Learning Seminar: SYZ Mirror Symmetry

Overview

After Kontsevich's famous proposal on Homological Mirror Symmetry [Kon94], the question of geometric construction of mirrors to a given variety has been an important question for mathematicians. One important proposal was made by Strominger-Yau-Zaslow [SYZ96], which stated that mirrors to a given Kähler manifold $X$ equipped with a Lagrangian torus fibration $X\to B$ should be a Kähler manifold $\check{X}\to B$ with a Lagrangian torus fibration over the same base $B$, and with "dual torus fiber".

This proposal has been successful firstly for elliptic curves whose corresponding Homological Mirror Symmetry was (partly) verified by Polishchuk-Zaslow [PZ98]. People quickly found finding a smooth Lagrangian torus fibration for any Kähler manifold $X$ is almost impossible: there can be toric singularities where the fibration has lower dimensional toric fibers, and there can be singular fibers. Cho and Oh [CO06] studied the effects of toric singularities for toric Fano varieties and showed that existence of toric singularities gives Landau-Ginzburg superpotentials $W$ of the mirror, via counting intersection numbers of holomorphic disks with the toric singularities. For toric varieties which admits canonical Lagrangian torus fibrations with only toric singularities, the corresponding Homological Mirror Symmetry statement was then proved by Fang-Liu-Treumann-Zaslow [FLTZ11] for smooth projective toric varieties and further generalized by Kuwagaki [Kuw20] to all toric stacks.

The simplest and the most generic case of singular fibers are non-degenerate singularities, similar to singular fibers of Lefschetz fibration, obtained by contracting a codimension $1$ subtorus to a point. This situation was studied by Auroux [Aur07] in the case when $X=\mathbb{C}^2\setminus\{xy=1\}$ and the Lagrangian torus fibration was constructed out of the Lefschetz fibration $\pi (x,y)=xy$.

In this case one can clearly see a wall-crossing phenomenon, where there's a single singular value for the torus fibration $X\to\mathbb{R}^2$ coming from singularities of the Lefschetz fibration, $\{xy=0\}\subseteq\mathbb{C}^2$, which corresponds to the wall $\{x=0\}\times\mathbb{R}\subseteq\mathbb{R}^2$. Auroux [Aur07] showed that for any $b=(b_1,b_2)$ with $b_1>1$ or $b_1 < 1$, disk countings computing mirror superpotential to $X\subseteq\mathbb{P}^2$ are different, so that the two moduli spaces of Lagrangian tori are glued together with non-trivial transition functions.

Although the phenomena seems mysterious, Fukaya proposed a possible explanation in [Fuk09] where we refine the previous picture by considering the Floer-theoretic deformation space of singular Lagrangian tori $L$ where the local systems play the role of deforming the Floer $A_{\infty}$-structure on $CF^{\ast} (L,L)$. For singular Lagrangian submanifolds $L\subseteq X$ the $A_{\infty}$-structure on $CF^{\ast} (L,L)$ is usually curved, meaning there exists a curvature term $\mu^0$ that prevents the identity $(\mu^1)^2=0$. However, there might exist an element $b\in H^1 (L,\lambda_0)$ where $\Lambda_0$ is the Novikov ring, such that the deformation $\mu_b$ satisfies $(\mu_b^1)^2=0$. The element $b$ must satisfy the Maurer-Cartan equation$$\sum_k\mathfrak{m}^k (b,b,\dotsb, b)=0,$$so we call elements satisfying this equation a Maurer-Cartan element, and use $\mathcal{MC} (L)$ for the space of all Maurer-Cartan elements (equivalently, the infinitesimal deformation space of $L$).

As for any choice of Maurer-Cartan element $b\in\mathcal{MC} (L)$, the pair $(L,b)$ defines an object in the Fukaya category of $X$, Fukaya proposed that the mirror to $X$ should be the quotient of a union $$\mathcal{MC} (\mathfrak{L}):=\bigcup_{u\in B}\mathcal{MC} (L_u)\big/H^1 (L_u;2\pi\sqrt{-1}\mathbb{Z})$$for a given family of Lagrangian submanifolds $\mathfrak{L}:=\{L_u\}_{u\in B}$ where $B$ is some smooth manifold with the same dimension as $H^1 (L;\mathbb{R})$. Moreover, for any given object $(L',b')\in\mathcal{F} (X)$, the Yoneda functor $$(u,b)\mapsto CF^{\ast}((L',b'),(L_u,b);\Lambda)$$should define a complex of coherent sheaves over $\mathcal{MC} (\mathfrak{L})$.

