$d{S}_4$ vacua from matter-coupled 4D N=4 gauged supergravity

Abstract

We study $d{S}_4$ vacua within matter-coupled $N=4$ gauged supergravity in the embedding tensor formalism. We derive a set of conditions for the existence of $d{S}_4$ solutions by using a simple ansatz for solving the extremization and positivity of the scalar potential. We find two classes of gauge groups that lead to $d{S}_4$ vacua. One of them consists of gauge groups of the form $G_{\text{e}}\times G_{\text{m}}\times H$ with $H$ being a compact group and $G_{\text{e}}\times G_{\text{m}}$ a non-compact group with $SO(3)\times SO(3)$ subgroup and dynonically gauged. These gauge groups are the same as those giving rise to maximally supersymmetric $Ad{S}_4$ vacua. The $d{S}_4$ and $Ad{S}_4$ vacua arise from different coupling ratios between $G_{\text{e}}$ and $G_{\text{m}}$ factors. Another class of gauge groups is given by $SO(2,1{)}_{\text{e}}\times SO(2,1{)}_{\text{m}}\times G_{\text{nc}}\times G_{\text{nc}}^{\prime }\times H$ with $SO(2,1)$, $G_{\text{nc}}$ and $G_{\text{nc}}^{\prime }$ dyonically gauged. We explicitly check that all known $d{S}_4$ vacua in $N=4$ gauged supergravity satisfy the aforementioned conditions, hence the two classes of gauge groups can accommodate all the previous results on $d{S}_4$ vacua in a simple framework. Accordingly, the results provide a new approach for finding $d{S}_4$ vacua. In addition, relations between the embedding tensors for gauge groups admitting $d{S}_4$ and $d{S}_5$ vacua are studied, and a new gauge group, $SO(2,1)\times SO(4,1)$, with a $d{S}_4$ vacuum is found by applying these relations to $SO(1,1)\times SO(4,1)$ gauge group in five dimensions.

38 pages, no figure, typos and errors corrected, more detail and a reference added