My new question here: has string theory been analyzed somewhere in the context of various quantization prescriptions formulated in a mathematically sound way? I mean something like geometric quantization, Klauder Quantization, Brownian quantization etc. It is important to know if string theory ever passed through all the mathematical requirements that define a quantum theory; and by the way, are all the mathematical prescriptions defined consistently? Do we have a mathematically sound (consistent) prescription for quantization that is free of ambiguities? People generally boast with the fact that string theory is "mathematically consistent" as a quantization of 1 dimensional objects. Is it so indeed? Is the "quantum" part in string theory really exactly the way it should be? and is the "way it should be" really known?
The core of perturbative string theory has a mathematically rigorous formulation. In fact much of mathematical physics and mathematical insight into quantum field theory as such has been gained from the study of the low-dimensional QFTs that constitute the worldvolume theories of the string and the various branes. For instance the axiomatization of QFT in the “FQFT” flavor (roughly dual to the AQFT picture) historically originates in insights gained in the study of (topological) string (namely the Moore-Seiberg axioms). On the other hand, the attempted implementations and applications of core string theory are vast and numerous, and when it finally comes to string phenomenology the usual level of rigor is just that common among practicing quantum field theorists. On the far end, deep aspects of string theory that are felt by many researchers to be of metaphysical relevance, such as the “landscape of string theory vacua” have led and are leading to speculations that are not anymore backed up by any disciplined reasoning.
More in detail:
The quantization of the string sigma-model may be obtained cleanly via the mathematical sound process of geometric quantization, see the references on the nLab at string – Symplectic geometry and Geometric quantization. The famous Weyl anomaly of the string is formally understood in terms of anomalous action functionals, see for instance (Freed 86, 2.). Various other obstructions to quantization (quantum anomalies) in the background fields for the string sigma-model such as notably the Freed-Witten-Kapustin anomaly, have been understood in fine detail in terms of obstructions in differential cohomology, see for instance (Distler-Freed-Moore 09).
Particularly well analyzed are the two special sectors of first quantized string theory, that of rational conformal field theory, which contains the example of strings propagating on Lie group manifolds – the Wess-Zumino-Witten model; as well as the example of topological strings. Rational conformal field theories indeed stand out as one non-trivial and rich class of QFTs which have been subject to complete mathematical classification (in the same sense in which mathematicians for instance do the classification of finite simple groups). For details on this classification see on the nLab at FRS formalism.
For the topological string much more is true. The topological string has effectively become a subject in pure mathematics, with its rigorous axiomatization via the TCFT version of the cobordism hypothesis-theorem, its formulation as mathematical homological mirror symmetry, its relation to geometric Langlands duality etc.
But the FQFT-axiomatics that serves to mathematically formalize the topological string is not restricted to the topological sector, it also applies to the physical string. For instance Huang’s theorem shows that the familiar description of physical string via vertex operator algebra is an instance of the FQFT-formalization. Indeed, in FRS formalism these two formalizations, vertex operator algebras (via their modular tensor categories of representations, and TQFT combined via the rigorous AdS3-CFT2 and CS-WZW correspondence give the classification of rational CFT). (In particular this says that in this low dimensionl holography and AdS-CFT duality is rigorous, of course this is far, far from true in higher dimensions.)
In summary this is a level of rigour with which the worldsheet 2d QFT of the string is understood which is well beyond of what one typically encounters for non-trivial interacting (non-free) QFT. And this is full non-perturbative quantum field theory (on the worldsheet!), not just the approximation in perturbation theory.
From here on, also string field theory (its action functional, that is), has a completely rigorous formulation in terms of operads and L-infinity algebras (Lie n-algebras for n→∞).
A snapshot of the state of the art of rigorous foundations of string theory as of 2011 is in (Sati-Schreiber 11).
The above text with hyperlinks for all technical terms is also on the nLab at string theory FAQ -- Is string theory mathematically rigorous?.