\draw[line width=1pt, postaction={decorate, decoration={
markings, mark=at position 0.5 with {\arrow[line width=1.2pt]{>}}}}]
(2,3) -- (1,0);
\end{tikzpicture} \vspace*{-1em}
\end{center}
The differential operator associated to $\Gamma_n$ is given by
\begin{center}$
B_{\Gamma_n,\pi}(f,g)(x) = \left.\displaystyle\prod_{k=1}^{n}\left( \displaystyle\sum_{i_k,j_k=1}^{2n}\pi^{i_k,j_k}\dfrac{\pa}{\pa x^{i_k}}\dfrac{\pa}{\pa y^{j_k}} \right)f(x)g(y)\right|_{x=y}.
The differential form $\omega_{\Gamma_n}$ is given by
\begin{center}$
\omega_{\Gamma_{n}} = \d\varphi_{(1,\bar 1)} \wedge \d\varphi_{(1,\bar 2)} \wedge \cdots \wedge \d\varphi_{(n,\bar 1)} \wedge \d\varphi_{(n,\bar 2)}
where, following \eqref{equ:III_angle_geom}, the differential 1-forms are given by $
\d\varphi_{(i,\bar j)}(p_1,...,p_n,q_1,q_2) = 2\d\arg(q_j-p_i)$, where $q_1 = 0$ and $q_2 = 1$ are fixed. The compactification $\bar H_n$ is parametrised as
\begin{center}$
\begin{aligned}
\bar H_n &\simeq \overline{\left\{ (\arg(0-p_1),\arg(1-p_1),...,\arg(0-p_n),\arg(1-p_n))\colon p_1,...,p_n \in \mathbb H \text{ distinct} \right\}} \\
&\simeq [-\pi,0]^{2n}.
\end{aligned}
Hence the star product is given by
\begin{align*}
(f \star g)(x) &= (fg)(x) + \displaystyle\sum_{n=1}^\infty \dfrac{\hbar^n}{2^n(2\pi)^{2n}n!}B_{{\Gamma_n},\pi}(f,g)(x)\cdot \displaystyle\int_{\bar H_n} \omega_{\Gamma} \\
&= (fg)(x) + \left.\displaystyle\sum_{n=1}^\infty\dfrac{\hbar^n}{2^n n!}\displaystyle\prod_{k=1}^{n}\left( \displaystyle\sum_{i_k,j_k=1}^{2n}\pi^{i_k,j_k}\dfrac{\pa}{\pa x^{i_k}}\dfrac{\pa}{\pa y^{j_k}} \right)f(x)g(y)\right|_{x=y}.
\end{align*}
We have thus shown that in this case the Kontsevich star product coincides with the Weyl--Moyal product \eqref{equ:I_Moyal}.
\end{example}

\section{Proof of Formality Theorem}\label{sec:proof_formality}

To show that the coefficients $U_n$ given in \eqref{equ:III_U_coeff} defines an $\linf$-quasi-morphism $U\colon \tpolyrd \to \dpolyrd$, it remains to show that
\begin{enumerate}[label = (\alph*)]
\item $U_n$ respects the grading on $\tpolyrd$ and $\dpolyrd$;
\item $U_1 = U_1^{(0)}$, which is the HKR map \eqref{equ:II_HKR} and is already proven to be a quasi-isomorphism;
\item $U_n$ satisfies the constraint equation \eqref{equ:II_formality_constr}.
\end{enumerate}

For (a), we show that for $\Gamma \in G_{n,m}$, $W_{\Gamma}B_\Gamma$ is a morphism of graded vector spaces of degree $1-n$ from $\bigwedge^n(\tpolyrd) \to \dpolyrd[m-1]$. Indeed, $W_\Gamma$ is non-zero only if
\begin{center}$
2n + m - 2 = \#(E_\Gamma) = \displaystyle\sum_{k=1}^n \#(\operatorname{Star}(k)) = \displaystyle\sum_{k=1}^n\left(\left| \xi_i \right| + 1\right) = \left| \xi_1 \otimes \cdots \otimes \xi_n \right| + n.
So $\deg U_n = (m-1)-(2n+m-2-n) = 1-n$ as claimed.

\subsection{Checking Quasi-Isomorphism}

For (b), the proof is simply a checking for the constructions.

\begin{lemma}[$\bm U$ is a Quasi-Isomorphism]{7}
The first Taylor coefficient $U_1\colon \tpolyrd \to \dpolyrd$ defined by \eqref{equ:III_U_coeff} satisfies $U_1 = U_1^{(0)}$, where $U_1^{(0)}$ is the HKR map defined in \eqref{equ:II_HKR}.
\end{lemma}
\begin{proof}
By \eqref{equ:III_U_coeff}, we have
\begin{center}$
U_1 = \displaystyle\sum_{m=0}^\infty \displaystyle\sum_{\Gamma \in G_{1,m}} W_{\Gamma}B_{\Gamma} = \displaystyle\sum_{m=0}^\infty W_{\Gamma_m}B_{\Gamma_m}
where $\Gamma_m$ is the unique graph in $G_{1,m}$ as shown below:
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=.65]
\draw (-5, 0) -- (0,0);
\draw [line width = 1pt, dotted] (0, 0) -- (2,0);
\draw  [->] (2, 0) -- (5,0);
\node at (5,0.3) {Re};

\draw[line width=1pt, postaction={decorate, decoration={
markings, mark=at position 0.8 with {\arrow[line width=1.2pt]{>}}}}]
(0,3) -- (-3,0);
\draw[line width=1pt, postaction={decorate, decoration={
markings, mark=at position 0.8 with {\arrow[line width=1.2pt]{>}}}}]
(0,3) -- (-2,0);
\draw[line width=1pt, postaction={decorate, decoration={
markings, mark=at position 0.8 with {\arrow[line width=1.2pt]{>}}}}]
(0,3) -- (-1,0);
\draw[line width=1pt, postaction={decorate, decoration={
markings, mark=at position 0.8 with {\arrow[line width=1.2pt]{>}}}}]
(0,3) -- (3,0);

\draw (-3,0) node[circle,fill,inner sep=0,minimum size=4pt]{} node[yshift=-10pt]{$\bar 1$};
\draw (-2,0) node[circle,fill,inner sep=0,minimum size=4pt]{} node[yshift=-10pt]{$\bar 2$};
\draw (-1,0) node[circle,fill,inner sep=0,minimum size=4pt]{} node[yshift=-10pt]{$\bar 3$};
\draw (3,0) node[circle,fill,inner sep=0,minimum size=4pt]{} node[yshift=-10pt]{$\bar m$};
\draw (0,3) node[circle,fill,inner sep=0,minimum size=4pt]{} node[yshift=10pt]{$1$};
\end{tikzpicture}
\label{fig:HKR_Graph} \vspace*{-2em}
\end{figure}
The differential operator associated to $\Gamma_m$ is given by
\begin{center}$
B_{\Gamma_m}(\xi)(f_1,...,f_m) = \displaystyle\sum_{i_1,...,i_m=1}^{d} \xi^{i_1,...,i_m}\pa_{i_1}f_1\cdots \pa_{i_m}f_m.
The computatuion of the weight integral is similar to that in Example \ref{eg:III_Moyal}:
\begin{center}$
W_{\Gamma_m} = \dfrac{1}{m!(2\pi)^m}\displaystyle\int_{\bar C^+_{1,m}} \d\varphi_{(1,\bar 1)} \wedge \cdots \wedge \d\varphi_{(1,\bar m)} = \dfrac{1}{m!(2\pi)^m} (2\pi)^m = \dfrac{1}{m!}.
The map $U_1$ thus takes the form
\begin{center}$
U_1(\xi)(f_1,...,f_m) = \dfrac{1}{m!}\displaystyle\sum_{i_1,...,i_m=1}^{d} \xi^{i_1,...,i_m}\pa_{i_1}f_1\cdots \pa_{i_m}f_m.
In particular, if $\xi = X_1 \wedge \cdots \wedge X_n$ for $X_1,...,X_n \in \Gamma(\R^d,\tanb \R^d)$, then
\begin{center}$
\xi^{i_1,...,i_m} = \displaystyle\sum_{\sigma \in S_m}\operatorname{sgn}(\sigma)X_1^{\sigma(i_1)}\cdots X_m^{\sigma(i_m)} \implies U_1(\xi)(f_1,...,f_m) = \dfrac{1}{m!}\det(X_i(f_j))_{i,j}.
We conclude that $U_1$ agrees with the HKR map $U_1^{(0)}$.
\end{proof}

\subsection{Formality Equation from Boundary Strata}

For (c), the construction in Section \ref{sec:Kont_con} allows us to put the formality equation \eqref{equ:II_graph_constr} into the form
\begin{equation}
\displaystyle\sum_{\substack{ \Gamma \in G_{n,m}\\ \#(E_\Gamma) = 2n + m -3 }} c_\Gamma B_\Gamma(\xi_1 \cdots \xi_n)(f_1 ,..., f_m) = 0. \label{equ:III_graph_constr}
\end{equation}
where we only sum over $\Gamma \in G_{n,m}$ for $\#(E_\Gamma) = 2n + m -3$, since only these graphs can have non-vanishing weights $W_\Gamma$ by definition. The equation \eqref{equ:III_graph_constr} is satisfied if we can prove that $c_\Gamma = 0$ for all such graphs $\Gamma$. In the following subsections, we shall prove that

\begin{lemma}[]{lem:III_strata_c}
Let $c_\Gamma \in \R$ be the coefficient associated to $\Gamma \in G_{n,m}$ from \eqref{equ:II_graph_constr} and \eqref{equ:III_graph_constr}. Then
\begin{center}$
c_\Gamma = \displaystyle\prod_{k=1}^n \dfrac{1}{\#(\operatorname{Star}(k))!} \dfrac{1}{(2\pi)^{\#(E_\Gamma)}}\displaystyle\int_{\pa \bar C^+_{n,m}} \omega_\Gamma.
\end{lemma}
Having proven this, we invoke the Stokes' theorem for manifolds with corners:
\begin{lemma}[Stokes' Theorem with Corners]{}
Let $X$ be an $n$-dimensional oriented smooth manifold with corners. Let $X_1$ be the union of strata of $X$ of codimension $0$ and $1$. Then $\pa X_1$ is a $(n-1)$-dimensional manifold with boundary with orientation induced from $X_1$. Let $\omega$ be a compactly supported differential $(n-1)$-form on $M$. We have
\begin{center}$
\displaystyle\int_X \d \omega = \displaystyle\int_{\pa X_1} \omega.
\end{lemma}
\begin{proof}
See \cite{conradDG}.
\end{proof}

Then $c_\Gamma = 0$ follows from that $\omega_\Gamma$ is a closed form:
\begin{center}$
c_\Gamma = \displaystyle\int_{\pa \bar C^+_{n,m}} \omega_\Gamma = \displaystyle\int_{\bar C^+_{n,m}} \d\omega_\Gamma = \displaystyle\int_{\bar C^+_{n,m}} \d \left( \bigwedge_{e \in E_\Gamma} \d \varphi_e \right) = 0.
To prove Lemma \ref{lem:III_strata_c}, we need to analyse the integral $\displaystyle\int_{\pa \bar C^+_{n,m}} \omega_\Gamma$, which has only contributions from boundary strata of codimension $1$. Informally, these strata represent the degenerate configurations where some points collapse together. They can be classified into the following two types:
\begin{enumerate}
\item[$\mathsf{S1}$:] The vertices $S := \{k_1,...,k_i\}$ ($i \geq 2$) of the first type collapse together to a point in the upper half plane $\mathbb H$. Such boundary strata can locally be expressed as a product $\pa_S \bar C^+_{n,m} \simeq C_i \times C^+_{n-i+1,m}$.

