\chapter{Implicit representations of codimension-2 submanifolds}\label{ch:implicit_representations}
In the previous chapter, we reviewed the \emph{explicit} representations of codimension-2 submanifolds as embeddings of a manifold into an ambient manifold. In this chapter, we introduce \emph{implicit} representations via a class of complex-valued functions and study the geometric structures of the space of these implicit representations.

This leads to further study of its relation with the Marsden--Weinstein symplectic structure of the codimension-2 shape space (\autoref{ch:prequantum_bundle}), and its exploitation for numerical simulation of curve dynamcis (\autoref{ch:hidden_degrees}). 


\section{Introduction}
Codimension-1 submanifolds---such as curves on a surface and surfaces in a three dimensional space---can be implicitly represented using a level set function. This representation naturally extends to multi-codimensional submanifolds using multiple level set functions.
Specifically, when the codimension is 2, submanifolds---such as curves in a three dimensional space and surfaces in a four dimensional space---can be expressed as the zero sets of complex-valued functions. 

Such implicit representations via complex functions are useful for studying the geometry and dynamics of submanifolds. 
For instance, prior work has exploited these implicit representations to study the dynamics of space curves, such as curve-shortening flow \cite{MBO2001diffusion} and quantum vortex filaments governed by the Gross-Pitaevskii equation \cite{ogawa2002study,villois2016GrossPitaevskii,jerrard2018leapfrogging}.

In the present and the next chapter, we investigate the space of these implicit representations for codimension-2 submanifolds. In particular, we study its geometric structures related to the Marsden--Weinstein (MW) structure.

As in the previous chapter, let us denote $\cO$ a $\Diff_0(M)$-orbit in the space of codimension-2 submanifolds, represented by embeddings modulo reparametrization. We call $\cO$ the \emph{explicit shape space}.
The implicit representation we define for each element in \(\cO\) is not unique, as multiple complex-valued functions can share the same zero set. This redundancy makes the \emph{implicit shape space} \(\cF_\cO\) a fiber bundle over the explicit shape space \(\cO\). \footnote{Similar to \(\cO\), the space \(\cF_\cO\) is more precisely described as a subset of the space of complex-valued functions, characterized by the orbits under a certain group action.}
Each element \(\psi\in\cF_\cO\) is a complex-valued function over \(M\), while each point \(\gamma\in\cO\) is a codimension-2 submanifold in \(M\). The projection from \(\cF_\cO\) to \(\cO\) is to extract the zero set and the orientation.

This chapter builds a foundation of implicit representations for our further investigation in the subsequent chapters, which leverages their non-uniqueness. In \autoref{ch:prequantum_bundle}, we  study the geometry of the implicit shape space related to the Marsden-Weinstein structure. Specifically, we reveal a prequantum structure on a certain quotient space of the implicit shape space $\cF_\cO$ over the explicit shape space $\cO$. In \autoref{ch:hidden_degrees}, we exploit this redundancy to develop a numerical method for space curve dynamics.

\paragraph{Conventions}
In the last chapter, we introduced the explicit shape space in terms of embeddings modulo reparametrization, and in terms of de Rham currents.  Throughout this and the following chapter, we identify the explicit shape space \(\cX\) of all the codimension-2 submanifolds with its counterpart \(\cD_{\cX}\) in terms of de Rham currents, as well as each \(\Diff_0(M)\)-orbit \(\cO\) with the corresponding currents \(\cD_\cO\), without explicitly restating this identification.

\section{Implicit representations of codimension-2 shapes}\label{sec:implicit_representations}

We define the space $\cF$ of implicit representations of submanifolds as a subset of $C^\infty(M,\CC)$ as follows. 
The space \(\cF\) consists of functions \(\psi\) such that the zero set \( \psi^{-1}(0)\) is nonempty, and the differential \(d\psi|_x : T_x M \to \CC\) is surjective at each \(x \in  \psi^{-1}(0)\). 

Define a fibration \(\Pi \colon \cF \to \cX\) by assigning to each \(\psi \in \cF\) a unique \(\gamma \in \cX\) such that $\im \gamma=\psi^{-1}(0)$, with the orientation of \(\psi\) determined as follows. 
For \(\gamma \in \UEmb(S,M)\) with some \(S\) having \(k\) connected components, there are \(2^{k}\) possible orientations. Among these, we select the unique one satisfying
\begin{align}
    \int_{\partial \Sigma} \Im \frac{d\psi}{\psi} = 2\pi \, \Sigma \cap \gamma(S)
\end{align}
for any oriented \((m-1)\)-dimensional topological disk \(\Sigma\) intersecting transversely with \(\gamma(S)\), where \(\Sigma \cap \gamma(S)\) denotes the signed intersection number between \(\Sigma\) and \(\gamma(S)\).

\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.33\columnwidth}
\includegraphics[width=\textwidth]{images_prequantum/solid_angle_field2.png}
\end{minipage}
\vrule
\begin{minipage}[b]{0.3\columnwidth}
\includegraphics[width=\textwidth]{images_prequantum/seifert1.png}
\end{minipage}
\begin{minipage}[b]{0.3\columnwidth}
\includegraphics[width=\textwidth]{images_prequantum/seifert3.png}
\end{minipage}
\caption{Left: Visualization of an implicit representation \(\psi \in \cF\) on the plane \(z=0\) in \(\RR^3\) for a Borromean ring \(\gamma \in \cX\). Each color represents a phase \ie, the preimage of an element \(s \in \SS^1\) under the phase map \(\phi= \psi/|\psi|\).  
Middle and right: Visualizations of the preimages of two distinct elements in \(\SS^1\) as surfaces bordered by \(\im \gamma = \psi^{-1}(0)\). These preimages have codimension 1, and their intersection has codimension 2, representing \(\im \gamma\).}
\label{fig:solid_angle_theta}
\end{figure}

% 
% by setting $\Pi^{-1} \gamma$ be a collection of functions whose zeros represent the configuration and non-zeros represent the direction of $\gamma$ as follows. We first require $\{\psi=0\}=\gamma(S)$.
% 

% For each $\gamma$, we can take a frame field $(n_1,n_2)$ on the normal bundle $N\gamma(S)$ so that
% % $(\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma, n_1,n_2) \in T_x \gamma(S) \oplus N_x \gamma(S)$ 
% at each zero $x\in \{\psi=0\}=\gamma(S)$ we have $\mu (\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma, n_1,n_2 )>0$ for any local coordinates $(s_1,\ldots,s_n)$ consistent with the orientation of $S$.
% Since $(\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma)$ are lying on the tangent space $T_x \gamma(S)$, the matrix representation of $d\psi|_x$ with respect to the basis \((\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma, n_1,n_2 )\) is given as the singular value decomposition, 
% \begin{align}\label{eq:matrix_dpsi}
%     \on{D\Psi}
%     &=   
%     \begin{pmatrix}
%     \mathbf{0} & \cdots & \mathbf{0} \on{D\Psi_N}
%     \end{pmatrix}
%     \begin{pmatrix}
%         \partial_{s_1}\gamma\\
%         \vdots\\
%         \partial_{s_n}\gamma\\
%         n_1 \\
%         n_2
%     \end{pmatrix}\\
%     &=\on{D\Psi_N}
%     \begin{pmatrix}
%     n_1\\n_2    
%     \end{pmatrix}
% \end{align}
% with the 2 by 2 matrix $\on{D\Psi_N}$ representing $d\psi$ restricted on the normal plane $N_x \gamma(S)$. Among the choices sharing the same configuration, there is a unique $\gamma$ that makes $\det \on{D\Psi_N}>0$ on every zero, for which we set $\Pi \psi\coloneqq \gamma$. 
% 
% 
% The tangent space $T_x M$ at each point $x$ of $\gamma(S)\subset M$ is decomposed into $T_x M=T_x \gamma(S) \oplus N_x \gamma(S)$ and we can take orthonormal frame fields $(t_1, \ldots, t_n, n_1,n_2) \in T_x \gamma(S) \oplus N_x \gamma(S)$ so that $\mu (t_1, \ldots, t_n, n_1,n_2 )=1$ and  $\mu (\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma, n_1,n_2 )>0$ for any oriented local charts.
% At each zero $x$, consider the restriction $\psi_N$ of $\psi$ onto a small normal disc $D_x$ with $D_x\cap \gamma(S)=\{x\}$. On $D_x$, we can take local holomorphic coordinates $z$ and $\bar z$ using $n_1$ and $n_2$. We impose $\psi$ to have the direction consistent with $\gamma$. That is, we require 
% \begin{align}\label{eq:orientation_condition}
%     \frac{\star\left({\psi_N} ^* ( dz\wedge d\bar z)\right)}{\star(dz\wedge d\bar z)}>0
% \end{align}
% at each zero where the Hodge star $\star$ is with respect to the local volume-form $dz\wedge d\bar z$.
% Note that this does not depend on the choice of the field $(n_1,n_2)$ and we did not use a Riemannian structure on $M$ as $n_1$ and $n_2$ do not need to have unit lengths with respect to a metric. \Albert{[The notion of normal plane sounds like metric dependent.]}
% \Albert{[I wonder if there is a more elegant way of defining the orientation for the zeros.  For example, \(\int_{\partial\Sigma} {d\psi\over\psi} = 2\pi i \Sigma\cap \gamma\) for each \(\Sigma\) transversally intersecting with \(\gamma\).]}

 % This condition is equivalent to the condition $\oint_c \lambda_\psi=2\pi$ at each $x\in\{\psi=0\}$ where $c$ is a curve parametrizing a small disc $D_x\subset N_x \gamma(S )$ whose orientation is consistent with the order of $n_1$ then $n_2$ and $ \lambda_\psi$ is the circle differential 1-form, which is locally $d\varphi$ using a local representation of $\psi=|\psi|e^{i\varphi}$. We will properly define  $ \lambda_\psi$ in the next subsection.
 
% \Albert{[There is a lot of metric structure of \(M\) used early on, which is concerning.  Would a change of metric, or even just a diffeomorphism flow, on \(M\) change the fibration \(\cF,\Pi\)?]}\Albert{[Probably only need the zeros of \(\psi\) be regular as the set up for the fibration.  The rest about having normal frame is a consequence of this regularity and a given Riemannian structure on \(M\), instead of part of the definition of the fibration.]}

% \Albert{[What is a natural motivation for considering the \(\SemiD\) group action?][Perhaps motivate that the entire function space \(\cF\) is too large.  Similar to the case for explicit representation where we restrict the space to an orbit of \(\SDiffp\) action, in this implicit representation, we restrict \(\cF\) to a subset characterized as an orbit of the following group.]}

\paragraph{Existence of implicit representations}
The fibration \(\Pi\colon \cF \to \cX\) is not surjective. More precisely, our implicit representation of shapes is limited to codimension-2 submanifolds that are exact in homology; that is, each \(\gamma\) must be the boundary of an oriented codimension-1 submanifold \(\Sigma\).

In this case, we can construct a map \(\theta \colon M \setminus \im\gamma \to \TT^1\) as the solution to the Dirichlet boundary problem:
\begin{align}
    &\Delta \theta(x) = 0 \quad \text{for } x \notin \Sigma,\\
    &\theta(x) \to 2\pi \quad \text{as } x \to \Sigma_{\mathrm{f}},\\
    &\theta(x) \to 0 \quad\ \ \text{as } x \to \Sigma_{\mathrm{b}},
\end{align}
where \(\Sigma_{\mathrm{f}}\) and \(\Sigma_{\mathrm{b}}\) denote the front and back sides of \(\Sigma\), respectively. Given any smooth non-negative function \(\rho\) such that \(\rho^{-1}(0)= \im \gamma\), we can define an implicit representation \(\psi\) by \(\psi(x) = \rho(x) e^{i\theta(x)}\). For \(M = \SS^m\), viewed as the one-point compactification of \(\RR^m\), the so-called solid angle field (visualized in \autoref{fig:solid_angle_theta}) is a special case of such a phase field \(e^{i\theta}\) \cite{borodzik2020solid, chernIshida2024sphericalPolygons}.



% \begin{example}[Solid angle fields]\label{rmk:solid_angle_field}
%     On $\SS^m$ as a one point compactification of $\RR^2$, some implicit representations are explicitly constructed.  We may choose illustrative representatives within a certain $\SemiD$-orbit on $\cO/\cQ$ and on $\cO/\cP$. Let us set $S=\bigsqcup_{i=1}^k \SS^n $ and $M=\RR^m$. Then we can define the so-called \emph{solid angle field}. Given $\gamma\in \UEmb(S,M)$, the solid angle field is an $\SS^1$-valued function whose value at each point $x\in\RR^m \setminus \gamma(S)$ is half of the solid angle subtended by $\gamma$ projected onto the unit sphere centered at \(x\). Explicitly, we consider the maps \(\gamma_x\colon
% S\to\SS^{m-1}\) given by projecting \(\gamma\) on the unit sphere centered at \(x\): 
% \begin{align}
%     \gamma_x(s) \coloneqq \frac{ \gamma(s)-x}{| \gamma(s)-x|},\quad s\in S.
% \end{align}
% The solid angle field $\zeta$ takes value in $\SS^1$ by
% \begin{align}
%     \zeta(x)=e^{2\pi i\on{Vol(\gamma_x)}/{\on{Vol}(\SS^{m-1})} }
% \end{align}
% where the signed volume of the spherical shape $\gamma_x$ and the sphere $\SS^{m-1}$ is given by the standard uniform measure on $\SS^{m-1}$,
% \begin{align}
%     \on{Vol}(\gamma_x)\coloneqq \int_{\SS^{m-1}} \tilde\gamma_x^* \sigma. 
% \end{align}
% where $\tilde \gamma_x\colon D^{m-1}\to\SS^{m-1}$ is any smooth map with $\tilde \gamma_x |_{\SS^n}=\gamma_x$.

