Dirac operators twisted by ramified Euclidean line bundles

Z/2 harmonic spinors arise naturally in gauge theory and calibrated geometry, notably as limiting objects in compactness problems. They are harmonic sections of Dirac operators twisted by Euclidean line bundles ramified along codimension-two subsets. Understanding the analysis of these operators is an important step towards developing a deformation theory of Z/2 harmonic spinors. In this talk, I will discuss the closed extensions of such Dirac operators and their characterization in terms of the Gelfand–Robbin quotient. When the branching locus is a smooth codimension-two submanifold, we construct a geometric realization of this quotient, allowing us to describe Fredholm extensions through residue conditions along the branching locus. I will also discuss elliptic regularity in adapted Sobolev spaces. This is joint work with Thomas Walpuski (arXiv:2503.01392).