We are having a small discussion group around Tamarkin’s paper “microlocal category”. There will be a blog post for each chapter, in which we collect together our understandings and misunderstandings. At the moment they are mostly empty. This is the post for chapter 1.

In the “microlocal category” paper: [Tam15], Tamarkin proposed a way to associate to a Fukaya-like category in the microlocal world, termed as microlocal category, to any compact symplectic manifold $M$ with symplectic form having integral periods. It’s expected to possess properties similar to the standard Fukaya category.

Here, following a secret reading group on Tamarkin’s paper, I will write my understandings and confusions about some of the preliminaries and chapter 1, concerning the local model ($Sh_q(X)$ and $Sh_{\epsilon}(X)$) in the construction of the microlocal category.

0. Preliminaries. Ref: [Tam08, section 2 & appendix 2]

“$Sh(X)$”:

Given a smooth manifold $X$, we can associate a sheaf category $Sh(X)$ (this is $D(X)$ in the “non-displaceability” paper), as follows:

Fix a base field $\mathbb{K}$. Consider the unbounded derived category $D(X\times \mathbb{R}_t)$ of complexes of sheaves of $\mathbb{K}$-vector spaces on $X\times \mathbb{R}$, and let $C_{\leq 0}(X)$ be the full subcategory of objects whose microsupport is non-positive in the $t$-direction. Then we define $Sh(X):=D(X\times\mathbb{R})/C_{\leq0}(X)$. It’s also equivalent to the left orthogonal complement to $C_{\leq 0}(X)$ in $D(X\times\mathbb{R})$ [Tam08, prop. 2.1]. So we can always view $Sh(X)$ as a full subcategory of $D(X\times\mathbb{R})$.

“Convolution in the $\mathbb{R}$-direction”:

Let $p_1, p_2$ be the projections of $(X\times\mathbb{R})\times\mathbb{R}$ onto the 1st and 2nd factors resp. Let $a:(X\times\mathbb{R})\times \mathbb{R} \rightarrow X\times\mathbb{R}$ be the map of “the addition of the last 2 factors”. We define

$$*_{\mathbb{R}}:D(X\times\mathbb{R})\times D(\mathbb{R})\rightarrow D(X\times\mathbb{R})$$

by $F*_{\mathbb{R}}S:=Ra_!(p_1^{-1}F\otimes p_2^{-1}S)$.

A good example to have in mind: Take $X=$ point, $F=\mathbb{K}_{(a,b)}$, $S=\mathbb{K}_{[c,\infty)}$, an easy calculation shows that $\mathbb{K}_{(a,b)}*_{\mathbb{R}}\mathbb{K}_{[c,\infty)} = \mathbb{K}_{[b+c,\infty)}[-1]$. Also, it’s easy to see that $\mathbb{K}_{(a,b)}*_{\mathbb{R}}\mathbb{K}_0=\mathbb{K}_{(a,b)}$.

Note: According to [Tam08, prop.2.1. 1)], any object in $D(X\times\mathbb{R})$ can be obtained from objects of the form $\mathbb{K}_{U\times(a,b)}$ ($U$ open in $X$) by taking direct limit. Though, I don’t know how to prove it?

Via this “convolution” operation, the sheaf category $Sh(X)$ has another characterization [Tam08, prop. 2.2]: An object $\mathcal{F}\in D(X\times\mathbb{R})$ is in $Sh(X)$ if and only if the natural map $F*_{\mathbb{R}}\mathbb{K}_{[0,\infty)}\rightarrow F*_{\mathbb{R}}\mathbb{K}_0=F$ is an isomorphism.

In the same example as above, we know $Sh(X=\text{point})$ is the category of sheaves on $\mathbb{R}$ microsupported away from negative co-directions. For example, $\mathbb{K}_{[a,b)}$ satisfies this microsupport condition. (Up to a degree shift, they give all the indecomposables). This is consistent with the characterization above, as $\mathbb{K}_{[a,b)}*_{\mathbb{R}}\mathbb{K}_{[0,\infty)}=\mathbb{K}_{[a,b)}$.(Confusion: what happens to $\mathbb{K}_{(-\infty,b)}$?)

