Tags: #my/talks #geomtop/symplectic-topology

Talbot Talk 2: Hiro

Part 1

Assign to a symplectic manifold a Fukaya category
- An $A_\infty$ category, slightly different than homotopy-theoretic notion. See A_infty category.
- A DG category
- A ${\mathbf{Z}}{\hbox{-}}$linear stable infinity category.
Replace with a functor ${\mathsf{Fuk}}$ that takes a category of symplectic manifolds to a stable infty-category over ${\mathbf{Z}}$.
Analogies:
- The “take modules” functor $(\mathsf{Ring})^{\operatorname{op}}\to {\operatorname{Stab}}{ \underset{\infty}{ \mathsf{Cat}} }{}{/ {{\mathbf{Z}}}} $ given by $R\mapsto {}{R}{\mathsf{Mod}} $
- ${\mathsf{Sch}}^{\operatorname{op}}\to {\operatorname{Stab}}{ \underset{\infty}{ \mathsf{Cat}} }{}_{/ {{\mathbf{Z}}}} $ where $X\mapsto { \mathsf{D} }^b{\mathsf{Coh}}(X)$.
Recent #open/conjectures: for certain $M$, make an ${\mathbb{S}}{\hbox{-}}$linear functor ${\mathsf{Fuk}}({-}, {\mathbb{S}})$ where ${\mathsf{Fuk}}(M, {\mathbb{S}})$ is a stable infty category
- Can get stable infty categories out of very geometric things: symplectic manifolds
- Hope to get an equivalence of categories between some infty category of symplectic manifolds and the infty category of stable infty categories
Morse theory recap
- Index: write $f$ locally as $\sum x_i^2 - \sum y_i^2$ and the number of negative components is the index of the critical point
Weinstein manifolds and sectors: special types of symplectic manifolds obtained from handle attachment (sectors: allowing boundaries)
- Allows some mild but controlled singularities making them non-manifolds
- Can construct interesting cosheaves of categories
Defining a symplectic manifold:
- $\omega^{\wedge 2}$ defines a volume form, or use $v\mapsto \omega(v, {-})$ is a non-degenerate 1-form, thinking of $\omega: TM \xrightarrow{\sim}{\mathbf{T}} {}^{ \vee }M$.
- The latter definition is useful in derived geometry.
- $d\omega = 0$, a flatness condition.
The most important example: for $Q$ any smooth manifold, the total cotangent space $T {}^{ \vee }Q, dp \wedge dq)$ is symplectic.
- Locally write coordinates ${ {q}_1, {q}_2, \cdots, {q}_{n}}$, get ${ {dq}_1, {dq}_2, \cdots, {dq}_{n}}$, then $\sum p_i dq_i\in{\mathbf{T}} {}^{ \vee }{\mathbf{R}}^n$. Take de Rham derivative to get $\sum dp_i \wedge dq_i \in \Omega^2({\mathbf{R}}^n)$.
Can make some symplectic manifolds out of Weinstein cells.
Taking a one form $\alpha = \omega({-}, X)$, it turns out $d\alpha = \omega$ so $\alpha$ is an antiderivative.
Fact: if $M$ is compact of dimension $d\geq 2$ then $M$ can not be Weinstein.
Some kind of “symplectic Pontryagin Thom” theorem
Note: need to distinguish between actual boundaries and “boundaries at infinity”

