Conformal field theory (CFT) was developed in the 1980s, and found immediate applications in string theory and two dimensional statistical mechanics, where critical exponents for many models (Ising, tricritical Ising, 3-state Potts, etc.) could be calculated exactly. The physical idea is that the principle of scale invariance is elevated from a global to a local invariance, which for reasons of consistency amounts to invariance under conformal transformations. This, in turn, yields a rich and fascinating mathematical structure for two dimensional systems (either two space or one time and one space dimension).

In later years, CFT has become relevant to many interesting areas of condensed matter physics, including Abelian and non-Abelianbosonization, quantized Hall states (where the bulk wave function is described in terms of conformal correlators, and the edge in terms 1+1 dimensional CFTs), the two-channel Kondo effect, fractional topological insulators, and in particular fault-tolerant topological quantum computing involving non-Abelian anyons. (Ising and Fibonacci anyons, for example, owe their names to the fusion rules of the associated conformal fields.)