The main goal of this learning seminar is to understand the current state of research concerning SYZ mirror constructions, in particular the work of Abouzaid [Abo21] on SYZ and Homological Mirror Symmetry for smooth Lagrangian torus fibrations (and hence for all abelian varieties) and Abouzaid-Auroux-Katzarkov [AEK16] on blow-ups of toric varieties along hypersurfaces near tropical limits.

Plan of the Seminar.

Introduction

In this talk I aim to provide a brief overview of SYZ mirror symmetry proposal from a more mathematical point of view. Recall that the Homological Mirror Symmetry proposal of Kontsevich [Kon94] states that there are pairs of (not necessarily smooth) varieties $(X,\omega ,J)$ and $(\check{X},\check{\omega},\check{J})$ such that $$ D^{\pi}\mathcal{Fuk} (X,\omega)\simeq D^b\operatorname{Coh} (\check{X},\check{J})\quad\text{and}\quad D^b\operatorname{Coh} (X,J)\simeq D^{\pi}\mathcal{Fuk} (\check{X},\check{\omega}), $$where $\mathcal{Fuk} (X,\omega)$ is a version of the "Fukaya category" for the symplectic variety $(X,\omega)$ and $D^b\operatorname{Coh} (X,J)$ is the bounded derived category of coherent sheaves on the complex variety $(X,J)$, whenever they're well-defined. On the algebraic (which we call B-model) side, there's a famous theorem of Bondal and Orlov [BO01] stating that we can re-construct a smooth projective variety $X$ from its derived category of coherent sheaves $D^b\operatorname{Coh} (X)$.

Based on the resconstruction theorem, we can roughly "re-construct" an algebraic variety from its derived category of coherent sheaves, assuming it's Fano or general type. Let's ignore the condition and use it as a heuristic. Now we have a source of category coming from symplectic geometry: for each symplectic manifold $(X,\omega)$ we can propose an $A_{\infty}$-category $\mathcal{Fuk} (X,\omega)$ which, if the mirror exists, is supposed to be an $A_{\infty}$-enhancement of the derived category of coherent sheaves of some algebraic variety $\check{X}$, and in particular we should be able to re-construct $X$ from the category $\mathcal{Fuk} (X,\omega)$. Assuming $\check{X}$ is separated, from [ND20] it suffices for us to determine the point objects and the invertible objects.

Let $p\in\check{X}$ be a smooth closed point of the algebraic variety $(\check{X},\check{J})$, and let $\mathcal{O}_p$ be the structure sheaf of the closed point $p$. Then we can directly compute that

In particular, an object $L\in\mathcal{Fuk} (X,\omega)$ behaves like a point object only when the endomorphism algebra of $L$ is equivalent to the exterior algebra $\Lambda\langle x_1,\dotsb, x_n\rangle$. Recall that for two Lagrangian submanifolds $L_1, L_2$ of $(X,\omega)$, the morphism $\Hom_{\mathcal{Fuk} (X,\omega)} (L_1, L_2)$ between $L_1$ and $L_2$ in $\mathcal{Fuk} (X,\omega)$ is provided by the Lagrangian Floer complex $CF^{\ast} (L_1, L_2)$ which is graded and is generated by intersection points between $L_1$ and $L_2$ once $L_1$ and $L_2$ are transversely intersecting (and the corresponding derived version in general, see e.g. [FOOO09]). When $L_1$ and $L_2$ are nice closed Lagrangian submanifolds and $(X,\omega)$ a nice symplectic manifold, the celebrated Floer's theorem holds:

In particular, when $L$ is a Lagrangian torus with $\Bbbk=\mathbb{R}$, we will get $HF^{\ast} (L,L;\mathbb{R})\cong H^{\ast} (L;\mathbb{R})\cong\Lambda_{\mathbb{R}}\langle x_1,\dotsb, x_n\rangle$ and is hence close to a point object up to changing the base coefficient field to $\mathbb{C}$. A geometrically natural way to extend the coefficient field is to equip $L$ with $U(1)$-local systems, equivalently trivial complex line bundle over $L$ with flat $U(1)$-connections.

