  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.
