The Gauss-Manin connection and noncommutative tori

TL;DR

This paper demonstrates the invariance of periodic cyclic cohomology in smooth deformations of noncommutative tori using Getzler's Gauss-Manin connection, and analyzes the behavior of the Chern-Connes pairing during deformation.

Contribution

It applies Getzler's Gauss-Manin connection to noncommutative tori, providing explicit calculations of parallel translation maps and insights into cyclic cohomology invariance.

Findings

Periodic cyclic cohomology remains invariant under smooth deformation.

Explicit formulas for parallel translation maps in noncommutative tori.

Behavior of the Chern-Connes pairing under deformation is characterized.

Abstract

We use Getzler's Gauss-Manin connection to prove the invariance of periodic cyclic cohomology for the smooth deformation of noncommutative tori. We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern-Connes pairing under this deformation.