= Spinor bundle
{title2=$S$}
A chosen <spin structure> on a <spin manifold> defines the complex spinor bundle associated to the spin representation. In dimension $2m$ its <vector bundle rank> is $2^m$. Its <chirality matrix> is $\Gamma=i^mc(e^1)\cdots c(e^{2m})$ in the negative <Clifford multiplication> convention. It satisfies $\Gamma^2=1$ and anticommutes with each $c(e^i)$, giving $S=S^+\oplus S^-$ and an odd <Dirac operator>. In odd dimension there is no such canonical chiral splitting of the irreducible spinor bundle.
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