% With some continuous real-valued function $\rho$ on $M$ that vanishes exactly on $\gamma(S)$, we define a solid-angle density field $\psi^{SA}$ by $\psi^{SA}(x)\coloneqq \rho(x) \zeta(x)$ as a representative of each element of an $\SemiD$-orbit on $\cQ$ containing $[\psi^{SA}]$. We denote this orbit by $\SemiD_{\psi^{SA}}$ and the $\Diff_0(M)$-orbit of $\gamma=\Pi{\psi^{SA}}$ by $\cO_\gamma$. Now we have a one-to-one correspondence between shapes $\cO_\gamma$ and Seifert hypersurfaces $\SemiD_{\psi^{SA}}$. 


% Using $\psi^{SA}$, we can define representatives in an $\Diff_0(M)$-orbit on $\cP$.  Globally constant phase-shift $\psi^{SA} \mapsto\psi^{SA} \cdot e^{ic}$ by $e^{ic}\in \SS^1$ as the circle action defines representatives on an-orbit in $\cP$ containing $[\psi^{SA}]$ which we denote by $\SemiD$ . By fixing element $e^{ic_0}\in\SS^1$, we can identify each element of  $[\psi^{SA}\cdot e^{ic}] \in \cP$ as a Seifert surface bounding $\gamma$ given as the preimage of $e^{ic_0}$ by $\tau\circ (\psi^{SA}\cdot e^{ic})$. We note that the preimage of $1\in \SS^1$ by $\psi^{SA}$ is unbounded while the classical definitino of Seifert surface requires the boundedness. 
% We also remark that each of $\psi^{SA} \cdot e^{ic}$ is an instance of open book decomposition.
% \end{example}
    

\begin{remark}\label{rmk:psi_for_unlabeled_shapes}
In this article, we consider \emph{unlabeled shapes} by taking the quotient of \(\Emb(S, M)\) by \(\Diff^+(S)\), rather than by \(\Diff_0(S)\), as noted in \autoref{rmk:labeled_and_unlabeled_shapes}. Our implicit representations are naturally suited to this setting, since 
the level set of each \(\psi\) is unlabeled and unparametrized, yet  having the orientation as explained above. 
% \Albert{[Not sure what the last sentence about symmetric products suppose to mean.  Maybe: ``Our implicit representations are naturally suited to this setting, since the zero set of any function is an unlabeled and unparameterized set.'']}
\end{remark}

\begin{remark}[Relation with open book decomposition]
Each \(\psi \in \cF\) can be interpreted as a (relaxed) instance of an \emph{open book decomposition} in contact geometry: a decomposition of the ambient manifold \(M\) into an \(m-2\) dimensional submanifold \(B\) called the \emph{binder}, and a family of \(m-1\) dimensional submanifolds \(\{\Sigma_s\}_{s \in \SS^1}\), called \emph{pages}, such that \(\partial \Sigma_s = B\) for each \(s\in\SS^1\) \cite{Geiges_2008contact,etnyre2005lecturesopenbookdecompositions}.

For a given \(\psi \in \cF\), if the phase map \(\phi\colon M\setminus\psi^{-1}(0) \to \SS^1\), defined by \(\phi(x) = \frac{\psi(x)}{|\psi(x)|}\) 
is a submersion, then the zero set \(\{\psi = 0\}\) serves as the binder, and the level sets \(\{\phi^{-1}(s)\}_{s \in \SS^1}\) form the pages. Even if \(\phi\) fails to be a submersion, its level sets still form a foliation of \(M\), and almost every \(s \in \SS^1\) is a binder by Sard's theorem.
\end{remark}


\paragraph{Orbit space}
The full space $\cF$ is large and, in particular, consists of multiple connected components. Just as in the case of the explicit representation space $\cX$, where each connected component is a $\Diff_0(M)$-orbit $\cO$, we focus here on a single orbit $\cU \subset \cF$ under the action of the following group.

We define the semi-direct product group
\begin{align}
    \SemiD \coloneqq \Diff_0(M) \ltimes \Exp(C^\infty(M,\CC)),
\end{align}
where $\Diff_0(M)$ acts on $\Exp(C^\infty(M,\CC)) \coloneqq \{ e^\varphi \mid \varphi \in C^\infty(M,\CC) \}$ via
\begin{align}
    f \rhd e^\varphi = e^\varphi \circ f^{-1}, \quad f \in \Diff_0(M), e^\varphi \in \Exp(C^\infty(M,\CC)).
\end{align}

The Lie group $\SemiD$ has a Lie algebra $\AlgSemiD = \diff(M) \ltimes C^\infty(M,\CC)$ with Lie bracket
\begin{align}\label{eq:LieBracekt_of_AlgSemiD}
    [(u, a), (v, b)] = ([u, v], -\cL_u b + \cL_v a), \quad u, v \in \diff(M),\; a, b \in C^\infty(M,\CC),
\end{align}
where $[u, v]$ denotes the Lie bracket of vector fields on $M$.

\begin{proposition}
The operation $\AlgSemiD \times \AlgSemiD \to \AlgSemiD$ given in \eqref{eq:LieBracekt_of_AlgSemiD} defines a Lie bracket \ie, it is a bilinear skew-symmetric form  satisfying the Jacobi identity.
\end{proposition}

\begin{proof}
These properties can be verified by a straightforward computation.
\end{proof}

We now define an $\SemiD$-action on $\cF$ by
\begin{align}
    (f, e^\varphi) \rhd \psi = (\psi \circ f^{-1}) \cdot e^\varphi, \quad (f, e^\varphi) \in \SemiD, \psi \in \cF.
\end{align}

Let \(\cU_\psi \subset \cF\) denote the \(\SemiD\)-orbit containing \(\psi \in \cF\). The tangent space at 
$\psi$ is 
\begin{align}
    T_\psi \cU_\psi = \{ -\cL_u \psi + \varphi \psi \mid (u, \varphi) \in \AlgSemiD \}.
\end{align}

Given $\gamma = \Pi\psi \in \cX$ and $\dot\psi = -\cL_u \psi + \varphi \psi \in T_\psi \cU_\psi$, we have \(d\Pi (\dot\psi) = u \circ \gamma \in T_\gamma \cX\).
Using the identification of $\cX$ with de Rham currents $\cD_\cX$, this becomes
\(d\Pi (\dot\psi) = -\cL_u \delta_\gamma \in T_{\delta_\gamma} \cD_\cX\).
Hence, we obtain
\begin{align}
    \Pi(\cU_\psi) &= \cO_{\Pi\psi}, \\
    d\Pi(T_\psi \cU_\psi) &= T_{\Pi\psi} \cO_{\Pi\psi},
\end{align}
where $\cO_{\Pi\psi}$ is the $\Diff_0(M)$-orbit of $\Pi\psi \in \cX$.

For the remainder of the present and the next chapter, we denote these orbits simply by $\cU \subset \cF$ and $\cO \subset \cX$ (or $\cD_\cO \subset \cD_\cX$ under the identification), without specifying a subscript \(\psi\) or $\gamma$ for indicating a particular orbit.
 
\begin{remark}
The semi-direct product group $\SemiD = \Diff_0(M) \ltimes \on{Exp}(C^\infty(M, \CC))$ is the simplest  extension of $\on{Exp}(C^\infty(M, \CC))$ by $\Diff_0(M)$. That is, there is a split short exact sequence of groups
\begin{align}
    1 \to \on{Exp}(C^\infty(M, \CC)) \to \SemiD \to \Diff_0(M) \to \id_M.
\end{align}
On $\cF$, the group $\SemiD$ acts analogously to Euclidean transformations, combining rotation and translation. The composition rule is
\begin{align}
    (f_2, e^{\varphi_2}) \circ (f_1, e^{\varphi_1}) = \left(f_2 \circ f_1, (e^{\varphi_1} \circ f_2^{-1}) \cdot e^{\varphi_2}\right),
\end{align}
and  $\id_M \ltimes \on{Exp}(C^\infty(M, \CC))$ is a normal subgroup in $\SemiD$.
\end{remark}

\section{Geometry of fibers}\label{sec:geometry_of_fibers}
In this subsection, we study the geometry of the fiber bundle $\Pi^{-1}\cO$ consisting of implicit representations over a $\Diff_0(M)$-orbit $\cO$ of explicit representations, as well as the geometry of a $\SemiD$-orbit $\cU$ within $\Pi^{-1}\cO$.\footnote{Clearly, $\Pi^{-1}\cO$ contains a $\SemiD$-orbit $\cU$, but at this point we do not know whether $\cU = \Pi^{-1}\cO$. For further discussion, see \autoref{conj:F_equals_U}.} In particular, we show that while $\cU$ is path-connected, each fiber may have multiple connected components, and the number of these components equals the rank of the first integral de Rham cohomology group of $M$.

We define a fiber-preserving group action on $\Pi^{-1}\cO$. For each $\gamma \in \cO$, let $\Diff_\gamma(M)$ be \rev{the connected component of the stabilizer of $\gamma$ containing $\id_M$. This} is the subgroup of $\Diff_0(M)$ consisting of diffeomorphisms $f$ for which there exists a path $\{f_t\}_{t \in [0,1]} \subset \Diff_0(M)$ such that $f_t \circ \gamma = \gamma$ for all $t \in [0,1]$, with $f_0 = \id_M$ and $f_1 = f$. Then $\SemiD_\gamma \coloneqq \Diff_\gamma(M) \ltimes \on{Exp}(C^\infty(M, \CC))$ acts on $\cU$ as a subgroup of $\SemiD$, preserving each fiber $\Pi^{-1}\gamma$.
This action is clearly free on each fiber, and it turns out to be transitive on each connected component of the fiber. 

We emphasize that both components of the \(\SemiD_\gamma\)-action,  $\Diff_\gamma(M)$ and $\on{Exp}(C^\infty(M, \CC))$,  must work together to achieve connected component-wise transitivity. Indeed, $\cU_\gamma \coloneqq \Pi^{-1}\gamma \cap \cU$ (and hence $\Pi^{-1}\gamma$) may not be attained by the action of either $\on{Exp}(C^\infty(M, \CC))$ or $\Diff_\gamma(M)$ alone, as illustrated in the following examples.



\begin{example}[Non-transitivity of $\Diff_\gamma(M)$-action]\label{eg:non_transitivity_twist}
    Let $\gamma$ be four points $\{z_k\}_{k=1}^4$ on $M=\RR^2$ where  where $z_1, z_2$ have positive orientations and $z_3,z_4$ have negative orientations.
     We then let $\psi_0$ be an implicit representation of $\gamma$ defined by
    \(\psi_0(z)=(z-z_1)(z-z_2)(\overline{z-z_3})(\overline{z-z_4})\),
     and  $\psi_1 \coloneqq \psi_0 e^{i\pi/2}$ as illustrated in \autoref{fig:BS_psis}.
    These two functions clearly lie in the same fiber $\Pi^{-1}\gamma$, but
    there is no diffeomorphism $f$ such that $\psi_1 = \psi_0 \circ f$ resolving the topological differences of the phase level sets. In contrast, the group action of $e^{i\pi/2} \in \on{Exp}(C^\infty(M, \CC))$ joins $\psi_0$ and $\psi_1$. 
\end{example}

\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.4\columnwidth}
\includegraphics[width=\textwidth]{images_prequantum/BSpsi_lines.png}
\end{minipage}
\begin{minipage}[b]{0.4\columnwidth}
\includegraphics[width=\textwidth]{images_prequantum/BSpsi_pi2_shift_lines.png}
\end{minipage}
% \caption{Implicit representations $\psi_0$ (left) and $\psi_1=\psi_0 e^{i\frac{\pi}{2}}$ (right) of four points on $\RR^2$. Each color corresponds to a phase given by \(\phi= \psi/|\psi|\), and the highlighted light-blue and red lines are $\phi^{-1}(1)$ and $\phi^{-1}(e^{i\pi})$. Clearly there is no diffeomorphism $f$ such that $\psi_0\circ f=\psi_1$  as topological changes of these level lines cannot be handled.}
\caption{Implicit representations $\psi_0$ (left) and $\psi_1 = \psi_0 e^{i\frac{\pi}{2}}$ (right) of four points in $\RR^2$. Each color indicates a phase value of \(\phi = \psi / |\psi|\). The highlighted light-blue and red curves correspond to the level sets $\phi^{-1}(1)$ and $\phi^{-1}(e^{i\pi})$, respectively. Clearly, there is no diffeomorphism $f$ such that $\psi_0 \circ f = \psi_1$ that can handle the topological changes in these level sets.}
\label{fig:BS_psis}
\end{figure}

\begin{example}[Non-transitivity of $\on{Exp}(C^\infty(M, \CC))$-action]\label{eg:non_transitivity_conformal}
Let $M = \RR^2$, and consider implicit representations $\psi_0, \psi_1$ defined by $\psi_0(x, y) = x + iy$ and $\psi_1(x, y) = x + y + iy$. These functions share the same zero at $(x, y) = (0, 0)$ and the same orientation. However, there is no nowhere-vanishing smooth function $\varphi$ such that $\psi_1 = \varphi \psi_0$. Indeed, the quotient 
\(\frac{\psi_1}{\psi_0} = 1 + \frac{xy}{x^2 + y^2} - i\frac{y^2}{x^2 + y^2}\)
is discontinuous at the origin. On the other hand, with the diffeomorphism $f(x, y) = (x - y, y)$, we attain $\psi_1 = \psi_0 \circ f$.
\end{example}
The non-transitivity of the $\Diff_\gamma(M)$-action and the $\on{Exp}(C^\infty(M, \CC))$-action separately holds on any ambient manifold \(M\), as it is caused solely by local information of implicit representations near the zero level set.
For instance, on \(M = \SS^2\) regarded as the one-point compactification of \(\RR^2\), the same choice of pairs \(\psi_0, \psi_1\) as in \autoref{eg:non_transitivity_twist} and \autoref{eg:non_transitivity_conformal} works, with the modification that \(|\psi_0|\), \(|\psi_1| \to 1\) at infinity so that they are smooth functions on $\SS^2$.

