| CFT | |
|---|---|
| Name | Conformal Field Theory |
| Caption | Schematic representation of conformal transformations acting on spacetime |
| Field | Theoretical physics |
| Institutions | Princeton University, Institute for Advanced Study, CERN, Perimeter Institute for Theoretical Physics |
| Notable people | Alexander Polyakov, Belavin, A. B. Zamolodchikov, Ludwig Faddeev, Paul Ginsparg |
| Introduced | 1970s |
| Related | Quantum field theory, String theory, Statistical mechanics |
CFT
Conformal Field Theory (CFT) is a class of quantum field theories invariant under conformal transformations, which locally preserve angles but not necessarily distances. CFTs provide exact descriptions of scale-invariant phenomena in low-dimensional Quantum Physics and play a central role in theoretical frameworks such as String theory and the AdS/CFT correspondence. Their constrained symmetry makes many quantities exactly computable and links disparate areas like critical phenomena and two-dimensional exactly solvable models.
Conformal Field Theory studies fields and operators on manifolds equipped with a conformal structure; infinitesimal generators form the conformal algebra. Historically, the subject matured through the work of physicists and mathematicians including Alexander Polyakov, Belavin, and A. B. Zamolodchikov, and became foundational in analyses of second-order phase transitions in statistical mechanics. In two dimensions, the conformal group enlarges to an infinite-dimensional Virasoro algebra, enabling classification of universality classes via central charge and conformal dimensions.
CFT rests on representation theory of the conformal group and associated algebras: in d>2 dimensions, the finite-dimensional algebra is isomorphic to so(d+1,1), while in d=2 the symmetry is governed by the Virasoro algebra and extended chiral algebras such as Kac–Moody algebras and W-algebras. Key invariants include the central charge, scaling dimensions, and fusion rules. Correlation functions are heavily constrained by conformal Ward identities and by the structure of primary and descendant operators. Modular properties on the torus involve the action of the modular group PSL(2,Z) and lead to consistency conditions like modular invariance.
In quantum physics, CFTs describe fixed points of the renormalization group (RG flow) and underlie continuum limits of lattice models such as the Ising model and Potts model. They furnish operator algebras and spectra that determine universal critical exponents measurable in experiments. In high-energy contexts, CFTs appear in the worldsheet description of strings in bosonic string theory and superstring theory, where worldsheet conformal symmetry ensures consistency and anomaly cancellation conditions related to central charge balance and the BRST quantization procedure.
Two-dimensional CFTs provide the richest solvable examples. The Ising model at criticality corresponds to a minimal model with central charge c=1/2. Minimal models classified by BPZ include a discrete series of rational CFTs solved by null-vector differential equations. Liouville field theory is a non-rational CFT relevant to 2D quantum gravity and noncritical string theory. Other central examples include free scalar and free fermion theories, the Wess–Zumino–Witten model based on affine Lie algebras (used to build current algebra realizations), and the c=1 compact boson describing Luttinger liquids and Kosterlitz–Thouless transitions.
CFTs classify universality classes of continuous phase transitions; quantities like central charge appear in finite-size scaling and entanglement entropy computations (e.g., Calabrese–Cardy formulas). In String theory, 2D CFTs define consistent string backgrounds: target-space geometry and fluxes correspond to deformations of the worldsheet CFT. The AdS/CFT correspondence (gauge/gravity duality), conjectured by Juan Maldacena, relates certain CFTs (notably N=4 supersymmetric Yang–Mills theory) to gravity in higher-dimensional anti-de Sitter space, enabling nonperturbative insights into strongly coupled quantum systems and black hole thermodynamics studied at CERN and in theoretical programs at Institute for Advanced Study and Perimeter Institute for Theoretical Physics.
Core computational tools include the Operator product expansion (OPE), which encodes short-distance operator algebra and converges inside correlation functions. Crossing symmetry and OPE associativity feed into the conformal bootstrap program, both numerical and analytic, pioneered in recent decades and applied to theories like the 3D Ising CFT using methods from semidefinite programming and spectral analysis. Modular invariance on compact Riemann surfaces constrains partition functions and spectra; in 2D this links to characters of the Virasoro and affine algebras. Additional techniques include bosonization, Coulomb gas methods, and integrability approaches connected to Bethe ansatz solvable models.
Recent progress includes numerical bootstrap bounds that accurately determine critical exponents for the 3D Ising and O(N) models, advances in conformal perturbation theory, and applications of entanglement measures in CFT contexts. Open problems span classification of higher-dimensional CFTs, rigorous proofs of the AdS/CFT duality beyond supersymmetric cases, and understanding non-unitary and logarithmic CFTs relevant to disorder and percolation. Connections to modern mathematics—such as vertex operator algebras, geometric representation theory, and the theory of modular tensor categories—continue to deepen, with active research at institutions like Princeton University and collaborative projects such as the Simons Foundation and major conferences including those organized by the American Physical Society.
Category:Quantum field theory Category:Conformal field theory