text:
\section{Introduction}

\subsection{Symplectic Embedding}

A symplectic manifold $(X,\omega)$ is a manifold with a non-degenerate closed $2$-form $\omega$. Classically it appears naturally as phase spaces in Hamiltonian Mechanics. An embedding $\varphi:(X,\omega) \hookrightarrow (X',\omega')$ is called symplectic if $\varphi^*\omega'=\omega$. A basic question in symplectic geometry is to decide when a symplectic embedding between two symplectic manifolds exists. The first result, the origin of the question, is the Gromov non-squeezing theorem:
\begin{iThm}[\cite{gromov1985}]Equip $\bR^{2d}$ with the linear symplectic form. Let $B_{\pi r^2}=\{(x,p)\in \bR^{2d} : \|x \|^2+\|p\|^2 < r^2\}$, and $Z_{\pi R^2}=\{(x,p)\in \bR^{2d} : x_1^2+p_1^2< R^2 \}$.

If there is a symplectic embedding $\varphi: B_{\pi r^2}\rightarrow \bR^{2d}$, such that $\varphi(B_{\pi r^2})\subset Z_{\pi R^2}$, then $r\leq R$.
\end{iThm}
A structure related to the embedding question is the so-called symplectic capacity.
\begin{iDef*}Fix a class of symplectic manifolds $\mathcal{C}$, we say that a function $c:\mathcal{C} \rightarrow [0,\infty]$ is a symplectic capacity if the followings are true.
\begin{itemize}[fullwidth]
    \item(Monotonicity) If there is a symplectic embedding $\phi:(X,\omega) \hookrightarrow (X',\omega')$, then $c(X,\omega)\leq c(X',\omega')$.
    \item(Conformality) $c(X,r\omega)= rc(X,\omega)$, for $r\in\bR_{>0}$.
\end{itemize}
\end{iDef*}

Different capacities are defined. One example is the Gromov width, which is defined as \[w(X,\omega)=\sup\{\pi r^2: B_{\pi r^2}\xhookrightarrow{symp} (X,\omega) \}.\] However it is essential a re-formulation of the embedding question, and is hard to compute. There are other capacities defined by generating functions \cite{viterbo1992symplectic}, Hamiltonian dynamics \cite{EkelandHoferII}, and $J$-holomorphic curves \cite{ECHhutchings2011,gutthutchings2018capacities, SiegelHigherCapacities}. A great survey about symplectic capacities is \cite{quantitivesymplectic.2010}.

By the definition of capacities, we can derive different numerical constrains of symplectic embeddings. When the dimension is 4, the ECH-capacity is a very effective tool. The ECH-capacity is defined by the embedded contact homology (ECH)\cite{ECHhutchings2011}, which is a $J$-holomorphic curve invariant. McDuff shows that the ECH-capacity provides us the sharp obstruction of embeddings between ellipsoids of dimension $4$ in \cite{mcduff2011}. In general, we may find in \cite{cristofaro-gardiner2019} more results about symplectic embeddings in dimension 4 using the ECH-capacity.

When the dimension is greater than 4, we know fewer results. The Ekeland-Hofer capacity $(c^{\text{EH}}_k)_{k\in\bZ_{\geq 1}}$ is a sequence of symplectic capacities defined for compact star-shaped domains in $\bR^{2d}$ for all $d$, which is defined using Hamiltonian dynamics. The computation of $c^{\text{EH}}_k$ is known for ellipsoids and poly-disks, say:
\begin{align*}
    &c_k^{\text{EH}}(E(a_1,\dots,a_d))=\min\left\{T: \sum_{i=1}^d \big\lfloor {\frac{T}{a_i}} \big\rfloor \geq k\right\},\\
    &c_k^{\text{EH}}(D(a_1,\dots,a_d))=ka_1,
\end{align*}
where
\begin{align*}
    &E(a_1,\dots,a_d)=\left\lbrace u\in \bC^d:\sum_{i=1}^d \frac{\pi|u_i|^2}{a_i} < 1 \right\rbrace,\\
    &D(a_1,\dots,a_d)=\left\lbrace u\in \bC^d:{\pi|u_i|^2} < a_i, \forall i=1,\dots,d \right\rbrace,\\
    &\text{with }0<a_1\leq \cdots \leq a_d.
\end{align*}
On the other hand, Gutt and Hutchings constructed a sequence of capacities $(c_k^{
\text{GH}})_{k\in\bZ_{\geq 1}}$ in \cite{gutthutchings2018capacities} based on the positive $S^1$-equivariant symplectic homology. They computed $c_k^{
\text{GH}}$ of convex toric domains and concave toric domains. Here a toric domain is an open set defined as follows:
\[X_{\Omega}=\left\{u\in \bC^d: (\pi|u_1|^2,\dots,\pi|u_d|^2)\in \Omega\right\}\]
where $\Omega \subset \bR^d_{\geq 0}$. We say $X_{\Omega}$ is convex if $\widehat{\Omega}=\left\{(x_1,\dots,x_d): (|x_1|,\dots,|x_d|)\in \Omega\right\}$ is convex, and is concave if $\bR^d_{\geq 0}\setminus \Omega$ is convex. For example, ellipsoids $E=X_{\Omega_E}$ and poly-disks $D=X_{\Omega_{D}}$ are convex toric domains, where
\[\Omega_E=\left\{\sum_{i=1}^d \frac{x_i}{a_i} < 1\right\},\quad \Omega_{D}=\prod_{i=1}^d [0,a_i),\quad 0 <a_1 \leq \cdots \leq a_d.\]
Gutt and Hutchings computed $c_k^{
\text{GH}}$ of both convex and concave toric domains. For example, when $X_{\Omega}$ is convex, they showed that
\begin{equation}
  c_k^{\text{GH}}(X_{\Omega})=\min\left\{\|v\|_\Omega^*: v\in \mathbb{N}^d, \sum_{i=1}^dv_i=k\right\}=\inf\left\{T\geq 0:\exists  {z}\in \Omega_T^\circ, I( {z})\geq k\right\} \label{capacities of convex toric domains}
\end{equation}
where
\begin{align}\label{convex toric domain notations}
\begin{aligned}
&\|v\|_\Omega^*=\max\{\langle v,w\rangle: w\in \Omega\},\\
&\Omega_T^\circ=\{z\in \bR^d: T+\langle z, \zeta \rangle \geq 0, \forall \zeta \in \Omega\},\\
&I(z)=\sum_{i=1}^d \big\lfloor {-z_i} \big\rfloor.
\end{aligned}
\end{align}
So one can observe that for ellipsoids and poly-disks, $c_k^{
\text{GH}}=c_k^{\text{EH}}$.

Unfortunately, even for ellipsoids, we know that the obstructions, given by Ekeland-Hofer capacities, and then the Gutt-Hutchings capacities, are not sharp. One new progress on higher dimensional embeddings is given by Siegel in \cite{SiegelHigherCapacities, CompSiegelHigherCapacities}. Siegel studied further on the chain level structures of the $S^1$-equivariant symplectic homology(or the linearized contact homology), and then constructed some higher capacities using $L_\infty$-structure on the corresponding chain complexes. He gave sharp obstructions for embeddings between some stabilized ellipsoids.

In this paper, we construct a sequence of capacities $(c_k)_{k\in\bZ_{\geq 1}}$ on the class of admissible open sets with non-degenerate boundaries. Here, we say an open set $U\subset T^*\bR^d$ is admissible if tools of sheaf theory work(see \autoref{defadmissibledomains}). Examples of admissible open sets are bounded open sets and toric domains, most of our interests are included in.

Our main ingredient is the complex $C_{T,\ell}(U,\bK)$ defined by Tamarkin and Chiu in \cite{mc-Tamarkin,chiu2017}, where $U$ is admissible, $T\geq 0$, and $\ell\in \mathbb{O}=\{\text{odd numbers}\}$. There is a structure of $\bK[u]$-module on $H^*(C_{T,\ell}(U,\bK))$, and a distinguished class $\eta_{\ell,T}(U,\bK) \in H^0(C_{T,\ell}(U,\bK)) $. As a clumsy imitation of the definition of $c^{\text{GH}}_k$\cite[Definition 4.1]{gutthutchings2018capacities}, we define (see \autoref{definition of capacities})
\begin{align*}
\begin{aligned}
 &\textnormal{Spec}(U,k)\\ \coloneqq &\{T\geq 0: \exists\ell_0\in \bO,\, \forall\ell \in \bO, \, p_\ell \geq p_{\ell_0},\, \exists \gamma_\ell  \in H^{*}(C_{T,\ell}(U,\bF_{p_\ell})),\, \eta_{\ell,T}(U,\bF_{p_\ell})=u^k\gamma_\ell \},
\end{aligned}
\end{align*}
and
\[c_k(U)\coloneqq \inf \textnormal{Spec}(U,k) \in [0,+\infty].\]
Then we show
\begin{iThm}[\autoref{capacity property symplectic}] The functions $c_k:\mathcal{C}_{\text{admissible}}\rightarrow (0,\infty]$ satisfy the following:
\begin{enumerate}[fullwidth]
    \item $c_k \leq c_{k+1}$ for all $k\in \bZ_{\geq 1}$.
    \item If there is an inclusion of admissible open sets $U_1 \subset U_2$, then $c_k(U_1) \leq c_k(U_2)$.
    \item For a compactly supported Hamiltonian isotopy $\varphi_z:\bR^{2d}\rightarrow \bR^{2d}$, we have
    $c_k(U)=c_k(\varphi_z(U)).$
    \item If $U=\{H<1\}$ is an admissible open set whose boundary $\partial U=\{H=1\}$ is a regular level set and $c_k(U)<\infty$, then $c_k(U)$ is represented by the action of a closed characteristic in the boundary  $\partial U$. Consequently, $c_k(rU)=r^2c_k(U)$ for $r>0$.
\end{enumerate}
\end{iThm}
Moreover, based on the structural theorem, \autoref{structure toric domian module}, of $H^*(C_{T,\ell}(X_{\Omega},\bF_{p_\ell}))$, where $X_{\Omega}$ is a convex toric domain, we can compute $c_k(X_{\Omega})$ as follows: \begin{equation}c_k(X_{\Omega})=\inf\left\{T\geq 0:\exists  {z}\in \Omega_T^\circ, I( {z})\geq k\right\}.\label{capacities of convex toric domains2}\end{equation}
 Therefore, $c_k(X_{\Omega})=c_k^{\text{GH}}(X_{\Omega})$ by \eqref{capacities of convex toric domains} and \eqref{capacities of convex toric domains2}.

Based on the computation on the convex toric domains, Gutt and Hutchings conjectured (\cite[Conjecture 1.9]{gutthutchings2018capacities}) that, for a bounded star-shaped domain $U$ and for all $k\in \bZ_{\geq 1}$,\[c_k^{\text{EH}}(U)=c_k^{\text{GH}}(U).\]
 In fact, the result $c_1^{\text{EH}}(U)=c_1^{\text{GH}}(U)=\text{Minimal action}$ has been proven by Irie\cite{irie2019symplectic} for convex body $U$. Comparing to our result, we hope the consistency could be extended to $c_k$ as well.
\begin{iConjecture}For a bounded star-shaped domain $U$ and for all $k\in \bZ_{\geq 1}$, there is\[c_k^{\text{EH}}(U)=c_k^{\text{GH}}(U)=c_k(U).\]
\end{iConjecture}
Finally, let us describe some evidence about the conjecture. We will see later that $C_{T,\ell}(U,\bK)$ is defined from a projector $P_U$, which is unique no matter how it is defined. In this paper, we will exhibit the existence of $P_U$ using the sheaf quantization of Guillermou-Kashiwara-Schapira, whose construction is dynamics natural\cite{GKS2012}. Therefore it may be related to the Ekeland-Hofer capacities. But on the other hand, Viterbo constructs a sheaf quantization of compact Lagrangian submanifold in cotangent bundles using the Lagrangian Floer theory\cite{viterbo2019sheafquantizationoflagrangian}. Hence, it is hopeful to construct a sheaf quantization using the Hamiltonian Floer theory, which is close to the theory of symplectic homology. In fact, the similarity of the Chiu-Tamarkin complex and the symplectic homology has been observed by Zhang in \cite[Section 4.8]{zhang2020quantitative}. Then one can expect that the capacities computed in this way are the Gutt-Hutchings capacities. So the uniqueness of $P_U$ could show that all the capacities are the same.

\subsection{Contact Embedding}
Contact geometry is the odd-dimensional cousin of symplectic geometry. A (co-oriented) contact manifold $(X,\alpha)$ consists of a manifold $X$ of dimension $2n+1$ and a $1$-form $\alpha$, such that $\alpha\wedge d\alpha^n \neq 0$. An embedding $\varphi: (X,\alpha) \hookrightarrow (X',\alpha')$ between two contact manifolds is called contact if $\varphi^*\alpha'=e^{f}\alpha$ for some function $f\in C^\infty(X)$. We also study the embedding question in contact geometry. The pioneering work of Eliashberg, Kim, and Polterovich\cite{eliashberg2006geometry} promote our understanding of the contact embedding question a lot. Let us explain here.

A naive attempt is to study the non-squeezing problem in the $1$-jet bundle $J^1(\bR^d)=T^*\bR^d\times \bR$ equipped with the contact form $\alpha=dz+\bq d\bp$. But the re-scaling map $(\bq,\bp,z)\mapsto (r\bq,r\bp,rz)$, which is a contactomorphism, squeezes any open set into an arbitrary small neighborhood of the origin when $r$ is big enough. This conformal naturality of $1$-jet space illustrates us that we can study the prequantized space $T^*\bR^d\times S^1$, where $S^1$ is a circle, equipped with a contact form $\alpha=d\theta+\frac{1}{2}(\bq d\bp-\bp d\bq)$. But there is a global contactomorphism $F_N: T^*\bR^d\times S^1\rightarrow T^*\bR^d\times S^1$ defined as follows: We use complex coordinates $T^*\bR^d \cong \bC^d$, and then $F_N(z,\theta)\coloneqq (\nu(\theta)e^{2\pi N \theta}z, \theta)$, where $\nu(\theta)=(1+N\pi|z|^2)^{-1/2}$.
One can compute directly that $F_N$ is still embedding any ball into arbitrary small neighborhood of $\{0\}\times S^1$ for $N$ big enough. However we notice that $F_N$ is not compactly supported.

So a better definition of contact squeezing is the following proposed in loc. cit.
\begin{iDef*}{\cite[p1636]{eliashberg2006geometry}} Let $(V,\alpha)$ be a contact manifold. If $U_1,U_2 \subset V$ are two open subsets, we say that $U_1$ is squeezed into $U_2$ if there exists a compactly support contact isotopy $\varphi_s: \overline{U_1}\rightarrow V$, $s\in [0,1]$, such that $\varphi_0=\textnormal{Id}$, and $\varphi_1(\overline{U_1}) \subset U_2$.
\end{iDef*}

An interesting phenomenon, which does not appear in the symplectic situation, is the scale of the ball will affect the validity of squeezing. Two results about both squeezing and non-squeezing of contact balls $B_{\pi R^2}\times S^1 $ are:

\begin{iThm}

\begin{enumerate}[fullwidth]
\item{{\cite[Theorem 1.3]{eliashberg2006geometry}}} Suppose $d \geq 2$. Then for all $0 <\pi r^2, \pi R^2<1$, one can squeeze the contact ball $B_{\pi R^2}\times S^1 $ into $B_{\pi r^2}\times S^1 $ whatever the relation between $r$ and $R$ is.
\item{{\cite[Theorem 1.2]{eliashberg2006geometry}}} If there exists an integer $m\in[\pi r^2,\pi R^2]$, then $B_{\pi R^2}\times S^1$ cannot be squeezed into $B_{\pi r^2}\times S^1$.\end{enumerate}
\end{iThm}

About the large scale phenomenon, Eliashberg, Kim, and Polterovich give a very nice physical explanation using the quantization process. Then the only case left about the contact non-squeezing is: what will happen if there is an integer $m$ such that $m<\pi r^2 < \pi R^2 <m+1$? It is solved by Chiu using the microlocal theory of sheaves\cite{chiu2017}, and by Fraser using technique of $J$-holomorphic curves\cite{fraser2016} in the spirit of \cite{eliashberg2006geometry}. They proved the following:
\begin{iThm}[{\cite{chiu2017,fraser2016}}]If $1\leq \pi r^2<\pi R^2$, then $B_{\pi R^2}\times S^1$ cannot be squeezed into $B_{\pi r^2}\times S^1$.
\end{iThm}

The second purpose of the paper is to explain how Chiu's work could be used for more embedding questions in the prequantization $T^*\bR^d\times S^1$. In fact, the notion of admissible open sets still makes sense. But in view of the scale features above, we need to consider the so called big admissible open sets. Concretely, we say a contact admissible open set $U \subset T^*\bR^d\times S^1$ is big if there is a ball $B_a\times S^1\subset U$ where $a>1$. For a contact convex toric domain $X_{\Omega}\times S^1$, it is big if $\|\Omega^\circ_1\|_{\infty}=\max_{z\in \Omega^\circ_1}\|z\|_\infty<1$. Besides, invariance needs to be re-considered. It forces we restrict to $T=\ell$ in the contact case. Then the geometric information defining the symplectic capacities $c_k$ could also be applied to define a version of ``contact capacities", and our computation on convex toric domains still works. Specifically, for each $k\in \bZ_{\geq 1}$, we define
\[[\textnormal{Spec}](U,k)\coloneqq \{\ell_0 \in \bO: \forall\ell \in \bO, \, p_\ell \geq p_{\ell_0},\, \exists \gamma_\ell  \in H^{*}(C_{\ell}(U,\bF_{p_\ell})),\, \eta_{\ell}(U,\bF_{p_\ell})=u^k\gamma_\ell \},\] and
\[[c]_k(U)\coloneqq \min [\textnormal{Spec}](U,k) \in \bO.\]Then we have, \autoref{contact capacities}, \autoref{computation of capacities of contact convex toric domian}:
\begin{iThm} The functions $[c]_k:\mathcal{C}_{\text{big admissible}}\rightarrow \bO $ satisfy the following:
\begin{enumerate}[fullwidth]
    \item $[c]_k \leq [c]_{k+1}$ for all $k\in \bZ_{\geq 1}$.
    \item If there is an inclusion of admissible open sets $U_1 \subset U_2$, then $[c]_k(U_1) \leq [c]_k(U_2)$.
    \item For a compactly supported contact isotopy $\varphi_z:\bR^{2d}\times S^1\rightarrow \bR^{2d}\times S^1$, we have
    $[c]_k(U)=[c]_k(\varphi_z(U)).$
    \item For a big contact convex toric domain $X_\Omega \times S^1 \subsetneqq T^*\bR^d \times S^1$, we have
\begin{align*}
[c]_k(X_\Omega\times S^1)=&\min\left\{\ell \in \bO: \ell/p_\ell <1/ \|\Omega^\circ_1\|_{\infty} ,\,\exists  {z}\in \Omega_\ell^\circ, I(z) \geq k\right\}\\
        =&\min\left\{\ell\in\bO:\ell/p_\ell <1/ \|\Omega^\circ_1\|_{\infty} ,\, \ell \geq c_k(E) \right\}.
\end{align*}
\end{enumerate}
\end{iThm}
A more concrete example is as follow. Suppose $X_\Omega\times S^1 = E(3.5,4)\times S^1$, we have:
\begin{center}
\begin{tabular}{c|cccccccccc}
\hline
  $k$   & 1 & 2 & 3& 4& 5& 6& 7& 8& 9 &10\\
\hline
 $c_k$  &3.5&4& 7 & 7.5 & 8& 10.5& 11& 11.5& 12&14\\
\hline
$[c]_k$ & 5 & 5 & 7& 9& 9& 11& 11& 13& 13 &17 \\
\hline
\end{tabular}
\end{center}

\subsection{Microlocal Theory of Sheaves and the Chiu-Tamarkin complex}
The main ingredient of our work is the microlocal theory of sheaves, introduced by Kashiwara and Schapira with motivation from algebraic analysis. We refer to \cite{KS90}. The main idea we use in the paper is the notion of microsupport or singular support, which is defined as follows: For a ground field $\bK$, let $D(X)$ be the derived category of complexes of sheaves of $\bK$-vector spaces over $X$.
\begin{iDef*}For $F\in D(X)$. The microsupport of $F$ is
\begin{align*}
    SS(F)= \overline{\left\lbrace(x,p)\in T^*X: \begin{gathered}
    \text{There is a }C^1\text{-function }f\text{ near }x\text{, such that,}\\
    f(x)=0\text{, }df(x)=p\text{, and }\left
({\textnormal{R}}\Gamma_{\{f\geq 0\}}F\right)_x\neq 0.
\end{gathered} \right\rbrace}.
\end{align*}
\end{iDef*}

It is proved in \cite{KS90} that $SS(F)$ is always a closed conic and coisotropic subset of $T^*X$. Moreover, when $X$ is real analytic, $SS(F)$ is Lagrangian if and only if $F$ is (weakly) constructible. This result inspires us that the sheaf theory plays its role in symplectic geometry and contact geometry. For instance, Tamarkin develops a new method to study displacibility of Lagrangians in \cite{tamarkin2013}. Guillermou gives sheafy proofs of Gromov-Eliashberg $C^0$-rigidity and of the result by Abouzaid and Kragh that closed exact Lagrangians in cotangent bundles are homotopically equivalent to the zero section. See \cite{conicexactlagrangianGuillermou,C0rigidityGuillermou,3cuspsconjectureGuillermou} and the survey \cite{guillermou2019sheaves} about these topics. Asano-Ike develop a lower bound for the symplectic displacement energy and a lower bound for the number of intersection point of some rational Lagrangian immersions using numerical information from the Tamarkin Category( \autoref{tamarkincategory}) in \cite{asano2017persistence,asano2020sheaf}. On the other hand, there are many works studying the category of sheaves from the point of view of the Fukaya category, see the work of Nadler and Zaslow on the compact Fukaya category \cite{nadler2009constructible,nadler2009microlocal}; and the work of Nadler\cite{Nadlerwarppedfukaya2016}, and of Ganatra, Pardon, and Shende on the wrapped Fukaya category\cite{GPS3}.