These discussions provide us a natural source of point objects in the Fukaya category: Lagrangian tori that are mutually disjoint. However, there're too many Lagrangian tori in a fixed symplectic manifold, and not all of them should be counted as the "points". To distinguish out the true "point objects", Strominger-Yau-Zaslow [SYZ96] suggested we should look at Lagrangian tori that are special, which are expected to be an equivalent notion of stable in modern algebro-geometric language [TY02].

One special case that we know this will work is when $(X,\omega)$ admits a smooth fibration over some smooth manifold $B$ of Lagrangian tori [Abo21], where we assume that $\pi_2 (B)=0$. In general, we can only expect that an open dense subset of $X$, $X^0$, admits a smooth Lagrangian fibration, and the complement consists of various singularities. There're worse issues arising: for a general symplectic manifold $(X,\omega)$, in order for Fukaya $A_{\infty}$-structure to be defined, we need to introduce the Novikov field $$\Lambda_{\Bbbk} :=\left\{\sum_{n=0}^{\infty} a_n T^{\alpha_n}\middle\vert a_n\in\Bbbk,\ \alpha_n\in\mathbb{R},\lim_{n\to\infty}\alpha_n =\infty\right\}$$ to bound energies of counted pseudo-holomorphic disks in order to achieve Gromov compactness. Therefore the constructed mirror space lives naturally over the coefficient field $\Lambda_{\Bbbk}$ instead of $\Bbbk$. Kontsevich-Soibelman [KS06] suggested that the mirror should be an analytic space over this non-archimedean field $\Lambda_{\Bbbk}$, and evaluating the formal variable $T$ at convergent complex values would reduce to usual complex mirror space.

Singularities of Lagrangian torus fibration can be separated into several different cases. Note that in order to make $CF^{\ast} (L,L)$ a cochain complex, one need to prove that $(\mathfrak{m}^1 )^2=0$, and in the compactification of moduli space of pseudo-holomorphic strips computing $(\mathfrak{m}^1)^2$ there're disk bubblings with a single output. This stratum would arise only when the Lagrangian submanifold $L$ bounds holomorphic disks of Maslov index $\leq 2$, so if all such disks do not appear, we should see a perfect mirror picture (as in [Abo21] or [PZ98]). If there exists holomorphic disks of Maslov index $2$, then we obtain an extra obstruction term $\mathfrak{m}^0$ in the $A_{\infty}$-structure of $CF^{\ast} (L,L)$, but if we assume that $L$ is monotone, then $\mathfrak{m}^0 (L)=0$ and hence $HF^{\ast} (L,L)$ can still be defined. $\mathfrak{m}^0 (L,\nabla)$ defines a smooth holomorphic function on the moduli space of pairs $(L,\nabla)$ [Aur07] and acts as the Landau-Ginzburg superpotential of the mirror. In other words, if we assume that $(Y,J)$ is the mirror moduli space and $W\colon Y\to\mathbb{C}$ is the corresponding superpotential, then we expect equivalences of categories $$D^{\pi}\mathcal{Fuk} (X,\omega)\simeq HMF(Y,W),$$ where $HMF$ is the homotopy category of matrix factorizations of $(Y,W)$, describing singularities of the superpotential $W$.

The case when $L$ bounds holomorphic disks of Maslov index $0$ is also slightly studied in [Aur07]. In nice geometric situations such disks come from singularities of the Lagrangian torus fibration $X\to\bar{B}$ where the fibers are singular Lagrangian tori, obtained by contracting some submanifolds into a point. In this case, what we can see is the subspace of pairs $(L,\nabla)$ where $L$ bounds no Maslov index $0$ disks is disconnected, and the space of pairs $(L,\nabla)$ where $L$ bounds Maslov index $0$ disks form real codimension $1$ submanifolds, called walls of the Lagrangian torus fibration. In the most generic case, when the singular fibers are one-pinched Lagrangian tori, we can explicitly understand the monodromy transformation and glue different connected components of $B$ together based on the monodromy information.

