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.