Rigid worldsheet supersymmetry boundary term ($\delta \mathcal{L}={\partial }_{\mu }{J}^{\mu }$)

In flat conformal gauge, take $\mathcal{L}=\partial X\cdot \partial X+i{\alpha }^{\prime }\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ and rigid transformations $\delta X=iA\overline{ϵ}\psi$, $\delta \psi =B{\gamma }^{\nu }{\partial }_{\nu }Xϵ$, with $A={\alpha }^{\prime }B$. The Majorana Grassmann bilinear interchange and Clifford algebra combine the variations intoThe identity is off shell. The integrated action is invariant when its boundary flux vanishes, for example on a closed worldsheet or with compatible supersymmetric endpoint conditions. Constant spinors and ordinary derivatives here refer to flat gauge; arbitrary curved worldsheets require covariant spinor derivatives and the locally supersymmetric completion.

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.

The local rigid worldsheet supersymmetry identity $\delta \mathcal{L}={\partial }_{\mu }{J}^{\mu }$ does not by itself prove preservation of every spin structure. If $X$ is periodic on a spatial circle and $\psi$ is antiperiodic, a constant $ϵ$ gives antiperiodic $\delta X=i\overline{ϵ}\psi$. It does not preserve the bosonic boundary condition. Compatible periodic fields admit the constant transformation; other sectors must be handled with the local superconformal formulation and their global spin data.