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Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties

Haniya Azam, Catherine Cannizzo, Heather Lee, Chiu-Chu Melissa Liu

Abstract

For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_τ$ together with a Strominger-Yau-Zalsow (SYZ) fibration, where $τ\in \mathcal{H}_g$ is the genus-$g$ Seigel upper half space and $P_g(\mathbb{Z}) \subset \mathrm{Sp}(2g,\mathbb{Z})$ is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli $\mathcal{H}_g$ and $\mathcal{A}^F_g$, relating the B-model on $V_τ$ and the A-model on its mirror, a compact $2g$-dimensional torus $\mathbb{T}^{2g}$ equipped with a complexified symplectic form. For each $V_τ$, we establish a homological mirror symmetry (HMS) result at the cohomological level over $\mathbb{C}$. This implies core HMS at the cohomological level over $\mathbb{C}$ and a graded $\mathbb{C}$-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups $HF^*(\hat{\ell}, \hat{\ell}')$ form coherent sheaves over a complex manifold parametrizing triples $(τ, \hat{\ell}, \hat{\ell}')$ where $τ\in \mathcal{H}_g$ defines a complexified symplectic form $ω_τ$ on $\mathbb{T}^{2g}$ and $\hat{\ell}$, $\hat{\ell} '$ are affine Lagrangian branes in $(\mathbb{T}^{2g}, ω_τ)$.

Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties

Abstract

For any positive integer , we introduce the moduli space parametrizing -dimensional principally polarized abelian varieties together with a Strominger-Yau-Zalsow (SYZ) fibration, where is the genus- Seigel upper half space and is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli and , relating the B-model on and the A-model on its mirror, a compact -dimensional torus equipped with a complexified symplectic form. For each , we establish a homological mirror symmetry (HMS) result at the cohomological level over . This implies core HMS at the cohomological level over and a graded -algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups form coherent sheaves over a complex manifold parametrizing triples where defines a complexified symplectic form on and , are affine Lagrangian branes in .

Paper Structure

This paper contains 40 sections, 31 theorems, 373 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Statements of main results
  4. Global SYZ mirror symmetry
  5. Categories of B-branes
  6. Categories of A-branes
  7. Homological Mirror Symmetry
  8. Ring of sections and Seidel's mirror map
  9. Family Floer cohomology and global HMS
  10. Acknowledgments.
  11. Principally polarized abelian varieties V tau and their SYZ mirrors (T2g Omega tau)
  12. Principally polarized abelian variety and their moduli
  13. The Riemann theta function
  14. Principally polarized tropical abelian varieties and their moduli
  15. The universal SYZ fibration
  16. ...and 25 more sections

Key Result

Theorem 1.1

For every $\tau\in \mathcal{H}$, the mirror functor sending $\hat{\ell}_{\mathsf{k}, [z]}[j]$ to its SYZ mirror $\mathcal{L}_{\mathsf{k}, [z]}[j]= \mathcal{L}^{\otimes \mathsf{k}}\otimes \mathbb L_{[z]} [j]$ (where $\mathsf{k}, j\in \mathbb Z$ and $[z]\in V_\tau^+$) is an equivalence of categories. Therefore, we have a fully faithful embedding of cohomological categories. In particular, the prod

Figures (3)

  • Figure 1: The $1\times 1$ square above, with the opposite sides identified, illustrates $\mathbb T^{2g}=\mathbb R^{2g}/\mathbb Z^{2g}$ in the case when $g=1$. The thick black line illustrates a circle that is the linear Lagrangian $\ell=\ell_{0,0}$ of slope zero. The red curve is the perturbed Lagrangian $\phi^1_H(\ell)$. These two circles, $\ell$ and $\phi^1_H(\ell)$, intersect at two points, $p$ and $q$, with $p$ of degree $0$ and $q$ of degree $1$. They bound two bigons $u_1$ and $u_2$ of equal area but in the opposite directions, shaded by gray and blue colors, respectively.
  • Figure 2: A triangle contributing to $\mathcal{M}(q,p_1, p_2)$.
  • Figure 3: A triangle contributing to $\mathcal{M}(q, p_{2,\infty}, p_{1,2})$.

Theorems & Definitions (52)

  • Theorem 1.1: HMS at the cohomological level
  • Corollary 1.2: core HMS at the cohomological level
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 3.1
  • ...and 42 more