Spinor bundle

A spinor bundle - also called a spin bundle - is a mathematical object from differential geometry or global analysis . It is a special kind of vector bundle over a manifold . Spinor bundles can only be defined for spin manifolds . These are special Riemannian manifolds with a spin structure on the tangential bundle . Whether a tangential bundle can be equipped with a spin structure can be measured by the second Stiefel-Whitney class .

The space of the smooth cuts of a spinor bundle is also known as the space of the spinors or spinor fields and serves as a natural definition set for the Dirac operator .

Spin structure

Let be a Riemannian manifold and an oriented Hermitian vector bundle of dimension . With denotes the spin group of . It can be understood as a two-leaf superposition of the orthogonal group . A spin structure is a - principal bundle along with a two-bladed overlay ${\displaystyle (M,g)}$ ${\displaystyle \pi _{E}\colon E\to M}$ ${\displaystyle n}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle C\ell _{n}}$ ${\displaystyle \tau _{0}\colon \operatorname {Spin} (n)\to \operatorname {SO} (n)}$ ${\displaystyle \operatorname {SO} (n)}$ ${\displaystyle E}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle P_{\operatorname {Spin} }(E)\to E}$

${\displaystyle \tau \colon P_{\operatorname {Spin} }(E)\to P_{\operatorname {SO} }(E)}$

of the main fiber bundle , so that applies to everyone and everyone . ${\displaystyle \operatorname {SO} (n)}$ ${\displaystyle P_{\operatorname {SO} }(E)}$ ${\displaystyle \tau (pg)=\tau (p)\tau _{0}(g)}$ ${\displaystyle p\in P_{\operatorname {Spin} }(E)}$ ${\displaystyle g\in \operatorname {Spin} (n)}$

Spin manifold

A spin manifold is an orientable Riemannian manifold that allows a spin structure on its tangential bundle .

Since the Stiefel-Whitney class of a manifold is defined as the Stiefel-Whitney class of its tangent bundle, this means that an orientable Riemannian manifold allows a spin structure if and only if holds. Then the different spin structures are determined by the elements of . ${\displaystyle \omega _{\text{2}}(X)=0}$ ${\displaystyle H^{1}(X;\mathbb {Z} _{\text{2}})}$

Definition of the spinor bundle

Let be a Riemannian manifold with even dimension and a spin structure on the tangential bundle , i.e. in short a spin manifold with even dimension. Let be the representation of the complex Clifford algebra (also called the spinor module). The group has a subset of also a representation . ${\displaystyle (M,g)}$ ${\displaystyle \pi \colon P_{\operatorname {Spin} }(M)\to TM}$ ${\displaystyle TM}$ ${\displaystyle S}$ ${\displaystyle \mathbb {C} l(n)}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle \mathbb {C} l(n)}$ ${\displaystyle \rho \colon \operatorname {Spin} (n)\to \operatorname {End} (S)}$

The spinor bundle over the manifold is defined as the associated complex vector bundle ${\displaystyle {\mathcal {S}}}$ ${\displaystyle M}$

${\displaystyle {\mathcal {S}}:=P_{\operatorname {Spin} }(M)\times _{\operatorname {Spin} (n)}S\,.}$

Here the fiber product denotes by with over . In this specific case, this means ${\displaystyle \times _{\operatorname {Spin} (n)}}$ ${\displaystyle P_{\operatorname {Spin} }(M)}$ ${\displaystyle S}$ ${\displaystyle \operatorname {Spin} (n)}$

${\displaystyle P_{\operatorname {Spin} }(M)\times _{\operatorname {Spin} (n)}S=P_{\operatorname {Spin} }(M)\times S/\{(p\cdot g,f)\sim (p,\rho (g)f\}}$

for , and . ${\displaystyle p\in P_{\operatorname {Spin} }(M)}$ ${\displaystyle g\in \operatorname {Spin} (n)}$ ${\displaystyle f\in S}$

literature

• Thomas Friedrich : Dirac operators in Riemannian geometry. With an outlook on the Seiberg-Witten theory. Friedr. Vieweg & Sohn, Braunschweig, 1997. ISBN 3-528-06926-0 .

Individual evidence

1. ^ Thomas Friedrich : Dirac operators in Riemannian geometry. With an outlook on the Seiberg-Witten theory. Friedr. Vieweg & Sohn, Braunschweig, 1997. ISBN 3-528-06926-0 , pp. 467-468.
2. ^ HB Lawson, M. Michelsohn: Spin Geometry . Princeton University Press, 1989, ISBN 978-0-691-08542-5 , p. 80.
3. ^ HB Lawson, M. Michelsohn: Spin Geometry . Princeton University Press, 1989, ISBN 978-0-691-08542-5 , p. 96.
4. ^ HB Lawson, M. Michelsohn: Spin Geometry . Princeton University Press, 1989, ISBN 978-0-691-08542-5 , pp. 96-97.
5. ^ Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin u. a. Springer 1992, ISBN 0-387-53340-0 , p. 111.