The rigid limit in Special Kahler geometry; From K3-fibrations to Special Riemann surfaces: a detailed case study

TL;DR

This paper investigates the rigid limit of special Kahler manifolds associated with Calabi-Yau moduli spaces, focusing on period calculations near singularities using K3 fibrations and Riemann surfaces, advancing methods for exact supergravity results.

Contribution

It provides a detailed analysis of periods in the rigid limit, utilizing K3 fibrations and Riemann surfaces to develop new computational methods for Calabi-Yau manifolds.

Findings

Periods identified as meromorphic forms on Riemann surfaces

Kahler potential reduces to that of a rigid special Kahler manifold

Methods developed enable exact supergravity calculations

Abstract

The limiting procedure of special Kahler manifolds to their rigid limit is studied for moduli spaces of Calabi-Yau manifolds in the neighbourhood of certain singularities. In two examples we consider all the periods in and around the rigid limit, identifying the nontrivial ones in the limit as periods of a meromorphic form on the relevant Riemann surfaces. We show how the Kahler potential of the special Kahler manifold reduces to that of a rigid special Kahler manifold. We extensively make use of the structure of these Calabi-Yau manifolds as K3 fibrations, which is useful to obtain the periods even before the K3 degenerates to an ALE manifold in the limit. We study various methods to calculate the periods and their properties. The development of these methods is an important step to obtain exact results from supergravity on Calabi-Yau manifolds.