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

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.