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

In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that local scale invariant theories have their currents given by ${\displaystyle T_{\mu \nu }\xi ^{\nu }}$ where ${\displaystyle \xi ^{\nu }}$ is a Killing vector and ${\displaystyle T_{\mu \nu }}$ is a conserved operator (the stress-tensor) of dimension exactly ⁠${\displaystyle d}$⁠. For the associated symmetries to include scale but not conformal transformations, the trace ${\displaystyle T_{\mu }{}^{\mu }}$ has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly ⁠${\displaystyle d-1}$⁠.

Under some assumptions it is possible to completely rule out this type of non-renormalization and hence prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.

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.

The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraining in two dimensions.^{[clarification needed]} All conformal field theories share the ideas and techniques of the conformal bootstrap. But the resulting equations are more powerful in two dimensions, where they are sometimes exactly solvable (for example in the case of minimal models), in contrast to higher dimensions, where numerical approaches dominate.

The development of conformal field theory has been earlier and deeper in the two-dimensional case, in particular after the 1983 article by Belavin, Polyakov and Zamolodchikov. The term conformal field theory has sometimes been used with the meaning of two-dimensional conformal field theory, as in the title of a 1997 textbook. Higher-dimensional conformal field theories have become more popular with the AdS/CFT correspondence in the late 1990s, and the development of numerical conformal bootstrap techniques in the 2000s.

The global conformal group of the Riemann sphere is the group of Möbius transformations ⁠${\displaystyle \mathrm {PSL} _{2}(\mathbb {C} )}$⁠, which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensional Witt algebra: the conformal Killing equations in two dimensions, ${\displaystyle \partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }=\partial \cdot \xi \eta _{\mu \nu },~}$ reduce to just the Cauchy-Riemann equations, ⁠${\displaystyle \partial _{\bar {z}}\xi (z)=0=\partial _{z}\xi ({\bar {z}})}$⁠, the infinity of modes of arbitrary analytic coordinate transformations ${\displaystyle \xi (z)}$ yield the infinity of Killing vector fields ⁠${\displaystyle z^{n}\partial _{z}}$⁠.