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 infinitedimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
Conformal field theory has important applications^{[1]} 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.
Contents
 Scale invariance vs. conformal invariance 1

Dimensional considerations 2
 Two dimensions 2.1
 More than two dimensions 2.2
 Conformal symmetry 3
 See also 4
 References 5
 Further reading 6
Scale invariance vs. conformal invariance
While it is possible for a quantum field theory to be scale invariant but not conformallyinvariant, examples are rare.^{[2]} 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.
Dimensional considerations
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.
Twodimensional CFTs are (in some way) invariant under an infinitedimensional 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 finitedimensional) PSL(2,C).
However, the infinitesimal conformal transformations^{[3]} form an infinitedimensional algebra, called the Witt algebra, but this infinity of conformal transformations do not have global inverses on ℂ. Only the primary fields (or chiral fields) are invariant with respect to this full infinitesimal conformal group. Its generators are indexed by integers n,
 L_n=\oint_{z=0} \frac{dz}{2\pi i } z^{n+1} T_{zz}~,
where T_{zz} is the holomorphic part of the nontrace piece of the energy momentum tensor of the theory. E.g., for a free scalar field,
 T_{zz}= \tfrac{1}{2} (\partial_z \phi)^2 ~.
In most conformal field theories, a conformal anomaly, also known as a Weyl anomaly, arises in the quantum theory. This results in the appearance of a nontrivial central charge, and the Witt algebra is extended to the Virasoro algebra.
In Euclidean CFT, one has both a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, one has a leftmoving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).
This symmetry makes it possible to classify twodimensional CFTs much more precisely than in higher dimensions. In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the central charge, c.
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of c. Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
A chiral field is a holomorphic field W(z) which transforms as
 L_n W(z)=z^{n+1} \frac{\partial}{\partial z} W(z)  (n+1)\Delta z^n W(z)
and
 \bar L_n W(z)=0~.
Analogously, mutatis mutandis, for an antichiral field. Δ is called the conformal weight of the chiral field W.
Furthermore, it was shown by Alexander Zamolodchikov that there exists a function, C, which decreases monotonically under the renormalization group flow of a twodimensional quantum field theory, and is equal to the central charge for a twodimensional conformal field theory. This is known as the Zamolodchikov Ctheorem, and tells us that renormalization group flow in two dimensions is irreversible.
Frequently, we are not just interested in the operators, but we are also interested in the vacuum state, or in statistical mechanics, the thermal state. Unless c=0, there can't possibly be any state which leaves the entire infinite dimensional conformal symmetry unbroken. The best we can come up with is a state which is invariant under L_{1}, L_{0}, L_{1}, L_{i}, i > 1. This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.
Twodimensional conformal field theories play an important role in statistical mechanics, where they describe critical points of many lattice models.
More than two dimensions
In d > 2 dimensions, the conformal group is isomorphic to SO(d+1, 1 ) in Euclidean signature, or SO(d, 2 ) in Minkowski space.
Higherdimensional conformal field theories are prominent in the AdS/CFT correspondence, in which a gravitational theory in antide Sitter space (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are d=4 N = 4 supersymmetric Yang–Mills theory, which is dual to Type IIB string theory on AdS_{5} x S^{5}, and d=3 N=6 superChern–Simons theory, which is dual to Mtheory on AdS_{4} x S^{7}. (The prefix "super" denotes supersymmetry, N denotes the degree of extended supersymmetry possessed by the theory, and d the number of spacetime dimensions on the boundary.)
Conformal symmetry
Conformal symmetry is a symmetry under scale invariance and under the special conformal transformations having the following relations.
 [P_\mu,P_\nu]=0,
 [D,K_\mu]=K_\mu,
 [D,P_\mu]=P_\mu,
 [K_\mu,K_\nu]=0,
 [K_\mu,P_\nu]=\eta_{\mu\nu}DiM_{\mu\nu},
where P generates translations, D generates scaling transformations as a scalar and K_\mu generates the special conformal transformations as a covariant vector under Lorentz transformation.
See also
 Logarithmic conformal field theory
 AdS/CFT correspondence
 Operator product expansion
 Vertex operator algebra
 WZW model
 Critical point
 Boundary conformal field theory
 Primary field
 Superconformal algebra
 Conformal algebra
 Conformal bootstrap
References
 ^ Paul Ginsparg (1989), Applied Conformal Field Theory. arXiv:hepth/9108028. Published in Ecole d'Eté de Physique Théorique: Champs, cordes et phénomènes critiques/Fields, strings and critical phenomena (Les Houches), ed. by E. Brézin and J. ZinnJustin, Elsevier Science Publishers B.V.
 ^ One physical example is the theory of elasticity in two and three dimensions (also known as the theory of a vector field without gauge invariance). See Riva V, Cardy J (2005). "Scale and conformal invariance in field theory: a physical counterexample". Phys. Lett. B 622: 339–342.
 ^ Since the conformal Killing equations in two dimensions, \partial_\mu \xi_\nu + \partial_\nu \xi_\mu =\partial \cdot\xi \eta_{\mu \nu},~ reduce to just the CauchyRiemann equations, \partial_{\bar{z}} \xi(z)=0=\partial_z \xi (\bar{z}) , the infinity of modes of arbitrary analytic coordinate transformations ξ(z) yield the infinity of Killing vector fields z^{n} ∂_{z} .
Further reading
 Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory, SpringerVerlag, Berlin, Heidelberg, 1997. ISBN 3540617531, 2nd edition 2008, ISBN 9783540686255.
 P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, SpringerVerlag, New York, 1997. ISBN 038794785X.
 Conformal Field Theory page in String Theory Wiki lists books and reviews.
 Slava Rychkov, Lectures on Conformal Field Theory in D≥ 3 Dimensions, 2012
