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

Definition

The spin group $\operatorname(n)$ is a double cover of the special orthogonal group $\operatorname(n)$, hence $\mathbb_2$ acts on it with $\operatorname(n)/\Z_2\cong\operatorname(n)$. Furthermore, $\mathbb_2$ also acts on the first unitary group $\operatorname(1)$ through the antipodal identification $y\sim -y$. The ''spinᶜ group'' is then:

: $\operatorname^\mathrm(n) :=\left( \operatorname(n)\times\operatorname(1) \right)/\mathbb_2$

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

: $\operatorname^k(n) :=\left( \operatorname(n)\times\operatorname(k) \right)/\mathbb_2.$

Low-dimensional examples

* $\operatorname^\mathrm(1) \cong\operatorname(1) \cong\operatorname(2)$, induced by the isomorphism $\operatorname(1) \cong\operatorname(1) \cong\mathbb_2$
* $\operatorname^\mathrm(3) \cong\operatorname(2)$,Nicolaescu, Exercise 1.3.9 induced by the exceptional isomorphism $\operatorname(3) \cong\operatorname(1) \cong\operatorname(2)$. Since furthermore $\operatorname(2) \cong\operatorname(1) \cong\operatorname(2)$, one also has $\operatorname^\mathrm(3) \cong\operatorname^\mathrm(2)$.
* $\operatorname^\mathrm(4) \cong\operatorname(2)\times_\operatorname(2)$, induced by the exceptional isomorphism $\operatorname(4) \cong\operatorname(2)\times\operatorname(2)$
* $\operatorname^\mathrm(6) \rightarrow\operatorname(4)$ is a double cover, induced by the exceptional isomorphism $\operatorname(6) \cong\operatorname(4)$

Properties

For all higher abelian homotopy groups, one has:

: $\pi_k\operatorname^\mathrm(n) \cong\pi_k\operatorname(n)\times\pi_k\operatorname(1) \cong\pi_k\operatorname(n)$

for $k\geq 2$.

See also

* Spinʰ group

Literature

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* {{cite book , author=Liviu I. Nicolaescu , url=https://www3.nd.edu/~lnicolae/new1.pdf , title=Notes on Seiberg-Witten Theory

References

Lie groups
Differential geometry