Index Theorems, Generalized Hall Currents, and Topology for Gapless Defect Fermions

We show how the index of the fermion operator from the Euclidean action can be used to uncover the existence of gapless modes living on defects (such as edges and vortices) in topological insulators and superconductors. The 1-loop Feynman diagram that computes the index reveals an analog of the quantum Hall current flowing on and off the defect-even in systems without conserved currents or chiral anomalies-and makes explicit the interplay between topology in momentum and coordinate space. We provide several explicit examples.