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A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. Conformal field theory is most often studied in two dimensions where there is a large group of local conformal transformations coming from holomorphic functions. Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the...
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
Conformal field theory has important applications in string theory, statistical mechanics, and condensed matter physics. A string theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ...
Conformal field theory can cover Riemann surfaces of arbitrary genus. Take, for example, the Riemann sphere. It has the Möbius transformations as the conformal group, which is isomorphic to PSL(2,C). However, if we disregard finite conformal transformations in favor of infinitesimal transformations, we have a much larger infinite dimensional algebra of conformal generators, called the Witt algebra. In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...
Geometry In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. ...
In mathematics, a Witt algebra is the Lie algebra of Killing vector fields defined on the Riemann sphere. ...
In most conformal field theories, a conformal anomaly, also known as a Weyl anomaly, arises. This results in the appearance of a nontrivial central charge resulting in the Virasoro algebra. In mathematics, the Virasoro group is a central extension of the orientation-preserving diffeomorphism group of the circle. ...
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 heighest weight modules of the Virasoro algebra.
A chiral field is a holomorphic field W(z) which transforms as
and
.
Similarly for an antichiral field. Δ is the conformal weight of the chiral field W.
See also
In mathematics, a vertex operator algebra (abbreviated: VOA) is a certain kind of algebra that plays a key part in conformal field theory and other fields of study in physics, and has also proven useful in purely mathematical contexts such as moonshine theory. ...
In theoretical physics and mathematics, the Wess-Zumino-Witten (WZW) model, also called the Wess-Zumino-Novikov-Witten model, is a simple model of conformal field theory whose solutions are realized by affine Kac-Moody algebras. ...
In quantum field theory, the operator product expansion (OPE) is a convergent expansion of the product of two fields at different points as a sum (possibly infinite) of local fields. ...
References and external links - Paul Ginsparg, Applied Conformal Field Theory (http://arxiv.org/abs/hep-th/?9108028).
- P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
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