Dirac operator on spinors and diffeomorphisms

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma ,g}$ of spinor fields on $M$ and a Hilbert space $\HH_{\sigma, g}= L^2(S_{\sigma, g},\vol{M}{g})$of$L^2$-spinorsof$S_{\sigma, g}$\mathrm{.}Thegroup$\diff{M}$ of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[\sigma ]$ (by a suitably defined pullback $f^{\ast }\sigma$). Any $f\in \diff{M}$liftsinexactlytwowaystoaunitaryoperator$U$from$\HH_{\sigma, g} $to$\HH_{f^*\sigma,f^*g}$. The canonically defined Dirac operator is shown to be equivariant with respect to the action of $U$, so in particular its spectrum is invariant under the diffeomorphisms.