Adams operations and periodic cyclic cohomology (Q2708735)

Given a connection \(\nabla\) on a vector bundle \(E\) over a smooth manifold \(X\), Bismut associated an equivariantly closed form on the free loop space of \(X\), \(X^{S^1}\); this the the equivariant Chern character \(\text{Ch}(E,\nabla)\). Its restriction to \(X\subseteq X^{S^1}\) is the usual Chern character of \(E\) and there are index formulæ for Dirac operators involving this. Getzler, Jones \& Petrack showed how this construction was related to cyclic cohomology and used an iterated integral map of Chen to describe \(\text{Ch}(E,\nabla)\) as an iterated integral. The present paper investigates the analogous situation in periodic cyclic cohomology and in particular computes certain of these groups and shows that there is an \(A_\infty\)-algebra structure on their underlying complexes. Adams operations induced from the power maps on the circle are used to show that a theorem of Chen fails to generalize from cyclic cohomology to periodic cyclic cohomology, thus highlighting the importance of the \(A_\infty\)-algebra structure.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].