| conformal field theory | |
|---|---|
| Name | Conformal field theory |
| Field | Quantum field theory |
| Introduced | Early 1970s |
| Based on | Conformal symmetry, Quantum mechanics, Special relativity |
| Notable works | BPZ (1984), AdS/CFT correspondence (1997) |
conformal field theory
Conformal field theory (CFT) is a class of Quantum field theorys invariant under conformal transformations, i.e., angle-preserving coordinate changes. CFTs play a central role in describing scale-invariant critical phenomena in statistical mechanics and provide tractable non-perturbative models in quantum field theory and string theory. Their symmetry structure enables exact results for correlation functions, operator spectra, and universality classes.
Conformal field theories are quantum systems with local operators organized by representations of the conformal group. In two dimensions the infinite-dimensional enhancement of symmetry renders many models exactly solvable, while in higher dimensions CFTs characterize fixed points of renormalization group flows studied by Wilson and others. Important physical examples include the critical Ising model, described by a minimal model CFT, and the N=4 supersymmetric Yang–Mills theory which is central to the AdS/CFT correspondence. CFTs determine universal scaling dimensions, critical exponents, and operator product expansion coefficients measurable in condensed matter experiments and numerical conformal bootstrap studies.
The defining structure of a CFT is invariance under the conformal group, which in d dimensions is isomorphic to SO(d+1,1) for Euclidean signature or SO(d,2) in Lorentzian signature. Primary operators transform covariantly under this group and generate descendant operators via the action of the conformal algebra. The operator product expansion (OPE) furnishes an associative algebraic structure that, together with unitarity and crossing symmetry, severely constrains correlation functions. In supersymmetric theories the conformal algebra is extended to superconformal algebras (e.g., superconformal algebras in 4d such as PSU(2,2|4)). Rigorous formulations use representations of Lie algebras and vertex operator algebras in two dimensions, and harmonic analysis on homogeneous spaces for higher-dimensional cases. Important mathematical tools include the theory of representation theory of Lie groups, complex analysis, and modular invariance on Riemann surfaces.
Two-dimensional CFTs exhibit an infinite-dimensional symmetry algebra generated by holomorphic and antiholomorphic currents; the central extension of the local conformal algebra is the Virasoro algebra. The seminal work by Belavin, Polyakov and Zamolodchikov (BPZ) classified minimal models and introduced null-vector decoupling equations allowing exact determination of correlators. Vertex operator algebras formalize chiral algebras appearing in these theories and are connected to string theory worldsheet descriptions pioneered by Murray Gell-Mann and others. Modular invariance on the torus constrains the spectrum via the Modular group and leads to relations exploited in rational CFTs. Key examples include the Ising CFT, minimal models, Wess–Zumino–Witten models based on affine Lie algebras (e.g., Kac–Moody algebras), and Liouville theory.
In d>2 dimensions the conformal algebra is finite-dimensional but still provides strong constraints. The modern conformal bootstrap program, revived by works of Rattazzi et al. and developed by groups at institutions such as Princeton University and Harvard University, uses unitarity, crossing symmetry and numerical optimization to bound operator dimensions and OPE coefficients. Prominent higher-dimensional examples include the three-dimensional Ising CFT, the O(N) vector models, and 4d N=4 super Yang–Mills. Analytic bootstrap methods exploit the light-cone expansion, large-spin perturbation theory, and the Mellin space formalism; numerical bootstrap techniques employ semidefinite programming and software like SDPB. Constraints from conformal collider bounds and the a-theorem (analogous to the c-theorem) further restrict allowed theories.
CFT methods compute exact critical exponents and correlation functions at second-order phase transitions in condensed matter physics and statistical models such as percolation and Potts models. In quantum field theory, CFTs provide endpoints of renormalization group flows and classification tools for universality classes. Supersymmetric localization and anomaly matching link CFT data to protected quantities in supersymmetric field theories studied by communities around N=4 super Yang–Mills, Seiberg–Witten theory, and SUSY dualities. Lattice models, Monte Carlo simulations, and experiments on cold atoms often compare results to CFT predictions. Mathematical physics benefits through connections to modular tensor categories and three-dimensional topological quantum field theory via the Verlinde formula.
A landmark application is the AdS/CFT correspondence conjectured by Juan Maldacena: a d-dimensional CFT can be dual to a (d+1)-dimensional theory of gravity on anti-de Sitter space (AdS). The prototype duality pairs 4d N=4 SYM with type IIB string theory on AdS5×S5 and relates CFT operator dimensions to masses of bulk fields. AdS/CFT provides a non-perturbative definition of certain quantum gravity models and tools for computing entanglement entropy via the Ryu–Takayanagi formula. Holographic CFTs are studied using large-N expansions, conformal Regge theory, and bulk reconstruction programs led by researchers at institutions like Institute for Advanced Study and Perimeter Institute.
Category:Quantum field theory Category:Mathematical physics