[Paper Review] Speculations on homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$

Denis Auroux|arXiv (Cornell University)|May 18, 2017

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper proposes an enriched homological mirror symmetry framework for hypersurfaces in $(\mathbb{C}^*)^n$, relating the wrapped Fukaya category of the hypersurface $H = f^{-1}(0)$ to the derived category of singularities of its mirror Landau-Ginzburg model, while also incorporating the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n \setminus H$ and the Fukaya-Seidel category of the mirror. The key contribution is a proof of homological mirror symmetry for higher-dimensional pairs of pants via $A_\infty$-structure constraints and exact triangle relations in the Fukaya category.

ABSTRACT

Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n\setminus H$ and the Fukaya-Seidel category of the Landau-Ginzburg model $((\mathbb{C}^*)^n,f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants.

Motivation & Objective

• To extend homological mirror symmetry to include the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n \setminus H$ alongside the wrapped Fukaya category of the hypersurface $H$ and the Fukaya-Seidel category of the mirror Landau-Ginzburg model.
• To establish a conjectural commutative diagram linking the derived categories of coherent sheaves on the singular fiber $Z = W^{-1}(0)$, the derived category of singularities $D^b_{\text{sg}}(Z)$, and the wrapped Fukaya categories of $H$ and $(\mathbb{C}^*)^n \setminus H$.
• To prove homological mirror symmetry for higher-dimensional pairs of pants by verifying the $A_\infty$-structure on the wrapped Floer cohomology of Lagrangian spheres under specific grading and product constraints.
• To demonstrate that the wrapped Fukaya category of the $n$-dimensional pair of pants $\Pi_n$ is split-generated by the Lagrangians $L_{\{i\}}$ for $i = 0, \dots, n+1$, using induction and symmetry.

Proposed method

• The mirror construction uses the tropicalization of the Laurent polynomial $f$, defining a piecewise linear function $\varphi(\xi) = \max_{\alpha \in A} \langle \alpha, \xi \rangle - \rho(\alpha)$, which generates the moment polytope $\Delta_Y \subset \mathbb{R}^{n+1}$ for the mirror toric variety $Y$.
• The mirror Landau-Ginzburg model is defined as $((\mathbb{C}^*)^n, f)$, with superpotential $W = -z^{(0,\dots,0,1)}$, whose zero locus $Z = W^{-1}(0)$ is the union of toric divisors $Z_\alpha$.
• The paper constructs a functor $j$ from the wrapped Fukaya category of $\Pi_{n-1}$ to $\mathcal{W}(\Pi_n)$ via an embedding into a fiber of $f = -x_0/x_{n+1}$, which maps generators $v_I$ to $u_I$ and respects exact triangle structures.
• The $A_\infty$-structure on $\bigoplus_{I,J} HW^*(L_I, L_J)$ is constrained by two conditions: (1) grading compatibility requiring outputs to differ from input sums by $(2-k)/2$ per variable, and (2) $\mu^3(u_I, u_J, u_K) = \pm \mathrm{id}$ for $I \sqcup J \sqcup K = \{0,\dots,n+1\}$.
• The proof relies on induction and $\mathfrak{S}_{n+2}$-symmetry to reduce to the case where $\{0, n+1\} \subset K$, and uses the fact that $j(v_{K'}) = u_K$ due to wrapping around the base locus of $f$.
• The split-generacy of $\mathcal{W}(\Pi_n)$ is established by showing that the images of the functors $j$ and $\alpha_\infty$ generate all $L_{\{i\}}$, and that exact triangles imply any two generators produce the third.

Experimental results

Research questions

• RQ1 How can homological mirror symmetry be extended to include the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n \setminus H$ in addition to the hypersurface $H$?
• RQ2 What is the precise relationship between the derived category of singularities $D^b_{\text{sg}}(Z)$ of the mirror Landau-Ginzburg model and the wrapped Fukaya category of $H$?
• RQ3 Can the $A_\infty$-structure on the wrapped Floer cohomology of the $n$-dimensional pair of pants be uniquely determined by grading constraints and the $\mu^3$-relation $\mu^3(u_I, u_J, u_K) = \pm \mathrm{id}$?
• RQ4 Is the wrapped Fukaya category of the $n$-dimensional pair of pants split-generated by the Lagrangians $L_{\{i\}}$ for $i = 0, \dots, n+1$?
• RQ5 How does the embedding of $\Pi_{n-1}$ into a fiber of $f = -x_0/x_{n+1}$ induce a functor between the wrapped Fukaya categories of $\Pi_{n-1}$ and $\Pi_n$?

Key findings

• The paper proves that the wrapped Fukaya category $\mathcal{W}(\Pi_n)$ of the $n$-dimensional pair of pants is split-generated by the Lagrangians $L_{\{i\}}$ for $i = 0, \dots, n+1$, using induction and symmetry.
• The $A_\infty$-structure on $\bigoplus_{I,J} HW^*(L_I, L_J)$ is uniquely determined by the grading condition and the $\mu^3$-relation $\mu^3(u_I, u_J, u_K) = \pm \mathrm{id}$ for $I \sqcup J \sqcup K = \{0,\dots,n+1\}$, as conjectured in Conjecture 9.6.
• The functor $j$ from $\mathcal{W}(\Pi_{n-1})$ to $\mathcal{W}(\Pi_n)$ maps generators $v_I$ to $u_I$ and respects exact triangle structures, with $j(v_{K'}) = u_K$ due to wrapping around the base locus of $f$.
• The proof of homological mirror symmetry for $\Pi_n$ relies on the commutative diagram involving $\mathcal{W}((\mathbb{C}^*)^n \setminus H)$, $D^b_{\text{sg}}(Z)$, and $\mathcal{W}(H)$, with the latter equivalence established via the $A_\infty$-structure and split-generation.
• The case $n=1$ is verified via Hochschild cohomology, and the higher-dimensional case is reduced to this via $\mathfrak{S}_{n+2}$-symmetry and induction.
• The mirror Landau-Ginzburg model $((\mathbb{C}^*)^n, f)$ has superpotential $W = -z^{(0,\dots,0,1)}$, which vanishes to order 1 on each toric divisor $Z_\alpha$, so $W^{-1}(0) = \bigcup_{\alpha \in A} Z_\alpha$.