Now, let us review some ideas of Tamarkin in \cite{tamarkin2013}. Tamarkin suggested to study the category of sheaves localized with respect to sheaves microsupported in non-positive direction, that is,
 the localization of $D(X\times \bR)$ with respect to the full thick subcategory $\{F:SS(F)\subset \{\tau\leq 0\} \}$.
This localization
is equivalent to the essential image of the functor $\star \bK_{[0,\infty)}:D(X\times \bR)\rightarrow D(X\times \bR)$,
where $\star: D(X\times \bR)\times D(X\times \bR)\rightarrow D(X\times \bR)$ is the convolution.
We denote these two equivalent categories by $\mathcal{D}(X)$ and call them the Tamarkin category of $X$.
The category $\mathcal{D}(X)$ is triangulated.

Because the microsupport is conic, Tamarkin considers the following conification
procedure: for a given closed set $A$ in the cotangent bundle $T^*X$ we set $\widehat{A}=\{(x,p,t,\tau)\in T^*(X\times \bR) :(x,p/\tau)\in A, \tau >0\}$. We are interested in the category of sheaves on $X\times \bR$ microsupported in $\widehat{A}$, that is $F\in \mathcal{D}(X)$ such that $SS(F) \cap \{\tau >0\} \subset \widehat{A}$, and we denote the category they form by $\mathcal{D}_A(X)$. Categorically, this category and its semi-orthogonal complement $^\perp \mathcal{D}_A(X)$ are completely determined by the projectors from $\mathcal{D}(X)$ onto them. Hopefully, we could understand the geometry of $A$ from these projectors. One way to study these projectors is to represent them as convolution functors defined by kernels, for example $\star \bK_{[0,\infty)}$ introduced by Tamarkin, or the {\em cut-off} functors of Kashiwara and Schapira \cite{KS90}. An open set $U$ whose projector $\mathcal{D}(X) \rightarrow \mathcal{D}_{U^c}(X)$ is represented by a convolution functor $P_U\star: \mathcal{D}(X)\rightarrow \mathcal{D}(X)$ is called admissible. We will see later that bounded open sets and toric domains are all admissible. One particularly interesting example is the open ball $U=B_{\pi R^2}$. Chiu constructed a kernel for $B_{\pi R^2}$ using the idea of generating functions in \cite{chiu2017}, which is the main ingredient of his proof of contact non-squeezing.

Another ingredient of Chiu's proof is a complex $C_{T,\ell}(U,\bK) \in D_G(\bK)$ defined using $P_U$,
where $G$ is the cyclic group $\bZ/\ell\bZ$, $\bK$ is some field
and $D_G(\bK)$ denotes the equivariant derived category (see  Burnstein and Lunts book\cite{BernsteinLunts}).
The complex $C_{T,\ell}(U,\bK)$ is an invariant of an admissible open set $U$ for $\ell \in \bZ_{\geq 1}$. We will see that $H^*(C_{T,\ell}(U,\bK))$ is a right graded module over the Yoneda algebra $A=\textnormal{Ext}^*_G(\bK,\bK)$, which is non-trivial
if $\text{char}(\bK)|\ell$. Let us assume $\ell$ to be odd due to a technical reason. Let $p_\ell$ to be the minimal prime factor of $\ell$, and assume $\bK=\bF_{p_\ell}$ to be the finite field of order $p_\ell$. Under these conditions, we know $\textnormal{Ext}^*_G(\bK,\bK)\cong \bK[u,\theta]$, where $|u|=2$, and $|\theta|=1$, $\theta^2=0$.

In \cite{chiu2017}
Chiu proved that $C_{T,\ell}(U,\bK)$ is a symplectic invariant of $U\subset  T^*X$ when $T$ is a non-negative real number, and is a contact invariant of $U \subset T^*X\times S^1$ when $T=\ell$. Besides he proved that $C_{T,\ell}(U,\bK)$ is functorial with respect to inclusions of admissible open sets. Then Chiu computed $C_{T,\ell}(B_{\pi R^2},\bK)$ when $\pi R^2 \geq 1$ by constructing a good kernel $P_{B_{\pi R^2}}$, and hence completed his proof of the contact non-squeezing theorem. Let us remark here that Chiu has used the $A=\textnormal{Ext}^*_G(\bK,\bK)$ module structure essentially.

We will develop the idea of Chiu further. The third main result of this paper is as follows. For a toric domain $X_{\Omega}\subset \bR^{2d}$, we construct a good kernel for $X_\Omega$ based on Chiu's construction and then study the Chiu-Tamarkin complex $C_{T,\ell}(X_\Omega,\bF_{p_\ell})$ when $X_\Omega$ is a convex toric domain. We will show the following structural theorem.
\begin{iThm}[See more precise statements in \autoref{structure toric domian module}]For a convex toric domain $X_\Omega \subsetneqq T^*\bR^d$.
Then, for  $\ell\in \bO$ such that $0\leq T <p_\ell /\|\Omega^\circ_1\|_{\infty}$, where  $\|\Omega^\circ_1\|_{\infty}=\max_{z\in \Omega^\circ_1}\|z\|_\infty$, we have:
\begin{itemize}[fullwidth]
    \item The minimal cohomology degree of $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is exactly $-2I(\Omega^\circ_T)$, i.e.,
    \[ H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))\cong  H^{\geq -2I(\Omega^\circ_T)}(C_{T,\ell}(X_\Omega,\bF_{p_\ell})) \]
    and
\[ H^{-2I(\Omega^\circ_T)}(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))\neq 0  \]
where $I(\Omega^\circ_T)=\max_{z\in \Omega^\circ_T}I(z)$( see \eqref{convex toric domain notations} for $I(z)$).
\item $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is a finitely generated $\bF_{p_\ell}[u]$ module, whose rank is $2$, and the torsion part is located exactly in $[-2I(\Omega^\circ_T),-1]$.
\end{itemize}
\end{iThm}

\subsection{Organization and Conventions of the Paper}We will review preliminary notions of sheaf theory in the first section. In the second section, we will the present main constructions, including kernels, the Chiu-Tamarkin complex, and the capacities; and demonstrate their basic properties. We will focus on toric domains, in the third section. We would like to exhibit all constructions and computations for toric domains therein. After then, we will state how our construction works for admissible open sets of the prequantized contact manifold $T^*\bR^d\times S^1$ in the fourth section.

At the end of this section, let us introduce some notations.

Usually we use subscripts to represent elements in sets. For example, $x\in X$ is denoted by $X_x$. In some situations, we use underline $\underline{x}=(x_1,\cdots,x_n)$ to record elements in Cartesian products $X^n$. The product set itself is denoted by $X^n_{\underline{x}}$. For the Cartesian product $X^n$, we define $\delta_{X^n}:X\rightarrow X^n$ to be the diagonal map and its image is denoted by $\Delta_{X^n}$.

Projection maps are always denoted by $\pi$, with a subscript that encode the fiber of the projection. For example, if there are two sets $X_x$ and $Y_y$, two projections are
$$\pi_Y=\pi_y: X_x\times Y_y \rightarrow X_x,\quad \pi_X=\pi_x: X_x\times Y_y \rightarrow Y_y.$$

If we have a trivial vector bundle $X\times V_v$, its summation map is $$\text{id}_X\times s_V^n =\text{id}_X\times s_v^n : X \times V^n \rightarrow X \times V,\, (x,v_1,\cdots,v_n) \mapsto (x,v_1+\cdots +v_n ).$$
In all cases, we will ignore the $\text{id}_X$ and only use $s_V^n=s_v^n$ for simplicity.

When you see an integer $k$, it either a positive integer $k\in \bZ_{\geq 1}$, or a periodic integer $k\in \bZ/\ell\bZ$. While for $\ell$, it could be a positive integer $\ell\in \bZ_{\geq 1}$; but in most of situations, we will assume $\ell$ to be odd, i.e., $\ell\in  \mathbb{O}=\{\text{odd numbers}\}$. For the odd number $\ell$, let us denote the minimal prime factor of $\ell$ by $p_\ell$, which is an odd prime number.

For a manifold $X$, we always use $\bq\in X$ to represent both points and local coordinates of $X$. Correspondingly, the canonical Darboux coordinate of $T^*X$ will be denoted by $(\bq,\bp)$.

One exception is $\bR_t$, its cocoordinate is denoted bt $\tau \in T_t^*\bR_t$. For a manifold $X$, the $1$-jet space is $J^1(X)=T^*X\times \bR_t$, which is a contact manifold equipped with the contact form $\alpha=dt+\bp d\bq$. The symplectization of $J^1(X)$ is identified with $T^*X\times T^*_{\tau>0}\bR_t=T^*X\times \bR_t\times \bR_{\tau>0}$, equipped with the symplectic form $\omega=d\bp\wedge d\bq+d\tau\wedge dt$. The symplectic reduction of $T^*X\times T^*_{\tau>0}\bR_t$ with respect to the hypersurface $\{\tau=1\}$ is denoted by $\rho$, which is identified with
\begin{equation}
  \rho:T^*X\times T^*_{\tau>0}\bR_t \rightarrow T^*X,\, (\bq,\bp,t,\tau) \mapsto (\bq,\bp/\tau).\label{Tamarkin conemap}
\end{equation}
We call it the Tamarkin's cone map. The map $\rho$ factors through the symplectization map $q$ tautologically:
\begin{center}
    \begin{tikzcd}
T^*X\times T^*_{\tau>0}\bR_t \arrow[r,"q"] \arrow[rr, "\rho", bend right] & J^1(X) \arrow[r] & T^*X.
\end{tikzcd}
\end{center}

\subsection*{Acknowledgements}I would like to thank St\'ephane Guillermou for many very helpful discussions, lots of important observations, and advice on the text; Claude Viterbo for pointing out the existence of capacities, and helpful discussions; Nicolas Vichery and Jun Zhang for many helpful comments and discussions; Shengfu Chiu, Octav Cornea for the encouragement. I also want to thank the encouragement of Shaoyun Bai, Honghao Gao, Xuelun Hou, Yichen Qin, Jian Wang, Xiaojun Wu, and suggestions for the improvement of the current text. The research is partially supported by the ANR project MICROLOCAL(ANR-15CE40-0007-01).

\section{Reminder on Sheaves and Equivariant Sheaves}

In this section, I would like to review notions and tools of sheaves we will use. Let $\bK$ be the ground field. For a manifold $X$, let us denote $D(X)$ the derived category of complexes of sheaves of $\bK$-vector spaces over $X$. Let us remark that we do not specify the boundedness of complexes we used in general. But in most of our applications, the complexes are locally bounded in the sense that their restrictions on relatively compact open sets are bounded.

\subsection{Microsupport of Sheaves and Functorial Estimate}We refer to \cite{KS90} as main reference of the section. For a locally closed inclusion $i:Z\subset X$ and $F\in D(X)$ we set
\[F_Z=i_!i^{-1}F, \quad {\textnormal{R}}\Gamma_ZF=i_*i^!F.\]
\begin{Def}[{\cite[Definition 5.1.2]{KS90}}]For $F\in D(X)$ the  microsupport of $F$ is
\begin{align*}
    SS(F)= \overline{\left\lbrace(\bq,\bp)\in T^*X: \begin{gathered}
    \text{There is a }C^1\text{-function }f\text{ near }\bq\text{, such that,}\\
    f(\bq)=0\text{, }df(\bq)=\bp\text{, and }\left
({\textnormal{R}}\Gamma_{\{f\geq 0\}}F\right)_\bq\neq 0.
\end{gathered} \right\rbrace}.\end{align*}\end{Def}
By definition, $SS(F)$ is a closed  subset of $T^*X$, conic with respect to the $\bR_{>0}$-action on the fibers.
There is a triangulated inequality for the microsupport: if there is a distinguished triangle $A\rightarrow B\rightarrow C\xrightarrow{+1}$, then $SS(A)\subset SS(B)\cup SS(C)$.

Let $f: X\rightarrow Y$ be a $C^\infty$ map of manifolds. Then there is a diagram of cotangent map:
\begin{center}
\begin{tikzcd}
T^*X \arrow[rd, "\pi"'] & X\times _Y T^*Y \arrow[l, "df^*"'] \arrow[r, "f_\pi"] \arrow[d, "\pi"] & T^*Y \arrow[d, "\pi"] \\
                        & X \arrow[r, "f"]                                                       & Y
\end{tikzcd}
\end{center}
\begin{Def}Let $f: X\rightarrow Y$ be a $C^\infty$ map of manifolds, and $\Lambda\subset T^*Y$ be a conic subset. One says $f$ is non-characteristic for $\Lambda$ if $\forall(\bq,\bp)\in \Lambda, \textnormal{ and } df^*_\bq(\bp)=0 \Rightarrow \bp=0.$\end{Def}
Then we list some functorial estimates we need.
\begin{Thm}[{\cite[Theorem 5.4]{KS90}}]\label{functional estimate}Let $f: X\rightarrow Y$ be a $C^\infty$ map of manifolds, $F\in D(X), G\in D(Y)$. Let $\omega_{X/Y}=f^!\bK_N$ be the dualizing complex.
\begin{enumerate}[fullwidth]
\item One has
\begin{align*}
&SS(F   \boxtimes G) \subset SS(F)\times SS(G),\\
&SS({ \textnormal{R}\mathcal{H}om  }(\pi_X^{-1}F, \pi_Y^{-1} G)) \subset SS(F)^a \times SS(G).
\end{align*}

\item Assume $f$ is proper on ${\textnormal{supp}}(F)$, then $SS({\textnormal{R}}f_!F)\subset f_\pi (df^*)^{-1}\left(SS(F) \right)$.

\item Assume $f$ is non-characteristic for $SS(G)$. Then the natural morphism $f^{-1}G \otimes \omega_{X/Y}  \rightarrow f^!G$ is an isomorphism, and $SS(f^{-1}G) \cup SS(f^{!}G)\subset df^*f^{-1} \left( SS(G)\right) $.

\item Assume $f$ is a submersion. Then $SS(F) \subset X \times _Y T^*Y$ if and only if $\forall j\in \bZ$, the sheaves $\mathcal{H}^j(F)$ are locally constant on the fibres of $f$.
\end{enumerate}
\end{Thm}

\begin{Coro} Let $F_1,F_2 \in D(X)$.
\begin{enumerate}[fullwidth]
\item Assume $SS(F_1)\cap (-SS(F_2))\subset 0_X$, then $SS(F_1 \otimes   F_2) \subset SS(F_1)+SS(F_2).$
\item Assume $SS(F_1)\cap SS(F_2)\subset 0_X$, then $SS({ \textnormal{R}\mathcal{H}om  }(F_2, F_1) ) \subset (-SS(F_2))+SS(F_1).$
\end{enumerate}
\end{Coro}
For the non-proper pushforward, we have
\begin{Thm}[{\cite[[Corollary 3.4]{tamarkin2013}}]Let $V$ be a $\bR$-vector space, $\pi_V:X\times V\rightarrow X$, and $\pi_V^\#: T^*X\times V\times V^*\rightarrow T^*V\times V^*$ be the corresponding projections, and $i:T^*X\rightarrow T^*V\times V^*$ be the inclusion. Then for $F\in D(X\times V)$, we have
\[SS(\pi_{V!}F),SS(\pi_{V*}F) \subset i^{-1}\overline{\pi_V^\#(SS(F))}.\]

\end{Thm}

\subsection{Convolution and Tamarkin Category}
Let $X_1,X_2,X_3$ be three manifolds, and $V$ be a $\bR$-vector space. Recall, $\pi_X: X\times Y \rightarrow Y$ is a projection whose fiber is $X$ for arbitrary $Y$.

\begin{Def}For $F\in D(X_1\times X_2\times V)$, $G\in D(X_2\times X_3\times V)$. The convolution is defined as
\[F\star G\coloneqq {\textnormal{R}}s_{V!}^2{\textnormal{R}}\pi_{X_2!}(\pi_{(X_3,v_2)}^{-1}F\otimes \pi_{(X_1,v_1)}^{-1}G)\in D(X_1\times X_3\times V) .\]
Similarly, for $F\in D(X_1\times X_2)$, $G\in D(X_2\times X_3)$, the composition is defined as
\[F\circ G\coloneqq {\textnormal{R}}\pi_{X_2!}(\pi_{X_3}^{-1}F  \otimes \pi_{X_1}^{-1}G  )  \in D(X_1\times X_3 ) .\]
\end{Def}
A basic example here is for $0\in V$, $F\in D(X\times V)$, then we have $F\star \bK_{0}\cong F$. So, the functor $\star \bK_{0}$ plays the role of the identity functor. Besides, $\star$ and $\circ$ satisfy the following monoidal identities:
\begin{align}\label{monoidal identities}
\begin{aligned}
&(F_1\star F_2)\star F_3 \cong F_1\star (F_2\star F_3),\quad (F_1\circ F_2)\circ F_3 \cong F_1\circ (F_2\circ F_3),\\
& F_1 \star F_2 \cong F_2 \star F_1, \quad F_1 \circ F_2 \cong F_2 \circ F_1,\\
&(F_1\star F_2)\circ F_3 \cong F_1\star (F_2\circ F_3).
\end{aligned}
\end{align}
Here, the commutative identities is induced by the factor permute map $X_1\times X_3 \cong X_3\times X_1$.

In this paper, we are mainly concerned with the situation of $V=\bR_t$.

Before going into further discussion, let us review the notion of semi-orthogonal decomposition of a triangulated category.

Let $\mathcal{T}$ be a triangulated category and $\mathcal{C}$ a thick full triangulated subcategory of $\mathcal{T}$. The left semi-orthogonal of $\mathcal{C}$ is defined by
\begin{equation}
    ^\perp \mathcal{C}  \coloneqq \{ X\in \mathcal{T} : { \textnormal{Hom}  }_{\mathcal{T}}(X,Y)=0, \forall Y\in \mathcal{C}   \}.\label{definitionofsemiorthcomplement}
\end{equation}

One can show that the following proposition holds, see \cite[Chapter 4 and Exercise 10.15.]{KS2006}.
\begin{Prop}\label{semiorthcomplement} Using the above notations, we have the followings three equivalent properties:
\begin{enumerate}[fullwidth]
    \item The inclusion $ \mathcal{C} \rightarrow \mathcal{T}$ admits a left adjoint functor $L:\mathcal{T}\rightarrow \mathcal{C}$.
    \item There is an isomorphism $\mathcal{T}/\mathcal{C} \xrightarrow{\cong}      {^\perp}\mathcal{C} $, where $\mathcal{T}/\mathcal{N}$ is the Verdier localization.
    \item There are two functors $P,Q:\mathcal{T} \rightarrow \mathcal{T}$, such that $\forall X\in\mathcal{T}$, we have the distinguished triangle:
    \[P(X)\rightarrow X \rightarrow Q(X) \xrightarrow{+1},\]
    such that $P(X)\in {^\perp} \mathcal{C}$, and $Q(X)\in \mathcal{C}$.
\end{enumerate}
In this situation, we say one of these data gives a left semi-orthogonal decomposition of $\mathcal{T}$. One can verify, if one of the conditions here is satisfied, that $P^2\cong P$, and $Q^2\cong Q$. $P,\,Q$ are called a pair of projectors associated to $\mathcal{C}$.
\end{Prop}
Now, let $\mathcal{T}=D(X\times\bR_t)$, and $\mathcal{C}=\left\{F: SS(F)\subset \{\tau \leq 0\}\right\}$. The triangulated inequality of microsupport shows that $\mathcal{C}$ is a thick full triangulated subcategory of $\mathcal{T}$. Tamarkin constructs a pair of projectors associated to $\mathcal{C}$ given by convolution:
\begin{Thm}[\cite{tamarkin2013}]
  The functors $F \mapsto F\star \bK_{[0,\infty)}$, $F \mapsto F\star \bK_{(0,\infty)}[1]$ on $D(X \times\bR_t)$
and  the excision triangle
\[ \bK_{[0,\infty)} \rightarrow \bK_{0} \rightarrow  \bK_{(0,\infty)}[1]\xrightarrow{+1}.\]
 give a left semi-orthogonal decomposition of $D(X\times\bR_t)$ associated to $\mathcal{C}$. Namely, for $F\in D(X \times\bR_t)$
we have the distinguished triangle
\begin{equation}
F\star \bK_{[0,\infty)} \rightarrow F \rightarrow F\star \bK_{(0,\infty)}[1]\xrightarrow{+1} \label{Tamarkinorthgonalcomplement}
\end{equation}
with $F\star \bK_{[0,\infty)}\in {^\perp} \mathcal{C}$, $F\star \bK_{(0,\infty)}[1] \in \mathcal{C}$.
\end{Thm}
One can also see \cite[Proposition 3.24]{GS2014} for a proof and some generalizations of the proposition.

\begin{Def}\label{tamarkincategory} We define the Tamarkin category as
\[\mathcal{D}(X)={^\perp} \left\{F: SS(F)\subset \{\tau \leq 0\}\right\}\cong D(X\times \bR) /\left\{F: SS(F)\subset \{\tau \leq 0\}\right\}.\]

By \autoref{semiorthcomplement} and \eqref{Tamarkinorthgonalcomplement} , $F\in D(X\times \bR)$ is in $\mathcal{D}(X)$ if and only if $F \cong F\star \bK_{[0,\infty)}\cong F\star \bK_{\Delta_{X^2}\times [0,\infty)}$.
\end{Def}
Consequently, the convolution functor $\star\bK_{\Delta_{X^2}\times [0,\infty)}$ of the Tamarkin category $\mathcal{D}(X)$ coincides with the identity functor.

For $F\in \mathcal{D}(X)$, one can show $SS(F)\subset \{\tau \geq 0\}$ using microsupport estimates we mentioned last section, see \cite[Proposition 3.17]{GS2014}. So it is helpful to define the Legendre microsupport and sectional microsupport of $F\in \mathcal{D}(X)$ as follows:
\begin{align}
\begin{aligned}
        {\mu}s_L(F)=q\left(SS(F)\cap \{\tau> 0\}\right)\subset J^1X,\\
    {\mu}s(F)=\rho\left(SS(F)\cap \{\tau> 0\}\right)\subset T^*X,
\end{aligned}
\end{align}
where $\rho, \,p$ are defined in \eqref{Tamarkin conemap}.

