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Conformal field theory
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.
Conformal field theory can cover Riemann surfaces of arbitrary genus. Take, for example, the Riemann sphere. It has the M÷bius 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 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.
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
Similarly for an antichiral field. Δ is the conformal weight of the chiral field W.
References and external links
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