4D de Sitter from 6D gauged supergravity with Green-Schwarz counterterm

Taking into account the Green-Schwarz anomaly counterterm in R-symmetry gauged $N=(1,0)$ supergravity in six dimensions, and the associated modification in the Maxwell kinetic term and potential, the theory admits half-supersymmetric Mink$_4\times S^2$ and non-supersymmetric dS$_4 \times S^2$ solutions with or without a monopole on $S^2$. The monopole charge and the anomaly coefficients play key roles in the vacuum structure and a diagonal gauging in which an admixture of an external $U(1)$ with the R-symmetry $U(1)_R$ is needed for the de Sitter solutions to exist. We determine the full Kaluza-Klein spectrum for both vacua. The spectrum is unitary in the Minkowski case but in the case of de Sitter vacuum, the dilaton and the breathing mode are tachyonic. We show that turning on small perturbations of the tachyonic modes around dS$_4\times S^2$ with a monopole on $S^2$ triggers a flow evolving towards Mink$_4\times S^2$ with the minimal potential energy. The diagonally gauged model also supports dS$_2\times S^4$ solution which avoids tachyons for certain values of the flux on dS$_2$. We also find nonsupersymmetric (A)dS$_4\times S^2$ solutions, when we turn on a flux associated with external $U(1)$ gauge field only, and show that they support the phenomenon of scale separation under certain conditions on anomaly coefficients.