Editing Conformal field theory (section)

=== Conformal field theories with a Virasoro symmetry algebra ===
{{main|Two-dimensional conformal field theory}}

In a conformally invariant two-dimensional quantum theory, the Witt algebra of infinitesimal conformal transformations has to be [[Lie algebra extension#Virasoro algebra|centrally extended]]. The quantum symmetry algebra is therefore the [[Virasoro algebra]], which depends on a number called the '''central charge'''. This central extension can also be understood in terms of a [[conformal anomaly]].

It was shown by [[Alexander Zamolodchikov]] that there exists a function which decreases monotonically under the [[renormalization group]] flow of a two-dimensional quantum field theory, and is equal to the central charge for a two-dimensional conformal field theory. This is known as the Zamolodchikov [[C-theorem]], and tells us that [[renormalization group flow]] in two dimensions is irreversible.<ref name="zam86"/>

In addition to being centrally extended, the symmetry algebra of a conformally invariant quantum theory has to be complexified, resulting in two copies of the Virasoro algebra. 
In Euclidean CFT, these copies are called holomorphic and antiholomorphic. In Lorentzian CFT, they are called left-moving and right moving. Both copies have the same central charge.

The [[state space (physics)|space of states]] of a theory is a [[Lie algebra representation|representation]] of the product of the two Virasoro algebras. This space is a [[Hilbert space]] if the theory is unitary.
This space may contain a vacuum state, or in statistical mechanics, a thermal state. Unless the central charge vanishes, there cannot exist a state that leaves the entire infinite dimensional conformal symmetry unbroken. The best we can have is a state that is invariant under the generators <math>L_{n\geq -1}</math> of the Virasoro algebra, whose basis is {{tmath|1=(L_n)_{n\in\mathbb{Z} } }}. This contains the generators <math>L_{-1},L_0,L_1</math> of the global conformal transformations. The rest of the conformal group is spontaneously broken.