\item[$\mathsf{S2}$:] The vertices $S := \{k_1,...,k_i\}$ ($i \geq 0$) of the first type and $S' := \{\overline{\ell+1},...,\overline{\ell+j}\}$ ($j \geq 0$) of the second type collapse together to a point on the real line $\R$, where $2i + j \geq 2$ and $i+j \leq n + m -1$. Such boundary strata can locally be expressed as a product $\pa_{S,S'} \bar C^+_{n,m} \simeq C^+_{i,j} \times C^+_{n-i,m-j+1}$.
\end{enumerate}
The products represent what Kontsevich \cite{kon97} called ``looking through a magnifying glass'', as shown in the following diagram:
\begin{figure}[H]
\centering
\includegraphics[width = 0.45\textwidth]{magnifying_S1.jpg}
\caption{Type $\mathsf{S1}$ boundary stratum through a magnifying glass.}
\end{figure}
Suppose that we start with an admissible graph $\Gamma$. Let $\Gamma_0$ be the subgraph of $\Gamma$ spanned by the collapsing vertices, and $\Gamma_1$ be the quotient of $\Gamma$ by $\Gamma_0$. There are cases where $\Gamma_1$ fails to be admissible:
\begin{itemize}[nosep]
\item $\Gamma_1$ may have multiple arrows.
\item $\Gamma_1$ may have a bad edge, as shown in the following diagram:
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[width = 0.6\textwidth]{S2_badedge.jpg}
\caption{Type $\mathsf{S2}$ boundary stratum with a bad edge.}
\end{figure}
In both cases, we note that the corresponding weight integrals vanish. Therefore we can safely neglect these cases. The decomposition of a stratum $F \simeq F_0 \times F_1$ induces the decomposition of the weight integral:
\begin{center}$
\displaystyle\int_{F} \omega_{\Gamma} = \pm \displaystyle\int_{F_0} \omega_{\Gamma_0} \displaystyle\int_{F_1} \omega_{F_1}.
It requires great caution when dealing with the orientations and signs in the integrals. For details refer to \cite{arnal02}.

\subsubsection[]{Type $\bm{\mathsf{S1}}$: $\bm{i = 2}$}

Now we consider type $\mathsf{S1}$ strata with $i = 2$. In this case, $\Gamma_0$ contains a single edge $e$. Suppose that $s(e) = p_i$ and $t(e) = p_j$. Then $\Gamma_1$ is obtained by identifying $p_i$ with $p_j$ and contracting the edge (if exists) from $p_i$ to $p_j$. The polydifferential operator associated to this stratum is given by
\begin{center}$
B_{\Gamma,R}(\xi_1\cdots\xi_n)(f_0,...,f_m) = B_{\Gamma_1}(\xi_i \diamond \xi_j\cdots \hat \xi_i \cdots \hat \xi_j \cdots \xi_n)(f_0,...,f_m).

\subsubsection[]{Type $\bm{\mathsf{S1}}$: $\bm{i\geq 3}$}

We prove that the integral vanishes for $i \geq 3$.
\begin{lemma}[Vanishing of Kontsevich Integrals]{lem:III_van_Kont_int}
For $n \geq 3$ and $\Gamma \in G_{n,0}$, the weight integral over the configuration space $C_n$ is zero:
\begin{center}$
\displaystyle\int_{\bar C_n} \omega_\Gamma = 0.
\end{lemma}
This result implies that type ${\mathsf{S1}}$ strata with $i \geq 3$ does not contribute to the coefficient $c_{\Gamma}$. For the proof we need a result from distribution theory, which is stated below:
\begin{lemma}[Distributional Forms]{}
Let $U$ be a complex manifold with compactification $\bar U$. For a differential form $\omega$ on $U$ such that the coefficients of $\omega$ and $\d\omega$ are locally integrable over $\bar U$, we denote by $\mathscr I(\omega)$ the corresponding distributional form, that is, differential form with coefficients in the space of distributions. Then $\mathscr I$ commutes with the exterior differential: $\d(\mathscr I(\omega)) = \mathscr I(\d \omega)$. In addition, the integral $\displaystyle\int_U \omega$ is absolutely convergent and is equal to $\displaystyle\int_{\bar U} \mathscr I (\omega)$.
\end{lemma}
\begin{proof}
See Lemma 6.6.1 of \cite{kon97}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:III_van_Kont_int}.]
We pick the edge $e_1$. Using the action of the Lie group $G'$, we may identify the configuration space as a subset of $\bar{\mathbb H}^n$ where $s(e_1)$ is fixed to the origin and $t(e_1)$ lies on the unit circle. This provides a decomposition $\bar C_n \simeq S^1 \times U$ where $U$ is a complex manifold. The weight integral factors accordingly:
\begin{center}$
\displaystyle\int_{\bar C_n} \omega_\Gamma = \displaystyle\int_{S^1 \times U} \d\varphi_{e_1} \wedge \bigwedge_{i=2}^{2n-3} \d(\varphi_{e_i}-\varphi_{e_1}) = 2\pi \displaystyle\int_{U} \bigwedge_{i=2}^{2n-3} \d\varphi_{e_i} = 2\pi \displaystyle\int_{U} \bigwedge_{i=1}^{2n-4} \d \arg z_i,
where $z_i$ is the difference in the complex coordinates of $s(e_{i+1})$ and $t(e_{i+1})$. Now we employ a trick using logarithm. We claim that
\begin{equation}
\displaystyle\int_{U} \bigwedge_{i=1}^{2n-4} \d \arg z_i = \displaystyle\int_{U} \bigwedge_{i=1}^{2n-4} \d \log |z_i|. \label{equ:III_log_trick}
\end{equation}
Indeed, both $\d \arg z_i$ and $\d \log |z_i|$ can be expressed as a sum of a holomorphic 1-form and an anti-holomorphic 1-form:
\begin{equation}
\d \arg z_i = \dfrac{1}{2\i}\left( \d \log z_i - \d \log \bar z_i \right); \qquad \d \log |z_i| = \dfrac{1}{2}\left( \d \log z_i + \d \log \bar z_i \right). \label{equ:III_log_arg}
\end{equation}
We know that the integration of a $(k,2N-k)$-form on a $2N$-dimensional complex manifold vanishes unless $k = N$. The non-vanishing contributions of \eqref{equ:III_log_arg} are the same in the integration, which justifies \eqref{equ:III_log_trick}. Finally, using the fact from distribution theory which is stated above, we have
\begin{align*}
\displaystyle\int_{U} \bigwedge_{i=1}^{2n-4} \d \log |z_i| &=  \displaystyle\int_{U} \d \left(\log |z_1|\bigwedge_{i=2}^{2n-4} \d \log |z_i|\right) = \displaystyle\int_{\bar U} \mathscr I\left(\d \left(\log |z_1|\bigwedge_{i=2}^{2n-4} \d \log |z_i|\right)\right) \\
&= \displaystyle\int_{\bar U} \d\left(\mathscr I \left(\log |z_1|\bigwedge_{i=2}^{2n-4} \d \log |z_i|\right)\right) = 0 \qedhere
\end{align*}
\end{proof}

\subsubsection[]{Type $\bm{\mathsf{S2}}$}

Now we consider type $\mathsf{S2}$ strata. Suppose that $\Gamma_0$ contains the vertices $k_1,...,k_i,\overline{\ell+1},...,\overline{\ell+j}$. The polydifferential operator associated to this stratum is given by
\begin{center}$
B_{\Gamma,F}(\xi_J)(f_0,...,f_m) = B_{\Gamma_1}(\xi_{\sigma(1)}\cdots \xi_{\sigma(i)})\left( f_0,...,f_{\ell},B_{\Gamma_0}(\xi_I)(f_{\ell+1},...,f_{\ell+j}),f_{\ell+j+1},...,f_m \right)
where the multi-indices $I = (k_1,...,k_i)$ and $J = \left\{ 1,...,n \right\} \setminus I$ (with increasing order).

These polydifferential operators match those appeared in the formality equation \ref{equ:II_formality_constr}. The matching of the coeffcients $c_{\Gamma,F}$ with $\displaystyle\int_F \omega_\Gamma$ is more subtle and we refer to \cite{arnal02}. This concludes the proof of Lemma \ref{lem:III_strata_c} and thus of the formality theorem \ref{thm:II_formality}.

% \begin{figure}[H]
%     \centering
%     \includegraphics[width = 0.7\textwidth]{S2_goodedge.jpg}
%     \caption{Type $\mathsf{S2}$ boundary stratum with no bad edges.}
% \end{figure}

\section{Globalisation of Star Product}

Kontsevich proved the formality theorem on general manifolds in \cite{kon97} and \cite{kontsevich01alg}. In this section we summarise a slightly different approach of the globalisation of Kontsevich star product following Cattaneo, Felder and Tomassini in \cite{cattaneo02global} and \cite{cattaneo01}. Note that this approach proves a weaker result, as the general formality theorem does not follow immediately from the existence of star products. Another approach due to Dolgushev (\cite{dolgushev05}, \cite{dolgushev05thesis}) directly globalises the formality theorem is stronger, but we will not cover it due to length limit.

\subsection{Formal Geometry}

The first step of globalisation is to construct two vector bundles over the Poisson manifold $M$, which fibre-wise represents the Poisson algebras and their quantisations. The idea is that two smooth functions with the same Taylor coeffcients are indistinguishable from the star product.

\begin{definition}
Let $M$ be a $d$-dimensional smooth manifold. The $k$-th jet bundle $\mathrm J^k(M)$ over $M$ is defined fibre-wise as follows: for $x \in M$, $\mathrm J^k_x(M)$ is the set of functions $f \in \smooth{M}$ quotient by the equivalence relation that $f \sim g$ if and only if the partial derivatives of $f$ and $g$ at $p$ agree up to order $k$. The \textbf{infinite jet bundle} $\mathrm J^\infty(M)$ is obtained by taking the projective limit\footnote{Technically we have to specify the topology with respect to which we are taking the limit. Heuristically we consider $\mathrm{J}^\infty(M)$ as a formal projective limit in the category of smooth manifolds.} of the forgetful maps $\mathrm J^{k+1}(M) \to \mathrm J^k(M)$. That is, for $f,g \in \smooth{M}$, $[f] = [g] \in \mathrm J^\infty(M)$ if and only if $f$ and $g$ have the same Taylor expansion at some $x \in M$.
\end{definition}

\newcommand{\Mcoor}{M^{\mathrm{coor}}}
\newcommand{\Maff}{M^{\mathrm{aff}}}
\newcommand{\phiaff}{\varphi^{\mathrm{aff}}}
We give a local characterisation of $\mathrm J^\infty(M)$. Let $\Mcoor := \mathrm J^\infty_0(\R^d,M)$ be the fibre at $0 \in \R^d$ of the infinite jet bundle $\mathrm J^\infty(\R^d,M)$. Note that $\operatorname{GL}(d,\R)$ acts on $\Mcoor$ by linear diffeomorphisms, which induces the quotient manifold $\Maff := \Mcoor/\operatorname{GL}(d,\R)$. In particular, $\Maff$ is a vector bundle over $M$ whose fibres are contractible, and hence admits a section $\phiaff\colon M \to \Maff$. We define the \textbf{associated bundle} $\tilde E_0$ over $\Maff$ by taking the fibre product of $\Mcoor$ with a formal power series:
\newcommand{\formpoww}[1]{{\left[ {}\!{\left[ #1 \right]}{}\! \right]}}
\begin{equation*}
\tilde E_0 := \Mcoor \times_{\operatorname{GL}(d,\R)} \R\formpow{y^1,...,y^d}.
\end{equation*}
Then the pull-back via $\phiaff$ provides a vector bundle $E_0 := (\phiaff)^* \tilde E_0$ over $M$, with fibres $\R\formpoww{y^1,...,y^d}$. Note that $E_0$ is isomorphic to $\mathrm J^\infty(M)$, given by identifying the jets of $f$ at $p$ with the Taylor expansion of $f \circ \varphi^{-1}$ at $0$, where $\varphi_x\colon M \to \R^d$ is a chosen coordinate chart around $x \in M$ and $\varphi_x(x) = 0$.

