s— date: 2022-09-20 10:18 modification date: Tuesday 20th September 2022 10:18:15 title: “2022-09-20” aliases: [2022-09-20] —

Last modified date: <%+ tp.file.last_modified_date() %>

• Tags:
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • #todo/create-links

Motivation

Goal: Classify Lagrangians \(L\hookrightarrow M\) up to Hamiltonian isotopy.

Potential approach

• Fix \((M, \omega)\), define a category \({\mathsf{Fuk}}(M, \omega)\) whose objects are Lagrangian submanifolds \(L \hookrightarrow M\).

• The moduli space \({\operatorname{Ob}}{\mathsf{Fuk}}(M, \omega)\) is infinite-dimensional, locally \(\ker d^1: \Omega^1_{/ {L}} \to {\mathcal{O}}_L\) (closed 1-forms); quotient by Hamiltonian isotopy to get \(\beta_1(L)\) (first Betti number).

• Known in special cases, e.g. Lagrangian \(S^2 \hookrightarrow S^2\times S^2\).

• Unknown: Lagrangian \(T^2 \hookrightarrow{\mathbf{P}}^3\)?

• Essentially known for \(X^1\), defined by:

  • Let \(X \hookrightarrow{\mathbf{C}}^N\) be a smooth affine algebraic variety
  • Take the projective closure to get a smooth projective \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\hookrightarrow{\mathbf{CP}}^N\)
  • Restrict the Fubini-Study-Kahler form \(\omega_{ \text{FS} }\) on \({\mathbf{CP}}^N\) to make \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) an exact symplectic manifold
  • Assume \(K_{\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu}\cong {\mathcal{O}}_{\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu}(- d)\) for some \(d\) so \(X\) is CY.
  • Take a hyperplane section \(X^1 \coloneqq X \cap{\mathbf{C}}^{N-1}\).
  • Theorem: For \(d\neq -2, n\geq 1, \operatorname{ch}{ \mathbf{F} }\neq 2\), a certain Fukaya-type category \(\Pi \mathsf{Tw}{\mathsf{Fuk}}(X^1, \omega_{{ \text{FS} }})\) is “computable” from combinatorial data arising from Picard-Lefschetz theory on \(X\), where \({\mathsf{Fuk}}(X^1, \omega_{ \text{FS} })\) is an \(A_\infty\) category.
    • Idea: embed \(\mathsf{A}\in A_\infty(\mathsf{Cat}) \hookrightarrow\mathsf{Tw}\mathsf{A} \in A_\infty(\mathsf{Cat})\) into a triangulated \(A_\infty{\hbox{-}}\)category, take derived category \(D(\mathsf{A}) \coloneqq H^0 \mathsf{Tw}\mathsf{A}\) to get a usual triangulated category, take a split closure/Karoubi-completion to get \(D^\pi(\mathsf{A})\) (close under taking summands of idempotent endomorphisms), lift this to a similar construction at the \(A_\infty\) level called \(\Pi \mathsf{Tw}\mathsf{A}\)

• New goal: find a (hopefully finite) set of Lagrangians \(L_i\hookrightarrow M\) which split-generate \(\Pi \mathsf{Tw}\mathsf{A}\): every object is obtained by taking iterated mapping cones and direct summands (weak property!).

• Conjecture (Arnold): if \(M\) is closed compact and \(L\hookrightarrow{\mathbf{T}}M\) is a closed compact exact Lagrangian with its standard Liouville form, then \(L\) is is Hamiltonian isotopic to the zero section of \({{\mathbf{T}}M} \to M\).

Defining \({\mathsf{Fuk}}\)

Main characterization of \(\mathsf{A} \coloneqq{\mathsf{Fuk}}(M, \omega)\): setting \(\mathsf{A}(L, L') = \operatorname{CF}(L, L')\) with differential \(\mu_1\), composition \(\mu_2\), and higher operations \(\mu_k\) makes \({\mathsf{Fuk}}(M, \omega)\) into a - \(\Lambda{\hbox{-}}\)linear - \({\mathbf{Z}}{\hbox{-}}\)graded - non-unital - but cohomologically unital - \(A_\infty\) category.

title: Definition
color: rgba(0,0,0)

