nLab spinʰ structure



Higher spin geometry



Special and general types

Special notions


Extra structure





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

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

and the Spin group

Spin(n)(Spin(n)×O(1))/(Z/2). 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)SO(n)×SO(3). Spin^h(n) \longrightarrow SO(n)\times SO(3) \,.

It induces canonical homomorphisms of Lie groups

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


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

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

Thus, in concrete terms, a spinh^h-structure on PP is a principal SO(3)SO(3)-bundle EE together with a principal Spin h(n)Spin^h(n)-bundle QQ and a double covering map QP×EQ\to P\times E equivariant with respect to the homomorphism Spin h(n)SO(n)×SO(3)Spin^h(n) \to SO(n)\times SO(3).

The canonical inclusions

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

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


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

Obstructions to existence

The homotopy fiber of BSpin h(n)BSO(n)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 spinh^h-structures.

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

In physics

Freed-Hopkins use spinh^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)SU(2) gauge theory that can appear when the theory is placed in spinh^h manifolds.


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:

Last revised on July 5, 2024 at 14:15:46. See the history of this page for a list of all contributions to it.