Self-adjoint local boundary problems on compact surfaces. II. Family index

TL;DR

This paper develops a family index theorem for self-adjoint boundary value problems on 2D manifolds with boundary, expressing the index in topological terms and establishing its universality as a homotopy invariant.

Contribution

It introduces the first family index theorem for self-adjoint elliptic operators with local boundary conditions on 2D manifolds, linking the index to boundary topology.

Findings

Computed the $K^1(X)$-valued index in terms of boundary data.

Proved the index's universality as a homotopy invariant.

Established the index theorem for families of operators on surfaces.

Abstract

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems. We address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space $X$. We compute the $K^1(X)$-valued index in terms of the topological data over the boundary. The second result is the universality of the index: we show that the index is a universal additive homotopy invariant for such families, if the vanishing on families of invertible operators is required.