**Definition**: the category $\mathsf{nu}\dash A_\infty(\Cat)$ of **non-unital $A_\infty$categories:** fix a field $\FF$, then $\cat A$ is the data of 

- A *set* of objects $\Ob \cat A$
- A $\ZZ\dash$graded vector space $\cat A(x, y)$ for every two objects
- For every $d\in \ZZ_{\geq 1}$, a composition map
$$\mu_{\cat A}^d: \cat{A}(x_{d-1}, x_d) \tensor_\FF \cat{A}(x_{d-2}, x_{d-1}) \tensor_\FF \cdots \tensor_\FF \cat{A}(x_{0}, x_1) \to \cat{A}(x_0, x_d)[2-d]
,$$
where $V[n]$ for a graded vector space denotes shifting the grading *down* by $n$.
![](attachments/Pasted%20image%2020220920105929.png)

- For every such $d$, $A_\infty$ associativity relations:
$$
\scriptstyle R_d\da \sum_{m=1}^d \sum_{n=0}^{d-m} (-1)^{ \eta_n } \mu_{\cat A}^{d-m+1}\qty{ a_d,\,\, a_{d-1}\,\, \cdots,\,\, a_{n+m+1}, \quad \star(n, m),\quad a_n,\,\, a_{n-1},\,\, \cdots,\,\, a_1} = 0
$$
where $\star(n, m) = \mu_{\cat A}^m(a_{n+m},\, a_{n+m-1}, \cdots, \, a_{n+1} )$ and $\eta_n \da \sum_{i=1}^n \abs{a_i} - n$.

> Keep $\eta$ on board.

NB: non-unital means “not necessarily unital”.

Remark: This category carries “higher products” coming from stringing together multiple morphisms. What this looks like in our case, at least when \([\omega].\pi_2(M, L_i) = 0\) (zero symplectic area for all spheres):

The map is defined by \begin{align*} \mu^k(p_k, \cdots, p_1) = \sum_{q\in L_0 \pitchfork L_k, \left\{{[u] {~\mathrel{\Big\vert}~}\operatorname{Ind}u = 2-k}\right\}} {\sharp}{\mathcal{M}}(p_1,\cdots, p_k, q; \, [u], J)T^{\omega([u])} q \end{align*}

For \(k=2\):

Remark: Why keep track of higher products? Compare cup-products (2-ary) to Massey products (3-ary): In the usual DGA setting, one can realize the Massey product \(\left\langle{x,y,z}\right\rangle_3\) as a “composition” \(\mu^3(x,y,z)\).

title: Definition


**Definition**: For $R$ a ring, a category $\cat A$ is **$R\dash$linear** iff it is *enriched* over the monoidal category $(\rmod, \tensor_R)$, i.e. $\cat A(x, y)\in \rmod$ and composition $\cat A(x, y)\tensor_R \cat A(y,z)\to \cat A(x, z)$ is a morphism in $\rmod$. The category is **$\ZZ\dash$graded** if $\cat A(x, y) = \oplus_{n\in \ZZ} \cat A(x, y)_n$, i.e. every hom set decomposes into $\ZZ\dash$graded pieces. It is a **differential $\ZZ\dash$graded category** if it is enriched over $(\Ch(\rmod), \tensor_{R, \gr} )$, i.e. there are differentials $\bd_{x,y,n}: \cat A(x, y)_n \to \cat A(x, y)_{n+1}$ of square zero.

Remark: Typically take \(R = { \mathbf{F} }\) or \(\Lambda\) a field to get vector spaces.

title: Definition

**Definition**: Let $\cat A\in \mathsf{nu}\dash A_\infty(\Cat)$. Its **cohomological category** $H(\cat A)$ has 

- the same objects as $\cat A$,
- morphisms given by taking cohomology of the morphisms of $\cat A$, i.e. $H(\cat A)(x, y) = H^*( \cat A(x, y), \mu_{\cat A}^1)$
- Composition defined by $[g] . [f] \da (-1)^{ \abs{g} } [ \mu^2_{\cat A}(g, f) ]$.

