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 $L\embeds M$ up to Hamiltonian isotopy. ## Potential approach - Fix $(M, \omega)$, define a category $\Fuk(M, \omega)$ whose objects are Lagrangian submanifolds $L \embeds M$. - The moduli space $\Ob\Fuk(M, \omega)$ is infinite-dimensional, locally $\ker d^1: \Omega^1\slice L \to \OO_L$ (closed 1-forms); quotient by Hamiltonian isotopy to get $\beta_1(L)$ (first Betti number). - Known in special cases, e.g. Lagrangian $S^2 \embeds S^2\times S^2$. - Unknown: Lagrangian $T^2 \embeds \PP^3$? - Essentially known for $X^1$, defined by: - Let $X \embeds \CC^N$ be a smooth affine algebraic variety - Take the projective closure to get a smooth projective $\bar X\embeds \CP^N$ - Restrict the Fubini-Study-Kahler form $\omega_\FS$ on $\CP^N$ to make $\bar X$ an exact symplectic manifold - Assume $K_{\bar X}\cong \OO_{\bar X}(- d)$ for some $d$ so $X$ is CY. - Take a hyperplane section $X^1 \da X \intersect \CC^{N-1}$. - **Theorem**: For $d\neq -2, n\geq 1, \characteristic \FF \neq 2$, a certain Fukaya-type category $\Pi \Tw \Fuk(X^1, \omega_{\FS})$ is "computable" from combinatorial data arising from Picard-Lefschetz theory on $X$, where $\Fuk(X^1, \omega_\FS)$ is an $A_\infty$ category. - Idea: embed $\cat A\in A_\infty(\Cat) \embeds \Tw\cat A \in A_\infty(\Cat)$ into a triangulated $A_\infty\dash$category, take derived category $D(\cat A) \da H^0 \Tw\cat A$ to get a usual triangulated category, take a split closure/Karoubi-completion to get $D^\pi(\cat A)$ (close under taking summands of idempotent endomorphisms), lift this to a similar construction at the $A_\infty$ level called $\Pi \Tw\cat A$ - **New goal**: find a (hopefully finite) set of Lagrangians $L_i\embeds M$ which *split-generate* $\Pi \Tw\cat A$: every object is obtained by taking iterated mapping cones and direct summands (weak property!). - **Conjecture (Arnold)**: if $M$ is closed compact and $L\embeds \T M$ is a closed compact exact Lagrangian with its standard Liouville form, then $L$ is is Hamiltonian isotopic to the zero section of $\TM \to M$. # Defining $\Fuk$ Main characterization of $\cat A \da \Fuk(M, \omega)$: setting $\cat{A}(L, L') = \CF(L, L')$ with differential $\mu_1$, composition $\mu_2$, and higher operations $\mu_k$ makes $\Fuk(M, \omega)$ into a - $\Lambda\dash$linear - $\ZZ\dash$graded - non-unital - but cohomologically unital - $A_\infty$ category. ```ad-col2 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 $$ \mu^k(p_k, \cdots, p_1) = \sum_{q\in L_0 \transverse L_k, \ts{[u] \st \ind u = 2-k}} \size \mcm(p_1,\cdots, p_k, q; \, [u], J)T^{\omega([u])} q $$ For $k=2$: ![](attachments/Pasted%20image%2020220918214036.png) **Remark**: Why keep track of higher products? Compare cup-products (2-ary) to Massey products (3-ary): ![](attachments/Pasted%20image%2020220918222129.png) In the usual DGA setting, one can realize the Massey product $\gens{x,y,z}_3$ as a "composition" $\mu^3(x,y,z)$. ```ad-col2 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 $R = \FF$ or $\Lambda$ a field to get vector spaces. ```ad-col2 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**: $H(\cat A)$ is generally an (ordinary) $R\dash$linear $\ZZ\dash$graded category, except it may not have identity morphisms. This the notion of isomorphism is delicate. The $A_\infty$ relations will imply that $\mu^2_{\cat A}$ descends to an associative composition on cohomology. If $\cat{A} \da \Fuk(M, \omega)$, then $H^0(\cat A)$ is sometimes called the **Donaldson-Fukaya category**. However, important information in the higher $\mu^i$ is lost. ```ad-col2 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. ``` ```ad-col2 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 $G$. ``` ```ad-col2 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 $F\homotopic G$ induce isomorphic functors on homological categories, $H(F) \cong H(G)$. ```ad-col2 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 $b \mapsvia{\del} (-1)^{\abs b}\mu^1_M(b)$. ``` ```ad-col2 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. ``` ```ad-col2 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 $e_x = \mu_{\cat A}^{0, (1)}$ is chain homotopy to the identity. The others say $\mu_{\cat A}^2(e_x, e_x) = e_x$ 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**: $\Fuk(M, \omega)$ 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.