“Translation”:

For $c\in\mathbb{R}$, let $T_c: X\times\mathbb{R}\rightarrow X\times\mathbb{R}$ be map of “add the second factor by $c$”. Denote by the same symbol $T_c: Sh(X)\rightarrow Sh(X)$ for the pushforward. The previous characterization of $Sh(X)$ shows also $T_cF=F*_{\mathbb{R}}\mathbb{K}_{[c,\infty)}$.

For $c\geq d$, we then immediately get a natural transformation $\tau_{d,c}:T_d\Rightarrow T_c$, induced by the natural map $\mathbb{K}_{[d,\infty)}\rightarrow\mathbb{K}_{[c,\infty)}$.

$Sh_q(X)$ and $Sh_{\epsilon}(X)$. (Warning: The following definition might be wrong…)

In [Tam15], the construction of the microlocal category involves some deformed version of $Sh(X)$:

“quantized sheaf category $Sh_q(X)$”:

The objects of $Sh_q(X)$ are the same as those of $Sh(X)$; The morphisms are modified as follows:

Let $\mathbb{Q}$-$mod$ be the category of complexes of $\mathbb{Q}$-vector spaces. Given $\mathcal{F}, \mathcal{G}$ in $Sh_q(X)$, $Hom_{Sh_q(X)}(\mathcal{F}, \mathcal{G}):=\oplus_{n\in\mathbb{Z}}\mathrm{gr}^{n\epsilon}Hom_{Sh_q(X)}(\mathcal{F}, \mathcal{G})$ (equipped with a Maurer-Cartan elemment $D$ ?), is a $\epsilon\mathbb{Z}$-graded object equiped with a $\mathbb{Z}[q]$-action in $\mathbb{Q}$-$mod$. For simplicity, we will write $Hom_{q}$ for $Hom_{Sh_q(X)}$. Note that now we have 2 gradings on $Hom_q(\mathcal{F}, \mathcal{G})$, the other one coming from the complex grading.

Here: $\mathrm{gr}^{n\epsilon}Hom_q(\mathcal{F},\mathcal{G}):=Hom(F,T_{-n\epsilon}(\mathcal{G}))$; $\epsilon>0$ is a small number, and $q$ is endowed with the grading $-\epsilon$; The action of $q$: $\mathrm{gr}^{-n\epsilon}Hom_q(\mathcal{F},\mathcal{G})\rightarrow \mathrm{gr}^{-(n+1)\epsilon}Hom_q(\mathcal{F},\mathcal{G})$ is induced by $\tau_{n\epsilon,(n+1)\epsilon}$;

A Maurer-Cartan element of a complex $(C,d)$ is a map $D\in Hom^1(C,C)$ (“connection”) satisfying the Maurer-Cartan equation $dD+D^2=0$ (“curvature=0”) (see also below, “ground categories”).

“The $\epsilon$-quasi-classical limit: $Sh_{\epsilon}(X)$”:

This is obtained from $Sh_q(X)$ by a ‘derived reduction modulo $q$’. The objects of $Sh_{\epsilon}(X)$ are the same as those in $Sh_q(X)$. The morphisms are defined as follows:

Given $\mathcal{F}, \mathcal{G}$, $Hom_{\epsilon}(\mathcal{F},\mathcal{G})=\oplus_{n\in\mathbb{Z}}\mathrm{gr}^{n\epsilon}Hom_{\epsilon}(\mathcal{F}, \mathcal{G})$ with

$\mathrm{gr}^{n\epsilon}Hom_{\epsilon}(\mathcal{F},\mathcal{G}):=\mathrm{cone}(q:\mathrm{gr}^{(n+1)\epsilon}Hom_q(\mathcal{F},\mathcal{G})\rightarrow \mathrm{gr}^{n\epsilon}Hom_q(\mathcal{F},\mathcal{G}))$.

