On the eta-invariant of certain nonlocal boundary value problems

TL;DR

This paper introduces a new class of nonlocal boundary value problems related to eta-invariants, deriving heat expansions and providing a new proof of the gluing formula, extending previous work on Atiyah-Patodi-Singer conditions.

Contribution

It generalizes existing boundary conditions for eta-invariants, deriving heat expansions and offering a new proof of the gluing formula for these invariants.

Findings

Derived full heat expansion for the new class of boundary problems

Provided a new proof of the eta invariant gluing formula

Unified local and nonlocal boundary conditions within this framework

Abstract

Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.