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The scalar fields $X^\mu$ in bosonic string theory have a clear physical interpretation - they describe the embedding of the string in spacetime.

Adding fermionic fields on the worldsheet is a generalization for sure, gives fermions in the spectrum, has a smaller critical dimension and no tachyons, that's all good - but I don't see how they can have any physical interpretation as nice as the above for the scalars - isn't everything about how a string moves in spacetime described by the $X^\mu$ part?

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Btw, there's no necessity for a clear geometric interpretation in string theory. As far as I understand, any 2D CFT with vanishing central charge can be regarded as a string theory and some CFTs have no clear interpretation as a sigma model. –  Squark Dec 15 '11 at 12:10

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The worldsheet fermions have to do with internal degrees of freedom, namely the spin -- therefore better name for the superstring is the more old-fashioned "spinning string" (since worldsheet SUSY should not be confused with spacetime SUSY). The worldsheet fermions generate multiplets of some internal symmetry group. If you want those internal degrees of freedom generated by WS fermions to transform under spacetime Lorentz Transformations, rather than an independent internal symmetry, you need to correlate the Lorentz transformations of the worldsheet bosons and fermions. This is what worldsheet SUSY does for you.

All of this is not specific to string theory. If you want to first-quantize a field theory, a "bosonic" worldline theory will give you a (free) scalar field theory. Adding fermions and the corresponding worldline supersymmetries will generate (free) higher spin fields. It is probably a useful exercise to get e.g. classical (free) Maxwell field from a (N=2 SUSY) worldline theory in order to appreciate precisely what the worldsheet structures mean precisely. Wish I had a good reference, but maybe someone can help me out.

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I think there's also an alternative point of view, namely that the string lives in a superspace and the fermions are the odd coordinates –  Squark Dec 11 '11 at 21:32
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Yes, that is the Green-Schwarz formulation of spacetime supersymmetric strings. But I think the question was about the slightly more familiar R-NS string, and in particular spacetime SUSY is not implied. –  user566 Dec 11 '11 at 21:44

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