Asymptotics of the Eta Invariant

Abstract

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.

The asymptotics of ECH capacities

“…We prove a stronger bound of this kind than any previous bound, see Proposition 6 and the discussion about the eta invariant below, and this is the key point which allows us to prove Opgr Q pσ j qq 2{5 asymptotics. Spectral flow bounds for families of Dirac operators were also considered in [20,21,26]. The main difference here is that in those works the bounds were proved on reducible solutions where the connections needed to define the relevant Dirac operators were explicitly given.…”

Dirac spectral flow on contact three manifolds II: Thurston–Winkelnkemper contact forms

“…Recently, Savale [13] improved this theorem by showing that the subleading order term is of O(r 3 2 ).…”

Sub-leading asymptotics of ECH capacities

“…The reason that we can not improve upon gr Q asymptotics is because we do not know how to strengthen the O r 3/2 spectral flow estimate on the irreducible solutions of Propositions 5 or 8. A better O(r ) estimate does however exist [35,37] for reducible solutions for which one understands the connection precisely in the limit r → ∞. However, the a priori estimates (3.23) are not strong enough to carry out the same for irreducibles.…”

“…We prove a stronger bound of this kind than any previous bound, see Proposition 5 and the discussion about the eta invariant below, and this is the key point which allows us to prove O j 2/5 asymptotics. Spectral flow bounds for families of Dirac operators were also considered in [34][35][36]42]. The main difference here is that in those works the bounds were proved on reducible solutions where the connections needed to define the relevant Dirac operators were explicitly given.…”