Complete classification of Minkowski vacua in generalised flux models

TL;DR

This work delivers a complete analytic and numerical classification of Minkowski vacua in ${\cal N}=1$, ${\rm D}=4$ supergravity from type II orientifolds with generalized fluxes. By leveraging a flux-algebra taxonomy and a careful mapping to type IIA duals, it identifies one fully stable Minkowski/dS combination in the semisimple ${\\frak{so(3,1)}^2}$ sector, while most other models yield Minkowski extrema with tachyons, all of which sit at singular loci in moduli space. The analysis shows that totally stable Minkowski and dS vacua require non-geometric fluxes and that the IIA energy partition clarifies how various flux components and localized sources contribute to stabilization. The results refine the landscape of flux vacua, link IIB and IIA perspectives, and point to the role of non-geometric fluxes in achieving fully stable, SUSY-broken vacua with phenomenological relevance.

Abstract

We present a complete and systematic analysis of the Minkowski extrema of the N=1, D=4 Supergravity potential obtained from type II orientifold models that are T-duality invariant, in the presence of generalised fluxes. Based on our previous work on algebras spanned by fluxes, and the so-called no-go theorems on the existence of Minkowski and/or de Sitter vacua, we perform a partly analytic, partly numerical analysis of the promising cases previously hinted. We find that the models contain Minkowski extrema with one tachyonic direction. Moreover, those models defined by the Supergravity algebra so(3,1)^2 also contain Minkowski/de Sitter minima that are totally stable. All Minkowski solutions, stable or not, interpolate between points in parameter space where one or several of the moduli go to either zero or infinity, the so-called singular points. We finally reinterpret our results in the language of type IIA flux models, in order to show explicitly the contribution of the different sources of potential energy to the extrema found. In particular, the cases of totally stable Minkowski/de Sitter vacua require of the presence of non-geometric fluxes.

Figures (11)

• Figure 1: Left: location of the Mkw solutions within the parameter space for the Supergravity models based on the $\,\mathfrak{iso(3)}\,$$B$-field reduction, highlighting the singular points. Right: the set of VEVs of the modulus $\,{\mathcal{Z}}\,$, reflecting its scaling nature. The points A and A' correspond to a singular limit $\,|{\mathcal{Z}}_{0}| \rightarrow \infty$.
• Figure 2: Left: location of the Mkw solutions within the parameter space for the Supergravity models based on the $\,\mathfrak{su(2) + u(1)^{3}}\,$$B$-field reduction, highlighting the singular points. Right: set of VEVs of the modulus $\,{\mathcal{Z}}\,$, reflecting its scaling nature. Again, the points A and A' are singular since $\,|{\mathcal{Z}}_{0}|\rightarrow \infty$.
• Figure 3: Left: location of the Mkw solutions within the parameter space for the Supergravity models based on the $\,\mathfrak{so(4)}\,$$B$-field reduction, highlighting the singular points. Right: set of VEVs of the modulus $\,{\mathcal{Z}}\,$. Note that, up to discrete transformations, the Mkw extrema describe closed curves in both plots.
• Figure 4: Left: location of the Mkw solutions within the parameter space for the Supergravity models based on the $\,\mathfrak{so(3,1)}\,$$B$-field reduction, highlighting the singular points. Right: set of VEVs of the modulus $\,{\mathcal{Z}}\,$. Notice that, up to discrete transformations, the Mkw extrema describe closed curves in both plots.
• Figure 5: This figure shows the narrow band above the line of stable Mkw vacua containing stable, dS vacua.