Spinc 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:

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

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 $\operatorname {Spin} (1)\cong \operatorname {O} (1)\cong \mathbb {Z} _{2}$
• $\operatorname {Spin} ^{\mathrm {c} }(3)\cong \operatorname {U} (2)$, 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 $k\geq 2$.

See also

• Spinʰ group

Literature

• Herbert Blaine Lawson, Jr. und Marie-Louise Michelsohn (1989). "Spin geometry". Princeton Mathematical Series. 38. Princeton: Princeton University Press. doi:10.1515/9781400883912.
• Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten. 201 (1).
• "Stable complex and Spinᶜ-structures" (PDF).
• Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory (PDF).