We now show that the action of $\SemiD_\gamma$ is indeed transitive on each connected component of the fiber $\Pi^{-1}\gamma$, and that the number of connected components is determined by the topology of the ambient manifold. Let $H^1_{dR}(M,\ZZ)$ be the first integral de Rham cohomology defined by
\[H^1_{dR}(M,\ZZ)=\left\{[\eta]\in H^1_{dR}(M,\RR) \;\middle|\; \int_\sigma \eta \in \ZZ  \ \text{ for any } [\sigma] \in H_1(M,\ZZ) \right\}.\]
Then we have the following relation;
\begin{proposition}\label{prop:cardinality_of_fiber_components}
     For each $\gamma \in \cO$, we have a bijection
    \[\Pi^{-1} \gamma \big/ \SemiD_\gamma \cong H^1_{dR}(M, \ZZ).\]
\end{proposition}


To prove the proposition and highlight the roles of $\Diff_\gamma(M)$-action and $\on{Exp}(C^\infty(M,\CC))$-action separately, we introduce two equivalence classes.
\begin{definition}[Twist class and conformal class]\label{def:conformal_and_twist}
    We say $\psi_0,\psi_1\in \Pi^{-1}\gamma$ are in the same \emph{twist class} if there is a diffeomorphism $f \in \Diff_\gamma(M)$ such that $\psi_1=\psi_0\circ f$. 
    
    We say $\psi_0,\psi_1\in \Pi^{-1}\gamma$ are in the same \emph{conformal class} if there is a nowhere vanishing function $\zeta\in C^\infty(M,\CC^\times)$ such that $\psi_1=\zeta\psi_0$.
\end{definition}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth,, trim={4cm 4cm 4cm 4cm},clip]{images_prequantum/ribbons.png}
\caption{A ribbon of an implicit representation $\psi$ for the Hopf link $\gamma$. For each regular value $s\in\SS^1$ of the phase field \( \phi = \psi/|\psi| \), the ribbon \( R_s \) (opaque cyan) is defined as the intersection of \( \phi^{-1}(s) \) (translucent blue) and a small tubular neighborhood of \( \im\gamma \).}
\label{fig:ribbon}
\end{figure}

Intuitively, $\psi_0$ and $\psi_1$ are in the same twist class if their \emph{ribbons} (\autoref{fig:ribbon}) have the same total twists and hence are diffeomorphic to each other. On the other hand, $\psi_0$ and $\psi_1$ are in the same conformal class if they share the same shear states (\ie, the rate of phase change) around the zeros so that $\psi_1/\psi_0$ exists and smooth on the zeros. 

\begin{proof}[Proof of \autoref{prop:cardinality_of_fiber_components}]
    We first claim that for each pair $\psi_0, \psi_1 \in \Pi^{-1}\gamma$, there exist functions $e^{\varphi_0}, e^{\varphi_1} \in \operatorname{Exp}(C^\infty(M,\CC))$ and a diffeomorphism $f \in \Diff_\gamma(M)$ such that $(e^{\varphi_0} \psi_0) \circ f$ and $e^{\varphi_1} \psi_1$ lie in the same conformal class.
    
    To see this, take some $\varphi_0, \varphi_1 \in C^\infty(M,\CC)$ so that, letting $\hat{\psi}_0 \coloneqq e^{\varphi_0} \psi_0$ and $\hat{\psi}_1 \coloneqq e^{\varphi_1} \psi_1$, we have $|\hat{\psi}_0| = |\hat{\psi}_1|$ in a small tubular neighborhood $B$ of $\operatorname{im} \gamma$, and the phase field $\hat\phi_j=\frac{\hat{\psi}_j}{ |\hat{\psi}_j|} \colon B \setminus \im\gamma \to \SS^1$ is a submersion for $j=0,1$.
    
    We can take coordinates $(p,z)$ in $B$ with $p \in \im\gamma$ and $z \coloneqq \hat{\psi}_0(p, \cdot)$. Then consider a tubular sub-neighborhood $B_R = \{ (p,z) \mid p \in \im \gamma, |z| \leq R \}$ small enough so that $\hat{\psi}_1$ is injective on each circle $S_p^\epsilon = \{(p,z) \mid |z| = \epsilon\}$ for $p\in\im\gamma$ and $\epsilon \leq R$.
    Then for a sufficiently small $r < R$, there exists $f \in \SemiD_\gamma$ such that $\hat{\psi}_0 \circ f = \tau \hat{\psi}_1$ inside the smaller tubular neighborhood $B_r = \{(p,z) \mid p \in \im \gamma, |z| \leq r \}$ and $f = \operatorname{id}$ outside $B_R$, as illustrated in \autoref{fig:psi_interpolation}. 
    Here $\tau$ is a smooth function on $B_r$ of the form $\tau(p,z) = e^{i \theta(p)}$ with some $\TT^1$-valued function $\theta$.
    
    Outside $B_r$, the function $\frac{\hat{\psi}_1}{\hat{\psi}_0 \circ f}$ is smooth and nowhere vanishing.
    Hence there exists a smooth function $\zeta \in C^\infty(M, \CC^\times)$ such that $\hat{\psi}_0 \circ f = \zeta \hat{\psi}_1$ and $\zeta = \tau$ inside $B_r$, showing that $\psi_0 \circ f$ and $\psi_1$ are in the same conformal class.
    
    We now note that the following is an exact sequence:
    \begin{align}
        0 \to \ZZ \xrightarrow[]{k \mapsto 2 \pi i k} C^\infty(M,\CC) \xrightarrow[]{\exp} C^\infty(M,\CC^\times) \xrightarrow[]{\frak{h}} H^1_{dR}(M, \ZZ) \to 0
    \end{align}
    where $\frak{h}\colon C^\infty(M,\CC^\times) \to H^1_{dR}(M, \ZZ) $ is defined by $\frak{h}(\kappa)= \left[\Im \frac{1}{2\pi} \frac{d\kappa}{\kappa} \right]$.
    Therefore $\zeta$ defines an element
    \( \frak{h}(\zeta) \in H^1_{dR}(M, \ZZ)\), and the functions $\hat{\psi}_0 \circ f$ and $\hat{\psi}_1$ are in the same twist class if and only if $\frak{h}(\zeta) = [0]$.
    
    On the other hand, each pair of elements $[\eta_1], [\eta_2] \in H^1_{dR}(M, \ZZ)$ define elements \([\zeta_1], [\zeta_2] \in C^\infty(M, \CC^\times)/\operatorname{Exp}(C^\infty(M, \CC))\) such that $\zeta_1 \psi$ and $\zeta_2 \psi$ for any choice of $\psi \in \Pi^{-1}\gamma$, and the representatives $\zeta_1, \zeta_2$ are in the same $\SemiD_\gamma$-orbit if and only if $[\eta_1] = [\eta_2]$. This completes the proof.
\end{proof}

% We notice that these results are valid for any pair $\psi_0,\psi_1$ in the same fiber $\Pi^{-1}\gamma$. 
Hence we have shown that,  while 
a $\SemiD$-orbit $\cU$ is path connected, each fiber $\Pi^{-1}\gamma$ consists of a disjoint copies of a $\SemiD_\gamma$-orbit associated with the discrete group $H^1_{dR}(M,\ZZ)$.
% 
% \rev{Since each fiber $\Pi^{-1}\gamma$ is an orbit of $\Diff_\gamma\ltimes C^\infty(M,\CC^\times)$, the entire fiber bundle $\cF_\cO\coloneqq \Pi^{-1}\cO$ over a fixed $\cO$ is a $\Diff_0(M)\ltimes C^\infty(M,\CC^\times)$-orbit.}

\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.33\columnwidth}
\includegraphics[width=\textwidth, trim={2cm 0 2cm 0},clip]{images_prequantum/psi_z.png}
\caption*{{\footnotesize $\hat\psi_0(p,z)=z$}}
\end{minipage}
\hspace{-0.5cm}
\begin{minipage}[b]{0.33\columnwidth}
\includegraphics[width=\textwidth, trim={2cm 0 2cm 0},clip]{images_prequantum/psi_spiral.png}
\caption*{{\footnotesize $\hat\psi_1(p,z)=ze^{i|z|^2}$}}
\end{minipage}
\hspace{-0.5cm}
\begin{minipage}[b]{0.33\columnwidth}
\includegraphics[width=\textwidth, trim={2cm 0 2cm 0},clip]{images_prequantum/psi_spiral_to_z.png}
\caption*{{\footnotesize $\hat\psi_0 \circ f$}}
\end{minipage}
\caption{Level curves $\hat\phi^{-1}(1), \hat\phi^{-1}(e^{i\frac{\pi}{2}}), \hat\phi^{-1}(-1), \hat\phi^{-1}(e^{i\frac{3\pi}{2}})$ of the phase maps $\hat\phi=\hat \psi/|\hat \psi|$ of implicit representations  $\hat\psi_0(p,z)=z$ (left), $\hat\psi_1(p,z)=ze^{ i|z|^2}$ (middle), and $\hat\psi_0\circ f$ (right), on the tubular domain $B$ sliced into a disc at fixed $p\in \im \gamma$. The diffeomorphism $f$ rotates these level lines on each circle $S^\epsilon_p=\{(p,z)\mid |z|=\epsilon\}$ for $\epsilon\leq R$ so that $\hat\psi_0 \circ f$ agrees with $\hat\psi_1$ inside the smaller domain $B_r$ (dark gray) and with $\hat \psi_0$ outside the larger domain $B_R$ (light gray). }
\label{fig:psi_interpolation}
\end{figure}

\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_0$}
\includegraphics[width=\textwidth, trim={0 -1cm 0 0},clip]{images_prequantum/psi0.png}
\end{minipage}
\quad
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_{t_1}$}
\includegraphics[width=\textwidth, trim={0 -1cm 0 0},clip]{images_prequantum/psi_t1.png}
\end{minipage}
\quad
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_{t_2}$}
\includegraphics[width=\textwidth, trim={0 -1cm 0 0},clip]{images_prequantum/psi_t2.png}
\end{minipage}
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_{t_3}$}
\includegraphics[width=\textwidth, trim={0 0 0 0},clip]{images_prequantum/psi_t3.png}
\end{minipage}
\quad
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_{t_4}$}
\includegraphics[width=\textwidth, trim={0 0 0 0}, clip]{images_prequantum/psi_t4.png}
\end{minipage}
\quad
\begin{minipage}[b]{0.28\columnwidth}
\caption*{$\psi_1$}
\includegraphics[width=\textwidth, trim={0 0 0 0},clip]{images_prequantum/psi1.png}
\end{minipage}
  \caption{A transition of the implicit representation from $\psi_0$ (top left) to $\psi_1$ (bottom right) for two points $\gamma$ on the torus $\TT^2$. In both $\psi_0$ and $\psi_1$, the colored curves represent level sets $\phi^{-1}(s)$ of some phases. Clearly, $\psi_0$ and $\psi_1$ belong to different connected components of the fiber $\pi^{-1}\gamma$. 
% 
Nevertheless, they lie on the same $\SemiD$-orbit. Starting from $\psi_0$, we first multiply by some $e^{\varphi_1} \in \Exp(\CC^\infty(M,\CC))$ to \emph{compress} the support of $d\phi$ into narrow bands, visualized as thick gray curves in $\psi_{t_1}$. 
% 
Next, multiplying $\psi_{t_1}$ by $e^{\varphi_2} \in \Exp(\CC^\infty(M,\CC))$ merges two level curves, producing $\psi_{t_2}$. Applying a diffeomorphism $f_1$ moves the level lines, yielding $\psi_{t_3}$, and a second diffeomorphism $f_2$ transforms $\psi_{t_3}$ into $\psi_{t_4}$. Finally, multiplying by another $e^{\varphi_3} \in \Exp(\CC^\infty(M,\CC))$ decompresses the support of $d\phi$, resulting in $\psi_1$.}

\label{fig:DC_orbit_U}
\end{figure}

% \rev{
% \begin{remark}
% \autoref{prop:cardinality_of_fiber_components} can be rephrased in terms of a principal bundle.  Let us fix  $\gamma_0\in \cO$ then $\cF_\cO$ is a principal $\Diff_{\gamma_0}\ltimes H^1_{dR}(M,\ZZ)$ bundle. On the fiber $\pi^{-1}\gamma$ over $\gamma=h\circ \gamma_0$ with some $h\in \Diff_0(M)$, $\Diff_{\gamma_0}$ acts by $\psi \circ h \circ f^{-1} \circ h^{-1}$. Note that if the entire action does not depend on the choice of $h$.
  


