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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 ϕ 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 ϕ 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 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 𝔏 V ) is proven to be equivalent to c.
MSC:
81T40Two-dimensional field theories, conformal field theories, etc.
81T05Axiomatic quantum field theory; operator algebras