Abstract
In this thesis we examine homogeneous spaces from the viewpoint of spin geometry, with a particular focus on the existence (or non-existence) of special spinor fields and their corresponding geometric structures. Recalling that a reductive homogeneous space M = G/H has an elegant description of its geometrically relevant bundles (e.g. the tangent bundle, frame bundle, bundles of tensors and differential forms, etc.) as homogeneous bundles associated to the H-principal bundle G → G/H, the natural objects of study are the (G-)invariant sections of these bundles. Under certain topological conditions on the isotropy representation, there exists a G-invariant spin structure and associated spinor bundle on M = G/H (see [DKL22, Prop. 1.3]), and we shall be interested in the invariant sections of the latter. By working at the origin o = eH ∈ G/H, finding invariant objects can be reduced to a purely representation-theoretic problem, which we approach using various results from classical invariant theory, among other methods.Chapter 3 is devoted to the exposition of [AHL23] (joint work with I. Agricola and M.-A. Lawn), where we have obtained a classification of the invariant spinors on the nine realizations of the sphere as a Riemannian homogeneous space. Partial results for a few of the simpler cases have appeared in, or may be deduced from, [Wan89], however a full classification and description of the invariant spinors and their related geometric structures has before now not been attempted. In each case we give an explicit basis for the space of invariant spinors, using the realization of the spin representation in terms of exterior forms, and describe the differential equations they satisfy (e.g. Killing, generalized Killing, etc.). Notably, we construct (to our knowledge) the first examples of generalized Killing spinors whose associated endomorphism field has four distinct eigenvalues. Where relevant, we also explore the relationships between the invariant spinors and certain invariant tensors and differential forms (and their related G-structures); these are compared with known results from the literature.
Chapter 4 presents the work [Hof22], which deals mainly with invariant spinors on homogeneous 3-Sasakian spaces, (M = G/H, g, ξi, ηi, φi)3i =1. The dimensions of the spaces of invariant forms of degree ≤ 3 on these spaces have appeared already in [DOP20]. We build on this to obtain a complete description of the invariant φ1-(anti-)holomorphic differential forms of all degrees, as well as an explicit description of the space of invariant spinors, which to the author’s knowledge has never appeared beyond the isolated case of Sp(n)/ Sp(n − 1) treated in [AHL23]. We show that the invariant spinors are spanned by the Clifford products of invariant differential forms with a certain invariant Killing spinor. It is well-known that a simply-connected 3-Sasakian manifold of dimension 4n − 1 admits n + 1 linearly independent Killing spinors [B¨ar93], and a partial construction of these spinors as sections of certain subbundles E−i , i = 1, 2, 3 of the spinor bundle is given in [FK90], however this description is incomplete for spaces of dimension > 19. We complete this description in the homogeneous case, giving an explicit basis for the space of Killing spinors carried by a homogeneous 3-Sasakian space. It follows from our result that any Killing spinor on a homogeneous 3-Sasakian space is invariant.
Chapters 5 and 6 are based on joint work with I. Agricola. We consider 3-(α, δ)-Sasaki spaces, which can be viewed as deformations of 3-Sasakian spaces [AD20]. The first half of the chapter contains a novel examination of the behaviour of certain Killing spinors on 3-Sasakian spaces under such deformations; we give a detailed proof of the new spinorial field equation satisfied by the deformed Killing spinors on the resulting 3-(α, δ)-Sasaki space. The second half of the chapter studies the dual compact/non-compact pairs of homogeneous 3-(α, δ)-Sasaki spaces described in [ADS21, Remark 3.1.1c]. We modify the dualization construction of Kath in [Kat00] to obtain an identification of the spinor bundles for these dual pairs, and show that there is a natural correspondence between deformed Killing spinors on the two spaces.
| Date of Award | 1 Jun 2023 |
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| Original language | English |
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| Supervisor | Konstanze Rietsch (Supervisor) & affiliated academic (Supervisor) |