Flat RNS Action Euler–Lagrange Equations

Given: Work locally with the Lorentzian metric and spinor conventions in RNS Flat Worldsheet Model, dimensionless $\tau ,\sigma$, and compactly supported variations. Put $\psi =({\psi }_{+},{\psi }_{-}{)}^T$.

RNS Flat Worldsheet Model: the adopted action is $-\frac{1}{8\pi }\int [(2\mathrm{/}{\alpha }^{\mathrm{\prime }}){\mathrm{\partial }}_{a}X\cdot {\mathrm{\partial }}^{a}X+2i\bar{\psi }{\rho }^a{\mathrm{\partial }}_a\psi ]d^2\sigma$.

Use the local classical field domain fixed in RNS Flat Worldsheet Model: on the bounded coordinate rectangle $\mathrm{\Omega }$, $X\in C^2(\bar{\mathrm{\Omega }};M)$ and $\psi \in C^1(\bar{\mathrm{\Omega }};{\mathrm{\Lambda }}_{\bar{1}}\otimes M\otimes {\mathbb{R}}^2)$. Take $\delta X\in C_c^2(\mathrm{\Omega };M)$. For odd variation tests extend $\mathrm{\Lambda }$ to ${\mathrm{\Lambda }}^{+}=\mathrm{\Lambda }\otimes \bigwedge (\theta )$ with one fresh odd generator $\theta$, using the graded tensor product; the fields remain in the original embedded $\mathrm{\Lambda }$. Allow independent compactly supported odd variations in ${\mathrm{\Lambda }}^{+}$, in particular $\delta {\psi }_{\pm }^{\mu }=\theta f$ for every real $f\in C_c^1(\mathrm{\Omega })$ in one component at a time. Stationarity means that every such first variation is zero in ${\mathrm{\Lambda }}^{+}{\otimes }_{\mathbb{R}}\mathbb{C}$; the complexification accommodates the factor $i$ in the action. This is the formal Grassmann variational convention, rather than testing only within a fixed finite odd algebra. Grassmann-valued integrals are taken coefficientwise. Use left Grassmann variations, so moving an odd variation past an odd field changes sign; compact support removes all boundary terms.