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{C}. An important application of spinc groups is for spinc structures, which are central for Seiberg–Witten theory.
Definition
The spin group \operatorname{Spin}(n) is a double cover of the special orthogonal group \operatorname{SO}(n), hence \mathbb{Z}_2 acts on it with \operatorname{Spin}(n)/\Z_2\cong\operatorname{SO}(n). Furthermore, \mathbb{Z}_2 also acts on the first unitary group \operatorname{U}(1) through the antipodal identification y\sim -y. The spinc group is then:[1][2][3][4]
\operatorname{Spin}^\mathrm{c}(n) :=\left( \operatorname{Spin}(n)\times\operatorname{U}(1) \right)/\mathbb{Z}_2
with (x,y)\sim(-x,-y). It is also denoted \operatorname{Spin}^\mathbb{C}(n). Using the exceptional isomorphism \operatorname{Spin}(2)
\cong\operatorname{U}(1), one also has \operatorname{Spin}^\mathrm{c}(n)
=\operatorname{Spin}^2(n) with:
\operatorname{Spin}^k(n) :=\left( \operatorname{Spin}(n)\times\operatorname{Spin}(k) \right)/\mathbb{Z}_2.
Low-dimensional examples
\operatorname{Spin}^\mathrm{c}(1) \cong\operatorname{U}(1) \cong\operatorname{SO}(2), induced by the isomorphism\operatorname{Spin}(1) \cong\operatorname{O}(1) \cong\mathbb{Z}_2\operatorname{Spin}^\mathrm{c}(3) \cong\operatorname{U}(2),[5] induced by the exceptional isomorphism\operatorname{Spin}(3) \cong\operatorname{Sp}(1) \cong\operatorname{SU}(2). Since furthermore\operatorname{Spin}(2) \cong\operatorname{U}(1) \cong\operatorname{SO}(2), one also has\operatorname{Spin}^\mathrm{c}(3) \cong\operatorname{Spin}^\mathrm{h}(2).\operatorname{Spin}^\mathrm{c}(4) \cong\operatorname{U}(2)\times_{\operatorname{U}(1)}\operatorname{U}(2), induced by the exceptional isomorphism\operatorname{Spin}(4) \cong\operatorname{SU}(2)\times\operatorname{SU}(2)\operatorname{Spin}^\mathrm{c}(6) \rightarrow\operatorname{U}(4)is a double cover, induced by the exceptional isomorphism\operatorname{Spin}(6) \cong\operatorname{SU}(4)
Properties
For all higher abelian homotopy groups, one has:
\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 k\geq 2.
See also
Literature
- Lawson, Herbert Blaine Jr. & Michelsohn, Marie-Louise (1989). Spin Geometry. Vol. 38. Princeton Mathematical Series. 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"
- Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory