nLab spinʰ structure

Definition

With Sp(1) denoting the quaternion unitary group, we define the spinʰ group

$$Spin^h(n) \coloneqq \big( Spin(n) \times Sp(1) \big)/ \big( \mathbf{Z}/2 \big) \,,$$

in complete analogy to the spinᶜ group

$$Spin^c(n) \coloneqq \big( Spin(n) \times U(1) \big)/ \big( \mathbf{Z}/2 \big) \,,$$

and the Spin group

$$Spin(n) \coloneqq \big( Spin(n) \times O(1) \big) / \big( \mathbf{Z}/2 \big) \,.$$

We have a canonical double covering, which is a homomorphism of Lie groups:

$$Spin^h(n) \longrightarrow SO(n)\times SO(3) \,.$$

It induces canonical homomorphisms of Lie groups

$$Spin^h(n) \to SO(n)$$

and

$$Spin^h(n) \to SO(3) \,.$$

A spin$^h$-structure on a principal bundle $P\to B SO(n)$ is a lift through the canonical map $B Spin^h(n) \to B SO(n)$.

Thus, in concrete terms, a spin$^h$-structure on $P$ is a principal $SO(3)$-bundle $E$ together with a principal $Spin^h(n)$-bundle $Q$ and a double covering map $Q\to P\times E$ equivariant with respect to the homomorphism $Spin^h(n) \to SO(n)\times SO(3)$.

The canonical inclusions

$$Spin(n)\to Spin^c(n)\to Spin^h(n)$$

allow promotions of spin-structures to spinᶜ structures to spinʰ structures.

Remark

The converse fails, for instance by the following counter-examples:

• The $\mathbb{CP}^2$ is a spin$^c$ manifold with no spin structure

• The Wu manifold $SU(3)/SO(3)$ is a spin$^h$ manifold with no spin$^c$ structure (cf. MO:q/304471).

Obstructions to existence

The homotopy fiber of $B Spin^h(n) \to B SO(n)$ is not an Eilenberg-MacLane space, so we cannot expect a single cohomological class to control the existence of spin$^h$-structures.

The first obstruction is the vanishing of the fifth integral Stiefel-Whitney class.

In physics

Freed-Hopkins use spin$^h$ invertible field theories to model and classify SPT phases in Altland-Zirnbauer class C.

Wang-Wen-Witten study an anomaly in 4d $SU(2)$ gauge theory that can appear when the theory is placed in spin$^h$ manifolds.

References

The original definition is due to

• Christian Bär, Elliptic symbols. Mathematische Nachrichten, 201(1), 7–35.

A survey is given in

• Michael Albanese, Aleksandar Milivojevic, Spin^h and further generalisations of spin. arXiv:2008.04934

Applications in physics:

• Dan Freed and Mike Hopkins, Reflection positivity and invertible topological phases (arXiv:1604.06527).

• Juven Wang, Xiao-Gang Wen, Edward Witten, A New SU(2) Anomaly, Journal of Mathematical Physics 60 (2019) 052301 [arXiv:1810.00844, doi:10.1063/1.5082852]