We recall that there exists a sheaf-theoretic Fourier transform called the Fourier-Sato transform. The Fourier-Sato transform defines a functor $D(V) \rightarrow D(V^*)$, where $V$ is a real vector space and $V^*$ is the dual of $V$. One can see \cite[Section 3.7, Section 5.5]{KS90} for more details. We want to mention that the Fourier-Sato gives an equivalence between $\bR_{> 0}$-equivariant sheaves. Tamarkin introduced a new version of the Fourier transform on the category $\mathcal{D}(V)$ which works also for not $\bR_{> 0}$-equivariant sheaves. We call it the Fourier-Sato-Tamarkin transform. For the relation between different versions of Fourier transforms, we refer \cite{radonHonghao, radonDAgnolo}.
\begin{Def}Let $FT=\lbrace (z,\zeta,t): t+\langle z,\zeta\rangle\geq 0 \rbrace \subset V\times V^*\times \bR_t$. Then the Fourier-Sato-Tamarkin transform is defined as
\begin{align*}
    &FT:  \mathcal{D}(V_z) \rightarrow  \mathcal{D}(V^*_{\zeta})\\
    &FT(F)=\widehat{F}\coloneqq F \star {\bK}_{FT}[\dim V],
\end{align*}
One can see that $FT$ is an equivalence in \cite[Theorem 3.5]{tamarkin2013}.
\end{Def}

The important thing for us is the microsupport estimate under the Fourier-Sato-Tamarkin transform.

\begin{Thm}[{\cite[Theorem 3.6]{tamarkin2013}}]Let $\phi\colon T^*(V\times \bR_t) \rightarrow T^*(V^*\times \bR_t)$ be the isomorphism given by $\phi(z,\zeta,t,\tau)=(\zeta,-z,t,\tau)$, where we identify $V^{**}$ with $V$ naturally. Then we have the microsupport relation:
\[SS(\widehat{F})\cap \{\tau>0\}=\phi( SS(F)   )\cap \{\tau>0\},\]
and consequently
\begin{equation}
  \mu s_L (\widehat{F})=\phi( \mu s_L (F)   ).  \label{microsupportfouriertransform}
\end{equation}

\end{Thm}

They play a very important role in the construction of kernels associated to admissible open sets.

\subsection{Guillermou-Kashiwara-Schapira Sheaf Quantization}\label{GKSsection}As a sheaf pattern of Hamiltonian actions, we introduce the Guillermou-Kashiwara-Schapira sheaf quantization as a basic tool here.

Consider $\dot{T}^*Y$ as a symplectic manifold equipped with the Liouville symplectic form and with a $\bR_{>0}$-action by dilation on cotangent fibers. If $\varphi:\dot{T}^*Y\times I \rightarrow \dot{T}^*Y$ is a $\bR_{>0}$-equivariant symplectic isotopy, one can show that it must be Hamiltonian with a $\bR_{>0}$-equivariant Hamiltonian function $H$.

Consider its total graph
\begin{equation}
  \Lambda_{{\varphi}}\coloneqq \left\lbrace (z, - {H_z}\circ{\varphi}_z(\bq,\bp)  ,(\bq,\bp),-{\varphi}_z(\bq,\bp) ) : (\bq,\bp) \in \dot{T}^*Y, z\in I \right \rbrace\subset \dot{T}^*(I\times Y). \label{totalgraph}
\end{equation}

Then Guillermou, Kashiwara, and Schapira proved the following theorem:
\begin{Thm}[{\cite[Theorem 3.7]{GKS2012}}]\label{GKS} Using the above notations there is a $K \in D(I\times Y \times Y )$ such that
\begin{enumerate}[fullwidth]
\item $SS(K)\subset \Lambda_{\widehat{\phi}}\cup 0_{I\times Y\times Y}$,
\item $K_0=\bK_{\Delta_{Y^2}}$.
\end{enumerate}
where $K_{z_0}=K|_{\{z=z_0\}}$.

If we set $K_z^{-1}=v^{-1}{ \textnormal{R}\mathcal{H}om  }(K_z, \omega_Y{\overset{L}\boxtimes } \bK_Y)$, $v(m,n)=(n,m),\,m,n\in Y,\,z\in I $, then
\begin{enumerate}[label=\emph{\alph*}$ )$]
\item $\textnormal{supp}(K)\rightrightarrows I \times Y $ are both proper,
\item $K_z\circ K_z^{-1}\cong  K_z^{-1}\circ K_z\cong \bK_{\Delta_{Y^2}}$,
\item $K$ is unique up to a unique isomorphism.
\end{enumerate}
Consequently, $F\mapsto K_z\circ F$, $D(Y)\rightarrow D(Y)$ is a category equivalence for all $z\in I$, whose quasi inverse is $K_z^{-1}\circ F$.
\end{Thm}

Let us describe two situations where we will use the theorem.
\begin{enumerate}[fullwidth]
\item[I)] Let $\varphi: I\times T^*X \rightarrow T^*X$ be a compactly supported Hamiltonian isotopy. Let $Y=X\times \bR_t$. Then one can lift $\widehat{\varphi}: I\times\dot{T}^*Y  \rightarrow \dot{T}^*Y $. Specifically, we have the following:
\begin{Prop}[{\cite[Proposition A.6]{GKS2012}}]\label{cone map}Let $\varphi: I\times T^*X  \rightarrow T^*X$ be a compactly supported Hamiltonian isotopy, whose Hamiltonian function is $H\in C^\infty( I\times T^*X )$.

There is a $\bR_{>0}$-equivariant Hamiltonian isotopy $\widehat{\varphi}:I\times\dot{T}^*Y  \rightarrow  \dot{T}^*Y $ such that:

\begin{enumerate}[fullwidth,label=\emph{\alph*}$ )$]
\item$\widehat{H}=\tau H(-,\rho(-))$ is a Hamiltonian function of $\widehat{\varphi}$.
\item The lifting $\widehat{\varphi}$ commutes with both the symplectization and the Tamarkin's cone map.
\item We can take
\begin{align*}
&\widehat{\varphi}(z,\bq,t,\bp,\tau)=(\tau.\varphi(z,\bq,\bp/\tau), t+u(z,\bq,\bp/\tau),\tau), &\tau \neq 0,\\
&\widehat{\varphi}(z,\bq,t,\bp,0)=(\bq,\bp, t+v(z),0), &\tau =0,
\end{align*}
where $u\in C^\infty( I\times T^*X ), v\in C^\infty(I) $. In fact, the proof shows $u(z,\bq,\bp)=S_{H}(z,\bq,\bp)=\int_0^z [\alpha(X_{H_x})-H_x]\circ \phi^x_{H}(\bq,\bp)dx$ is the symplectic action function.
\end{enumerate}
We call this $\widehat{\varphi}$ or $\widehat{\varphi}_z$ the conification of $\varphi$.
\end{Prop}
\begin{RMK}Noticed, it is easy to lift $\varphi$ to $T^*X\times T^*_{\tau>0}\bR_t$ without the compactly supported assumption, but this is not enough to apply the Guillermou-Kashiwara-Schapira theorem. If we want to lift $\varphi$ to $\dot{T}^*(X\times \bR_t)$, we need the compactly supported condition.
\end{RMK}

Now, apply \autoref{GKS} to $\widehat{\varphi}$, there is a sheaf $K(\widehat{\varphi}) \in D(I\times X^2\times \bR^2_t)$. In our later application, we prefer to use only one $t$-variable. This is possible. Consider $m(t_1,t_2)=t_1-t_2$, then \cite[Corollary 2.3.2]{guillermou2019sheaves} shows there is a unique $\mathcal{K}(\widehat{\varphi}) \in D(I\times X^2 \times \bR_t)$ such that $K(\widehat{\varphi})=m^{-1}\mathcal{K}(\widehat{\varphi})$, and $\mathcal{K}(\widehat{\varphi})={\textnormal{R}}m_!K(\widehat{\varphi})$.
Then we can take $\mathcal{K}=\mathcal{K}(\widehat{\varphi})$ as the sheaf quantization of $\varphi$.

By the commutativity of the lifting with symplectization, we have the following estimates for the Legendrian microsupport and sectional microsupport of $\mathcal{K}(\widehat{\varphi})$:
\begin{align}\label{microsupportofsheafquantization}
\begin{aligned}
    {\mu}s_L(\mathcal{K}(\widehat{\varphi})) &\subset \lbrace (z,- H(\bq,\bp), \bq, \bp,-\varphi_z(\bq,\bp) ,-S_{H}(z,\bq,\bp) ):(z,\bq,\bp)\in  I\times T^*X\rbrace, \\
   \mu s(\mathcal{K}(\widehat{\varphi}))&\subset \lbrace (z,- H(\bq,\bp), \bq, \bp,-\varphi_z(\bq,\bp) ):(z,\bq,\bp)\in  I\times T^*X \rbrace.
\end{aligned}
\end{align}
\item[II)] Let $\varphi: I\times T^*X\times S^1 \rightarrow T^*X\times S^1$ be a contact isotopy. One can lift $\varphi$ to a $\bZ$-equivariant contact isotopy of $J^1(X)=T^*X\times \bR_t$, where $\bZ$ acts by shifting $t$. Then it is easy to lift $\varphi$ to the symplectization, $T^*X\times T^*_{\tau>0}\bR_t$, of $J^1(X)$ to a $\bZ\times \bR_{>0}$ equivariant Hamiltonian isotopy $\widehat{\varphi}: I\times T^*X\times T^*_{\tau>0}\bR_t\rightarrow T^*X\times T^*_{\tau>0}\bR_t$. Similarly
to the symplectic case, the compactly supported condition is necessary to extend $\widehat{\varphi}$ to whole $\dot{T}^*(X\times \bR_t)$.

In this case, we still take the sheaf quantization $\mathcal{K}=\mathcal{K}(\widehat{\varphi})$ of $\widehat{\varphi}$ as sheaf quantization of $\varphi$. The only difference here is, the $\bZ$-equivariant condition is inherited by the sheaf $\mathcal{K}(\widehat{\varphi})$. For more about this point, we refer \cite[Formula (26)]{chiu2017} for readers.

\end{enumerate}

\subsection{Equivariant Sheaves for Finite Group Actions.}
Here, we review basic notions of equivariant sheaves. We refer to \cite{BernsteinLunts} for all details about the general theory of equivariant sheaves and equivariant derived categories. In this paper, we only use the theory of equivariant sheaves and equivariant category for a finite group (in fact, a cyclic group). So we assume $G$ to be a finite group equipped with the discrete topology.

For a manifold $X$ with a $G$ action $\rho: G\times X \rightarrow X$, a $G$-equivariant sheaf is a pair $(F, \theta)$ where $F\in Sh(X)$ and $\theta: \rho^{-1}F \cong \pi^{-1}_G F$ is an isomorphism of sheaves satisfying the cocycle conditions:
\[d_0^{-1}\theta \circ d_2^{-1}\theta=d_1^{-1}\theta, \quad s_0^{-1}\theta=\text{Id}_F.\]
Here
\begin{align*}
  d_0(g,h,x)&=(h,g^{-1}x),\\
  d_1(g,h,x)&=(gh,x),\\
  d_2(g,h,x)&=(g,x),\\
  s_0(x)&=(e,x).
\end{align*}
A sheaf morphism between two $G$-equivariant sheaves is equivariant if it commutes with the $\theta$'s.
We let $Sh_G(X)$ be the category of $G$-equivariant sheaves.
For example, when $X={\textnormal{pt}}$, $Sh_G(X)\simeq
{\bK}[G]-Mod$, the category of all $G$-modules. The category of $G$-equivariant sheaves $Sh_G(X)$ is Abelian. Moreover, Grothendieck proved in the Tohoku paper that when $G$ is finite, $Sh_G(X)$ admits enough injective objects. Therefore, the derived category $D(Sh_G(X))$ makes sense, which is treated as a naive version of equivariant derived category of sheaves.

For general topological groups, the naive version is not good as our expectation. A basic difference is the hom space ${ \textnormal{RHom}  }_{D(Sh_G(X))}(\bK_X,\bK_X)$ is not isomorphic to the equivariant cohomology of $X$. A more serious problem is how to define 6-operations with correct adjunction properties.

To resolve these problems, we must use the equivariant derived category $D_G(X)$ defined by Burnstein-Lunts, in where the expected isomorphism holds, and the correct 6-operations live. But for discrete groups, both the naive  and  advanced versions are equivalent, i.e., $D(Sh_G(X))\simeq D_G(X)$. In particular, $D({\bK}[G]-Mod)\simeq D_G(\text{pt}) $.  More luckily, there is a forgetful functor $D_G(X) \rightarrow D(X) $, which commutes with Grothendieck 6-operations. So for us, $D(Sh_G(X))$ is enough for our applications. In practice, we always write everything in the usual derived category and run the machine of equivariant derived category implicitly. As a rule of convenience, we only write a lower subscript $G$ for all possible places to indicate that we are working on some version of equivariant categories without mentioning which one we are really working on.

\section{Projectors, Chiu-Tamarkin complex, and Capacities.}

\subsection{Projectors Associated to Open Sets in \texorpdfstring{$T^*X$}{} }\label{section:UniquenessandExistence}
In this subsection, we are going to study the categories related to sheaves microsupported in an open set $U\subset T^*X$. Next, we will construct kernels of the projectors onto these categories.

For a {\em closed} subset $Z \subset T^*X$ we define $\mathcal{D}_Z(X)$ as the full subcategory of $\mathcal{D}(X)$ consisting of the sheaves satisfying $\mu s(F) \subset Z$. For an {\em open} subset $U \subset T^*X$  we define  $\mathcal{D}_{U}(X)$ to be the left semi-orthogonal complement of $\mathcal{D}_{T^*X\setminus U}(X)$ in  $\mathcal{D}(X)$, i.e., $\mathcal{D}_{U}(X)={^\perp}\mathcal{D}_{T^*X\setminus U}(X)$.

Now we have a diagram of inclusions
\begin{equation}
 \mathcal{D}_{T^*X\setminus U}(X)
\hookrightarrow \mathcal{D}(X) \hookleftarrow \mathcal{D}_{U}(X) \label{inclusion diagram of categories supported in U}
\end{equation}

Following Tamarkin, we are looking for a convolution kernel to prove these inclusions admit suitable adjoint functors and give corresponding semi-orthogonal decomposition.

\begin{Def}\label{defadmissibledomains} We say $U$ is $\bK$-admissible if there is a distinguished triangle
\[ P_U\rightarrow {\bK}_{\Delta_{X^2} \times[0,\infty)} \rightarrow Q_U \xrightarrow{+1},\]
in $D(X^2 \times \bR_t)$ such that the convolution functor $P_U\star $ is right adjoint to $\mathcal{D}_{U}(X)
\hookrightarrow \mathcal{D}(X) $ and $Q_U\star $
is left adjoint to  $\mathcal{D}_{T^*X\setminus U}(X)
\hookrightarrow \mathcal{D}(X) $, i.e.,
\[\mathcal{D}_{T^*X\setminus U}(X)  \xleftarrow{Q_U\star }\mathcal{D}(X) \xrightarrow{P_U\star }\mathcal{D}_{U}(X),    \]
are two projectors.

Such a pair of sheaves $(P_U,Q_U)$ together with the distinguished triangle give an orthogonal decomposition of $\mathcal{D}(X)$ by \autoref{semiorthcomplement}. We also call the pair $(P_U,Q_U)$ as (the kernels of) projectors associated with $U$.

We say $U$ is admissible if $U$ is $\bK$-admissible for all fields $\bK$.
\end{Def}

\begin{RMK}\begin{enumerate}[fullwidth]\label{kernel remarks}
    \item We define the $\bK$-admissibility of $U$ at the beginning. But the field dependence seems redundant because all our existence results in the following works for all fields $\bK$. Moreover, we can define the $\bZ$-admissibility of open sets. All our existence results hold for $\bZ$-admissibility, and one can show that if $U$ is $\bZ$-admissible, $U$ is $\bK$-admissible for all fields $\bK$( proven by taking the tensor product $ \bK{\overset{L}\otimes }_\bZ$ with kernels). From this point of view, we do not emphasize the ground field $\bK$ for the kernels $(P_U,Q_U)$. But we will see later that the field $\bK$ does affect the computation of the Chiu-Tamarkin complex.
    \item The adjoint functors of the inclusion functor $\mathcal{D}_{T^*X\setminus U}(X)
\hookrightarrow \mathcal{D}(X) $ is also studied in \cite{kuo2021wrapped}. They construct the left and right adjoint of the inclusion functor, which are called infinite wrapping functors. Same with our existence results below( e.g. \autoref{opensetsareadmissible}), they also use the Guillermou-Kashiwara-Schapira sheaf quantization as a fundamental tool. Our point here is the existence of the kernel $P_U$.
\end{enumerate}\end{RMK}

In the following, we will present the functorial property, uniqueness of kernels, and existence of kernels for bounded open sets.
\begin{Prop}[Functorial property, {Theorem 4.7(2) of \cite{chiu2017}}]\label{functorial}Any inclusion $U_1 \subset U_2 \subset T^*X$ between $\bK$-admissible open subsets induces a morphism of distinguished triangle
\begin{center}
    \begin{tikzcd}
P_{U_1} \arrow[r] \arrow[d] & {\bK}_{\Delta_{X^2}\times[0,\infty)} \arrow[r] \arrow[d,Equal] & Q_{U_1} \arrow[r, "+1"] \arrow[d,] & {} \\
P_{U_2} \arrow[r]             & {\bK}_{\Delta_{X^2}\times[0,\infty)} \arrow[r]                    & Q_{U_2} \arrow[r, "+1"]             & {.}
\end{tikzcd}
\end{center}
These morphisms are natural with respect to inclusions of open sets. In particular, when $U_1=U_2$ (but $P_{U_1}$ and $P_{U_2}$ are not necessarily the same), the morphism is an isomorphism of distinguished triangle(See {\cite[Section 4.6]{zhang2020quantitative}} for example). \end{Prop}

Next, let us study the existence of admissible open sets $U\subset T^*X$. In general, we can take a smooth Hamiltonian function $H$ such that $U=\{H<1\}$. Our tools to construct kernels are sheaf quantizations and the Fourier-Sato-Tamarkin transform.

For our later application for toric domains, let us state our idea in a more general form. Suppose there is a Hamiltonian $\bR^m_{z}$-action  on $T^*X$, i.e., a symplectic action $\varphi:\bR^m_{z}\times T^*X\rightarrow T^*X$ with a moment map $\mu: T^*X\rightarrow (\bR^m_{z} )^*=\bR^m_{\zeta}$.

We assume there is a sheaf quantization $\mathcal{K}\in D(\bR^m_{z} \times X^2\times \bR_t)$ associated with the Hamiltonian action in the sense:
\begin{align}\label{sheafquantization}
\begin{aligned}
 &\mathcal{K}|_{z=0} \cong {\bK}_{\{0\}\times \Delta_{X^2 }\times \{0\}}\\
&\mu s(\mathcal{K}) \subset \{ (z,- \mu(\bq,\bp), \bq,\bp,-\varphi_z(\bq,\bp) ):(z,\bq,\bp)\in  \bR^m_{z}\times T^*X\}.
\end{aligned}
\end{align}
\begin{RMK}
One can see that when $m=1$, it is exactly the single Hamiltonian situation, whose existence is discussed in \autoref{GKSsection}. In fact, as \cite[Remark 3.9]{GKS2012} discussed, it exists for general $m$. A situation of particular interest is the standard torus action on $\bC^d$, which will be discussed in \autoref{GFmodelToricdomain}.
\end{RMK}

First of all, let us apply the Fourier-Sato-Tamarkin transform to the $z$-variable. That is $\widehat{\mathcal{K}}=\mathcal{K}\star \bK_{FT}[m]$. So by \eqref{sheafquantization} and \eqref{microsupportfouriertransform}, we have
\begin{equation}
\mu s(\widehat{\mathcal{K}})\subset \{ (\mu(\bq,\bp),z, \bq, \bp,-\varphi_z(\bq, \bp)):(z,\bq,\bp)\in  \bR^m_{z}\times T^*X\}.
\end{equation}
Now, take $\Omega \subset \bR^m_{\zeta}$ open. Intuitively, if we want to restrict the microsupport of some sheaves into $\{\mu \in \Omega \}$, we should cut-off the support of $\mathcal{K}$ into $\Omega$ in some way. The  correct formulation of the idea is as follows. Consider the excision triangle:\[{\bK}_{\Omega} \rightarrow {\bK}_{\bR^m_{{\zeta}}} \rightarrow {\bK}_{\bR^m_{{\zeta}}\setminus\Omega}\xrightarrow{+1}.\]
After composition with $\widehat{\mathcal{K}}$, we obtain a distinguished triangle
\[\widehat{\mathcal{K}}\circ {\bK}_{\Omega} \rightarrow \widehat{\mathcal{K}}\circ {\bK}_{\bR^m_{{\zeta}}} \rightarrow \widehat{\mathcal{K}}\circ {\bK}_{\bR^m_{{\zeta}}\setminus\Omega}\xrightarrow{+1} .\]

By the associativity of convolutions and compositions(formula\eqref{monoidal identities}), we have $\widehat{\mathcal{K}}\circ F = (\mathcal{K} \star \bK_{FT}[m])\circ F\cong\mathcal{K} \star (\bK_{FT}[m] \circ F) $. So because $\bK_{FT}[m] \circ \bK_{\bR^m_\zeta}=\bK_{\lbrace z=0,\,t \geq 0\rbrace}$, we have
\[(\widehat{\mathcal{K}}\circ \bK_{\bR^m_\zeta}) \cong \mathcal{K} \star \bK_{\lbrace z=0,\,t \geq 0\rbrace}\cong \bK_{\Delta_{X^2} \times [0,\infty)},\]
the last isomorphism comes from \eqref{sheafquantization}, i.e., $\mathcal{K}|_{z=0} \cong {\bK}_{\Delta_{X^2}\times \{0\}}$. Therefore, we have the distinguished triangle
\[\widehat{\mathcal{K}}\circ {\bK}_{\Omega} \rightarrow \bK_{\Delta_{X^2} \times [0,\infty)}\rightarrow \widehat{\mathcal{K}}\circ {\bK}_{\bR^m_{{\zeta}}\setminus\Omega}\xrightarrow{+1} .\]

\begin{Prop}\label{Existence}Suppose there is Hamiltonian $\bR^m_{{z}}$-action $\varphi$ on $T^*X$ with a moment map $\mu: T^*X\rightarrow \bR^m_{{\zeta}} $. We assume there is a sheaf quantization $\mathcal{K}\in D(\bR^m_{{z}} \times X^2\times \bR_t)$ of the Hamiltonian action in the sense of \eqref{sheafquantization}. For an open subset $\Omega \subset  \bR^m_{ {\zeta}}$ such that for any $ {\zeta} \in \Omega$, $\mu^{-1}( {\zeta})$ is compact, the open set $U=\mu^{-1}(\Omega)\subset T^*X$ is admissible.

