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

Marco Billo, Frederik Denef, Pietro Fre, Igor Pesando, Walter Troost, Antoine Van Proeyen, Daniela Zanon

TL;DR

The paper tackles how local special Kähler moduli spaces of Calabi–Yau manifolds approach rigid special geometry near chosen singularities, using two explicit K3-fibered CY examples. It develops and applies methods to compute all CY and K3 periods, monodromies, and intersection matrices, showing that in the rigid limit the local Kahler potential reduces to the rigid one and that CY periods organize into Seiberg–Witten–type data via an auxiliary Riemann surface. By exploiting K3-fibration structures, the authors obtain exact period expressions and demonstrate the appearance of SW curves (SU(2) and SU(3) cases) as branches of a higher-genus base surface. The work provides concrete routes to extract rigid-limit actions from full supergravity on CY manifolds, connecting geometric period data to low-energy gauge dynamics and suggesting broader applications in exact supergravity expansions, with ties to M-theory brane pictures. Overall, the study advances exact techniques for rigid limits in string compactifications and clarifies the interplay between local and rigid special geometry through detailed, computable examples.

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.

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

TL;DR

The paper tackles how local special Kähler moduli spaces of Calabi–Yau manifolds approach rigid special geometry near chosen singularities, using two explicit K3-fibered CY examples. It develops and applies methods to compute all CY and K3 periods, monodromies, and intersection matrices, showing that in the rigid limit the local Kahler potential reduces to the rigid one and that CY periods organize into Seiberg–Witten–type data via an auxiliary Riemann surface. By exploiting K3-fibration structures, the authors obtain exact period expressions and demonstrate the appearance of SW curves (SU(2) and SU(3) cases) as branches of a higher-genus base surface. The work provides concrete routes to extract rigid-limit actions from full supergravity on CY manifolds, connecting geometric period data to low-energy gauge dynamics and suggesting broader applications in exact supergravity expansions, with ties to M-theory brane pictures. Overall, the study advances exact techniques for rigid limits in string compactifications and clarifies the interplay between local and rigid special geometry through detailed, computable examples.

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.

Paper Structure

This paper contains 83 sections, 284 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Ingredients
  3. Special geometry
  4. Special Kähler geometry in moduli spaces
  5. The rigid limit
  6. Monodromies and the Picard--Lefshetz formula
  7. $K3$ fibrations, singularities and periods
  8. Description of the CY and $K3$ in weighted projective spaces
  9. $K3$ and its Picard and transcendental lattices
  10. A class of $K3$ fibrations
  11. Using the $K3$ fibration for the periods
  12. Strategies for obtaining $K3$ periods
  13. Introducing the examples: fibrations and singularities
  14. Description and fibration of $X^*_{8}[1,1,2,2,2]$
  15. Description and fibrations of $X^*_{24}[1,1,2,8,12]$
  16. ...and 68 more sections

Figures (8)

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  • ...and 3 more figures