Dirac operators

Suppose we are given a compact oriented (boundaryless) Riemannian manifold $(M, \eps)$ and a spinor module with charge conjugation $(\sS,C)$, together with a Riemannian metric $g$, so that the Clifford action $c \: \sB \to \End_{\sA}(\sS)$ has been specified. We can also write it as $\hat c \in \Hom_{\sA}(\sB \ox_{\sA} \sS, \sS)$ by setting $\hat c(\ka \ox \psi) := c(\ka) \psi$.

The $(-i)$ is included in the definition to make $\Dslash$ symmetric (instead of skewsymmetric) as an operator on a Hilbert space, because we have chosen $g$ to be positive definite, that is, $\ga^\al \ga^\bt + \ga^\bt \ga^\al = +2 \dl^{\al\bt}$. Historically, $\Dslash$ was introduced as $-i\ga^\mu \dl_\mu = \ga^\mu p_\mu$ where the $p_\mu$ are components of a $4$-momentum, but in the Minkowskian signature.

Using local (coordinate or orthonormal) bases for $\gX(M)$ and $\sA^1(M)$, we get nicer formulas:

$$\Dslash\psi = -i \hat c(\nb^S\psi) = -i c(dx^j) \nb^S_{\del_j} \psi = -i \ga^\al \nb^S_{E_\al} \psi.$$ (12)

The essential algebraic property of $\Dslash$ is the commutation relation:

$$[\Dslash, a]= -i c(da), \words{for all} a \in \sA = \Coo(M).$$ (13)

Indeed,
$$\begin{align*}[\Dslash, a] \psi &= -i \hat c(\nb^S(a\psi)) + ia \hat c(\nb^S\... ... -i \hat c(da \ox\psi) = -i c(da) \psi, \word{for} \psi \in \sS. \end{align*}$$