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The analysis of elliptic families. II: Dirac operators, êta invariants, and the holonomy theorem. (English) Zbl 0657.58038
In this paper we specialize the results obtained in part I to the case of a family of Dirac operators. We first calculate the curvature of the unitary connection on the determinant bundle which we introduced in part I [ibid. 106, 159-176 (1986); see the preceding review]. We also calculate the odd Chern forms of Quillen for a family of self-adjoint Dirac operators and give a simple proof of certain results of Atiyah- Patodi-Singer on êta invariants. We finally give a heat equation proof of the holonomy theorem, in the form suggested by E. Witten [Commun. Math. Phys. 100, 197-229 (1985; Zbl 0581.58038); Anomalies, Geometry, Topology, Symp. Chicago/Ill. 1985, 61-99 (1985; Zbl 0643.53071)].

MSC:
58J10 Differential complexes
58J20 Index theory and related fixed-point theorems on manifolds
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