The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary
| dc.creator | Kirk, Paul | |
| dc.creator | Lesch, Matthias | |
| dc.date | 2000-12-15 | |
| dc.date | 2002-03-11 | |
| dc.date.accessioned | 2026-07-07T06:21:26Z | |
| dc.date.available | 2026-07-07T06:21:26Z | |
| dc.description | Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderon projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott-Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Bruning-Lesch to the mod Z gluing formula. Our presentation includes careful constructions of the Maslov index and triple index in a symplectic Hilbert space. As a byproduct we give intuitively appealing proofs of two theorems of Nicolaescu on the spectral flow of Dirac operators. As an application of our methods, we carry out a detailed analysis of the eta-invariant of the odd signature operator coupled to a flat connection using adiabatic methods. This is used to extend the definition of the Atiyah-Patodi-Singer rho-invariant to manifolds with boundary. We derive a ``non-additivity'' formula for the Atiyah-Patodi-Singer rho-invariant and relate it to Wall's non-additivity formula for the signature of even-dimensional manifolds. | |
| dc.description | 64 pages, 1 figure; v2 Nov 2001, several improvements in Sec. 8, minor polishing in the other sections | |
| dc.identifier | https://arxiv.org/abs/math/0012123 | |
| dc.identifier | http://arxiv.org/abs/math/0012123 | |
| dc.identifier | Forum Math. 16 (2004), no 4, 553--629 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/95601 | |
| dc.subject | Differential Geometry | |
| dc.subject | Geometric Topology | |
| dc.subject | 58J28; 58J30; 58J32 | |
| dc.title | The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary | |
| dc.type | text |