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Open/closed Correspondence and Extended LG/CY Correspondence for Quintic Threefolds

Konstantin Aleshkin, Chiu-Chu Melissa Liu

TL;DR

The work builds a unified GLSM framework that ties Walcher’s open disk potential for the quintic Calabi–Yau threefold to a central charge of an extended quintic GLSM, thereby realizing an open/closed correspondence in genus-zero GLSM invariants. It develops both CY and LG sectors, including GW/FJRW theories, CY/LG mirror theorems, open mirror symmetry, and extended Picard–Fuchs equations, then shows how wall-crossing between phases encodes the open/closed data. The authors introduce an extended GLSM (with additional fields) to capture the real quintic and reveal a precise open/closed correspondence between disk invariants and closed GLSM periods, plus explicit Mellin–Barnes representations via the Higgs–Coulomb correspondence. On the B-model side, they formulate an extended open/closed B-model and demonstrate how the extended periods reproduce the disk potentials under mirror symmetry, thereby providing a coherent open/closed and LG/CY picture across both A- and B-models and across CY and LG phases.

Abstract

We show that Walcher's disk potential for the quintic threefold can be represented as a central charge of a specific Gauged Linear Sigma Model which we call the extended quintic GLSM. This representation provides an open/closed correspondence for the quintic threefold since the central charge is a generating function of closed genus-zero GLSM invariants. We also explain how open Landau-Ginzburg/Calabi-Yau correspondence and open mirror symmetry for the quintic are compatible with wall-crossing and mirror symmetry of the extended GLSM, respectively.

Open/closed Correspondence and Extended LG/CY Correspondence for Quintic Threefolds

TL;DR

The work builds a unified GLSM framework that ties Walcher’s open disk potential for the quintic Calabi–Yau threefold to a central charge of an extended quintic GLSM, thereby realizing an open/closed correspondence in genus-zero GLSM invariants. It develops both CY and LG sectors, including GW/FJRW theories, CY/LG mirror theorems, open mirror symmetry, and extended Picard–Fuchs equations, then shows how wall-crossing between phases encodes the open/closed data. The authors introduce an extended GLSM (with additional fields) to capture the real quintic and reveal a precise open/closed correspondence between disk invariants and closed GLSM periods, plus explicit Mellin–Barnes representations via the Higgs–Coulomb correspondence. On the B-model side, they formulate an extended open/closed B-model and demonstrate how the extended periods reproduce the disk potentials under mirror symmetry, thereby providing a coherent open/closed and LG/CY picture across both A- and B-models and across CY and LG phases.

Abstract

We show that Walcher's disk potential for the quintic threefold can be represented as a central charge of a specific Gauged Linear Sigma Model which we call the extended quintic GLSM. This representation provides an open/closed correspondence for the quintic threefold since the central charge is a generating function of closed genus-zero GLSM invariants. We also explain how open Landau-Ginzburg/Calabi-Yau correspondence and open mirror symmetry for the quintic are compatible with wall-crossing and mirror symmetry of the extended GLSM, respectively.
Paper Structure (43 sections, 9 theorems, 241 equations)

This paper contains 43 sections, 9 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. LG/CY correspondence and mirror symmetry
  3. LG/CY correspondence
  4. LG/CY correspondence for quintic threefolds
  5. GW invariants
  6. The CY mirror theorem
  7. FJRW invariants
  8. The LG mirror theorem
  9. Picard-Fuchs equation, analytic continuation, and symplectic transform
  10. Open mirror symmetry
  11. The CY open mirror theorem
  12. The inhomogeneous and extended Picard-Fuchs equations
  13. The LG open mirror conjecture
  14. Gauged Linear Sigma Models
  15. Input data
  16. ...and 28 more sections

Key Result

Theorem 2.1

Theorems & Definitions (18)

  • Theorem 2.1: mirror theorem for the CY threefold $X_5$
  • Theorem 2.2: mirror theorem for the LG model $(W_5,\mu_5)$
  • Theorem 2.3: Open mirror theorem for $(X_5, \mathbb{R} X_5)$
  • Conjecture 2.4: open mirror conjecture for $(W_5, \mu_5)$
  • Definition 3.1: GLSM quasimap central charges of matrix factorizations
  • Remark 3.2
  • Definition 3.3: non-equivariant quasimap central charges
  • Remark 3.4
  • Proposition 3.5: comparison of quasimap central charges
  • Lemma 3.6
  • ...and 8 more