Rational Conformal Field Theory in Four Dimensions

We use the recently established rationality of correlation functions in a globally conformally invariant quantum field theory satisfying Wightman axioms to construct a family of solvable models in the four-dimensional Minkowski space–time. We consider the model of a neutral two-dimension scalar field $\phi$ in detail. It depends on a positive real parameter $c$, an analogue of the Virasoro central charge; for all (finite) $c$, it admits an infinite number of conserved symmetric tensor currents. The operator product algebra of $\phi$ coincides 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 depend only on the central charge $c$. The Wightman positivity (i.e. unitarity of the representations of $\mathfrak {L}_V$ ) is proved equivalent to $c \in \mathbb {N}$.