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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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