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 Spin$^h$ group
in complete analogy to the Spin$^c$ 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 spin^c-structures to spin^h-structures. The converse is not true: just as $\mathbb{CP}^2$ 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 (MathOverflow discussion).
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 November 1, 2023 at 08:00:35. See the history of this page for a list of all contributions to it.