Now consider $M$ as a Poisson manifold. Let $E_0 = \mathrm J^\infty(M)$ be its infinite jet bundle. The canonical map $\smooth{M} \to E_0$ transports the Poisson structure of $\smooth{M}$ to $E_0$.

\begin{proposition}[Grothendieck Connection]{prop:III_groth_con}
There exists a canonical connection $D_0$ on $E_0$ such that
\begin{enumerate}[nosep]
\item $D_0$ is a derivation: $D_0(fg) = f D_0g+ g D_0f$ for $f, g \in \Gamma(M,E_0)$.
\item $D_0$ is flat: $D_0^2 = 0$. This implies that the $E_0$-valued differential forms
\begin{center}
$\Omega^\bullet(E_0) := \Omega^\bullet(M) \otimes_{\smooth{M}} \Gamma(M,E_0)$
\end{center}
is a cochain complex with respect to the connection $D_0$. It provides the cohomology groups $\mathrm H^\bullet(E_0,D_0)$.
\item The cohomology is concentrated at zeroth degree: $\mathrm H^n(E_0,D_0)$ for $n>0$.
\item The canonical map $\smooth{M} \to E_0$ induces a Poisson isomorphism between $\smooth{M}$ and the algebra of $D_0$-horizontal sections $\mathrm H^0(E_0,D_0)$.
\end{enumerate}
We call $D_0$ the \textbf{Grothendieck connection} of $E_0$.
\end{proposition}
\begin{proof}
See \S 2 of \cite{cattaneo01}.
\end{proof}
% This lemma summarises the classical part of the story on the Poisson manifold. The task is to deform the connection $D_0$ to

% this map induces a Poisson isomorphism $\smooth{M} \cong \mathrm H^0(E_0, D_0)$, where $D_0$ is a canonical flat connection (i.e. the curvature $F_{D_0}$ of $D_0$ vanishes, \emph{cf.\ }\S 2 of \cite{cattaneo02global}) of $E_0$, and $\mathrm H^0(E_0, D_0)$ is the space of $D_0$-horizontal sections of $E_0$ (i.e. sections $s$ with $D_0s = 0$).

On the other hand, let $\tilde E$ be the associated bundle of $\R\formpow{\hbar}$-modules
\begin{center}
$\tilde E := \Mcoor \times_{\operatorname{GL}(d,\R)} \R\formpow{y^1,...,y^d,\hbar}$
\end{center}
over $\Maff$. And let $E := (\phiaff)^*\tilde E = E_0\formpow{\hbar}$ be a vector bundle over $M$. We shall regard $E$ as the fibre-wise quantisation of $E_0$. More explicitly we would like to prove that
\begin{theorem}[Deformation Quantisation on a Poisson Manifold]{thm:defo_quan_Poi}
Let $(M,\pi)$ be a Poisson manifold and $E$ be the vector bundle over $M$ constructed above. There exists a flat connection $\bar D = D_0 + \displaystyle\sum_{n=1}^\infty \hbar^n D_n$ of $E$ such that the $\R\formpow{\hbar}$-algebra $\mathrm H^0(E,\bar D)$ is a quantisation of $\smooth{M}$.
\end{theorem}

Following the spirit of Fedosov's construction (\cite{fedosov94}), the next step is to construct a connection of $E$ and transport the Kontsevich star product to the sections $\Gamma(M,E)$ of $E$. The formality theorem in $\R^d$ provides such constructions.

\subsection{Star Product and Connection on Deformed Bundle}

Recall that Kontsevich's construction \eqref{equ:III_U_coeff} provides an $\linf$-quasi-isomorphism $U\colon \tpolyrd \to \dpolyrd$. We make the following definition:

\begin{definition}
Let $\pi$ be a Poisson bivector field and $\xi,\eta \in \Gamma(\R^d,\tanb \R^d)$. Define:
\begin{align*}
P(\pi) &:= \displaystyle\sum_{k=0}^\infty \dfrac{\hbar^k}{k!}U_k(\pi\cdots\pi), \\
A(\xi,\pi) &:= \displaystyle\sum_{k=0}^\infty \dfrac{\hbar^k}{k!}U_{k+1}(\xi\cdot\pi\cdots\pi), \\
F(\xi,\eta,\pi) &:= \displaystyle\sum_{k=0}^\infty \dfrac{\hbar^k}{k!}U_{k+2}(\xi\cdot\eta\cdot\pi\cdots\pi).
\end{align*}
\end{definition}
Note that $P(\pi) \in \dpolyrd[1]\formpow{\hbar}$ is exactly the star product.
% $A(\xi,\pi) \in \dpolyrd[0]\formpow{\hbar}$, and $F(\xi,\eta,\xi) \in \dpolyrd[-1]\formpow{\hbar}$.
We expect that $A(-,\pi)$ and $F(-,-,\pi)$ behave like a connection $1$-form and its curvature $2$-form.

% To study the relations among $P,A,F$, we construct a Chevalley--Eilenberg complex as follows.
% \begin{definition}
%     We say that $V\colon \tpolyrd[1] \to \dpolyrd$ is a \textbf{local polynomial map}, if for $\alpha \in \tpolyrd[1]$,
%     \begin{center}
%         $V(\alpha) \in \displaystyle\bigoplus_{r=0}^\infty \smooth{\R^d} \otimes \R[\pa_1,...,\pa_d]^{\otimes r}.$
%     \end{center}
%     That is, the coefficients of $V(\alpha)$ at $y \in \R^d$ are polynomials in the partial derivatives of the coefficients $\alpha^{ij}$ at $y$. Let $\mathscr{U}$ denotes the space of local polynomial maps.
% \end{definition}

\begin{definition}
The Lie algebra $\Gamma(\tanb \R^d)$ of vector fields acts on the space of \emph{local polynomial maps} $\mathscr U$ (\emph{cf.\ }\S 3 of \cite{cattaneo02global}). We consider the \textbf{Chevalley--Eilenberg complex}:
\footnote{
In general, let $\frg$ be a Lie algebra and $V$ be a $\frg$-module. We can similar construct the Chevalley--Eilenberg complex $C^\bullet_{\mathsf{CE}}(\frg,V)$. The cohomology of this complex is called the \emph{Lie algebra cohomology of $\frg$ with value in $V$}. See Chapter 7 of \cite{weibel} for details.

\begin{center}
$C_{\mathsf{CE}}^\bullet(\tanb \R^d,\mathscr U) := \operatorname{Hom}_{\R}(\bigwedge^\bullet\Gamma(\tanb \R^d),\mathscr U)$,
\end{center}
where $S \in C_{\mathsf{CE}}^p(\tanb \R^d,\mathscr U)$ sends $X_1 \wedge \cdots \wedge X_p$ to $S(X_1 \wedge \cdots \wedge X_p,\pi) \in \dpolyrd$ which depends polynomially on $\pi$. Therefore $P \in C_{\mathsf{CE}}^0(\tanb \R^d,\mathscr U)\formpow{\hbar}$, $A \in C_{\mathsf{CE}}^1(\tanb \R^d,\mathscr U)\formpow{\hbar}$, and $F \in C_{\mathsf{CE}}^2(\tanb \R^d,\mathscr U)\formpow{\hbar}$.
\end{definition}
The Chevalley--Eilenberg differential $\delta\colon C_{\mathsf{CE}}^p(\tanb \R^d,\mathscr U) \to C_{\mathsf{CE}}^{p+1}(\tanb \R^d,\mathscr U)$ is defined as
\begin{equation*}
\begin{aligned}
(\delta S)\left(X_{1}, \ldots, X_{k}+1, \pi\right):=&\sum_{i=1}^{k+1}(-1)^{i} \left.\frac{\mathrm{d}}{\mathrm{d} t} S{\left(X_{1}, \ldots, \hat{X}_{i}, \ldots, X_{k+1},\left(\Phi_{X}^{t}\right)_{*} \pi\right)}\right|_{t=0} \\
&+\sum_{i<j} (-1)^{i+j} S{\left(\left[X_{i}, X_{j}\right], X_{1}, \ldots, \hat{X}_{i}, \ldots, \hat{X}_{j}, \ldots, X_{k+1}, \pi\right)},
\end{aligned}
\end{equation*}
where $\Phi_{X}^{t}$ is the flow along the vector field $X$. From this differential and the formality equation \eqref{equ:II_graph_constr} follows the lemma:
\begin{lemma}[]{}
\begin{enumerate}[label = (\roman*), nosep]
\item $P \circ (P \otimes \id - \id \otimes P) = 0$;
\item $(\delta P)(\xi, \pi) = P(\pi) \circ(A(\xi, \pi) \otimes \id +\id \otimes A(\xi, \pi))-A(\xi, \pi) \circ P(\pi)$;
\item $(\delta A)(\xi, \eta, \pi) = P(\pi) \circ(F(\xi, \eta, \pi) \otimes \id-\id \otimes F(\xi, \eta, \pi))-A(\xi, \pi) \circ A(\eta, \pi)+A(\eta, \pi) \circ A(\xi, \pi)$;
\item $(\delta F)(\xi, \eta, \zeta, \pi) = -A(\xi, \pi) \circ F(\eta, \zeta, \pi)-A(\eta, \pi) \circ F(\zeta, \xi, \pi)-A(\zeta, \pi) \circ F(\xi, \eta, \pi)$.
\end{enumerate}
\end{lemma}
In the lemma, (i) is the associativity equation of the star product, and (ii),(iii),(iv) describe the changes of $P,A,F$ under a coordinate transformation induced by the vector field $\xi$. These identifies are useful in the construction of the Fedosov connection $D$ in Proposition \ref{prop:III_fed_conn}. In addition, the following lowest order properties of $P,A,F$ are required.

\begin{lemma}[]{lem:III_PAF_lower}
\begin{enumerate}[label = (\roman*), nosep]
\item $P(\pi)(f \otimes g)=f g+  \pi(\d f, \d g)\hbar + \mathcal O(\hbar^2)$.
\item $A(\xi,\pi) = \xi + \mathcal O(\hbar)$, and $A(\xi,\pi) = \xi$ if $\xi$ is a linear vector field.
\item $F(\xi,\eta,\pi) = \mathcal O(\hbar)$.
\item $P(\pi)(1 \otimes f)=P(\pi)(f \otimes 1)=f$.
\item $A(\xi,\pi)1 = 0$.
\end{enumerate}
\end{lemma}
With these tools in hand, we can now lift the star product to the deformed bundle $E$. A key observation in \cite{cattaneo02global} is that, since the Kontsevich star product is $\operatorname{GL}(d,\R)$-equivariant, after taking an open cover of $M$ by contractible coordinate charts and fixing the representatives of $\operatorname{GL}(d,\R)$-equivalence classes on each chart, we may assume that $E$ is a \emph{trivial bundle} over $M$, with fibres isomorphic to $\R\formpow{y^1,...,y^d,\hbar}$. A section $f \in \Gamma(M,E)$ is a map $x \mapsto f_x$, where $f_x = f_x(y) \in \R\formpow{y^1,...,y^d,\hbar}$.

\begin{definition}
Let $\star = P(\pi)$ be the Kontsevich star product on $\R^d$. It induces an associative product on the space $\Gamma(M,E)$ of sections of $E$ by:
\begin{center}
$(f \star g)_x := P(\pi_x)(f_x \otimes g_x) = P((\varphi^{-1}_x)_*\pi)(f_x \otimes g_x)$,
\end{center}
where $\varphi_x\colon U \to \R^d$ is the coordinate chart around $x$.
\end{definition}

\begin{definition}
We define a connection $D\colon \Gamma(M,E) \to \Omega(M) \otimes_{\smooth{M}} \Gamma(M,E)$ of $E$ by
\begin{center}
$ (Df)_x := \d_x f + A^{M}_xf $,
\end{center}
where $\d_x$ is the de Rham differential of $x \mapsto f_x$ with value in $\R\formpow{y^1,...,y^d,\hbar}$, and $A^{M}_x$ is a connection 1-form satisfying $A^{M}_x(\xi) := A(\hat \xi_x, \pi_x)$ for $\xi \in \Gamma(M,\tanb M)$. Here $\hat \xi_x$ is defined by
\begin{equation*}
\hat \xi_x(y) = -\sum_{i,j,k=1}^{\dim M}\xi^i\left( \left( \dfrac{\pa \varphi_x}{\pa y} \right)^{-1} \right)^k_j \dfrac{\pa \varphi_x^j}{\pa x^i}\dfrac{\pa}{\pa y^k},
\end{equation*}
which is a formal vector field in $y$ and depends on the coordinate chart $\varphi_x$. Furthermore, it can be shown from Lemma \ref{lem:III_PAF_lower} that $D$ is independent of the choice of $\varphi_x$ and hence is a well-defined global connection of $E$.
\end{definition}

