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I've noticed a general phenomenon in compactifying on a circle where if you start with, say, an NS field, then the KK fields with an index along the circle will be in the R sector, and those without will of course remain in the NS sector.

I am wondering how to understand this phenomenon. What is the story for fermions? Their behavior under compactification is quite different, with the spinor bundle tending to split as a tensor product rather than as a direct sum.

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It's a good sketch for a question. However, it's a bit confusing. Are you talking about the (anti)periodicity on the spacetime circle (then you shouldn't call it R/NS) or the periodicity along a closed string i.e. world sheet? In the latter case, I may imagine that you're asking why the antiperiodic NS sector is given by simpler operators than the R sector which contains spin fields with spinor indices. Is that what you're asking? Or are you asking about some orbifolds that correlate NS/R with the momentum along the spacetime circle? –  LuboŇ° Motl Mar 23 '13 at 10:57

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