Integrable Conformal Field Theory in Four Dimensions and Fourth-Rank Geometry

TL;DR

This paper explores the conformal properties of higher-rank geometries, introduces null-flat spaces, and constructs an integrable conformal field theory in four dimensions based on fourth-rank geometry.

Contribution

It demonstrates the necessity of fourth-rank geometry for integrable conformal theories in 4D and introduces a new model with ${Vir}^4$ symmetry group.

Findings

Null-flat spaces have a critical dimension equal to their rank.

A specific 4D conformal model is shown to be integrable.

The symmetry group of the model is ${Vir}^4$.

Abstract

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 ${ds}^r=r!d\zeta_1\cdots d\zeta_r$. 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 4 dimensions we need to rely on fourth-rank geometry. We consider the simple model ${\cal L}={1\over 4} G^{\mu\nu\lambda\rho}\partial_\mu\phi\partial_\nu\phi\partial_\lambda\phi \partial_\rho\phi$ and show that it is an integrable conformal model in 4 dimensions. Furthermore, the associated symmetry group is ${Vir}^4$.