Conformal Field Theory

At a fixed point of the renormalization group the coupling stops running, the theory loses all scales, and scale invariance is promoted to the larger conformal symmetry. A conformal field theory (CFT) is a QFT with this enhanced symmetry. CFTs are the fixed points that organize all RG flows, describe critical phenomena, and — via holography — connect to quantum gravity. They are also among the few QFTs that can be solved exactly.

Conventions: $\hslash =c=1$, Euclidean signature is standard for CFT.

From scale to conformal invariance

A theory at an RG fixed point is scale invariant: no parameter sets a length, so physics looks the same at all magnifications. In a local, unitary QFT this scale invariance almost always enlarges to the full conformal group — transformations that preserve angles but not lengths. In $d$ dimensions the conformal group is $SO(d,2)$ (Euclidean $SO(d+1,1)$), extending Poincaré by:

• dilatations $x\to \lambda x$ (scale), and
• special conformal transformations (inversions composed with translations).

The energy–momentum tensor becomes traceless, $T^{\mu }{}_{\mu }=0$ — the operator statement of scale invariance, whose quantum failure is the trace anomaly.

Primary operators and scaling dimensions

CFT reorganizes the theory around operators rather than particles (there is no S-matrix — no mass gap, no asymptotic states, so the LSZ framework does not apply). Operators are classified by how they transform under the conformal group:

• Primary operators $O(x)$ are annihilated by the special conformal generators and labeled by a scaling dimension $\Delta$ and a spin: under a dilatation $x\to \lambda x$, $O\to {\lambda }^{-\Delta }O$. The dimension $\Delta ={\Delta }_{\text{classical}}+\gamma$ includes the anomalous dimension $\gamma$ from interactions.
• Descendants are derivatives $\partial \cdots \partial O$ of primaries.

Conformal symmetry fixes the form of low-point correlators completely:

$$⟨O(x)O(y)⟩=\frac{1}{|x-y{|}^{2\Delta }},⟨O_1{O}_2{O}_3⟩=\frac{C_{123}}{|x_{12}{|}^{{\Delta }_1+{\Delta }_2-{\Delta }_3}\cdots }.$$

Two- and three-point functions are determined up to constants; the OPE coefficients $C_{ijk}$ and dimensions ${\Delta }_i$ are the theory's defining data.

The operator product expansion and the bootstrap

The operator product expansion (OPE) states that a product of two nearby operators equals a convergent sum over the operator spectrum:

$$O_i(x)O_j(0)=\sum _k\frac{C_{ijk}}{|x{|}^{{\Delta }_i+{\Delta }_j-{\Delta }_k}}{O}_k(0)+\cdots .$$

In a CFT the OPE converges (unlike the asymptotic OPE of a generic QFT), so the data $\{{\Delta }_i,C_{ijk}\}$ determine all correlators. Demanding that different ways of applying the OPE to a four-point function agree — crossing symmetry — yields the conformal bootstrap: a set of consistency equations that constrain, and sometimes uniquely determine, the allowed CFTs. The bootstrap has produced the world's most precise values for the 3D Ising critical exponents, purely from symmetry and consistency, with no Lagrangian at all.

Two dimensions: an infinite symmetry

In $d=2$ the conformal algebra is infinite-dimensional (the Virasoro algebra), making 2D CFTs extraordinarily constrained and often exactly solvable. The key data is the central charge $c$, which counts degrees of freedom (a free boson has $c=1$, a free Majorana fermion $c=1/2$) and controls the trace anomaly on a curved worldsheet. 2D CFT is the mathematical backbone of string theory (the worldsheet is a 2D CFT) and of exactly solved critical statistical models (Ising, Potts). Zamolodchikov's $c$-theorem — that $c$ decreases monotonically under RG flow — makes precise the intuition that RG flow loses degrees of freedom from UV to IR.

Why CFT matters

• Fixed points of the RG are CFTs; every RG flow runs between a UV CFT and an IR CFT, so CFTs are the "endpoints" organizing the space of all QFTs.
• Critical phenomena: continuous phase transitions are described by CFTs, and critical exponents are combinations of scaling dimensions — the deep QFT–statistical-mechanics link.
• Holography (AdS/CFT): a CFT in $d$ dimensions is dual to a quantum gravity theory in $d+1$-dimensional anti-de Sitter space — the most concrete realization of quantum gravity we have, and a tool for strongly coupled QFT.

Summary

• A CFT is a QFT at an RG fixed point: scale invariance enlarges to conformal symmetry $SO(d,2)$; $T^{\mu }{}_{\mu }=0$.
• Organized by primary operators with scaling dimensions $\Delta$; conformal symmetry fixes 2- and 3-point functions.
• The OPE + crossing give the conformal bootstrap, solving CFTs from consistency.
• 2D CFT has infinite (Virasoro) symmetry and a central charge; underlies string theory and exact critical models. CFTs also power AdS/CFT holography.

Where this leads

• The RG flows CFTs bound: the renormalization group, Wilsonian RG.
• Holography and quantum gravity: QFT in curved spacetime.
• The trace anomaly: the chiral anomaly.

References

• Di Francesco, Mathieu & Sénéchal, Conformal Field Theory.
• Belavin, Polyakov & Zamolodchikov, Nucl. Phys. B 241, 333 (1984).
• Rychkov, EPFL Lectures on Conformal Field Theory in $D\ge 3$.
• Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) (AdS/CFT).