INTEGRABLE CONFORMAL FIELD THEORY IN FOUR DIMENSIONS AND FOURTH-RANK GEOMETRY

We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as dsr=r!dζ1 … dζr. We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in four dimensions we need to rely on fourth-rank geometry. We consider the simple model ℒ = ¼ Gμνλρ ∂μ ϕ ∂ν ϕ ∂λ ϕ ∂ρϕ and show that it is an integrable conformal model in four dimensions. Furthermore, the associated operators satisfy a Vir4 algebra.