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Extending the Picard-Fuchs system of local mirror symmetry

Brian Forbes, Masao Jinzenji

TL;DR

This work introduces extended Picard–Fuchs operators to complete local mirror symmetry for noncompact Calabi–Yau threefolds, especially when no 4-cycles exist. By coupling a Gauss–Manin framework with a conjectured intersection theory, the authors derive PF extensions that recover the full prepotential and Yukawa data, including open-string disc invariants, across multiple toric examples (local P^1, toric trees, Hirzebruch surfaces, and del Pezzo surfaces). They show that ordinary PF systems can be incomplete and that extending PF operators (often via powers of the Euler operator) and incorporating additional period data from associated Riemann surfaces yields a coherent, largely canonical description. Open-string scenarios require analogous PF extensions to account for disc counting, suggesting a unified extension principle for both closed and open local mirror symmetry. The results provide concrete extended PF systems for X1,X2, K_{F_n}, and K_{dP_2}, linking discriminants, Yukawa couplings, and intersection-theory data, and they highlight areas where a deeper geometric interpretation remains to be developed.

Abstract

We propose an extended set of differential operators for local mirror symmetry. If $X$ is Calabi-Yau such that $\dim H_4(X,\Z)=0$, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such $X$ is uncovered. We also find new operators on several examples of type $X=K_S$ through similar techniques. In addition, open string PF systems are considered.

Extending the Picard-Fuchs system of local mirror symmetry

TL;DR

This work introduces extended Picard–Fuchs operators to complete local mirror symmetry for noncompact Calabi–Yau threefolds, especially when no 4-cycles exist. By coupling a Gauss–Manin framework with a conjectured intersection theory, the authors derive PF extensions that recover the full prepotential and Yukawa data, including open-string disc invariants, across multiple toric examples (local P^1, toric trees, Hirzebruch surfaces, and del Pezzo surfaces). They show that ordinary PF systems can be incomplete and that extending PF operators (often via powers of the Euler operator) and incorporating additional period data from associated Riemann surfaces yields a coherent, largely canonical description. Open-string scenarios require analogous PF extensions to account for disc counting, suggesting a unified extension principle for both closed and open local mirror symmetry. The results provide concrete extended PF systems for X1,X2, K_{F_n}, and K_{dP_2}, linking discriminants, Yukawa couplings, and intersection-theory data, and they highlight areas where a deeper geometric interpretation remains to be developed.

Abstract

We propose an extended set of differential operators for local mirror symmetry. If is Calabi-Yau such that , then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such is uncovered. We also find new operators on several examples of type through similar techniques. In addition, open string PF systems are considered.

Paper Structure

This paper contains 24 sections, 4 theorems, 260 equations, 7 figures.

Table of Contents

  1. Introduction.
  2. The Main Strategy: Overview of the Gauss-Manin System.
  3. Mirror symmetry for local $\mathbb{P}^1$.
  4. A Picard-Fuchs operator for local $\mathbb{P}^1$.
  5. PF extensions and Riemann surfaces.
  6. Geometric interpretation through the Riemann surface.
  7. Period integrals for local $\mathbb{P}^1$.
  8. A period integral from $(H_1(\Sigma,\mathbb{Z}))_c.$
  9. Discussion.
  10. Mirror Symmetry for Toric Trees.
  11. Ordinary Picard-Fuchs systems.
  12. Two building blocks of solutions.
  13. Mirror Symmetry when $\dim H_4(X,\mathbb{Z})=0$.
  14. Extended Picard-Fuchs System for $X_{1}$ and $X_{2}$
  15. Adding open strings to $\mathcal{O}(-1)\oplus \mathcal{O}(-1)\rightarrow \mathbb{P}^1$.
  16. ...and 9 more sections

Key Result

Proposition 1

The GKZ operators associated to the geometry $\hat{X}$ are the same as those associated to $\Sigma$.

Figures (7)

  • Figure 1: 1-cycles on $\Sigma$. $\{\gamma_0, \tau_+, \tau_-\}$ is a basis for $H_1(\Sigma, \mathbb{Z})$.
  • Figure 2: Toric diagram for $X_1$.
  • Figure 3: Toric diagram for $X_2$.
  • Figure 4: $X$ containing a string of curves.
  • Figure 5: Real curves defining open string periods on $\Sigma$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Conjecture 1
  • Proposition 1
  • Definition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Example 1
  • Example 2
  • Conjecture 2
  • Definition 2
  • ...and 1 more