  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 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.
