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Equivariant mirror symmetry for footballs

Zhuoming Lan

TL;DR

This work develops equivariant mirror symmetry for footballs $\mathcal{F}(m,r)$, extending the known results from projective lines to weighted projective lines by unifying A-model equivariant Gromov-Witten theory with B-model Eynard-Theory via a common $R$-matrix framework. It establishes all genus equivariant mirror symmetry by matching graph sums derived from Givental's formalism and Eynard-Orantin recursion, and proves that the $R$-matrices agree at large radius and general radius. In the large radius limit, the theory connects to orbifold Hurwitz numbers and yields a generalized Bouchard-Mariño conjecture; in the non-equivariant limit, it provides a Norbury-Scott type statement for the corresponding Hodge-theoretic invariants. The results fuse Frobenius manifold, quantum cohomology, and topological recursion techniques to extend equivariant mirror symmetry to footballs and to reveal deep links with Hurwitz theory and orbifold geometry.

Abstract

In this paper, we establish equivariant mirror symmetry for footballs $\mathcal{F}(m,r)$. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [{\it Geometry \& Topology} 24:2049-2092, 2017], and the results by D. Tang of weighted projective lines, on [arXiv:1712.04836]. More precisely, we prove the equivalence of the $R$-matrices for A-model and B-model at large radius limit, and establish isomorphism for $R$-matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for footballs. In last two sections the large radius limit and equivariant limit are considered, resulting a generealized Bouchard-Mariño conjecture and Norbury-Scott conjecture respectively.

Equivariant mirror symmetry for footballs

TL;DR

This work develops equivariant mirror symmetry for footballs , extending the known results from projective lines to weighted projective lines by unifying A-model equivariant Gromov-Witten theory with B-model Eynard-Theory via a common -matrix framework. It establishes all genus equivariant mirror symmetry by matching graph sums derived from Givental's formalism and Eynard-Orantin recursion, and proves that the -matrices agree at large radius and general radius. In the large radius limit, the theory connects to orbifold Hurwitz numbers and yields a generalized Bouchard-Mariño conjecture; in the non-equivariant limit, it provides a Norbury-Scott type statement for the corresponding Hodge-theoretic invariants. The results fuse Frobenius manifold, quantum cohomology, and topological recursion techniques to extend equivariant mirror symmetry to footballs and to reveal deep links with Hurwitz theory and orbifold geometry.

Abstract

In this paper, we establish equivariant mirror symmetry for footballs . This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [{\it Geometry \& Topology} 24:2049-2092, 2017], and the results by D. Tang of weighted projective lines, on [arXiv:1712.04836]. More precisely, we prove the equivalence of the -matrices for A-model and B-model at large radius limit, and establish isomorphism for -matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for footballs. In last two sections the large radius limit and equivariant limit are considered, resulting a generealized Bouchard-Mariño conjecture and Norbury-Scott conjecture respectively.

Paper Structure

This paper contains 33 sections, 22 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Frobenius Manifold
  3. Geometry of $\mathcal{F}(m,r)$ as a toric orbifold
  4. Equivariant cohomology of $\mathcal{F}(m,r)$
  5. The $\mathbb{T}$-equivariant superpotential and its Jacobian ring
  6. Equivariant Gromov-Witten invariants
  7. Generating functions
  8. Mirror theorem of $\mathcal{F}(m,r)$ as a toric stack
  9. Equivariant quantum cohomology
  10. Basis
  11. The A-model canonical coordinates and the $\Psi$-matrix
  12. The $\mathcal{S}$-operator
  13. Graph Sum Formula and the $R$-matrix
  14. Graph sum formula
  15. Givental's formula and the A-model graph sum
  16. ...and 18 more sections

Key Result

Lemma 2.1

It holds that $\mathcal{P}_d(I(q,z))=0,\;\forall d\in\mathbb{N}$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • ...and 35 more