Dirac operator spectrum on a nilmanifold

We obtain the spectrum of the Dirac operator on the three-dimensional Heisenberg nilmanifold ${\mathcal{M}}_3$, and its complete dependence on the metric moduli. As an application, we construct the four-dimensional low-energy effective action obtained by compactification of a seven-dimensional gauge-fermion theory on ${\mathcal{M}}_3$.