Spectral flow and family index for Dirac operators on surfaces with boundary

A one-parameter family of self-adjoint Fredholm operators has a well-known integer-valued invariant, the spectral flow. It counts (with signs) the number of eigenvalues changing their sign along the way. For loops of elliptic operators on a closed manifold, the spectral flow was computed by Atiyah, Patodi, and Singer (1976) in terms of topological data of a loop. But if a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes more complicated. In the talk I will explain how to compute the spectral flow for loops of Dirac type operators with classical boundary conditions in two-dimensional case (that is, for a compact surface with boundary). A particular case of this result may be interpreted as the Aharonov-Bohm effect for a graphene sheet with holes.

More generally, if operators and boundary conditions are parametrized by points of a compact space X, then the relevant invariant takes values in the odd K-group of X and is called the analytical index. I will show how this index is computed in terms of the topological data of the family over the boundary.

The talk is based on my papers arXiv:1703.06105 and 1809.04353.