Transversals, duality, and irrational rotation

An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the 2 2 -torus T 2 \mathbb {T}^2 , which induces a Poincaré self-duality for T 2 \mathbb {T}^2 , can be ‘quantized’ to give a spectral triple and a K-homology class in K K 0 ( A θ ⊗ A θ , C ) \mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C}) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ A_\theta for any θ ∈ R ∖ Q \theta \in \mathbb {R}\setminus \mathbb {Q} . Connes’ proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b b , a finitely generated projective module L b \mathcal {L}_{b} over A θ ⊗ A θ A_\theta \otimes A_\theta by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ \theta and θ + b \theta + b , using the fact that these flows are transverse to each other. We then compute Connes’ dual of [ L b ] [\mathcal {L}_{b}] and prove that we obtain an invertible τ b ∈ K K 0 ( A θ , A θ ) \tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta ) , represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘ b b -twists’ and this allows us to describe the unit of Connes’ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit – a kind of ‘quantized Thom class’ for the diagonal embedding of the noncommutative torus.