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Four-dimensional conformal field theory models with rational correlation functions. (English) Zbl 1041.81097
Summary: Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field $\phi$ of dimension two. It depends on a positive real parameter $c$, an analogue of the Virasoro central charge, and admits for all (finite) $c$ an infinite number of conserved symmetric tensor currents. The operator product algebra of $\phi$ is shown to coincide with a simpler one, generated by a bilocal scalar field $V(x_1,x_2)$ of dimension (1,1). The modes of $V$ together with the unit operator span an infinite dimensional Lie algebra $\frak L_V$ whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge $c$. Wightman positivity (i.e. unitarity of the representations of $\frak L_V$) is proven to be equivalent to $c\in \Bbb N$.

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. 81T05 Axiomatic quantum field theory; operator algebras
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