Spinor bundle

 ( Algebraic Topology ) 

In differential geometry, given a spin structure on an $n$-dimensional orientable Riemannian manifold $(M,\; g),\backslash ,$ one defines the spinor bundle to be the complex vector bundle $\backslash pi\_\{\backslash mathbf\; S\}\backslash colon\{\backslash mathbf\; S\}\backslash to\; M\backslash ,$ associated to the corresponding principal bundle $\backslash pi\_\{\backslash mathbf\; P\}\backslash colon\{\backslash mathbf\; P\}\backslash to\; M\backslash ,$ of spin frames over $M$ and the spin representation of its spin group $\{\backslash mathrm\; \{Spin\}\}(n)\backslash ,$ on the space of $\backslash Delta\_n$.

A section of the spinor bundle $\{\backslash mathbf\; S\}\backslash ,$ is called a spinor field.

---

Formal definition Let $(\{\backslash mathbf\; P\},F\_\{\backslash mathbf\; P\})$ be a spin structure on a Riemannian manifold $(M,\; g),\backslash ,$that is, an equivariant lift of the oriented orthonormal frame bundle $\backslash mathrm\; F\_\{SO\}(M)\backslash to\; M$ with respect to the double covering $\backslash rho\backslash colon\; \{\backslash mathrm\; \{Spin\}\}(n)\backslash to\; \{\backslash mathrm\; \{SO\}\}(n)$ of the special orthogonal group by the spin group.

The spinor bundle $\{\backslash mathbf\; S\}\backslash ,$ is defined page 53 to be the complex vector bundle $$\{\backslash mathbf\; S\}=\{\backslash mathbf\; P\}\backslash times\_\{\backslash kappa\}\backslash Delta\_n\backslash ,$$ associated to the spin structure $\{\backslash mathbf\; P\}$ via the spin representation $\backslash kappa\backslash colon\; \{\backslash mathrm\; \{Spin\}\}(n)\backslash to\; \{\backslash mathrm\; U\}(\backslash Delta\_n),\backslash ,$ where $\{\backslash mathrm\; U\}(\{\backslash mathbf\; W\})\backslash ,$ denotes the group of acting on a Hilbert space $\{\backslash mathbf\; W\}.\backslash ,$ The spin representation $\backslash kappa$ is a faithful and unitary representation of the group $\{\backslash mathrm\; \{Spin\}\}(n).$ pages 20 and 24

---

See also

• Clifford bundle
• Clifford module bundle
• Orthonormal frame bundle
• Spin geometry
• Spinor
• Spinor representation

---

Notes

---

Further reading

• (1989). 9780691085425, Princeton University Press. ISBN 9780691085425

|

Post Comment