Spin^c group

In spin geometry, a spinᶜ 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 $\mathbb {C}$ . An important application of spinᶜ groups is for spinᶜ structures, which are central for Seiberg–Witten theory.

Definition

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

\operatorname {Spin} ^{\mathrm {c} }(n):=\left(\operatorname {Spin} (n)\times \operatorname {U} (1)\right)/\mathbb {Z} _{2}

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

\operatorname {Spin} ^{k}(n):=\left(\operatorname {Spin} (n)\times \operatorname {Spin} (k)\right)/\mathbb {Z} _{2}.

Low-dimensional examples

$\operatorname {Spin} ^{\mathrm {c} }(1)\cong \operatorname {U} (1)\cong \operatorname {SO} (2)$ , induced by the isomorphism $\operatorname {Spin} (1)\cong \operatorname {O} (1)\cong \mathbb {Z} _{2}$
$\operatorname {Spin} ^{\mathrm {c} }(3)\cong \operatorname {U} (2)$ ,^[5] induced by the exceptional isomorphism $\operatorname {Spin} (3)\cong \operatorname {Sp} (1)\cong \operatorname {SU} (2)$ . Since furthermore $\operatorname {Spin} (2)\cong \operatorname {U} (1)\cong \operatorname {SO} (2)$ , one also has $\operatorname {Spin} ^{\mathrm {c} }(3)\cong \operatorname {Spin} ^{\mathrm {h} }(2)$ .
$\operatorname {Spin} ^{\mathrm {c} }(4)\cong \operatorname {U} (2)\times _{\operatorname {U} (1)}\operatorname {U} (2)$ , induced by the exceptional isomorphism $\operatorname {Spin} (4)\cong \operatorname {SU} (2)\times \operatorname {SU} (2)$
$\operatorname {Spin} ^{\mathrm {c} }(6)\rightarrow \operatorname {U} (4)$ is a double cover, induced by the exceptional isomorphism $\operatorname {Spin} (6)\cong \operatorname {SU} (4)$

Properties

For all higher abelian homotopy groups, one has:

\pi _{k}\operatorname {Spin} ^{\mathrm {c} }(n)\cong \pi _{k}\operatorname {Spin} (n)\times \pi _{k}\operatorname {U} (1)\cong \pi _{k}\operatorname {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 Spinᶜ-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 Spinᶜ-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 Spinᶜ-structures, section 2.1

[4] Nicolaescu, page 30

[5] Nicolaescu, Exercise 1.3.9

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