\subsection{Fedosov's Construction}
Similar to Fedosov's construction (\cite{fedosov94}) for a star product on the symplectic manifold, we would like to construct a flat Fedosov connection $\bar D$ of $E$ so that the $\bar D$-horizontal sections are isomorphic to the $\R\formpow{\hbar}$-algebra $\smooth{M}\formpow{\hbar}$.
\newcommand{\starbra}[1]{{\left[ #1 \right]_\star}}
\begin{proposition}[]{prop:III_fed_conn}
Let $F^M \in \Omega^2(E)$ be the $E$-valued 2-form $x \mapsto F^M_x$, where $F^M_x(\xi,\eta) := F(\hat \xi_x, \hat\eta_x,\pi_x)$ for $\xi,\eta \in \Gamma(M,\tanb M)$. Then $D$ is a \textbf{Fedosov connection} with \textbf{Weyl curvature} $F^M$. More specifically, for $f,g \in \Gamma(M,E)$,
\begin{enumerate}[nosep]
\item $D(f \star g) = Df \star g + f \star Dg$;
\item $D^2f = \starbra{F^M,f} := F^M \star f - f \star F^M$;
\item Bianchi identity: $DF^M = 0$.
\end{enumerate}
\end{proposition}

To prove Theorem \ref{thm:defo_quan_Poi}, we need to deform $D$ into a \emph{flat connection} $\bar D$ which is still Fedosov. We need the following lemma:
\begin{lemma}[]{}
Suppose that $D$ is a Fedosov connection on $E$ with Weyl curvature $F$. Then for $\gamma \in \Omega^1(E)$, $\bar D := D + \starbra{\gamma,-}$ is a Fedosov connection with Weyl curvature $F + D\gamma + \gamma \star \gamma$.
\end{lemma}

The construction of a flat Fedosov connection $\bar D$ is now reduced to finding a solution to the differential equation $F^M + D\gamma + \gamma \star \gamma = 0$. It can be proven that $\gamma$ can be constructed order by order as a result of the vanishing of higher cohomology of $E_0$:
\begin{lemma}[]{}
Suppose that $D_0$ is a flat connection of $E_0$ with $\mathrm H^2(E_0,D_0) = 0$. Then there exists $\gamma \in \hbar\Omega^1(E)$ such that $\bar D := D + \starbra{\gamma,-}$ has zero Weyl curvature.
\end{lemma}

Finally, we would like to construct a quantisation from $\mathrm H^0(E_0,D_0)$ to $\mathrm H^0(E,\bar D)$. Let $\operatorname{End}(E_0)$ be vector bundle over $M$ of the fibre endomorphisms of $E_0$. Then $\Omega^\bullet(\operatorname{End}(E_0))$ is a differential graded algebra, with the differential given by the super-commutator:
\begin{center}
$D_0(\Phi) := D_0 \circ \Phi - (-1)^p\Phi \circ D_0, \qquad \Phi \in \Omega^p(E_0)$.
\end{center}
Note that if $D = D_0 + \displaystyle\sum_{n=1}^\infty \hbar^n D_n$ is a connection on $E = E_0 \formpow{\hbar}$, then $D_n \in \Omega^1(\operatorname{End}(E_0))$ for $n \in \Z_{>0}$.

\begin{lemma}[]{lem:III_quan_H0}
Suppose that $D = D_0 + \smash{\displaystyle\sum_{n=1}^\infty \hbar^n D_n}$ is a flat Fedosov connection on $E$, and $\mathrm H^1(\operatorname{End}(E_0),D_0) = 0$. Then there exists a formal power series $\rho = \id + \displaystyle\sum_{n=1}^\infty \hbar^n \rho_n \in \Omega^0(\operatorname{End}(E_0))\formpow{\hbar}$ which induces an $\R\formpow{\hbar}$-isomorphism $\mathrm H^0(E_0,D_0)\formpow{\hbar} \to \mathrm H^0(E,\bar D)$.
\end{lemma}

The proof of the lemma is similar to the previous one, where $\rho$ is constructed recursively as a result of the vanishing of cohomology. Combining Proposition \ref{prop:III_groth_con} and Lemma \ref{lem:III_quan_H0}, we obtain an isomorphism $\tau\colon \smooth{M}\formpow{\hbar} \to \mathrm H^0(E,\bar D)$. An explicit star product on $\smooth{M}$ is given by
\begin{equation*}
(f \star g)(x) = \left[\tau^{-1}{\left( \tau(f)_x(y) \star \tau(g)_x(y)  \right)}\right]_{y=0}.
\end{equation*}
It remains to check that $f \star g$ satisfies the conditions for a star product on $\smooth{M}$ and that it recovers the Poisson bivector $\pi$ in the first order of $\hbar$. For the details refer to \cite{cattaneo02global}. This concludes the proof of Theorem \ref{thm:defo_quan_Poi}.

\chapter*{Conclusion}
\addcontentsline{toc}{chapter}{Conclusion}

We covered the basic aspects of deformation quantisation and deformation theory which culminates in the $\linf$-quasi-isomorphism theorem. We have proven the formality theorem in $\R^d$ following Kontsevich's paper \cite{kon97}. This provides a classification theorem for deformation quantisation at least locally on any Poisson manifold. The globalisation of star product is outlined following Cattaneo, Felder and Tomassini's work \cite{cattaneo02global}. For the complete proof of formality on general Poissons we refer to Dolgushev \cite{dolgushev05}.

There are some aspects in \cite{kon97} and the subsequent paper \cite{kontsevich99op} we did not cover in this dissertation. An important point is the non-uniqueness of Kontsevich star product. The moduli space of deformation quantisations can be identified with a principal homogeneous space of the Grothendieck--Teichm\"uller group. This perspective is studied in \cite{kontsevich99op}. On the other hand, Kontsevich proved in \cite{kon97} that the formality quasi-isomorphism he constructed is canonical in the sense that it preserves the cup products on the differential graded Lie algebras. In addition, we deliberately avoid using the language from formal geometry in the dissertation. It is not an essential ingredient of the formality theorem, but it does provide geometric intuitions for $\linf$-morphisms.

Some problems regarding deformation quantisation remains open after Kontsevich's work. For example, his approach to the problem was entirely based on formal deformation quantisation. One may postulate under what condition can we obtain a finite radius of convergence for the star product. For such strict deformation quantisation, it is possible to study the representation theory, which has a closer relation to quantum mechanics, in a similar way to geometric quantisation. A comprehensive theory for strict deformation quantisations and their representations on a general Poisson manifold is yet to be discovered.

In summary, it is clear that Kontsevich's work not only concludes a long-standard conjecture in deformation quantisation, but, more importantly, generates new insights into geometry and mathematical physics. In particular, we have seen the extensive utilisation of homological methods and higher algebras in quantum field theory in recent studies. This demonstrates once more the tight connection between pure mathematics and theoretical physics.

% Known problems:
% \begin{itemize}
%     \item \S 2.3: Details of proof of HKR theorem (\emph{cf.\ }\cite{gutt99})
%     \item \S 2.4: Definition of reduced symmetric algebra
%     \item \S 2.4: The signs of $Q_n$ and $\ell_n$ (\emph{cf.\ }\cite{doubek07}, \cite{arnal02})
%     \item \S 2.4: Details of proof of minimal model theorem (\emph{cf.\ }\cite{arnal02})
%     \item \S 2.4: Interpretation of $\linf$-quasi-isomorphism theorem
%     \item \S 2.5: Derivation of constraints for $U_n$ (\emph{cf.\ }\cite{arnal02})
%     \item \S 3.2: Orientations and signs concerning boundary strata (\emph{cf.\ }\cite{arnal02})
%     \item \S 3.3: Add a section on globalisation on Poisson manifold
% \end{itemize}

\nocite{atiyah}
\nocite{arnold}
\nocite{takhtajan}
\nocite{kontsevich02defo}
\nocite{schlessinger68}
\nocite{waldmann05}
\nocite{huybrechts05}
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\field{title}{Choix des signes pour la formalit{\'e} de {M.} {Kontsevich}}
\field{volume}{203}
\field{journaltitle}{Pacific journal of mathematics}
\field{year}{2002}
\endentry

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\field{volume}{111}
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\endentry

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\endentry

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\field{pages}{012002}
\field{title}{Deformation quantization: a survey}
\field{volume}{103}
\field{year}{2008}
\endentry

\entry{chern99}{book}{}
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\endentry

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Product and the Perturbative {Poisson} Sigma Model}
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\field{year}{2001}
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\field{pages}{329\bibrangedash 352}
\field{title}{From local to global deformation quantization of {Poisson}
\field{volume}{115}
\field{journaltitle}{Duke Mathematical Journal}
\field{year}{2002}
\endentry

\entry{cattaneo04notes}{article}{}
\name{author}{2}{}{%
given={Alberto\bibnamedelima S.},
giveni={A\bibinitperiod\bibinitdelim S\bibinitperiod},
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\field{labeltitlesource}{title}
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\field{sortinit}{C}
\field{sortinithash}{C}
\field{title}{Formality and star products}
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/math/0403135}{\normalfont{\texttt{math/0403135}}}}
\field{year}{2004}
\endentry

\entry{conradDG}{misc}{}
\name{author}{1}{}{%
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\strng{fullhash}{CB1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Con}
\field{sortinit}{C}
\field{sortinithash}{C}
\field{title}{Stokes' Theorem with corners, Differential Geometry handouts}
\verb{url}
\verb http://math.stanford.edu/~conrad/diffgeomPage/handouts/stokescorners.
\verb pdf
\endverb
\endentry

\entry{deligne87letter}{misc}{}
\name{author}{1}{}{%
\strng{namehash}{DP1}
\strng{fullhash}{DP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Del87}
\field{sortinit}{D}
\field{sortinithash}{D}
\field{title}{A letter to Millson}
\field{journaltitle}{Unpublished manuscript}
\field{year}{1987}
\endentry

\entry{deligne95}{article}{}
\name{author}{1}{}{%
\strng{namehash}{DP1}
\strng{fullhash}{DP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
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\field{sortinithash}{D}
\field{number}{4}
\field{pages}{667\bibrangedash 697}
\field{title}{D{\'e}formations de l'alg{\`e}bre des fonctions d'une
vari{\'e}t{\'e} symplectique: comparaison entre Fedosov et De Wilde, Lecomte}
\field{volume}{1}
\field{journaltitle}{Selecta Mathematica}
\field{year}{1995}
\endentry

\entry{doubek07}{article}{}
\name{author}{3}{}{%
\strng{namehash}{DMMMZP1}
\strng{fullhash}{DMMMZP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
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\field{sortinithash}{D}
\field{title}{Deformation theory (lecture notes)}
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/math/0705.3719}{\normalfont{\texttt{math/0705.3719}}}}
\field{year}{2007}
\endentry

\entry{dolgushev05}{article}{}
\name{author}{1}{}{%
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\strng{fullhash}{DV1}
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\field{labeltitlesource}{title}
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\field{sortinit}{D}
\field{sortinithash}{D}
\field{number}{1}
\field{pages}{147\bibrangedash 177}
\field{title}{Covariant and equivariant formality theorems}
\field{volume}{191}
\field{journaltitle}{Advances in Mathematics}
\field{year}{2005}
\endentry

