RNS bosonized picture charge convention

In each chiral RNS sector use the bosonization convention $\beta =e^{-\varphi }\mathrm{\partial }\xi$, $\gamma =\eta e^{\varphi }$, $\varphi (z)\varphi (w)\sim -\log (z-w)$, and $T_{\varphi }=-\frac{1}{2}(\mathrm{\partial }\varphi {)}^2-{\mathrm{\partial }}^2\varphi$. An operator $e^{p\varphi }$ has picture charge $p$. The small Hilbert space is $\ker {\eta }_0$; picture number is a grading distinct from the $\beta \gamma$ ghost number. In this convention the canonical NS and R representatives have pictures $-1$ and $-\frac{1}{2}$, respectively.

For formal chiral-correlator bookkeeping on a compact connected closed genus-$g$ worldsheet with Euclidean metric $h$, write $K_h$ for its Gaussian curvature and $d{A}_h$ for its Riemannian area form, and set the scalar curvature convention to $R^{(2)}(h):=2K_h$. Thus Gauss–Bonnet gives ${\int }_{\mathrm{\Sigma }}{R}^{(2)}(h)d{A}_h=4\pi \chi (\mathrm{\Sigma })$. Adopt the bosonized background term $S_{\varphi ,\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}}=-\frac{Q_{\varphi }}{8\pi }{\int }_{\mathrm{\Sigma }}{R}^{(2)}(h)\varphi d{A}_h,Q_{\varphi }=2.$ Split $\varphi ={\varphi }_0+\tilde{\varphi }$. Define the formal zero-mode Ward functional to retain a chiral correlator only when the total coefficient of ${\varphi }_0$ vanishes. This is local CFT correlation data; it is not a functional measure over moduli or a claim that a global path integral exists.

The picture-charge selection rule is proved conditionally in RNS chiral picture-charge selection rule. The local PCO operation is defined in RNS picture-changing insertion on a local superstring chart; neither item chooses global PCO locations or constructs a measure.