Remark: \(H(\mathsf{A})\) is generally an (ordinary) \(R{\hbox{-}}\)linear \({\mathbf{Z}}{\hbox{-}}\)graded category, except it may not have identity morphisms. This the notion of isomorphism is delicate. The \(A_\infty\) relations will imply that \(\mu^2_{\mathsf{A}}\) descends to an associative composition on cohomology. If \(\mathsf{A} \coloneqq{\mathsf{Fuk}}(M, \omega)\), then \(H^0(\mathsf{A})\) is sometimes called the Donaldson-Fukaya category. However, important information in the higher \(\mu^i\) is lost.

title: Definition


**Definition**: For $\cat A, \cat B\in \mathsf{nu}\dash A_\infty(\Cat)$, define **non-unital $A_\infty$ functors** $F\in \mathsf{nu}\dash\Fun(\cat A, \cat B)$ as 

- A map $F: \Ob \cat A\to \Ob \cat B$
- For every $d\geq 1$, 
$$
F^d: \cat{A}(x_{d-1}, x_d) \tensor_\FF \cdots \cat A(x_0, x_1) \to \cat{B}(Fx_0, Fx_d)
$$
- Relations
$$
\begin{align}
\sum_{r=1}^\infty \sum_{s_1+\cdots + s_r = d} \mu_{\cat B}^r (\, F^{s_r}(a_d, \cdots, a_{d-s_r+1}),\,\, \cdots, \,\,F^{s_1}(a_{s_1}, \cdots, a_1 )\, ) )
\\ =
\sum_{m, n} (-1)^{\eta_n} F^{d-m+1}(a_d, \cdots, a_{n+m+1}, \,\, \mu_{\cat A}^m(a_{n+m}, \cdots, a_{n+1} ), \,\,\cdots, a_n, \cdots, a_1)
\end{align}
$$

$H(F): H(\cat A)\to H(\cat B)$ is an ordinary linear graded non-unital functor whose action on morphisms is $[f] \mapsto [F^1(f)]$. We say $F$ is **cohomologically full (resp. faithful)** if $H(F)$ is full (resp. faithful), and $F$ is a **quasi-isomorphism** if $H(F)$ is an isomorphism. Two $A_\infty$ categories are **quasi-isomorphic** iff there exists a quasi-isomorphism.

title: Definition

**Definition:** the category $\cat Q\da \mathsf{nu}\dash A_\infty(\Cat)$ has objects $F$ as above.
Its morphisms are chain complexes, an element $T\in \cat{Q}(F, G)_g$ is a sequence $(T^0, T^1,\cdots)$ where each $T^d$ is a family of multinear maps of degree $(g-d)$:
$$
\cat{A}(x_{d-1}, x_d)\tensor_\FF \cdots \cat{A}(x_0, x_1) \to \cat{B}(Fx, Gx_d)[g-d] \qquad\forall (x_0,\cdots, x_d)\in \cat{A}
$$
E.g. $T^0$ is a family of maps in $\cat B(F x, Gx)_g$ for each objects $x\in \cat{A}$.
We call $T$ a **pre-natural transformation** from $F$ to $G$.

title: Definition

**Definition**: Say $F, G\in \Ob \cat Q$ are **homotopic** if the following holds: let $D = F- G \in \cat{Q}(F, G)_1$ be the pre-natural transformation defined by

- $D^0 = 0$
- $D^d = F_0^d - G_1^d$ for $d>0$

This yields an ordinary natural transformation where $\mu^1_{\cat Q}(D) = 0$.
We say that $F, G$ are **homotopic** if $D = \mu^1_{\cat Q}(T)$ for some $T\in \cat{Q}(F, G)_0$. where $T^0 = 0$.

Remark: homotopic functors \(F\simeq G\) induce isomorphic functors on homological categories, \(H(F) \cong H(G)\).

title: Definition

**Definition**: For a fixed $\cat A\in \mathsf{nu}\dash A_\infty(\Cat)$, the category of **right $A_\infty\dash$modules over $\cat A$** is defined as $\mathsf{nu}\dash\mods{A} \da \mathsf{nu}\Fun(A\op, \Ch(\mods{\FF} ))$. An object $M \in \mathsf{nu}\dash\mods{A}$ is a graded $\FF\dash$modules $M(x)$ for each $x\in \cat A$, along with maps
$$
\mu^d_M: M(x_{d-1}) \tensor_\FF \cat{A}(x_{d-2}, x_{d-1}) \tensor_\FF \cdots \tensor_\FF \cat{A}(x_0, x_1) \to M(x_0)[2-d]
$$
This induces $H(M) \in \mathsf{nu}\dash\mods{H(\cat A)}$, i.e. a functor $H(M) \in \mathsf{nu}\dash\Fun(H(A), \Ch(\mods \FF))$, which for every $x\in A$ is the cohomology of $M(x)$ with respect to the differential $b \mapsvia{\del} (-1)^{\abs b}\mu^1_M(b)$.

