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twisted spin structure

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cohomology

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Idea

An ordinary spin structure on a special orthogonal group-principal bundle is a lift of the corresponding cocycle g:XBSOg : X \to \mathbf{B} SO through the spin group fibration BSpinBSO\mathbf{B} Spin \to \mathbf{B} SO. The obstruction for this to exist is a cohomology class w 2H 2(X, 2)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[w_2(g)] = 0.

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

Definition

Let

2SpinSOB 2 \mathbb{Z}_2 \to Spin \to SO \to \mathbf{B} \mathbb{Z}_2

be the fiber sequence in H=\mathbf{H} = ETop∞Grpd or H=\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:BSOB 2 2 w_2 : \mathbf{B }SO \to \mathbf{B}^2 \mathbb{Z}_2

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

w 2:H(X,BSO)H(X,B 2 2) w_2 : \mathbf{H}(X,\mathbf{B}SO) \to \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)
[w 2]:SOBund(X)H 2(X, 2). [w_2] : SO Bund(X) \to H^2(X,\mathbb{Z}_2) \,.

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

SpinStruc tw(X) H 2(X, 2) H(X,BSO) w 2 H(X,B 2 2), \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)SpinStruc_{tw}(X) are twisted Spin-bundles.

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

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

Revised on May 28, 2012 09:18:17 by Urs Schreiber (82.113.121.232)