= On-shell closure of rigid worldsheet supersymmetry
{title2=$[\delta_1,\delta_2]=2\bar\epsilon_1\rho^\alpha\epsilon_2\,\partial_\alpha$}
For $\delta X=-\bar\epsilon\psi$ and $\delta\psi=\rho^\alpha\partial_\alpha X\epsilon$, with constant Grassmann-odd Majorana parameters, the commutator on $X$ is the displayed translation. Set $M=\epsilon_1\bar\epsilon_2-\epsilon_2\bar\epsilon_1$. In two dimensions $\{\rho^\alpha,M\}=2\bar\epsilon_1\rho^\alpha\epsilon_2$ times the identity, so the fermion commutator is the translation minus $M\rho^\alpha\partial_\alpha\psi$. The remainder vanishes on the fermion field equation. This distinguishes on-shell closure from the off-shell total-derivative invariance of the action.
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