Special geometry of Calabi-Yau compactifications near a rigid limit

TL;DR

This paper explores the special geometry of Calabi-Yau compactifications near a rigid limit, illustrating how gravity decouples and how Seiberg-Witten curves emerge as degenerations within a higher genus surface, providing insights into gravitational corrections.

Contribution

It explicitly demonstrates the decoupling of gravity and the transition from local to rigid special Kahler geometry without noncompact approximations, linking to Seiberg-Witten theory.

Findings

Decoupling of gravity in Calabi-Yau compactifications

Derivation of Seiberg-Witten curves as degenerations of higher genus surfaces

Calculation of gravitational corrections to Seiberg-Witten central charge

Abstract

We discuss, in the framework of special Kahler geometry, some aspects of the "rigid limit" of type IIB string theory compactified on a Calabi-Yau threefold. We outline the general idea and demonstrate by direct analysis of a specific example how this limit is obtained. The decoupling of gravity and the reduction of special Kahler geometry from local to rigid is demonstrated explicitly, without first going to a noncompact approximation of the Calabi-Yau. In doing so, we obtain the Seiberg-Witten Riemann surfaces corresponding to different rigid limits as degenerating branches of a higher genus Riemann surface, defined for all values of the moduli. Apart from giving a nice geometrical picture, this allows one to calculate easily some gravitational corrections to e.g. the Seiberg-Witten central charge formula. We make some connections to the 2/5-brane picture, also away from the rigid limit, though only at the formal level.