From states in string theory to differential forms This question is probably really easy and isn't usually discussed in textbooks.
So suppose when we construct states in string theory we obtain something like $\left|- \right\rangle \otimes \left|+ \right\rangle$ (In RR sector, BBS page 137 for reference). This object decomposes in certain way, but right now this is just some collection of states (64 of them). Next step is to get differential forms on our space-time, thats the part I dont understand - logic is clearly that for every point of spacetime we will have this decomposition and it's varying in a smooth way, but what is the formal way to do it? 
 A: To my knowledge, there is no formal map from string states to spacetime fields. It is merely a (well-motivated) heuristic.
The first "hint" of a spacetime formulation of string theory appears when we observe that the massless states of string theory neatly fall into representations of the massless little group of the (26/10) dimensional spacetime, hence are prefectly suited to be, in the standard QFT sense, the states associated to fields in those representations. And when you see a massless antisymmetric 3-tensor representation, it comes from a 3-form on spacetime with gauge symmmetry in a standard QFT. So, if there is a spacetime QFT view of string theory, then it will contain such a field in order to generate states transforming in this representation.
The second and stronger way of associating such forms to the states comes from the low energy effective string action (for type IIa/b), which is a 10d SUGRA action matched with the tree-level massless string amplitudes such that the action, viewed as an ordinary QFT, reproduces the string amplitudes as its usual QFT scattering amplitudes. For the QFT to have the same massless spectrum as the string theory, of course we have to associate to every representation of the 10d Poincaré group the appropriate field. There are remarkable "uniqueness" properties of these SUGRA actions (maximal possible supersymmetry in the absence of higher spin, for one), indicating that associating spacetime differential forms to the string states is not a completely crazy thing to do.
