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A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

Kwokwai Chan, Ziming Nikolas Ma, Hao Wen

Abstract

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following Li-Li-Saito, we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.

A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

Abstract

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following Li-Li-Saito, we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.
Paper Structure (16 sections, 44 theorems, 180 equations)

This paper contains 16 sections, 44 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. The log LG model mirror to a projective toric manifold
  3. Constructions
  4. The state space isomorphism
  5. A homotopy formula
  6. Integration and the higher residue pairing
  7. Grading structure and good basis
  8. Deformation theory of log LG models
  9. Unfolding
  10. Gauss-Manin connection and the higher residue pairing
  11. Logarithmic Frobenius manifold
  12. Log LG mirrors of semi-Fano toric manifolds
  13. The period map for log LG mirror of a semi-Fano toric manifold
  14. Explicit computations for the Hirzebruch surface $\mathbb{F}_2$
  15. Dolbeault resolution
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(X^\dag, \phi, f)$ be the logarithmic Landau-Ginzburg model mirror to a projective toric manifold $Y$. Then there is a perturbative construction of primitive forms which yields a logarithmic Frobenius manifold structure on the base space $\mathscr{S}$ of the universal unfolding of $(X^\dag, \ph

Theorems & Definitions (84)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Logarithmic Landau-Ginzburg model
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: State space isomorphism
  • proof
  • Theorem 2.5: Hodge-to-de Rham degeneration
  • Remark 2.6
  • Corollary 2.7
  • ...and 74 more