title: Definition


**Definition**: A usual category is **unital** if it has identity morphisms for every object. A category $\cat A\in \mathsf{nu}\dash A_\infty(\Cat)$ is **strictly unital** if for each $x\in \cat A$ there is a unique $e_x \in \cat{A}(x, x)_0$ such that

- $\mu_{\cat A}^1(e_x) = 0$
- For every $a\in \cat{A}(x_0, x_1)$, 
  $$(-1)^{\abs a} \mu_{\cat A}^2(e_{x_1}, a) = \mu_{\cat A}^2(a, e_{x_0}) = a$$
- For $a_k \in \cat{A}(x_{k-1}, x_k)$ and any $d>2$ and $0\leq n < d$,
$$
\mu_{\cat A}^d(a_{d-1}, \cdots, a_{n+1}, e_{x_n}, a_n, \cdots, a_1) = 0
$$

We say $\cat A$ is **cohomologically unital** or **$c\dash$unital** iff $H(A)$ is a unital, making it an ordinary graded linear category.

title: Definition

**Definition**: A $\cat{A}\in \mathsf{nu}\dash A_\infty(\Cat)$ is **homotopy unital** if 

- $\Ob(\cat A)$ forms a set.
- Homs $\cat{A}(x_0, x_1)$ are graded vector spaces,
- There are multilinear maps
$$
\mu_{\cat A}^{d, (i_d,\cdots, i_0)} : \cat{A}(x_{d-1}. x_d) \tensor_\FF \cdots \tensor_\FF \cat{A}(x_0, x_1) \to \cat{A}(x_0, x_d)[2-d-2\sum_k i_k]
$$
- Satisfying generalized associativity equations which reduce to the usual ones when $i_1=\cdots = i_d=0$:
      - $\mu_{\cat A}^1(\mu_{\cat A}^{0, (1) } ) = 0$
      - $$(-1)^{|a|-1} \mu_{\cat{A}}^2\left(\mu_{\cat{A}}^{0,(1)}, a\right)+\mu_{\cat{A}}^1\left(\mu_{\cat{A}}^{1,(1,0)}(a)\right)+\mu_{\cat{A}}^{1,(1,0)}\left(\mu_{\cat{A}}^1(a)\right)=a$$
      - $$\mu_{\cat{A}}^2\left(a, \mu_{\cat{A}}^{0,(1)}\right)+\mu_{\cat{A}}^1\left(\mu_{\cat{A}}^{1,(0,1)}(a)\right)+(-1)^{|a|-1} \mu_{\cat{A}}^{1,(0,1)}\left(\mu_{\cat{A}}^1(a)\right)=-a,$$
      - $$\mu_{\cat{A}}^{1,(1,0)}\left(\mu_{\cat{A}}^{0,(1)}\right)+\mu_{\cat{A}}^{1,(0,1)}\left(\mu_{\cat{A}}^{0,(1)}\right)+\mu_{\cat{A}}^1\left(\mu_{\cat{A}}^{0,(2)}\right)=0 .$$

Remark: equations 1 and 2 say that multiplication with the cocycle \(e_x = \mu_{\mathsf{A}}^{0, (1)}\) is chain homotopy to the identity. The others say \(\mu_{\mathsf{A}}^2(e_x, e_x) = e_x\) up to a coboundary, and the difference of any two such coboundaries is a cohomologically trivial cocycle. Continuing these equations yields higher such coherens.

Remark: \({\mathsf{Fuk}}(M, \omega)\) will not be strictly unital but will be cohomologically unital and homoopty unital, and homotopy units can be constructed geometrically. Moreover any homotopy unital \(A_\infty\) category is quasi-isomorphic to a strictly unital \(A_\infty\) category in a canonical way, and there is a general procedure to equip a c-unital category with homotopy units. So we can just work with c-unital categories.