\entry{dolgushev05thesis}{thesis}{}
\name{author}{1}{}{%
given={Vasiliy\bibnamedelima A.},
giveni={V\bibinitperiod\bibinitdelim A\bibinitperiod},
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\strng{fullhash}{DVA1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Dol05}
\field{extraalpha}{2}
\field{sortinit}{D}
\field{sortinithash}{D}
\field{note}{arXiv
\href{https://arxiv.org/abs/math/0504420}{\normalfont{\texttt{math/0504420}}}}
\field{title}{A proof of {Tsygan's} formality conjecture for an arbitrary
smooth manifold}
\list{institution}{1}{%
{Massachusetts Institute of Technology}%
\field{type}{phdthesis}
\field{year}{2005}
\endentry

\entry{lecomte83symp}{article}{}
\name{author}{2}{}{%
family={De\bibnamedelima Wilde},
familyi={D\bibinitperiod\bibinitdelim W\bibinitperiod},
\list{publisher}{1}{%
\strng{namehash}{DWMLP1}
\strng{fullhash}{DWMLP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{DWL83}
\field{extraalpha}{1}
\field{sortinit}{D}
\field{sortinithash}{D}
\field{number}{6}
\field{pages}{487\bibrangedash 496}
\field{title}{Existence of star-products and of formal deformations of the
Poisson Lie algebra of arbitrary symplectic manifolds}
\field{volume}{7}
\field{journaltitle}{Letters in Mathematical Physics}
\field{year}{1983}
\endentry

\entry{lecomte83cotan}{article}{}
\name{author}{2}{}{%
family={De\bibnamedelima Wilde},
familyi={D\bibinitperiod\bibinitdelim W\bibinitperiod},
\list{publisher}{1}{%
\strng{namehash}{DWMLP1}
\strng{fullhash}{DWMLP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{DWL83}
\field{extraalpha}{2}
\field{sortinit}{D}
\field{sortinithash}{D}
\field{number}{3}
\field{pages}{235\bibrangedash 241}
\field{title}{Star-products on cotangent bundles}
\field{volume}{7}
\field{journaltitle}{Letters in Mathematical Physics}
\field{year}{1983}
\endentry

\entry{fedosov94}{article}{}
\name{author}{1}{}{%
given={Boris\bibnamedelima V.},
giveni={B\bibinitperiod\bibinitdelim V\bibinitperiod},
\list{publisher}{1}{%
{Lehigh University}%
\strng{namehash}{FBV1}
\strng{fullhash}{FBV1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Fed94}
\field{sortinit}{F}
\field{sortinithash}{F}
\field{number}{2}
\field{pages}{213\bibrangedash 238}
\field{title}{A simple geometrical construction of deformation
\field{volume}{40}
\field{journaltitle}{Journal of differential geometry}
\field{year}{1994}
\endentry

\entry{Fukaya01}{inproceedings}{}
\name{author}{1}{}{%
\strng{namehash}{FK1}
\strng{fullhash}{FK1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Fuk01}
\field{sortinit}{F}
\field{sortinithash}{F}
\field{booktitle}{{Geometry and Physics of Branes}}
\field{pages}{121\bibrangedash 209}
\field{title}{{Deformation theory, homological algebra and mirror
\field{month}{05}
\field{year}{2001}
\endentry

\entry{getzler09}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{GE1}
\strng{fullhash}{GE1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Get09}
\field{sortinit}{G}
\field{sortinithash}{G}
\field{pages}{271\bibrangedash 301}
\field{title}{Lie theory for nilpotent $\mathrm{L}_\infty$-algebras}
\field{journaltitle}{Annals of mathematics}
\field{year}{2009}
\endentry

\entry{gutt99}{article}{}
\name{author}{2}{}{%
\list{publisher}{1}{%
\strng{namehash}{GSRJ1}
\strng{fullhash}{GSRJ1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{GR99}
\field{sortinit}{G}
\field{sortinithash}{G}
\field{number}{4}
\field{pages}{347\bibrangedash 392}
\field{title}{{Equivalence of star products on a symplectic manifold; an
introduction to Deligne's \v Cech cohomology classes}}
\field{volume}{29}
\field{journaltitle}{Journal of geometry and physics}
\field{year}{1999}
\endentry

\entry{grassi99}{article}{}
\name{author}{1}{}{%
\strng{namehash}{GM1}
\strng{fullhash}{GM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Gra99}
\field{sortinit}{G}
\field{sortinithash}{G}
\field{pages}{49\bibrangedash 66}
\field{title}{{DG} (co)algebras, {DG} {Lie} algebras and
\field{journaltitle}{Seminari di geometria algebrica}
\field{year}{1999}
\endentry

\entry{groenewold46}{incollection}{}
\name{author}{1}{}{%
given={Hilbrand\bibnamedelima Johannes},
giveni={H\bibinitperiod\bibinitdelim J\bibinitperiod},
\list{publisher}{1}{%
\strng{namehash}{GHJ1}
\strng{fullhash}{GHJ1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Gro46}
\field{sortinit}{G}
\field{sortinithash}{G}
\field{booktitle}{On the principles of elementary quantum mechanics}
\field{pages}{1\bibrangedash 56}
\field{title}{On the principles of elementary quantum mechanics}
\field{year}{1946}
\endentry

\entry{gutt01notes}{article}{}
\name{author}{1}{}{%
\strng{namehash}{GS1}
\strng{fullhash}{GS1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Gut01}
\field{sortinit}{G}
\field{sortinithash}{G}
\field{pages}{171\bibrangedash 220}
\field{title}{Deformation Quantisation of {Poisson} Manifolds}
\field{volume}{17}
\field{journaltitle}{Geom. Topol. Monogr}
\field{year}{2001}
\endentry

\entry{hall}{book}{}
\name{author}{1}{}{%
given={Brian\bibnamedelima C.},
giveni={B\bibinitperiod\bibinitdelim C\bibinitperiod},
\list{publisher}{1}{%
\strng{namehash}{HBC1}
\strng{fullhash}{HBC1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Hal13}
\field{sortinit}{H}
\field{sortinithash}{H}
\field{series}{Graduate Texts in Mathematics}
\field{title}{Quantum Theory for Mathematicians}
\field{volume}{267}
\field{year}{2013}
\endentry

\entry{hinich03}{article}{}
\name{author}{1}{}{%
\strng{namehash}{HV1}
\strng{fullhash}{HV1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Hin03}
\field{sortinit}{H}
\field{sortinithash}{H}
\field{pages}{591\bibrangedash 614}
\field{title}{Tamarkins proof of Kontsevich formality theorem}
\field{volume}{15}
\field{journaltitle}{Forum Mathematicum}
\field{year}{2003}
\endentry

\entry{HKR}{article}{}
\name{author}{3}{}{%
\strng{namehash}{HGKBRA1}
\strng{fullhash}{HGKBRA1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{HKR62}
\field{sortinit}{H}
\field{sortinithash}{H}
\field{number}{3}
\field{pages}{383\bibrangedash 408}
\field{title}{Differential forms on regular affine algebras}
\field{volume}{102}
\field{journaltitle}{Transactions of the American Mathematical Society}
\field{year}{1962}
\endentry

\entry{humphreys}{book}{}
\name{author}{1}{}{%
given={James\bibnamedelima E.},
giveni={J\bibinitperiod\bibinitdelim E\bibinitperiod},
\list{publisher}{1}{%
\strng{namehash}{HJE1}
\strng{fullhash}{HJE1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Hum80}
\field{sortinit}{H}
\field{sortinithash}{H}
\field{series}{Graduate Texts in Mathematics}
\field{title}{Introduction to {Lie} algebras and representation theory}
\field{volume}{9}
\field{year}{1980}
\endentry

\entry{huybrechts05}{book}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{HD1}
\strng{fullhash}{HD1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Huy05}
\field{sortinit}{H}
\field{sortinithash}{H}
\field{title}{Complex Geometry: An Introduction}
\field{year}{2005}
\endentry

\entry{jurvco19}{article}{}
\name{author}{4}{}{%
\list{publisher}{1}{%
{Wiley Online Library}%
\strng{namehash}{JB+1}
\strng{fullhash}{JBRLSCWM1}
\field{shorthand}{Jur19}
\field{label}{Jur19}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Jur19}
\field{sortinit}{J}
\field{sortinithash}{J}
\field{number}{7}
\field{pages}{1900025}
\field{title}{$\mathrm{L}_\infty$-Algebras of Classical Field Theories and
the {Batalin--Vilkovisky} Formalism}
\field{volume}{67}
\field{journaltitle}{Fortschritte der Physik}
\field{year}{2019}
\endentry

\entry{keller03}{unpublished}{}
\name{author}{1}{}{%
\strng{namehash}{KB1}
\strng{fullhash}{KB1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Kel03}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{note}{Preprint}
\field{title}{Notes for an Introduction to {K}ontsevich's Quantization
\verb{url}
\verb https://webusers.imj-prg.fr/~bernhard.keller/publ/emalca.pdf
\endverb
\field{year}{2003}
\endentry

\entry{kontsevich01alg}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{KM1}
\strng{fullhash}{KM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Kon01}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{number}{3}
\field{pages}{271\bibrangedash 294}
\field{title}{Deformation quantization of algebraic varieties}
\field{volume}{56}
\field{journaltitle}{Letters in Mathematical Physics}
\field{year}{2001}
\endentry

\entry{kon97}{article}{}
\name{author}{1}{}{%
\strng{namehash}{KM1}
\strng{fullhash}{KM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Kon97}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{title}{Deformation Quantization of {Poisson} Manifolds}
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/q-alg/9709040}{\normalfont{\texttt{q-alg/9709040}}}}
\field{year}{1997}
\endentry

\entry{kon97conjecture}{article}{}
\name{author}{1}{}{%
\strng{namehash}{KM1}
\strng{fullhash}{KM1}
\field{shorthand}{Kon97a}
\field{label}{Kon97a}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Kon97a}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{pages}{139\bibrangedash 156}
\field{title}{Formality Conjecture}
\field{volume}{128}
\field{journaltitle}{Deformation theory and symplectic geometry}
\field{year}{1997}
\endentry

\entry{kontsevich99op}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{KM1}
\strng{fullhash}{KM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Kon99}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{number}{1}
\field{pages}{35\bibrangedash 72}
\field{title}{Operads and motives in deformation quantization}
\field{volume}{48}
\field{journaltitle}{Letters in Mathematical Physics}
\field{year}{1999}
\endentry

\entry{kontsevich02defo}{unpublished}{}
\name{author}{2}{}{%
\strng{namehash}{KMSY1}
\strng{fullhash}{KMSY1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{KS02}
\field{sortinit}{K}
\field{sortinithash}{K}
\field{note}{Book in preparation}
\field{title}{Deformation theory}
\field{year}{2002}
\endentry

\entry{manetti05}{article}{}
\name{author}{1}{}{%
\strng{namehash}{MM1}
\strng{fullhash}{MM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Man05}
\field{sortinit}{M}
\field{sortinithash}{M}
\field{title}{Deformation theory via differential graded {L}ie algebras}
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/math/0507284}{\normalfont{\texttt{math/0507284}}}}
\field{year}{2005}
\endentry

\entry{moyal}{inproceedings}{}
\name{author}{1}{}{%
given={Jos{\'e}\bibnamedelima E.},
giveni={J\bibinitperiod\bibinitdelim E\bibinitperiod},
\list{organization}{1}{%
{Cambridge University Press}%
\strng{namehash}{MJE1}
\strng{fullhash}{MJE1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Moy49}
\field{sortinit}{M}
\field{sortinithash}{M}
\field{booktitle}{Mathematical Proceedings of the Cambridge Philosophical
\field{number}{1}
\field{pages}{99\bibrangedash 124}
\field{title}{Quantum mechanics as a statistical theory}
\field{volume}{45}
\field{year}{1949}
\endentry

\entry{markl02}{book}{}
\name{author}{3}{}{%
given={James\bibnamedelima D.},
giveni={J\bibinitperiod\bibinitdelim D\bibinitperiod},
\list{publisher}{1}{%
{American Mathematical Society Providence, RI}%
\strng{namehash}{MMSSSJD1}
\strng{fullhash}{MMSSSJD1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{MSS02}
\field{sortinit}{M}
\field{sortinithash}{M}
\field{title}{Operads in algebra, topology and physics}
\field{volume}{96}
\field{year}{2002}
\endentry

