Spin Holonomy Research Articles

Linear perturbations of metrics with holonomy Spin(7)

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.

Complete noncompact Spin(7) manifolds from self-dual Einstein 4–orbifolds

We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk\"ahler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold.

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds

Suppose (X,Ω,g) is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin(7), or a Calabi–Yau 4-fold. Let G be U(m) or SU(m), and P→X be a principal G-bundle. We show that the infinite-dimensional moduli space BP of all connections on P modulo gauge is orientable, in a certain sense. We deduce that the moduli space MPSpin(7)⊂BP of irreducible Spin(7)-instanton connections on P modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung [9] and Muñoz and Shahbazi [42].If X is a Calabi–Yau 4-fold, the derived moduli stack M of (complexes of) coherent sheaves on X is a −2-shifted symplectic derived stack (M,ω) by Pantev–Toën–Vaquié-Vezzosi [46], and so has a notion of orientation by Borisov–Joyce [7]. We prove that (M,ω) is orientable, by relating algebro-geometric orientations on (M,ω) to differential-geometric orientations on BP for U(m)-bundles P→X, and using orientability of BP.This has applications to defining Donaldson–Thomas type invariants counting semistable coherent sheaves on a Calabi–Yau 4-fold, as in Donaldson and Thomas [15], Cao and Leung [8], and Borisov and Joyce [7].

On mirror maps for manifolds of exceptional holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

Non-supersymmetric F-theory compactifications on Spin(7) manifolds

We propose a novel approach to obtain four-dimensional effective actions coupled to supersymmetry-breaking boundaries by considering F-theory on manifolds with special holonomy Spin(7). To perform such studies we suggest that a duality relating M-theory on a certain class of Spin(7) manifolds with F-theory on the same manifolds times an interval exists. The Spin(7) geometries under consideration are constructed as quotients of elliptically fibered Calabi-Yau fourfolds by an anti-holomorphic and isometric involution. The three-dimensional minimally supersymmetric effective action of M-theory on a general Spin(7) manifold with fluxes is determined and specialized to the aforementioned geometries. This effective theory is compared with an interval Kaluza-Klein reduction of a non-supersymmetric four-dimensional theory with definite boundary conditions for all fields. Using this strategy a minimal set of couplings of the four-dimensional low-energy effective actions is obtained in terms of the Spin(7) geometric data. We also discuss briefly the string interpretation in the Type IIB weak coupling limit.

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang-Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian-Einstein connections on the Calabi-Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian-Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.

Riemannian Spin holonomy manifold carries octonionic-Kähler structure

We prove that Riemannian $Spin(7)$ holonomy manifolds carry octonionic-K\{a}hler structure.

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M16 we write explicitly the matrix ψ of its Kahler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of ΦSpin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).

IIB black hole horizons with five-form flux and KT geometry

We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-form flux preserving at least two supersymmetries. We demonstrate that there are three classes of solutions distinguished by the choice of Killing spinors. We find that the spatial horizon sections of the class of solutions with an SU(4) invariant pure Killing spinor are hermitian manifolds and admit a hidden Kahler with torsion (KT) geometry compatible with the SU(4) structure. Moreover the Bianchi identity of the 5-form, which also implies the field equations, can be expressed in terms of the torsion H as d(\omega\wedge H)=\partial\bar\partial \omega^2=0, where omega is a Hermitian form. We give several examples of near horizon geometries which include group manifolds, group fibrations over KT manifolds and uplifted geometries of lower dimensional black holes. Furthermore, we show that the class of solutions associated with a Spin(7) invariant spinor is locally a product R^{1,1} x S, where S is a holonomy Spin(7) manifold.

New examples of compact manifolds with holonomy Spin(7)

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds. We find that different hypersurfaces within the same family of Calabi-Yau 4-orbifolds may result in different Spin(7)-manifolds.

Heterotic black horizons

We show that the supersymmetric near horizon geometry of heterotic black holes is either an AdS 3 fibration over a 7-dimensional manifold which admits a G 2 structure compatible with a connection with skew-symmetric torsion, or it is a product $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ , where $ {\mathcal{S}^8} $ is a holonomy Spin(7) manifold, preserving 2 and 1 supersymmetries respectively. Moreover, we demonstrate that the AdS 3 class of heterotic horizons can preserve 4, 6 and 8 supersymmetries provided that the geometry of the base space is further restricted. Similarly $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ horizons with extended supersymmetry are products of $ {\mathbb{R}^{1,1}} $ with special holonomy manifolds. We have also found that the heterotic horizons with 8 supersymmetries are locally isometric to AdS 3 × S 3 × T 4, AdS 3 × S 3 × K 3 or $ {\mathbb{R}^{1,1}} \times {T^4} \times {K_3} $ , where the radii of AdS 3 and S 3 are equal and the dilaton is constant.

IIB backgrounds with five-form flux

We investigate all N = 2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups Spin ( 7 ) ⋉ R 8 , SU ( 4 ) ⋉ R 8 and G 2 . In the SU ( 4 ) ⋉ R 8 case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the SU ( 4 ) ⋉ R 8 structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The G 2 backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic G 2 structure. The Spin ( 7 ) ⋉ R 8 backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy Spin ( 7 ) . In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the AdS 5 black holes, lift to SU ( 4 ) ⋉ R 8 pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one G 2 manifold which has principal orbit S 3 × S 3 .