% that $\cF_\cO$ as a principal bundle. Let us fix some $\gamma_0\in\cO$ and define the action of the semi-direct product group $\SemiD_{\gamma_0}$ on the entire $\cF_\cO$ by
% \begin{align}
%     (f\circ )
% \end{align}

%     The fiber bundle $\cF_\cO$ is actually a principal . $\cF_\cO$ is a principal circle bundle 
% \end{remark}
% }


\paragraph{Is the fiber bundle $\cF_\cO\coloneqq \Pi^{-1}\cO$ larger than a single orbit $\cU$?}

Since each fiber $\Pi^{-1}\gamma$ is an orbit of $\Diff_\gamma\ltimes C^\infty(M,\CC^\times)$, the entire fiber bundle $\cF_\cO\coloneqq \Pi^{-1}\cO$ over a fixed $\cO$ is a $\Diff_0(M)\ltimes C^\infty(M,\CC^\times)$-orbit. As each fiber has multiple connected components when $H^1_{dR}(M,\ZZ)\neq 0$, the fiber bundle $\cF_\cO$ appears to be strictly larger than $\cU$, a single orbit under the action of $\SemiD=\Diff_0(M)\ltimes \Exp(C^\infty(M,\CC))$. It, however, turns out that $\cF_\cO=\cU$ at last for the case of the base dimension 2 as illustrated in \autoref{fig:DC_orbit_U}. By moving around zeros while multiplying functions in $\on{Exp}(C^\infty(M,\CC))$, we can overcome topological changes of phase level lines. We expect that a similar argument holds in higher dimensional cases and the result remains valid:
\begin{conjecture}\label{conj:F_equals_U}
Let $\cO$ be a $\Diff_0(M)$-orbit in the explicit shape space $\cX$,  $\cF_\cO\coloneqq \Pi^{-1}\cO$ be the entire fiber bundle over $\cO$, and $\cU$ be a $\SemiD$-orbit in $\cF_\cO$. 
Suppose each connected component of $M$ intersects with $\im \gamma$ for $\gamma\in \cO$. Then \(\cF_\cO=\cU\).
\end{conjecture}

% \begin{figure}[htbp]
%     \centering
%     \vspace{5pt}
% \begin{overpic}[width=1.0\textwidth,keepaspectratio]{images_prequantum/DC_orbit_U.png}
%     \put(0,14){{\small $\psi_0$}}
%     \put(26,14){{\small $\psi_{t_a}=e^{\varphi}\psi_0$}}
%     \put(52,14){{\small $\psi_{t_b}=(e^{\varphi}\psi_0)\circ f$}}
%     \put(78,14){{\small $\psi_1=(e^{\varphi}\psi_0)\circ f\circ h$}}
% \end{overpic}
%     \caption{Level curves $\phi^{-1}(s)$ of a fixed phase $s \in \SS^1$ for implicit representations $\psi_t$ of two points on $\TT^2$. The leftmost and rightmost functions, $\psi_0$ and $\psi_1$, clearly belong to different connected components of a fiber. Nevertheless, they lie on the same $\SemiD$-orbit. Starting from $\psi_0$, we can first merge its two level lines into one by multiplying by $e^\varphi$, yielding $\psi_{t_a}$. Then, applying a diffeomorphism $f$ moves the level line, producing $\psi_{t_b}$. Finally, another diffeomorphism $h$ transforms $\psi_{t_b}$ into $\psi_1$.}
%     \label{fig:DC_orbit_U}
% \end{figure}


We conclude this subsection by observing that an $\Diff_0(M)\ltimes C^\infty(M,\CC^\times)$-orbit $\cF_\cO$ is locally a $\SemiD$-orbit $\cU$ and they share the same tangent space at each  $\psi$ even if $\cU\subsetneq\cF_\cO$. In this section and the next section, we still focus on local arguments on $\cU$, which are all available on $\cF_\cO$ when we consider prequantum $G$-bundles in \autoref{sec:prequantum_bundle}.





\chapter{Prequantum structure on the space of implicit representations}\label{ch:prequantum_bundle}

In this chapter, we further study the geometry of the implicit shape space $\cF_\cO$, related to the Marsden--Weinstein symplectic structure on the explicit shape space $\cO$. In particular, we construct a prequantum bundle over $\cO$ as a certain quotient space of $\cF_\cO$. This reveals a new geometric interpretation of the Marsden--Weinstein form.

% In this chapter, we further study the geometry of the implicit shape space $\cF_\cO$, related to the Marsden--Weinstein symplectic structure defiened on the explicit shape space $\cO$. In particular, we construct a prequantum bundle as a certain quotient space of $\cF_\cO$. 
% 
% A prequantum bundle $P$ is a principle $G$-bundle over a symplectic manifold $(B,\beta)$ equipped with a connection form $\alpha$ whose curvature agrees with $\beta$. Intuitively, it attaches the information of quantum phase to each point of $B$ in such a way that Hamiltonian flows on $B$ is uniquely lifted onto the bundle $P$. In this way, it is a preliminary set up for converting a space for classical mechanics into a space for quantum mechanics. 

% We, however, do not perform quantum mechanical analysis. We, rather, focus on geometric insights that our prequantum bundle offers. In particular, we present a new geometric interpretation of the Marsden--Weinstein form.

\section{Overview}

As we saw in \autoref{ch:implicit_representations}, the fiber bundle $\cF_\cO$ of implicit representations has a natural geometric interpretation.
Each complex function $\psi\in\cF_\cO$ representing a codimension-2 submanifold $\gamma\in\cO$ carries additional information in the form of its complex phase. The level sets of this phase function constitute a $\SS^1$-family of hypersurfaces in the ambient manifold \(M\), that are bordered by $\gamma$. For the motion of each of these hypersurfaces, we can consider the swept volume. Hence any motion of $\psi$ induces the average volume swept out by these phase hypersurfaces.

From this observation, we introduce a quotient space of $\cF_\cO$. In each fiber, we define two hypersurface configurations as equivalent (\(\sim\)) if they can be continuously deformed into each other while keeping their boundaries fixed and maintaining zero net volume change throughout the motion.  
The space of equivalence classes is denoted by \(\cP\coloneqq\cF_\cO/\sim\), which is still a fiber bundle over \(\cO\).
We call \(\cP\) the \emph{volume class} for the implicit shape space.

It turns out that the volume class $\cP$ over $\cO$ carries a \emph{prequantum bundle} structure, where each fiber consists of multiple copies of a circle, reflecting the first integral de Rham cohomology $H^1_{dR}(M, \ZZ)$. \rev{This is a relaxation of a standard definition of a prequantum bundle which usually requires each fiber to be a complex line or a circle, while each fiber of our bundle is still one dimensional.}

The connection form $\Th_\cP$ on this bundle is defined such that the horizontal lift of any path in $\cO$ corresponds to a motion of phase hypersurfaces that sweep out zero net volume along the path. With this setup, we have the following theorem.

\begin{theorem}[\autoref{th:prequantum_circle_bundle}]\label{th:informal:prequantum_circle_bundle}
    The fibration $\Pi_\cP\colon(\cP,\Th_\cP)\to (\cO,\omega)$ is a prequantum $G$-bundle
    where \(\omega\) is the MW symplectic form  
   and the structure group is $G=\SS^1\times H^1_{dR}(M,\ZZ)$.\footnote{\rev{Precisely, we normalize  $\Theta$ and $\omega$ in \autoref{th:prequantum_circle_bundle} so that the normalized $\Theta$ is a connection form in a standard definition of principal bundle. For details, see \autoref{sec:prequantum_bundle}.}} 
 
    % The bundle \textit{prequantum circle bundle} equipped with a connection, whose curvature coincides with the MW form on the base space.
\end{theorem}
In other words, the curvature of the connection \(\Theta_{\cP}\) agrees with the MW symplectic form.
As a consequence, this prequantum bundle allows the following geometric interpretation of the MW form:

% \begin{theorem}[Informal version of Theorem \ref{...}]
%     Consider a horizontal lift of hypersurfaces in $\cF_\cO$ over a closed path $\partial \Sigma$ bounding a 2-dimensional disc $\Sigma$ on $\cO$.
%     % of the underlying codimension-2 submanifolds on the base space.
%     Then the volume enclosed by the initial and the end hypersurfaces of the path agrees with the integral of the MW-form over $\Sigma$. 
% \end{theorem}
% \begin{theorem}
%     Consider a closed path \(\partial\Sigma\) bounding a 2-dimensional disk \(\Sigma\) in \(\cO\), which corresponds to a cyclic motion of a codimension-2 submanifold \(\gamma_t\subset M\), \(0\leq t\leq 1\),  \(\gamma_0 = \gamma_1\).  Suppose at each \(0\leq t\leq 1\) the submanifold \(\gamma_t\) is the boundary of a hypersurface \(\gamma_t = \partial\sigma_t\), and the total volume swept out by \(\sigma_t\) is zero at each \(t\); that is, \(\iota_{\dot\sigma_t}\mu = 0\) where \(\mu\) is the volume form on \(M\).  Then the volume enclosed by \(\sigma_0\) and \(\sigma_1\) equals to \(\iint_{\Sigma}\omega\), where \(\omega\) is the MW-form on \(\cO\).
% \end{theorem}

\begin{corollary}[\autoref{cor:average_swept_volume}]\label{cor:intro_average_swept_volume}
Consider a closed path \(\partial\Sigma\) in \(\cO\) that bounds a 2-dimensional disk \(\Sigma\), representing a cyclic motion of a codimension-2 submanifold \(\gamma_t \subset M\) for \(0 \leq t \leq 1\), with \(\gamma_0 = \gamma_1\). Let $[\psi_t]_\cP$ be a horizontal lift over $\gamma_t$ and  $\phi_t\coloneqq \psi_t/|\psi_t|\colon M\setminus \psi^{-1}_t(0) \to \SS^1$ be the phase map of a representative $\psi_t\in [\psi_t]_\cP$. \rev{That is,} \(\gamma_t\) bounds a family of hypersurfaces \(\{\sigma_t^s\}_{s\in \SS^1}\), defined by $\sigma_t^s = \phi_t^{-1}(s)$ and the average volume swept out by \(\sigma_t^s\) remains zero at each \(t\), i.e.,
\begin{align}
    \oneOverTwoPi\int_{\SS^1}\int_{\sigma_t^s}\iota_{\partial_t{\sigma}_t^s} \mu \ ds = 0,
\end{align}
where \(\mu\) is the volume form on \(M\).

Then, the volume enclosed between \(\sigma_0^s\) and \(\sigma_1^s\), averaged over \(s\in \SS^1\), equals to \(\iint_{\Sigma} \omega\), where \(\omega\) is the MW form on \(\cO\).
\end{corollary}

In the limiting case of \autoref{cor:intro_average_swept_volume}, where the phase of $\psi_t$ becomes constant except a \(2\pi\) jump at a single hypersurface \(\sigma_t\) bounding \(\gamma_t\), the result simplifies to the following corollary. This version has no explicit reference to the complex function $\psi_t$.

\begin{corollary}[\autoref{cor:single_surface_swept_volume}]
Let \(\Sigma\) be a disc and \(\{\gamma_t\}_{t \in [0,1]}\) be a path in $\cO$ as in \autoref{cor:intro_average_swept_volume}. Suppose that each \(\gamma_t\) bounds a hypersurface, i.e., \(\gamma_t = \partial \sigma_t\), and that the volume swept out by \(\sigma_t\) remains zero at each \(t\).
Then, the volume enclosed between \(\sigma_0\) and \(\sigma_1\) agrees with \(\iint_{\Sigma} \omega\).
\end{corollary}

\paragraph{Related work on prequantum bundles over the codimension-2 shape spaces}
There are other prequantum bundles over $(\cO,\omega)$. In \cite{HallerVizman2003NonlinearGA}, Haller and Vizman constructed a prequentum circle bundle by covering $\cO$ with a collection of open sets $\{U_i\}_i$ and gluing the sets $\{\SS^1\times U_i\}_i$ using carefully chosen transition functions that ensure a prequantum bundle structure. Another approach, utilizing differential characters, was introduced in \cite{Diez2020InducedDC}.
Presently we are not aware of any explicit relationships between their proposed prequantum bundles and ours. 

However, we emphasize that, unlike these previously established prequantum bundle structures, our framework is inherently local. Specifically, we focus on submanifolds that are exact in homology, meaning they must bound codimension-1 domains. Consequently, the existence of our prequantum bundle is restricted to such exact submanifolds as explained in \autoref{sec:implicit_representations}.


\paragraph{Future work}
The codimension-2 shape space admits not only the MW structure but also other symplectic structures, as recently shown in \cite{bauerMichorIshida2024symplectic}. In this article, we focus specifically on the MW structure. Extending our framework to a wider class of symplectic structures is an interesting direction for future work.

Secondly, we restrict our attention to the space of embeddings modulo reparametrization, meaning that self-intersecting shapes are outside the scope of this study, even though the MW form is defined on the space of immersions. On the other hand, the tools we employ from geometric measure theory, such as currents, are capable of handling such singularities. In the future, we aim to extend our framework to accommodate topological changes and reconnection events in shapes.

\paragraph{Organization of the chapter}

In \autoref{sec:Liouville_form_prequantum}, we define a 1-form $\Th$ on $\cF_\cO$ and show that it defines a Liouville form for the MW form $\omega \in \Omega^2(\cO)$ in the prequantum sense, i.e., $\Pi^*\omega = d\Th$ via the fibration $\Pi \colon \cF_\cO \to \cO$. Our proof, using geometric measure theory, offers a geometric interpretation of $\Th$ in terms of the swept volumes of phase hypersurfaces.

In \autoref{sec:prequantum_bundle}, we define the volume class $\cP$ as explained above. We show that $\Th$ factored onto $\cP$ becomes a connection form $\Th_\cP$ forming a prequantum $G$-bundle $\Pi_\cP\colon (\cP,\Th_\cP)\to(\cO,\omega)$ with the structure group $G=\SS^1 \times H^1_{dR}(M,\ZZ)$.

In \autoref{sec:additional_structures}, we study additional structures that implicit representations and prequantum structures reveal, such as horizontal Hamiltonian vector fields on $\cP$ with respect to the connection $\Theta_\cP$, and a degenerate formal K\"ahler structure on $\cF_\cO$.

