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 group
in complete analogy to the Spin 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-structure on a principal bundle is a lift through the canonical map .
Thus, in concrete terms, a spin-structure on is a principal -bundle together with a principal -bundle and a double covering map equivariant with respect to the homomorphism .
The canonical inclusions
allow promotions of spin-structures to spin^c-structures to spin^h-structures. The converse is not true: just as is a spin manifold with no spin structure, the Wu manifold is a spin manifold with no spin structure (MathOverflow discussion).
The homotopy fiber of is not an Eilenberg-MacLane space, so we cannot expect a single cohomological class to control the existence of spin-structures.
The first obstruction is the vanishing of the fifth integral Stiefel-Whitney class.
Freed-Hopkins use spin invertible field theories to model and classify SPT phases in Altland-Zirnbauer class C.
Wang-Wen-Witten study an anomaly in 4d gauge theory that can appear when the theory is placed in spin 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, and Edward Witten, A New SU(2) Anomaly, Journal of Mathematical Physics 60, 052301 (2019) (arXiv:1810.00844).
Last revised on February 19, 2021 at 14:49:00. See the history of this page for a list of all contributions to it.