However, non-generic cases do arise, when we want certain symmetry to be preserved, and this requires us to have a better understanding of this "gluing of moduli spaces". Recall that moduli spaces of paris $(L,\nabla)$ locally comes from the spectrum $\operatorname{Spec}\operatorname{Sym}^{\ast} H^1 (L,\nabla;\Lambda_{\mathbb{R}})$ for smooth unobstructed tori $L$, and for singular $L$ its Lagrangian Floer cochain is naturally obstructed, i.e. $\mathfrak{m}^0\neq 0$ and hence $(\mathfrak{m}^1)^2\neq 0$. In this case we can deform it by introducing cycles $b\in CF^1 (L,\nabla ;\Lambda)$ such that $\mathfrak{m}^1 (b)=\mathfrak{m}^0$, called bounding cochain, and the deformed $A_{\infty}$-structure $\mathfrak{m}^i_b$ will satisfy the desired $A_{\infty}$-relations, and hence we still have some well-defined Floer cohomology. This can be geometrically imagined as deforming the singular Lagrangian submanifold $L_0$ into a nearby smooth Lagrangian $L_{\eta}$, so that its Floer cohomology is well-defined. Therefore we can consider the moduli space of all such deformations, called Maurer-Cartan space $\mathcal{MC} (L_0)$, which should serve as the interpolation between different disconnected moduli spaces.

This finishes the rough theoretical framework of how SYZ mirror symmetry construction should work and result in the mirror space we want. However, as the audiences would see in this talk, there're still a lot of mysteries to understand. In later talks, we will see in greater details some of the computations and several rigorous statements concerning SYZ and Homological Mirror Symmetry.

Lagrangian Torus Fibrations

In this talk we will review some basic geometric facts concerning Lagrangian torus fibrations, particularly regular ones.

Regular Lagrangian fibrations.

• They're locally modelled on completely integrable hamiltonian systems;
• Fibers of regular Lagrangian fibrations are Lagrangian tori;
• Basis of regular Lagrangian fibrations carry integral affine structures.

Let's start with the definition of Lagrangian fibrations.

Now we are ready to give definitions on Lagrangian fibrations.

Integrable hamiltonian systems.

Frobenius theorem immediately tells us that

Let $c$ be a regular value of $\vec{H}$, then it's immediate from the definition of completely integrable hamiltonian system that $TH^{-1} (c)=\espan\left\{X_{H_1},\dotsb,X_{H_n}\right\}$ and $H^{-1} (c)$ is a Lagrangian orbit of the $\mathbb{R}^n$-action. Now choose a local section $\sigma\colon B\to M$ of $\pi$, where $B\subseteq\vec{H}(M)\subseteq\mathbb{R}^n$ is an open subset of regular values.

Locally near a Lagrangian section $\sigma\colon B\to M$, we can construct a map $\psi\colon B\times\mathbb{R}^n\to M$ by $\psi (b,t)=\phi_H^t (\sigma(b))$, which can be verified to be an immersion and submersion with $\psi^{\ast}\omega =\displaystyle\sum_{i=1}^n db_i\wedge dt_i$. This map is called the Liouville coordinate.

We call $a$ the action coordinate, and $t$ the angle coordinate.

Now we can show the first important property for Lagrangian fibrations:

The action-angle coordinates are local, but we can patch them together to glue regular Lagrangian fibrations over contractible spaces into some smooth manifold $B$, which carries a specific rigid structure and we call such manifolds integral affine manifolds.

Equivalences.

We are ready to prove a "reconstruction theorem" for regular Lagrangian fibrations similar to Theorem 1.1.

In fact, under these assumptions we can understand the space $M$ explicitly: the integral affine structure on $B$ induces a locally free sheaf of lattices $\Lambda$ in the cotangent bundle $T^{\ast} B$, and we can naturally form the symplectic manifold $T^{\ast} B/\Lambda$ where fibers of the projection $T^{\ast} B/\Lambda\to B$ are Lagrangian tori. Note that $T^{\ast} B/\Lambda\to B$ always has a global Lagrangian section provided by the zero section, and therefore reconstruction theorem 2.20 applies to yield that $M$ is fibered symplectomorphic to $T^{\ast} B/\Lambda$.

Flux.

This definition does not depend on the choice of the primitive $1$-form $\lambda$, as different choices differ by shifting by a constant which is the integral affine transformation, and also not on the choice of the basis $\{c_i\}$ for any choice differs by an element in $GL(n,\mathbb{Z})$, again absorbed into the integral affine structure.

