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 phenomenologically admissible potentials for the description of
inflation and in particular alpha-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
alpha-attractor, so that neither type of isometry can be a priori excluded. The
requirement of regularity of the manifold Sigma poses instead 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.
