Axial Symmetric Kahler manifolds, the D-map of Inflaton Potentials and
  the Picard-Fuchs Equation
  In this paper we provide a definition of the D-map, namely of the
mathematical construction implicitly utilized by supergravity that associates
an axial symmetric Kahler surface to every positive definite potential function
V(phi). The properties of the D-map are discussed in general. Then the D-map is
applied to the list of integrable cosmological potentials classified by us in a
previous publication with A. Sagnotti. Several interesting geometrical and
analytical properties of the manifolds in the image of this D-map are
discovered and illustrated. As a by-product of our analysis we demonstrate the
existence of (integrable) Starobinsky-like potentials that can be embedded into
supergravity. Some of them follow from constant curvature Kahler manifolds. In
the quest for a microscopic interpretation of inflaton dynamics we present the
Ariadne's thread provided by a new mathematical concept that we introduce under
the name of axial symmetric descendants of one dimensional special Kahler
manifolds. By means of this token we define a clearcut algorithm that to each
potential function V(phi) associates a unique 4th order Picard-Fuchs equation
of restricted type. Such an equation encodes information on the chiral ring of
a superconformal field theory to be sought for, unveiling in this way a
microscopic interpretation of the inflaton potential. We conjecture that the
physical mechanism at the basis of the transition from a special manifold to
its axial symmetric descendant is probably the construction of an Open String
descendant of a Closed String model.
