15:30
The noncommutative Gauss-Manin connection is a flat connection on the periodic cyclic homology of a family of dg algebras (or more generally, A-infinity categories), introduced by the speaker in 1991.
The problem now arises of lifting this connection to the complex of periodic cyclic chains. Such a lift was provided in 2007 by Tsygan, though without an explicit formula. In this talk, I will explain how this problem is simplified by considering a new A-infinity structure on the de Rham complex of a derived scheme, which we call the Fedosov product; in joint work with Jones in 1990, the speaker showed that this product plays a role in a multiplicative version of the Hochschild-Kostant-Rosenberg theorem, and the point of the present talk is that it seems to be the correct product on the de Rham complex for derived geometry.
Let be an open subset of a derived affine space parametrizing a family of
-algebras
. We will construct a chain level lift
of the Gauss-Manin connection that satisfies a new equation that we call the Fedosov equation:
.