On fiber and base decompositions in the Fukaya category of a symplectic Landau-Ginzburg model

TL;DR

The paper addresses the challenge of computing disc weights and Lagrangian gradings in symplectic Landau-Ginzburg models without assuming exactness or Lefschetz structure. It develops a base/fiber decomposition framework, via vertical/horizontal splittings, framed and unitary frame bundles, and squared phase data, to separate base and fiber contributions in both disc weights and gradings. It proves that disc areas contributing to $\mu^2$ and higher products decompose into vertical and horizontal parts and that the $\mathbb{Z}$-grading on morphisms likewise splits into base and fiber components, enabling more tractable Fukaya-Seidel category computations. These results provide evidence for Kontsevich's Homological Mirror Symmetry by enabling base/fiber analyses across a broader class of LG models and their associated Fukaya categories.

Abstract

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.

Figures (4)

• Figure 1: An example of two U-shaped curves in the base of $v:Y\to \mathbb C$ (with the black dot being a critical value), over which fibered Lagrangians $L_0$ and $L_1$ are defined. The radial ends of the base curve $v(L_0)$ are positioned counterclockwise away from those of $v(L_1)$. The intersections between $L_0$ and $L_1$ are in the fibers $Y_{c_+}$ and $Y_{c_-}$ above the points $c_+, c_- \in v(L_0)\cap v(L_1)$. The shaded gray area is the image under $v$ of a $J$-holomorphic bigon contributing to $\mu^1$.
• Figure 2: The three figures in the bottom illustrate the base of $(Y,v)$. The shaded triangle in the leftmost figure is the image under $v$ of a $J$-holomorphic triangle $u$ contributing to $\mu^2$. The Lagrangian isotopy $\psi$ deforms $L_0$ by letting $\delta$ contract to the point $\tilde{\delta}\equiv -\epsilon$, where all three base curves intersect. Theorem \ref{['cor:triangle disc area']} relates the symplectic area of $u$ to that of $u"$, which is a $J$-holomorphic triangle contained in $Y_{-\epsilon}$ with boundary on the three fibered Lagrangians over the three base curves in the rightmost figure. The pyramid drawn at the top is a schematic illustration of the discs involved in the total space. The shaded triangle on the right face of the pyramid depicts $u$. As we deform $L_0$ by the Lagrangian isotopy, $u$ deforms to $u'$. The disc $u'$ is not $J$-holomorphic and it's depicted as the union of the shaded triangle on the right and the top shaded hexagon in the pyramid. The shaded triangle on the left side of the pyramid depicts a $J$-holomorphic disc that is in the same homotopy class as $u'$.
• Figure 3: The leftmost disc in this figure is the domain for maps $u$ and $u'$, with $u$ in the form of Equation \ref{['eq: map u in D']} with $k+1=3$. As we deform $L_0$ by a Lagrangian isotopy $\psi$, the map $u$ deforms to $u':(\mathbb D, \partial'_0\mathbb D\cup \partial'_1\mathbb D\cup \partial'_2\mathbb D, \{z_0', z_1', z_2\})\to (Y, \psi_1(L_0)\cup L_1 \cup L_2, \{\psi_1(p_0), \psi_1(p_1), p_2\})$. We use the piecewise smooth construction for $u'$ given in ACLLb. To very briefly summarize, this construction involves first deforming the domain disc $\mathbb D$ using maps $\phi_1$ and $\phi_2$. The map $\phi_1$ sends the region shaded by lines in the leftmost figure to the entire $\mathbb D$ in the middle figure. The map $\phi_2$ is orientation reversing, and it sends the solidly shaded region in the leftmost figure to $\partial_0\mathbb D\times [-1,0]$ as shown in the middle figure. Both maps agree on the line segment $\overline{z_0z_1}$, which is the dotted vertical line in the leftmost figure, so $\phi_1\cup_{\overline{z_0z_1}}\phi_2$ is a continuous map. The map $u'$ is then defined by composing $\phi_1$ by $u$ and composing $\phi_2$ by $\psi_s\circ \partial_0 u$. In the case of Theorem \ref{['cor:triangle disc area']}, the image of $u'$ is the union of the faces on the right side and top of the pyramid shown in Figure \ref{['fig: prism_complex_paper5']} (the illustrations in the first two images in Figure \ref{['fig:move_L2_paper5']} are the domains). The rightmost picture in this Figure \ref{['fig:move_L2_paper5']} shows the image of $u'$ under $v$.
• Figure 4: Grading of points in a bigon

Theorems & Definitions (42)

• Definition 1.1: Symplectic Landau-Ginzburg model $(Y,v)$
• Remark 1.2
• Remark 1.3
• Definition 1.4: Fibered Lagrangians
• Remark 1.5
• Remark 1.6
• Lemma 1.7
• Remark 1.8
• Theorem 1.9: Disc areas in $\mu^2$ decompose into vertical and horizontal contributions
• Theorem 1.10: Gradings for Floer complexes decompose into vertical and horizontal contributions