On the Gauged Kahler Isometry in Minimal Supergravity Models of Inflation

In this paper we address the question how to discriminate whether the gauged isometry group G_Sigma of the Kahler manifold Sigma 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 Sigma.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 Kahler 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 all phenomenologically admissible potentials for the description of inflation and in particular alpha-attractors are mostly obtained from the gauging of a parabolic isometry. The requirement of regularity of the manifold Sigma poses strong constraints on the alpha-attractors and reduces their space considerably. Curiously there is a unique integrable alpha-attractor corresponding to a particular value of this parameter.