arXiv subjects
Publications and source records attributed to Catherine Cannizzo.
We describe the formulation of Fukaya categories of symplectic manifolds with $B$-fields. In addition, we give a formula for how the $A_\infty$ structure maps change as we deform an object by a Lagrangian isotopy.
We prove evidence of Kontsevich's homological mirror symmetry conjecture (HMS) for a blow-up of an abelian surface times the complex plane, on the complex side, and its symplectic Landau-Ginzburg mirror. Specifically, the first author proved evidence of HMS for a 1-parameter family of genus 2 curves on the complex side, as a hypersurface in an abelian surface. The generalized SYZ mirror to the hypersurface is then the SYZ mirror to the Landau-Ginzburg model given by the blow-up of the abelian surface times the complex plane, along the hypersurface times zero, with superpotential given by projection to the complex plane. The mirror to the blow-up - without the superpotential - is obtained by removing a generic smooth fiber from the generalized SYZ mirror superpotential. We prove a categorical HMS result for the latter pair, between categories expected to split-generate. To do so, we equip the punctured superpotential with a Fukaya category which involves both partial and full wrapping in the base of the symplectic Landau-Ginzburg model due to the removal of a generic fiber. Semi-orthogonality appears in the categorical invariants on both sides of HMS.
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 $(\mathbb{T}^{2g}, ω_τ)$.
In mirror symmetry, symplectic Landau-Ginzburg models are mirror to a large class of examples, in particular to Fano varieties and hypersurfaces of many Calabi-Yau and Fano varieties. When studying their Fukaya categories on the A-model in homological mirror symmetry, one needs to calculate the weights of pseudo-holomorphic discs bounded by Lagrangian branes. While these calculations simplify for exact and Lefschetz fibrations, we generalize the machinery for computing these weights by dropping the exact and Lefschetz assumptions. For a general symplectic Landau-Ginzburg model, a singular symplectic fibration, we prove that the weights and Lagrangian gradings split into base and fiber components. This is used in many calculations of Fukaya-Seidel categories to provide evidence of Kontsevich's homological mirror symmetry conjecture.
In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds $(M^{2n}, ω)$ equipped with an effective Hamiltonian $\mathbb{T}^n\cong (S^1)^n$-action. We summarize the construction of $M$ as a symplectic quotient of $\mathbb{C}^d$, the $\mathbb{T}^n$-actions on $M$ and their moment maps, and Guillemin's Kähler potential on $M$. While the theories presented in this paper are for compact toric manifolds, they do carry over for some noncompact examples as well, such as the canonical line bundle $K_M$, which is one of our main running examples, along with the complex projective space $\mathbb{P}^n$ and its canonical bundle $K_{\mathbb{P}^n}$. One main topic explored in this article is how to write the moment map in terms of the complex homogeneous coordinates $z\in \mathbb{C}^d$, or equivalently, the relationship between the action-angle coordinates and the complex toric coordinates. We end with a brief review of homological mirror symmetry for toric geometries, where the main connection with the rest of the paper is that $K_M$ provides a prototypical class of examples of a Calabi-Yau toric manifold $Y$ which serves as the total space of a symplectic fibration $W: Y \to \mathbb{C}$ with a singular fiber above $0$, known as a Landau-Ginzburg model in mirror symmetry. Here we write $W$ in terms of the action-angle coordinates, which will prove to be useful in understanding the geometry of the fibration in our forthcoming work [ACLL].
Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce symplectic information about $Y$ from known complex properties of $X$. Strominger-Yau-Zaslow arXiv:hep-th/9606040 described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich arXiv:alg-geom/9411018 conjectured that a complex invariant on $X$ (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of $Y$ (the Fukaya category, see references in article abstract). This is known as homological mirror symmetry. In this project, we first use the construction of "generalized SYZ mirrors" for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov arXiv:1205.0053v4, in order to obtain $X$ and $Y$ as manifolds. The complex manifold is the genus 2 curve $Σ_2$ (so of general type $c_1<0$) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model $(Y,v_0)$ equipped with a holomorphic function $v_0:Y \to \mathbb{C}$ which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of $D^bCoh(Σ_2)$ into a cohomological Fukaya-Seidel category of $Y$ as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations and in Abouzaid-Seidel.