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\(\eta\)-invariants and their adiabatic limits. (English) Zbl 0671.58037

The Atiyah-Patody-Singer eta invariant is a spectral invariant which measures the symmetry of the spectrum of a self-adjoint elliptic differential operator. The eta invariant was introduced in connection with the index theorem for manifolds X with boundary \(\partial X\). The invariant \(\eta\) (A) is not locally computable. That is, it cannot be obtained by integrating over \(\partial X\), any differential form which is given in local coordinates by a canonical expression derived from the symbol of A.
This article is devoted to study the limiting value of the \(\eta\)- invariant for Dirac operators, in situations in which the metric (or part of the metric) on the underlying manifold is multiplied by the factor \(\epsilon^{-1}\) and \(\epsilon\) \(\to 0\). It is shown that in the presence of an additional invertibility hypothesis, the \(\eta\)-invariant approaches a limiting value which is locally computable (or partly locally computable).
Reviewer: V.Deundjak

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58J70 Invariance and symmetry properties for PDEs on manifolds
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