\section{Liouville form in the prequantum sense}\label{sec:Liouville_form_prequantum}
In this section, we present one of our main results: the MW symplectic form $\omega$ on $\cO$ over a general closed manifold $M$ admits a Liouville form in the \emph{prequantum sense}. In the rest of the chapter, we assume that the ambient manifold $M$ is closed, in contrast to the setting in \autoref{sec:explicit_symplectic} for the explicit shape space.

\begin{definition} 
Define the following 1-form on each \(\SemiD\)-orbit \(\cU\) of \(\cF\):
\begin{align}\label{eq:Th_integral} \Th_\psi(\dot \psi) = \oneOverTwoPi \int_{M} \Im  \frac{\dot \psi \bar\psi}{|\psi|^2} \  \mu, 
\end{align} where $\dot \psi = -\cL_u \psi + \varphi \psi \in T_\psi \cU$ for some $(u, \varphi) \in \AlgSemiD$.
\end{definition}

At first glance, the integral in \eqref{eq:Th_integral} may appear divergent as the integrand is unbounded near the zeros of \(\psi\). However, we will show that it is in fact finite. We also demonstrate that the value $\Th_\psi(\dot\psi)$ has a clear physical interpretation (\autoref{rmk:meaning_of_Th}).


\begin{definition}[Formal prequantization and Liouville form]\label{def:formal_prequantization}
    Let $(B, \beta)$ be a symplectic manifold, and let $E$ be a manifold equipped with a 1-form $\alpha$. We say that a fibration $\pi \colon E \to B$ is a \emph{formal prequantization}, and that $\alpha$ is a \emph{Liouville form} of $\beta$ in the \emph{prequantum sense}, if
\begin{align}\label{eq:prequantum_condition}
        d\alpha = \pi^* \beta.
    \end{align}
\end{definition}

\begin{theorem}[Liouville form for the MW structure in the prequantum sense] \label{th:quantum_liouville_form}
    Suppose $M$ is a closed manifold equipped with a volume form $\mu$.
    % where $B_\epsilon\subset M$ is the $\epsilon$-tubular neighborhood of $\{\psi=0\}$ with respect to any Riemannian distance on $M$. 
    Then the fibration $\Pi \colon (\cU, \Th) \to (\cO, \omega)$ is a formal prequantization.
\end{theorem}
\paragraph{Proof strategy}
To prove the theorem, that is, to show \(d\Th = \Pi^*\omega\), we combine a standard approach in differential geometry with the coarea formula from geometric measure theory.
% 
We compute the exterior derivative of \(\Theta_\psi\) by the formula
\begin{align}\label{eq:exterior_derivative_in_U}
d\Theta_\psi(\dot \psi, \ring \psi)
= \cL_{Y_1} \iota_{Y_2} \Th_\psi - \cL_{Y_2} \iota_{Y_1} \Th_\psi - \iota_{[Y_1, Y_2]} \Th_\psi
\end{align}
 for each \(\psi\in\cU\) and \(\dot\psi,\mathring \psi\in T_\psi\cU\).  Here, the vector fields \(Y_1,Y_2\in\Gamma(T\cU)\) are arbitrary extensions of the vectors \(\dot\psi,\mathring \psi\) respectively; that is, \(Y_1|_{\psi} = \dot\psi\) and \(Y_2|_\psi = \mathring \psi\).

To perform this calculation, we first choose specific vector fields \(Y_1\) and \(Y_2\) and explicitly compute each term in \eqref{eq:exterior_derivative_in_U}. This calculation leads to a simple expression of $d\Theta$ (\autoref{lem:dTheta}).  As the final step of the proof, we show that the fibration  \(\Pi: \cU \to \cD_\cO\subset \cD_{m-2}(M)\) is decomposed into $\Pi=\partial \circ \Lambda$  (\autoref{lem:lam_psi_in_fiber}) using the coarea formula,
 % as in the following sequence
% \[
% \begin{tikzcd}
% \psi \in \cU = \SemiD \rhd \psi \subset \cF 
% \arrow[d,"\Lambda"]\\
% \Lambda(\psi)
% \in
%  \cD_{m-1}(M)
% \arrow[d,"\partial"]\\
% \delta_\gamma \in \cD_\cO = \Diff_0(M) \rhd \delta_\gamma 
% \subset \cD_{m-2}(M).
% \end{tikzcd}
% \]
where the map \(\Lambda\colon \cU\to \cD_{m-1}(M)\) assigns to each $\psi$ its \emph{circle differential \((m-1)\)-current}, as defined below (\autoref{def:circle_differential}), and $\partial\colon \cD_{m-1}(M)\to \cD_{m-2}(M)$ is the boundary operator for currents. 

\begin{definition}[Circle differential 1-form and $(m-1)$-current]\label{def:circle_differential}
Using the normalization map $\tau\colon\CC^\times\to \SS^1$ given  by $z\mapsto \frac{z}{|z|}$ and the standard Haar measure \(\sigma\in\Omega^1(\SS^1)\) on \(\SS^1\) with 
 \(\int_{\SS^1}\sigma = \rev{2\pi}\), define \emph{the circle differential 1-form \(\lambda_{\psi}\in \Omega^1(M\setminus\psi^{-1}(0))\)} by
\begin{align}
    \lambda_\psi \coloneqq \psi^*\tau^*\sigma.
\end{align}
This induces the map $\Lambda\colon \cF\to \cD_{m-1}(M)$ by
\begin{align}\label{eq:integral_lambda_psi}
    \langle \Lambda(\psi), \beta \rangle\coloneqq
    % \int_{\SS^1} \langle\delta_{\phi^{-1}(s)}, \alpha\rangle \  ds,
    \int_{M} \lambda_\psi \wedge \beta, 
    \quad \beta \in \Om^{m-1}(M).
\end{align}
We call $\Lambda(\psi)$ \emph{the circle differential current}.
\end{definition}
The circle differential \(\lambda_\psi\) represents the gradient of the phase of \(\psi\). In fact, there is a small neighborhood $N_x$ around each point $x\in M\setminus\psi^{-1}(0)$ such that $\psi$ is locally represented by $\psi(y)=r(y)\cdot e^{i\phi(y)}$ with some function $r\colon N_x\to \RR_{>0}$ and $\phi\colon N_x \to \RR$ 
% where $U$ is an open interval in $\RR$ with the length strictly small than $2\pi$ 
with $d\phi=\lambda_\psi$. 

The integral \eqref{eq:integral_lambda_psi} is well-defined because the integrand $\lambda_\psi\wedge \alpha$ is defined almost everywhere on $M$ with respect to 
$\mu$, in particular, except on the codimension-2 set \(\psi^{-1}(0)\). 
In fact, $\Lambda(\psi)$ is indeed a current \ie, a continuous linear functional on $\Omega^{m-1}(M)$, and namely the value \eqref{eq:integral_lambda_psi} is always finite;
\begin{lemma}\label{lem:finiteness_of_circle_differential}
    Let $M$ be a closed manifold and $\psi\in \cF$. Then $\Lambda(\psi)\in \cD_{m-1}(M)$.
\end{lemma}
This result is a direct consequence of the coarea formula from geometric measure theory, which is a nonlinear version of the Fubini theorem. 
\begin{proposition}[Smooth coarea formula]\label{prop:coarea_formula}
Let $X$ be an oriented smooth manifold and $Y$ be an oriented compact smooth manifold with dimension $n_X$ and $n_Y<n_X$ respectively. For a submersion $f\colon X\to Y$ and forms $\beta \in \Om^{n_X-n_Y}(Y), \sigma \in \Om^{n_Y}(Y)$, we have
\begin{align}
    \int_X f^* \sigma \wedge \beta 
    = \int_{Y} \langle \delta_{f^{-1}(\cdot )}, \beta\rangle \  \sigma.
    % = \int_{Y} \langle \delta_{f^{-1}(\cdot )\cap \{df \text{ subjective}\}}, \alpha \rangle \  \sigma 
    % = \int_{Y_{\on{reg}}} \langle \delta_{f^{-1}(\cdot )}, \alpha \rangle \  \sigma.
\end{align}
% We may also write The last equality is due to Sard's theorem.
% Due to Sard's theorem, for almost every $y\in Y$, the inverse image $f^{-1}(y)$ is a $n$-dimenional submanifold and $f^{-1}(y)=f^{-1}(y)\cap \{x\in M \mid df|_x \neq 0\}$. Hence we may formally write the domain of integral $Y$ instead of $Y_{\on{reg}}$.
\end{proposition}

\begin{remark}
There are different versions of the coarea formula. Here, we use the version from \cite[Chapter I.3]{demailly1997complex}, as it is formulated in terms of differential forms and does not rely on Hausdorff measure or a Riemannian metric, making it well-suited for our context. 
% While the statement in \cite{knnoeppel2023randomzerocurrentssections} is given for a Riemannian manifold $X$, we see by carefully inspecting the proof that it is actually independent of the choice of the Riemannian metric. Hence we can drop the assumption of having a Riemannian metric. 

For standard references on the coarea formula, readers may find works like \cite{federer2014geometric} and \cite{nicolaescu2011coarea} helpful.
\end{remark}
\begin{proof}[Proof of \autoref{lem:finiteness_of_circle_differential}]
Clearly $\Lambda(\psi)$ is linear in $\cD_{m-1}(M)$. We now show that $\Lambda(\psi)$ is a bounded functional.
      By the coarea formula (\autoref{prop:coarea_formula}), we have for any $\beta\in \Omega^{m-1}(M)$ that,
    \begin{align}
       \langle\Lambda(\psi),\beta\rangle
        &=\int_{M} \lambda_\psi \wedge \beta
        =\int_{\Mreg} \lambda_\psi \wedge \beta
        =\int_{\SS^1} \langle \delta_{(\phi^{-1}(\cdot )\cap \Mreg)}, \beta\rangle \sigma
    \end{align}
    where $\phi=\tau\circ \psi\colon {M\setminus\psi^{-1}(0)}\to \SS^1 $ and $\Mreg$ denotes the set of regular points of $\phi$.
    Note that the set $\phi^{-1}(s)\cap \Mreg$ for each $s\in \SS^1$ is a $m-1$ dimensional submanifold and namely $\delta_{\partial(\phi^{-1}(s )\cap \Mreg)}$ is a $m-1$-current. In particuluar, the operator norm $\| \delta_{\partial(\phi^{-1}(s )\cap \Mreg)} \|_{\cD_{m-1}(M)}$ is bounded. Therefore, 
    \[\|\Lambda(\psi)\|_{\cD_{m-1}(M)}\leq \int_{\SS^1}\sigma \sup_{s\in \SS^1}\| \delta_{\partial(\phi^{-1}(s )\cap \Mreg)} \|_{\cD_{m-1}(M)} = 2\pi \sup_{s\in \SS^1}\| \delta_{\partial(\phi^{-1}(s )\cap \Mreg)} \|_{\cD_{m-1}(M)}\]
    is bounded, from which we conclude that $\Lambda(\psi)$ is a current.
\end{proof}
As we see from the proof, the current $\Lambda(\psi)$ is in fact the superposition of a $\SS^1$-family of hypersurfaces given as the phase level sets of $\psi$.


\paragraph{Vector fields $Y_1, Y_2 \in \Gamma(T\cU)$}
We now choose the vector fields \(Y_1\) and \(Y_2\) that extend \(\dot\psi\) and \(\mathring\psi\) to evaluate \eqref{eq:exterior_derivative_in_U}. As in our approach for \autoref{th:liouville_classical}, we  use the fundamental vector field associated with the action of \(\SemiD=\Diff_0(M)\ltimes \Exp(C^\infty(M,\CC))\) on \(\cU\).
Recalling that the Lie 
algebra of $\SemiD$ is $\AlgSemiD = \diff(M) \ltimes C^\infty(M,\CC)$, we define $Y^{(u,a)}\in\Gamma(T\cU)$ for each $(u,a) \in \AlgSemiD$ by
\begin{align}\label{eq:fundamental_vector_field_DC}
    Y^{(u,a)} (\psi)
    \coloneqq
       \frac{d}{dt} \big|_{t=0} {\on{Fl}_u^{-t}}^*\psi \cdot e^{ta}
    = -\cL_u\psi + a \psi.
\end{align}

We first explicitly compute their Lie bracket; 
\begin{lemma}\label{lem:YIsLieAlgebraAntiHomomorphism}

    For $(u,a),(v,b)\in \AlgSemiD$ and $\psi \in \cU$, we have
    \begin{align}
       [Y^{(u,a)}, Y^{(v,b)}](\psi)=-\cL_{[u,v]}\psi + (\cL_u b - \cL_v a)\psi.
    \end{align}
\end{lemma}