Use the short exact sequence $0\rightarrow \mathbb{K}_{[-(n+1)\epsilon,-n\epsilon)}\rightarrow \mathbb{K}_{[-(n+1)\epsilon,\infty)}\rightarrow\mathbb{K}_{[-n\epsilon,\infty)}\rightarrow 0$, we can rewrite

$\mathrm{gr}^{n\epsilon}Hom_{\epsilon}(\mathcal{F},\mathcal{G})=Hom(F,G*_{\mathbb{R}}\mathbb{K}_{[-(n+1)\epsilon,-n\epsilon)})[1]$.

From the definition, we get a natural functor $red:Sh_q(X)\rightarrow Sh_{\epsilon}(X)$, which may be called the ‘$\epsilon$-quasi-classical reduction’.

“Ground categories”:

The sheaf categories $Sh_q(X)$ (resp. $Sh_{\epsilon}(X)$) is enriched over $Quant(\epsilon)$ (resp. $Classic(\epsilon)$).

$Quan(\epsilon)$:

The objects of $Quant(\epsilon)$ are the $\epsilon\mathbb{Z}$-graded objects $X$ equipped with a Maurer-Cartan element $D_X$, in $\mathbb{Q}$-$mod$. The morphisms are:

$Hom_{q}((X,D_X), (Y,D_Y)):=(Hom_Q(X,Y),D_{XY})$, with $Hom_Q(X,Y):=\prod_{m\leq n}Hom(\mathrm{gr}^{m\epsilon}X,\mathrm{gr}^{n\epsilon}Y)$ as a vector space, and the differential $D_{XY}$ defined by

$D_{XY}f:=df+ad(D)(f)$ (“covariant derivative”)

where $d$ is the usual differential on $Hom_Q(X,Y)$, and as always $ad(D)f=[D, f]$ is the supercommutator.

One can check that $D_{XY}$ indeed gives a differential. By the (graded) Leibniz Rule, we have

\begin{eqnarray}D_{XY}^2f&=&(d+ad(D))^2f=(d^2+d\circ ad(D)+ad(D)\circ d+ad(D)^2)f\\&=&d[D,f]+[D,df]+ad(D)[D,f]\\&=&([dD,f]-[D,df])+[D,df]+ad(D)[D,f]\end{eqnarray}

Note: $ad(D)[D,f]=[ad(D)D,f]-[D,ad(D)f]$ implies $ad(D)[D,f]=\frac{1}{2}[ad(D)D, f]=[D^2, f]$. It follows that:

\begin{equation}D_{XY}^2f=[dD+D^2,f]\end{equation}

So it vanishes precisely when $d_XD_X+D_X^2=0$ and $d_YD_Y+D_Y^2=0$. This probably explains where the Maurer-Cartan element comes from.

Note: For $(X,D_X)$ in $Quant(\epsilon)$, let $D_{nm}\in Hom^1(\mathrm{gr}^{n\epsilon}X, \mathrm{gr}^{m\epsilon}X), n\leq m$ be the components. In the paper, it says that, “every object $(X,D_X)$ is isomorphic to that with $D_{nn}=0$”. So, we will always assume this. Again, I don’t how how to prove it?

$Classic(\epsilon)$:

This is simply the category of $\epsilon\mathbb{Z}$-graded objects in $\mathbb{Q}$-$mod$, with morphisms $Hom_{\epsilon}(X,Y):=\prod_nHom(\mathrm{gr}^{n\epsilon}X,\mathrm{gr}^{n\epsilon}Y)$.

By definition, we get a natural functor $red: Quant(\epsilon)\rightarrow Classic(\epsilon)$ via $(X,D)\rightarrow X$.

The following is something I haven’t understood at all:

The goal of the paper: Consider a family of symplectic balls in $M$ $I:F\times B_R\rightarrow M$, and define an $A_{\infty}$-algebra $A$ with curvature in the monoid category $Sh_q((F\times B_R)^2)$. The latter acts on $Sh_q(F\times B_R)$ by convolutions, this allows us to define the desired microlocal category as that of $A$-modules in $Sh_q(F\times B_R)$. The construction of $A$ starts with an algebra $A_0$ in the ‘$\epsilon$-quasi-classical limit’ $Sh_{\epsilon}(F\times B_R)$, then deals with the lifting into $Sh_q(F\times B_R)\xrightarrow[]{red}Sh_{\epsilon}(F\times B_R)$.

