Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras

TL;DR

This paper establishes an index theorem for boundary value problems on manifolds with boundary, using continuous fields of C*-algebras and deformation techniques to connect analytic and topological indices.

Contribution

It introduces a new index theorem for boundary value problems via continuous fields of C*-algebras and deformation theory, extending Connes' tangent groupoid approach to manifolds with boundary.

Findings

The tangent semigroupoid generalizes Connes' tangent groupoid for boundary cases.

The analytic index map is identified via deformation theory.

The topological index map is constructed and shown to coincide with the analytic index.

Abstract

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid $\cT^-X$ generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C*_r(\cT^-X) of C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.