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 \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.

43 pages

References in corpus (7)

• Topological Field Theory and Rational Curves
• Special geometry, cubic polynomials and homogeneous quaternionic spaces
• Picard-Fuchs equations and mirror maps for hypersurfaces
• Periods and Duality Symmetries in Calabi-Yau Compactifications
• Picard-Fuchs equations and mirror maps for hypersurfaces
• Flat Holomorphic Connections and Picard-Fuchs Identities From $N=2$ Supergravity
• On the Moduli Space of the $T^6/Z_3$ Orbifold and Its Modular Group

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