## 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 $$\mathfrak L_V$$ 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 $$\mathfrak L_V$$) is proven to be equivalent to $$c\in \mathbb N$$.

### MSC:

 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81T05 Axiomatic quantum field theory; operator algebras
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