Spinorial classification of Spin(7) structures

Abstract

We describe the different classes of Spin(7) structures in terms of spinorial equations. We relate them to the spinorial description of G 2 structures in some geometrical situations. Our approach enables us to analyze invariant Spin(7) structures on quasi abelian Lie algebras. IntroductionBerger's list [2] (1955) of possible holonomy groups of simply connected, irreducible and non-symmetric Riemannian manifolds contains the so-called exceptional holonomy groups, G 2 and Spin (7), which occur in dimensions 7 and 8 respectively. Non-complete metrics with exceptional holonomy were given by Bryant in [3], complete metrics were obtained by Bryant and Salamon in [4], but compact examples were not constructed until 1996, when Joyce published [12], [13] and [14].The remaining groups of Berger's list different from SO(n), called special holonomy groups, are U(n), SU(n), Sp(n) and Sp(n) · Sp(1). If the holonomy of a Riemannian manifold is contained in a group G, the manifold admits a G structure, that is, a reduction to G of its frame bundle. Therefore, holonomy is homotopically obstructed by the presence of G structures. Examples of manifolds endowed with G structures for some of the holonomy groups in the Berger list are not only easier to obtain than manifolds with holonomy in G, but also relevant in M-theory, especially if they admit a characteristic connection [10], that is, a metric connection with totally skew-symmetric torsion whose holonomy is contained in G. It is worth mentioning that Ivanov proved in [11] that each manifold with a Spin(7) structure admits a unique characteristic connection. Moreover, Friedrich proved in [9] that Spin (7) is the unique compact simple Lie group G such that all the G structures admit a unique characteristic connection.The Lie group G 2 is compact, simple and simply connected. It consists of the endomorpisms of R 7 which preserve the cross product from the imaginary part of the octonions [22]. Hence, a G 2 structure on a manifold Q determines a 3-form Ψ, a metric and an orientation. In [7], Fernández and Gray classify G 2 structures into 16 different classes in terms of the G 2 irreducible components of ∇Ψ. Related to this, the analysis of the intrinsic torsion in [5] allowed to obtain equations involving dΨ and d( * Ψ) for each of the 16 classes, determined by the G 2 irreducible components of Λ 4 T * Q and Λ 5 T * Q. In particular, one obtains that the holonomy of Q is contained in G 2 if and only if dΨ = 0 and d( * Ψ) = 0. The Lie group Spin (7) is also compact, simple and simply connected. It is the group of endomorphisms of R 8 which preserve the triple cross product from the octonions [22]. Thus, a Spin(7) structure on a manifold M determines 4-form Ω, a metric and an orientation. In [6], Fernández classifies Spin(7) structures into 4 classes in terms of differential equations for dΩ, which are determined by the Spin(7)-irreducible components of Λ 5 T * M . Parallel structures verify dΩ = 0, locally conformally parallel structures satisfy dΩ = θ ∧ Ω for a closed 1-form θ and ...

Spin-harmonic structures and nilmanifolds

“…As for spin-harmonic structures, the following result was proved by the second author in [21]: Theorem 3.5. The spinor η determines a positive spin-harmonic structure if and only if the induced Spin(7) structure is balanced.…”

“…Spin-harmonic structures have already appeared, under disguise, in the papers [1] and [21]; we proceed to review the relevant results and to relate spin-harmonic structures with the different kinds of Spin (7), G 2 and SU(3) structures. There is no spinorial description of SU(2) structures in dimension 5; we will carry out this classification in Section 4.…”

“…A systematic spinorial approach to Spin (7), along the lines of [1], was taken by the second author in [21]. In particular, the observation that balanced Spin (7) structures are equivalent to unitary harmonic spinors was exploited in [21] to construct examples of balanced Spin (7) structures on 8-dimensional nilmanifolds and solvmanifolds. There it became clear that the spinorial approach has some practical advantages on the "classical" one, which uses the 4-form.…”

The analysis of the harmonic-Spin(7) flow

“…1. A classification of Spin( 7)-structures was given by Fernandez in [14] and a formulation in terms of spinors can be found in [27].…”