\entry{schlessinger68}{article}{}
\name{author}{1}{}{%
\strng{namehash}{SM1}
\strng{fullhash}{SM1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Sch68}
\field{sortinit}{S}
\field{sortinithash}{S}
\field{number}{2}
\field{pages}{208\bibrangedash 222}
\field{title}{Functors of Artin rings}
\field{volume}{130}
\field{journaltitle}{Transactions of the American Mathematical Society}
\field{year}{1968}
\endentry

\entry{schlessinger12}{article}{}
\name{author}{2}{}{%
\strng{namehash}{SMSJ1}
\strng{fullhash}{SMSJ1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{SS12}
\field{sortinit}{S}
\field{sortinithash}{S}
\field{title}{Deformation theory and rational homotopy type}
\field{journaltitle}{\href{https://arxiv.org/pdf/1211.1647.pdf}{\normalfont{\texttt{arXiv:1211.1647}}}}
\field{year}{2012}
\endentry

\entry{takhtajan}{book}{}
\name{author}{1}{}{%
given={Leon\bibnamedelima A.},
giveni={L\bibinitperiod\bibinitdelim A\bibinitperiod},
\list{publisher}{1}{%
{American Mathematical Society}%
\strng{namehash}{TLA1}
\strng{fullhash}{TLA1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Tak08}
\field{sortinit}{T}
\field{sortinithash}{T}
\field{series}{Graduate Studies in Mathematics}
\field{title}{Quantum mechanics for mathematicians}
\field{volume}{95}
\field{year}{2008}
\endentry

\entry{tamarkin98}{article}{}
\name{author}{1}{}{%
given={Dmitry\bibnamedelima E.},
giveni={D\bibinitperiod\bibinitdelim E\bibinitperiod},
\strng{namehash}{TDE1}
\strng{fullhash}{TDE1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Tam98}
\field{sortinit}{T}
\field{sortinithash}{T}
\field{title}{Another Proof of {M}.\ {Kontsevich} Formality Theorem}
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/math/9803025}{\normalfont{\texttt{math/9803025}}}}
\field{year}{1998}
\endentry

\entry{konwork}{inproceedings}{}
\name{author}{1}{}{%
given={Clifford\bibnamedelima Henry},
giveni={C\bibinitperiod\bibinitdelim H\bibinitperiod},
\strng{namehash}{TCH1}
\strng{fullhash}{TCH1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Tau98}
\field{sortinit}{T}
\field{sortinithash}{T}
\field{booktitle}{Documenta Mathematica, Extra volume ICM 1998}
\field{pages}{119\bibrangedash 126}
\field{title}{The Work of Maxim Kontsevich}
\field{volume}{I}
\field{year}{1998}
\endentry

\entry{vey75}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{VJ1}
\strng{fullhash}{VJ1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Vey75}
\field{sortinit}{V}
\field{sortinithash}{V}
\field{number}{1}
\field{pages}{421\bibrangedash 454}
\field{title}{{D\'e}formation du crochet de {Poisson} sur une
vari{\'e}t{\'e} symplectique}
\field{volume}{50}
\field{journaltitle}{Commentarii Mathematici Helvetici}
\field{year}{1975}
\endentry

\entry{waldmann05}{article}{}
\name{author}{1}{}{%
\strng{namehash}{WS1}
\strng{fullhash}{WS1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Wal05}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{number}{01}
\field{pages}{15\bibrangedash 75}
\field{title}{States and representations in deformation quantization}
\field{volume}{17}
\field{journaltitle}{Reviews in Mathematical Physics}
\field{year}{2005}
\endentry

\entry{weibel}{book}{}
\name{author}{1}{}{%
given={Charles\bibnamedelima A.},
giveni={C\bibinitperiod\bibinitdelim A\bibinitperiod},
\list{publisher}{1}{%
{Cambridge university press}%
\strng{namehash}{WCA1}
\strng{fullhash}{WCA1}
\field{shorthand}{Weib95}
\field{label}{Weib95}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Weib95}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{title}{An introduction to homological algebra}
\field{year}{1995}
\endentry

\entry{weinstein95}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
{[Paris: Societe mathematique de France}%
\strng{namehash}{WA1}
\strng{fullhash}{WA1}
\field{shorthand}{Wein95}
\field{label}{Wein95}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Wein95}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{pages}{389\bibrangedash 409}
\field{title}{Deformation quantization}
\field{volume}{227}
\field{journaltitle}{Ast{\'e}risque}
\field{year}{1995}
\endentry

\entry{weyl}{article}{}
\name{author}{1}{}{%
\strng{namehash}{WH1}
\strng{fullhash}{WH1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Wey27}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{number}{1}
\field{pages}{1\bibrangedash 46}
\field{title}{Quantenmechanik und gruppentheorie}
\field{volume}{46}
\field{journaltitle}{Zeitschrift f\"{u}r Physik}
\field{year}{1927}
\endentry

\entry{wigner}{article}{}
\name{author}{1}{}{%
\strng{namehash}{WE1}
\strng{fullhash}{WE1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Wig32}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{number}{5}
\field{pages}{749\bibrangedash 759}
\field{title}{On the Quantum Correction For Thermodynamic Equilibrium}
\field{volume}{40}
\field{journaltitle}{Physical Review}
\field{year}{1932}
\endentry

\entry{weinstein97}{article}{}
\name{author}{2}{}{%
\strng{namehash}{WAXP1}
\strng{fullhash}{WAXP1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{WX97}
\field{sortinit}{W}
\field{sortinithash}{W}
\field{title}{Hochschild cohomology and characteristic classes for
\field{journaltitle}{arXiv
\href{https://arxiv.org/abs/q-alg/9709043}{\normalfont{\texttt{q-alg/9709043}}}}
\field{year}{1997}
\endentry

\entry{yekutieli05}{article}{}
\name{author}{1}{}{%
\list{publisher}{1}{%
\strng{namehash}{YA1}
\strng{fullhash}{YA1}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{labelalpha}{Yek05}
\field{sortinit}{Y}
\field{sortinithash}{Y}
\field{number}{1}
\field{pages}{383\bibrangedash 432}
\field{title}{Deformation quantization in algebraic geometry}
\field{volume}{198}
\field{journaltitle}{Advances in Mathematics}
\field{year}{2005}
\endentry
\enddatalist
\endinput

\end{document}


# Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum

Ygal Y. Klein  0009-0004-1914-5821<EMAIL_ADDRESS>Boaz Katz
0000-0003-0584-2920 Dept. of Particle Phys. & Astrophys., Weizmann Institute
of Science, Rehovot 76100, Israel

###### Abstract

The gradual evolution of the restricted hierarchical three body problem is
analyzed analytically, focusing on conditions of Kozai-Lidov Cycles that may
lead to orbital flips from prograde to retrograde motion due to the octupole
(third order) term which are associated with extremely high eccentricities. We
revisit the approach described by Katz, Dong and Malhotra (Phys. Rev. Lett.
107, 181101 (2011)) and show that for most initial conditions, to an excellent
approximation, the analytic derivation can be greatly simplified and reduces
to a simple pendulum model allowing an explicit flip criterion. The resulting
flip criterion is much simpler than the previous one but the latter is still
needed in a small fraction of phase space. We identify a logical error in the
earlier derivation but clarify why it does not affect the final results.

††preprint: APS/123-QED

## Introduction

The dynamics of the restricted, hierarchical three-body problem (a test
particle orbiting a central mass on a Keplerian orbit which is perturbed by a
distant mass) involves oscillations of eccentricity and inclination on a
timescale much longer than the orbital periods. Expanding the perturbing
potential up to leading order in the small parameter of the ratio of semi
major axes (quadrupole order) results in periodic oscillations which have been
solved analytically (Kozai-Lidov Cycles, KLCs) [1, 2]111See recent historical
overview including earlier relevant work by von Zeipel [30] in Ito and Ohtsuka
[31].. The octupole term allows for the generation of extremely high
eccentricities of the inner orbit and the possibility of a ”flip”, i.e a
change from prograde to retrograde orbits, for perturbers on an eccentric
orbit (Eccentric Kozai Lidov, EKL) [4, 5, 6, 7, 8, 9, 10, 11] (for a review
see [12]). Very close approaches of the inner binary members due to the EKL
have been argued to play a major role in a wide range of astrophysical
phenomena, including satellites, planets, and black hole mergers [13, 14, 15,
16, 17, 18, 19, 20].

An analytical approximation for the EKL was derived in [4] by averaging the
secular equations of motion over KLCs obtaining effective equations for the
evolution of slow variables allowing for an analytical flip criterion to be
derived. Recently, [21] and [11] studied analytically the EKL dynamics through
perturbative methods and mentioned a ”pendulum”-like structure in the
resulting maps. Additionally, [22] constructed an analytic approach for the
secular descent and the timescale of the octupole term effect.

In this Letter the analysis of [4] is revisited. It is shown that for most
initial conditions, to an excellent approximation, the effective equations
reduce to those of a simple pendulum clarifying the dynamics and allowing the
derivation of a flip criterion which is much simpler than that of [4]. For a
small region of phase space the complex analytical criterion in [4] gives
better reconstructions of the numerical results. A logical error in the
derivation of [4] is identified and resolved.

## Coordinate System

Consider a test particle orbiting a central mass $M$ on an inner orbit with
semimajor axis $a$ and eccentricity $e$ and a distant mass $m_{per}$ on an
outer orbit with $a_{per},e_{per}$ where $a/a_{per}\ll 1$. Following [4] we
align the z axis along the direction of the total angular momentum vector
(which is the angular momentum vector of the outer orbit since the inner orbit
is of a test particle). The x axis is directed towards the pericenter of the
(constant) outer orbit. The dynamics of the test particle can be parameterized
by two dimensionless orthogonal vectors $\mathbf{j}=\mathbf{J}/\sqrt{GMa}$,
where $\mathbf{J}$ is the specific angular momentum vector, and $\mathbf{e}$ a
vector pointing in the direction of the pericenter with magnitude $e$. Using
the variables $i_{e}$ and $\Omega_{e}$ as in [4] the eccentricity vector
$\mathbf{e}$ is given by $\mathbf{e}=e\left(\sin i_{e}\cos\Omega_{e},\sin
i_{e}\sin\Omega_{e},\cos i_{e}\right)$. We note that the usage of $\Omega_{e}$
as one of the canonical variables was suggested in [23] (denoted $\theta_{a}$
therein) and the connection between $\Omega_{e}$ and the critical argument
studied in [10] ($\sigma_{1}$ therein) is discussed in the appendix of [10].

Expanding the perturbing potential to the third order term in the small
parameter $a/a_{per}$ (the octupole term) and averaging over the two orbits
results with the following potential (e.g, [4, 24, 19, 25, 26]):

$\Phi_{\text{per}}=\Phi_{0}\left(\phi_{\text{quad}}+\epsilon_{\text{oct}}\phi_{\text{oct}}\right)$
(1)

where

$\phi_{\text{quad}}=\frac{3}{4}\left(\frac{1}{2}j_{z}^{2}+e^{2}-\frac{5}{2}e_{z}^{2}-\frac{1}{6}\right),$
(2)
$\phi_{\text{oct}}=\frac{75}{64}\left(e_{x}\left(\frac{1}{5}-\frac{8}{5}e^{2}+7e_{z}^{2}-j_{z}^{2}\right)-2e_{z}j_{x}j_{z}\right),$
(3)
$\Phi_{0}=\frac{Gm_{per}a^{2}}{a_{per}^{3}\left(1-e_{per}^{2}\right)^{\frac{3}{2}}},\,\,\,\,\,\,\,\epsilon_{\text{oct}}=\frac{a}{a_{per}}\frac{e_{per}}{1-e_{per}^{2}}$
(4)

with $\Phi_{0}$ and $\epsilon_{\text{oct}}$ constant. Time and its derivatives
are expressed in units of the secular timescale

$t_{sec}=\sqrt{GMa}/\Phi_{0}$ (5)

using $\tau\equiv t/t_{sec}$.

