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Mirror Symmetry for Two Parameter Models -- II

Philip Candelas, Anamaria Font, Sheldon Katz, David R. Morrison

TL;DR

This work extends mirror symmetry to a two-parameter Calabi–Yau family, precisely describing the two Kähler moduli of IP4^(1,1,1,6,9)[18] by analyzing the mirror’s complex structure space. It constructs a symplectic period basis, determines the $Sp(6,\mathbb{Z})$ action, and derives the explicit mirror map to compute Yukawa couplings and the generalized $N=2$ index, yielding genus-zero and genus-one instanton data. The paper also performs a thorough monodromy analysis around discriminant components, identifies the large complex structure limit, and unveils an $SL(2,\mathbb{Z})$ symmetry acting on a boundary component, with a Zariski–van Kampen-based description of the duality group generated by $\mathbf{A}$ and $\mathbf{T}$. Moreover, it reports nontrivial instanton numbers, including negative genus-zero contributions from degenerate families, and provides a geometric interpretation via families of singular rational curves on the resolved manifold. Overall, the results advance explicit multi-parameter mirror maps, period computations, and instanton enumerations in a nontrivial two-parameter setting, with implications for higher-dimensional moduli and duality structures.

Abstract

We describe in detail the space of the two Kähler parameters of the Calabi--Yau manifold $¶_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.

Mirror Symmetry for Two Parameter Models -- II

TL;DR

This work extends mirror symmetry to a two-parameter Calabi–Yau family, precisely describing the two Kähler moduli of IP4^(1,1,1,6,9)[18] by analyzing the mirror’s complex structure space. It constructs a symplectic period basis, determines the action, and derives the explicit mirror map to compute Yukawa couplings and the generalized index, yielding genus-zero and genus-one instanton data. The paper also performs a thorough monodromy analysis around discriminant components, identifies the large complex structure limit, and unveils an symmetry acting on a boundary component, with a Zariski–van Kampen-based description of the duality group generated by and . Moreover, it reports nontrivial instanton numbers, including negative genus-zero contributions from degenerate families, and provides a geometric interpretation via families of singular rational curves on the resolved manifold. Overall, the results advance explicit multi-parameter mirror maps, period computations, and instanton enumerations in a nontrivial two-parameter setting, with implications for higher-dimensional moduli and duality structures.

Abstract

We describe in detail the space of the two Kähler parameters of the Calabi--Yau manifold by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an symmetry that acts on a boundary of the moduli space.

Paper Structure

This paper contains 19 sections, 114 equations, 6 figures.

Table of Contents

  1. intro
  2. model
  3. Linear systems
  4. Chern classes
  5. moduli1
  6. Basic facts
  7. Considerations of toric geometry
  8. The resolved moduli space
  9. vps
  10. The fundamental period
  11. The function $U_\nu$
  12. Expansions for large $\psi$
  13. monodromy
  14. Generalities
  15. Monodromy about $\psi=0$ and $\psi=\infty$
  16. ...and 4 more sections

Figures (6)

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  • ...and 1 more figures