Geometry and quantum transport

Abstract

This is abrief and informal introduction to a differential geometric interpetation of adiabatic charge transport in quantum mechanics. It involves the study of afamily of Schrodinger operators. For compact multiply connected surfaces the charge transported around the “holes” is related to the first Chern character of spectral bundles. For noncompact surfaces the charge transported to infinity is related to the index of a certain Fredholm operator which involves the comparison of appropriate spectral projections. There are also relations to Connes noncommutative differential geometry. Simple examples are given.