Spin^h and further generalisations of spin

Abstract

The question of which manifolds are spin or spin c has a simple and complete answer. In this paper we address the same question for spin h manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spin h is the fifth integral Stiefel-Whitney class W 5 . Moreover, we show that every orientable manifold of dimension 7 and lower is spin h , and that there are orientable manifolds which are not spin h in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we determine an answer to the following question: is there an integer k such that every manifold embeds in a spin manifold with codimension k?

Spin^h Manifolds

“…In fact the work in [1] goes much further than my Spin h discussion. They define a notion of a Spin k -manifold for all integers k > 0 where k = 1, 2, 3 correspond to the three cases above.…”

“…See, for example, [9,21]. Recent math papers include [1,2,12,13,18]. Now in fact, it turned out that physicists had also found Spin h -manifolds, and were interested in them for physical reasons.…”

“…Jiahao then defined a Thom class for Spin h -vector bundles in symplectic K-theory, using (∆ + − ∆ − ) ⊗ R H where ∆ ± are the real irreducible spinor bundles for Spin(8n) and H is the canonical representation of Sp (1). He constructed the Thom spectrum MSpin h for Spin hcobordism, and defined a map ind h : MSpin h −→ KSp, which allowed him to define the cycles needed for real K-theory.…”

Differential geometry of ${\mathsf{SO}}^{*}(2n)$-structures and ${\mathsf{SO}}^{*}(2n){\mathsf{Sp}}(1)$-structures - Part II

“…In particular, by the inclusion SO * (2n) Sp(1) ⊂ Sp(4n, R) we conclude that the vanishing of w 2 (M ) for n = 2m, guarantees the reduction of the metaplectic structure to a certain 2-fold covering of SO * (2n) Sp(1). Such a (unique) 2-fold covering is given by Spin * (2n) Sp(1), and the corresponding lifts of the Gstructure can been seen as a generalization of the so-called Spin q -structures, discussed in [N95, B99,AlM21]. We plan to examine Spin * (2n) Sp(1)-structures on 8n-dimensional almost qs-H manifolds (M, Q, ω) in a forthcoming paper in this series.…”