Description:

Title: Noncommutative calculus and the Gauss-Manin connection in noncommutative geometry, revisited.

Abstract: Noncommutative calculus is a version of the Cartan calculus of differential forms and multivectors on a manifold when one replaces a manifold with an associative algebra. The De Rham complex is replaced by the Hochschild chain complex, vector fields by derivations, and multivector fields by Hochschild cochains. In this context, the Gauss-Manin connection is a way to start with a derivation of degree one of a graded algebra and, using Cartan calculus, to extend it to a differential on periodic cyclic chains.

The topic has been extensively studied during the last forty years. We will present it in a revised form. We will start by explaining what precise form of the Cartan identities should hold. Then we will give explicit formulas for the Gauss-Manin connection. We find it notable for two reasons. First, they seem to have reasonable convergence properties, both classically and p-adically. Second, they are rich in allusions to expressions from mathematical physics (where the Planck constant is replaced by the formal parameter u from cyclic homology theory) and also to the filtered circle of Moulinos-Ronaldo-Toën and Raksit.