A Ghost-Free dS/AdS Correspondence in $\mathcal{N}=1$ Supergravity

Abstract

Reversing the sign of the scalar potential exchanges de Sitter and anti-de Sitter vacua and, through the domain-wall/cosmology correspondence, maps flat domain walls onto flat FLRW cosmologies, but its supersymmetric version has ghosts on the cosmological side. We show that this pointwise correspondence is realized between two ordinary, ghost-free four-dimensional $\mathcal{N}=1$ supergravities with spontaneously broken supersymmetry. Within a single superconformal Kähler cone, a $\mathbb{Z}_2$ reflection of the nilpotent goldstino Kähler metric through the flat limit $V=0$ reverses $V\leftrightarrow-V$ pointwise while preserving the visible Kähler metric, the superpotential, and the unitary-gauge fermion Lagrangian. The paired vacua obey a sharp dictionary with $H=1/L$, $m^2_{\rm dS}=-m^2_{\rm AdS}$, equal conformal weights, and $m_{3/2}>H$, and are simultaneously stable precisely within the window $0\le m^2_{\rm dS}\le\frac94H^2$ set by the Breitenlohner--Freedman bound. Applied to string compactifications and cosmology, this correspondence pairs over-uplifted with under-uplifted anti-D3-brane vacua, confines the de Sitter swampland conjecture to the goldstino (anti-brane) sector, and maps slow-roll inflation onto a healthy AdS domain wall obeying the holographic $c$-theorem.