More precisely, the pair of sheaves
\begin{equation}\label{standard model of kernels}
  P_U\coloneqq \widehat{\mathcal{K}}\circ {\bK}_{\Omega},\,Q_U\coloneqq \widehat{\mathcal{K}}\circ {\bK}_{\bR^m_{ {\zeta}}\setminus\Omega},
\end{equation}
 and the distinguished triangle
 \begin{equation}\label{dt of admissible open set}
    \widehat{\mathcal{K}}\circ {\bK}_{\Omega} \rightarrow {\bK}_{\Delta_{X^2}\times [0,\infty)} \rightarrow \widehat{\mathcal{K}}\circ {\bK}_{\bR^m_{ {\zeta}}\setminus\Omega}\xrightarrow{+1},
 \end{equation}

provide the kernels of projectors and the semi-orthogonal decomposition associated to $U$.
\end{Prop}
\begin{proof}
Our construction is a straight forward generalization of Chiu's result \cite[Theorem 3.11]{chiu2017}. One only need to notice that we consider a Hamiltonian $\bR^m_z$-action more than a single Hamiltonian function, and we replace $(-\infty,R)$ in Chiu's paper by $\Omega$. The properness condition of $\Omega$ is a technical condition which is automatically true in the situation of Chiu. One can check the proof of Chiu to confirm that our condition is enough to make sure the virtue of the semi-orthogonal decomposition without any other modification.

Besides, Chiu's argument works for all field $\bK$. So we obtain not only $\bK$-admissibility but also admissibility.
\end{proof}

Now let us state some corollaries.

\begin{Coro}\label{opensetsareadmissible}Bounded open sets $U$ are admissible.
\end{Coro}
\begin{proof}$T^*X \setminus U$ is a closed subset of $T^*X$. The smooth Urysohn's lemma(which is a corollary of the partition of unity) implies that there is a smooth function $H: T^*X
\rightarrow [0,1]$, such that $U=\{H<1 \}$ and $T^*M \setminus U =\{H=1\}=\{H\geq 1\}$.

Since $U$ is bounded, the subsets $\{H=a\}\subset U$ with $a<1$ are compact. Moreover $dH$ has compact support. So we can take the GKS quantization $\mathcal{K}(\widehat{\varphi^H})$. Then the result follows from the \autoref{Existence} by taking $\Omega=(-\infty,1)$.
\end{proof}

The second corollary here is about the kernel of products of open sets.

\begin{Coro}\label{product}Suppose we have two bounded open sets $U_i \subset T^*X_i$, with two pairs of kernels
$ (P_{U_i}, Q_{U_i}) ,\, i=1,2. $

Then $U_1\times U_2$ is admissible and $P_{U_1\times U_2}\cong  {\textnormal{R}}s_{t!}^2(P_{U_1}\boxtimes P_{U_2})$.
\end{Coro}
\begin{proof}
Same as the last corollary, assume $U_i=\{H_i<1\}$ for two Hamiltonian functions $H_i \in C^\infty( T^*X_i)$.
So by the uniqueness of projectors, we can assume projectors associated to $U_i$
are obtained from the GKS quantization $\mathcal{K}_i$, i.e., \[ (P_{U_i}, Q_{U_i})= (\widehat{\mathcal{K}}_i\circ {\bK}_{(-\infty,1)},\widehat{\mathcal{K}}_i\circ {\bK}_{[1,\infty)} ),\, i=1,2.\]
Now, consider the product Hamiltonian  $\bR_{{z}}^2$-action on $T^*(X_1\times X_2)$ whose moment maps is $\mu=(H_1, H_2)$. Then $\mathcal{K}={\textnormal{R}}s^2_{t!}(\mathcal{K}_1\boxtimes \mathcal{K}_2)$ is a sheaf quantization of the Hamiltonian action in the sense of \eqref{sheafquantization}.

Observe that if we take $\Omega=\{\zeta=(\zeta_1,\zeta_2):\zeta_1<1, \zeta_2<1 \}$, then we have $U_1\times U_2=\mu^{-1}(\Omega)$. Consequently, the \autoref{Existence} implies that $U_1\times U_2$ is admissible by the following distinguished triangle
\[\widehat{\mathcal{K}}\circ {\bK}_{\Omega} \rightarrow {\bK}_{\Delta\times \{t\geq 0\}} \rightarrow \widehat{\mathcal{K}}\circ {\bK}_{\bR^2_{{\zeta}}\setminus\Omega}\xrightarrow{+1}.\]

Subsequently, let us compute $\widehat{\mathcal{K}}\circ {\bK}_{\Omega}$.

Recall \[\widehat{\mathcal{K}}\circ {\bK}_{\Omega}\cong  \mathcal{K}\star {\bK}_{\{({z}, {\zeta},t): t+{z} \cdot {\zeta}\geq 0\}}[2] \circ {\bK}_{\Omega}.
\]
Noticed $\Omega$ is an open convex set. Therefore ${\bK}_{\{({z}, {\zeta},t): t+{z} \cdot {\zeta}\geq 0\}}[2] \circ {\bK}_{\Omega}$ is the constant sheaf ${\bK}_{\Omega^\circ}$ supported on the $\Omega^\circ$ polar cone of  $\Omega$, where
\[\Omega^\circ =\{({z},t): t+{z} \cdot {\zeta}\geq 0, \forall \zeta \in \Omega \}.\]
In fact, ${\bK}_{\{({z}, {\zeta},t): t+{z} \cdot {\zeta}\geq 0\}}[2] \circ {\bK}_{\Omega}$ is the Fourier-Sato transform of the constant sheaf of the conification of $\Omega$. The conification is a convex cone. The Fourier-Sato transform of a constant sheaf supported on a convex cone is the constant sheaf supported on the polar cone of the given cone. And a direct computation shows that the polar cone of the conification of $\Omega$ is exactly $\Omega^\circ$. Then our computation follows.

In particular, when $\Omega=\{\zeta_1<1, \zeta_2<1 \}$, we have ${\Omega}^\circ=\{({z},t): z=(z_1,z_2),,z_1\leq 0, z_2\leq 0, t\geq -(z_1+z_2)\geq 0\}$. Moreover, ${\bK}_{{\Omega}^\circ} \cong {\textnormal{R}}s_{t!}^2( \bK_{\gamma_1 \times \gamma_2})$, where $\gamma_i = \{(z_i,t) :t \geq -z_i\geq 0\}$.

Now we have
\begin{align*}
 \widehat{\mathcal{K}}\circ {\bK}_{\Omega}&\cong  \mathcal{K}\star {\bK}_{\{({z}, {\zeta},t): t+{z} \cdot {\zeta}\geq 0\}}[2] \circ {\bK}_{\Omega}\\
   &\cong \mathcal{K} \star \bK_{\Omega^\circ}\\
 &\cong \mathcal{K} \star {\textnormal{R}}s_{t!}^2( \bK_{\gamma_1 \times \gamma_2})\\
 &\cong {\textnormal{R}}s^2_{t!}(\mathcal{K}_1\boxtimes \mathcal{K}_2) \star {\textnormal{R}}s_{t!}^2( \bK_{\gamma_1 \times \gamma_2})\\
 &\cong {\textnormal{R}}s_{t!}^2((\mathcal{K}_1 \star {\bK}_{\gamma_1}) \boxtimes (\mathcal{K}_2 \star {\bK}_{\gamma_2}))
 \end{align*}

Finally, noticed that $ {\bK}_{\{(z,t) :t \geq -z\geq 0\}}\cong {\bK}_{\{(z, \zeta,t): t+z\zeta\geq 0\}}[1] \circ {\bK}_{(-\infty,1)} $, one can conclude that
\[ P_{U_1\times U_2}\cong \widehat{\mathcal{K}}\circ {\bK}_{\Omega}  \cong {\textnormal{R}}s_{t!}^2((\mathcal{K}_1 \star \gamma_1) \boxtimes (\mathcal{K}_2 \star \gamma_2)) \cong {\textnormal{R}}s_{t!}^2(P_{U_1}\boxtimes P_{U_2}).\]

\end{proof}

\subsection{Chiu-Tamarkin complex}
Let $G=\bZ/\ell\bZ$ be the finite cyclic group of order $\ell\in \bZ_{\geq 1}$, $X$ be a smooth orientable manifold of dimension $d$ with a fixed orientation. In the following, we are going to study a $G$-equivariant invariant constructed from $P_U$.

Now take an admissible open set $U\subset T^*X$, and let $P_U$ be the kernel associated to $U$.
The manifold $(X^2\times \bR_t)^{\ell}$ admits a $G$-action induced by the cyclic permutation of $\ell$ factors.

There is an adjoint pair $(\alpha_T,\beta_T)$:
\begin{center}
    \begin{tikzcd}
F\in D_G((X^2\times \bR_t)^\ell) \arrow[rr, "\alpha_T", shift left] &  & D_G({\textnormal{pt}}) \ni F', \arrow[ll, "\beta_T"]
\end{tikzcd}
\end{center}
defined by:
\begin{align}\label{definition of adjoint loop functor}
\begin{aligned}
    &\alpha_T(F)=i_T^{-1}{\textnormal{R}}\pi_{\underline{\bq}!}{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(F\right)\\
    &\beta_T(F')=\Tilde{\Delta}_* s_t^{\ell!}\pi_{\underline{\bq}}^!i_{T!} F',
\end{aligned}
\end{align}
where $i_T:\{T\} \rightarrow \bR$, and $\Tilde{\Delta}(\underline{\bq},\underline{t})=(\bq_n,\bq_1,\bq_1,\cdots,\bq_{n-1},\bq_{n-1},\bq_n,\underline{t})$ is a twisted diagonal.

\begin{Def}Under these notations, we define the Chiu-Tamarkin complex as the derived Hom:
\[C_{T,\ell}(U,\bK)={ \textnormal{RHom}  }_G\left(\alpha_T(P_U^{\boxtimes \ell}),\bK[-d]  \right)\cong{ \textnormal{RHom}  }_G\left(P_U^{\boxtimes \ell},\beta_T\bK[-d]   \right) .\]
We see $C_{T,\ell}(U,\bK)$ as an object of $D({\bK}[G]-Mod)\simeq D_G(\text{pt})$.  We set $A = \text{Ext}_G^*(\bK,\bK)$. Then $H^*(C_{T,\ell}(U,\bK))$ is a graded module over $A \cong \text{Ext}_G^*(\bK[-d],\bK[-d])$ via the Yoneda product.
\end{Def}
\begin{RMK}
\begin{enumerate}[fullwidth]
    \item For our convenience, the adjoint isomorphism here is denoted by :\begin{equation}\label{adjoint isomorphism}
    N:{ \textnormal{RHom}  }_G\left(\alpha_T(P_U^{\boxtimes \ell}),\bK[-d] \right) \xrightarrow{\cong} { \textnormal{RHom}  }_G \left(P_U^{\boxtimes \ell}, \beta_T\bK[-d] \right) .  \end{equation}
    \item $C_{T,\ell}(U,\bK)$ is mentioned by Tamarkin in \cite{mc-Tamarkin}, and is defined explicitly by  Chiu in \cite{chiu2017}. It looks slightly different from what Chiu concerned. But one can check directly that, because $X$ is orientable, $ \beta_T\bK[-d] $ is exactly the constant sheaf supported on the twisted diagonal with a degree shift depending only on $\ell$ and $\dim
X$. So the complex $C_{T,\ell}(U,\bK)$ is essentially the same as what Chiu defined.
\end{enumerate}
\end{RMK}

Now, denote \begin{equation}\label{definition of F}
 F_{\ell}(U,\bK)={\textnormal{R}}\pi_{\underline{\bq}!}{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell}\right).
\end{equation}
Then we establish
\begin{equation}
 C_{T,\ell}(U,\bK)={ \textnormal{RHom}  }_G\left((F_{\ell}(U,\bK))_T, \bK[-d]\right).
\end{equation}

Let us compute an example here.
Recall $P_{T^*X}=\bK_{\Delta_{X^2}\times [0,\infty)}$. So
\[\Tilde{\Delta}^{-1}\left(P_{T^*X}^{\boxtimes \ell}\right)= \bK_{\Tilde{\Delta}^{-1}[ (\Delta_{X^2}\times [0,\infty))^\ell]}=\bK_{\Delta_{X^{\ell}}\times [0,\infty)^\ell }.\]
Then
\[F_{\ell}(T^*X,\bK)= {\textnormal{R}}\pi_{\underline{\bq}!}{\textnormal{R}}s_{t!}^\ell(\bK_{\Delta_{X^{\ell}}\times [0,\infty)^\ell })={\textnormal{R}}\pi_{\underline{\bq}!}(\bK_{\Delta_{X^{\ell}}\times [0,\infty) })= {\textnormal{R}}\Gamma_c(\Delta_{X^{\ell}},\bK)_{[0,\infty)} \]
and $G$ acts on ${\textnormal{R}}\Gamma_c(\Delta_{X^{\ell}},\bK)\cong {\textnormal{R}}\Gamma_c(X,\bK)$ trivially. It also shows us $C_{T,\ell}$ is non-trivial only when $T\geq 0$.

Since $G$ acts on ${\textnormal{R}}\Gamma_c(\Delta_{X^{\ell}},\bK)\cong {\textnormal{R}}\Gamma_c(X,\bK)$ trivially, we have
\begin{equation}\label{F of the cotangent bundle}
  \begin{split}
    C_{T,\ell}(T^*X,\bK)&\cong { \textnormal{RHom}  }_G({\textnormal{R}}\Gamma_c(X,\bK),\bK[-d]) \\
    &\cong  { \textnormal{RHom}  }_G(\bK,\bK)\otimes{ \textnormal{RHom}  }({\textnormal{R}}\Gamma_c(X,\bK),\bK[-d]) \\
  &\cong  { \textnormal{RHom}  }_G(\bK,\bK)\otimes {\textnormal{R}}\Gamma(X,\bK)\\
  &\cong  { \textnormal{RHom}  }_G(\bK,\bK)\otimes {\textnormal{R}}\Gamma_X(T^*X,\bK)[d]\\
\end{split}
\end{equation}
where the last two isomorphisms are Poincar\'e-Verdier duality (since $X$ is orientable).
Finally, for $T\geq 0$
\begin{equation}\label{C of the cotangent bundle}
H^*(C_{T,\ell}(T^*X,\bK))\cong   A\otimes H^{*+d}(T^*X,T^*X\setminus X,\bK).
\end{equation}

One of the most important theorems about the Chiu-Tamarkin complex is
\begin{Thm}[{{Theorem 4.7 of} \cite{chiu2017}}]\label{invariance1} Let $U,U_1,U_2$ be admissible open sets and let $U_1\xhookrightarrow{i} U_2$ be an inclusion. Then one has, for $T\geq 0$,
\begin{enumerate}[fullwidth]
    \item There is a morphism $C_{T,\ell}(U_2,\bK) \xrightarrow{i^*} C_{T,\ell}(U_1,\bK)$, which is functorial with respect to inclusions of admissible open sets.
    \item Suppose there is a compactly supported Hamiltonian isotopy $\varphi:T^*X \times I \rightarrow T^*X $, then there is an isomorphism $\Phi_{\ell,T,z }:C_{T,\ell}\left(\varphi_z(U),\bK\right) \cong C_{T,\ell}(U,\bK)$, for all $z\in I$. When $\varphi_z=\text{id}$ for all $z\in I$, $\Phi_{\ell,T,z }=\text{id}$.
\end{enumerate}
\end{Thm}

For our application later, let us review the constructions here. The notations are the same as in \autoref{invariance1}.
\begin{enumerate}[fullwidth]
    \item \label{remarkChiustheorem} Recall, \autoref{functorial} shows that we have a natural morphism $P_{U_1}\rightarrow P_{U_2}$. We deduce morphisms $P_{U_1}^{\boxtimes \ell}\rightarrow P_{U_2}^{\boxtimes \ell}$, then  $\alpha_T(P_{U_1}^{\boxtimes \ell})\rightarrow \alpha_T(P_{U_2}^{\boxtimes \ell})$ and then $i^*$. In fact, we have\begin{equation}\label{functoriality of F}
     F_{ \ell}(U_1,\bK)\xrightarrow{F_{\ell}(i,\bK)}F_{\ell}(U_2,\bK).
    \end{equation}
    \item We take the GKS sheaf quantization $\mathcal{K}\coloneqq\mathcal{K}(
\widehat{\varphi})\in D(I\times X^2\times \bR_t)$ of $\varphi$.

We denote by $\kappa_z$ the ``conjugation'' functor on $D((X^2\times \bR_t)^l)$:
\begin{equation}\label{definition of kappa}
\kappa_z(F)\coloneqq \mathcal{K}_z^{\boxtimes\ell}\star F \star  \mathcal{K}_z^{-\boxtimes\ell} ,
\end{equation}
where $\mathcal{K}_z^{-\boxtimes\ell}\coloneqq (\mathcal{K}_z^{-1})^{\boxtimes\ell}$. Chiu showed that
\begin{itemize}[fullwidth]
\item $P_{\varphi_z(U)}\cong \mathcal{K}_z^{-1}\star P_U \star \mathcal{K}_z$, where $\mathcal{K}_z=\mathcal{K}|_z$.

Hence $\mathcal{K}_z^{\boxtimes\ell}\star P_{\varphi_z(U)}^{\boxtimes\ell} \star  \mathcal{K}_z^{-\boxtimes\ell}\cong  P_U^{\boxtimes\ell} $.

\item There exists a unique isomorphism $g_z:\mathcal{K}_z^{\boxtimes\ell}\star \beta_T\bK[-d] \star \mathcal{K}_z^{-\boxtimes\ell}\xrightarrow{\cong} \beta_T\bK[-d] $ such that $g_0=\text{id}$.

Specifically, $g_z$ comes from the following construction: Consider the whole family $[\beta_T\bK[-d]\boxtimes \bK_{I}]^{-1}\star_{I} \delta_z^{-1}\mathcal{K}^{\boxtimes\ell} \star_{I} [\beta_T\bK[-d]\boxtimes \bK_{I}] $, where $\delta_z:\bR_z\rightarrow \bR^\ell_z$ is the diagonal map, $\star_{I}$ is the convolution with respect to $\bR^\ell_{\underline{t}}$ and relative to $I$. We refer \cite[Section 1.6]{GKS2012} for the definition of this relative convolution. One can estimate the microsupport to verify that the family defines a sheaf quantization of the conic lifting of the Hamiltonian isotopy ${\varphi_z}^{\times \ell}: (T^*X)^\ell \circlearrowleft$. On the other hand, $\delta_z^{-1}\mathcal{K}^{\boxtimes\ell}$ is obviously a sheaf quantization of the conic lifting of ${\varphi_z}^{\times \ell}$. So the uniqueness part of \autoref{GKS} shows there is a unique isomorphism \[\Tilde{g}: \delta_z^{-1}\mathcal{K}^{\boxtimes\ell}\xrightarrow{\cong} [\beta_T\bK[-d]\boxtimes \bK_{I}]^{-1}\star_{I} \delta_z^{-1}\mathcal{K}^{\boxtimes\ell} \star_{I} [\beta_T\bK[-d]\boxtimes \bK_{I}]. \]

    One can take convolution to obtain
    \begin{align}\label{definition of g}
    \begin{aligned}
        g\coloneqq& (\delta_z^{-1}\mathcal{K}^{\boxtimes\ell})^{-1}\star_{I}(\beta_T\bK[-d]\boxtimes \bK_{I})\star_{I}\Tilde{g}:\\
        &(\delta_z^{-1}\mathcal{K}^{\boxtimes\ell})^{-1}\star_{I}(\beta_T\bK[-d]\boxtimes \bK_{I})\star_{I}(\delta_z^{-1}\mathcal{K}^{\boxtimes\ell})\xrightarrow{\cong} \beta_T\bK[-d]\boxtimes \bK_{I},
    \end{aligned}\end{align}
    such that $g_z=g|_z$.
\end{itemize}
The $\Phi_{\ell,T,z}$ is defined as the composition $\Phi_{\ell,T,z}=N^{-1}\circ (g_{z*}\circ {\kappa_z})\circ  N$, i.e. $\Phi_{\ell,T,z}(f)=N^{-1}((g_{z}\circ {\kappa_z})( N(f)))$, where $N$ is given by the adjoint isomorphism \eqref{adjoint isomorphism}, and $g_{z*}=g_z\circ -$. Sometimes, we use $\Phi_{\ell,T,z}=g_{z*}\circ {\kappa_z}$ too as an abuse of notation.
\end{enumerate}

Taking into account the structure of $A=\text{Ext}^*_G(\bK,\bK)$-modules, then we have
\begin{Prop}\label{moduleisomorphism}Under the notations of \autoref{invariance1}, we have:
\begin{enumerate}[fullwidth]
\item $H^*(i^*)$ is a morphism of $A$-modules.
\item $H^*(\Phi_{\ell,T,z })$ is an isomorphism of $A$-modules.
\end{enumerate}
\end{Prop}

\begin{proof}The first statement is direct from the definition of the Yoneda product. We prove the second statement. For $f\in H^*(C_{T,\ell}(\varphi_z(U)))$, $a\in \textnormal{Ext}_G^*(\bK,\bK)$, (ignoring degrees to simplify), we have
\[\Phi_{\ell,T,z}(a.f)=g_{z*}\left({\kappa_z}(\beta_T(a)\circ f )\right)=g_{z*}({\kappa_z}(\beta_T(a))\circ {\kappa_z}(f))=g_{z} \circ {\kappa_z}(\beta_T(a))\circ {\kappa_z}(f),\]
and
\[a.\Phi_{\ell,T,z}(f)=\beta_T(a)\circ \Phi_{\ell,T,z}(f)=\beta_T(a)\circ  g_{z*}({\kappa_z}(f))=\beta_T(a)\circ  g_{z}\circ{\kappa_z}(f).\]
Then the problem is reduced to show $g_{z} \circ {\kappa_z}(\beta_T(a))=\beta_T(a)\circ  g_{z}$ for all $z\in I$.

