Toroidal orbifolds of Z3 and Z6 symmetries of noncommutative tori

Abstract

The Hexic transform ρ of the noncommutative 2-torus Aθ is the canonical order 6 automorphism defined by ρ(U)=V, ρ(V)=e−πiθU−1V, where U, V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUV. The Cubic transform is κ=ρ2. These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C⁎-algebras Aθρ, Aθκ are noncommutative Z6, Z3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ, a projection e of trace q2θ−pq is constructed in Aθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθe contains a Hexic invariant q×q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C∞-projections of the continuous field {At}0<t<1 of noncommutative tori C⁎-algebras such that ρ(E(t))=E(t). It turns out that the projection E(t) is the support projection of a canonical C∞-positive element that has the appearance of a noncommutative 2-dimensional Theta function. The topological invariants (or ‘quantum’ numbers) of E(t), e, and related projections are computed by a new and quicker method than in previous works. (They would also give topological invariants for finitely generated projective modules over noncommutative orbifolds associated to Z6 and Z3 symmetries of noncommutative tori.) We remark that these results have some bearing on research work related to noncommutative orbifolds used in string theory.