Four Dimensional CFT Models with Rational Correlation Functions

TL;DR

This paper constructs and analyzes a family of exactly solvable four-dimensional conformal field theory models with rational correlation functions, revealing a connection between unitarity and the integer-valued central charge parameter.

Contribution

It introduces a new class of 4D CFT models with rational correlators, featuring an infinite-dimensional Lie algebra structure and establishing conditions for unitarity based on the central charge.

Findings

Correlation functions are rational in these models.

Unitarity holds only when the central charge c is a positive integer.

The operator algebra simplifies to a bilocal scalar field structure.

Abstract

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 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field $\phi$ of dimension 2. 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 $L_V$ 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 $c \in N$.