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On the \(\eta\)-invariant of certain nonlocal boundary value problems. (English) Zbl 0956.58014
The authors define a class of families of self-adjoint boundary conditions for first-order elliptic differential operators. Their analysis of a family in this class includes an asymptotic expansion of the heat trace and a variation formula for the eta invariant of the associated family of operators. Applied to the case of a compact manifold cut by a hypersurface, one such family interpolates between continuous transmission across the hypersurface and an Atiyah-Patodi-Singer boundary condition on the boundary components arising from the cut. In this setting the variation formula provides a new proof of the gluing law for eta invariants.
In this paper, Section one and the end of Section two clearly describe the context in which the authors’ work fits and provide relevant references (too many to mention in a short review). Briefly, there can be no local formula for the eta invariant. The gluing law is a natural substitute, of interest both for its relationship with topological quantum field theory and for its potential in helping extend index theory to manifolds with corners. E. Witten’s covariant anomaly formula helped make eta invariants on manifolds with boundary the focus of an approach to spectral invariants that is based on investigating adiabatic limits of these invariants. Such adiabatic limits have revealed new relationships among index-theoretic invariants.

MSC:
58J28 Eta-invariants, Chern-Simons invariants
58J32 Boundary value problems on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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