Sur la limite adiabatique des fonctions êta et zêta

In this Note we prove the existence of the adiabatic limit of the η(s) function of an operator on the total space of a fibration over S¹, constructed from an invertible family of first-order differential operators. We identify this limit as the holonomy of a meromorphic family of connections in the trivial bundle. In the same context, the ζ function diverges. We give a formula for the first two terms of the asymptotic expansion. The first result remains true for a non-invertible family if we restrict ourselves to s=0. For a family of Dirac operators, we retrieve the holonomy formula of Bismut–Freed. To cite this article: S. Moroianu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 131–134