RNS Flat Worldsheet Model

Adopt the flat-target RNS model on a Lorentzian two-dimensional worldsheet with dimensionless coordinates and metric of signature $(-,+)$. Let $D\in {\mathbb{Z}}_{>0}$, set $M={\mathbb{R}}^{1,D-1}$ with $\eta =\mathrm{diag}(-,+,\dots ,+)$, and represent a local coordinate patch by a bounded open rectangle $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with coordinates $(\tau ,\sigma )$ and measure $d^2\sigma$. Fix a finite-dimensional unital real Grassmann algebra $\mathrm{\Lambda }$ with odd subspace ${\mathrm{\Lambda }}_{\bar{1}}$. The bosonic field is a map $X:\bar{\mathrm{\Omega }}\to M$ of class $C^2$, with components $X^{\mu }$ of length dimension; the fermionic field is a map $\psi :\bar{\mathrm{\Omega }}\to {\mathrm{\Lambda }}_{\bar{1}}\otimes {\mathbb{R}}^{1,D-1}\otimes {\mathbb{R}}^2$ of class $C^1$, with dimensionless components ${\psi }^{\mu }$. The factor ${\mathbb{R}}^2$ is the real worldsheet Majorana representation fixed below. Grassmann-valued integrals are taken coefficientwise. For $p\in \mathrm{\Omega }$, $d{X}_p:{\mathbb{R}}^2\to M$ denotes the derivative of $X$ and $(X^{\ast }\eta {)}_p(u,v)=\eta (d{X}_{p}u,d{X}_{p}v)$ for $u,v\in {\mathbb{R}}^2$ is its induced bilinear form. The adopted RNS field domain imposes neither rank two on $d{X}_p$ nor nondegeneracy of $(X^{\ast }\eta {)}_p$. Set ${\alpha }^{\mathrm{\prime }}=1\mathrm{/}(2\pi T)>0$ with dimension $L^2$, where $T$ is the string tension of dimension $L^{-2}$ in units $\mathrm{\hslash }=c=1$; with the stated coordinate and field dimensions, the action is dimensionless. In superconformal gauge adopt

$S=-\frac{1}{8\pi }{\int }_{\mathrm{\Omega }}{d}^2\sigma [\frac{2}{{\alpha }^{\mathrm{\prime }}}{\mathrm{\partial }}_a{X}^{\mu }{\mathrm{\partial }}^a{X}_{\mu }+2i{\bar{\psi }}^{\mu }{\rho }^a{\mathrm{\partial }}_a{\psi }_{\mu }],$

with $\{{\rho }^a,{\rho }^b\}=2{\eta }^{ab}I$ and the real representation ${\rho }^0=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, ${\rho }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, $\bar{\psi }={\psi }^T{\rho }^0$. Write ${\psi }^{\mu }=({\psi }_{+}^{\mu },{\psi }_{-}^{\mu }{)}^T$ in this ordered representation basis. Define $x^{\pm }=\tau \pm \sigma$ and ${\mathrm{\partial }}_{\pm }=\frac{1}{2}({\mathrm{\partial }}_{\tau }\pm {\mathrm{\partial }}_{\sigma })$. Adopt the residual stress and supercurrent conventions $T_{++}=-\frac{1}{{\alpha }^{\mathrm{\prime }}}{\mathrm{\partial }}_{+}X\cdot {\mathrm{\partial }}_{+}X-\frac{i}{2}{\psi }_{+}\cdot {\mathrm{\partial }}_{+}{\psi }_{+},T_{--}=-\frac{1}{{\alpha }^{\mathrm{\prime }}}{\mathrm{\partial }}_{-}X\cdot {\mathrm{\partial }}_{-}X-\frac{i}{2}{\psi }_{-}\cdot {\mathrm{\partial }}_{-}{\psi }_{-},J_{\pm }=-\frac{1}{2}\sqrt{\frac{2}{{\alpha }^{\mathrm{\prime }}}}{\psi }_{\pm }\cdot {\mathrm{\partial }}_{\pm }X\mathrm{.}$ The worldsheet metric and gravitino equations are retained as residual constraints $T_{++}=T_{--}=J_{+}=J_{-}=0$. These formulas are adopted gauge-fixed model data matched to Weigand, Chapter 6.1. Their Noether derivation from variations of the parent metric/gravitino action is not claimed here. This physical-model postulate asserts neither a spacetime supersymmetry theorem nor a global functional measure.

Timo Weigand, Introduction to String Theory, Chapter 6.1

The bosonic normalization uses the ${\alpha }^{\mathrm{\prime }}$ and tension convention in String Tension Alpha Prime, but the bosonic supplier's $D\ge 3$ immersive-map branch is not imported into this gauge-fixed RNS field domain. The dimensionless $\psi$ convention here matches the dimensionless worldsheet coordinates: if source coordinates $\xi$ have length dimension and ${\psi }_{\xi }$ has mass dimension $1\mathrm{/}2$, then for a chosen coordinate length unit ${\mathrm{\ell }}_0>0$ with $[{\mathrm{\ell }}_0]=L$, the rescaling $\sigma =\xi \mathrm{/}{\mathrm{\ell }}_0$ and $\psi ={\mathrm{\ell }}_0^{1\mathrm{/}2}{\psi }_{\xi }$ gives the dimensionless coefficient used above and preserves the displayed kinetic normalization. The Euler–Lagrange equations are derived conditionally from this action in Flat RNS Action Euler–Lagrange Equations. The stress and supercurrent formulas are adopted source-backed conventions; their parent metric/gravitino Noether derivation is not attributed to the gauge-fixed action here. The local Euclidean OPE normalization used by the quantum calculations is fixed separately in Local RNS Free-Field Operator Conventions. This is a classical local field domain; global Euclidean spinor bundles and sectors are supplied separately, and quantum OPE items define their formal operator fields independently.