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.