Conformal field theories have been long known to describe the universal physics of scale invariant critical points, such those occurring at regions near to continuous phase transitions in fluids, ferromagnets and quantum field theories. Studying conformal field theories would help us to understand those universal characteristics that relate several seemingly unrelated physical systems. Also from a renormalization group perspective, studying the space of conformal field theories amounts to studying the space of all well-defined (or UV complete) quantum field theories.
On the other hand, a long standing desire in physics, in particular in quantum field theory, is the development of a general and practical framework to study strongly coupled theories nonperturbatively. A very promising idea, called the bootstrap, consists in carving out the space of physical theories, by imposing explicitly expected symmetries and physical consistency conditions, without the need of controlling particular values for the couplings.
This idea has found the most success when applied to conformal field theories in several dimensions. In this talk, I plan to give a quick overview on some of the analytical as well as numerical techniques leading to these encouraging results.