We consider the whole families $g_{z} \circ {\kappa_z}(\beta_T(a))$,  $\beta_T(a)\circ  g_{z}$ and introduce the functor $\kappa=\delta_z^{-1}\mathcal{K}^{\boxtimes\ell}\star_{I} -\star_{I} \delta_z^{-1}\mathcal{K}^{-\boxtimes\ell}$. Then applying \cite[Proposition 3.12]{GKS2012}, we have the following isomorphism induced by the restriction $|_z$ for all $z\in I$:
\begin{equation}\label{restrition isomorphism}
{ \textnormal{Hom}  }_{D_G}({\kappa}(\beta_T\bK[-d]\boxtimes \bK_{I}), \beta_T\bK[-d]\boxtimes \bK_{I})\xrightarrow{\cong}{ \textnormal{Hom}  }_{D_G}({\kappa_z}(\beta_T\bK[-d]), \beta_T\bK[-d]).
\end{equation}

Now, one can verify directly that $g_0\circ {\kappa_0}(\beta_T(a))=\beta_T(a)\circ g_0$ because $\mathcal{K}^{\boxtimes \ell}_0=\bK_{\Delta_{X^{2\ell}}\times [0,\infty)^d}$. Then the isomorphism \eqref{restrition isomorphism} shows that $g_z\circ {\kappa_z}(\beta_T(a))=\beta_T(a)\circ g_z$ for all $z\in I$.\end{proof}

\subsection{Capacities}\label{capacities}
In this section, we assume that $\ell \in \bO$ is odd, $p_\ell$ is the minimal prime factor of $\ell$, and $\bK=\bF_{p_\ell}$ is the finite field of order $p_\ell$. Under these conditions, the Yoneda algebra $A=\text{Ext}^*_G(\bF_{p_\ell},\bF_{p_\ell})$ is isomorphic to ${\bF_{p_\ell}}[u,\theta]$, where $|u|=2, |\theta|=1$, and $\theta^2=0$. Let $V$ be a vector space of dimension $d$.

Now, we are going to define a sequence of capacities using the Chiu-Tamarkin complex for the class of admissible open sets in $T^*V$. For $U\xhookrightarrow{i_U} T^*V$ an admissible open subset, the morphism \eqref{functoriality of F} induces a morphism of $G$-equivariant sheaves:
\[F_{\ell}(U,\bF_{p_\ell}) \xrightarrow{F_{\ell}( i_U,\bF_{p_\ell})} F_{\ell}(T^*V, \bF_{p_\ell}).\]
Now, we first take the stalk at $T \geq 0$, and then use the orientation of $V$. There is a composition of morphisms:\[\eta_{\ell,T}(U,\bF_{p_\ell}): (F_{\ell}(U,\bF_{p_\ell}))_T \rightarrow (F_{\ell}(T^*V, \bF_{p_\ell}))_T\cong {\textnormal{R}}\Gamma_c(V,\bF_{p_\ell})\cong \bF_{p_\ell}[-d].\]
So \begin{align*}
    \eta_{\ell,T}(U,\bF_{p_\ell})\in& { \textnormal{Hom}  }_{D_G}((F_{\ell}(U,\bF_{p_\ell}))_T,\bF_{p_\ell}[-d])\\
    =&H^0({ \textnormal{RHom}  }_G(F_{\ell}(U,\bF_{p_\ell}))_T,\bF_{p_\ell}[-d]))\\\cong& H^0(C_{T,\ell}(U,\bF_{p_\ell})).
\end{align*}
\begin{Def}\label{definition of capacities}For an admissible open set $U$ and $k\in \bZ_{\geq 1}$ we define
\begin{align}\label{definition of c_k}
\begin{aligned}
 &\textnormal{Spec}(U,k)\\ \coloneqq &\{T\geq 0: \exists\ell_0\in \bO,\, \forall\ell \in \bO, \, p_\ell \geq p_{\ell_0},\, \exists \gamma_\ell  \in H^{*}(C_{T,\ell}(U,\bF_{p_\ell})),\, \eta_{\ell,T}(U,\bF_{p_\ell})=u^k\gamma_\ell \},
\end{aligned}
\end{align}
and
\begin{equation}
 c_k(U)\coloneqq \inf \textnormal{Spec}(U,k) \in [0,+\infty ].
 \end{equation}

\end{Def}
In the following, we will prove step by step that $(c_k)_{k\in\bZ_{\geq 1}}$ defines a sequence of non-trivial symplectic capacities.
\begin{Thm}\label{capacity property symplectic} The functions $c_k:\mathcal{C}_{\text{admissible}}\rightarrow (0,\infty]$ satisfy the following:
\begin{enumerate}[fullwidth]
    \item $c_k \leq c_{k+1}$ for all $k\in \bZ_{\geq 1}$.
    \item If there is an inclusion of admissible open sets $U_1 \subset U_2$, then $c_k(U_1) \leq c_k(U_2)$.
    \item For a compactly supported Hamiltonian isotopy $\varphi_z:\bR^{2d}\rightarrow \bR^{2d}$, we have
    $c_k(U)=c_k(\varphi_z(U)).$
    \item If $U=\{H<1\}$ is an admissible open set whose boundary $\partial U=\{H=1\}$ is a regular level set and if  $c_k(U)<\infty$, then $c_k(U)$ is represented by the action of a closed characteristic in the boundary  $\partial U$. Consequently, $c_k(rU)=r^2c_k(U)$ for $r>0$.
\end{enumerate}
\end{Thm}
\begin{proof}(1) and (2) are direct consequence of the \autoref{definition of capacities} and \eqref{functoriality of F}. The invariance (3) follows from \eqref{definition of c_k} by the \autoref{moduleisomorphism} and the following \autoref{invariance20} below. (4) is exactly \autoref{representing property} below.

We will see later, in \autoref{nonzeroellisoid}, that for a ball $B_a$, one has $c_k(B_a)=\lceil k/d \rceil a$. Consequently, $c_k(U)>0$ for all admissible subsets.
\end{proof}
\begin{RMK}We also see from the proof that if $U$ is a bounded open set( which are admissible by \autoref{opensetsareadmissible}), then $c_k(U)< \infty$.
\end{RMK}

\begin{Lemma}\label{invariance20} Let $\varphi:T^*V \times I \rightarrow T^*V$ be  a compactly supported Hamiltonian isotopy and $U$ an admissible open set. Recall the $A$-module isomorphism, defined in \autoref{invariance1},  \[H^*(\Phi_{\ell,T,z}): H^*(C_{T,\ell}(\varphi_z(U),\bK)) \xrightarrow{\cong} H^*(C_{T,\ell}(U,\bF_{p_\ell})).\]
Then for all filed $\bK$, we have $H^*(\Phi_{\ell,T,z})(\eta_{\ell,T}(\varphi_z(U),\bK))=\eta_{\ell,T}(U,\bK)$ for all $z\in I$.
\end{Lemma}
\begin{proof}Recall $\eta_{\ell,T}(\varphi_z(U),\bK)$ is the composition
\begin{center}
\begin{tikzcd}
(F_{\ell}(\varphi_z(U),\bK))_T \arrow[r] \arrow[Equal]{d}             & (F_{T*V,\ell}(T*V, \bK))_T \arrow[r, "\cong"] \arrow[Equal]{d}    & \textnormal{R}\Gamma_c(V,\bK) \arrow[r, "or"] & {\bK[-d],} \\
\alpha_T(P^{\boxtimes\ell}_{\varphi_z(U)}) \arrow[r] & \alpha_T(P^{\boxtimes\ell}_{T^*V}) &                                        &
\end{tikzcd}
\end{center}
where $or$ is the isomorphism given by taking an orientation of $\Delta_{V^{2\ell}} \cong V$.
Then
\[N^{-1}(\eta_{\ell,T}(\varphi_z(U),\bK))=\left[P_{\varphi_z(U)}^{\boxtimes\ell} \rightarrow \beta_T\alpha_T(P^{\boxtimes\ell}_{T^*V}) \xrightarrow{\beta_T(or)}   \beta_T\bK[-d]  \right]  \]
For the image $H^*(\Phi_{\ell,T,z})(N^{-1}(\eta_{\ell,T}(\varphi_z(U),\bK)))$, we embed it into a commutative diagram.
\begin{center}
\begin{tikzcd}
P_{U}^\ell \arrow[d, "\cong"] \arrow[r]                                                                   & \mathcal{K}_z^{-\boxtimes \ell}\star \beta_T\alpha_T(P_{T^*V}^\ell) \star \mathcal{K}_z^{\boxtimes \ell}  \arrow[r, "\kappa_z(or)"] \arrow[d, "g_z"] & {\mathcal{K}_z^{\boxtimes \ell}\star \beta_T\bK[-d]\star\mathcal{K}_z^{-\boxtimes \ell}} \arrow[d, "g_z"] \\
\mathcal{K}_z^{-\boxtimes \ell}\star P_{\varphi_z(U)}^\ell \star \mathcal{K}_z^{\boxtimes \ell} \arrow[r] & \beta_T\alpha_T(P_{T^*V}^{\boxtimes \ell}) \arrow[r, "or"]                  & \bK[-d]
\end{tikzcd}
\end{center}

The left square is commutative by the natural isomorphism $\kappa_z $(See \eqref{definition of kappa}), and commutativity of the right is essentially the same with \autoref{moduleisomorphism}. On the other hand, the lower-left  arrow is $N^{-1}(\eta_{\ell,T}(U,\bK))$, and the upper-right arrow is $N^{-1}H^*(\Phi_{\ell,T,z})(\eta_{\ell,T}(\varphi_z(U),\bK))$.
Then the commutativity of the diagram gives us $N^{-1}H^*(\Phi_{\ell,T,z})(\eta_{\ell,T}(\varphi_z(U),\bK))=N^{-1}(\eta_{\ell,T}(U,\bK))$. That is $H^*(\Phi_{\ell,T,z})(\eta_{\ell,T}(\varphi_z(U),\bK))=\eta_{\ell,T}(U,\bK)$. \end{proof}

\begin{Prop}\label{representing property}Suppose $U=\{H<1\}$ is admissible such that $\partial U=\{H=1\}$ is a regular level set of a Hamiltonian function $H$. If $c_k(U) < \infty $, then
\begin{equation}
c_k(U)\in \left\lbrace \bigg|\int_\gamma \bp d\bq \bigg|: \gamma \text{ is a closed orbit of }\varphi_z^H \right\rbrace.
\end{equation}
Consequently, for this $U$, we have $c_k(rU)=r^2c_k(U)$ for $r\in\bR_{>0}$.
\end{Prop}
\begin{proof}Let $T=c_k(U)$. Suppose it is not given by the action of a closed characteristic.

The Liouville 1-form is non-degenerated here, so there is only finitely many closed characteristics with action less than $2T$. So there is a small $\varepsilon>0$ such that there is no action happening in $[T-\varepsilon,T+\varepsilon]$.

However, one can estimate the microsupport of $F_{\ell}(U,\bK)$ to see that \[{\mu}s_L(F_{\ell}(U,\bK))\subset\left\lbrace t\in \bR: t= \bigg|\int_\gamma \bp d\bq \bigg|\text{ for some closed orbit }  \gamma\text{ of }\varphi_z^H \right\rbrace.\]

Therefore $F_{\ell}(U,\bK)$ is constant on $[T-\varepsilon,T+\varepsilon]$. Consequently, $(F_{\ell}(U,\bK))_{T-\varepsilon}\cong (F_{\ell}(U,\bK))_T$, and then $\eta_{T-\varepsilon,\ell}(U,\bK)=\eta_{\ell,T}(U,\bK)$ for all $\ell$. All assertions here are independent with the field $\bK$. Then we have that $c_k(U)\leq T-\varepsilon$, contradiction!\end{proof}
Finally, let us remark about the computability of $c_k$. We define $c_k(U)$ using an arbitrary (but unique) kernel $P_U$. But in practice, we will choose a particular kernel. Usually, these kernels will admit some properties that are not so obvious from general existence results like \autoref{Existence}, and \autoref{opensetsareadmissible}.

We will see, in the section 3, that how to use a kernel associated to toric domains constructed from generating functions to compute capacities of convex toric domains.

\subsection{Geometry of \texorpdfstring{$F_{\ell}(U,\bK)$}{}}\label{Discrete Hamiltonian loop section}
Here, we make a pause to discuss the geometry of the construction of the Chiu-Tamarkin complex. Following ideas of Chiu, we can resolve $F_{\ell}(U,\bK)\cong {\textnormal{R}}\pi_{\underline{\bq}!}{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell}\right)$ step in step.

\begin{Lemma}\label{step1}We assume $P_U\cong \widehat{\mathcal{K}}\circ {\bK}_{\Omega} \cong \mathcal{K}\star \widehat{{\bK}_{\Omega}}$, where $\mathcal{K}$ is a sheaf quantization of some Hamiltonian $\bR^m_{ {z}}$ action on $T^*X$ with  moment map $\mu$, $\Omega \subset \bR_{ {\zeta}}^m$ and $U=\mu^{-1}(\Omega)$. Then we have
    \[{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell  }\right) \cong {\textnormal{R}}\pi_{{ {z}}!}{\textnormal{R}}s_{t!}^2 {\textnormal{R}}s_{{ {z}}!}^\ell  \left( \pi_{t_2}^{-1}\left(  {\textnormal{R}}s_{ {t_1}!}^\ell  \Tilde{\Delta}^{-1}\mathcal{K}^{\boxtimes \ell  }   \right) \otimes \pi_{( {\bq},t_1)}^{-1}  {\textnormal{R}}s_{t_2!}^\ell\widehat{{\bK}_{\Omega}}^{\boxtimes \ell  }     \right).\]

\end{Lemma}
\begin{proof}
It is a direct application of the proper base change and the projection formula.
\end{proof}

The lemma shows, as the construction itself, we can consider separately the Hamiltonian action and the cut-off by $\Omega$. Let us study the Hamiltonian action first.
In view of the lemma, it is convenient to define
\begin{equation}
  \label{eq:defLell}
  \begin{split}
    L_\ell(\mathcal{K}) &\coloneqq {\textnormal{R}}s_{t!}^\ell(\Tilde{\Delta}^{-1}(\mathcal{K}^{\boxtimes \ell})) ,\\
   \mathcal{L}_\ell(\mathcal{K}) &\coloneqq {\textnormal{R}}s_{{z}!}^{\ell}\, L_\ell(\mathcal{K}) .
  \end{split}
\end{equation}

The sheaf $L_\ell(\mathcal{K})$ encodes the information of discrete Hamiltonian loops. Precisely, we have
\begin{Prop}\label{Discrete Hamiltonian loop} With the notations~\eqref{eq:defLell} we have
\begin{enumerate}[fullwidth]
\item $\mu s(L_\ell(\mathcal{K}))$ is
\[
\left\lbrace (\underline{z}, \underline{\zeta}, \bq_j, \bp_j):
    \begin{aligned}
    &\text{There exist } (\underline{z}, \underline{\zeta}, \bq'_j, \bp'_j,\bq_j'', \bp_j'')\text{ such that }
(\bq''_{j}, -\bp''_{j})= {z_j}.(\bq'_j,\bp'_j), \\
&\bq_j'=\bq_{j+1},\,
\bp_j=\bp_j'+\bp_{j+1}'',\zeta=-\mu(\bq_j',\bp_j')\,
j\in \bZ/\ell\bZ
    \end{aligned}
   \right\rbrace
\]

\item $L_\ell(\mathcal{K})\cong (s_{ {z}}^{\ell})^{-1}{\textnormal{R}}s_{ {z}*}^{\ell}L_\ell(\mathcal{K})\cong (s_{ {z}}^{\ell})^{-1} \mathcal{L}_\ell(\mathcal{K})$.
\end{enumerate}
\end{Prop}
\begin{proof}
\begin{enumerate}[fullwidth]
\item It follows directly from the functorial estimate of microsupport. The equality follows that the righthand side is a connected Lagrangian submanifold.
\item The first isomorphism follows the microsupport estimate. The second isomorphism follows from $s_z^\ell|_{ supp(L_\ell(K))}$ is proper, which could be proven similarly to \cite[Proposition 3.2(i)]{GKS2012}.\end{enumerate}\end{proof}

We call both $L_\ell(\mathcal{K})$ and $\mathcal{L}_\ell(\mathcal{K})$ the sheaves of discrete Hamiltonian loops, and then the result of \autoref{step1} is written as
\begin{align}\label{loop sheaf rep F}
\begin{aligned}
{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell  }\right)
&\cong {\textnormal{R}}\pi_{{ {z}}!}{\textnormal{R}}s_{t!}^2   \left( \pi_{t_2}^{-1}  \mathcal{L}_\ell(\mathcal{K})   \otimes \pi_{( {\bq},t_1)}^{-1}  {\textnormal{R}}s_{(z,t_2)!}^\ell\widehat{{\bK}_{\Omega}}^{\boxtimes \ell  }\right) \\
&\cong {\textnormal{R}}\pi_{{ {z}}!} \left( \mathcal{L}_\ell(\mathcal{K})\star {\textnormal{R}}s_{(z,t_2)!}^\ell \widehat{{\bK}_{\Omega}}^{\boxtimes \ell  } \right)  . \end{aligned}
\end{align}

Next, let us study ${\textnormal{R}}s_{(z,t_2)!}^\ell\widehat{{\bK}_{\Omega}}^{\boxtimes \ell  }$. In general, the Fourier-Sato transformation is not easy to compute. But when $\Omega$ is an open convex set, the result is known. This condition is satisfied for two situations: (i) When we just consider one single Hamiltonian function $H$, and $\Omega=(-\infty,1)$ such that $U=H^{-1}(\Omega)$. (ii) In the next section, we will assume $\Omega$ to be convex to study convex toric domains.

So, here, it is not so bad to assume convexity of $\Omega$ as our first glimpse.
In this case $\widehat{{\bK}_{\Omega}}\cong  {\bK}_{\Omega^{\circ}}$(see the same argument with \autoref{product}), where \[\Omega^\circ =\{({z},t): t+{z} \cdot {\zeta}\geq 0, \forall \zeta \in \Omega \}.\]
Moreover, $\Omega^\circ$ is a closed convex cone containing the origin. So
\begin{equation}\label{cut-offer}
{\textnormal{R}}s_{({z},t)!}^{\ell} \widehat{{\bK}_{\Omega}}^{\boxtimes \ell} \cong {\textnormal{R}}s_{({z},t)!}^{\ell} {\bK}_{\Omega^{\circ}}^{\boxtimes \ell} \cong {\bK}_{\Omega^{\circ}}.
\end{equation}

Consequently, \eqref{loop sheaf rep F} could be read as
\begin{equation}
{\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell}\right)\cong {\textnormal{R}}\pi_{ {z}!}{\textnormal{R}}s_{t!}^2   \left(\pi_{t_2}^{-1}  \mathcal{L}_\ell(\mathcal{K}) \otimes\pi_{(\underline{\bq},t_1)}^{-1}  {\bK}_{{\Omega}^{\circ}}  \right).
\end{equation}
Therefore,
\begin{align}\label{reformulation of F}
\begin{aligned}
F_{\ell}(U,\bK) &\cong {\textnormal{R}}\pi_{\underline{\bq}!} {\textnormal{R}}s_{t!}^\ell   \Tilde{\Delta}^{-1}\left(P_U^{\boxtimes \ell}\right)\\
&\cong {\textnormal{R}}\pi_{\underline{\bq}!} {\textnormal{R}}\pi_{{z}!}{\textnormal{R}}s_{t!}^2   \left(\pi_{t_2}^{-1}  \mathcal{L}_\ell(\mathcal{K}) \otimes\pi_{(\underline{\bq},t_1)}^{-1}  {\bK}_{{\Omega}^{\circ}}  \right)\\
&\cong {\textnormal{R}}\pi_{{z}!}{\textnormal{R}}s_{t!}^2   \left(\pi_{t_2}^{-1}  {\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{K}) \otimes\pi_{t_1}^{-1}  {\bK}_{{\Omega}^{\circ}}  \right)\\
&\cong {\textnormal{R}}\pi_{{z}!}\left( {\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{K}) \star   {\bK}_{{\Omega}^{\circ}}  \right)
\end{aligned}
\end{align}

From this formula, the study of $F_{U,\ell}$ is reduced to understand ${\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(K)$, which could be thought as a cohomology sheaf of discrete loop space.

So far, we find two different ways to understand $F_{\ell}(U,\bK)$. Initially, from the definition of $F_{\ell}(U,\bK)$, we first cut off the energy of a Hamiltonian isotopy to obtain the kernels and then use the functor $\alpha_T$ to obtain cohomology of some discrete loop space. On the other hand, the results of this section shows, we can study discrete loops of a Hamiltonian isotopy first, and then cut off energy. The result of the section clarifies that these two ways are the same.

The second way is more direct than the first in many cases; we will see more about this point of view when doing computation for toric domains.

\section{Toric domains}
In this section, we would like to study toric domains.

The $2$-dimensional rotation $\varphi_z(u)=e^{-2i\pi z}u$ on $\bC_u$ is the Hamiltonian flow of the Hamiltonian function $H(u)=\pi|u|^2 $.

