Axial Symmetric Kahler manifolds, the D-map of Inflaton Potentials and the Picard-Fuchs Equation

Abstract

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.

59 pages, LaTex, 18 eps figures, 2 tables; v2: misprints corrected, one reference added and some credit assessments reformulated

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Cited by in corpus (9)

• Integrable Scalar Cosmologies I. Foundations and links with String Theory
• Complexified Starobinsky Inflation in Supergravity in the Light of Recent BICEP2 Result
• On the Topology of the Inflaton Field in Minimal Supergravity Models
• On the Gauged Kahler Isometry in Minimal Supergravity Models of Inflation
• Integrable Scalar Cosmologies II. Can they fit into Gauged Extended Supergavity or be encoded in N=1 superpotentials?
• Generalized two-field $\alpha$-attractor models from geometrically finite hyperbolic surfaces
• Integrable Cosmological Potentials
• The $c$-map, Tits Satake subalgebras and the search for $\mathcal{N}=2$ inflaton potentials
• The Integral Form of D=3 Chern-Simons Theories Probing ${\mathbb{C}}^n/\Gamma$ Singularities