Noncommutative Geometry and Quantum Physics Vietri sul Mare, August 31 - September 5, 2009
Home page Speakers Participants Program Picture Lodging Travel Equipment Registration Sponsors	Speaker: Giuseppe De Nittis, SISSA Title: A family of vector bundle representations of the rational NCT: theory and applications Abstract: In this talk we will show the correspondence between a countable family of representation of the Non-Commutative Torus (NCT) with rational deformation parameter M/N and a countable family of rank N vector bundles over the 2-dimensional torus. These representations are labelled by an integer which can be identified with the first Chern number of the related vector bundle. This analysis uses as main mathematical tool the "direct integral decomposition theorem" of von Neumann which is a manifestation of the existence of a class of symmetries for each representation of the C*-algebra of the NCT. The notion of symmetry can be introduced in the game in terms of the notion of "commutant" which is a another way (different language) to highlight the existence of a "Morita-equivalence". The main result of this approach is to write a family of Diophantine equations (generalized TKNN equations) which describe in terms of Chern numbers some relevant spectral properties of the selfadjoint elements of the NCT. These results have some interesting applications in the explanation of the Quantum Hall Effect (QHE)
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