Complex spin structure

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In mathematics a complex spin group SpinC(n) is a generalized form of a spin group. Although not all manifolds admit a spin group, all 4-manifolds admit a complex spin group.[1]

The complex spin group can be defined by the exact sequence

1 \to \mathbb{Z}_2 \to \operatorname{Spin}^{C}(n) \to \operatorname{SO}(n)\times\operatorname{U}(1) \to 1.

On a 4-manifold M with a complete set of open neighborhoods {Ua}, the 2nd Stiefel-Whitney class w_2 (T_M)\in H^2 (M; \mathbb{Z}_2) is the obstruction to finding a global spin structure. In other words, if w2=0 then one can find a global spin structure Spin(4) by lifting a cocycle \{g_{ab}:U_a \cup U_b \to \operatorname{SO}(4)\} to the simply-connected group Spin(4). These lifted cocycles (as well as the original cocycles) habsatisfy the cocycle condition,

h_{ab}\circ h_{bc} \circ h_{ca}= 1.

However, if w_2\neq 0, the cocycle condition must be expanded to include the opposite 'orientation',

h_{ab}\circ h_{bc} \circ h_{ca}= \pm 1.

In this case the concept of a spin structure must be generalized to a complex spin structure, and the original cocycles gab must be lifted to this new structure. In four dimensions, this complex spin group can be formally defined as

\operatorname{Spin}^{C}(4)= \operatorname{U}(1)\times\operatorname{Spin}(4) / \pm 1.

In the same manner that Spin(4) is a double cover of SO(4), SpinC(4) admits the double-cover projection

\operatorname{Spin}^{C}(4)\to\operatorname{U}(1)\times\operatorname{SO}(4).

[edit] Notes

  1. ^ Scorpan, A., 2005 The Wild World of 4 Manifolds

[edit] References

  • Scorpan, Alexandru (2005), The Wild World of 4-Manifolds, Providence, Rhode Island: American Mathematical Society 
  • Asselmeyer-Maluga, Torsten & Brans, Carl H (2007), Exotic Smoothness and Physics: Differential Topology and Spacetime Models, New Jersey, London: World Scientific