CFTs

Conformal field theories (CFTs) are quantum field theories that are invariant under conformal transformations, which are angle-preserving spacetime maps that extend usual dilations and translations. This enhanced symmetry imposes strong constraints on the structure of the theory, including how fields transform and how correlation functions behave. Local operators in a CFT fall into primary fields, characterized by their scaling dimensions, and their descendants obtained through the action of the symmetry generators. The operator product expansion encodes how operators behave when brought close together, organizing the theory into a highly constrained algebraic framework.

In two dimensions, the conformal symmetry is infinite-dimensional, described by the Virasoro algebra. This makes 2D CFTs particularly powerful and well understood. Correlation functions are fixed up to a small set of data, such as the central charge c and the spectrum of primary fields. The central charge measures the number of effective degrees of freedom and appears in the commutation relations of the Virasoro generators. Two-dimensional CFTs also include important families known as minimal models, which have a finite number of primary fields and are labeled by pairs of integers (p, q). For unitary theories, p and q are consecutive (p = q+1), yielding a discrete set of exactly solvable models with c = 1 − 6/(p(p+1)).

In higher dimensions, CFTs are explored using the conformal bootstrap, which exploits crossing symmetry and unitarity to bound operator dimensions and OPE data. CFTs describe critical points in statistical mechanics, universality classes of phase transitions, and play a central role in string theory and the AdS/CFT correspondence, which relates certain CFTs to gravitational theories in higher-dimensional anti-de Sitter space. Examples include free scalar and free fermion theories, and more complex interacting models studied through nonperturbative methods.