Geometry on the quantum Heisenberg manifold

Abstract

A class of C^∗-algebras called quantum Heisenberg manifolds were introduced by Rieffel in (Comm. Math. Phys. 122 (1989) 531) as strict deformation quantization of Heisenberg manifolds. Using the ergodic action of Heisenberg group we construct a family of spectral triples. It is shown that associated Kasparov modules are homotopic. We also show that they induce cohomologous elements in entire cyclic cohomology. The space of Connes-deRham forms have been explicitly calculated. Then we characterize torsionless/unitary connections and show that there does not exist a connection that is simultaneously torsionless and unitary. Explicit examples of connections are produced with negative scalar curvature. This part illustrates computations involving some of the concepts introduced in Frohlich et al. (Comm. Math. Phys. 203 (1999) 119), for which to the best of our knowledge no infinite-dimensional example is known other that the noncommutative torus.