Laplacian spectrum on a nilmanifold, truncations and effective theories

Motivated by low energy effective theories arising from compactification on curved manifolds, we determine the complete spectrum of the Laplacian operator on the three-dimensional Heisenberg nilmanifold. We first use the result to construct a finite set of forms leading to an $ \mathcal{N}=2 $ gauged supergravity, upon reduction on manifolds with SU(3) structure. Secondly, we show that in a certain geometrical limit the spectrum is truncated to the light modes, which turn out to be left-invariant forms of the nilmanifold. We also study the behavior of the towers of modes at different points in field space, in connection with the refined swampland distance conjecture.