PICARD-FUCHS EQUATIONS AND SPECIAL GEOMETRY

Abstract

We investigate the system of holomorphic differential identities implied by special Kählerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on Calabi-Yau 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 nonvanishing of w4. 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.