Michael Gregory, University of Waterloo
Dirac Operators on Orbifold Resolutions
In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families (X_t,g_t), which collapse to the orbifold (X_0,g_0) as t goes to 0.
The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of X_t into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for D_t, and derive an index additivity formula.
The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free G-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.
MC 5403