Dirac operator on spinors and diffeomorphisms

TL;DR

This paper investigates how the Dirac operator on spinors behaves under diffeomorphisms of a manifold, demonstrating that its spectrum remains invariant under the action of orientation-preserving diffeomorphisms.

Contribution

It establishes the equivariance of the Dirac operator with respect to diffeomorphisms and clarifies the transformation properties of spin structures and associated Hilbert spaces.

Findings

Dirac operator spectrum is invariant under diffeomorphisms.

Diffeomorphisms lift to unitary operators on spinor Hilbert spaces.

The action of diffeomorphisms preserves the Dirac operator's spectrum.

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 $\HH_{\sigma, g}= L^2(S_{\sigma, g},\vol{M}{g})$ 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 $f^*\sigma$). Any $f\in \diff{M}$ lifts in exactly two ways to a unitary operator $U$ 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.