# Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric

We prove that the Atiyah-Singer Dirac operator $\mathrm{D}_g$ in $L^2$ depends Riesz continuously on $L^\infty$ perturbations of a complete metric $g$. The Lipschitz bound for the map $g \to \mathrm{D}_g/\sqrt{1 + \mathrm{D}_g^2}$ depends on curvature bounds and a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles. This is joint work with Alan McIntosh (Australian National University) and Andreas Rosén (Gothenburg University).