I am a PhD student and I am choosing the topic to work on in the graduate school. I am interested in various non-perturbative aspects of String Theory. Does anybody know if there were any attempts to generalize AdS/CFT to the case of String Field Theory instead of usual low-energy description involving SUGRA fields on the "gravity" side? If no, what the problem with it?
Light-cone superstring field theory in the pp-wave background has been useful for understanding a certain subsector of operators with large R-charge in N=4 super Yang-Mills---the so-called BMN limit. But by "string field theory" I assume you ask about covariant formulations. There is no serious work applying covariant string field theories in the context of the AdS/CFT correspondence.
When people talk about "string theory in AdS," I think implicitly they have in mind some kind of covariant closed string field theory. However, there are major obstacles to describing this closed string field theory explicitly.
First of all, you would need to solve the closed string sigma model in AdS. Quite a bit has been learned about this in the last decade or so, especially due to the miracles of integrability. But this remains a major open problem.
Then you would need to understand the sigma model in a covariant form realizing worldsheet BRST invariance. One can try to do this in the RNS formalism, which realizes worldsheet supersymmetry, or the pure spinor formalism, which is better at realizing spacetime supersymmetry. The RNS formalism does not deal very well with strings in the presence of Ramond-Ramond flux, but on the other hand no one has formulated a satisfactory string field theory based on the pure spinor formalism. This is perhaps related to lack of clarity about how the pure spinor formalism emerges from a gauge-fixed worldsheet path integral. There has been some recent progress on this question, but the implications for string field theory are unclear at present.
Let's assume it's possible to deal with the Ramond-Ramond flux in some way, and we had an explicit covariant quantization of the closed superstring on AdS_5*S^5 in the RNS formalism. Then one can finally start thinking about the construction of the closed string field theory. The major challenge at this point is not so much related to AdS/CFT, but with providing a sufficiently rigorous definition of superstring perturbation theory in the RNS formalism. This is a long standing and subtle problem, but there has been some renewed interest in the subject.
Supposing you succeeded in constructing closed string field theory on AdS, what could you do with it? If history is any guide, not much. Very few computations have been performed using closed string field theories. The formalism is just too unwieldy. There is little evidence that closed string field theory contains information about nonpertubative string theory, but this may partly be because the theory is so poorly understood.
Probably the most significant impact of constructing a covariant closed string field theory in AdS would be resolving all of the deep technical issues required just to write the action. But it is always possible that new insights could improve the prospects of this approach.
Independently of the AdS/CFT correspondence, string field theory becomes problematic for closed strings. Closed string dynamics is the stringy extension of quantum gravity which is really different from a local quantum field theory.
On the other hand, open string dynamics is a stringy tower extension of Yang-Mills theory, so it preserves its proximity to local quantum field theory.
Consequently, all the examples of consistent string field theories that "really seem to work" are only good for calculating scattering amplitudes with open string external states. The closed string intermediate states are automatically included but the closed string external states and the corresponding fields are not included.
The consistent string field theories include Witten's cubic open string field theory (SFT of the Chern-Simons type) and the non-polynomial boundary string field theory (BSFT). Both focus on open string external states. Both of them seem simpler for bosonic string theory. The extension to the superstring case is "more non-polynomial" but the required added difficulties seem surmountable.
There is a general, non-stringy reason why closed string dynamics – and its massless sector, the quantum gravity – is unlikely to be described as a local quantum field theory. The reason is that quantum gravity can't be "quite" local. There are various reasons for that. For example, the Hawking radiation must be able to get the information out of the black hole interior so it violates the locality. No "closed string field theory" could achieve such a result.
One should point out one more problematic assumption included in the question. Even though string field theory was believed to be relevant as a nonperturbative definition of string theory (much like lattice QCD may be used as a non-perturbative definition of QCD), it turned out not to be the case. As we understand it today, string field theory is just another formalism to calculate the scattering amplitudes as perturbative expansions. Perhaps it may make off-shell Green's functions more accessible than the conventional, e.g. first-quantized approaches to string theory; but it doesn't make the observables beyond the perturbative expansions more accessible.
In particular, the strong coupling limits such as M-theory or the S-dualities and string-string dualities have never been derived from string field theory and they probably cannot be derived from string field theory. String field theory is a formalism to get the result as power law expansions – and go "slightly" beyond those in the sense that string field theory is extremely good for studying various states of D-branes (it's been great to study the tachyon condensation on D-branes and the confirmation of Sen's hypotheses about these tachyons). They're nonperturbative objects (infinitely heavy ones in the weakly coupled limit) but all of their properties are encoded in perturbative open string dynamics.
To understand string theory nonperturbatively, one has to use different tools than string field theory, such as matrix string theory (a variation of the BFSS matrix theory optimized for type IIA or heterotic HE string theory), a subject I co-discovered, or AdS/CFT itself. Matrix string theory is a good exact definition of type IIA (or some similar) string theory in the flat Minkowski space for any value of the coupling constant and one can prove that for a very small coupling constant, it reduces to the usual perturbative rules (while it has the 11D limit for very strong couplings). There's no known way to "deform" the matrix string theory definition to a general background, e.g. an AdS background. Quite generally, such nonperturbative definitions only allow specific enough superselection sectors (demand some particular asymptotic conditions in the spacetime).
Similarly, AdS/CFT may help one to define string theory in the AdS (instead of the flat backgrounds of matrix theory) nonperturbatively exactly because the bulk AdS string theory is equivalent to a boundary CFT and this boundary CFT may arguably be defined in a nonperturbative way, e.g. by some lattice gauge theory definition (when some problems with SUSY, fermion doubling on the lattice etc. are solved, which seems possible).
So the experts' opinions – the summary of existing literature – is that string field theory isn't helpful for obtaining a "more nonperturbative" perspective into the structure of string theory, whether it's in the AdS background or any other background. There may be a loophole that everyone has overlooked but you should be warned that the goal of the project you are outlining seems to contradict at least some generally believed modern lore about string field theory.
There is still a lot of perturbative stringy physics that may be seen both in the boundary and bulk side of the AdS/CFT correspondence. The BMN (Berenstein-Maldacena-Nastase) pp-wave limit of the AdS/CFT correspondence is a good place to start to investigate those. There are also known dual descriptions of the bulk wrapped D-brane states (Witten) and other things. But none of those seems to become more comprehensible in the language of a string field theory – there is really no good string field theory one could reliably use for the bulk at all.