Polynomial identities for hypermatrices

We develop a method to construct algebraic invariants for hypermatrices. We then construct hyperdeterminants and exhibit a generalization of the Cayley-Hamilton theorem for hypermatrices.Comment: 65 pages. Several results expanded. Title changed to better reflect the content and agree with published preprint versio

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 ${ds}^r=r!d\zeta_1\cdots d\zeta_r$. 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 ${Vir}^4$.Comment: 17 pages, plain TE