Local RNS Free-Field Operator Conventions

Use one chiral Euclidean coordinate $z$ and write $u=z-w$. The target metric is ${\eta }_{\mu \nu }=\mathrm{diag}(-,+,\dots ,+)$, with target indices contracted using $\eta$. Fix the free-field OPEs $X^{\mu }(z)X^{\nu }(w)\sim -\frac{{\alpha }^{\mathrm{\prime }}}{2}{\eta }^{\mu \nu }\log u,{\psi }^{\mu }(z){\psi }^{\nu }(w)\sim \frac{{\eta }^{\mu \nu }}{u}\mathrm{.}$ The corresponding derivative contraction is $\mathrm{\partial }{X}^{\mu }(z)\mathrm{\partial }{X}^{\nu }(w)\sim -\frac{{\alpha }^{\mathrm{\prime }}}{2}{\eta }^{\mu \nu }{u}^{-2}$. Normal ordering is radial point-splitting with these singular parts subtracted. Define $T_m=T_X+T_{\psi },T_X=-\frac{1}{{\alpha }^{\mathrm{\prime }}}:\mathrm{\partial }X\cdot \mathrm{\partial }X:,T_{\psi }=-\frac{1}{2}:\psi \cdot \mathrm{\partial }\psi :,G_m=i\sqrt{\frac{2}{{\alpha }^{\mathrm{\prime }}}}:\psi \cdot \mathrm{\partial }X:\mathrm{.}$ The $N=1$ current convention is $G_m(z)G_m(w)\sim D{u}^{-3}+2T_m(w)u^{-1}$, matched to the declared $N=1$ mode-algebra central normalization. The matter stress convention is $T(z)T(w)\sim \frac{c}{2}{u}^{-4}+2T(w)u^{-2}+\mathrm{\partial }T(w)u^{-1}$.

For the reparametrization ghosts use anticommuting $b,c$ fields of weights $(2,-1)$ with OPE $b(z)c(w)\sim (z-w{)}^{-1}$, and stress tensor $T_{bc}=-2:b\mathrm{\partial }c:-:\mathrm{\partial }bc:$. This is the local free-field convention consumed by the RNS criticality calculation; closed strings use an independent barred copy. It defines neither a global determinant nor a functional measure.

RNS Flat Worldsheet Model

Flat RNS Action Euler–Lagrange Equations

Bosonic Bc Ghost System

$N=1$ superconformal mode algebra in NS and Ramond sectors

Timo Weigand, Introduction to String Theory, Chapter 6

Paul Ginsparg, Applied Conformal Field Theory, §5.4

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.1.5–2.1.6

The local criticality theorem RNS Local Central-Charge Cancellation derives the matter charge $c_m=3D\mathrm{/}2$ from these OPE conventions. The Majorana OPE and stress normalization are compatible with the single-real-fermion component convention in Free two-dimensional fermion CFT with spin sectors. That supplier's nonchiral common-spin-structure sum is not the independent left/right RNS sector sum. The Lorentzian free-field equations in Flat RNS Action Euler–Lagrange Equations identify the chiral on-shell restriction; these Euclidean OPEs and the $N=1$ current are the quantized free-field conventions, with the factor of $i$ fixed by the latter's normalization. Schlotterer's §2.1.6 current is $G_{\mathrm{S}\mathrm{c}\mathrm{h}}=i(2{\alpha }^{\mathrm{\prime }}{)}^{-1\mathrm{/}2}:\psi \cdot \mathrm{\partial }X:$, while $G_m=2G_{\mathrm{S}\mathrm{c}\mathrm{h}}$ here. Ginsparg's component OPE $T_F(z)T_F(w)\sim c\mathrm{/}6(z-w{)}^{-3}+\frac{1}{2}T(w)(z-w{)}^{-1}$ therefore gives $G_m(z)G_m(w)\sim (2c\mathrm{/}3)(z-w{)}^{-3}+2T(w)(z-w{)}^{-1}$; with $c=3D\mathrm{/}2$ the leading coefficient is $D$, as stated above.