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spin^h structure

Contents

Context

Higher spin geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

With Sp(1) denoting the quaternion unitary group, we define the Spin h ^h group

Spin h(n)=(Spin(n)×Sp(1))/(Z/2),Spin^h(n) = (Spin(n) \times Sp(1))/(\mathbf{Z}/2),

in complete analogy to the Spin c ^c group

Spin c(n)=(Spin(n)×U(1))/(Z/2),Spin^c(n) = (Spin(n) \times U(1))/(\mathbf{Z}/2),

and the Spin group

Spin(n)=(Spin(n)×O(1))/(Z/2).Spin(n) = (Spin(n) \times O(1))/(\mathbf{Z}/2).

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

Spin h(n)SO(n)×SO(3).Spin^h(n) \to SO(n)\times SO(3).

It induces canonical homomorphisms of Lie groups

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

and

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 spin^c-structures to spin^h-structures. The converse is not true: just as ℂℙ 2\mathbb{CP}^2 is a spinc^c manifold with no spin structure, the Wu manifold SU(3)/SO(3)SU(3)/SO(3) is a spinh^h manifold with no spinc^c structure (MathOverflow discussion).

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.

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:

Last revised on February 19, 2021 at 09:49:00. See the history of this page for a list of all contributions to it.