With the flux map $I\colon\tilde{B}\to\mathbb{R}^n$, we naturally obtain an integral affine structure on $\tilde{B}$ by pulling back the standard one on $\mathbb{R}^n$ via $I$. This integral affine structure then descends to $B$ and gives the same integral affine structure on $B$ coming from regular Lagrangian fibrations.

SYZ Mirror Symmetry for Toric Fano Varieties

In this talk we will discuss the first type of non-regular singularities of Lagrangian fibrations, which we call toric singularities. These are good singularities in that the torus action from the Lagrangian torus fibration extends to the singular fibers, where the stabilization groups are non-trivial. One of the simplest case is the so-called toric varieties, where the regular stratum of the base $B$ is $\mathbb{R}^n$, and we will quickly see that the singular stratum form a polyhedral complex partially compactifying $\mathbb{R}^n$.

Symplectic toric varieties.

A hamiltonian $G$-action on $M$ induces a map of Lie algebras $\rho\colon\mathfrak{g}\to C^{\infty} (M)/\mathbb{R}$, equivalently a map $\mu_G\colon M\to\mathfrak{g}^{\ast}$ up to translation in $\mathfrak{g}^{\ast}$ where $\mu_G(x)(X):=\rho (X)(x)$, called the moment map of $G$. When $G=\mathbb{T}^n$ is a torus, the theorem of Atiyah-Guillemin-Sternberg [Ati82,GS82] states that

We call the image of the moment map the moment polytope of the toric symplectic manifold $M$. Let $\Sigma$ be the polytope in $\mathfrak{g}^{\ast}$, it naturally admits a stratification by facets. There is only one top-dimensional facet $\Sigma^{\circ}$ corresponding to values of $\mu_G$ where $T^n$ acts freely transitively on the fiber, which are Lagrangian tori in $M$. Lower dimensional facets correspond to values of $\mu_G$ where $T^n$ acts with stabilizer subgroups of rank equal to the codimension of the corresponding facet. They're naturally isotropic tori obstructing the fibration being regular.

Now we proceed to the Fanoness on toric varieties. Recall that a symplectic manifold is monotone or Fano if there exists a positive constant $\tau$ such that $[\omega]=\tau c_1 (X)$, where $c_1 (X)$ is the first Chern class of $X$. As $c_1 (X)$ is integral and algebro-geometrically it's equal to the sum of the representatives of all the toric divisors appearing on the boundary strata, the Fano condition can be read-off combinatorially as the following:

Homological Mirror Symmetry.

This theorem also tells us that cotangent fiber of $T^{\ast}\mathbb{T}^n$ is mirror to the structure sheaf of $\left(\mathbb{C}^{\ast}\right)^n$, and hence we can try to recover all the point objects from this homological mirror equivalence, leading to the SYZ construction. Note that as $\mathbb{T}^n$ is a Lie group, $T^{\ast}\mathbb{T}^n\cong\mathbb{T}^n\times\mathbb{R}^n$ is trivial, and we obtain a family of Lagrangian tori $\{L_b\}_{b\in \mathbb{R}^n}$ coming from the product decompostion.

We now investigate Lagrangian Floer theories of these Lagrangian tori. Firstly it's easy to see that $HF^{\ast} (L_b, L_{b'})=0$ if $b\neq b'$ as they have no geometric intersections, and $HF^{\ast} (L_b,T^{\ast}_q\mathbb{T}^n)\cong\mathbb{R}$ as they have a unique transversal intersection point. The only interesting complex is $CF^{\ast} (L_b,L_b)$, for $b\in\mathbb{R}^n$.

Recall that the standard symplectic structure on $T^{\ast}\mathbb{T}^n$ is given by $-d\lambda_{can}$ where $\lambda_{can}=\sum_{i=1}^n (\log r_i)d\theta_i$ where $(\log r_i)_{i=1}^n$ is the cotangent coordinate and $(\theta_i)$ is the torus coordinate. Recall that

Exactness of Lagrangian submanifolds $L$ prevent the existence of any non-constant pseudo-holomorphic disks with boundaries on $L$, and hence the Floer cohomology is easy to compute: it's identical to the Morse cohomology and hence we have $$HF^{\ast} (L_b,L_b)\cong H^{\ast} (L_b;\mathbb{R})$$for exact Lagrangian tori $L_b$.

