Dirac Operators on Orbifold Resolutions: Uniform Elliptic Theory

Abstract

We study Dirac operators on resolutions of Riemannian orbifolds by developing a uniform elliptic theory. The key idea is to view orbifolds as conically fibred singular (CFS) spaces and resolve them by gluing asymptotically conical fibrations (ACF) into the singular strata. This yields smooth Gromov--Hausdorff resolutions that preserve the large--scale structure of the orbifold while replacing its singularities with well--understood local models. Dirac bundles are resolved compatibly with this construction, which allows us to analyse entire families of Dirac operators in a uniform way. Inspired by the linear gluing framework of Hutchings--Taubes, we build uniformly bounded right inverses and describe precisely how kernels and cokernels behave as the geometry degenerates. The analysis relies on weighted spaces adapted to the conically fibred, conically fibred singular and asymptotically conical fibred regions. These allow us to prove Fredholm properties, uniform bounds and to establish exactness of the linear gluing sequence. Consequently, we obtain an index formula decomposing the index into contributions from the orbifold and the ACF models. This framework provides a direct and flexible analytic approach to Dirac operators on orbifold resolutions, avoiding the full edge calculus, and setting the stage for applications to special holonomy metrics and gauge theory.