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As far as I understand, both in bosonic and superstring theory one considers initially a free string propagating through D-dimensional Minkowskispace. Regardless of what quantization one uses, at the end one arrives at a spectrum, where the excitations are classified among other things by the representation of the Poincaré group (or maybe a cover) they are in.

This enables one to speak of their masses, but seems to make the construction very dependent on the background. Do massless modes remain massless in an other background, for example? And how does one define the mass of excitations in this case? As far as I can tell even after compactification, at least 4 dimensions are usually left to be some symmetric space (so there is probably some notion of mass) and the space of harmonic forms on the calabiyau is supposed to correspond to the space of massless excitations (of a low energy effective theory?).

Moreover D'Hoker states in his lectures in Quantum Fields and Strings, that he is not aware of a proof that in a general background a Hilbertspace can be constructed. Is there still none?

I apologize if these questions are too elementary or confused. They were not adressed in the string theory lectures I attended.

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Cross-posted from physics.stackexchange.com/q/22159/2451 –  Qmechanic Nov 21 '12 at 0:26
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up vote 4 down vote accepted

The massless spectrum depends on the background. What one usually means by masses in a background describing a compactification are masses in the non-compact flat space (most often 4D). That is, you take the momentum in the non-compact directions, you square it and you find the mass.

About your second question, whenever the background has a light-like Killing vector you can build a light-cone Hamiltonian and a Hilbert space. I'm not sure if there are more general situations where a Hilbert space description is available. For a general time-dependent background, I don't think much is known.

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Good answer, +1. I still think that one may expand and/or re-emphasize some points. When one determines masses, they're not just "defined". The mass spectrum of particles contains much of the information about the theory and it has to be calculated; of course that this calculation depends on the background. The massless spectrum depends on the background in field theory, too. It's not a "disadvantage" in any sense; it's a tautology. We're just saying that the most important feature of the dynamical laws, the spectrum, depends on the dynamical laws governing individual particles/strings. –  Luboš Motl Mar 24 '12 at 7:12
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