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What is the difference between worldsheet supersymmetry and spacetime supersymmetry? For worldline formulation of fermions quantum mechanics, there is a supersymmetry. But the corresponding spacetime theory (namely, QFT) is totally void of supersymmetry. Now to have fermionic excitations for strings, we need to add supersymmetry to the worldsheet, then how does the worldsheet supersymmetry induce spacetime supersymmetry?

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Worldsheet supersymmetry is the fermionic symmetry of the worldsheet RNS action under the worldsheet supsresymmetry transformations that look like $$ \sqrt{\frac{2}{\alpha'}}X \mapsto \sqrt{\frac{2}{\alpha'}}X + \mathrm{i}\bar{\epsilon}\psi^\mu$$ and which I'm too lazy to type out for all fields (and which also depend on whether or not we're looking at the gauge-fixed or non-gauge-fixed version). $\epsilon(\sigma,\tau)$ here is an unconstrained worldsheet Majorana fermion parametrizing the transformation. It is a symmetry of the classical action of the superstring.

On the other hand, spacetime supersymmetry is the symmetry of the low-energy effective 10d SUGRA action that describes the massless spectrum of its corresponding string theory. It is one of the "accidents" of string theory that the GSO projection works exactly in such a way that the retained bosonic and fermionic states organize into 10d supersymmetry multiplets, since the 2d supersymmetry of the worldsheet theory does not a priori force this to be the case.

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  • $\begingroup$ Once we have gravitinos in the spectra, unitarity dictates that we must have a local susy in the target space, so it is not an accident, it is a consequence from unitarity. $\endgroup$ – Nogueira Apr 28 '18 at 17:32

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