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 $K{K}_0(A_{\theta }\otimes A_{\theta },\mathbb{C})$ providing the co-unit for a Poincaré self-duality for the irrational rotation algebra $A_{\theta }$ for any $\theta \in \mathbb{R}\setminus \mathbb{Q}$. This spectral triple has been extensively studied since. 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-trivial element $g$ of the modular group, a finitely generated projective module ${\mathcal{L}}_g$ 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 $g(\theta )$, using the fact that these flows are transverse to each other. We then compute Connes' dual of $[{\mathcal{L}}_g]$ for $g$ upper triangular, and prove that we obtain an invertible in $K{K}_0(A_{\theta },A_{\theta })$, represented by what one might regard as a noncommutative bundle of Dirac-Schrödinger operators. An application of $\mathbb{Z}$-equivariant Bott Periodicity proves that twisting the module by the family gives the requisite spectral cycle for the unit, thus proving self-duality for $A_{\theta }$ with both unit and co-unit represented by spectral cycles.

substantially revised from first version

References in corpus (4)

• Sums of regular selfadjoint operators in Hilbert-C*-modules
• On the Unbounded Picture of $KK$-Theory
• Lefschetz numbers for C*-algebras
• The Godbillon-Vey invariant in equivariant $KK$-theory

Cited by in corpus (2)

• A geometric representative for the fundamental class in KK-duality of Smale spaces
• The Imprimitivity Fell Bundle