However, it's not hard to check that the only exact Lagrangian tori $L_b$ in $T^{\ast}\mathbb{T}^n$ is the zero section $L_0$. For all other Lagrangian tori $L_b$, $b\neq 0$, areas of holomorphic disks with boundaries on $L_b$ are determined by the integral of the boundary curve class in $L_b$ against the Liouville form $\lambda$, or more explicitly, against the $1$-form $\lambda\vert_{L_b}$ of $\lambda$ restricting to the torus $L_b$.

Let's write down the $A_{\infty}$-structure to see the effect of these non-trivial holomorphic curves: write $CF^{\ast} (L_b,L_b)=\Omega(L_b)$ be the de Rham algebra of $L_b$, then the $A_{\infty}$-structure $\mathfrak{m}^d\colon CF^{\ast} (L_b,L_b)\otimes\dotsb\otimes CF^{\ast} (L_b, L_b)\to CF^{\ast} (L_b,L_b)[2-d]$ is given by $$\mathfrak{m}^d (x_1,\dotsb, x_d)=\mu^d (x_1,\dotsb, x_d)+\sum_{\beta}\sum_{\deg (x_0)=\sum\deg (x_i)+2-k+\mu(\beta)}\left(\int_{[\mathcal{M}_{d+1} (\beta)]^{vir}}\ev_0^{\ast} x_0\cup\ev_1^{\ast} x_1\cup\dotsb\cup\ev_d^{\ast} x_d\right)T^{\lambda (\partial\beta)} x_0,$$where $\mu^d$ is the Morse $A_{\infty}$-structure on the de Rham algebra $\Omega^{\ast} (L_0)$ and $\beta\in\pi_2 (X,L_b)$ is a disk class and $\partial\beta\in\pi_1 (L_b)$ is the corresponding boundary class. Note that the value $\lambda(\partial\beta)$ only depends on the cohomology class of $\left[\lambda\vert_{L_b}\right]\in H^1 (L_b;\mathbb{R})\cong H^1 (L_0;\mathbb{R})\cong\mathbb{R}^2$ and $[\partial\beta]\in\pi_1 (L_b)\cong\pi_1 (L_0)\cong\mathbb{Z}^2$, we see that the $A_{\infty}$-algebra $CF^{\ast} (L_b,L_b)$ is a deformation of the $A_{\infty}$-algebra $CF^{\ast} (L_0,L_0)$ by an element $[\lambda_b]\in H^1_{dR} (L_0;\mathbb{R})$, where the generators of $CF^{\ast}$ are the same with deformed $A_{\infty}$-structure $\mathfrak{m}^d_b$.

In this case it's not hard to see topologically that the deformation is in fact trivial: as $T^{\ast}\mathbb{T}^n\to\mathbb{R}^n$ is a fibration, it follows that $\pi_2 (T^{\ast}\mathbb{T}^n,L_b)\cong\ker (\pi_1 (L_b)\to\pi_1 (T^{\ast}\mathbb{T}^n))=0$, and hence all pseudo-holomorphic disks with boundaries on $L_b$ must be trivial. Therefore all algebras $CF^{\ast} (L_b,L_b;\mathbb{R})$ can be canonically identified as their de Rham algebras, with the trivial Morse $A_{\infty}$-structure. In particular, they correspond all to smooth closed points in the mirror $\left(\mathbb{C}^{\ast}\right)^n$, and we see that the moduli space of all pairs $(L_b,\nabla_b)$ where $\nabla_b$ is a falt connection on the trivial complex line bundle on $L_b$ is isomorphic to $\left(\mathbb{C}^{\ast}\right)^n$.

Now let's proceed to general projective toric varieties. As all toric varieties contain a Zariski open dense algebraic torus, it follows that all toric varieties have to be compactifications of $\left(\mathbb{C}^{\ast}\right)^n$. In these cases, we still get a $\left(\mathbb{C}^{\ast}\right)^n$ family of Lagrangian tori equipped with local systems, but if we now look at the $A_{\infty}$-structure of $CF^{\ast} (L_b,L_b)$, there are possibly pseudo-holomorphic disks with boundaries on $L_b$ intersecting with the toric boundary divisor. As we see from the formula, existence of extra disks corresponds to deformation of the previous $A_{\infty}$-structures, and therefore we expect that Fukaya categories of toric varieties are deformations of Fukaya categories of $\left(\mathbb{C}^{\ast}\right)^n$. The mirror algebraic category should also behave the same way, and we will go back to this in the next section.

