Spinc structure

In spin geometry, a spinᶜ structure (or complex spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinᶜ manifolds. C stands for the complex numbers, which are denoted $\mathbb{C}$ and appear in the definition of the underlying spinᶜ group. In four dimensions, a spinᶜ structure defines two complex plane bundles, which can be used to describe negative and positive chirality of spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifolds.

Definition

Let $M$ be a $n$ -dimensional orientable manifold. Its tangent bundle $TM$ is described by a classifying map $M\rightarrow\operatorname{BSO}(n)$ into the classifying space $\operatorname{BSO}(n)$ of the special orthogonal group $\operatorname{SO}(n)$ . It can factor over the map $\operatorname{BSpin}^\mathrm{c}(n)\rightarrow\operatorname{BSO}(n)$ induced by the canonical projection $\operatorname{Spin}^\mathrm{c}(n)\twoheadrightarrow\operatorname{SO}(n)$ on classifying spaces. In this case, the classifying map lifts to a continuous map $M\rightarrow\operatorname{BSpin}^\mathrm{c}(n)$ into the classifying space $\operatorname{BSpin}^\mathrm{c}(n)$ of the spinᶜ group $\operatorname{Spin}^\mathrm{c}(n)$ , which is called spinᶜ structure.Stable complex and Spinᶜ-structures, Definition D.28

Let $\operatorname{BSpin}^\mathrm{c}(M)$ denote the set of spinᶜ structures on $M$ up to homotopy. The first unitary group $\operatorname{U}(1)$ is the second factor of the spinᶜ group and using its classifying space [[Principal U(1)-bundle| $\operatorname{BU}(1)
\cong\operatorname{BSO}(2)$ ]], which is the infinite complex projective space $\mathbb{C}P^\infty$ and a model of the Eilenberg–MacLane space $K(\mathbb{Z},2)$ , there is a bijection:Mellor 1995, Theorem 5

: $\operatorname{BSpin}^\mathrm{c}(M)
\cong[M,\operatorname{BU}(1)]
\cong[M,\mathbb{C}P^\infty]
\cong[M,K(\mathbb{Z},2)]
\cong H^2(M,\mathbb{Z}).$

Due to the canonical projection $\operatorname{BSpin}^\mathrm{c}(n)\rightarrow\operatorname{U}(1)/\mathbb{Z}_2
\cong\operatorname{U}(1)$ , every spinᶜ structure induces a principal $\operatorname{U}(1)$ -bundle or equvalently a complex line bundle.

Properties

Every spin structure induces a canonical spinᶜ structure.Mellor 1995, Theorem 2Nicolaescu, Example 1.3.16 The reverse implication doesn't hold as the complex projective plane $\mathbb{C}P^2$ shows.
Every spinᶜ structure induces a canonical spinʰ structure. The reverse implication doesn't hold as the Wu manifold $\operatorname{SU}(3)/\operatorname{SO}(3)$ shows.{{Citation needed|date=March 2025}}
An orientable manifold $M$ has a spinᶜ structure iff its third integral Stiefel–Whitney class $W_3(M)$

\in H^2(M,\mathbb{Z}) vanishes, hence is the image of the second ordinary Stiefel–Whitney class

w_2(M)
\in H^2(M,\mathbb{Z})

under the canonical map

H^2(M,\mathbb{Z}_2)\rightarrow H^2(M,\mathbb{Z})

.Stable complex and Spinᶜ-structures, Proposition D.31

Every orientable smooth manifold with four or less dimensions has a spinᶜ structure.
Every almost complex manifold has a spinᶜ structure.Mellor 1995, Theorem 3

The following properties hold more generally for the lift on the Lie group $\operatorname{Spin}^k(n)
:=\left(
\operatorname{Spin}(n)\times\operatorname{Spin}(k)
\right)/\mathbb{Z}_2$ , with the particular case $k=2$ giving:

If $M\times N$ is a spinᶜ manifold, then $M$ and $N$ are spinᶜ manifolds.Albanese & Milivojević 2021, Proposition 3.6.
If $M$ is a spin manifold, then $M\times N$ is a spinᶜ manifold iff $N$ is a spinᶜ manifold.
If $M$ and $N$ are spinᶜ manifolds of same dimension, then their connected sum $M\# N$ is a spinᶜ manifold.Albanese & Milivojević 2021, Proposition 3.7.
The following conditions are equivalent:Albanese & Milivojević 2021, Proposition 3.2.
$M$ is a spinᶜ manifold.
There is a real plane bundle $E\twoheadrightarrow M$ , so that $TM\oplus E$ has a spin structure or equivalently $w_2(TM\oplus E)$

=0.

$M$ can be immersed in a spin manifold with two dimensions more.
$M$ can be embedded in a spin manifold with two dimensions more.

Literature

{{cite web |author=Blake Mellor |date=1995-09-18 |title=Spinᶜ manifolds |url=https://www.maths.ed.ac.uk/~v1ranick/papers/mellor.pdf |language=en}}
{{cite web |title=Stable complex and Spinᶜ-structures |url=https://ncatlab.org/nlab/files/StableComplexSpinC.pdf |language=en}}
{{cite book |author=Liviu I. Nicolaescu |url=https://www3.nd.edu/~lnicolae/new1.pdf |title=Notes on Seiberg-Witten Theory |language=en}}
{{cite journal |author=Michael Albanese und Aleksandar Milivojević |date=2021 |title=Spinʰ and further generalisations of spin |language=en |volume=164 |pages=104–174 |arxiv=2008.04934 |doi=10.1016/j.geomphys.2022.104709 |periodical=Journal of Geometry and Physics}}

References

External links

spinᶜ structure on nLab

Category: Differential geometry