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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 \(\mathfrak L_V\) 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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