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## Some important natural operators that come with Atiyah-Singer analysis on spin manifolds

This is from Lawson-Michelson’s book.

$Cl(X)$ carries a canonical bundle mapping $\alpha: Cl(X) \rightarrow Cl(X)$ that on each fiber extends -1 on the tangent space.  The +/-1 eigenspaces of $\alpha$ decomposes $Cl(X) = Cl^0(X)+Cl^1(X)$ corresponding to even and odd degree forms.  Another mapping $L(\phi) = \sum e_j \phi e_j$ a globally diagonizable map producing the degree decomposition $Cl(X) = \oplus \Lambda^p(X)$ and $L = (-1)^p(n-2p)$ on $\Lambda^p(X)$.  Finally $\omega = e_1 \cdots e_n$ the Clifford product of basis elements and one defines a map $\lambda_\omega(\phi) = \omega \cdot \phi$. The last map carries over to Dirac bundles and the decomposition $S = S^+ \oplus S^'-$ corresponds to $(1\pm \omega)S$.

These maps relate to $D$ as:

$\alpha L - L \alpha = 0$

$\alpha \lambda_\omega + (-1)^{n-1}\lambda_\omega \alpha = 0$

$L\lambda_\omega + (-1)^n \lambda_\omega L = 0$