% We then have the following relation.
% \begin{lemma}\label{lem:YIsLieAlgebraAntiHomomorphism
% The map
% \begin{align}
%         \AlgSemiD &\to \Gamma(T\cU) \\
%         (u,a) &\mapsto Y^{(u,a)}
% \end{align}
% is a Lie algebra homomorphism. That is,
% \begin{align}
% \label{lem:YIsLieAlgebraAntiHomomorphism}
%          Y^{[(u,a),(v,b)]} = [Y^{(u,a)}, Y^{(v,b)}]
% \end{align}
% for all $(u,a), (v,b)\in \AlgSemiD$.
% \end{lemma}

\begin{proof}
     To compute the Lie derivative of vector fields $[Y^{(u,a)}, Y^{(v,b)}]=\cL_{Y^{(u,a)}}Y^{(v,b)}$, we use the formula:
    \begin{align}
        \cL_{Y^{(u,a)}}Y^{(v,b)}(\psi)=\frac{d}{dt}\Big|_{t=0} D\Phi^{-t}_{Y^{(u,a)}}\left( Y^{(v,b)}(\Phi^{t}_{Y^{(u,a)}}(\psi))\right), \quad \psi\in \cU
    \end{align}
    where $\Phi^{t}_{Y}\colon \cU \to \cU$ is the flow map along $Y\in\Gamma(T\cU)$ defined by the ODE
    \begin{align}
        &\frac{d}{dt}(\Phi^{t}_{Y}(\psi))=Y(\Phi^{t}_{Y}(\psi)),\\
        &\Phi^{0}_{Y}=\id_{\cU},
    \end{align}
    and $D\Phi^t_Y|_\psi\colon T_\psi \cU\to T_{\Phi^t_Y(\psi)}\cU$ is its differential at $\psi$. For the fundamental vector field $Y^{(u,a)}$ associated to $(u,a)\in \AlgSemiD$, we have by definition   $\Phi^t_{Y^{(u,a)}}(\psi)={\on{Fl}_u^{-t}}^*\psi\cdot e^{ta}$. 
    Since
    \begin{align}
        Y^{(v,b)}(\Phi^t_{Y^{(u,a)}}(\psi))
        % =\frac{d}{ds}\Big|_{s=0} {\on{Fl}_v^{-s}}^* (\Phi^t_{Y^{(u,a)}}(\psi))\cdot e^{sb} 
        &= \frac{d}{ds}\Big|_{s=0} \Phi^s_{Y^{(v,b)}} \left( \Phi^t_{Y^{(u,a)}}(\psi)\right),  
        % &=  \frac{d}{ds}\big|_{s=0}  {\on{Fl}_v^{-s}}_*{\on{Fl}_u^{-t}}_*\psi \cdot {\on{Fl}_v^{-s}}_*e^{ta} \cdot e^{sb}, 
    \end{align}
   we have
    \begin{align}
        \frac{d}{dt}\Big|_{t=0} D\Phi^{-t}_{Y^{(u,a)}}\left( Y^{(v,b)}(\Phi^{t}_{Y^{(u,a)}}(\psi))\right)
        &= \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} \Phi^{-t}_{Y^{(u,a)}} \left( \Phi^{s}_{Y^{(v,b)}} \left( \Phi^{t}_{Y^{(u,a)}}(\psi) \right) \right)\\
        &= \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} \Phi^{-t}_{Y^{(u,a)}} \left( \Phi^{s}_{Y^{(v,b)}} (\psi) \right)+\Phi^{s}_{Y^{(v,b)}} \left( \Phi^{t}_{Y^{(u,a)}} (\psi) \right)
    \end{align}
    using the product rule.
    We then compute the first term:
    \begin{align}
        \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} \Phi^{-t}_{Y^{(u,a)}} \left( \Phi^{s}_{Y^{(v,b)}} (\psi) \right)
        &= \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} {\on{Fl}^{t}_u}^* \left({\on{Fl}^{-s}_v}^* \psi \cdot e^{sb} \right)\cdot e^{-ta}\\
        &=\frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} {\on{Fl}^{t}_u}^* \left({\on{Fl}^{-s}_v}^* \psi \right)\cdot {\on{Fl}^{t}_u}^* e^{sb}  \cdot e^{-ta}\\
        &=\frac{d}{ds}\Big|_{s=0} \cL_u \left({\on{Fl}^{-s}_v}^* \psi \right)\cdot e^{sb} + \left({\on{Fl}^{-s}_v}^* \psi \right)\cL_u ( e^{sb})-\left({\on{Fl}^{-s}_v}^* \psi \right)\cdot e^{sb}\cdot a\\
        &=-\cL_u\cL_v \psi + \cL_u\psi\cdot b-\cL_v \psi \cdot 0+\psi \cL_u b+\cL_v \psi\cdot a - \psi \cdot b\cdot a\\
        &=-\cL_u\cL_v \psi + b\cL_u \psi +a\cL_v\psi+\psi \cL_u b-ab\psi
    \end{align}
    where we used $\cL_u (e^{sb})=s\cL_ub \cdot e^{sb}$. Similarly, we have 
    \begin{align}
         \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0}\Phi^{s}_{Y^{(v,b)}} \left( \Phi^{t}_{Y^{(u,a)}} (\psi) \right)
         =\cL_v \cL_u\psi-a\cL_v\psi - b \cL_u\psi - \psi \cL_v a + ab \psi.
    \end{align}
    Summing these results, we have
    \begin{align}
         \frac{d}{dt}\Big|_{t=0} \frac{d}{ds}\Big|_{s=0} \Phi^{-t}_{Y^{(u,a)}} \left( \Phi^{s}_{Y^{(v,b)}} (\psi) \right)+\Phi^{s}_{Y^{(v,b)}} \left( \Phi^{t}_{Y^{(u,a)}} (\psi) \right)
         &=-\cL_u\cL_v \psi +\cL_v \cL_u \psi +\psi\cL_v b - \psi\cL_v a\\
         &=-\cL_{[u,v]} \psi +(\cL_u b - \cL_v a)\psi.
    \end{align}
    We thus obtained the stated expression.
\end{proof}

\begin{remark}
The fundamental vector field mapping $\AlgSemiD\to \Gamma(T\cU)$ given by \eqref{eq:fundamental_vector_field_DC}
is in fact a Lie algebra anti-homomorphism.
To see this, notice that the mapping $ \diff(M)\to  \on{Der}(C^\infty(M,\CC))$ by $u\to \cL_u$ is an anti-homomorphism as noted in \autoref{rmk:Lie_algebra_homomorphism}. Hence we have
\begin{align}
     Y^{[(u,a),(v,b)]} 
     = Y^{([u,v], -\cL_u b +\cL_v a )} 
     = \cL_{[u,v]}\psi + (\cL_v a-\cL_u b )\psi
     = - [Y^{(u,a)}, Y^{(v,b)}]
\end{align}
 where the last equality is the application of \autoref{lem:YIsLieAlgebraAntiHomomorphism}.
\end{remark}

\paragraph{Evaluation of $\Theta$ and $d\Theta$}
We now evaluate each term in \eqref{eq:exterior_derivative_in_U} with \(Y_1 = Y^{(u,a)}\) and \(Y_2 = Y^{(v,b)}\).
First, we express the term $\Th_\psi(Y^{(u,a)}) = \Th_\psi(-\cL_u \psi + a\psi)$ more explicitly.

\begin{lemma}[Evaluation of $\Th_\psi$]\label{lem:evaluation_of_Theta}
For any $(u,a) \in \AlgSemiD$, we have
\begin{align}
   \Th_\psi(-\cL_u \psi + a\psi)
   &= \Th_\psi(-\cL_u \psi) +  \Th_\psi(a \psi) \\
   &= \oneOverTwoPi \left( - \langle \Lambda(\psi), \iota_u \mu \rangle + \int \Im a \ \mu \right),
\end{align}
where $\Lambda(\psi)$, defined in \eqref{eq:integral_lambda_psi}, is the current associated to the circle differential of \(\psi\), and \(\mu\) is the volume form on \(M\).
\end{lemma}




\begin{proof}
By direct computation we get
\begin{align}
    \Th_{\psi}(a\psi)
        = \oneOverTwoPi \int \Im \frac{a\psi \bar\psi}{ |\psi|^2}\mu
        = \oneOverTwoPi  \int \Im a\ \mu.
\end{align}
We then compute $\Th_{\psi}(-\cL_u \psi)$.
Using the local expression $\psi=r e^{i\phi}$ with $r=|\psi|$ and a function $\phi$ around each $x\in M\setminus \psi^{-1}(0)$,
 we have locally
\begin{align}
    d\psi= e^{i\phi}  dr + r e^{i\phi} i d\phi
\end{align}
and 
\begin{align}
     \Im \frac{\iota_u d \psi  \cdot \bar \psi}{|\psi^2|}
     = \Im \frac{\iota_u  dr}{r}+ \Im (i \iota_v  d\phi) = 0+\iota_u \lambda_\psi.
\end{align}
Hence, we obtain using the product rule for the interior product that,
\begin{align}
     \Th_{\psi}(-\cL_u \psi)
&=- \oneOverTwoPi \int \Im \frac{\iota_u d \psi  \cdot \bar \psi} {|\psi^2|}\mu
= - \oneOverTwoPi \int \iota_u \lambda_\psi \ \mu
\\& = - \oneOverTwoPi \int \lambda_\psi \wedge \iota_u \mu
=- \oneOverTwoPi  \langle\Lambda(\psi), \iota_u \mu\rangle.
\end{align}
\end{proof}

We next need an auxiliary result describing how $\Th$ varies under the $\SemiD$-action. 

\begin{lemma}\label{lem:theta_under_S_action}
Let $\psi\in\cU$, $(f,e^\varphi)\in\SemiD$, and $\hat \psi\coloneqq (f,e^\varphi)\rhd \psi=\psi\circ f^{-1}\cdot e ^\varphi$, For  the fundamental vector field of $Y^{(u,a)}\in \Gamma(T\cU)$ of $(u,a)\in \AlgSemiD$, we have
\begin{align}
    \Theta_{\hat \psi}(Y^{(u,a)})
    = \oneOverTwoPi \left( -\int \iota_u ({f^{-1}}^* \lambda_\psi + \Im d\varphi) \mu + \int a\,\mu \right).
\end{align}

    % Under the $\SemiD$-action $(f,e^\varphi)\rhd \psi = \psi\circ f^{-1} \cdot e^\varphi\eqqcolon \tilde \psi $ and the lifted action
    % \begin{eqnarray}
    %     &\SemiD \times T\cU &\to T\cU\\
    %     & (f,\varphi), (\psi,-\cL_u \psi+ a\psi) &\mapsto (\tilde\psi, -\cL_u\tilde \psi+a \tilde \psi),
    % \end{eqnarray}
    % we have 
    % \begin{align}
    %     \Th_{\tilde\psi}(-\cL_u\tilde \psi+\varphi \tilde \psi)
    %     &=\Th_{\tilde\psi}(-\cL_u\tilde \psi)+ \Th_{\tilde\psi}(a \tilde \psi)\\
    %     &=-\int \iota_u ({f^{-1}}^* \lambda_\psi + \Im d\varphi)\ \mu+\int a\ \mu.
    % \end{align}

    Additionally, if $u$ is divergence-free, it simplifies to
    \begin{align}
         \Th_{\hat\psi}(Y^{(u,a)})
         =\oneOverTwoPi \left( -\int \iota_u {f^{-1}}^*  \lambda_\psi + \int a \, \mu \right).
    \end{align}
    
\end{lemma}

\begin{proof}
Clearly we have $ \Th_{\hat\psi}(a \hat \psi)=\int a \ \mu$. 
We now compute  $ \Th_{\hat\psi}(-\cL_u \hat \psi)$.
From 
    \begin{align}
        d\hat \psi
        &= d(\psi\circ {f^{-1}} \cdot e^\varphi)
        =e^\varphi \cdot d(\psi\circ {f^{-1}} ) + \psi\circ {f^{-1}} \cdot  e^\varphi d\varphi
        =e^\varphi d(\psi\circ {f^{-1}} ) + \hat \psi d\varphi
         % \\
         % &=d|\psi\circ f| +e^\varphi \cdot\psi\circ f \cdot \lambda_{\psi\circ f} + e^\varphi \cdot \psi\circ f \cdot d\varphi
         % = \hat \psi (\lambda_{\psi\circ f} + d\varphi)
    \end{align}
    it follows that,
    \begin{align}
\twoPi\  \Th_{\hat \psi}(-\cL_u\hat\psi)
&= -\int \Im \frac{\iota_u d\hat\psi \cdot \bar{\hat \psi}}{ |\hat \psi|^2}\mu
        =-\int \Im \frac{e^\varphi \iota_u d(\psi\circ f^{-1})  + \bar{\hat \psi} \hat \psi  \iota_u d\varphi}{ |\hat \psi|^2}\mu\\
        &=-\int \Im \frac{\bar{\hat \psi} \hat \psi i \iota_u  \lambda_{\psi\circ f}+|\hat \psi|^2 \iota_u d\varphi}{|\hat \psi|^2}\mu
        % &=\Im \frac{\iota_u d\psi \bar\psi}{ |\psi|^2}\cdot \frac{e^\varphi\cdot e^{-\varphi}}{|e^\varphi|^2}+ \Im \frac{\psi_a \iota_u d\varphi \bar\psi_\varphi}{ |\psi_\varphi|^2}\\
        =-\int \left(\iota_u \lambda_{\psi\circ f}+\iota_u \Im d \varphi \right) \mu.
    \end{align}
    We have by definition that $\lambda_{\psi\circ f}= (\psi\circ f) ^*\tau^*\sigma=f^*\psi^*\tau^*\sigma=f^*\lambda_{\psi} $, which gives the stated expression.
   % Let us set $\psi$ $\psi_\varphi\coloneqq \psi e^{\varphi}$.
    
    Finally, if $u$ is divergence-free, we get by integral by parts that
    \begin{align}
        \int \iota_u d \Im \varphi \ \mu 
        = -\int \Im \varphi \ \cL_u \mu 
        =0.
    \end{align}
    
\end{proof}

Using these results, we can now explicitly evaluate each term in \eqref{eq:exterior_derivative_in_U} and compute \(d\Theta\).
\begin{lemma}\label{lem:dTheta}
    We have
%     \begin{align}
% &\cL_{Y^{(u,a)}}\iota_{Y^{(v,b)}}\Theta_\psi=\int \iota_v \cL_u \lambda_\psi -(\cL_v \Im a) \psi \  \mu,\\
% &\cL_{Y^{(v,b)}}\iota_{Y^{(u,a)}}\Theta_\psi=\int \iota_u \cL_v \lambda_\psi -(\cL_u \Im b) \psi \  \mu,\\
% &\iota_{[Y^{(u,a)},Y^{(v,b)}]}\Theta_\psi=\int  -\iota_{[u,v]}\lambda_\psi +(\cL_u \Im b -\cL_v \Im a)\psi\  \mu.
%     \end{align}
\begin{align}
\label{eq:dThetaLemma}
    d\Theta_\psi(Y^{(u,a)},Y^{(v,b)}) =\oneOverTwoPi\langle \partial\Lambda(\psi),  \iota_v\iota_u \mu \rangle.
\end{align}
where $\partial\colon \cD_{m-1}(M)\to \cD_{m-2}(M)$ is the boundary operator for currents.
\end{lemma}
\begin{proof}
    First, we have 
    \begin{align}
       \Theta_\psi([Y^{(u,a)},Y^{(v,b)}])
       =\oneOverTwoPi\int  \left( -\iota_{[u,v]}\lambda_\psi +\cL_u \Im b -\cL_v \Im a\right) \mu
    \end{align}
   using the expressions of $[Y^{(u,a)},Y^{(v,b)}]$  (\autoref{lem:YIsLieAlgebraAntiHomomorphism}) and of $\Theta_\psi$ (\autoref{lem:evaluation_of_Theta}). 
    
