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Twisted Sectors in Calabi-Yau Type Fermat Polynomial Singularities and Automorphic Forms

Dingxin Zhang, Jie Zhou

TL;DR

The paper reveals a deep link between twisted sectors in vanishing cohomology of one-parameter Fermat CY-type singularities and automorphic forms on triangular groups, enabling automorphic interpretations of genus-zero GW invariants for corresponding Fermat CY varieties. It develops a unified D-module and mixed Hodge theoretic framework for geometric sections, showing they become components of automorphic forms under Schwarzian uniformization, and demonstrates this explicitly in Fermat cubic, quartic, and quintic examples. The results connect LG/CY correspondence, period integrals, and mirror symmetry to automorphic structures, and establish that the genus-zero GW I-function is an automorphic form component valued in cohomology for appropriate groups, with special cases yielding elliptic modular forms. A notable contribution is the Yamaguchi–Yau ring for the quintic, shown to be a differential ring of automorphic generators with algebraic independence, which provides new tools for understanding higher-genus GW theory and its modular properties.

Abstract

We study one-parameter deformations of Calabi-Yau type Fermat polynomial singularities along degree-one directions. We show that twisted sectors in the vanishing cohomology are components of automorphic forms for certain triangular groups. We prove consequentially that genus zero Gromov-Witten generating series of the corresponding Fermat Calabi-Yau varieties are components of automorphic forms. The main tools we use are mixed Hodge structures for quasi-homogeneous polynomial singularities, Riemann-Hilbert correspondence, and genus zero mirror symmetry.

Twisted Sectors in Calabi-Yau Type Fermat Polynomial Singularities and Automorphic Forms

TL;DR

The paper reveals a deep link between twisted sectors in vanishing cohomology of one-parameter Fermat CY-type singularities and automorphic forms on triangular groups, enabling automorphic interpretations of genus-zero GW invariants for corresponding Fermat CY varieties. It develops a unified D-module and mixed Hodge theoretic framework for geometric sections, showing they become components of automorphic forms under Schwarzian uniformization, and demonstrates this explicitly in Fermat cubic, quartic, and quintic examples. The results connect LG/CY correspondence, period integrals, and mirror symmetry to automorphic structures, and establish that the genus-zero GW I-function is an automorphic form component valued in cohomology for appropriate groups, with special cases yielding elliptic modular forms. A notable contribution is the Yamaguchi–Yau ring for the quintic, shown to be a differential ring of automorphic generators with algebraic independence, which provides new tools for understanding higher-genus GW theory and its modular properties.

Abstract

We study one-parameter deformations of Calabi-Yau type Fermat polynomial singularities along degree-one directions. We show that twisted sectors in the vanishing cohomology are components of automorphic forms for certain triangular groups. We prove consequentially that genus zero Gromov-Witten generating series of the corresponding Fermat Calabi-Yau varieties are components of automorphic forms. The main tools we use are mixed Hodge structures for quasi-homogeneous polynomial singularities, Riemann-Hilbert correspondence, and genus zero mirror symmetry.
Paper Structure (25 sections, 9 theorems, 176 equations, 2 figures, 7 tables)

This paper contains 25 sections, 9 theorems, 176 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Cohomology bundles and automorphic forms
  3. Twisted sectors for quasi-homogeneous singularities
  4. Mixed Hodge structure on vanishing cohomology
  5. Twisted sectors in vanishing cohomology
  6. Ordinary differential equations for geometric sections
  7. Real structure and oscillating integrals
  8. Automorphic bundles corresponding to D-modules
  9. Examples
  10. Fermat cubic
  11. $D$-modules for geometric sections
  12. Schwarzian uniformization and elliptic modular forms
  13. Real structure
  14. Fermat quartic
  15. Automorphic bundles
  16. ...and 10 more sections

Key Result

Theorem 1.3

Consider the Dwork family of polynomial singularities eqnDworksingularity. For each $\bold{m}$, there exists a triple $\bold{\ell}(\bold{m})=(\ell_{0}(\bold{m})=n+1, \ell_{1}(\bold{m}),\ell_{\infty}(\bold{m}))$, such that the corresponding flat bundle $D_{\bold{m}}$ over $\mathcal{M}$ is an automorp

Figures (2)

  • Figure 1: Automorphic bundles arise in both A-model and B-model. Identification of the corresponding D-modules arising in the two models provides a mirror map.
  • Figure 2: A deformation of singularity.

Theorems & Definitions (27)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['thmautomorphicform']}
  • Theorem 1.4: Theorem \ref{['thmgenuszeroGWautomorphicform']}
  • Theorem 1.5: Proposition \ref{['propYYdifferentialring']}, Theorem \ref{['thmalgebraicindependence']}
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 17 more