Metastable de Sitter vacua in N = 2 to N = 1 truncated supergravity

We study the possibility of achieving metastable de Sitter vacua in general N=2toN=1truncatedsupergravitieswithoutvectormultiplets,andcomparewiththe situations arising in N = 2 theories with only hypermultiplets and N = 1 theories with only chiral multiplets. In N = 2 theories based on a quaternionic manifold and a graviphoton gauging, de Sitter vacua are necessarily unstable, as a result of the peculiar properties of the geometry. In N = 1 theories based on a Kähler manifold and a superpotential, de Sitter vacua can instead be metastable provided the geometry satisfies some constraint and the superpotential can be freely adjusted. In N = 2 to N = 1 truncations, the crucial requirement is then that the tachyon of the mother theory be projected out from the daughter theory, so that the original unstable vacuum is projected to a metastable vacuum. We study the circumstances under which this may happen and derive general constraints for metastability on the geometry and the gauging. We then study in full detail the simplest case of quaternionic manifolds of dimension four with at least one isometry, for which there exists a general parametrization, and study two types of truncations defining Kähler submanifolds of dimension two. As an application, we finally discuss the case of the universal hypermultiplet of N = 2 superstrings and its truncations to the dilaton chiral multiplet of N = 1 superstrings. We argue that de Sitter vacua in such theories are necessarily unstable in weakly coupled situations, while they can in principle be metastable in strongly coupled regimes.