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In the strong coupling limit of type IIA and heterotic E8 string theory, we get 11 dimensional M-theory in which we have no strings. Instead, we have M2 branes.

Are there any other backgrounds in string theory which can't be described by first quantized string theory?

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the strings always cease to be the main light degrees of freedom when you go to the strong coupling. In type IIB, you get an isomorphic type IIB string theory at strong coupling but the fundamental strings are what used to be called D1-branes at weak coupling, and vice versa.

We have mentioned three 10-dimensional superstring backgrounds. The remaining 10-dimensional supersymmetric string vacua are those with the $SO(32)$ gauge group - the HO heterotic theory and type I string theory - and they're S-dual to each other. The heterotic strings at the strong coupling become type I D1-branes which are heavy. The type I strings, because they're unoriented, can't carry any string-like conserved charge, become unstable, and they don't leave any "visible traces" in the heterotic string spectrum at all.

The same thing holds for lower-dimensional vacua, too. The first quantized string theory is only a good description when the coupling is weak - when the dilaton goes to minus infinity. Whether we use the "first quantized" description of the strings or "string field theory" is just a matter of choice of the formalism; both of them are only valid perturbatively. It used to be believed that string field theory knew "much more" if not "everything" about non-perturbative physics but we no longer believe so. It's just a different way to formulate the methods for perturbative (and transperturbative, I mean calculations of D-brane properties) calculations.

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