Parallel Pure Spinors on Pseudo-Riemannian Manifolds

Kath, Ines

doi:10.1142/9789812792051_0008

Cited by 41 publications

(62 citation statements)

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“…Proof. This well-known fact is usually proved using Killing spinors; see [7,24,29]. We give a proof relying exclusively on the framework of stable forms and the Hitchin flow.…”

Section: Cones Over Nearly ε-Kähler Manifoldsmentioning

confidence: 93%

Half-flat structures and special holonomy

Cortés

Leistner

Schäfer

et al. 2010

Proceedings of the London Mathematical Society

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It was proved by Hitchin that any solution of his evolution equations for a half-flat SU(3)structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G 2. We give a new proof, which does not require the compactness of M . More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N , for any real form G of SL(3, C). If G is non-compact, then the holonomy group of N is a subgroup of the non-compact form G * 2 of G C 2 . Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G 2-or G * 2 -structures, as well as for the extension of cocalibrated G2-and G * 2 -structures by parallel Spin(7)-and Spin 0 (3, 4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G 2-or G * 2 -structure. For the group H3 × H3, where H 3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G 2-or G * 2 -structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G 2 and G * 2 . Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (ω, ρ) on H 3 × H3 satisfying ω(z, z) = 0, where z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special Kähler manifolds and one special para-Kähler manifold.Proposition 1.1. Let V be an n-dimensional real or complex vector space. The general linear group GL(V ) has an open orbit in Λ k V * , with 0 k n/2, if and only if k 2 or if k = 3 and n = 6, 7 or 8.Proof. The representation of GL(V ) on Λ k V * is irreducible. In the complex case the result thus follows, for instance, from the classification of irreducible complex prehomogeneous vector spaces [32]. The result in the real case follows from the complex case, since the complexification of the GL(n, R)-module Λ k R n * is an irreducible GL(n, C)-module. Remark 1.2. An open orbit is unique in the complex case, since an orbit that is open in the usual topology is also Zariski-open and Zariski-dense (see [31, Proposition 2.2]). Over the reals, the number of open orbits is finite by a well-known theorem of Whitney.Proposition 1.4. Let V be an oriented real vector space of dimension n and assume that k ∈ {2, n − 2} with n even, or k ∈ {3, n − 3} with n = 6, 7 or 8. There is a GL + (V )-equivariant mappinghomogeneous of degree n/k, which assigns a volume form to a stable k-form and which vanishes on non-stable forms. Given a stable k-form ρ, the derivative of φ in ρ defines a dual (n − k)-form ρ ∈ Λ n−k V * by the property(1.1)The dual formρ is also stable and satisfiesA stable form, it...

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“…Proof. This well-known fact is usually proved using Killing spinors; see [7,24,29]. We give a proof relying exclusively on the framework of stable forms and the Hitchin flow.…”

Section: Cones Over Nearly ε-Kähler Manifoldsmentioning

confidence: 93%

Half-flat structures and special holonomy

Cortés

Leistner

Schäfer

et al. 2010

Proceedings of the London Mathematical Society

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“…The main examples we will be dealing with are (1) Ω(X) = λ ⋅ X ♭ for λ ∈ C, which leads to the equation for geometric Killing spinors (cf. [10,13]). In this case (5) is vacuous.…”

Section: Holonomy Of Left-invariant Spinor Connectionsmentioning

confidence: 99%

“…These backgrounds have been studied in detail in [4]. However, it is yet unclear which of the classical backgrounds presented there admit solutions to (13). We want to study this problem using the results from the previous section, i.e.…”

Section: Application To Symmetric M-theory Backgroundsmentioning

confidence: 99%

(M-theory-)Killing spinors on symmetric spaces

Hustler

Lischewski

2015

Journal of Mathematical Physics

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Abstract. We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form D = ∇ + Ω in a purely algebraic and algorithmic way, whereis a left-invariant homomorphism. Specialising this to the case of symmetric M −theory backgrounds (i.e. (M, g, F ) with (M, g) a symmetric space and F an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric symmetric M −theory backgrounds.

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“…Refs. [23,13] give a local normal form of the metric in this situation: For (M, h) a pseudo-Riemannian spin manifold of split signature (m + 1, m) admitting a real pure parallel spinor field in Γ(M, S h ), one can find for every point in M local coordinates (x, y, z), x = (x 1 , . .…”

Section: Application To Twistor Spinorsmentioning

confidence: 99%

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

Lischewski

2015

Differential Geometry and its Applications

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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.

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Parallel Pure Spinors on Pseudo-Riemannian Manifolds

Cited by 41 publications

References 0 publications

Half-flat structures and special holonomy

Half-flat structures and special holonomy

(M-theory-)Killing spinors on symmetric spaces

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

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