RNS Beta-Gamma Superghost and Picture System

For one RNS chirality, take a commuting first-order $\beta ,\gamma$ system of weights $h_{\beta }=3\mathrm{/}2$ and $h_{\gamma }=-1\mathrm{/}2$, action $S_{\beta \gamma }=(4\pi {)}^{-1}\int d^{2}z\beta \bar{\mathrm{\partial }}\gamma$, and ordered OPE $\gamma (z)\beta (w)\sim (z-w{)}^{-1}$. Its modes are $\beta (z)={\sum }_r{\beta }_r{z}^{-r-3\mathrm{/}2},\gamma (z)={\sum }_r{\gamma }_r{z}^{-r+1\mathrm{/}2},$ with $r\in \mathbb{Z}+\frac{1}{2}$ in NS and $r\in \mathbb{Z}$ in R, $[{\gamma }_r,{\beta }_s]={\delta }_{r+s,0}I$, and all same-species mode commutators zero. The stress tensor is $T_{\beta \gamma }=-\frac{3}{2}:\mathrm{\partial }\gamma \beta :-\frac{1}{2}:\gamma \mathrm{\partial }\beta :\mathrm{.}$

Fix the bosonized convention $\beta =e^{-\varphi }\mathrm{\partial }\xi$, $\gamma =\eta e^{\varphi }$, with $\varphi (z)\varphi (w)\sim -\log (z-w)$ and $\eta (z)\xi (w)\sim (z-w{)}^{-1}$. The fermionic $(\eta ,\xi )$ weights are $(1,0)$ and $T_{\eta \xi }=-:\eta \mathrm{\partial }\xi :$; the scalar stress tensor is $T_{\varphi }=-\frac{1}{2}:(\mathrm{\partial }\varphi {)}^2:-{\mathrm{\partial }}^2\varphi$, giving $h(e^{p\varphi })=-p(p+2)\mathrm{/}2$. Define ghost number by $j_{\mathrm{g}\mathrm{h}}=-:\beta \gamma :$ and picture by $j_{\mathrm{p}\mathrm{i}\mathrm{c}}=-\mathrm{\partial }\varphi$, so $\beta ,\gamma$ have ghost numbers $-1,+1$ and $e^{p\varphi }$ has picture $p$. The canonical NS and R vacua have pictures $-1$ and $-1\mathrm{/}2$. The small Hilbert space is $\ker {\eta }_0$, excluding the independent ${\xi }_0$ zero-mode partner. Closed strings use an independent antiholomorphic copy.

RNS Flat Worldsheet Model

RNS Spin Structures and Sectors

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.3.2–2.3.6

The displayed stress tensor and bosonization are formulation conventions. Their local central charge is calculated separately in RNS Local Central-Charge Cancellation. Nilpotence of an RNS BRST charge and equivalence of pictures at exceptional momenta are not supplied by this definition.