Solvmanifolds and noncommutative tori with real multiplication

We prove that the Shimizu $L$-function of a real quadratic fieldis obtained from a (Lorentzian) spectral triple on anoncommutative torus with real multiplication, as an adiabaticlimit of the Dirac operator on a 3-dimensional solvmanifold. TheDirac operator on this 3-dimensional geometry gives, via theConnes–Landi isospectral deformations, a spectral triple forthe noncommutative tori obtained by deforming the fiber tori tononcommutative spaces. The 3-dimensional solvmanifold is thehomotopy quotient in the sense of Baum–Connes of thenoncommutative space obtained as the crossed product of thenoncommutative torus by the action of the units of the realquadratic field. This noncommutative space is identified withthe twisted group $C^*$-algebra of the fundamental group of the3-manifold. The twisting can be interpreted as the cocyclearising from a magnetic field, as in the theory of the quantumHall effect. We prove a twisted index theorem that computes therange of the trace on the $K$-theory of this noncommutativespace and gives an estimate on the gaps in the spectrum of theassociated Harper operator.

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Published 1 January 2008