Analysis on Vector Bundles over Noncommutative Tori

Abstract

Noncommutative geometry is the study of noncommutative algebras, especially C*-algebras, and their geometric interpretation as topological spaces. One C*-algebra particularly important in physics is the noncommutative n-torus, the irrational rotation C*-algebra AΘ with n unitary generators U1, . . . , Un which satisfy UkUj = e2πiθj,kUjUk and Uj* = Uj-1, where Θ ∈ Mn(ℝ) is skew-symmetric with upper triangular entries that are irrational and linearly independent over ℚ. We focus on two projects: an analytically detailed derivation of the pseudodifferential calculus on noncommutative tori, and a proof of an index theorem for vector bundles over the noncommutative two torus. We use Raymond's definition of an oscillatory integral with Connes' construction of pseudodifferential operators to rederive the calculus in more detail, following the strategy of the derivations in Wong's book on pseudodifferential operators. We then define the corresponding analog of Sobolev spaces on noncommutative tori, for which we prove analogs of the Sobolev and Rellich lemmas, and extend all of these results to vector bundles over noncommutative tori. We extend Connes and Tretkoff's analog of the Gauss-Bonnet theorem for the noncommutative two torus to an analog of the McKean-Singer index theorem for vector bundles over the noncommutative two torus, proving a rearrangement lemma where a self-adjoint idempotent e appears in the denominator but does not commute with the k2 already there from the rearrangement lemma proven by Connes and Tretkoff.