In spin geometry, a spinc structure (or complex spin structure) is a generalization of a spin structure. In mathematics, these are used to describe spinor bundles and spinors, which in physics are used to describe spin, an intrinsic angular momentum of particles after which they have been named. Since spinc structures also exist under weakened conditions, which might not allow spin structures, they provide a suitable alternative for such situations. Orientable manifolds with a spinc structure are called spinc manifolds.[1] 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). Its homotopy class is called spinc structure.[2][3]

Assume M has a spinc structure. Let then \operatorname{Spin}^\mathrm{c}(M) denote the set of spinc structures on M. The first unitary group \operatorname{U}(1) is the second factor of the spinc group and using its classifying space \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:[4]

\operatorname{Spin}^\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}).

The former isomorphism follows from the Puppe sequence for the fibration \mathbb{C}P^\infty\hookrightarrow\operatorname{BSpin}^\mathrm{c}(n)\twoheadrightarrow\operatorname{BSO}(n) (when applying [M,-]).[5]

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 equivalently a complex line bundle.

Properties

  • Every spin structure induces a canonical spinc structure.[6][7] 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]
  • 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}).[8][9]
  • Every orientable smooth manifold with four or less dimensions has a spinc structure.[7]
  • Every almost complex manifold has a spinc structure.[10][7]
  • For a compact spinc manifold M, for which a torsion class c\in H^2(M,\mathbb{Z}) with w_2(M)=c\operatorname{mod}2 exists and which has a Riemannian metric of overall positive scalar curvature, its  genus vanishes, hence \widehat{A}(M)=0.[11]

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.[12]
  • If M is a spin manifold, then M\times N is a spinc manifold iff N is a spinc manifold.[12]
  • If M and N are spinc manifolds of same dimension, then their connected sum M\# N is a spinc manifold.[13]
  • The following conditions are equivalent:[14]
    • 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.

Cohomology of infinite classifying space

The cohomology ring of the infinite classifying space \operatorname{BSpin}^\mathrm{c} :=\lim_{n\rightarrow\infty}\operatorname{BSpin}^\mathrm{c}(n) with coefficients in \mathbb{Z}_2 can be expressed using Steenrod squares and Wu classes:[15][16]

H^*(\operatorname{BSpin}^\mathrm{c},\mathbb{Z}_2) \cong H^*(\operatorname{BSO},\mathbb{Z}_2)/(\operatorname{Sq}^1\nu_{2^r},r\geq 1).

See also

Literature

References

  1. ^ Lawson & Michelson 90, Definition D.3
  2. ^ Albanese & Milivojević 2021, Definition 3.1
  3. ^ Stable complex and Spinc-structures, Definition D.28
  4. ^ Mellor 1995, Theorem 5
  5. ^ Albanese & Milivojević 2021, p. 6
  6. ^ Mellor 1995, Theorem 2
  7. ^ Nicolaescu, Example 1.3.16
  8. ^ Lawson & Michelson 90, Theorem D.2 und Corollary D.4
  9. ^ Stable complex and Spinc-structures, Proposition D.31
  10. ^ Mellor 1995, Theorem 3
  11. ^ Lawson & Michelson 90, Corollary D.16
  12. ^ Albanese & Milivojević 2021, Proposition 3.6.
  13. ^ Albanese & Milivojević 2021, Proposition 3.7.
  14. ^ Albanese & Milivojević 2021, Proposition 3.2.
  15. ^ Lawson 2023, p. 8
  16. ^ Hu 2023, Rem. 4.30