The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Details The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

2000-12-15

arxiv.org/abs/math/0012123v2

Forum Math. 16 (2004), no 4, 553--629

Mathematics

Differential Geometry

64 pages, 1 figure; v2 Nov 2001, several improvements in Sec. 8, minor polishing in the other sections

Scientific paper

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.

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