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

TL;DR

This work shows that Green-Schwarz anomaly counterterms in 6D ${N}=(1,0)$ supergravity, together with diagonal gauging that mixes external $U(1)$ with $U(1)_R$, admit a half-supersymmetric Mink$_4 imes S^2$ vacuum and a non-supersymmetric dS$_4 imes S^2$ vacuum with a monopole on $S^2$. The authors compute the full KK spectrum around both vacua, finding unitary behavior for Mink$_4 imes S^2$ but two tachyonic scalar modes in dS$_4 imes S^2$, arising from the dilaton and internal volume. A small perturbation of these tachyons drives a dynamical flow from dS$_4 imes S^2$ to the Mink$_4 imes S^2$ vacuum, indicating the Minkowski vacuum as an attractor. They also exhibit an AdS$_4 imes S^2$ solution with external flux and discuss scale separation, as well as the implications for higher-derivative corrections and 4D effective descriptions. The results illuminate how GS terms can evade some no-go constraints on de Sitter vacua in higher-dimensional theories and guide future explorations of anomaly-free diagonal gaugings and quantum corrections in such setups.

Abstract

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, we observe that the theory admits half-supersymmetric Minkowski$_4\times S^2$ and non-supersymmetric dS$_4 \times S^2$ as exact solutions with a monopole on $S^2$. The monopole charge and the anomaly coefficients play key roles in the vacuum structure and it turns out that 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, two scalars violate the Higuchi unitarity bound, and they represent two tachyonic states, one from the dilaton and the other from the volume of the internal space. We show that turning on small perturbations of the tachyonic modes around the dS$_4\times S^2$ triggers a flow evolving towards Mink$_4\times S^2$ with the minimal potential energy. We also find a nonsupersymmetric AdS$_4\times S^2$ solution, when we turn on a flux associated with external $U(1)$ gauge field only, and show that the phenomenon of scale separation is realized.

Figures (5)

• Figure 1: The phase-space diagram of ($\varphi,\varphi'$), the initial point is $(10^{-4},0)$. Here, $\tau\in(0,150)$.
• Figure 2: The phase-space diagram of ($B,B'$), the initial point is $(\frac{1}{2}\ln\frac{3}{2}-2\times10^{-3},0)$.
• Figure 3: $\varphi$ and $B$ approach constant quickly after several damping oscillations, and $(2e^{2B}-\cosh{\varphi})$ approaches zero.
• Figure 4: Scalar potential in 4D. The local maximum corresponds to the de Sitter, and the $ve^{\varphi}+\Tilde{v} e^{\sigma}=2$ valley to the Minkowski vacua.
• Figure 5: The scalar potential in 4D for $\Tilde{v}=0$. The valley at $ve^{\varphi}=2$ denotes the Minkowski vacuum.