Consider the product action of single $2$-dimensional rotations, i.e., the  $\bR^d_{ {z}}$ action on $\bC^d_{u}$, which is indeed a torus action, given by
\[{ {z}}.(u_1,\dots, u_n) =(e^{-2i\pi z_1}u_1,\dots, e^{-2i\pi z_d} u_d).\]
The moment map of the standard Hamiltonian torus action is \begin{equation}\label{definition of the moment map of rotation}
 \mu(u_1,\dots, u_n)=(\pi|u_1|^2,\dots,\pi|u_d|^2).
\end{equation}

\begin{Def}For $\Omega \subset \bR^d_{{\zeta}\geq 0}$ open, we call $X_{\Omega}\coloneqq \mu^{-1}(\Omega)$ an (open) toric domain. We say $X_\Omega$ is a convex toric domain if $\widehat{\Omega}\coloneqq \{{\zeta}\in \bR^d:(|\zeta_1|,\dots ,|\zeta_d|) \in \Omega\}$ is convex.
\end{Def}

\begin{RMK}\label{toricdomainremark}For any open subset $\Omega \subset \bR^d_{ {\zeta}\geq 0}$, one can take an open set $\Omega^+$ such that $-\bR^d_{ {\zeta}\geq 0} \subset\Omega^+ \subset \bR^d$ and $\Omega^+\cap \bR^d_{ {\zeta}\geq 0}=\Omega$. One can see $X_{\Omega}=\mu^{-1}(\Omega^+)$, and it is independent of choices of $\Omega^+$. Moreover, when $X_{\Omega}$ is a convex toric domain, one can take $\Omega^+$ to be convex (in the usual sense). In the following, we prefer to use $\Omega^+$ instead of $\Omega$, we will not distinguish them.\end{RMK}
For example, we can take a sequences of real numbers $0<a_1\leq \cdots \leq a_d \leq +\infty$, let $\Omega_D= \{{\zeta}: \zeta_i < a_i, i=1,\cdots,d\}$ and $\Omega_E=\{{\zeta}: \frac{\zeta_1}{a_1}+\cdots +\frac{\zeta_d}{a_d}<1\}$. Then
$X_{\Omega_D}=D(a_1,\cdots,a_d)$ is an open poly-disc and $X_{\Omega_E}=E(a_1,\cdots,a_d)$ is an open ellipsoid. They are all convex toric domains.
\subsection{Generating Function Model for Kernel of Toric Domains}\label{GFmodelToricdomain}

In \cite[Proposition 3.10]{chiu2017}, Chiu constructs a sheaf quantization of Hamiltonian rotation in all dimensions, particularly for the $2$-dimensional  $\varphi_z$, say
$\mathcal{S} \in D(\bR_z\times \bR_{q_1}\times\bR_{q_2} \times \bR_t)$. This quantization possesses one more property than we stated for general sheaf quantizations (see \eqref{sheafquantization}), namely
\begin{equation}\label{formula of GF sheaf quantization}
 \mathcal{S}\cong \textnormal{R}\pi_{(q_2,\cdots,q_M)!}{\bK}_{\{(z,q_1,\cdots,q_{M+1},t):t+\sum_{j=1}^MS_{H}(z/M,q_j,q_{j+1})\geq 0\}},
\end{equation}
where $M$ is big enough such that $z/M \in (0,1/4)$ (we have the same property in the negative direction), and $S_H$ is the generating function of the Hamiltonian rotation:
\begin{equation}\label{formula of S_H}
S_H(z,q_1,q_{2})= \frac{q_1^2+q_{2}^2}{2\tan(2\pi z)} -\frac{q_1 q_{2}}{\sin(2\pi z)}.
\end{equation}
The formula \eqref{formula of GF sheaf quantization} is essential when doing the computation of Chiu-Tamarkin complexes of convex toric domains.

Let
\begin{equation}\label{sheafquantization of Hamiltonian torus action}
 \mathcal{T}\coloneqq \textnormal{R}s_{t!}^d(\mathcal{S}^{\boxtimes d}) \in D(\bR_{{z}}^d\times \bR^d_{\bq_1}\times\bR^d_{\bq_2} \times \bR_t).
\end{equation}
The microsupport estimates show $\mathcal{T}$ is a sheaf quantization of the standard torus action in the sense of \eqref{sheafquantization}.

As a corollary of \autoref{Existence}, we have
\begin{Prop}Toric domains $X_\Omega$ are admissible by the distinguished triangle
\begin{equation}
\widehat{\mathcal{T}}\circ {\bK}_{\Omega} \rightarrow {\bK}_{\Delta_{V^2}\times \{t\geq 0\}} \rightarrow \widehat{\mathcal{T}}\circ {\bK}_{\bR^d_{\underline{\zeta}}\setminus\Omega}\xrightarrow{+1} ,
\end{equation}
and the pair of kernels
\begin{equation}
    P_{X_{\Omega}}\coloneqq \widehat{\mathcal{T}}\circ {\bK}_{\Omega} ,\qquad Q_{X_{\Omega}}\coloneqq \widehat{\mathcal{T}}\circ {\bK}_{\bR^d_{\underline{\zeta}}\setminus\Omega}.
\end{equation}
\end{Prop}

This pair of kernels $(P_{X_{\Omega}}, Q_{X_{\Omega}}) $ constructed from $\mathcal{T}$ is called the generating function model of the kernel associated to toric domains.
As we studied in \autoref{Discrete Hamiltonian loop section}, when $X_{\Omega}$ is a convex toric domain, we have
\begin{equation}\label{GFmodel convex case}
    \widehat{\mathcal{T}}\circ {\bK}_{\Omega} \cong \mathcal{T} \star {\bK}_{\Omega^\circ}
\end{equation}
as the generating function model of the kernel of convex toric domains.
\begin{eg}Take a sequence of real numbers $0<a_1\leq \cdots \leq a_d$.
\begin{enumerate}[fullwidth]
\item Suppose $\Omega_D= \{ {\zeta}: \zeta_i < a_i, i=1,\dots,d\}$, then $X_{\Omega_D}=D(a_1,\dots,a_d)$ is the open poly-disc.

Let $P_{r}$ be the kernel of the open disc $\{ \pi |u|^2 < r \}$ in $\bC$, then \autoref{product} applies and $P_{D}\cong {\textnormal{R}}s^d_{t!}(P_{a_1}\boxtimes\cdots \boxtimes P_{a_d})$.

\item Suppose $\Omega_E=\{ {\zeta}: \frac{\zeta_1}{a_1}+\cdots +\frac{\zeta_d}{a_d}<1\}$, then $X_{\Omega_E}=E(a_1,\dots,a_d)$ is the open ellipsoid, and $\Omega_E^\circ=\{( {z},t): t\geq -a_1z_1=\cdots=-a_dz_d\geq 0\}$.

Let $e: \bR_z\times \bR_t \rightarrow \bR_{  {z}}^d\times \bR_t, (z,t)\mapsto (a_1z,\dots,a_dz,t)$, then ${\bK}_{\Omega_E^\circ}={\textnormal{R}}e_!{\bK}_{\{t\geq -z\geq 0\}}$. Therefore, the projection formula shows
\[P_{E}\coloneqq P_{X_{\Omega_E}} \cong \mathcal {T}\star {\textnormal{R}}e_!{\bK}_{\{t\geq -z\geq 0\}} \cong (e^{-1}\mathcal{T}) \star {\bK}_{\{t\geq -z\geq 0\}} \cong \widehat{e^{-1}\mathcal{T}} \circ {\bK}_{(-\infty,1)}.\]

One can check directly that $e^{-1}\mathcal{T}$ is the sheaf quantization of the diagonal Hamiltonian rotation $\varphi_z({u})=(e^{ \frac{-2i\pi z}{a_1}}u_1,\dots, e^{ \frac{-2i\pi z}{a_d}} u_d)$ in the sense of \eqref{sheafquantization}.

In particular, when $a_1=\dots=a_d=\pi R^2>0$, the construction is the same as Chiu's for balls.
\end{enumerate}
\end{eg}

\subsection{Chiu-Tamarkin complex and Capacities of Convex Toric Domains}
In this section, we focus on convex toric domains, that is, $X_{\Omega}=\mu^{-1}(\Omega)$, where $\Omega \subset \bR^d_{\geq 0}$ is an open set such that $\{\underline{\zeta}\in \bR^d:(|\zeta_1|,\cdots ,|\zeta_d|) \in \Omega\}$ is convex. In particular, $\Omega$ itself is convex too.

For such an $\Omega$, we can a take open set $\Omega^+$ such that $-\bR^d_{ {\zeta}\geq 0} \subset\Omega^+ \subset \bR^d$ and $\Omega^+\cap \bR^d_{ {\zeta}\geq 0}=\Omega$. We will see later in the \autoref{rmkgammatopology} that such a choice does not affect on the computation of $C_{T,\ell}(X_\Omega,\bF_{p_\ell})$. So, we always choose one $\Omega^+$ and we will not distinguish $\Omega$ and $\Omega^+$.

One can verify that, under such conditions, the polar cone satisfies $\{O\}\times \bR_{\geq 0}\subset \Omega^\circ \subset \bR^d_{\leq 0}\times \bR_{\geq 0}$. For  $T\geq 0$, we set
\begin{align*}
  &{\Omega}^\circ_T \coloneqq {\Omega}^\circ\cap \{t=T\}=\{z\in \bR^d: T+\langle z, \zeta \rangle \geq 0, \forall \zeta \in \Omega\},\\
  &\|\Omega^\circ_T\|_{\infty}=\max_{z\in \Omega^\circ_T}\|z\|_{\infty} .
\end{align*}
Then $\|\Omega^\circ_T\|_{\infty}=T\|\Omega^\circ_1\|_{\infty}$.
We also define
\begin{equation*}
I(\Omega^\circ_T)=\max_{z\in \Omega^\circ_T}I(z) ,
\qquad \text{ where }
  I(z)=\sum_{i=1}^d \big\lfloor {-z_i} \big\rfloor \text{ for $z\in \bR^d$.}
\end{equation*}

So far, we are in a position to state our structural theorem of $H^*(C_{T,\ell}(X_{\Omega},\bF_{p_\ell}))$, where $X_{\Omega}$ is a convex toric domain:
\begin{Thm}\label{structure toric domian module}For a convex toric domain $X_\Omega \subsetneqq T^*V$,  $\ell\in \bO$, and $0\leq T <p_\ell /\|\Omega^\circ_1\|_{\infty}$, we have:
\begin{itemize}[fullwidth]
    \item The minimal cohomology degree of $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is exactly $-2I(\Omega^\circ_T)$, i.e.,
    \[ H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))\cong  H^{\geq -2I(\Omega^\circ_T)}(C_{T,\ell}(X_\Omega,\bF_{p_\ell})) \]
    and
\[ H^{-2I(\Omega^\circ_T)}(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))\neq 0 . \]

\item $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is a finitely generated $\bF_{p_\ell}[u]$-module. The free part is isomorphic to $A=\bF_{p_\ell}[u,\theta]$, so $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is of rank $2$ over $\bF_{p_\ell}[u]$. The torsion part is located exactly in cohomology degree $[-2I(\Omega^\circ_T),-1]$.  $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is torsion free when $X_\Omega$ is an open ellipsoid.
\item For each $Z\in \Omega^\circ_T$, the inclusion $\overline{OZ} \subset \Omega^\circ_T$ induces a decomposition $\eta_{\ell,T}(X_{\Omega},\bF_{p_\ell})=u^{I(Z)}\gamma_{Z,\ell}$ for a non-torsion element $\gamma_{Z,\ell}\in H^{-2I(Z)}(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$. In particular, $\eta_{\ell,T}(X_{\Omega},\bF_{p_\ell})$ is non-zero.

\end{itemize}
\end{Thm}

\begin{RMK}Quillen studied a more general localization question about $\bZ/\ell\bZ$-equivariant cohomology in \cite{Quillen_1971}. In fact, he considered a localization with respect to an element in $A$, in which the generator $\theta$ is considered. The result of Quillen shows again that, modulo torsion, $H^*(C_{T,\ell}(X_\Omega,\bF_{p_\ell}))$ is isomorphic to $A$ as an $A$-module, which improves parts our result. But his result can not provide us the information about minimal cohomology degree, that is why we provide this proof, which is essentially independent of the result of Quillen.

We also hope to know what geometric information can be seen from $\theta$. In this direction, let us mention the work of Granja, Karshon, Pabiniak, and Sandon\cite{Granja_2020}, where they study some non-linear Maslov index involve $\theta$.
\end{RMK}

Before proving it, let us use it to compute the capacities $c_k(X_\Omega)$.
\begin{Thm}\label{computation of capacities of convex toric domian}For a convex toric domain $X_\Omega \subsetneqq T^*V$, we have
\[c_k(X_\Omega)=\inf\left\{T\geq 0:\exists  {z}\in \Omega_T^\circ, I( {z})\geq k\right\}.\]
\end{Thm}
\begin{proof}Let $S=\left\{T\geq 0:\exists  {z}\in \Omega_T^\circ, I( {z})\geq k\right\}$, $L=\inf(S)$.

For $T\in S$, there is $ {Z}\in \Omega_T^\circ$ such that $I( {Z})= k$. Consider the closed inclusion of the segment $\overline{OZ} \subset \Omega_T^\circ$. Then \autoref{structure toric domian module} shows the closed inclusion induces a decomposition $\eta_{\ell,T}(X_\Omega,\bF_{p_\ell})=u^k\gamma_{Z,\ell}$, if we choose $\ell_0\in \bO$ with $ p_{\ell_0} >T\|\Omega^\circ_1\|_{\infty}$, and $p_\ell \geq p_{\ell_0}$. So $T\in \textnormal{Spec}(X_{\Omega},k)$, and $L\geq c_k(X_{\Omega})$.

Conversely, if $T\in \textnormal{Spec}(X_{\Omega},k)$, there is an $\ell_0\in\bO$, such that for all $\ell\in \bO$ with $p_\ell \geq p_{\ell_0}$ there is a $\gamma_\ell \in H^*(C_{T,\ell}(X_{\Omega},\bF_{p_\ell}))$, such that $\eta_{\ell,T}(X_\Omega,\bF_{p_\ell})=u^k\gamma_\ell$. Now, we can take an odd prime $\ell=p_\ell$ big enough such that $T<\ell /\|\Omega^\circ_1\|_{\infty}$, then $\eta_{\ell,T}(X_\Omega,\bF_{p_\ell})$, and $\gamma_\ell$ are non-zero. Hence $0=|\eta_{\ell,T}(X_\Omega,\bF_{p_\ell})|=2k+|\gamma_\ell|$ shows $2k=-|\gamma_\ell|$. Therefore, the \autoref{structure toric domian module} shows $2k=-|\gamma_\ell| \leq 2I(\Omega^\circ_T)$. Hence $T\in S$, and $c_k(X_{\Omega}) \geq L$.
\end{proof}

Here, let us test the result by the example of ellipsoids.
\begin{Coro}\label{nonzeroellisoid}Let $X_{\Omega}=E(a_1,\cdots,a_d)$ be an ellipsoid. For $0\leq T <p_\ell a_1$, set $T/\underline{a}=(T/a_1,\cdots, T/a_d)$. We have $H^*(C_{T,\ell}(E,\bF_{p_\ell}))\cong u^{-I(T/\underline{a})}\bF_{p_\ell}[u,\theta]$, $\eta_{\ell,T}(E,\bF_{p_\ell})\neq 0$, and $c_k(E)=\min\{T\geq 0: \sum_{i=1}^d\lfloor {T/a_i} \rfloor\geq k\}$. In particular, $c_k(B_a)=\lceil k/d \rceil a$.
\end{Coro}
\begin{proof}All statements follow directly from the \autoref{structure toric domian module}.
\end{proof}

The rest part of the section is devoted to the proof of \autoref{structure toric domian module}.

Let us define some notions here. Set $\gamma=(-\infty,0]^d$ and $\gamma^a=-\gamma=[0,\infty)^d$. A subset $C\subset \bR^d$ is called $O$ star-shaped, if $\forall z\in C$, $\overline{Oz} \subset C$. So unions and intersections of $O$ star-shaped sets are still $O$ star-shaped. Examples of $O$ star-shaped are convex sets containing the original $O$, and $\gamma\setminus(\mathring{\gamma}+z)$ for $z\in \gamma $. More generally, for any $O$ star-shaped set $C\subset \bR^d$, let $C_\gamma=(C+\gamma^a)\cap \gamma$, then $C_\gamma$ is still $O$ star-shaped. Notice that $(C_{\gamma})_\gamma=C_\gamma$, so let us call an $O$ star-shaped set $C$ is {\em $\gamma$-saturated} if $C=C_\gamma$.

\begin{RMK}A related notion is the $\gamma$-topology, see \cite[Section 3.5]{KS90} and \cite{KS2018} for more about the definition and sheaf theory related to $\gamma$-topology. In fact, here we are considering the $\gamma^a$-topology on $\gamma$. But we will not use related results. \end{RMK}

Now we recall the results of \autoref{GFmodelToricdomain}, and \autoref{Discrete Hamiltonian loop section}. We have $P_{X_{\Omega}} \cong \mathcal{T}\star {\bK}_{\Omega^{\circ}} $ (formula \eqref{GFmodel convex case}) and $ F_{\ell}(X_{\Omega},\bK) \cong {\textnormal{R}}\pi_{ {z}!}{\textnormal{R}}s_{t!}^2   \left(\pi_{t_2}^{-1}  {\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{T}) \otimes\pi_{t_1}^{-1}  {\bK}_{\Omega^{\circ}}  \right)$ (formula \eqref{reformulation of F}).

So let us study ${\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{T})$ following ideas of Chiu. Let us study ${\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{T})$ following ideas of Chiu. Recall that $\mathcal{T}\cong {\textnormal{R}}s_{t!}^d(\mathcal{S}^{\boxtimes d})$, where $\mathcal{S}$ is the sheaf quantization of Hamiltonian rotation in dimension $2$. Using the K\"unneth formula, we have
\begin{align*}
    \mathcal{L}_\ell(\mathcal{T}) &\cong {\textnormal{R}}s_{t!}^d \mathcal{L}_\ell(\mathcal{S})^{\boxtimes d},\\
    {\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{T}) &\cong {\textnormal{R}}s_{t!}^d \left( {\textnormal{R}}\pi_{\underline{q}!}\mathcal{L}_\ell(\mathcal{S})\right)^{\boxtimes d}.
\end{align*}

Luckily, Chiu has computed ${\textnormal{R}}\pi_{\underline{q}!}\mathcal{L}_\ell(\mathcal{S})$:
\begin{Prop}\textnormal{(\cite[Formula (38)]{chiu2017})} For all fields $\bK$, there is a (unique) sheaf ${\mathcal{E}_{\ell}}\in D^+_G(\bR_z)$ such that we have  an isomorphism in $D^+_G(\bR_z\times \bR_t)$
\begin{equation}\label{loop sheaf of GF model of rotation}
 {\textnormal{R}}\pi_{\underline{q}!}\mathcal{L}_\ell(\mathcal{S}) \cong {{\mathcal{E}_{\ell}}} \boxtimes {\bK}_{[0,\infty)}.
\end{equation}
Moreover, for any integer $M>0$,
\begin{equation}\label{E sheaf of GF model of rotation}
\left.{\mathcal{E}_{\ell}}\right|_{(-M\ell /4,0)}\cong {\textnormal{R}}\pi_{\underline{q}!}{\bK}_{\mathcal{W}},
\end{equation}
with
\[\mathcal{W}=\{(z,\underline{q})\in (-M\ell /4,0) \times \bR^{M\ell }:\sum_{k\in \mathbb{Z}/M\ell\mathbb{Z}} S_H(z/M\ell,q_k,q_{k+1}) \geq 0    \},\]
and
\[S_H(z,q_k,q_{k+1})= \frac{q_k^2+q_{k+1}^2}{2\tan(2\pi z)} -\frac{q_k q_{k+1}}{\sin(2\pi z)}. \]
The $G$-action on ${\mathcal{E}_{\ell}}$ is induced by the linear action $(q_k)\mapsto (q_{k+M})$ of $G$ on $\bR^{M\ell}$, and $G$ acts trivially on $\bR_z\times \bR_t$.
\end{Prop}

For our convenience, let us make some remarks on ${\mathcal{E}_{\ell}}$ and $\mathcal{W}$.
Notice that $z/M\ell\in (-1/4,0)$, so $\sin(2\pi z/M\ell)<0$. One can rewrite $\mathcal{W}$ as follow:
\begin{equation*}\label{original definition of W}
\mathcal{W}=\left\{(z,\underline{q})\in (-M\ell/4,0) \times \bR^{M\ell}:\cos(2\pi z/M\ell)\sum_{k\in \mathbb{Z}/M\ell\mathbb{Z}} q_k^2 \leq \sum_{k\in \mathbb{Z}/M\ell\mathbb{Z}} q_k q_{k+1}\right\}.
\end{equation*}
Let us define \begin{equation}
Q(z,\underline{q})\coloneqq  \sum_{k\in \mathbb{Z}/M\ell\mathbb{Z}} \left(q_k q_{k+1}-\cos(2\pi z/M\ell)q_k^2.   \right)
\end{equation}
Since $Q(0,\underline{q})$ is well defined, we can extend the definition of $\mathcal{W}$ (using the same notation) to
\begin{equation}\label{definition of W}
\mathcal{W}=\{(z,\underline{q})\in (-M\ell/4,0] \times \bR^{M\ell}:Q(z,\underline{q})\geq 0\}.
\end{equation}

For a given $z\in (-M\ell/4,0]$, let us denote $\mathcal{W}\cap(\{z\}  \times \bR^{M\ell})$ by $\mathcal{W}(z)$, and $Q_z=Q(z,-)$. Notice that the map $z \mapsto \mathcal{W}(z)$ is a decreasing map with respect to the inclusion order.

By the fundamental inequality, we have $\sum_{k} q_k^2 \geq \sum_{k} q_k q_{k+1}$, and it takes equality when $q_1=\cdots=q_{M\ell}$. So  \[\mathcal{W}(0)=\{\underline{q}\in  \bR^{M\ell}:q_1=\cdots=q_{M\ell}\}=\Delta_{\bR^{M\ell}}.\]

Finally, one can check $\mathcal{L}_\ell(\mathcal{S})|_{\{z=0\}}={\bK}_{\Delta_{\bR^{M\ell}}}\boxtimes {\bK}_{\{t\geq 0\}} $ because $\mathcal{S}|_{\{z=0\}}={\bK}_{\Delta_{\bR^{2}}}\boxtimes {\bK}_{\{t\geq 0\}} $.

So one can extend the isomorphism \eqref{E sheaf of GF model of rotation} to $z=0$,  say ${\mathcal{E}_{\ell}}|_{(-M\ell/4,0]}\cong {\textnormal{R}}\pi_{\underline{q}!}{\bK}_{\mathcal{W}}$, which is compatible with the definition of ${\mathcal{E}_{\ell}}$.

Consequently, if we take $\bq=(q_1,\cdots,q_d)\in \bR^d \eqqcolon V_{\bq}$, we have
\begin{equation}\label{loop sheaf of torus action}
{\textnormal{R}}\pi_{\underline{\bq}!}\mathcal{L}_\ell(\mathcal{T}) \cong {\mathcal{E}_{\ell}}^{\boxtimes d}\boxtimes {\bK}_{\{t\geq 0\}},
\end{equation}
and
\begin{equation}\label{E^d sheaf of torus action}
\left.{\mathcal{E}_{\ell}}^{\boxtimes d}\right|_{(-M\ell/4,0]^d} \cong {\textnormal{R}}\pi_{\underline{\bq}!}{\bK}_{\prod_{i=1}^d\mathcal{W}_i},
\end{equation}
here $\mathcal{W}_i$ means the $i.$th copy of one $\mathcal{W}$.

