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

Abstract

Abstract. Several proofs have been published of the mod Z gluing formula for the η-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the η-invariant is left obscure in the literature. In this article we present a gluing formula for the η-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón projectors of the two parts of the decomposition. The main ingredients of our presentation are the ScottWojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Brüning-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 η-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 ρ-invariant to manifolds with boundary. We derive a "non-additivity" formula for the AtiyahPatodi-Singer ρ-invariant and relate it to Wall's non-additivity formula for the signature of even-dimensional manifolds.

Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

“…The Maslov triple index τ µ (B, P, g) is an integer [39]. We do not have a physical explanation of τ µ (B, P, g) yet and it possibly relates to the central charge of the theory [40].η is the reduced eta-invariant.…”

“…where g ∈GL(N, C) (or a subgroup like SU(N )); R X and R L are the curvatures of the background manifold X and L;η[B, g] is a reduced eta-invariant and τ µ , which is an integer, is the Maslov triple index [39]. We do not have a physical explanation of τ µ yet and it possibly relates to the central charge of the theory [40].…”

The Mathematical Work of V. K. Patodi

“…A solution to the problem of generalizing (3.19) to the case of manifolds with boundary, without any condition on g, is later given in [70], where the results in [87] and [101] play important roles. The result proved in [70] may be thought of as an odd dimensional analogue of Theorem 3.1.…”

The Calderón projector for the odd signature operator and spectral flow calculations in 3-dimensional topology

“…More subtly, spectral invariants such as spectral flow and η invariants also have topological properties for which no general analysis free definitions exist. One of the purposes of this article is to catalog some of the analytical properties of the Calderón projector which are special to the odd signature operator developed in the articles by Daniel and Kirk [14] and Kirk and Lesch [24].…”

“…Roughly speaking it says that if one replaces a closed manifold with the two pieces in a decomposition along a separating hypersurface with infinite collars attached, then the L 2 -solutions to D A φ = 0 on each piece do not interact. The precise statement is key to the proof of a splitting theorem for η and ρ α invariants in [24] and simplifies the spectral flow calculations of [8].…”