The Gauss–Manin connection for the cyclic homology of smooth deformations, and noncommutative tori

Abstract

Given a smooth deformation of topological algebras, we define Getzler’s Gauss–Manin connection on the periodic cyclic homology of the corresponding smooth field of algebras. Basic properties are investigated including the interaction with the Chern–Connes pairing with $K$-theory. We use the Gauss–Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension. Then we illustrate the Gauss–Manin connection for the deformation of noncommutative tori. We use the Gauss–Manin connection to identify the periodic cyclic homology of a noncommutative torus with that of the commutative torus via a parallel translation isomorphism.We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern–Connes pairing under this deformation.