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

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!dζ 1 · · · dζ 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 L = 1 4 G µνλρ ∂ µ φ∂ ν φ∂ λ φ∂ ρ φ and show that it is an integrable conformal model in 4 dimensions. Furthermore, the associated symmetry group is V ir 4 . ICP. 04.50.+h Unified field theories and other theories of gravitation. *

Invariants and polynomial identities for higher rank matrices

“…Higher rank tensors, which look similar to higher rank matrices, appear in several contexts such as in Finsler geometry [2,17] and in fourth-rank gravity [19,21,22]. The results presented here are a first step for the construction of differential invariants for higher-rank tensors.…”

Riemann-Eddington theory: Incorporating matter, degravitating the cosmological constant

“…generalizes the usual determinant in (4) to rank-four tensors [19,20]. There are four ǫ symbols here yet the covariant-looking and contravariant-looking symbols contract as…”

“…The identity (20) ensures that Q βνα µ vanishes identically unless c 1 = 1. With c 5 = −c 4 = c 1 = 1, however, the identity (19) never conforms to the Bianchi identities for curvature tensors.…”

Towards an alternative gravitational theory

“…Moreover, we attempt to generalize the formalism to the relativistic scenario. We argue that the Spiric sections may eventually lead to a gravitational theory with a four-rank metric [13]- [18]. We comment about the possible geometric object implied by such a four-rank metric such as a generalized eight-rank Riemann curvature tensor.…”

“…In Refs. [13]- [18] have already consider a solution for this question. The general idea is to associate a four-rank Riemann tensor for g µ 1 ...µ 4 .…”