THE PROPER LANDAU–GINZBURG POTENTIAL, INTRINSIC MIRROR SYMMETRY AND THE RELATIVE MIRROR MAP - Generating function of relative Gromov–Witten invariants

Sep 10, 2026

Given a smooth log Calabi–Yau pair (X, D), intrinsic mirror symmetry defines a mirror proper Landau–Ginzburg potential and shows it acts as a generating function for two-point relative Gromov–Witten invariants of (X, D). Relative invariants are computed with several negative contact orders, then evaluated via the relative mirror theorem. When D is nef, the proper potential equals the inverse of the relative mirror map. For toric X, this yields the proper potential as the open mirror map; for Fano X, it relates to the anti-derivative of the regularized quantum period.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages45

Reading Time1 h 53 min

AccessFree to download

Introduction

Intrinsic mirror symmetry and theta functions

Relative mirror maps and computing genus zero relative invariants

The proper Landau–Ginzburg potential

Acknowledgement

Relative Gromov–Witten invariants with negative contact orders

General theory

A special case

The proper Landau–Ginzburg potential from intrinsic mirror symmetry

The Landau–Ginzburg potential

Structure constants

Theta functions

arXiv :2209 . 15371v3 [math .AG] 26 Mar 2024 THE PROPER LANDAU–GINZBURG POTENTIAL, INTRINSIC MIRROR SYMMETRY AND THE RELATIVE MIRROR MAP FENGLONG YOU Abstract. Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to de􀀌ne the mirror proper Landau– Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D) . We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of [FTY19] to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of [GRZ22] that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period. Contents 1. Introduction 2 1.1. Intrinsic mirror symmetry and theta functions 3 1.2. Relative mirror maps and computing genus zero relative invariants. 4 1.3. The proper Landau–Ginzburg potential 7 1.4. Acknowledgement 9 2. Relative Gromov–Witten invariants with negative contact orders 10 2.1. General theory 10 2.2. A special case 14 3. The proper Landau–Ginzburg potential from intrinsic mirror symmetry 15 3.1. The Landau–Ginzburg potential 15 3.2. Structure constants 16 3.3. Theta functions 19 4. A mirror theorem for smooth pairs 23 4.1. Relative mirror theorem 23 4.2. Relative invariants with several negative contact orders 26 4.3. Degree zero relative invariants 31 4.4. Relative mirror map 35 Date: March 27, 2024 . 2 FENGLONG YOU 5. Theta function computation via relative mirror theorem 35 5.1. Extracting the coeﬃcient of the J-function 36 5.2. Extracting the coeﬃcient of the I-function 38 5.3. Matching 39 6. Toric varieties and the open mirror map 40 6.1. Toric Fano varieties 41 6.2. Toric varieties with a smooth, nef anticanonical divisor 41 6.3. Beyond the toric case 42 7. Fano varieties and quantum periods 42 References 44 1. Introduction Mirror symmetry is originally stated as a duality between Calabi–Yau manifolds. Mirror symmetry predicts that the symplectic geometry (or the complex geometry) of a Calabi–Yau manifold is equivalent to the complex geometry (or the symplectic geometry, respectively) of the mirror Calabi– Yau manifold. The mirror duality has been generalized to Fano varieties, more generally, to log Calabi–Yau pairs. The mirror of a smooth log Calabi– Yau pair (X, D) is a Landau–Ginzburg model, which is a variety X∨ with a proper map, called the superpotential, W ∶ X∨ → C. One further expects that the generic ﬁber of the superpotential W is mirror to a smooth anticanonical divisor D of X. We call the mirror duality between a smooth log Calabi–Yau pair (X, D) and a proper Landau–Ginzburg model relative mirror symmetry. To construct the mirror of the smooth log Calabi–Yau pair (X, D), one needs to construct the variety X∨ and the proper Landau– Ginzburg potential W. The variety X∨ is considered as the mirror of the complement X ∖ D. A general construction of the variety X∨ is through intrinsic mirror symmetry [GS19] in the Gross–Siebert program. One considers a maximally unipotent degeneration g ∶ Y → S, where S is an aﬃne curve, of the pair (X, D) . The mirror is constructed as the projective spectrum of the degree zero part of the relative quantum cohomology QH0log (Y, D′) of (Y, D′), where D′ is a certain divisor that contains g−1(0) . It remains to compute the proper Landau–Ginzburg potential W. Following the Gross–Siebert program, the Landau–Ginzburg potentials are given by the theta functions. The theta functions are usually diﬃcult to compute. Recently, [GRZ22] computed the proper Landau–Ginzburg potentials for toric del Pezzo surfaces. They considered a toric degeneration of the

FAQs

What does the paper prove about the proper Landau–Ginzburg potential for a smooth log Calabi–Yau pair (X, D)?

It constructs the proper Landau–Ginzburg potential using intrinsic mirror symmetry and shows it is a generating function of two-point relative Gromov–Witten invariants of (X, D).

How are relative invariants computed when several negative contact orders are involved?

The paper computes certain relative invariants with several negative contact orders and then applies the relative mirror theorem of FTY19 to compute two-point relative invariants.

What special relationships are obtained under extra geometric assumptions on D or on the variety X?

If D is nef, the proper potential is the inverse of the relative mirror map. For toric X, the proper potential becomes the open mirror map; for Fano X, it connects to the anti-derivative of the regularized quantum period.

Sep 10, 2026

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