The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary
By Paul A. Kirk, M. Lesch. 2000
115 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/25ac33717768527103e28e0c3ecec8d519c9aace
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Additivity and non-additivity for perverse signatures (2009, methodology): “For manifolds with boundary, the Maslov triple index term in Wall’s non-additivity formula has been interpreted analytically in the context of analytic signature theorems for manifolds with corners of codimension two in [28] and in terms of a gluing formula for the η-invariant and the spectral flow ”