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Concept# Two-dimensional conformal field theory

Summary

A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method.
Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.
Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions.
While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces.
Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain the CFT on the glued surface.
On the other hand, some CFTs exist only on the sphere.
Unless stated otherwise, we consider CFT on the sphere in this article.
Given a local complex coordinate , the real vector space of infinitesimal conformal maps
has the basis , with . (For example, and generate translations.) Relaxing the assumption that is the complex conjugate of , i.e. complexifying the space of infinitesimal conformal maps, one obtains a complex vector space with the basis .
With their natural commutators,
the differential operators generate a Witt algebra.
By standard quantum-mechanical arguments, the symmetry algebra of conformal field theory must be the central extension of the Witt algebra, i.e. the Virasoro algebra, whose generators are , plus a central generator. In a given CFT, the central generator takes a constant value , called the central charge.
The symmetry algebra is therefore the product of two copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators , and the right-moving or antiholomorphic algebra, with generators .

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Minimal model (physics)

In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra. In minimal models, the central charge of the Virasoro algebra takes values of the type where are coprime integers such that .

Liouville field theory

In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge of its Virasoro symmetry algebra, but it is unitary only if and its classical limit is Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically.

Virasoro algebra

In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. The Virasoro algebra is spanned by generators Ln for n ∈ Z and the central charge c. These generators satisfy and The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.

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