    We then compute $\cL_{Y^{(u,a)}}\iota_{Y^{(v,b)}}\Theta_\psi$. Applying \autoref{lem:theta_under_S_action} to  the time-$t$ flow map of $Y^{(u,a)}$ given by $\Phi^t_{Y^{(u,a)}}(\psi)={\on{Fl}_u^{-t}}^*\psi\cdot e^{ta}$, we have
    \begin{align}
        \cL_{Y^{(u,a)}}\Theta(Y^{(v,b)}(\psi))
        &=\frac{d}{dt}\Big|_{t=0}\Theta(Y^{(v,b)}(\Phi^t_{Y^{(u,a)}}(\psi)))\\
        &=\oneOverTwoPi\left(\frac{d}{dt}\Big|_{t=0}\int -\left(\iota_v {\on{Fl}_{u}^{-t}}^*\lambda_\psi + \iota_v d\Im (t a)\right) \mu + \int \Im b\, \mu \right) \\
        &=\oneOverTwoPi\int \left(\iota_v \cL_u\lambda_\psi - \cL_v \Im a\right) \mu
    \end{align}
    and \(\cL_{Y^{(v,b)}}\Theta(Y^{(u,a)}(\psi))=\oneOverTwoPi\int \left(\iota_u \cL_v \lambda_\psi -\cL_u \Im b  \right)  \mu \) in the same way.

    We can now compute 
    \begin{align}
       \iota_{Y^{(v,b)}} \iota_{Y^{(u,a)}}d\Theta_\psi 
       &=  \cL_{Y^{(u,a)}}\iota_{Y^{(v,b)}} \Theta_\psi - \cL_{Y^{(v,b)}} \iota_{Y^{(u,a)}}\Theta_\psi - \iota_{[Y^{(u,a)},Y^{(v,b)}]}\Theta_\psi\\
       &=\oneOverTwoPi\int \Big(\iota_v \cL_u\lambda_\psi - \cL_v \Im a - \iota_u \cL_v \lambda_\psi + \cL_u \Im b \\
       &\qquad\quad\quad + \iota_{[u,v]}\lambda_\psi + \cL_v \Im a - \cL_u \Im b \Big) \mu\\
       &=\oneOverTwoPi\int \left( \iota_v \cL_u\lambda_\psi - \iota_u \cL_v \lambda_\psi + \iota_{[u,v]}\lambda_\psi\right) \mu.
    \end{align}
    We then simplify the integrand:
\begin{align}
% -\iota_v \cL_u \lambda_\psi + \iota_u \cL_v \lambda_\psi +\iota_{[v,u]}\lambda_\psi
% &= (-\iota_v \cL_u + \iota_u \cL_v  +\cL_v\iota_u - \iota_u \cL_v ) \lambda_\psi\\
% &= (-\iota_v \cL_u + \cL_v\iota_u) \lambda_\psi\\
% &=(-\iota_v d \iota_u - \iota_v\iota_u d +d\iota_v \iota_u + \iota_v d\iota_u)\lambda_\psi\\
% &= (-\iota_v\iota_u d + d\iota_v \iota_u)\lambda_\psi\\
% &= \iota_u\iota_v d \lambda_\psi -0.
\iota_v \cL_u \lambda_\psi - \iota_u \cL_v \lambda_\psi -\iota_{[v,u]}\lambda_\psi
&= (\iota_v \cL_u - \iota_u \cL_v  -\cL_v\iota_u + \iota_u \cL_v ) \lambda_\psi\\
&= (\iota_v \cL_u - \cL_v\iota_u) \lambda_\psi\\
&=(\iota_v d \iota_u + \iota_v\iota_u d -d\iota_v \iota_u - \iota_v d\iota_u)\lambda_\psi\\
&= (\iota_v\iota_u d - d\iota_v \iota_u)\lambda_\psi\\
&= \iota_v\iota_u d \lambda_\psi -0.
\end{align}
Hence we obtain by the product rule that, 
\begin{align}
    % -d\Th_\psi(\dot \psi,\ring \psi)
    % &= \int \iota_u\iota_v d \lambda_\psi \ \mu
    % = \int d \lambda_\psi \wedge \iota_v\iota_u\mu
    % = - \int \lambda_\psi \wedge d\iota_v\iota_u\mu \\
    % &=-\langle \Lambda(\psi), d\iota_v\iota_u \mu\rangle
    % =-\langle \partial \Lambda(\psi), \iota_v\iota_u \mu\rangle.
     \iota_{Y^{(v,b)}} \iota_{Y^{(u,a)}}d\Theta_\psi 
    &= \oneOverTwoPi\int \iota_v\iota_u d \lambda_\psi \ \mu
    = -\oneOverTwoPi\int d \lambda_\psi \wedge \iota_u\iota_v\mu
    = - \oneOverTwoPi\int \lambda_\psi \wedge d\iota_u\iota_v\mu \\
    &=\oneOverTwoPi\langle \Lambda(\psi), d\iota_v\iota_u \mu\rangle
    =\oneOverTwoPi\langle \partial \Lambda(\psi), \iota_v\iota_u \mu\rangle
\end{align}
and get the stated expression.
\end{proof}

We are now one step away from proving \autoref{th:quantum_liouville_form}. To complete the proof, we show the next lemma asserting that \(\oneOverTwoPi\partial\Lambda(\psi)\) in \eqref{eq:dThetaLemma} is \(\delta_{\Pi\psi}\).

% \begin{lemma}\label{lem:perturbed_circle_differential} 
% For $\dot \psi = \cL_u \psi + \varphi\psi \in T_\psi \cF$ with $(u,\varphi)\in \diff(M) \ltimes C^\infty(M,\CC)$, we have
%     \begin{align}
%        \cL_{\dot\psi} \lambda_\psi 
%        =\cL_u \lambda_\psi + \Im d\varphi.
%     \end{align}
% \end{lemma}

% \begin{proof}
%     Let us set $\on{Fl}_t^u\colon M\to M$ the time-$t$ flow along the vector field $u$ and $\psi_t \coloneqq (\on{Fl}^u_t,e^{t\varphi})\rhd \psi= \psi\circ \on{Fl}_t^u \cdot e^{t \varphi} $ so that we have $\psi_0=\psi$ and $\partial_t|_{t=0} \psi_t = \dot\psi$. Our goal is to compute $\cL_{\dot \psi} \lambda_\psi[v]=\frac{d}{dt}\Big|_{t=0} 
%        \lambda_{\psi_t}[v]$ for any $v\in T_x M$ at $x$ in $\{\psi \neq 0\}\subset M$.

%        First notice that for a fixed $t$ we have
%     \begin{align}\label{eq:explicit_th_psit}
%        \tau\circ \psi_t 
%        = \tau \circ (\psi \circ \on{Fl}_u^t \cdot e^{t\varphi})
%        = (\tau \circ \psi \circ \on{Fl}_u^t )\cdot e^{ti\phi}
%     \end{align}
%     where we write $\phi=\Im\varphi$ for simplicity.
    
%     With this the differential $d(\tau\circ \psi_t)|_x\colon T_x M \to T_{\tau\circ \psi_t(x)}\SS^1$ is given as,
%     \begin{align}
%         d(\tau\circ \psi_t)
%         = e^{it\phi} d(\tau \circ \psi \circ \on{Fl}_u^t )+(\tau \circ \psi \circ \on{Fl}_u^t )\cdot e^{it\phi} it d\phi.
%     \end{align}
% Hence we get
% \begin{align}
%     \psi_t^*\tau^*\sigma|_x(v)
%     &=\sigma|_{\tau\circ \psi_t(x)}(d(\tau\circ \psi_t) [v])\\
%     &=\sigma|_{\tau\circ \psi_t(x)} (e^{it\phi} d(\tau \circ \psi \circ \on{Fl}_u^t ) [v])
%     + \sigma|_{\tau\circ \psi_t(x)} ((\tau \circ \psi \circ \on{Fl}_u^t )\cdot e^{it\phi} it d\phi [v]). \label{eq:two_terms_circle_differential}
% \end{align}

% For the first term, we have due to \eqref{eq:explicit_th_psit} and the rotation invariance of $\sigma$ that
% \begin{align}
%     \sigma|_{(\tau \circ \psi \circ \on{Fl}_u^t(x) )\cdot e^{ti\phi(x)}} (e^{it\phi} d(\tau \circ \psi \circ \on{Fl}_u^t ) [v])
%     &=\sigma_{\tau\circ \psi\circ \on{Fl}_u^t(x)}(d(\tau \circ \psi \circ \on{Fl}_u^t ) [v])\\
%     &=\iota_v{\on{Fl}_u^t }^* \psi^* \tau^*\sigma|_x.
% \end{align}
% By taking the time derivative, we get
% \begin{align}
%     \frac{d}{dt}\Big|_{t=0} \iota_v{\on{Fl}_u^t }^* \psi^* \tau^*\sigma
%     &= \iota_v{\on{Fl}_u^t }^* \cL_u \psi^* \tau^*\sigma  \Big|_{t=0} = \iota_v \cL_u \psi^* \tau^*\sigma.
% \end{align}

% For the second term of \eqref{eq:two_terms_circle_differential}, we again apply the rotation invariance of $\sigma$ to get
% \begin{align}
%      \sigma|_{(\tau \circ \psi \circ \on{Fl}_u^t(x) )\cdot e^{ti\phi(x)}} ((\tau \circ \psi \circ \on{Fl}_u^t )\cdot e^{it\phi} it d\phi [v])
%      &= \sigma|_1 (it d\phi[v]) 
%      = t d\phi[v].
% \end{align}
% By taking the time-derivative we get $\iota_v d\phi$. 

% Summing the two terms, we obtain
% \begin{align}
%     \frac{d}{dt}\Big|_{t=0}\iota_v \psi_t^*\tau^*\sigma= \iota_v (\cL_v \psi^*\tau^*\sigma + d\phi),
% \end{align}
% which concludes the proof.
% \end{proof}

\begin{lemma}\label{lem:lam_psi_in_fiber}
Let $\Lambda\colon \cU\to \cD_{m-1}(M)$ be the circle differential map as defined in \autoref{def:circle_differential} and $\partial\colon \cD_{m-1}(M)\to \cD_{m-2}(M)$ be the boundary operator for currents.
Then we have $\oneOverTwoPi\partial\circ \Lambda=\Pi\colon \cU\to  \cD_{\cO}$. That is, for any $\psi\in \Pi^{-1}\gamma$ for some $\gamma\in \cO$, we have $\oneOverTwoPi \partial (\Lambda(\psi))=\delgam$.
\end{lemma}
\begin{proof}
    We have for $\alpha\in \Om^{m-2}(M)$ that
    \begin{align}
        \langle \partial \Lambda(\psi), \alpha\rangle
        =\langle\Lambda(\psi), d\alpha\rangle
        =\int_{M} \lambda_\psi\wedge d\alpha.
    \end{align}
    As in the proof of \autoref{lem:finiteness_of_circle_differential}, we have by the coarea formula (\autoref{prop:coarea_formula}) that,
    \begin{align}
        \int_{M} \lambda_\psi \wedge d\alpha
        % &=\int_{\{d\phi\neq 0\}} \lambda_\psi \wedge d\alpha
        % =\int_{\SS^1_{\on{reg} \phi}} \langle \delta_{\phi^{-1}(\cdot )\setminus B_\epsilon}, d\alpha\rangle \sigma 
        =\int_{\SS^1} \langle \delta_{(\phi^{-1}(\cdot )\cap \Mreg)}, d\alpha\rangle \sigma
        =\int_{\SS^1} \langle \delta_{\partial(\phi^{-1}(\cdot )\cap \Mreg)}, \alpha\rangle \sigma
    \end{align}
    where $\phi\colon {M\setminus\psi^{-1}(0)}\to \SS^1 $ is the phase map and $\Mreg$ is the set of regular points of $\phi$ as in \autoref{lem:finiteness_of_circle_differential}.
    % Note here that the set $\phi^{-1}(s)\setminus \{d\phi=0\}$ for each $s$ is a $m-1$ dimensional submanifold and $\langle \delta_{\partial(\phi^{-1}(\cdot )\setminus \{d\phi=0\})}, \alpha\rangle$ is finite since $\alpha$ is continuos.
    
