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

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

1993-03-03

arxiv.org/abs/hep-th/9303019v1

Int. J. Mod. Phys. D2 (1993) 413-430

Physics

High Energy Physics

High Energy Physics - Theory

17 pages, plain TEX

Scientific paper

10.1142/S0218271893000295

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 ${Vir}^4$.

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