Chiral proof of rigid worldsheet supersymmetry ($\delta \mathcal{L}=i{ϵ}_2{\partial }_{-}({\psi }_1{\partial }_{+}X)-i{ϵ}_1{\partial }_{+}({\psi }_2{\partial }_{-}X)$)

Let ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$ and $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X+i{\psi }_1\cdot {\partial }_{+}{\psi }_1+i{\psi }_2\cdot {\partial }_{-}{\psi }_2$. For constant Grassmann-odd parameters, set $\delta X=i({ϵ}_2{\psi }_1-{ϵ}_1{\psi }_2)$, $\delta {\psi }_1=-{ϵ}_2{\partial }_{-}X$, $\delta {\psi }_2={ϵ}_1{\partial }_{+}X$. In the first chiral sector the variation of the fermionic term cancels $i{ϵ}_2{\partial }_{+}{\psi }_1\cdot {\partial }_{-}X$ from the bosonic term, leaving $i{ϵ}_2{\partial }_{-}({\psi }_1\cdot {\partial }_{+}X)$. The second sector gives the other total derivative. This proves off-shell invariance when the boundary flux vanishes. Closure on fermions uses their free equations of motion. A constant parameter must also preserve the chosen boundary conditions and spin structure.