Reducible conformal holonomy in any metric signature and application to twistor spinors in low dimension

Abstract

We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally isotropic subspace of arbitrary dimension, there is, w.r.t. a local metric in the conformal class defined off a singular set, a parallel, totally isotropic distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant isotropic lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification of geometries admitting twistor spinors described in terms of parallel spin tractors using conformal spin tractor calculus. As an example we obtain together with already known results about generic distributions in dimensions 5 and 6 a complete geometric description of local geometries admitting real twistor spinors in signatures (3, 2) and (3,3). In contrast to the generic case where generic geometric distributions play an important role, the underlying geometries in the non-generic case without zeroes turn out to admit integrable distributions.

The ambient obstruction tensor and conformal holonomy

“…Finally, if actually H ⊂ R p+1,q+1 is totally degenerate, of dimension k + 1 ≥ 2 and holonomy invariant, there exists -again locally and off a singular set -a metric g ∈ c admitting a ∇ g -invariant and totally degenerate distribution L ⊂ T M of rank k which additionally satisfies Im(Ric g ) ⊂ L, as has been shown in [38,40,45].…”

“…Proof. The existence of a parallel distribution L ⊂ T M containing the image of Ric g was proven in [45]. To see that at each x ∈ M , the fibre L…”

“…where f = f (x 1 , x 2 , x 3 ) is a differentiable function of the variables (x 1 , x 2 , x 3 ) only. It was shown that, whenever f depends only on x 3 and x 1 , the corresponding conformal class contains a metric for which the image of the Schouten tensor lies in a parallel rank 3 distribution, which implies [45] that the conformal holonomy is contained in spin (3,4) H . In addition, these conformal structures turned out to have vanishing obstruction tensor, and therefore they admit ambient metrics.…”

Explicit ambient metrics and holonomy

“…. , p. Moreover, the image of the Ricci endomorphism is contained in V. Then, on the other hand, the results in [17, Theorem 1.1], in the case p = 1, and the generalization in [19,Theorem 1] imply that the normal conformal tractor bundle admits a totally null subbundle of rank p + 1 that is parallel for the normal conformal tractor connection of [g], i.e., its fibres are invariant under the conformal holonomy. On the other hand, in the case of odd-dimensional analytic pp-waves, the general theory in [12] ensures that 3 Note that the converse is not true: a tensor whose stabilizer contains the holonomy algebra, must not be parallel.…”

“…Remark 4.3. Since the conformal holonomy is contained in the ambient holonomy, the result in [19,Theorem 1] implies that a conformal classes [g D f ] with f = f (x 1 , x 3 ) contains a certain preferred metric g 0 . This metric admits a parallel totally null rank 3 distribution which contains the image of the Ricci-tensor (or equivalently, of the Schouten tensor).…”