The spin connection

We now leave the topological level and introduce differential structure. Thus we replace $A = C(M)$ by $\sA = \Coo(M)$, and continuous sections $\Ga_{\mathrm{cont}}$ by smooth sections $\Ga_{\mathrm{smooth}}$. Thus $\sS = \Ga(M,S)$ will henceforth denote the $\sA$-module of smooth spinors.

Our treatment of Morita equivalence of unital algebras passes without change to the smooth level. We can go back with the functor $- \ox_{\Coo(M)} C(M)$, if desired.

Employing the usual contraction of vector fields with forms in $\sA^\8(M)$, namely,

$$\iota_X\om(Y_1,\dots, Y_k) := \om(X, Y_1,\dots, Y_k) \words{for} \om \in \sA^{k+1}(M),$$

extended to $\sA^\8(M) \ox_{\sA} \sE$ as $\iota_X \ox \id_{\sE}$ --but still written $\iota_X$-- we get operators $\nb_X$ on $\sA^\8(M, E)$ of degree 0 by defining

$$\nb_X := \iota_X \circ \nb + \nb \circ \iota_X.$$

This is $\sA$-linear in $X$. Moreover, if $\om \in \sA^\8(M)$ and $s \in \sE$, one can check that $\nb_X(\om \ox s) = \sL_X\om \ox s + \om \nb_X s$, where $\sL_X = \iota_X d + d \iota_X$ is the Lie derivative of forms with respect to $X$.

We mention two natural constructions for connections, on tensor products of $\sA$-modules and on dual $\sA$-modules. Firstly, if $\nb' \: \sF \to \sA^1(M) \ox_{\sA} \sF$ is another connection in another $\sA$-module, then

$$\Tilde{\nb}(s \ox t) := \nb s \ox t + s \ox \nb' t$$

(extneded by linearity, as usual) makes $\Tilde{\nb}$ a connection on $\sE \ox_{\sA} \sF$.

Next, if $\sE^\3 = \Hom_{\sA}(\sE,\sA)$, then the dual connection $\nb^\3$ on $\sE^\3$ is determined by
$$\begin{align*} d(\ze(s)) &=: (\nb^\3\ze)(s) + \ze(\nb s) \word{in} \sA^1(M); \... ...X \ze)(s) + \ze(\nb_X s) \word{in} \sA, \word{for} X \in \gX(M), \end{align*}$$
whenever $\ze \in \sE^\3$ and $s \in \sE$.

If $\nb, \nb'$ are connections on $\sE$, then $\nb' - \nb$ is an $\sA$-module map: $(\nb' - \nb)(fs) = f(\nb' - \nb)s$, so that locally, over $U \subset M$ for which $E\bigr\vert _U \to U$ is trivial, we can write

$$\nb = d + \al, \word{where} \al \in \sA^1(U, \End E).$$

The explicit formula for this connection is

It is called Levi-Civita connection associated to $g$. (The proof of existence consists in showing that the right hand side of this expression is $\sA$-linear in $Y$ and $Z$, and obeys a Leibniz rule with respect to $X$, so it gives a connection; and uniqueness is obtained by checking that metric compatibility and torsion freedom make the right hand side automatic.)

The dual connection on $\sA^1(M)$ will also be called the ``Levi-Civita connection''. At the risk of some confusion, we shall use the same symbol $\nb$ for both of these Levi-Civita connections.