A Tour of Exceptional Geometry

A discussion of G 2 and its manifestations is followed by the definition of various groups acting on R 8 . Calculation of exterior and covariant derivatives is carried out for a specific metric on a 7-manifold, as a means to illustrate their dependence on the underlying Lie algebra. This example is used to construct an explicit metric with holonomy Spin(7), which is reduced so as to obtain both G 2 and SU(3) structures. Special categories of such structures are investigated and related to metrics with holonomy G 2 . A final section describes the orbifold construction leading to a known hyperkahler 8-manifold.

New cohomogeneity one metrics with Spin(7) holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A 8 , is complete and non-singular on R 8 . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S 4, and are denoted by B 8 + , B 8 − and B 8 . The metrics on B 8 + and B 8 − occur in families with a non-trivial parameter. The metric on B 8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A 8 . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold.

Self-dual N=(1,0) supergravity in eight dimensions with reduced holonomy Spin(7)

We construct chiral N=(1,0) self-dual supergravity in Euclidean eight dimensions with reduced holonomy Spin(7), including all the higher-order interactions in a closed form. We first establish the non-chiral N=(1,1) superspace supergravity in eight dimensions with SO(8) holonomy without self-duality, as the foundation of the formulation. In order to make the whole computation simple, and the generalized self-duality compatible with supersymmetry, we adopt a particular set of superspace constraints similar to the one originally developed in ten-dimensional superspace. The intrinsic properties of octonionic structure constants make local supersymmetry, generalized self-duality condition, and reduced holonomy Spin(7) all consistent with each other.

Coset construction of Spin(7), G2 gravitational instantons

We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy Spin(7), G2. We concentrate on the metrics which are defined on R × G/H. If the homogeneous coset spaces G/H have weak G2, SU(3) holonomy, the manifold R × G/H may have Spin(7), G2 holonomy metrics. Using the formulation with vector fields, we investigate the metrics with Spin(7) holonomy on R × Sp(2)/Sp(1), R × SU(3)/U(1). We have found the explicit volume-preserving vector fields on these manifolds using the elementary coordinate parametrization. This construction is essentially dual for solving the generalized self-duality condition for spin connections. We present the most general differential equations for each coset. Then, we develop a similar formulation in order to calculate metrics with G2 holonomy.

New complete noncompact Spin(7) manifolds

We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A 8 , is topologically R 8 and another, which we denote by B 8 , is the bundle of chiral spinors over S 4. Unlike the previously-known complete noncompact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S 4, our new metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L 2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A 8 and B 8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In Appendix A we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B 8 + and B 8 − similar to B 8 .

Spin Holonomy of Einstein Manifolds

Berger's Theorem classifies the linear holonomy groups of irreducible, simply connected Riemannian manifolds. For physical applications, however, it is at least as important to have a classification of the possible spin holonomy groups (defined by the parallel transport of spinors) of non-simply-connected manifolds. We establish a complete classification of the spin holonomy groups of all compact, locally irreducible, Einstein Riemannian spin manifolds of non-negative scalar curvature.

New Examples of Compact 8-manifolds of Holonomy Spin(7)

New Examples of Compact 8-manifolds of Holonomy Spin(7)

Compact 8-manifolds with holonomySpin(7)

In Berger’s classification [4] of the possible holonomy groups of a nonsymmetric, irreducible riemannian manifold, there are two special cases, the exceptional holonomy groups G2 in 7 dimensions and Spin(7) in 8 dimensions. Bryant [6] proved that such metrics exist locally, using the theory of exterior differential systems, and gave some explicit examples. Later, Bryant and Salamon [7] constructed explicit, complete metrics with holonomy G2 and Spin(7). In two previous papers [12], [13], the author constructed many examples of compact riemannian 7-manifolds with holonomy G2. This paper will construct examples of compact riemannian 8-manifolds with holonomy Spin(7), using similar methods. We believe that these are the first examples known. Since metrics with holonomy Spin(7) are ricci-flat, these are also new examples of compact, ricci-flat riemannian 8-manifolds. Let M be an 8-manifold. A Spin(7)structure on M can be encoded in a 4-form Ω on M , a special 4-form satisfying the condition that the stabilizer of Ω at each point should be isomorphic to Spin(7). By an abuse of notation, we usually identify the Spin(7)structure with its associated 4-form Ω. Since Spin(7) ⊂ SO(8), the Spin(7)structure also induces a riemannian metric g and an orientation on M . It turns out that the holonomy group Hol(g) of g is a subgroup of Spin(7), with Spin(7)structure Ω, if and only if dΩ = 0. The quantity dΩ is called the torsion of the Spin(7)structure Ω, and Ω is called torsion-free if dΩ = 0. Very briefly, the plan of the paper splits into four steps, as follows. Firstly, a compact 8-manifold M is given. Secondly, a Spin(7)structure Ω on M is found, with small torsion, i.e. dΩ is small. Thirdly, Ω is deformed to a nearby Spin(7)structure Ω with dΩ = 0. Thus Ω is torsion-free, and if g is the associated metric then Hol(g) ⊂ Spin(7). Fourthly, it is shown that Hol(g) is Spin(7), and not some proper subgroup. The structure of the paper is designed around this division of the construction into four steps. There are six chapters. This first chapter is of introductory material. The second chapter has the full statements of the main results. In