Before going into further computations, let us describe the $G=\bZ/\ell\bZ$-action. $G$ acts on $\bR^{M\ell}$ linearly by $(q_l)\mapsto (q_{l+M})$, and the quadratic form $Q_z$ is invariant under this action for every $z$. So $G$ acts on $\mathcal{W}$, and acts diagonally on $\prod_{i=1}^d\mathcal{W}_i$. The action of $G$ on $({z},t)\in \bR^d_z\times \bR_t$ is trivial.

Now, let us compute $(F_{\ell}(X_{\Omega},\bK))_T$.  We recall the notation  $F_C=i_!i^{-1}F$, for a locally closed inclusion $i:C\subset X$ and for $F\in D(X)$. We also recall that ${\textnormal{R}}\Gamma_c$ stands for the derived compactly supported global section.
\begin{Lemma}\label{cohomology of E^d}For a compact $O$ star-shaped set $C$, there is an integer $M>0$ (recall that $\mathcal{W}$ depends on an integer $M$) such that
\begin{equation}\label{section of E^d}
{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C}   \right) \cong  {\textnormal{R}}\Gamma_c\left(\mathcal{U}(C), {\bK}    \right),
\end{equation}
where
\begin{equation}\label{definition of U(C)}
\mathcal{U}(C)=\bigcup_{{z} \in C} \prod_{i=1}^d \mathcal{W}_i(z_i),
\end{equation}
$\mathcal{W}_i(z_i)=\mathcal{W}_i\cap \{z_i\}$.  ${\textnormal{R}}\Gamma_c$ stands for the derived compactly supported global section. The section is taken as the usual cohomology and equips the $G$-action induced from the $G$-action on $\mathcal{W}$ we given above to be a $G$-module.
\end{Lemma}
\begin{proof}The compactness of $C$ implies that there is $M>0$ such that $C \subset (-M\ell/4,0]^d$. Now we have ${\mathcal{E}_{\ell}}^{\boxtimes d}|_{(-M\ell/4,0]^d} \cong {\textnormal{R}}\pi_{\underline{\bq}!}{\bK}_{\prod_{i=1}^d\mathcal{W}_i} $, and then we obtain
\begin{align*}
   {\textnormal{R}}\Gamma\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_C    \right) &\cong  {\textnormal{R}}\pi_{ {z}!} \left[ ({\textnormal{R}}\pi_{\underline{\bq}!}{\bK}_{\prod_{i=1}^d\mathcal{W}_i})_{C}\right]\\
   &\cong {\textnormal{R}}\pi_{ {z}!}  {\textnormal{R}}\pi_{\underline{\bq}!}{\bK}_{(\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C )}\\
   &\cong {\textnormal{R}}\pi_{\underline{\bq}!}{\textnormal{R}}\pi_{ {z}!}  {\bK}_{(\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C )}.
\end{align*}
Claim: When restricted to $(\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C)$, the fibers of $\pi_{ {z}}$ are compact and contractible.

Indeed, Chiu has proved that the fiber of $\pi_{z_i}$ over $\bR^{M\ell}$ is a closed interval (See \cite[Lemma 4.10]{chiu2017}. So the fiber of $\pi_{ {z}}$ over $\prod_{i=1}^d\mathcal{W}_i$ is a closed cube containing the origin $O$. Hence, the fiber of $\pi_{ {z}}$ over $(\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C)$ is the intersection of a closed cube containing the origin $O$ with a compact $O$ star-shaped set $C$, which is compact and contractible.

Consequently, the Vietoris-Begel theorem implies
\[{\textnormal{R}}\pi_{{z}!}  {\bK}_{(\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C})\cong {\bK}_{\pi_{{z}}((\prod_{i=1}^d\mathcal{W}_i)\cap(V^{M\ell}\times C ))}={\bK}_{\mathcal{U}(C)}.\]

Therefore, ${\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_C   \right)={\textnormal{R}}\pi_{\underline{\bq}!}({\bK}_{\mathcal{U}(C)})\cong  {\textnormal{R}}\Gamma_c\left(\mathcal{U}(C), {\bK}    \right).$
\end{proof}
Now \eqref{reformulation of F} and \eqref{loop sheaf of torus action} show
\begin{align}\label{F of convex toric domains}
\begin{aligned}
F_{\ell}(X_{\Omega},\bK)&\cong {\textnormal{R}}\pi_{{z}!}{\textnormal{R}}s_{t!}^2\left( {\mathcal{E}_{\ell}}^{\boxtimes d}\boxtimes {\bK}_{\{t_1\geq 0\}} \otimes  \pi_{t_1}^{-1}  {\bK}_{\Omega^{\circ}}  \right)\\
  &\cong {\textnormal{R}}\pi_{{z}!} \left[ ({\mathcal{E}_{\ell}}^{\boxtimes d} \boxtimes {\bK}_{\{t\geq 0\}})_{ \Omega^\circ}\right]
\end{aligned}
 \end{align}
As $\Omega_T^\circ$ is compact and $O$ star-shaped, \autoref{cohomology of E^d} and \eqref{F of convex toric domains} show:
\begin{Coro}\label{stalks of F of toric domain}For $T \geq 0$, we have \begin{equation}\label{stalks of F for convex toric domains}
(F_{\ell}(X_{\Omega},\bK))_T\cong {\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{\Omega_T^\circ}   \right)\cong  {\textnormal{R}}\Gamma_c\left(\mathcal{U}(\Omega_T^\circ), {\bK}    \right).
\end{equation}

\end{Coro}
Another ingredient we need to take into account is the morphism \[\eta_{\ell,T}(X_\Omega,\bK): (F_{\ell}(X_{\Omega},\bK))_T\rightarrow (F_{\ell}(T^*X,\bK))_T\cong \bK[-d].\] We can embed it into an excision triangle:
\[{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{{\Omega}^\circ_T\setminus O }   \right) \rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{{\Omega}^\circ_T} \right)\xrightarrow{\eta_{\ell,T}(X_\Omega,\bK)} {\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{O}   \right)  \xrightarrow{+1}.\]

A similar argument shows the cocone of $\eta_{\ell,T}(X_\Omega,\bK)$ can be computed by the cohomology of a sheaf. In fact, let $X=\{(z,\underline{\bq})\in \bR_z^d\times V^{M\ell}:  \bq\in \mathcal{W}(z)\setminus\Delta_{V^{M\ell}} \}$ and
\begin{equation}\label{eq:def_Fd}
\mathcal{F}^d={\textnormal{R}}\pi_{\bq !}\bK_{X},
\end{equation}
viewed as an object of $D_G(\bR^d_z)$.Then we have:

\begin{Lemma}\label{nonpositivedegree}
For a compact $O$ star-shaped set $C  $, there is an isomorphism
\begin{equation}
{\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C\setminus O}  \right)\cong {\textnormal{R}}\Gamma_c\left(\bR^d, \mathcal{F}^d_C   \right) .
\end{equation}
\end{Lemma}
\begin{proof}

 A similar argument with the proof of \autoref{cohomology of E^d} shows that
 \begin{align*}
  {\textnormal{R}}\Gamma_c\left(\bR^d, \mathcal{F}^d_C   \right) &\cong {\textnormal{R}}\Gamma_c\left(\bigcup_{z\in C}(\mathcal{W}(z)\setminus \Delta_{V^{M\ell}}), \bK   \right) \\
   &= {\textnormal{R}}\Gamma_c\left(\mathcal{U}(C)\setminus \Delta_{V^{M\ell}}, \bK   \right)\\& \cong {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C\setminus O}   \right),
 \end{align*}
where the last isomorphism comes from \autoref{cohomology of E^d} and the excision sequence:
\[{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{C\setminus O }   \right) \rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{C} \right)\xrightarrow{\eta_{\ell,T}(X_\Omega,\bK)} {\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{O}   \right)  \xrightarrow{+1}.\]\end{proof}

\begin{RMK}\label{rmkgammatopology}Because $z\mapsto \mathcal{W}(z)$ is a decreasing map, one can see that
\[\mathcal{U}(C)=\mathcal{U}(C_\gamma)\text{ and }\mathcal{U}(C)\setminus \Delta_{V^{M\ell}}=\mathcal{U}(C_\gamma)\setminus \Delta_{V^{M\ell}}.\]
Then, later, we will mainly focus on $\gamma$-saturated $O$ star-shaped sets.
\end{RMK}

Now, to understand topology of $\mathcal{U}(C)$(see \eqref{definition of U(C)}), let us spend some time on the quadratic form $Q_z=Q(z,-)$, for fixed $z\in (-\infty,0]$.
The matrix of $Q_z$ is a circulant matrix
\begin{center}
$A_z=\left(\begin{matrix}
     -\cos(\frac{2\pi z}{M\ell})       & \frac{1}{2}       & 0 &\cdots     & \frac{1}{2} \\
     \frac{1}{2} & -\cos(\frac{2\pi z}{M\ell})       & \frac{1}{2} &\cdots     & 0 \\
     0 & \frac{1}{2} & -\cos(\frac{2\pi z}{M\ell}) &\cdots     & 0 \\
      \vdots & \vdots &\vdots &                 & \vdots \\
      \frac{1}{2}       & 0       & 0 &\cdots     & -\cos(\frac{2\pi z}{M\ell})
\end{matrix}\right)$
\end{center}
So one can diagonalize $A_z$ unitarily using the discrete Fourier transform \[(\omega^{ij})_{i,j=0,\cdots,M\ell-1},\]
where $\omega$ is a primitive $M\ell.$th root of unity.
In particular, the eigenvalues of $A_z$ are
\begin{equation}\label{eigenvalue of $A_z$}
\lambda_k(z)=\text{Re}\left(\exp\left(\frac{2\pi k \sqrt{-1}}{M\ell}\right)\right)-\cos\left(\frac{2\pi z}{M\ell}\right)=\cos\left(\frac{2\pi k}{M\ell}\right)-\cos\left(\frac{2\pi z}{M\ell}\right),
\end{equation}
where $k\in \mathbb{Z}/M\ell\mathbb{Z}$.

So $A_z$ admits $\#\{k\in \mathbb{Z}/M\ell\mathbb{Z}: \lambda_k  \geq 0\}=1+2\big\lfloor {-z} \big\rfloor$ non-negative eigenvalues if $M\ell$ is odd, that is, if $M$ is odd, since we already assumed $\ell$ to be odd.

Now, let $u_0=\langle \underline{q},(1,1,\dots,1)\rangle\in \bR$ and $u_k=\langle \underline{q},(1,\omega^k,\omega^{2k},\dots)\rangle\in \bC$, $k=1,2,\dots, M(\ell-1)$. Then the diagonal form of $Q_z$ is
\begin{equation}\label{diagonal form of Q}
Q_z(u_0,\underline{u_k})=\lambda_0(z)u_0^2+\sum_{k=1}^{(M\ell-1)/2} \lambda_k(z)|u_k|^2,\, (u_0,\underline{u_k})\in \bR\times \bC^{\frac{M\ell-1}{2}}\cong \bR^{M\ell}.
\end{equation}

Therefore, $\mathcal{W}(z)=\{ Q_z\geq 0\}$ is a quadratic cone of index $1+2\big\lfloor {-z} \big\rfloor$. In particular, $\mathcal{W}(0)=\Delta_{\bR^{M\ell}}=\{(u_0,0,0):u_0\in \bR\}$.

Next, we need to study the $G$-action on $\mathcal{W}(z)$. The action is easier to understand under the diagonal form, where the $G$-action is as follows: if we take $\mu=\omega^M$ a primitive $\ell$th. root of unity, then \begin{equation}\label{action formula under diagonal form}
 \mu.(u_0,\underline{u_k})=(u_0, \underline{\mu^k u_k}).
\end{equation}

Consequently, the fixed point sets $\mathcal{W}(z)^G$ is again a quadratic cone, whose index is $1+2\big\lfloor {-z/\ell} \big\rfloor$.

For  $z\in \bR^d_{\geq 0}$, let $C_z=\prod_{i=1}^d(-z_i-1,-z_i]$ and $Z_z=\prod_{i=1}^d(-z_i,0]$. Let $\mathbbm{1}=(1,\dots,1)$ and $e_i=(\delta_{ij})_{j=1}^d$. Then $\mathbbm{1},e_i\in \bZ^d_{\geq 0}$.

For a compact $\gamma$-saturated $O$ star-shaped set
$C$, let $J_C=(-C) \cap \bZ^d_{\geq 0}=\lbrace v\in \bZ^d_{\geq 0}: C\cap C_v \neq \varnothing \rbrace$, and $\partial J_C=\{v\in J_C: \forall i, v+e_i \not\in J_C\}= \{v\in J_C: C\cap (\overline{C_v}\setminus C_v)=\varnothing\}$. The compactness of $C$ shows that both $J_C$ and $\partial J_C$ are finite sets.

\begin{Lemma}\label{cornerlemme}Let $C$ be a compact $\gamma$-saturated $O$ star-shaped set. Then $\partial J_C=\{v\}$ for some $v\in \bZ^d_{\geq 0}$ if and only if $\overline{Z}_v \subset C \subset  {Z}_{v+\mathbbm{1}}$ for the same $v\in \bZ^d_{\geq 0}$.
\end{Lemma}
\begin{proof}When $\overline{Z}_v \subset C \subset  {Z}_{v+\mathbbm{1}}$, we can obtain $\partial J_C=\{v\}$ directly from
\[ \overline{Z}_v \cap (\overline{C_w}\setminus C_w)\subset C \cap(\overline{C_w}\setminus C_w) \subset  {Z}_{v+\mathbbm{1}} \cap (\overline{C_w}\setminus C_w).  \]
Conversely, when $\partial J_C=\{v\}$, then we have $v\in C$, so $C=C_\gamma$ implies $\overline{Z}_v=\{v\}_\gamma \subset C$. Now, suppose $C \nsubseteq Z_{v+\mathbbm{1}}$, then there is a $z\in C$ such that $z_i=-v_i-1$. Therefore, $v+e_i\in J_C$. Besides, $\partial J_C=\{v\}$, then $v+e_i\notin  \partial J_C$. Hence we get a contradiction! Then $C \subset Z_{v+\mathbbm{1}}$.\end{proof}

\begin{Lemma}\label{inductionstep of minimaldegree} We recall the notation \eqref{eq:def_Fd}. We assume that $\partial J_C=\{v\}$. Then we have ${\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C) \cong {\textnormal{R}}\Gamma(S^{2I(v)-1},\bK)[-d-1]$, when $I(v)>0$, and ${\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C) \cong 0$ when $I(v)=0$.
\end{Lemma}
\begin{proof}By \autoref{nonpositivedegree}, we have ${\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C)\cong {\textnormal{R}}\Gamma_c(\mathcal{U}(C)\setminus \Delta_{V^{M\ell}},\bK)$. Let
\begin{equation*}\label{definition of V}
 \mathcal{V}\coloneqq \mathcal{U}(C)\setminus \Delta_{V^{M\ell}}=\left\lbrace (u_i)=(u_{0,i},\underline{u_{k,i}}):u_{0,i}\neq 0, \; \exists z\in C, \forall i,\, Q_{z_i}(u_i)\geq 0  \right\rbrace,
\end{equation*}
where $u_i$ are the coordinates diagonalizing $Q_{z_i}$ to \eqref{diagonal form of Q}.
We are going to study the proper homotopy type of $\mathcal{V}$.

When $I(v)>0$. For $1\leq m \leq (M\ell-1)/2$, consider $h_m:\bR\times \bC^{\frac{M\ell-1}{2}}\times [0,1] \rightarrow \bR\times \bC^{\frac{M\ell-1}{2}}$,
\[h_m(u_0,u_+,u_-,t)=h_{m,t}(u_0,u_+,u_-)=(u_0,u_+,t u_-),\]
where $u_+=(u_1,\dots,u_m)$, $u_-=(u_{m+1},u_{m+2},\dots)$.

Now, define $H_v:\left(\bR\times \bC^{\frac{M\ell-1}{2}}\right)^d \times [0,1] \rightarrow \left(\bR\times \bC^{\frac{M\ell-1}{2}}\right)^d$ by
\[H_{v,t}=h_{v_1,t}\times \cdots \times h_{v_d,t}. \]

Then we have following:
\begin{itemize}[fullwidth]
    \item $H_{v,t}(\mathcal{V})\subset \mathcal{V}$.
Indeed    $u\in \mathcal{V}$ implies there is $z\in C$, such that for all $i=1,\dots,d$, we have $Q_{z_i}(u_i) \geq 0$. That is, in the diagonal form \eqref{diagonal form of Q},
    \[\sum_{k=0}^{v_i}\lambda_k(z_i)|u_{k,i}|^2 \geq \sum_{k\geq v_i+1}(-\lambda_k(z_i))|u_{k,i}|^2.\]
    $\overline{Z}_v \subset C \subset  {Z}_{v+\mathbbm{1}}$ implies that $z_i <v_{i}+1$ for all $i=1,\dots,d$, then $\lambda_k(z_i)<0$ for $k\geq v_i+1$ and for all $i=1,\dots,d$. So
        \[\sum_{k=0}^{v_i}\lambda_k(z_i)|u_{k,i}|^2 \geq\sum_{k\geq v_i+1}(-\lambda_k(z_i))|u_{k,i}|^2 \geq t^2\sum_{k\geq v_i+1}(-\lambda_k(z_i))|u_{k,i}|^2,\]
    i.e., $Q_{z_i}(h_{v_i,t}(u_i))\geq 0$ for all $i=1,\dots,d$. Hence $H_{v,t}(u)\in \mathcal{V}$.

    \item $\left.H_v\right|_{\mathcal{V}}$ is proper. Indeed, take  $u\in \mathcal{V}$, with $\sum_{k=0}^{v_i}|u_{k,i}|^2+\sum_{k\geq v_i+1}|t u_{k,i}|^2 \leq R^2$, for all $i=1,\dots,d$. Obviously, $\sum_{k=0}^{v_i}|u_{k,i}|^2 \leq R^2$, for all $i=1,\dots,d$, and
    \[\max_{\substack{k=1,\dots,v_i\\z\in C}}|\lambda_k(z_i)|R^2\geq \sum_{k=0}^{v_i}\lambda_k(z_i)|u_{k,i}|^2 \geq \sum_{k\geq v_i+1}(-\lambda_k(z_i))|u_{k,i}|^2\geq \min_{\substack{k\geq v_i+1\\z\in C}}|\lambda_k(z_i)|\sum_{k\geq v_i+1}|u_{k,i}|^2.\]
    Because $\lambda_k(z_i)<0$ for $k\geq v_i+1$, and $z\in C$. We have $\min_{\substack{k\geq v_i+1\\z\in C}}|\lambda_k(z_i)|>0$. Consequently,
    \[\sum_{k\geq v_i+1}|u_{k,i}|^2 \leq \frac{\max\limits_{\substack{k=1,\dots,v_i\\z\in C}}|\lambda_k(z_i)|}{\min\limits_{\substack{k\geq v_i+1\\z\in C}}|\lambda_k(z_i)|} R^2.\]
    That is $\left(\left.H_v\right|_{\mathcal{V}}\right)^{-1}(\text{compact sets})\subset \text{compact sets}$.
  \item $H_{v,0}(\mathcal{V})=\bR^d\times \left(\bC^{I(v)} \setminus \{0\}\right)\times \{0\}$. Indeed
    \[H_{v,0}(\mathcal{V})=\lbrace(\underline{u_0},\underline{u_+},0)\in \bR^d\times \bC^{d\frac{M\ell-1}{2}}: \exists z\in C,  \forall i,\, Q_{z_i}(u_i)\geq 0,\; (u_+,0)\neq 0 \rbrace \]
    and, because $v\in C$, the form $Q_{v_i}(u_0,u_+,0)$ is semi-positive, for each $i$.
\end{itemize}

Therefore, $H_{v,0}(\mathcal{V})\cong \bR^d \times S^{2I(v)-1}\times \bR_{>0}$ is a proper strong deformation retraction of $\mathcal{V}$.

When $I(v) = 0$, that is, $v=0$, the same idea shows $\mathcal{U}(C)$ is properly homotopic to $\Delta_{V^{M\ell}}$. Then ${\textnormal{R}}\Gamma(C,\mathcal{F})\cong 0$ by excision.\end{proof}

\begin{Lemma}\label{minimaldegree}For $\bK=\bF_{p_\ell}$, $C$ be a compact $\gamma$-saturated $O$ star-shaped set and $I(C)=\max_{z\in C}I(z)$. We assume that $C\subset Z_{p_\ell\mathbbm{1}}$. Then
\[\text{Ext}^*_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C),\bK[-d]\right),\]is zero when $*\notin [-2I(C),-1]$, and it is finite dimensional as a graded $\bK$-vector space.
\end{Lemma}
\begin{proof}Let us do an induction on $|J_C|$. Notice that, if $v\in J_C$ satisfies $I(v)=I(C)$, then $v\in \partial J_C$. Because $I$ is upper-semi continuous, the maximum $I(C)$ can be achieved by some $v$.

If $|J_C|=1$, that is $J_C=\{0\}$. Then the result follows from the \autoref{inductionstep of minimaldegree}.

Now, suppose the result is true for $C'$ such that $|J_{C'}|< |J_C|$.

For $C$ given at the beginning, we need consider two possibility.
\begin{enumerate}[fullwidth]
    \item\label{1stpossibility}If $\partial J_C = \{v\}$ is a singleton, we apply the \autoref{inductionstep of minimaldegree}. The result is trivial when $I(v)=0$. Now we assume $I(v)>0$ and we have \[{\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C)\cong {\textnormal{R}}\Gamma_c(\mathcal{U}(C)\setminus \Delta_{V^{M\ell}},\bK)\cong {\textnormal{R}}\Gamma(S^{2I(v)-1},\bK)[-d-1].\] Now, because $C\subset Z_{p_\ell\mathbbm{1}}$, the one can see from the formula \eqref{action formula under diagonal form} that $G$ acts on $S^{2I(v)-1}$ freely. Accordingly,
\[\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C),\bK[-d]\right)\cong\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma(S^{2I(v)-1},\bK),\bK[1]\right)\cong \text{Ext}^{*}\left({\textnormal{R}}\Gamma(L^{2I(v)-1},\bK),\bK\right)[1],\]
where $L^{2I(v)-1}=S^{2I(v)-1}/G$ is a lens space of dimension $2I(v)-1$.

