Two-dimensional conformal field theory

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 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. 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 ${\displaystyle z}$, the real vector space of infinitesimal conformal maps has the basis ${\displaystyle (\ell _{n}+{\bar {\ell }}_{n})_{n\in \mathbb {Z} }\cup (i(\ell _{n}-{\bar {\ell }}_{n}))_{n\in \mathbb {Z} }}$, with ${\displaystyle \ell _{n}=-z^{n+1}{\frac {\partial }{\partial z}}}$. (For example, ${\displaystyle \ell _{-1}+\ell _{-1}}$ and ${\displaystyle i(\ell _{1}-\ell _{-1})}$ generate translations.) Relaxing the assumption that ${\displaystyle {\bar {z}}}$ is the complex conjugate of ${\displaystyle z}$, i.e. complexifying the space of infinitesimal conformal maps, one obtains a complex vector space with the basis ${\displaystyle (\ell _{n})_{n\in \mathbb {Z} }\cup ({\bar {\ell }}_{n})_{n\in \mathbb {Z} }}$.

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