## Slow variables

When $\epsilon_{\text{oct}}=0$ (the periodic analytically solved KLCs) the
perturbing potential is axisymmetric (with respect to the z axis) admitting a
constant of the motion, $j_{z}$, which limits the eccentricity through the
constraint $j>j_{z}$. KLCs are classified by the values of the constants
$j_{z}$ and

$\displaystyle C_{K}$
$\displaystyle=\frac{4}{3}\phi_{\text{quad}}+\frac{1}{6}-\frac{1}{2}j^{2}_{z}$
(6) $\displaystyle=e^{2}-\frac{5}{2}e^{2}_{z}.$ (7)

When $C_{K}<0$, the argument of pericenter of the inner orbit, $\omega$,
librates around $\frac{\pi}{2}$ or $-\frac{\pi}{2}$ (librating cycles), and
when $C_{K}>0$, it goes through all values $\left[0,2\pi\right]$ (rotating
cycles).

In the problem we study here with $\epsilon_{\text{oct}}>0$, $C_{K}$ and
$j_{z}$ are no longer constant. The three slow variables $j_{z},C_{K}$ and
$\Omega_{e}$ change slowly on a timescale of $\sim
t_{sec}/\sqrt{\epsilon_{\text{oct}}}$ (see Eqs. 4-5) [27]. As in [4] we focus
on the regime of high eccentricity, i.e $\left|j_{z}\right|\ll 1$.

## Simple Pendulum

Up to the leading order in $j_{z}$ and $\epsilon_{\text{oct}}$ and averaging
over rotating KLCs (librating KLCs with $\left|j_{z}\right|\ll 1$ accumulate
change in $j_{z}$ on a much longer timescale [4]), the averaged equations for
the long term evolution of $\Omega_{e}$ and $j_{z}$ are [4]

$\displaystyle\dot{\Omega}_{e}$
$\displaystyle=\left<f_{\Omega}\right>j_{z},\quad$ $\displaystyle\dot{\phi}$
$\displaystyle=\frac{1}{L}v$ (8) $\displaystyle\dot{j}_{z}$
$\displaystyle=-\epsilon_{\text{oct}}\left<f_{j}\right>\sin\Omega_{e},\quad$
$\displaystyle\dot{v}$ $\displaystyle=-g\sin\phi$

where

$\displaystyle\left\langle f_{\Omega}\right\rangle$
$\displaystyle=\frac{6E\left(x\right)-3K\left(x\right)}{4K\left(x\right)},$
(9) $\displaystyle\left\langle f_{j}\right\rangle$
$\displaystyle=\frac{15\pi}{128\sqrt{10}}\frac{1}{K\left(x\right)}\left(4-11C_{K}\right)\sqrt{6+4C_{K}},$
$\displaystyle x$ $\displaystyle=3\frac{1-C_{K}}{3+2C_{K}},$

and $K\left(m\right)$ and $E\left(m\right)$ are the complete elliptic
functions of the first and second kind, respectively. For comparison, the
equations of motion of a simple pendulum with angle $\phi$ and velocity $v$
along with constants $g$ and $L$ are provided on the right side of Equations
8. Since $\Phi_{\text{per}}$ is constant and $\phi_{\text{oct}}$ and $j_{z}$
are small - $C_{K}$ and therefore $\left\langle
f_{\Omega}\right\rangle,\left\langle f_{j}\right\rangle$ are approximately
constant (see discussion of exceptions below). By comparing the structures of
the two sides of Equations 8 we see that the averaged equations for
$\Omega_{e}$ and $j_{z}$ are equivalent to those of a simple pendulum.

The precise correspondence to a simple pendulum depends on the sign of the
product $\left\langle f_{\Omega}\right\rangle\left\langle f_{j}\right\rangle$
which in turn depends on the initial conditions through $C^{0}_{K}$ having
three regions bounded by the zero crossing of $\left<f_{\Omega}\right>$ at
$C^{0}_{K}\approx 0.112$ and $\left<f_{j}\right>$ at $C^{0}_{K}=\frac{4}{11}$.
The angle $\phi$ of the pendulum is given by the following:
For $0.112\lesssim C_{K}^{0}<\frac{4}{11}$

$\left<f_{j}\right>\left<f_{\Omega}\right>>0\Rightarrow\phi=\Omega_{e}$ (10)

and otherwise

$\left<f_{j}\right>\left<f_{\Omega}\right><0\Rightarrow\phi=\Omega_{e}+\pi.$
(11)

The velocity of the pendulum is $\left(+j_{z}\right)$ for $C^{0}_{K}\gtrsim
0.112$ where $\left<f_{\Omega}\right>$ is positive and $\left(-j_{z}\right)$
otherwise.

An example of a numerical integration of the full double averaged equations
(Equations 4 in [4], for explicit terms see Equations C3 in [26])) compared
with the solution of Equations 8 with the approximation of constants
$\left\langle f_{\Omega}\right\rangle,\left\langle f_{j}\right\rangle$
(evaluated at $C^{0}_{K}$) is shown in Figure 1. As can be seen - for the
example shown - the long term evolution of $j_{z}$ is successfully
approximated by equations of a simple pendulum with velocity
$\left(-j_{z}\right)$.

Figure 1: Results of a numeric integration of the double averaged equations
(blue) along with the result of a simple pendulum (Equations 8, red). The
values of the initial conditions and $\epsilon_{\text{oct}}$ are shown above
the plots. Upper panel: $j_{z}$ as a function of (normalized) time. The two
green horizontal lines are the maximum and minimum values of $j_{z}$ of the
simple pendulum (Equation 12). Lower panel: $j_{z}$ vs. $\Omega_{e}$.

## flip criterion

A flip - zero crossing of $j_{z}$ \- is equivalent to the velocity of the
pendulum changing sign which occurs only if the pendulum is librating. Given
$\epsilon_{\text{oct}}$ and initial values $C^{0}_{K},\Omega^{0}_{e}$ the
maximal value of $j^{0}_{z}$ where a flip occurs is given by

$j^{0}_{z,\text{max}}\left(C^{0}_{K},\Omega^{0}_{e}\right)=\sqrt{2\epsilon_{\text{oct}}\left|\frac{\left\langle
f_{j}\right\rangle}{\left\langle
f_{\Omega}\right\rangle}\right|\left(1\pm\cos\Omega_{e}^{0}\right)}$ (12)

where the $\pm$ sign is positive if $\left\langle
f_{j}\right\rangle\left\langle f_{\Omega}\right\rangle>0$ and negative
otherwise. A global flip criterion is thus

$j^{0}_{z,\text{max}}\left(C^{0}_{K}\right)=2\sqrt{\epsilon_{\text{oct}}\left|\frac{\left\langle
f_{j}\right\rangle}{\left\langle f_{\Omega}\right\rangle}\right|}$ (13)

or equivalently

$\epsilon_{\text{oct}}>\frac{1}{4}\left|\frac{\left\langle
f_{\Omega}\right\rangle}{\left\langle
f_{j}\right\rangle}\right|\left(j_{z}^{0}\right)^{2}.$ (14)

A comparison between a numerical flip map and the analytic prediction of
Equations 13 and 9 as a function of $j^{0}_{z}$ and $C^{0}_{K}$ is shown in
Figure 2 for $\epsilon_{\text{oct}}=0.001$. Each point represents a set of
numerical simulations (up to $\tau=10/\epsilon_{\text{oct}}$) with the same
$j^{0}_{z}$ and $C^{0}_{K}$, $\omega^{0}=0$ and a range of values for the
longitude of ascending node, $\Omega^{0}$ (which is equal to $\Omega_{e}$ when
$\omega=0$) equally spaced by $10^{\circ}$ between $0$ and $360^{\circ}$. The
values of $e^{0}$ and $i^{0}$ are set by the choice of the other parameters. A
point is marked red if for some $\Omega^{0}$ a flip occurred and blue
otherwise. A black line marks the analytic prediction of Equations 13 and 9.
As can be seen, the black line follows the border between red and blue points
with some exceptions discussed below.

Figure 2: Parameter space of orbital flips from the solution of the double
averaged secular equations (up to $\tau=10/\epsilon_{\text{oct}}$). Each point
represents $36$ numerical integrations with $\omega^{0}=0$ and different
$\Omega^{0}$ scanned between $0$ and $2\pi$. Red points mark integrations
where $j_{z}$ crossed zero during the evolution for some $\Omega^{0}$ and blue
points mark those where $j_{z}$ kept its sign for all $\Omega^{0}$. The black
line is the analytic maximal value of $j^{0}_{z}$ allowing a flip obtained by
the simple pendulum model (Equations 13 and 9).

The maximal and minimal values of $j_{z}$ obtained in $391$ numerical
simulations with $j^{0}_{z}=0$ and randomly chosen initial conditions
(uniformly distributed in $e_{x},e_{y}$ and $j_{x}$) is plotted using black
crosses in Figure 3 for $\epsilon_{\text{oct}}=0.001$. The maximal deviation
from $j^{0}_{z}=0$, $j^{\text{max}}_{z}$, represents the maximal $j^{0}_{z}$
that lead to a flip assuming $C_{K}$ is constant (see below discussion of
changes in $C_{K}$). The black line in Figure 3 comes from Equations 13 and 9
and is identical to the black line in Figure 2. As can be seen, the black line
agrees with the envelope of the numerically available $j^{\text{max}}_{z}$.
The open circles in Figure 3 denote the analytical predicted values using
Equations 12 and 9222In contrast to the maximal value of $j^{0}_{z}$ allowing
a flip, for the maximal and minimal values of $j_{z}$ starting from
$j^{0}_{z}=0$ the $\pm$ sign of Equation 12 is positive if $\left\langle
f_{j}\right\rangle\left\langle f_{\Omega}\right\rangle<0$ and negative
otherwise. As can be seen, the predictions of Equation 12 for different values
of $\Omega^{0}_{e}$ agree with the numerical results to an excellent
approximation.

As can be seen in Equation 13 and in the black lines in Figures 2-3 at
$C^{0}_{K}\approx 0.112$ where $\left\langle f_{\Omega}\right\rangle=0$,
$j^{\text{max}}_{z}$ diverges. At this point, $\Omega_{e}$ is constant so the
accumulation in $j_{z}$ continues indefinitely [4]. The numerical results do
have a significant increase in that vicinity of $C^{0}_{K}$ but the divergence
is saturated and some points deviate from the analytic prediction. A
saturation is expected when taking into account the small change in $C_{K}$
during the evolution allowing $\left\langle f_{\Omega}\right\rangle$ to
deviate from zero. The maximal (and minimal) values of $j_{z}$ starting at
$j^{0}_{z}=0$ based on Equations 14 and 15 from [4] is shown in Figure 3 using
a magenta line. As can be seen, the magenta line does not diverge and agrees
with the location and value of the maximal saturation of $j_{z}$. Note that
when $C_{K}$ changes, the location of this maximum does not correspond to the
location in the flip map and requires a (similar) separate criterion.

Figure 3: $j^{\text{max}}_{z}$ and $j^{\text{min}}_{z}$ vs. $C^{0}_{K}$ when
$j^{0}_{z}=0$ and all other components randomly chosen (uniformly distributed
in $e_{x},e_{y}$ and $j_{x}$). The results of direct numerical integrations of
the double averaged equations (up to $\tau=10/\epsilon_{\text{oct}}$) are
shown using black crosses. The analytic results of the simple pendulum model
are marked with open red circles (through Equation 12) and a black line
(envelope, Equations 13 and 9). The expected envelope in the (more
complicated) analytic model of [4] (using Equations 14 and 15 therein) is
shown with a magenta line for $C_{K}<\frac{4}{11}$.

