Spinc group

Short description: Twisted spin 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 $\mathbb{ℂ}$. 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 ${\mathbb{\mathbb{Z}}}_2$ acts on it with $\mathrm{Spin}(n)/{\mathbb{\mathbb{Z}}}_2\cong \mathrm{SO}(n)$. Furthermore, ${\mathbb{\mathbb{Z}}}_2$ also acts on the first unitary group $\mathrm{U}(1)$ through the antipodal identification $y\sim -y$. The spin^c group is then:

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

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

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

Low-dimensional examples

• ${\mathrm{Spin}}^{\mathrm{c}}(1)\cong \mathrm{U}(1)\cong \mathrm{SO}(2)$, induced by the isomorphism $\mathrm{Spin}(1)\cong \mathrm{O}(1)\cong {\mathbb{\mathbb{Z}}}_2$
• ${\mathrm{Spin}}^{\mathrm{c}}(3)\cong \mathrm{U}(2)$, 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

• Lawson, Herbert Blaine Jr.; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton Mathematical Series. 38. Princeton: Princeton University Press. doi:10.1515/9781400883912. ISBN 978-1-4008-8391-2. 
• Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten 201 (1). https://www.researchgate.net/publication/280877898. 
• "Stable complex and Spin^c-structures". https://ncatlab.org/nlab/files/StableComplexSpinC.pdf. 
• Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory. https://www3.nd.edu/~lnicolae/new1.pdf.