If we switch our symplectic and algebraic structures, then Homological Mirror Symmetry for general toric stacks has been proven by Fang-Liu-Treumann-Zaslow [FLTZ11] and Kuwagaki [Kuw20], in terms of constructible sheaves. The mirror statement is usually referred to as coherent-constructible correspondence.

When $X_{\Sigma}$ is toric Fano, [GPS24] showed that $\operatorname{Sh}_c (\mathbb{T}^n,\Lambda_{\Sigma})$ is equivalent to the partially wrapped Fukaya category $\mathcal{W}\left(\left(\mathbb{C}^{\ast}\right)^n,W_{\Sigma}\right)$ where $W_{\Sigma}$ is the so-called Hori-Vafa superpotential which is also determined by the combinatorial structure of $\Sigma$.

SYZ mirror construction for toric Fano varieties.

The first result is that for toric compactifications, Maslov indices of pseudo-holomorphic disks are well-understood:

In particular, there exists pseudo-holomorphic disks of Maslov index $2$ that contributes to the curvature term $\mathfrak{m}^0$ of the $A_{\infty}$-structure, potentially obstructing Floer cohomologies of Lagrangian tori to be well-defined. To compute this term, we need classifications of these pseudo-holomorphic disks. This is the following main theorem of [CO06]:

In particular, the numbers $\mu_i$ are intersection multiplicities of the pseudo-holomorphic disk $u(\mathbb{D}^2)$ with the toric boundary component $D_i$. Using this classification, we can completely compute contributions of the pseudo-holomorphic disks with arbitrary Maslov indices to $\mathfrak{m}^0$.

So only disks of Maslov index $2$ contribute non-trivially to the curvature $\mathfrak{m}^0$. Note that they correspond to holomorphic disks with a single intersection with one of the toric boundary components, and we can in fact determine all such holomorphic disks.

As all such counts are weighted by the area of the holomorphic disks, we also need to determine the symplectic area of these Maslov index $2$ holomorphic disks.

With all these preparations, we are able to write down $\mathfrak{m}^0$ explicitly as follows: note that $\mathfrak{m}^0$ counts only holomorphic disks with one output, and therefore representing an element in $CF^{\ast} (L_b,\nabla;L_b,\nabla)$ of the form $$\mathfrak{m}^0 (L_b,\nabla)=\sum_{i=1}^N T^{\omega (D(v_i))}PD\left(\ev_{\ast}\left[\mathcal{M}_1 (D(v_i))\right]^{vir}\right)\hol_{\nabla} (\partial D(v_i))=\sum_{i=1}^N T^{2\pi(\langle b,v_i\rangle -\lambda_i)}\hol_{\nabla} (\partial D(v_i)),$$ and the function $z_i(L_b,\nabla)\colon =\exp\left(-\int_{D(v_i)}\omega\right)\hol_{\nabla} (\partial D(v_i))$ for $1\leq i\leq N$ can be shown to be holomorphic.

Choose a basis $\{w_i\}_{i=1}^w$ of the torus $\left(\mathbb{C}^{\ast}\right)^n$, the curvature term then defines a Laurent polynomial on $\left(\mathbb{C}^{\ast}\right)^n$ which we call the Hori-Vafa superpotential. The other part of Homological Mirror Symmetry states that

Going back to the computation of Fukaya $A_{\infty}$-algebra $CF^{\ast}(L_b,L_b)$, we want to firstly verify that the Floer cohomology is well-defined, i.e. $\left(\mathfrak{m}^1\right)^2=0$. This can be achieved by the following Proposition:

In particular, we have $\left(\mathfrak{m}^1\right)^2=\mathfrak{m}^2 (\mathfrak{m}^0,x)+(-1)^{\deg (x)+1}\mathfrak{m}^2 (x,\mathfrak{m}^0)=((-1)^{\deg (x)}+(-1)^{\deg (x)+1})C=0$.

Therefore $HF^{\ast} ((L_b,\nabla),(L_b,\nabla))$ for all Lagrangian brane $(L_b,\nabla)$ is well-defined.

References

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