Locally stable non-supersymmetric Minkowski vacua in supergravity

TL;DR

The paper addresses the problem of finding locally stable, non-supersymmetric Minkowski vacua in supergravity with only chiral multiplets by deriving a simple necessary bound on the Kähler geometry and the Goldstino direction: $R_{i \bar{j} p \bar{q}} G^i G^{\bar{j}} G^p G^{\bar{q}} < 6$, together with the Minkowski condition $G^k G_k = 3$. In the separable Kähler case, this bound reduces to $\sum_i R_i (\frac{G_i G_{\bar{i}}}{G_{i\bar{i}}})^2 < 6$, allowing an explicit Goldstino-angle parametrization and leading to exact curvature constraints in one- and two-field models. For one field, stability requires $R_X < 2/3$; for two fields, $\cos^4\theta R_X + \sin^4\theta R_Y < 2/3$, with a derived curvature-sum condition $R_X^{-1} + R_Y^{-1} > 3/2$, and generalizes to $n$ fields as $\sum_i \Theta_i^4 R_i < 2/3$ under $\sum_i \Theta_i^2 = 1$. The results have implications for moduli stabilization in string compactifications, constraints on SUSY-breaking soft terms, and the viability of uplifting AdS vacua to Minkowski/dS vacua, with extensions to warped geometries and uplift scenarios discussed. Overall, the work provides practical, geometry-driven criteria to assess and construct non-supersymmetric Minkowski vacua in supergravity.

Abstract

We perform a general study about the existence of non-supersymmetric minima with vanishing cosmological constant in supergravity models involving only chiral superfields. We study the conditions under which the matrix of second derivatives of the scalar potential is positive definite. We show that there exist very simple and strong necessary conditions for stability that constrain the Kahler curvature and the ratios of the supersymmetry-breaking auxiliary fields defining the Goldstino direction. We then derive more explicitly the implications of these constraints in the case where the Kahler potential for the supersymmetry-breaking fields is separable into a sum of terms for each of the fields. We also discuss the implications of our general results on the dynamics of moduli fields arising in string compactifications and on the relative sizes of their auxiliary fields, which are relevant for the soft terms of matter fields. We finally comment on how the idea of uplifting a supersymmetric AdS vacuum fits into our general study.