Relative Pairing in Cyclic Cohomology and Divisor Flows

We show that Melrose’s divisor flow and its generalizations by Lesch and Pflaum are invariants of K-theory classes for algebras of parametric pseudodifferential operators on a closed manifold, obtained by pairing the relative K-theory modulo the symbols with the cyclic cohomological character of a relative cycle constructed out of the regularized operator trace together with its symbolic boundary. This representation gives a clear and conceptual explanation to all the essential features of the divisor flow – its homotopy nature, additivity and integrality. It also provides a cohomological formula for the spectral flow along a smooth path of self-adjoint elliptic first order differential operators, between any two invertible such operators on a closed manifold.