Flat Holomorphic Connections and Picard-Fuchs Identities From $N=2$ Supergravity

TL;DR

This paper demonstrates that in $N=2$ supergravity, the gauge-variant part of the connection is holomorphic and flat, leading to Picard-Fuchs identities related to Calabi-Yau periods, revealing new geometric and differential equation structures.

Contribution

It establishes the holomorphic and flat nature of the gauge-variant connection in special Kähler geometry and links Picard-Fuchs identities to Calabi-Yau period differential equations.

Findings

Gauge-variant connection is holomorphic and flat.

Derivation of Picard-Fuchs identities in this context.

Connection to Calabi-Yau three-form period equations.

Abstract

We show that in special K\"ahler geometry of $N=2$ space-time supergravity the gauge variant part of the connection is holomorphic and flat (in a Riemannian sense). A set of differential identities (Picard-Fuchs identities) are satisfied on a holomorphic bundle. The relationship with the differential equations obeyed by the periods of the holomorphic three form of Calabi-Yau manifolds is outlined.