Part 2

Constructing the (wrapped) Fukaya category
A half-dimensional submanifold $L$ of a symplectic manifold is Lagrangian iff $\omega{ \left.{{}} \right|_{{L}} } = 0$.
- Example: any curve in $M$, since a two-form restricted to a one-manifold is trivial - $Q \hookrightarrow{\mathbf{T}} {}^{ \vee }Q$ - Any cotangent fiber $T_q {}^{ \vee }Q$
Almost complex structure used to define a differential equation
Informal definition of ${\mathsf{Fuk}}(M)$: it’s like a DG category
- Objects are Lagrangians
- $\mathop{\mathrm{Hom}}(L_0, L_1)$ is like a chain complex: a graded abelian group $\bigoplus_{z\in L_2 \pitchfork L_1} {\mathbf{Z}}/2[d]$ for some shift $d$ with differential ${\partial}$ whose coefficients are given by counting holomorphic discs from $x$ to $y$.
- Composition is given by $y\otimes x\mapsto \sum ? z$ where the count is given by counting holomorphic discs mapping to the triangle $x,y,z$. Note: non-associative, need to consider discs filling in punctured $n{\hbox{-}}$gons for all $n$
- Can recover presentation of Stasheff associahedra.
There is an equivalence ${\mathsf{Fuk}}(M) \xrightarrow{\sim} {\mathsf{Fuk}}(M \times{\mathbf{T}} {}^{ \vee }{\mathbf{R}}^N)$ where $L\mapsto L \times T_0 {}^{ \vee }{\mathbf{R}}^N$, take colim to replace $N$ with $\infty$.
Need to do wrapping, but we won’t get into it.
In the category of Weinstein manifolds, a morphism is a codimension 0 embedding $j: M\hookrightarrow(N, X_N)$ where we convert $X_N$ to a one-form $\theta_N$ using $\omega$, such that $j^* \Theta_N = \Theta_M + df$ for $f$ some compactly supported function.
Theorem: the wrapped Fukaya category defines a functor from the category of Weinstein manifolds to $A_\infty{\hbox{-}}\mathsf{Cat}_{/ {{\mathbf{Z}}}} $ which factors through taking $M\times{\mathbf{T}} {}^{ \vee }{\mathbf{R}}^N$.
- The target has an infinity category structure.
Ways to improve this to an ${\mathbb{S}}{\hbox{-}}$linear category:
- Reformulate ${\mathsf{Fuk}}(M)$ as the solution to a deformation problem. Very difficult!!
- From $M$ construct a stable $\infty{\hbox{-}}$category of Lagrangian cobordisms ${\mathsf{Lag}}(M)$ (already enriched in spectra). Conjecturally: ${\mathsf{Lag}}(M) \otimes_{\mathbb{S}}{H{\mathbf{Z}}}= {\mathsf{Fuk}}(M)$
- Microlocal special sheaves.
All three are conjecturally thought to work.
Question: can one symplectically construct certain $E_\infty$ maps, e.g. ${\mathbb{S}}, {\mathbb{S}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }$.? See E_infty.
- Yes, if we localize ${\mathsf{Wein}}$ in a certain way
There is a known class of maps $W$ where $M\to N$ induces ${\mathsf{Fuk}}(M) \xrightarrow{\sim} {\mathsf{Fuk}}(N)$.
Theorem: ${\mathsf{Wein}}{ \left[ { \scriptstyle \frac{1}{W} } \right] }$ is symmetric monoidal
- Can construct a symplectic manifold $D_p$ which is an $E_\infty$ algebra in ${\mathsf{Wein}}{ \left[ { \scriptstyle \frac{1}{W} } \right] }$ where ${\mathsf{Fuk}}(D_p)^\otimes\xrightarrow{\sim} {}_{{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}{\mathsf{Mod}}^\otimes$.
First case of a purely symplectic construction of an $E_\infty{\hbox{-}}$algebra! See E_infty algebra
- Which ones can we construct?
#open/conjectures: $\mathop{\mathrm{Hom}}_{{\mathsf{Wein}}{ \left[ { \scriptstyle \frac{1}{W} } \right] }}({\operatorname{pt}}, {\operatorname{pt}}) \simeq$ to the groupoid of finite spectra, or equivalently the space of functors from finite spectra to itself (since all are given by smash against a specific spectrum)
- Here ${\operatorname{pt}}\cong{\mathbf{T}} {}^{ \vee }{\mathbf{R}}^{\infty}$.
A way to make “functors are bimodules” concrete in this category.