De-Higgsing In Eleven-Dimensional Supergravity On The Squashed $S^7$

TL;DR

This paper analyzes the modes relevant for compactifying eleven-dimensional supergravity on a squashed $S^7$, exploring their properties, supermultiplet structure, de-Higgsing phenomena, and implications for stability under different boundary conditions.

Contribution

It identifies and characterizes the singlet modes on squashed $S^7$, relating their properties to previous work and exploring their role in supermultiplet formation and stability considerations.

Findings

Modes related to squashed $S^7$ are explicitly constructed and analyzed.

Supermultiplet structures and de-Higgsing phenomena are elucidated.

Boundary conditions affecting stability in non-supersymmetric compactifications are discussed.

Abstract

In this paper we construct the subset of modes on $S^7$ that are relevant in the compactification of eleven-dimensional supergravity on a squashed $S^7$ when restricted to the sector that comprises singlets under the $Sp(1)\times Sp(2)$ isometry of the squashed sphere. Some of the properties of these modes, connected to the transition from the round $S^7$ to the squashed $S^7$, are analysed in detail. Special features of the Rarita-Schwinger operator, described in earlier work by Buchdahl, are explained and related to properties of the squashed $S^7$ operator spectrum obtained in previous work by the authors. We then discuss how the singlet modes give rise to supermultiplets in the left-squashed case, the phenomenon of de-Higgsing, and what happens to the AdS$_4$ fields in these supermultiplets under an orientation reversal (``skew-whiffing'') of the squashed $S^7$. Finally, we consider the possible choices of boundary conditions that appear for some of these fields in AdS$_4$ in the case of the right-squashed non-supersymmetric compactification, and how these choices may affect the stability of the gravity theory.