Picard-Fuchs Equations and Special Geometry

TL;DR

This paper explores the mathematical structure of special Kähler geometry in N=2 supergravity, linking it to Picard-Fuchs equations, instanton corrections, and topological field theory, revealing deep geometric and physical connections.

Contribution

It establishes a detailed relationship between special geometry, Picard-Fuchs equations, and topological field theory, including the role of instanton corrections in Yukawa couplings.

Findings

Picard-Fuchs equations characterize periods in special geometry.

Instanton corrections influence Yukawa couplings via non-vanishing w4.

Yukawa couplings relate to derivatives of a holomorphic function F.

Abstract

We investigate the system of holomorphic differential identities implied by special K\"ahlerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on \cy threefolds these identities are equivalent to the Picard-Fuchs equations of algebraic geometry that are obeyed by the periods of the holomorphic three-form. For one variable they reduce to linear fourth-order equations which are characterized by classical $W$-generators; we find that the instanton corrections to the Yukawa couplings are directly related to the non-vanishing of $w_4$. We also show that the symplectic structure of special geometry can be related to the fact that the Yukawa couplings can be written as triple derivatives of some holomorphic function $F$. Moreover, we give the precise relationship of the Yukawa couplings of special geometry with three-point functions in topological field theory.