Laplacian spectrum on a nilmanifold, truncations and effective theories

TL;DR

This paper computes the Laplacian spectrum on a nilmanifold and explores its implications for effective theories, including mode truncation and connections to the swampland conjecture.

Contribution

It provides the complete Laplacian spectrum on the Heisenberg nilmanifold and analyzes mode truncation in effective supergravity theories derived from it.

Findings

Spectrum explicitly determined for the nilmanifold

Mode truncation to light modes in a geometrical limit

Insights into the swampland distance conjecture

Abstract

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.