On the gauged Kähler isometry in minimal supergravity models of inflation

In this paper we address the question how to discriminate whether the gauged isometry group 𝒢 Σ of the Kähler manifold Σ that produces a D‐type inflaton potential in a Minimal Supergravity Model is elliptic, hyperbolic or parabolic. We show that the classification of isometries of symmetric cosets can be extended to non symmetric Σ.s if these manifolds satisfy additional mathematical restrictions. The classification criteria established in the mathematical literature are coherent with simple criteria formulated in terms of the asymptotic behavior of the Kähler potential 𝒦(C) = 2 J(C) where the real scalar field C encodes the inflaton field. As a by product of our analysis we show that phenomenologically admissible potentials for the description of inflation and in particular α‐attractors are mostly obtained from the gauging of a parabolic isometry, this being, in particular the case of the Starobinsky model. Yet at least one exception exists of an elliptic α‐attractor, so that neither type of isometry can be a priori excluded. The requirement of regularity of the manifold Σ poses instead strong constraints on the α‐attractors and reduces their space considerably. Curiously there is a unique integrable α‐attractor corresponding to a particular value of this parameter.