Conformal Field Theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.

Conformal field theory has important applications to string theory, statistical mechanics, and condensed matter physics. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

While it is possible for a quantum field theory to be scale invariant but not conformally-invariant, examples are rare. For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the scale symmetry group is smaller.

In some particular cases it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.

There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to statistical mechanics, and the latter to quantum field theory. The two versions are related by a Wick rotation.

Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on the Riemann sphere. It has the Möbius transformations as the conformal group, which is isomorphic to (the finite-dimensional) PSL(2,C).

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