Transversals, duality, and irrational rotation

Abstract:

An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb {T}^2$, which induces a Poincaré self-duality for $\mathbb {T}^2$, can be ‘quantized’ to give a spectral triple and a K-homology class in $\mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C})$ providing the co-unit for a Poincaré self-duality for the irrational rotation algebra $A_\theta$ for any $\theta \in \mathbb {R}\setminus \mathbb {Q}$. 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 $b$, a finitely generated projective module $\mathcal {L}_{b}$ over $A_\theta \otimes A_\theta$ by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope $\theta$ and $\theta + b$, using the fact that these flows are transverse to each other. We then compute Connes’ dual of $[\mathcal {L}_{b}]$ and prove that we obtain an invertible $\tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta )$, 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 ‘$b$-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.