On the Vacuum Structure of the $\mathcal{N}=4$ Conformal Supergravity

TL;DR

This paper analyzes the vacuum structure of ${\cal N}=4$ conformal supergravity with a scalar-dependent function, classifying vacua by scalar eigenvalues and spacetime geometry, and discusses supersymmetry breaking and scalar fluctuations.

Contribution

It provides a detailed classification of vacua in ${\cal N}=4$ conformal supergravity with arbitrary scalar functions, including spectrum analysis and supersymmetry breaking conditions.

Findings

Vacua classified by scalar eigenvalues and spacetime type.

Presence of tachyonic states in scalar spectrum, removable by function choice.

$S$-supersymmetry always broken; $Q$-supersymmetry only on Minkowski.

Abstract

We consider ${\cal N}=4$ conformal supergravity with an arbitrary holomorphic function of the complex scalar $S$ which parametrizes the $SU(1,1)/U(1)$ coset. Assuming non-vanishings vevs for $S$ and the scalars in a symmetric matrix $E_{ij}$ of the $\overline{\bf 10}$ of $SU(4)$ R-symmetry group, we determine the vacuum structure of the theory. We find that the possible vacua are classified by the number of zero eigenvalues of the scalar matrix and the spacetime is either Minkowski, de Sitter or anti-de Sitter. We determine the spectrum of the scalar fluctuations and we find that it contains tachyonic states which however can be removed by appropriate choice of the unspecified at the supergravity level holomorphic function. Finally, we also establish that $S$-supersymmetry is always broken whereas $Q$-supersymmetry exists only on flat Minkowski spacetime.