# Spinor bundles

A spin structure on an orientable Riemannian manifold $${M}$$ is a principal bundle map

$$\displaystyle \Phi_{P}\colon(P,M^{n},\mathrm{Spin}(n))\rightarrow(F{}_{SO},M^{n},SO(n))$$

from the spin frame bundle (AKA bundle of spin frames) $${P}$$ to the orthonormal frame bundle $${F_{SO}}$$ with respect to the double covering map $${\Phi_{G}\colon\mathrm{Spin}(n)\rightarrow SO(n)}$$. The equivariance condition on the bundle map is then $${\Phi_{P}(U(p))=\Phi_{G}(U)(\Phi_{P}(p))}$$, so that the right action of a spinor transformation $${U\in\mathrm{Spin}(n)}$$ on a spin basis corresponds to the right action of a rotation $${\Phi_{G}(U)}$$ on the corresponding orthonormal basis $${\Phi_{P}(p)}$$. On a time and space orientable pseudo-Riemannian manifold, a spin structure is a principal bundle map with respect to the double covering map $${\Phi_{G}\colon\mathrm{Spin}(r,s)^{e}\rightarrow SO(r,s)^{e}}$$ (except in the case $${r=s=1}$$, which is not a double cover).

The above depicts the spin structure, a principal bundle map that gives a global 2-1 mapping from the fibers of the spin frame bundle to the fibers of the orthonormal frame bundle. The existence of a spin structure means that a change of frame can be smoothly and consistently mapped to changes of spin frame, permitting the existence of spinor fields.

If a spin structure exists for $${M}$$, then $${M}$$ is called a spin manifold (one also says $${M}$$ is spin; sometimes a spin manifold is defined to include a specific spin structure). Any manifold that can be defined with no more than two coordinate charts is then spin, and therefore any parallelizable manifold and any $${n}$$-sphere is spin. As we will see, the existence of spin structures can be related to characteristic classes. It also can be shown that any non-compact spacetime manifold with signature $${(3,1)}$$ is spin iff it is parallelizable. Finally, a vector bundle $${(E,M^{n},\mathbb{C}^{m})}$$ associated to the spin frame bundle $${(P,M,\mathrm{Spin}(r,s)^{e})}$$ under a rep of $${\mathrm{Spin}(r,s)^{e}}$$ on $${\mathbb{C}^{m}}$$ is called a spinor bundle, and a section of this bundle is a spinor field on $${M}$$.

For a charged spinor field taking values in $${U(1)\otimes\mathbb{C}^{m}}$$, where $${\mathbb{C}^{m}}$$ is acted on by a rep of $${\mathrm{Spin}(r,s)^{e}}$$, the action of $${(e^{i\theta},U)\in U(1)\times\mathrm{Spin}(r,s)^{e}}$$ and $${(-e^{i\theta},-U)}$$ are identical, so that the structure group is reducible to

\displaystyle \begin{aligned}\mathrm{Spin^{\mathbb{\mathit{c}}}}(r,s)^{e} & \equiv U(1)\times_{\mathbb{Z}_{2}}\mathrm{Spin}(r,s)^{e}\\ & \equiv\left(U(1)\times\mathrm{Spin}(r,s)^{e}\right)/\mathbb{Z}_{2}, \end{aligned}

where the quotient space collapses all points in the product space which are related by changing the sign of both components. The superscript refers to the circle $${U(1)}$$. A spinc structure on an orientable pseudo-Riemannian manifold $${M}$$ is then a principal bundle map

$$\displaystyle \Phi_{P}\colon(P,M^{n},\mathrm{Spin^{\mathbb{\mathit{c}}}}(r,s)^{e})\rightarrow(F{}_{SO},M^{n},SO(r,s)^{e})$$

with respect to the double covering map $${\Phi_{G}\colon\mathrm{Spin^{\mathbb{\mathit{c}}}}(r,s)^{e}\rightarrow SO(r,s)^{e}}$$ where the $${U(1)}$$ factor is ignored. For spinor matter fields that take values in $${V\otimes\mathbb{C}^{m}}$$ for some internal space $${V}$$ with structure (gauge) group $${G}$$ with $${\mathbb{Z}_{2}}$$ in its center (e.g. a matrix group where the negative of every element remains in the group), we can analogously define a spinG structure. It can be shown (see [1]) that spinG structures exist on any four dimensional $${M}$$ if such a $${G}$$ is a compact simple simply connected Lie group, e.g. $${SU(2i)}$$; therefore the spacetime manifold has no constraints due to spin structure in the standard model, or in any extension that includes $${SU(2)}$$ gauged spinors.