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

Definition

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

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

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

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

Low-dimensional examples

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

Properties

For all higher abelian homotopy groups, one has:

${\displaystyle \pi _{k}\operatorname {Spin} ^{\mathrm {c} }(n)\cong \pi _{k}\operatorname {Spin} (n)\times \pi _{k}\operatorname {U} (1)\cong \pi _{k}\operatorname {SO} (n)}$

for ${\displaystyle k\geq 2}$.

See also

• Spinʰ group

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).