Higher geometric class field theory and moduli spaces of zero cycles

Global class field theory provides a universal abelian reciprocity laws for 1-dimensional global fields. It has been generalized in a number of different directions. In the geometric setting, it manifests as a duality between (different kinds of) sheaves on the moduli spaces of line bundles and 1-dimensional representations of the fundamental group of a curve. In a different direction, Kato and Saito gave a generalization to higher dimensional varieties. I will describe a categorification of the Kato-Saito higher class field theory, in which the moduli space of line bundles is generalized to the moduli space of zero cycles. This is joint work in progress with Elmanto.