Spinc 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 $\mathrm{Spin}(n)$ is a double cover of the special orthogonal group $\mathrm{SO}(n)$, hence ${\mathbb{Z}}_2$ acts on it with $\mathrm{Spin}(n)/{\mathbb{Z}}_2\cong \mathrm{SO}(n)$. Furthermore, ${\mathbb{Z}}_2$ also acts on the first unitary group $\mathrm{U}(1)$ through the antipodal identification $y\sim -y$. The spin^c group is then:

${\mathrm{Spin}}^{\mathrm{c}}(n):=(\mathrm{Spin}(n)\times \mathrm{U}(1))/{\mathbb{Z}}_2$

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

${\mathrm{Spin}}^k(n):=(\mathrm{Spin}(n)\times \mathrm{Spin}(k))/{\mathbb{Z}}_2.$

Low-dimensional examples

• ${\mathrm{Spin}}^{\mathrm{c}}(1)\cong \mathrm{U}(1)\cong \mathrm{SO}(2)$, induced by the isomorphism $\mathrm{Spin}(1)\cong \mathrm{O}(1)\cong {\mathbb{Z}}_2$
• ${\mathrm{Spin}}^{\mathrm{c}}(3)\cong \mathrm{U}(2)$, induced by the exceptional isomorphism $\mathrm{Spin}(3)\cong \mathrm{Sp}(1)\cong \mathrm{SU}(2)$. Since furthermore $\mathrm{Spin}(2)\cong \mathrm{U}(1)\cong \mathrm{SO}(2)$, one also has ${\mathrm{Spin}}^{\mathrm{c}}(3)\cong {\mathrm{Spin}}^{\mathrm{h}}(2)$.
• ${\mathrm{Spin}}^{\mathrm{c}}(4)\cong \mathrm{U}(2){\times }_{\mathrm{U}(1)}\mathrm{U}(2)$, induced by the exceptional isomorphism $\mathrm{Spin}(4)\cong \mathrm{SU}(2)\times \mathrm{SU}(2)$
• ${\mathrm{Spin}}^{\mathrm{c}}(6)\to \mathrm{U}(4)$ is a double cover, induced by the exceptional isomorphism $\mathrm{Spin}(6)\cong \mathrm{SU}(4)$

Properties

For all higher abelian homotopy groups, one has:

${\pi }_k{\mathrm{Spin}}^{\mathrm{c}}(n)\cong {\pi }_k\mathrm{Spin}(n)\times {\pi }_k\mathrm{U}(1)\cong {\pi }_k\mathrm{SO}(n)$

for $k\ge 2$.

See also

• Spin^h group

Literature

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