Dirac operator on spinors and diffeomorphisms

Abstract

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 $\HH_{σ, g}= L^2(S_{σ, g},\vol{M}{g})$ of $L^2$-spinors of $S_{\sigma ,g}$. The group $\diff{M}$ of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[\sigma ]$ (by a suitably defined pullback $f^{\ast }\sigma$). Any $f\in \diff{M}$ lifts in exactly two ways to a unitary operator $U$ from $\HH_{σ, g} $ to $\HH_{f^*σ,f^*g}$. 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.

13 pages