Above all, is there any interpretation of the idea behind the construction?

Here are some of my questions and understandings for chapter $1$.

1. What is a “traditional quantization setting”? If I understand correctly, what Tamarkin does in this paper is a dg sheaf theory analogue of it. So I guess it might be helpful if we know it? There is still some difference. Instead of using objects over the ring $\mathbb{Q}[[q]]$, we want to, naively, use the category of $\epsilon . \mathbb{Z}$-graded objects in the category of complexes of $\mathbb{Q}$-vector spaces, with a $\mathbb{Q}[q]$ action. Tamarkin denotes this category by $\mathcal{G}$ for this chapter and $\mathbb{Q}[q]$ can be regarded as a ring object in $\mathcal{G}$ by $\mathbf{gr}^{k} = \mathbb{Q} \langle q^k \rangle $ with $\mathbb{Q} \langle q^k \rangle $ regarded as a chain complex centered at degree $0$.

2. Tamarkin says that instead of using the category of $\mathbb{Q}[q]$-modules, we will want to use a full sub-category of “semi-free” objects, $\mathbf{Quant} \langle \epsilon \rangle$. The only reason he gives here is that we can avoid using derived tensor product if we use this category. My question is why, in general, we want to use semi-free objects when we dealing with dg categories? (Sorry, I know dg categories no more than definitions.)

3. For Tau’s question, “Note for $(X,D_X)$ in $\mathbf{Quant} \langle \epsilon \rangle$, $\cdots$ with $D_{nn} = 0$. So, we will always assume this. Again, I don’t how how to prove it?”, I think the reason is the same as why the dg category of $\mathbb{K}$-complexes is $D$-closed, $\mathbb{K} = \mathbb{Q}$ or $\mathbb{Z}$. Because you can absorb the $nn$-th piece of the MC-element $D$ into the chain complex $\mathbf{gr}^{n} X$.

4. I just want to point out that the notations and sometimes even the objects that Tamarkin use in the introduction are different from the text. Denote $\mathfrak{A}$ the symmetric monoidal category of complexes of $\mathbb{K}$-modules.

First, the $\mathbf{Quant} \langle \epsilon \rangle$ and $ \mathbf{Classic} \langle \epsilon \rangle$ he defines in the introduction is denoted as $\mathbf{Com} (\mathfrak{A}) \langle \epsilon \rangle$ and $ \mathbf{Gr} (\mathfrak{A}) $ in chapter 4.

Then, for a $D$-closed dg category $\mathcal{C}$ enriched over $\mathfrak{A}$, he defines categories $\mathbf{Quant} (\mathcal{C}) \langle \epsilon \rangle$ and $ \mathbf{Classic} (\mathcal{C}) \langle \epsilon \rangle$ enriched over $\mathbf{Com} \langle \epsilon \rangle$ and $ \mathbf{Gr} \langle \epsilon \rangle$ whose objects are $\mathbb{R}$-graded objects in $\mathcal{C}$. Also, he can drop the $\epsilon$ to define $\mathbf{Quant} (\mathcal{C})$ enriched over just $\mathfrak{A}$. I haven’t figured out exactly how he does that and what are the relations of those categories with $sh(X)$, $sh_q(X)$, and $sh_\epsilon(X)$ so I won’t post here for now. What I want to mention is that the $\epsilon$ here might be associated with the convexity radius of the compact symplectic manifold $M$ (associated with a pseudo-Kaehler in the later chapters).

1. “traditional quantization setting” means either geometric quantization, or deformation quantization, I am not sure exactly which, and maybe both. Naively it’s closer to the latter, which usually means deforming the sheaf of functions on a manifold to a sheaf of noncommutative algebras in a parameter h, such that to first order, the commutator is h times the poisson bracket.

2. He’s just saying that to compute a derived tensor product you should first take a projective resolution, so to avoid worrying about it, just assume all items are projective to begin with. (to learn about dg categories etc. read Keller’s “On dg categories”)