spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
With Sp(1) denoting the quaternion unitary group, we define the spinʰ group
in complete analogy to the spinᶜ group
and the Spin group
We have a canonical double covering, which is a homomorphism of Lie groups:
It induces canonical homomorphisms of Lie groups
and
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
allow promotions of spin-structures to spinᶜ structures to spinʰ structures.
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).
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.
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.
The original definition is due to
A survey is given in
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]
Last revised on July 5, 2024 at 14:15:46. See the history of this page for a list of all contributions to it.