GAMP/QMATH Seminar: Spectral flow and family index for elliptic boundary value problems on compact surfaces
Title: Spectral flow and family index for elliptic boundary value problems on compact surfaces
Abstract: 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 operatorsʼ eigenvalues passing through zero with the change of parameter. For loops of elliptic operators on a closed manifold, the spectral flow was computed by Atiyah, Patodi, and Singer 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 much more complicated.
My talk is devoted to simplest non-trivial manifolds with boundary, namely two-dimensional manifolds. I will explain how to compute the spectral flow for loops of first order self-adjoint elliptic operators with classical boundary conditions. I will also present an index theorem for families of such operators parametrized by points of a compact space X. The analytical index of such a family takes values in the group K^1(X), and I compute this index in terms of the topological data of the family over the boundary.
The talk is based on my preprints arXiv:1703.06105 and 1809.04353.