Quantum Integrable models and string theory I’ve seen some recent papers on integrability by string theorists like Ed Witten or Kevin Costello but I don’t know why quantum integrable models are of interest to the string community. I would appreciate it if anyone can explain why this is an active field of research and what are the most interesting questions to string theorists in this field?
 A: The thing is that you perceive the situation upside down. The main interest in this topic is not because integrable models are useful for string theory. It's integrable models themselves that get much interest and it "just happens" that string theory is intimately related to some important QFTs.
In most quantum field theories one can rarely go beyond the perturbation theory using Feynman diagrams. Even that quickly becomes untenable as you try to calculate more loops or processes involving many particles. It's usually very hard to say much about strongly coupled regime e.g. QCD. As result much effort was put into understanding of the strongly coupled QFT. The supersymmetric gauge theories, their dualities etc are studied mainly because they provide much more tools to study nonperturbative dynamics.
When we talk about integrability you should remember that it usually means that certain quantities may be exactly solved in some sense. That integrability appears in some gauge theories is highly nontrivial and important as you can go beyond limitations of the usual approaches.
The string theory happens to be connected to various gauge theories. One important point of connection is AdS/CFT but there's also a relation between scattering amplitudes in gauge theories and scattering amplitudes of strings on twistor spaces etc. Of course that goes both ways - results in those QFTs can be used to obtain knowledge about string theory itself especially in the regime where string perturbation theory doesn't help much. However most of the people perceived as members of the "string community" actually work on the better understanding of the quantum field theories treating string theory more as a tool.
