Linear perturbations of metrics with holonomy Spin(7)

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Abstract

We apply the method of linear perturbations to the case of Spin(7)-structures, showing that the only nontrivial perturbations are those determined by a rank one nilpotent matrix.

We consider linear perturbations of the Bryant-Salamon metric on the spin bundle over S4 that retain invariance under the action of Sp(2), showing that the metrics obtained in this way are isometric.

Introduction

Riemannian metrics with holonomy Spin(7) have been studied in differential geometry since the celebrated theorem of Berger [3], listing the possible holonomy groups of an irreducible, nonsymmetric simply connected Riemannian manifold. Metrics with holonomy contained in Spin(7) are known to be Ricci-flat [4], and they imply the presence of a parallel spinor [24]. They are also relevant for string theory (see [14]).

The first local examples of metrics with holonomy Spin(7) were constructed in [5], and the first complete metric was obtained in [6]; the latter takes the form of an explicit Sp(2)-invariant metric on the spinor bundle over S4. It was later shown in [10] that this metric belongs to a one-parameter family of invariant metrics.

We note that the metrics of [6] are of cohomogeneity one; other cohomogeneity one metrics with holonomy contained in Spin(7) have been constructed in [11], [18], [14], [23], [9], [7], [1], [2]. Outside of the cohomogeneity one setting other constructions exist, but the metrics they determine are not explicit (see [16], [17], [13], [20]).

As observed in [5], a metric with holonomy contained in Spin(7) is defined by a closed form Ω which is pointwise linearly equivalent to a reference 4-form on R8 with stabilizer Spin(7). It is then possible to define perturbations of a Spin(7)-metric by replacing Ω with a perturbed form Ω+δ which remains pointwise linearly equivalent to Ω. Notice that for the parallel 3-forms φ arising in the context of holonomy G2 the form φ+δ is always linearly equivalent to φ for δ sufficiently small; in other terms, φ is stable in the sense of [15]. The Spin(7) form Ω is not stable, however, so more work is needed in order to obtain a perturbation.

One possible approach was considered in [19, Section 5.2] by takingδ=v(wΩ)w(vΩ), for v,w vector fields on M. In terms of the infinitesimal action ρ of gl(TxM) on Λ4TxM, this amounts to setting δ=ρ(A)Ω, where A is the skew-symmetric endomorphism A=vwwv. We recall that under Spin(7) the bundle of four-forms splits asΛ14Λ74Λ274Λ354; the skew-symmetric A determines a perturbation term δ in Λ74. Whilst this construction gives nontrivial perturbations of the original metric in the case of G2 (mutatis mutandis: the relevant decomposition is Λ13Λ73Λ273 and the perturbation δ an element of Λ73), it turns out that in the Spin(7) case the perturbed form never defines a Spin(7)-structure ([19]).

A different ansatz was considered in [8] in the context of Sp(2)Sp(1)-structures, which amounts to imposing that A be nilpotent, rather than skew-symmetric. The key observation, working at a point, is that whenρ(A)(ρ(A)Ω)=0, the formΩ+tδ,δ=ρ(A)Ω is always in the same GL(8,R)-orbit as Ω for any t; one then says that δ is a linear perturbation of Ω. It turns out (see [8]) that one can assume A to be nilpotent without loss of generality.

In this paper we study nilpotent perturbations of the Spin(7)-form Ω. By a case-by-case analysis of the possible Jordan forms of a nilpotent matrix in gl(8,R), and making use of Spin(7)-invariance of (3), we prove that any linear perturbation of the Spin(7) form Ω is defined by a rank one nilpotent matrix, i.e. it has the formδ=v(wΩ), with v,w orthogonal vector fields. In terms of (2), the resulting perturbations of the Spin(7) form turn out to be elements of Λ74Λ354.

We apply the method of linear perturbations to the Bryant-Salamon metric; we construct a family of linear perturbations parameterized by three functions of one variable. However, it turns out that the resulting metrics are isometric; due to the fact that nilpotent perturbations preserve volumes, we do not recover the squashed deformations of [10].

Our result complements the result of [21], stating that the Bryant-Salamon is rigid in the class of asymptotically conical Spin(7) metrics.

Acknowledgments. This work is partly based on the second author's master thesis [22]. We thank Thomas Madsen for useful discussions.

Section snippets

Linear perturbations

In this section we classify linear perturbation at a point of 4-forms defining a Spin(7)-structure, proving that they are in one-to-one correspondence with nilpotent matrices of rank one in gl(8,R).

We first recall some results from [8]. For a lighter notation, we shall write Rn instead of (Rn). Denote bygl(Rn)×ΛkRnΛkRn,(A,ω)ρ(A)ω the natural action of gl(Rn) on ΛkRn. We shall write ρ(A)2ω for ρ(A)(ρ(A)ω).

Proposition 1.1

[8]

Fix ωΛkRn and a solution Agl(Rn) ofρ(A)2ω=0. Thenβt=ω+tρ(A)ω lies in the same

A cohomogeneity one description of the Bryant-Salamon metric

Recall from [6] that the spinor bundle S over S4 carries a cohomogeneity one metric with holonomy Spin(7); this metric has cohomogeneity one under the action of Sp(2). In this section we give a description of these metrics in terms of cohomogeneity one actions which will be needed in order to study the linear perturbations.

Explicitly, the Lie group Sp(2)={gGL(2,H)|gg=I} contains two copies of Sp(1), i.e.Sp(1)+={(p001)|pSp(1)},Sp(1)={(100q)|qSp(1)}. At the Lie algebra level,sp(2)={(abbc)},

Linear perturbations of the Bryant-Salamon metric

In this section we study Sp(2)-invariant linear perturbations of the Bryant-Salamon metric.

By Theorem 1.7, a linear perturbation of a Spin(7)-structure is obtained by the choice of a rank one nilpotent endomorphism of the tangent bundle at each point. Thus, the global data for a linear perturbation is given by the choice of a vector field X and a one form α with α(X)=0. Since we work in the Sp(2)-invariant setting, we will require both vector field and form to be invariant.

Thus, the first step

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