Integrable models are a class of quantum and classical theories that possess rich symmetries which constrain their dynamics, often leading to their exact solution. Such models also appeared in AdS/CFT, and
allowed e.g. to study the spectrum of non-protected operators in planar N=4 super Yang-Mills with stunning precision and efficiency.
Indeed in recent years integrability manifested itself in several AdS/CFT dual pairs, and could be exploited in the computation of a broad set of observables through the bootstrap of appropriate algebraic
objects, such as the S-matrix or certain form factors. I would like to outline this integrable bootstrap approach and advocate the opportunity of systematically applying it to a large class of observables and
AdS/CFT setups.
In my talk I will work out a relatively simple example: the computation of the spectrum of AdS3xS3xT4 and AdS3xS3xS3xS1 super-strings in the planar limit. This saw remarkable progress in recent years and is
interesting in itself, as these backgrounds are related to the D1-D5 system, to CFT2s with N=(4,4) super-conformal symmetry, and in a certain limit to higher-spin theories. I will then discuss how the
approach used there can be adapted to other backgrounds and observables such as three-point functions and one-point functions in presence of defects.
I will conclude by presenting a rich list of open problems that are within our grasp by integrable bootstrap techniques.
