 Spin-structure obstruction to constant worldsheet supersymmetry

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.