nLab Spinᶜ(4)

Idea

$Spin^\mathrm{c}(4)$ is the fourth spinᶜ group. It is used to describe spinᶜ structures on orientable 4-manifolds, which are studied in Seiberg-Witten theory.

Properties

Proposition

One has an exceptional isomorphism:

$$Spin^\mathrm{c}(4) \cong U(2)\times_{U(1)}U(2) =\{(U_-,U_+)\in U(2)\times U(2)|det(U_-)=det(U_+)\}.$$

Proof

Using the exceptional isomorphism $Spin(4)\cong SU(2)\times SU(2)$ yields:

$$Spin^\mathrm{c}(4) \cong(SU(2)\times SU(2)\times U(1))/\mathbb{Z}_2 \twoheadrightarrow U(2)\times_{U(1)}U(2), (U_-,U_+,z)\mapsto(U_-z,U_+z)$$

(In general, one has $(SU(n)\times U(1))/\mathbb{Z}_n\cong U(n)$ using the homomorphism theorem? on the group homomorphism $SU(n)\times U(1)\rightarrow U(n),(U,z)\mapsto Uz$, which is surjective and has $\{(\zeta_n^k\mathbf{1}_n,\zeta_n^{-k})|[k]\in\mathbb{Z}_n \}\cong\mathbb{Z}_n$ as kernel.)

Related concepts

• U(2) (Spinᶜ(3), Spinh(2))
• Spinᶜ(6)

References

• Liviu Nicolaescu, Notes on Seiberg-Witten theory, American Mathematical Society (2000) [ISBN:978-0-8218-2145-9, pdf]