Family index for self-adjoint elliptic boundary value problems

An index theory for elliptic operators on a closed manifold was developed by Atiyah and Singer. For a family of such operators parametrized by points of a compact space X, they computed the K^0(X)-valued analytical index in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by Atiyah, Patodi, and Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.

If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes much more complicated. The integer-valued index of a single boundary value problem was computed by Atiyah and Bott. This result was recently generalized to K^0(X)-valued family index by Melo, Schrohe, and Schick. The self-adjoint case, however, remained open.

In the talk I shall present a family index theorem for self-adjoint elliptic operators on a surface with boundary. I compute the K^1(X)-valued analytical index in terms of the topological data of the family over the boundary. The talk is based on my preprint arXiv:1809.04353.