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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 has important applications in string theory, statistical mechanics, and condensed matter physics.

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 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

$L_n W(z)=-z^{n+1} \frac{\partial}{\partial z} W(z) - \Delta z^n W(z)$

and

$\bar L_n W(z)=0$.

Similarly for an antichiral field. Δ is the conformal weight of the chiral field W.