supply a complex super vector space ${\mathcal{D}}_L$ (the large algebraic state domain), odd endomorphisms $Q_L,{\eta }_0,\xi (z)$ on that common invariant domain, and the identities $Q_L^2={\eta }_0^2=0$, ${\eta }_0{Q}_L+Q_L{\eta }_0=0$, and ${\eta }_0\xi (z)+\xi (z){\eta }_0=I$. Products here are algebraic endomorphism compositions; a field realization may use formal Laurent coefficients or separated, smeared insertions, and must supply that realization and its invariant domain. Set ${\mathcal{D}}_S:=\ker {\eta }_0$ and require $Q_L{\mathrm{\mid }}_{{\mathcal{D}}_S}=Q_B$ on the supplied small complex. This is an additional extension hypothesis, not a consequence of a differential given only on ${\mathcal{D}}_S$. Define $X_L(z):=Q_L\xi (z)+\xi (z)Q_L$ on ${\mathcal{D}}_L$ and the local small-space insertion $X(z):=X_L(z){\mathrm{\mid }}_{{\mathcal{D}}_S}$; the restriction is justified below.

For this operation supply a total picture grading $P$ on ${\mathcal{D}}_L$ with $Q_L$ of degree $0$, ${\eta }_0$ of degree $-1$ and $\xi (z)$ of degree $+1$. It agrees with the scalar exponent $p$ on $e^{p\varphi }$ and assigns degrees $-1,+1$ to $\eta ,\xi$, respectively; the remaining matter and $bc$ fields have degree $0$. This refines the scalar-exponent bookkeeping used in the bosonization supplier: the scalar exponent alone is not a homogeneous grading for all BRST terms. Indeed $\gamma =\eta e^{\varphi }$ and $\beta =e^{-\varphi }\mathrm{\partial }\xi$ have total picture $0$, so the displayed BRST charge can have degree $0$. For external eigenoperators with additional $\eta ,\xi$ factors, use their total $P$ eigenvalue, not just the $\varphi$ exponent. The charge-neutrality assumption for this local correlator is that its total picture equals $2g-2$, extending the supplied exponential-insertion rule. When only exponential picture carriers occur externally it is precisely that rule. Compatibility with any relative or GSO restrictions must be supplied for the intended external representatives; no picture isomorphism on those restricted cohomologies is inferred. In each chiral sector, if external total pictures are $p_1,\dots ,p_n$ and $N_{\mathrm{P}\mathrm{C}\mathrm{O}}=2g-2-{\sum }_i{p}_i\in {\mathbb{Z}}_{\ge 0},$ then this is the number of local picture-changing insertions required by the charge-selection rule. The right-moving sector is counted independently. This definition does not choose insertion points, resolve collisions or spurious singularities, define a vertical integration chain, or construct a global superstring amplitude form.

Wang–Yin construct local closed-Type-II/on-shell PCO interpolation chains over the interior of ordinary moduli in their stated setup. Their paper explicitly sets aside ordinary-moduli boundary treatment. The present definition does not misclassify that interior result as unresolved; the remaining all-genus question about regulator-independent finite values across compactification boundary strata is stated in Global compactified Type II RNS amplitude prescription ‡.