An additional region of initial conditions where $\left\langle
f_{\Omega}\right\rangle$ and $\left\langle f_{j}\right\rangle$ can
significantly change due to the small change in $C_{K}$ is around
$C^{0}_{K}\ll 1$ which is the focus of the analysis of [4]. In this region the
small change in $C_{K}$ can be of the order of $C^{0}_{K}$ itself. In Figures
2 and 3 where $\epsilon_{\text{oct}}=0.001$ the deviation is not apparent. In
Figure 4 the dependence of the maximal available value of $j^{\text{max}}_{z}$
(starting from $j^{0}_{z}=0$) on $\epsilon_{\text{oct}}$ is presented for low
and high values of $C^{0}_{K}$, $C^{0}_{K}=0.05$ and $C^{0}_{K}=0.6$
respectively. As can be seen, at $C^{0}_{K}=0.05$ and for high values of
$\epsilon_{\text{oct}}$ the numerical results (solid black line) deviate from
the simple pendulum model (solid red line,
$\propto\epsilon^{\frac{1}{2}}_{\text{oct}}$). The prediction of [4] (solid
magenta) agrees with the numerical results to much higher values of
$\epsilon_{\text{oct}}$. In contrast, for the high value of $C^{0}_{K}=0.6$
shown (dashed lines) the numerical results follow the simple pendulum
dependence across a wide range of $\epsilon_{\text{oct}}$ values.

Figure 4: The maximal attainable value of $j^{\text{max}}_{z}$ vs.
$\epsilon_{\text{oct}}$ for initial conditions with $j^{0}_{z}=0$ for two
values of $C^{0}_{K}$ ($C^{0}_{K}=0.05$ solid lines and $C^{0}_{K}=0.6$ dashed
lines). Shown are results from a numerical solution of the double averaged
equations using black lines. Red lines mark the prediction of the simple
pendulum model (Equations 13 and 9,
$\propto\epsilon^{\frac{1}{2}}_{\text{oct}}$). For $C^{0}_{K}=0.05$ the
analytic prediction of [4] (using Equations 14 and 15 therein) is plotted in
magenta.

## Discussion

The simple pendulum model is valid for most initial conditions because
$C_{K}$, $\left\langle f_{\Omega}\right\rangle$ and $\left\langle
f_{j}\right\rangle$ are constant for most values of $C^{0}_{K}$ (see Equation
7 and discussion below). There are two small regions of phase space where the
approximation of constant $C_{K}$ fails to reconstruct the dynamics: (i) At
the vicinity of $C_{K}\approx 0.112$ where $\left\langle
f_{\Omega}\right\rangle$ approaches zero the simple pendulum model predicts
diverging change in $j_{z}$. (ii) For $C^{0}_{K}\ll 1$ where $C_{K}$ can
change by the order of itself. As shown, the more complicated analytic
solution constructed in [4] manages to approximately take into account the
change in $C_{K}$ in these cases.

Examining the derivation of the change in $C_{K}$ presented in [4] reveals a
logical error that appears to contradict the successful agreement with
numerical results. The change in $C_{K}$ is approximated to be equal to the
change in $\left(-\frac{1}{2}j^{2}_{z}\right)$ based on approximating
$\phi_{\text{quad}}$ as constant (in Equation 7). This is not justified since
$\phi_{\text{quad}}$ is constant only up to $\sim\epsilon_{\text{oct}}$ while
$j^{2}_{z}$ is comparable or smaller. A closer inspection reveals the reason
why this error does not affect the results. For low enough values of
$e_{\text{min}}$ (and $\left|j_{z}\right|\ll 1$) - applicable for the two
regions where $\left\langle f_{\Omega}\right\rangle$ and $\left\langle
f_{j}\right\rangle$ change - $\phi_{\text{oct}}$ (Equation 3) is much smaller
than $1$ throughout most of the time during each KLC. The small time spent at
high $e$ where $\phi_{\text{oct}}\sim 1$ is not sufficient for accumulation of
a significant change in $j_{z}$ and $\Omega_{e}$. In contrast, for regions of
$e_{\text{min}}\sim 1$ ($C^{0}_{K}\sim 1$), the change in $C_{K}$ is
significantly different from the change in
$\left(-\frac{1}{2}j^{2}_{z}\right)$ but in these regions it is not important
since the functions $\left\langle f_{\Omega}\right\rangle,\left\langle
f_{j}\right\rangle$ do not change significantly.

Finally, we note that beyond a simpler and more intuitive analytical model for
EKL, the approach presented in this Letter allows an extension [29] to the
analytic solution that includes corrections to the double averaging that are
important in the presence of a massive perturber (e.g [26, 25]).

We thank Smadar Naoz, Scott Tremaine and Chris Hamilton for useful
discussions.

## References

  * Kozai [1962] Y. Kozai, Secular perturbations of asteroids with high inclination and eccentricity, The Astronomical Journal 67, 591 (1962).
  * Lidov [1962] M. Lidov, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies, Planetary and Space Science 9, 719 (1962).
  * Note [1] See recent historical overview including earlier relevant work by von Zeipel [30] in Ito and Ohtsuka [31].
  * Katz  _et al._ [2011] B. Katz, S. Dong, and R. Malhotra, Long-Term Cycling of Kozai-Lidov Cycles: Extreme Eccentricities and Inclinations Excited by a Distant Eccentric Perturber, Phys. Rev. Lett. 107, 181101 (2011), arXiv:1106.3340 [astro-ph.EP] .
  * Naoz  _et al._ [2011] S. Naoz, W. M. Farr, Y. Lithwick, F. A. Rasio, and J. Teyssandier, Hot Jupiters from secular planet-planet interactions, Nature (London) 473, 187 (2011), arXiv:1011.2501 [astro-ph.EP] .
  * Lithwick and Naoz [2011] Y. Lithwick and S. Naoz, The eccentric kozai mechanism for a test particle, The Astrophysical Journal 742, 94 (2011).
  * Naoz  _et al._ [2013] S. Naoz, W. M. Farr, Y. Lithwick, F. A. Rasio, and J. Teyssandier, Secular dynamics in hierarchical three-body systems, Monthly Notices of the Royal Astronomical Society 431, 2155 (2013), https://academic.oup.com/mnras/article-pdf/431/3/2155/4890577/stt302.pdf .
  * Ford  _et al._ [2000] E. B. Ford, B. Kozinsky, and F. A. Rasio, Secular evolution of hierarchical triple star systems, The Astrophysical Journal 535, 385 (2000).
  * Li  _et al._ [2014] G. Li, S. Naoz, B. Kocsis, and A. Loeb, Eccentricity growth and orbit flip in near-coplanar hierarchical three-body systems, The Astrophysical Journal 785, 116 (2014).
  * Lei [2022] H. Lei, A systematic study about orbit flips of test particles caused by eccentric von zeipel–lidov–kozai effects, The Astronomical Journal 163, 214 (2022).
  * Lei, Hanlun and Gong, Yan-Xiang [2022] Lei, Hanlun and Gong, Yan-Xiang, Dynamical essence of the eccentric von zeipel-lidov-kozai effect in restricted hierarchical planetary systems, A&A 665, A62 (2022).
  * Naoz [2016] S. Naoz, The eccentric kozai-lidov effect and its applications, Annual Review of Astronomy and Astrophysics 54, 441 (2016).
  * Naoz  _et al._ [2012] S. Naoz, W. M. Farr, and F. A. Rasio, On the formation of hot jupiters in stellar binaries, The Astrophysical Journal Letters 754, L36 (2012).
  * Teyssandier  _et al._ [2013] J. Teyssandier, S. Naoz, I. Lizarraga, and F. A. Rasio, Extreme orbital evolution from hierarchical secular coupling of two giant planets, The Astrophysical Journal 779, 166 (2013).
  * Stephan  _et al._ [2016] A. P. Stephan, S. Naoz, A. M. Ghez, G. Witzel, B. N. Sitarski, T. Do, and B. Kocsis, Merging binaries in the Galactic Center: the eccentric Kozai–Lidov mechanism with stellar evolution, Monthly Notices of the Royal Astronomical Society 460, 3494 (2016), https://academic.oup.com/mnras/article-pdf/460/4/3494/8117332/stw1220.pdf .
  * Liu and Lai [2018] B. Liu and D. Lai, Black hole and neutron star binary mergers in triple systems: Merger fraction and spin–orbit misalignment, The Astrophysical Journal 863, 68 (2018).
  * Angelo  _et al._ [2022] I. Angelo, S. Naoz, E. Petigura, M. MacDougall, A. P. Stephan, H. Isaacson, and A. W. Howard, Kepler-1656b’s extreme eccentricity: Signature of a gentle giant, The Astronomical Journal 163, 227 (2022).
  * Melchor  _et al._ [2023] D. Melchor, B. Mockler, S. Naoz, S. C. Rose, and E. Ramirez-Ruiz, Tidal disruption events from the combined effects of two-body relaxation and the eccentric kozai–lidov mechanism, The Astrophysical Journal 960, 39 (2023).
  * Petrovich [2015] C. Petrovich, Steady-state planet migration by the kozai–lidov mechanism in stellar binaries, The Astrophysical Journal 799, 27 (2015).
  * Stephan  _et al._ [2021] A. P. Stephan, S. Naoz, and B. S. Gaudi, Giant Planets, Tiny Stars: Producing Short-period Planets around White Dwarfs with the Eccentric Kozai-Lidov Mechanism, Astrophys. J. 922, 4 (2021), arXiv:2010.10534 [astro-ph.EP] .
  * Sidorenko [2018] V. V. Sidorenko, The eccentric Kozai-Lidov effect as a resonance phenomenon, Celestial Mechanics and Dynamical Astronomy 130, 4 (2018), arXiv:1708.06001 [astro-ph.EP] .
  * Weldon  _et al._ [2024] G. C. Weldon, S. Naoz, and B. M. S. Hansen, Analytical models for secular descents in hierarchical triple systems (2024), arXiv:2405.20377 [astro-ph.EP] .
  * Tremaine [2001] S. Tremaine, Canonical Elements for Collision Orbits, Celestial Mechanics and Dynamical Astronomy 79, 231 (2001), arXiv:astro-ph/0012278 [astro-ph] .
  * Liu  _et al._ [2014] B. Liu, D. J. Muñoz, and D. Lai, Suppression of extreme orbital evolution in triple systems with short-range forces, Monthly Notices of the Royal Astronomical Society 447, 747 (2014), https://academic.oup.com/mnras/article-pdf/447/1/747/4907993/stu2396.pdf .
  * Tremaine [2023] S. Tremaine, The Hamiltonian for von Zeipel–Lidov–Kozai oscillations, Monthly Notices of the Royal Astronomical Society 522, 937 (2023), https://academic.oup.com/mnras/article-pdf/522/1/937/50009201/stad1029.pdf .
  * Luo  _et al._ [2016] L. Luo, B. Katz, and S. Dong, Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction, Monthly Notices of the Royal Astronomical Society 458, 3060 (2016), https://academic.oup.com/mnras/article-pdf/458/3/3060/13772044/stw475.pdf .
  * Antognini [2015] J. M. O. Antognini, Timescales of Kozai–Lidov oscillations at quadrupole and octupole order in the test particle limit, Monthly Notices of the Royal Astronomical Society 452, 3610 (2015), https://academic.oup.com/mnras/article-pdf/452/4/3610/18240079/stv1552.pdf .
  * Note [2] In contrast to the maximal value of $j^{0}_{z}$ allowing a flip, for the maximal and minimal values of $j_{z}$ starting from $j^{0}_{z}=0$ the $\pm$ sign of Equation 12 is positive if $\left\langle f_{j}\right\rangle\left\langle f_{\Omega}\right\rangle<0$ and negative otherwise.
  * Klein and Katz [2024] Y. Y. Klein and B. Katz, Eccentric kozai-lidov cycles with a massive perturber are approximately analytically solved, In prep. (2024).
  * von Zeipel [1910] H. von Zeipel, Sur l’application des séries de M. Lindstedt à l’étude du mouvement des comètes périodiques, Astronomische Nachrichten 183, 345 (1910).
  * Ito and Ohtsuka [2019] T. Ito and K. Ohtsuka, The Lidov-Kozai Oscillation and Hugo von Zeipel, Monographs on Environment, Earth and Planets 7, 1 (2019), arXiv:1911.03984 [astro-ph.EP] .