Spin^c group

In spin geometry, a spin^c group (or complex spin group) is a Lie group obtained by the spin group through twisting with the first unitary group. C stands for the complex numbers, which are denoted $ℂ$ . An important application of spin^c groups is for spin^c structures, which are central for Seiberg–Witten theory.

Definition

The spin group $Spin (n)$ is a double cover of the special orthogonal group $SO (n)$ , hence $ℤ_{2}$ acts on it with $Spin (n) / ℤ_{2} ≅ SO (n)$ . Furthermore, $ℤ_{2}$ also acts on the first unitary group $U (1)$ through the antipodal identification $y \sim - y$ . The spin^c group is then:^[1]^[2]^[3]^[4]

{Spin}^{c} (n) := (Spin (n) \times U (1)) / ℤ_{2}

with $(x, y) \sim (- x, - y)$ . It is also denoted ${Spin}^{ℂ} (n)$ . Using the exceptional isomorphism $Spin (2) ≅ U (1)$ , one also has ${Spin}^{c} (n) = {Spin}^{2} (n)$ with:

{Spin}^{k} (n) := (Spin (n) \times Spin (k)) / ℤ_{2} .

Low-dimensional examples

${Spin}^{c} (1) ≅ U (1) ≅ SO (2)$ , induced by the isomorphism $Spin (1) ≅ O (1) ≅ ℤ_{2}$
${Spin}^{c} (3) ≅ U (2)$ ,^[5] induced by the exceptional isomorphism $Spin (3) ≅ Sp (1) ≅ SU (2)$ . Since furthermore $Spin (2) ≅ U (1) ≅ SO (2)$ , one also has ${Spin}^{c} (3) ≅ {Spin}^{h} (2)$ .
${Spin}^{c} (4) ≅ U (2) \times_{U (1)} U (2)$ , induced by the exceptional isomorphism $Spin (4) ≅ SU (2) \times SU (2)$
${Spin}^{c} (6) \to U (4)$ is a double cover, induced by the exceptional isomorphism $Spin (6) ≅ SU (4)$

Properties

For all higher abelian homotopy groups, one has:

π_{k} {Spin}^{c} (n) ≅ π_{k} Spin (n) \times π_{k} U (1) ≅ π_{k} SO (n)

for $k \geq 2$ .

Literature

Lawson, Herbert Blaine Jr.; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton Mathematical Series. Vol. 38. Princeton: Princeton University Press. doi:10.1515/9781400883912. ISBN 978-1-4008-8391-2.
Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten. 201 (1).
"Stable complex and Spin^c-structures" (PDF).
Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory (PDF).

References

↑ Lawson & Michelson 1989, Appendix D, Equation (D.1)
↑ Bär 1999, page 14
↑ Stable complex and Spin^c-structures, section 2.1
↑ Nicolaescu, page 30
↑ Nicolaescu, Exercise 1.3.9

[1] Lawson & Michelson 1989, Appendix D, Equation (D.1)

[2] Bär 1999, page 14

[3] Stable complex and Spin^c-structures, section 2.1

[4] Nicolaescu, page 30

[5] Nicolaescu, Exercise 1.3.9

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[2]

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Definition

Low-dimensional examples

Properties

See also

Literature

References