nLab twisted spin structure

Idea

An ordinary spin structure on a special orthogonal group-principal bundle is a lift of the corresponding cocycle $g : X \to \mathbf{B} SO$ through the spin group fibration $\mathbf{B} Spin \to \mathbf{B} SO$. The obstruction for this to exist is a cohomology class $w_2 \in H^2(X, \mathbb{Z}_2)$ – the second Stiefel-Whitney class: it exists precisely if this class is trivial, $[w_2(g)] = 0$.

Conversely, one can ask for an $SO$-cocycle $g$ with prescribed non-trivial obstruction $[w_2(g)] = \alpha \in H^2(X, \mathbb{Z}_2)$. These may usefully be understood as $\alpha$-twisted $spin$-structures, following the general logic of twisted cohomology.

Definition

Let

$$\mathbb{Z}_2 \to Spin \to SO \to \mathbf{B} \mathbb{Z}_2$$

be the fiber sequence in $\mathbf{H} =$ETop∞Grpd or $\mathbf{H} =$Smooth∞Grpd given by the spin group extension of the special orthogonal group (regarded as a topological group or as a Lie group, respectibely). Its delooping defines the second Stiefel-Whitney class

$$w_2 : \mathbf{B }SO \to \mathbf{B}^2 \mathbb{Z}_2$$

so that for any $X \in \mathbf{X}$ we have a characteristic class

$$w_2 : \mathbf{H}(X,\mathbf{B}SO) \to \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)$$

$$[w_2] : SO Bund(X) \to H^2(X,\mathbb{Z}_2) \,.$$

For $X$ a manifold define the groupoid of twisted spin-structures $SpinStruc_{tw}(X)$ to be the (∞,1)-pullback

$$\array{ SpinStruc_{tw}(X) &\to& H^2(X, \mathbb{Z}_2) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}SO) &\stackrel{w_2}{\to}& \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2) } \,,$$

where the right vertical morphism picks one cocycle representative in each cohomology class.

The cocycles in $SpinStruc_{tw}(X)$ are twisted Spin-bundles.

The obstruction cocycles in $\mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)$ are $\mathbf{B}\mathbb{Z}_2$-principal 2-bundles. These may be modeled by $\mathbb{Z}_2$-bundle gerbes. In this incarnation the obstruction cocycles $w_2(g)$ above have been discussed as spin gerbes in (MurraySinger).

Related concepts

• cohomology, twisted cohomology, Whitehead tower, twisted smooth cohomology in string theory

• orientation

• twisted differential c-structure

  • spin structure, twisted spin structure

  • string structure, differential string structure

  • fivebrane structure, differential fivebrane structure

References

A model of twisted spin structures by bundle gerbes is discussed in

• Michael Murray, Michael Singer, Gerbes, Clifford modules and the index theorem Annals of Global Analysis and Geometry

  Volume 26, Number 4, 355-367, DOI: 10.1023/B:AGAG.0000047514.71785.96 (arXiv:math/0302096)

• Atsushi Tomoda, A relation of spin-bundle gerbes and Mayer’s Dirac operator (pdf)

The general abstract discussion given above appears as an example in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Differential Structures