On the squashed seven-sphere operator spectrum

TL;DR

This paper computes key parts of the eigenvalue spectrum of operators on the squashed seven-sphere, which are crucial for understanding the mass spectrum and supermultiplet structure in eleven-dimensional supergravity compactifications.

Contribution

It advances previous work by deriving significant portions of the operator spectrum on the squashed seven-sphere relevant to supergravity compactification.

Findings

Determined eigenvalue spectra relevant to $AdS_4$ fields.

Connected spectra to supermultiplet structures.

Comments on $G_2$ holonomy implications.

Abstract

We derive major parts of the eigenvalue spectrum of the operators on the squashed seven-sphere that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in $AdS_4$ and are important for the corresponding ${\mathcal N} =1$ supermultiplet structure. This work is a continuation of the work in [1] where the complete spectrum of irreducible isometry representations of the fields in $AdS_4$ was derived for this compactification. Some comments are also made concerning the $G_2$ holonomy and its implications on the structure of the operator equations on the squashed seven-sphere.