Besides, $\text{Ext}^{-q}\left({\textnormal{R}}\Gamma(L^{2I(v)-1},\bK),\bK\right)$ computes the (Borel-Moore) homology $H_q(L^{2I(v)-1})$ and we have: $H_q(L^{2I(v)-1})=\bK$, when $q\in [0,2I(v)-1]$, and $H_q(L^{2I(v)-1})=0$ for other $q$. Converting to cohomology degree, we obtain: $\text{Ext}^{*}\left({\textnormal{R}}\Gamma(L^{2I(v)-1},\bK),\bK\right)[1]$ is concentrated in $ [-2I(v),-1]$.

The proof of this part is independent of our induction, so it can be applied to the second case.
\item If $|\partial J_C|\geq 2$, take $v\in \partial J_C$ such that $I(v)=I(C)$. Then we can take $\epsilon>0$ such that $C\cap (\gamma+(\epsilon \mathbbm{1}-v)) \cap (\overline{Z}_{v+\mathbbm{1}}\setminus Z_v) =\varnothing$. This is possible due to compactness of $C$. Define:
\begin{align}\label{construcionofAB}
\begin{aligned}
    &A=[C\cap (\gamma+(\epsilon \mathbbm{1}-v))]_\gamma,\\
    &B=C\cap [\mathring{\gamma} \setminus (\gamma+(\epsilon \mathbbm{1}-v))].
\end{aligned}
\end{align}
Then we have a closed covering $C=A\cup B$. Moreover, both $A$ and $B$ are compact $\gamma$-saturated $O$ star-shaped sets, and so is $A\cap B$.
\begin{figure}[htbp]
\begin{center}

    \tikzset{every picture/.style={line width=0.5pt}}
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\end{tikzpicture}
  \caption{The picture illustrate the construction of $A,B$. $C$ is the background blue set.}
    \label{ConstructionofC}
    \end{center}
\end{figure}

Then we have the Mayer-Vietoris triangle,
\[{\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C) \rightarrow {\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_A)\oplus {\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_B) \rightarrow {\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{A\cap B})\xrightarrow{+1}.\]
Next, we apply the $\text{Ext}^{*}_G\left(-,\bK[-d]\right)$ to obtain a long exact sequence
\begin{align}\label{exact sequence}
    \text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{A\cap B}),\bK[-d]\right) \rightarrow\,
    \begin{gathered}
        \text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_A),\bK[-d]\right)\\\bigoplus\\ \text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_B),\bK[-d]\right)
    \end{gathered}
    \rightarrow \text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_C),\bK[-d]\right)\xrightarrow{+1}.
\end{align}

By our construction \eqref{construcionofAB}, we have
\begin{itemize}[fullwidth]
    \item $|\partial J_A|=1$. Hence we can apply the result of \autoref{1stpossibility}. So that $\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{A}),\bK[-d]\right)$ is concentrated in $[-2I(A),-1]\subset [-2I(C),-1]$.
    \item $|J_B|<|J_C|$. We can use the induction hypothesis, hence $\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{ B}),\bK[-d]\right)$ is concentrated in $[-2I(B),-1]\subset [-2I(C),-1]$.
    \item $|I(A\cap B)| < I (C)$, and $|J_{A\cap B}|<| J_C|$, because $A\cap B \subset A$, but $v\notin \partial J_{A\cap B}$. Then we can use the induction hypothesis, that $\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{A\cap B}),\bK[-d]\right)$ is concentrated in $[-2I(A\cap B),-1]\subset [-2I(C)+2,-1]$.
\end{itemize}
\end{enumerate}

Therefore, it follows from the  sequence \eqref{exact sequence}  that $\text{Ext}^{*}_G\left({\textnormal{R}}\Gamma_c(\bR^d,\mathcal{F}^d_{C}),\bK[-d]\right)$ is concentrated in $[-2I(C),-1]$. \end{proof}

Now, we are in the position to prove the \autoref{structure toric domian module}. Its proof is essentially a refinement of the computation for the case of a ball given by Chiu.

\begin{proof}[Proof of \autoref{structure toric domian module}]
Here, we take $\bK=\bF_{p_\ell}$. The \autoref{stalks of F of toric domain} says that \[(F_{\ell}(X_{\Omega},\bK))_T \cong {\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{\Omega_T^\circ}   \right).\]
Besides, we embed $\eta_{\ell,T}(X_\Omega,\bK)$ into the excision triangle
\begin{equation}\label{dt1}{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{{\Omega}^\circ_T\setminus O }   \right) \rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{{\Omega}^\circ_T} \right)\xrightarrow{\eta_{\ell,T}(X_\Omega,\bK)} {\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{O}   \right)  \xrightarrow{+1}.
\end{equation}
When $0\leq T <p_\ell/\|\Omega^\circ_1\|_{\infty}$, we have $\Omega^\circ_T\subset Z_{p_\ell\mathbbm{1}}$ (a technical condition needed below). By \autoref{minimaldegree},  $\textnormal{Ext}_G({\textnormal{R}}\Gamma_c\left( \bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{{\Omega}^\circ_T\setminus O}    \right),\bK[-d])$ is a finite dimensional  graded $\bK$ vector space which is concentrated in degrees $[-2I(\Omega^\circ_T),-1]$.

One the other hand, $\textnormal{Ext}_G({\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_O    \right),\bK[-d]) \cong A $ is concentrated in $[0,\infty)$. Then the long exact sequence associate with \eqref{dt1} implies that $H^*(C_{T,\ell}(X_\Omega,\bK))$ is concentrated in $[-2I(\Omega^\circ_T),\infty)$.

Moreover, this estimate is sharp. Indeed, we take $Z\in \Omega_T^\circ$ such that $I(Z)=I(\Omega^\circ_T)$. Let us consider the morphism induced by the closed inclusion $j: \overline{OZ}\hookrightarrow \Omega_T^\circ$, and we embed it into an excision triangle:
\begin{equation}\label{dt2}
{\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{{\Omega}^\circ_T\setminus \overline{OZ}}   \right) \rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{{\Omega}^\circ_T}   \right)\xrightarrow{{\textnormal{R}}\Gamma_c\left(\bR^d , j  \right)} {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right)  \xrightarrow{+1}.
\end{equation}
 It means, the cocone of ${\textnormal{R}}\Gamma_c\left(\bR^d , j  \right)$ is ${\textnormal{R}}\Gamma\left({\Omega}^\circ_T\setminus \overline{OZ} , {\mathcal{E}_{\ell}}^{\boxtimes d}   \right)$. But we can embed the cocone of ${\textnormal{R}}\Gamma_c\left(\bR^d , j  \right)$ into another excision triangle, say:
\begin{equation}\label{dt3}
{\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{{\Omega}^\circ_T\setminus \overline{OZ}}   \right) \rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{{\Omega}^\circ_T \setminus O}   \right)\rightarrow {\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}\setminus O}   \right)  \xrightarrow{+1}.
\end{equation}
 Consequently, the \autoref{minimaldegree} and the long exact sequence associated with \eqref{dt3} show that ${\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}\setminus O}   \right)$ is finite dimensional. After tensoring with $\otimes_{\bK[u]}\bK(u)$, we see that the closed inclusion induces an isomorphism $\text{Ext}^*_G\left(j,\bK[-d] \right)\otimes_{\bK[u]}\bK(u)$ between $\bK(u)$-modules:
 \[H^*(C_{T,\ell}(X_\Omega,\bK))\otimes_{\bK[u]}\bK(u) \xrightarrow{\cong} \text{Ext}^*_G\left({\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right),\bK[-d] \right)\otimes_{\bK[u]}\bK(u).\]

Now we compute $\text{Ext}^*_G\left({\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right),\bK[-d] \right)$.  By \eqref{section of E^d} we have ${\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right)\cong {\textnormal{R}}\Gamma_c\left(\mathcal{U}(\overline{OZ}), {\bK}    \right) $ and  \eqref{definition of U(C)} shows $\mathcal{U}(\overline{OZ})=\prod_{i=1}^d\mathcal{W}_i(Z_i)$.

So ${\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right)\cong \bK[-d-2I(Z)] $ and
\[\text{Ext}^*_G\left({\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right),\bK[-d] \right)\cong u^{-I(\Omega^\circ_T)}\bK[u,\theta].\]
Then
\[\text{Ext}^*_G\left({\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right),\bK[-d] \right)\otimes_{\bK[u]}\bK(u) \cong u^{-I(\Omega^\circ_T)}\bK(u,\theta).\]

All our discussion show that $H^*(C_{T,\ell}(X_\Omega,\bK))$ is a finitely generated $\bK[u]$-module, whose free part is $\bK[u,\theta]$, and torsion part is concentrated exactly in degrees $[-2I(\Omega^\circ_T),-1]$.

When $X_\Omega$ is an ellipsoid, $\Omega^\circ_T$ itself is a segment( Here, we take a half space $\Omega^+$). So $H^*(C_{T,\ell}(X_\Omega),\bK)$ is torsion free.

The decomposition follows from the same idea for all $Z\in \Omega^\circ_T$. In fact, we have a commutative diagram
\begin{center}
    \begin{tikzcd}
{{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{{\Omega}^\circ_T }   \right)} \arrow[rd, "{\gamma_{Z,\ell}}"] \arrow[rr, "{\eta_{\ell,T}(X_\Omega,\bK)}"] \arrow[equal, "\eqref{F of convex toric domains}"]{d} &                                                                                & {{\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d} )_{  O }   \right)} \arrow[equal, "\eqref{F of convex toric domains}"]{d} \\
(F_{\ell}(X_{\Omega},\bK))_T                                                                                & {{\textnormal{R}}\Gamma_c\left(\bR^d , ({\mathcal{E}_{\ell}}^{\boxtimes d})_{ \overline{OZ}}   \right)} \arrow[ru, "u^{I(Z)}"] & (F_{\ell}(X_{\Omega},\bK))_T
\end{tikzcd}
\end{center}

Now $\text{Ext}^*_G\left(\gamma_{Z,\ell},\bK[-d] \right)\otimes_{\bK[u]}\bK(u)$ is an isomorphism of $\bK(u)$-vector spaces in the same way. So $\gamma_{Z,\ell}$ is non-zero.
\end{proof}

At the end of the section, let us make a romantic description of the structure of
\[\text{Ext}^*_G \left({\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C}   \right),\bK[-d] \right),\]
where $C$ is a $\gamma$-saturated $O$ star-shaped set, and $C\subset Z_{\ell \mathbbm{1}}$.

Due to our structural theorem, we know that $\text{Ext}^*_G \left({\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C}   \right),\bK[-d] \right)$ could be given as follow: For each $v\in \partial J_C$, we have a copy of $A=\bK[u,\theta]$, degree shifted by $-2I(v)$, we can think of it as a flower, and treat the non-negative degree part as the stem of the flower.

Next, we can imagine we have many flowers. Then we tie their stems, so we have a beautiful bouquet now. That is $\text{Ext}^*_G \left({\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C}   \right),\bK[-d] \right)$. Moreover, the $A$-module structure seems like a water-drop is draping down from the top of blooms into the earth (maybe never reach).

\begin{center}
\begin{figure}[htbp]

\tikzset{every picture/.style={line width=0.5pt}}

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw    (186.9,75.8) -- (547.9,73.8) ;
\draw    (69.9,45.8) -- (186.9,75.8) ;
\draw    (58.9,103.8) -- (186.9,75.8) ;
\draw    (110.9,131.8) -- (186.9,75.8) ;
\draw    (70.9,77.8) -- (121.4,58.8) ;
\draw    (87.9,117.8) -- (122.9,89.8) ;
\draw    (98.9,19.8) -- (186.9,75.8) ;
\draw  [dash pattern={on 0.84pt off 2.51pt}]  (185.9,13.05) -- (187.9,149.8) ;
\draw    (36.9,149.8) -- (590.9,145.81) ;
\draw [shift={(592.9,145.8)}, rotate = 539.5899999999999] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw   (147.9,106.06) -- (156.3,106.06) -- (161.9,113.43) -- (156.3,120.8) -- (147.9,120.8) -- cycle ;

\draw   (343.9,59.69) -- (352.3,59.69) -- (357.9,67.06) -- (352.3,74.43) -- (343.9,74.43) -- cycle ;

\draw   (127.9,21.69) -- (136.3,21.69) -- (141.9,29.06) -- (136.3,36.43) -- (127.9,36.43) -- cycle ;

\draw   (87.9,80.69) -- (96.3,80.69) -- (101.9,88.06) -- (96.3,95.43) -- (87.9,95.43) -- cycle ;

\draw (91,152.4) node [anchor=north west][inner sep=0.75pt]    {$H^{< } 0$};
\draw (358,150.4) node [anchor=north west][inner sep=0.75pt]    {$H^{\geq } 0$};
\draw (147.5,109) node [anchor=north west][inner sep=0.75pt]    {$u$};
\draw (343.5,62.5) node [anchor=north west][inner sep=0.75pt]    {$u$};
\draw (127.5,24.5) node [anchor=north west][inner sep=0.75pt]    {$u$};
\draw (87.5,83.5) node [anchor=north west][inner sep=0.75pt]    {$u$};

\end{tikzpicture}

    \caption{The shape of $\text{Ext}^*_G\left({\textnormal{R}}\Gamma_c\left(\bR^d, ({\mathcal{E}_{\ell}}^{\boxtimes d})_{C}   \right),\bK[-d]\right)$}
    \label{shapeofchiucomplex}

\end{figure}
\end{center}

\section{Contact Gleanings}
In this section, let us review how to obtain embedding obstructions of admissible open sets in $T^*X\times S^1$.

For an open set $U\subset T^*X\times S^1$, we can lift it to $\Tilde{U}\subset J^1(X)$ as a $\bZ$-invariant open subset. In this way, we can discuss sheaves microsupported in $\mu_L(F)\subset J^1(X)\setminus \Tilde{U}$. Then $\mathcal{D}_{J^1(X)\setminus \Tilde{U}}(X)$ and its left semi-orthogonal complement are all well-formulated and the notion of admissibility is well-defined. In particular, our results in \autoref{section:UniquenessandExistence} work as well. They show the uniqueness, functoriality and  existence of kernels associated to admissible open sets in $T^*X\times S^1$.

Therefore the Chiu-Tamarkin complex can be  defined, for admissible open sets $U\subset T^*V\times S^1$, without any modifications as:
\[C_{T,\ell}(U,\bK)={ \textnormal{RHom}  }_G(P_U^{\boxtimes \ell}, \Tilde{\Delta}_* s_t^{\ell!}\pi_{\underline{\bq}}^! {\bK}_{\{t=T\}}[-d]).\]
However, a problem here is the invariance. As Chiu discussed in \cite{chiu2017} $C_{T,\ell}(U,\bK)$ is a contact invariant only when $T=\ell\in \bZ_{>0}$. That is

 \begin{Thm}[ {\cite[Theorem 4.7]{chiu2017}}]\label{contactinvariance1} Let $U,U_1,U_2$ be contact admissible open sets and let $U_1\xhookrightarrow{i} U_2$ be an inclusion. Then one has, for $\ell\in \bZ_{>0}$,
\begin{enumerate}[fullwidth]
    \item There is a morphism $C_{\ell,\ell}(U_2,\bK) \xrightarrow{i^*} C_{\ell,\ell}(U_1,\bK)$, which is natural with respect to inclusions of admissible open sets.
    \item Suppose $\varphi$ is a compactly supported contact isotopy of $T^*V\times S^1$. Then there is an isomorphism $\Phi_{\ell,\ell }:C_{\ell,\ell}\left(\varphi_z(U),\bK\right) \cong C_{\ell,\ell}(U,\bK)$, for all $z\in \mathbb{R}$.
\end{enumerate}
\end{Thm}
The proof is based on the fact that the lifting of  $\varphi$ on $J^1(X)$ commutes with the $\bZ$-action (but not with the $\bR$-action, which is in fact used to prove the isomorphisms after \eqref{definition of kappa}). Then the sheaf quantization of $\varphi$ is $\bZ$-equivariant. Therefore, the isomorphism $\Phi_{\ell,T,z}$ can be defined only when $T=\ell$. Consequently, we can only use $T=\ell$ information of $F_{\ell}(U,\bK)$ as contact invariants. Now, for the same reason as in the symplectic case, we assume $\ell\in \bO$, i.e., $\ell$ is odd.  From now on, we only need one lower-subscript for contact admissible open sets
$U$. That is $C_{\ell}(U,\bK)=C_{\ell,\ell}(U,\bK)$, and $\eta_{\ell,\ell}(U,\bK)=\eta_{\ell}(U,\bK)$.

For the definition of capacities, it is reasonable to require discrete spectrum now. That is
\begin{Def}For an admissible open set $U\subset T^*V\times S^1$, $k\in \bZ_{\geq 1}$. Define
\[[\textnormal{Spec}](U,k)\coloneqq \{\ell_0 \in \bO: \forall\ell \in \bO, \, p_\ell \geq p_{\ell_0},\, \exists \gamma_\ell  \in H^{*}(C_{\ell}(U,\bF_{p_\ell})),\, \eta_{\ell}(U,\bF_{p_\ell})=u^k\gamma_\ell \},\]
and
\[[c]_k(U)\coloneqq \min [\textnormal{Spec}](U,k) \in \bO.\]
\end{Def}
Besides, we also need to take care of properties about $[c]_k$.The invariance is still true. But to obtain non-trivial information, we must address some restrictions on the size of domains. In fact, in the computation of $[c]_k(B_a)$, we need to take $T=\ell < a p_\ell$ to make sure $\eta_{\ell}(U,\bF_{p_\ell})$ is non-zero torsion. Here, the constrain is read as $a>\ell/p_\ell$ for all $\ell\in \bO$, then
a necessary constraint is $a>1$. This fits into the framework of \cite{eliashberg2006geometry} that a small contact ball can be squeezed into smaller contact balls. Therefore, we define
\begin{Def}For a contact admissible open set $U\subset T^*V\times S^1$, we say it is big if there is a contact ball $B_a\times S^1 \subset U$ such that $a>1$.
\end{Def}
Then, we can make sure that the proofs of monotonicity of $[c]_k$ run well under the condition of big admissible open sets.

Unluckily, the proof of representing property is invalid now. In fact, it is even worse, because actions of closed characteristics are usually non-integer. In summary, we organize our discussions as the following theorem.
Maybe in the contact case, the spectrum sets could provide us more interesting obstructions. So we state results of spectrum sets as well.
\begin{Thm}\label{contact capacities} The functions $[c]_k:\mathcal{C}_{\text{big admissible}}\rightarrow \bO$ satisfy the following:
\begin{enumerate}[fullwidth]
    \item $[c]_k \leq [c]_{k+1}$ and $[\textnormal{Spec}](U,k+1)\subset [\textnormal{Spec}](U,k)$, for all $k\in \bZ_{\geq 0}$.
    \item If there is an inclusion of admissible open sets $U_1 \subset U_2$, then $[c]_k(U_1) \leq [c]_k(U_2)$ and $[\textnormal{Spec}](U_2,k)\subset [\textnormal{Spec}](U_1,k)$.
    \item For a compactly supported contact isotopy $\varphi_z:T^*V\times S^1 \rightarrow T^*V\times S^1$, we have $[c]_k(U)=[c]_k(\phi_z(U))$ and $[\textnormal{Spec}](U,k)= [\textnormal{Spec}](\phi_z^H(U),k)$.
\end{enumerate}
\end{Thm}
\begin{RMK}Maybe in the contact case, spectrum sets could provide us more interesting obstructions. So I state results of spectrum sets here.\end{RMK}
Finally, let us discuss the problem of toric domains. In fact, here we do not need to change too much, because we have set everything up well now. We only need to change the statement of the structural theorem slightly.
\begin{Thm}\label{structure contact toric domian module}For any big contact convex toric domain $X_\Omega \times S^1 \subsetneqq T^*V\times  S^1$ (that means $\|\Omega^\circ_1\|_{\infty}<1$), we have:
\begin{itemize}[fullwidth]
    \item The minimal cohomology degree of $H^*(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))$ is exactly $-2I(\Omega^\circ_\ell)$, i.e.,
    \[ H^*(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))\cong  H^{\geq -2I(\Omega^\circ_\ell)}(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell})) \]
    and
\[ H^{-2I(\Omega^\circ_\ell)}(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))\neq 0.  \]

\item $H^*(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))$ is a finitely generated $\bK[u]$ module. The free part is isomorphic to $A=\bK[u,\theta]$, $H^*(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))$ is of rank 2. The torsion part is located in cohomology degree $[-2I(\Omega^\circ_\ell),-1]$.  $H^*(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))$ is torsion free when $X_\Omega$ is an open ellipsoid.
\item For each $Z\in \Omega^\circ_\ell$, the inclusion $\overline{OZ} \subset \Omega^\circ_\ell$ induces a decomposition $\eta_{\ell}(X_{\Omega}\times S^1,\bF_{p_\ell})=u^{I(Z)}\gamma_{Z,\ell}$ for a non-torsion element $\gamma_{Z,\ell}\in H^{-2I(Z)}(C_{\ell}(X_\Omega\times S^1,\bF_{p_\ell}))$. In particular, $\eta_{\ell}(X_{\Omega}\times S^1,\bF_{p_\ell})$ is non-zero.
\end{itemize}
\end{Thm}
And,
\begin{Thm}\label{computation of capacities of contact convex toric domian}For a big contact convex toric domain $X_\Omega \times S^1 \subsetneqq T^*V \times S^1$, we have
\begin{align*}
[c]_k(X_\Omega\times S^1)=&\min\left\{\ell\in\bO: \ell/p_\ell <1/\|\Omega^\circ_1\|_{\infty} ,\,\exists  {z}\in \Omega_\ell^\circ, I( {z})\geq k\right\}\\
    =&\min\left\{\ell\in\bO: \ell/p_\ell <1/\|\Omega^\circ_1\|_{\infty} ,\, \ell \geq c_k(X_\Omega) \right\}
\end{align*}
\end{Thm}
The result is much weaker than the symplectic case, but it is still interesting. For example, when we consider ellipsoids, we have
\begin{align*}
    [c]_k(E\times S^1)=&\min\left\{\ell \in \bO:  \ell/p_\ell <a_1 ,\, \sum_{i=1}^d \left\lfloor\frac{\ell}{a_i}\right\rfloor \geq k\right\}\\
    =&\min\left\{\ell\in\bO:\ell/p_\ell <a_1 ,\, \ell \geq c_k(E) \right\}
\end{align*}

where $1<a_1\leq a_2\leq\cdots\leq a_d$.

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redpajama_set_name: RedPajamaArXiv