Yang-Mills connections and sigma-models on quantum Heisenberg manifolds

We construct a spectral triple on a quantum Heisenberg manifold, which generalizes the results of Chakraborty and Shinha, and associate to it an energy functional on the set of projections, following the approach of Mathai-Rosenberg to non-linear sigma models. The spectral triples that we construct extend the We derive a lower bound for this energy functional that is linked on the topological charge of the projection which depends on the curvature of a compatible connection. A detailed study of this lower bound is given for the Kang projection in quantum Heisenberg manifolds. These results display an intriguing interplay between non-linear sigma models and Yang-Mills theory on quantum Heisenberg manifolds, unlike in the well-studied case of noncommutative tori.