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Mirror symmetry for singular double cover Calabi--Yau varieties: quantum test

Tsung-Ju Lee, Bong H. Lian, Shing-Tung Yau

TL;DR

The paper extends mirror symmetry to singular Calabi--Yau double covers arising from nef-partitions on semi-Fano toric manifolds, formulating a quantum mirror test between the $A$-model and the $B$-model in this singular setting. Using gauge-fixed double covers and Batyrev--Borisov duality, it constructs mirror pairs $(Y,Y^ abla)$ and analyzes their periods via GKZ $A$-hypergeometric systems, matching them to untwisted genus-zero orbifold Gromov--Witten invariants through cohomology-valued $B$-series. For two explicit branching configurations on $P^3$, the authors prove a mirror theorem: the $A$-model correlators of $Y$ coincide with the $B$-model correlators of $Y^ abla$ after a mirror map, with Gamma-factor effects playing a crucial role in normalization and instanton counts (e.g., $n_1=64$ for the first case). Morrison's conjecture is also tested in this framework, showing extremal transitions are reversed under the singular mirror correspondence and linking through dual polytopes. Overall, the work provides concrete computational evidence and a robust toolkit for quantum mirror symmetry in singular CYs within toric and GLSM-inspired contexts, enriching the dialogue between topological and geometric aspects of mirror symmetry.

Abstract

We continue our study on the pairs of singular Calabi--Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of \(\mathbb{P}^{3}\) branched along (1) a union of eight hyperplanes in general position, and (2) a union of four hyperplanes and a quartic in generation. Our previous construction produces hypothetical singular mirror partners. We prove that they are mirror pairs in the sense that the \(B\)-model of one (variation of Hodge structure) is equivalent to the \(A\)-model of another (the untwisted part of the genus zero orbifold Gromov--Witten invariants). The technique can be generalized and applied to the case when the nef-partition is trivial. As a byproduct, we also verify Morrison's conjecture in certain circumstances.

Mirror symmetry for singular double cover Calabi--Yau varieties: quantum test

TL;DR

The paper extends mirror symmetry to singular Calabi--Yau double covers arising from nef-partitions on semi-Fano toric manifolds, formulating a quantum mirror test between the -model and the -model in this singular setting. Using gauge-fixed double covers and Batyrev--Borisov duality, it constructs mirror pairs and analyzes their periods via GKZ -hypergeometric systems, matching them to untwisted genus-zero orbifold Gromov--Witten invariants through cohomology-valued -series. For two explicit branching configurations on , the authors prove a mirror theorem: the -model correlators of coincide with the -model correlators of after a mirror map, with Gamma-factor effects playing a crucial role in normalization and instanton counts (e.g., for the first case). Morrison's conjecture is also tested in this framework, showing extremal transitions are reversed under the singular mirror correspondence and linking through dual polytopes. Overall, the work provides concrete computational evidence and a robust toolkit for quantum mirror symmetry in singular CYs within toric and GLSM-inspired contexts, enriching the dialogue between topological and geometric aspects of mirror symmetry.

Abstract

We continue our study on the pairs of singular Calabi--Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of branched along (1) a union of eight hyperplanes in general position, and (2) a union of four hyperplanes and a quartic in generation. Our previous construction produces hypothetical singular mirror partners. We prove that they are mirror pairs in the sense that the -model of one (variation of Hodge structure) is equivalent to the -model of another (the untwisted part of the genus zero orbifold Gromov--Witten invariants). The technique can be generalized and applied to the case when the nef-partition is trivial. As a byproduct, we also verify Morrison's conjecture in certain circumstances.

Paper Structure

This paper contains 33 sections, 37 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Statements of the main results
  4. Preliminaries
  5. The Batyrev--Borisov duality construction
  6. Calabi--Yau double covers
  7. Notation and conventions
  8. GKZ A-hypergeometric systems
  9. Stacky fans and Chen--Ruan cohomology
  10. Double covers of P3 with the nef-partition -K=h+h+h+h
  11. Picard--Fuchs equations for
  12. An instanton prediction from mirror symmetry
  13. An instanton calculation
  14. The toric varieties $\mathbb{P}^{7}\slash G$ and $\mathbb{P}^{7}\slash G'$
  15. Genus zero Gromov--Witten invariants for the orbifold $[\mathbb{P}^{7}\slash G']$
  16. ...and 18 more sections

Key Result

Corollary 2.1

The predicted instanton numbers $n_{d}$ of $Y$ for small $d$ are given by

Theorems & Definitions (80)

  • Remark 1
  • Remark 2
  • Remark 3
  • definition 1.1
  • Remark 1.2
  • definition 1.3
  • Remark 2.1
  • Corollary 2.1
  • Corollary 2.2
  • proof
  • ...and 70 more