The noncommutative geometry of the Landau Hamiltonian: differential aspects

Abstract

In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R². This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.