    Due to Sard's theorem, almost $\sigma$-every point of $\SS^1$ is a regular value of $\phi$. Hence we have,
    \begin{align}
    \int_{\SS^1} \langle \delta_{\partial(\phi^{-1}(\cdot )\cap \Mreg)}, \alpha\rangle \sigma
    &=\int_{\SS^1_{\on{reg} \phi}} \langle \delta_{\partial(\phi^{-1}(\cdot ))}, \alpha\rangle \sigma\\
    &=\int_{\SS^1_{\on{reg} \phi}} \langle \delta_\gamma, \alpha\rangle \sigma= \langle \delta_\gamma, \alpha\rangle \cdot \int_{\SS^1_{\on{reg} \phi}}\sigma=2\pi \langle \delta_\gamma, \alpha\rangle
    \end{align}
    where $\SS^1_{\on{reg} \phi}$ is the set of regular values of $\phi$.
\end{proof}

We now complete the proof of the theorem.
\begin{proof}[Proof of \autoref{th:quantum_liouville_form}]
By \autoref{lem:dTheta} and \autoref{lem:lam_psi_in_fiber} we have for $\psi\in \Pi^{-1}\gamma$ and $\dot \psi=-\cL_u \psi + a \psi, \ring \psi=-\cL_v\psi+ b\psi$ that,
\begin{align}
    d\Theta_\psi(\dot \psi, \ring \psi)
    =\oneOverTwoPi\langle \partial\Lambda(\psi), \iota_v \iota_u \mu\rangle
    =\langle \delgam, \iota_v \iota_u \mu\rangle
    = \omega_\delgam(-\cL_u \delgam, -\cL_v \delgam)
    = \Pi^*\omega_\psi(\dot \psi, \ring \psi).
\end{align}
\end{proof}

\autoref{lem:lam_psi_in_fiber} also gives a physical interpretation of the Liouville form $\Theta$ in the following sense.
\begin{corollary}[Liouville form as flux]\label{cor:Liouville_form_flux}
We have for $\psi\in \cU, u\in \diff(M)$, 
    \begin{align}
        \Th_\psi(-\cL_u \psi)
        =-\oneOverTwoPi\fluxMone(\Lambda(\psi),u)
         = -\oneOverTwoPi\int_{\SS^1_{\on{reg} \phi}} \fluxMone(\delta_{\phi^{-1}(\cdot)},u) \sigma.
    \end{align}

    Moreover, if $u$ is exact divergence-free, we have 
    \begin{align}
         \Th_\psi(-\cL_u \psi)= - \fluxMone(\delta_{\phi^{-1}(s)},u) =- \fluxMtwo(\delta_{\Pi\psi},u), \quad \forall s\in \SS^1_{\on{reg}\phi}.
    \end{align}
    Here, $\fluxMone$ and $\fluxMtwo$ are flux for $m-1$ currents and $m-2$ exact currents respectively (\autoref{def:flux}).
\end{corollary}
\begin{proof}
    The first equality is immediate from \autoref{lem:lam_psi_in_fiber}. The second equality follows from the exactness of the $m-1$ form $\iota_u \mu$.
\end{proof}

\begin{remark}[Physical interpretation of $\Th_\psi$]\label{rmk:meaning_of_Th}
By Sard's theorem, $\sigma$-almost every point of \(\SS^1\) is a regular value of the phase map \(\phi\), and each such point \(s\) defines a hypersurface \(\phi^{-1}(s)\). The first expression in \autoref{cor:Liouville_form_flux} indicates that \(\Th(-\cL_u \psi)\) measures the average flux of the vector field \(u\) through these phase level hypersurfaces. The second expression in the corollary shows that if the vector field $u$ is exact divergence-free, the flux through all these level surfaces are actually the same.

Together with \autoref{lem:evaluation_of_Theta}, we now observe that \(\Th(\dot \psi)\) captures the infinitesimal phase shift of $\psi$ over the space and the flux of vector fields through phase hypersurfaces. In other words, it is the average of the swept volume by the $\SS^1$-family of hypersurfaces $\{\phi^{-1}(s)\}_s$.  
\end{remark}

% We are now ready to prove \autoref{th:quantum_liouville_form}.
% \begin{proof}[Proof of \autoref{th:quantum_liouville_form}]
%     We compute $\tilde Z_\psi=d\Th_\psi$ using the formula
%     \begin{align}
%         d\Theta(\dot \psi,\ring \psi)=   \cL_{\dot \psi} \Th(\ring \psi) - \cL_{\ring \psi} \Th(\dot \psi)- \Theta ([\dot \psi,\ring \psi]).
%     \end{align}
% % 
% Here we regard $\dot \psi, \ring \psi$, by slightly abusing notation, as vector fields $\cU$ defined by
%     \begin{align}
%         \dot \psi(\psi)&=-\cL_u \psi + a\psi,\\
%         \ring \psi(\psi)&=-\cL_v \psi + b\psi
%     \end{align}
% at each $\psi\in \cU$, with fixed $u,v \in \frak{X}(M)$ and $a,b \in C^\infty(M,\CC)$.

%     We first compute $\dot \psi \circ \ring \psi$. 
%     For an arbitrary smooth function $\cA\colon \cU\to \RR$, we see 
%     \begin{align}
%       ( \dot \psi \circ \ring \psi) \cA(\psi)
%       &=  \dot \psi \left(\ring \psi \cA\right)(\psi)
%       = \frac{d}{dt} \Big|_{t=0} (\ring\psi \cA)(\on{Fl}_{\dot \psi}^t\circ  \psi)
%       = \frac{d}{dt} \Big|_{t=0} \frac{d}{d\epsilon} \Big|_{\epsilon=0} \cA(\on{Fl}_{\ring \psi}^\epsilon \circ (\on{Fl}_{\dot \psi}^t\circ  \psi))
%     \end{align}
%     where $\on{Fl}_{\dot \psi}^t\colon \cU\to \cU$ is the time-$t$ flow map along $\dot \psi$, which is given as
%     \begin{align}
%        \on{Fl}_{\dot \psi}^t (\psi) =(\on{Fl}^t_u, e^{at})\rhd \psi = \psi\circ {\on{Fl}^t_u}^{-1} \cdot e^{at}
%     \end{align}
%     with the time-$t$ flow map $\on{Fl}^t_u\colon M\to M$ along $u\in \frak{X}(M)$, and $\on{Fl}^\epsilon_{\ring\psi}$ is given in the same way.
%     % 
%     From this we get,
% \begin{align}
%     (\dot \psi \circ \ring \psi)(\psi)
%     &= \frac{d}{dt} \Big|_{t=0} \frac{d}{d\epsilon} \Big|_{\epsilon=0} \on{Fl}_{\ring \psi}^\epsilon \left( \on{Fl}_{\dot \psi}^t(\psi)\right)\\
%     &= \frac{d}{dt} \Big|_{t=0} \frac{d}{d\epsilon} \Big|_{\epsilon=0} \on{Fl}_{\ring \psi}^\epsilon \left(\psi\circ {\on{Fl}_u^t}^{-1} \cdot e^{ta}\right)\\
%     &= \frac{d}{dt} \Big|_{t=0} \frac{d}{d\epsilon} \Big|_{\epsilon=0} \left(\psi\circ {\on{Fl}_u^t}^{-1} \circ {\on{Fl}_v^\epsilon}^{-1} \right) \cdot \left(e^{t a }\circ {\on{Fl}_v^\epsilon}^{-1} \right)\cdot e^{\epsilon b}\\
%     &=\frac{d}{dt} \Big|_{t=0} -\cL_v \left(\psi\circ{\on{Fl}_u^t}^{-1} \right)\cdot e^{ta}- \left(\psi\circ {\on{Fl}_u^t}^{-1}\right) \cdot \cL_v e^{t a}+ \left(\psi\circ{\on{Fl}_u^t}^{-1} \right) \cdot e^{ta}\cdot b\\
%     &=\cL_v \cL_u \psi - a\cL_v \psi - \psi \cL_v a - b \cL_u \psi + ab\psi.
% \end{align}
% Here we used $\cL_v e^{ta}=\cL_v (ta)\cdot e^{ta}= t \cL_v a\cdot e^{ta}$ and $\frac{d}{dt} \Big|_{t=0} \cL_v e^{ta}=\cL_v a$. In the same way, we get
% \begin{align}
%     \ring \psi \circ \dot \psi 
%     = \cL_u \cL_v \psi - b\cL_u \psi - \psi \cL_u b - a \cL_v \psi + ab\psi. 
% \end{align}
% From this we get,
% \begin{align}
%     [\dot \psi, \ring \psi]
%     &=\dot \psi \circ \ring \psi - \ring \psi \circ \dot \psi
%     =\cL_v \cL_u \psi - \cL_u \cL_v \psi -\psi \cL_v a + \psi \cL_u b \\
%     &=\cL_{[v,u]} \psi + ( \cL_u b-\cL_v a )\psi.
% \end{align}
% Note that neither $ \ring \psi \circ \dot \psi$ nor $\dot \psi \circ \ring \psi $ is in $\frak{S}=\diff(M)\ltimes C^\infty (M,\CC)$ but $[\dot \psi, \ring \psi]\in \frak{S}$.
% Hence we have
% \begin{align}
%     \Theta_\psi ([\dot \psi, \ring \psi])= \int \iota_{[v,u]}\lambda_\psi + \Im(\cL_u b-\cL_v  a ) \ \mu
%     % =\int \iota_{[v,u]}\lambda_\psi.
% \end{align}

% Now we compute $\cL_{\dot \psi}\Th_\psi(\ring \psi)$.
% By setting
% \begin{align}
%     \psi_t &\coloneqq (\on{Fl}_u^{t}, e^{at} ) \rhd \psi=\psi\circ {\on{Fl}_u^{t}}^{-1}\cdot e^{at}\\
%     \ring \psi_t&\coloneqq \cL_v \psi_t + a \psi_t,
% \end{align}
% and applying \lemref{lem:theta_under_S_action}, we have
% \begin{align}
%     \cL_{\dot \psi} \Th_\psi (\ring \psi)
%     &= \frac{d}{dt} \Big|_{t=0}\Th_{\psi_t}(\ring \psi_t)\\ 
%     &= - \frac{d}{dt} \Big|_{t=0}   \int \iota_v \psi_t^* \tau^*\sigma +  \Im a \ \mu\\
%     &= - \int \iota_v \cL_u \psi^*\tau^* \sigma + \iota_v d\Im a + 0 \ \mu\\
%     &= -\int \iota_v \cL_u \lambda_\psi + \Im \cL_v  a \ \mu.
% \end{align}
% % \begin{align}
% %     \cL_{\ring \psi} \Th_\psi (\dot \psi)
% %     &= \frac{d}{d\epsilon} \Big|_{\epsilon=0}\Th_{\psi_\epsilon}(\dot \psi_\epsilon)\\ 
% %     &= \frac{d}{d\epsilon} \Big|_{\epsilon=0} \lim_{\varepsilon\to 0}  \int_{M\setminus B^\varepsilon_{\psi_\epsilon}} \iota_u \arg |_{\psi_\epsilon} +  \Im a \ \mu\\
% %     &= \frac{d}{d\epsilon} \Big|_{\epsilon=0} \lim_{\varepsilon\to 0}  \int_{M\setminus B^\varepsilon_{\psi_\epsilon}} \iota_u \sigma^*\arg |_{\psi_\epsilon} +0 \ \mu\\
% %     &= \frac{d}{d\epsilon} \Big|_{\epsilon=0}\lim_{\varepsilon\to 0}  \int_{M\setminus B^\varepsilon_{\psi_\epsilon}}  \iota_u \{\sigma^* (\cL_v \arg|_\psi ) +  \Im db\} \mu\\
% %     &=\frac{d}{d\epsilon} \Big|_{\epsilon=0}\lim_{\varepsilon\to 0}  \int_{M\setminus B^\varepsilon_{\psi_\epsilon}}  \iota_u \{\cL_v \sigma^* \arg |_\psi +  \Im d b\}\mu\\
% %     &=\lim_{\varepsilon\to 0}  \int_{M\setminus B^\varepsilon}  \iota_u \cL_v \lambda_\psi + \Im \cL_u b \ \mu.
% % \end{align}
% In the same way, we get
% \begin{align}
%     \cL_{\ring \psi} \Th_\psi (\dot \psi)
%     =-\int \iota_u \cL_v \lambda_\psi + \Im \cL_u b \ \mu.
% \end{align}
% Hence we obtain
%     \begin{align}
%     %    \cL_{\ring \psi} \Th(\dot \psi)  - \cL_{\dot \psi} \Th(\ring \psi)  + \Theta ([\dot \psi,\ring \psi])
%     %    &=\int - \iota_u \cL_v \lambda_\psi - \Im \cL_u b
%     %    +\iota_v \cL_u \lambda_\psi + \Im \cL_v  a 
%     %    \\&\quad\quad +\iota_{[v,u]}\lambda_\psi + \Im( \cL_u b-\cL_v  a )\mu\\
%     %    &=\int -\iota_u \cL_v \lambda_\psi +\iota_v \cL_u \lambda_\psi +\iota_{[v,u]}\lambda_\psi \  \mu.
%      \cL_{\dot \psi} \Th(\ring \psi) -  \cL_{\ring \psi} \Th(\dot \psi) - \Theta ([\dot \psi,\ring \psi])
%     &=\int - \iota_v \cL_u \lambda_\psi - \Im \cL_v  a
%     + \iota_u \cL_v \lambda_\psi + \Im \cL_u b 
%     \\&\quad\quad -\iota_{[v,u]}\lambda_\psi - \Im( \cL_u b-\cL_v  a )\mu\\
%     &=\int  \iota_v \cL_u \lambda_\psi -\iota_u \cL_v \lambda_\psi -\iota_{[v,u]}\lambda_\psi \  \mu.
%     \end{align}
% Here we get by a direct computation that
% \begin{align}
% % -\iota_v \cL_u \lambda_\psi + \iota_u \cL_v \lambda_\psi +\iota_{[v,u]}\lambda_\psi
% % &= (-\iota_v \cL_u + \iota_u \cL_v  +\cL_v\iota_u - \iota_u \cL_v ) \lambda_\psi\\
% % &= (-\iota_v \cL_u + \cL_v\iota_u) \lambda_\psi\\
% % &=(-\iota_v d \iota_u - \iota_v\iota_u d +d\iota_v \iota_u + \iota_v d\iota_u)\lambda_\psi\\
% % &= (-\iota_v\iota_u d + d\iota_v \iota_u)\lambda_\psi\\
% % &= \iota_u\iota_v d \lambda_\psi -0.
% \iota_v \cL_u \lambda_\psi - \iota_u \cL_v \lambda_\psi -\iota_{[v,u]}\lambda_\psi
% &= (\iota_v \cL_u - \iota_u \cL_v  -\cL_v\iota_u + \iota_u \cL_v ) \lambda_\psi\\
% &= (\iota_v \cL_u - \cL_v\iota_u) \lambda_\psi\\
% &=(\iota_v d \iota_u + \iota_v\iota_u d -d\iota_v \iota_u - \iota_v d\iota_u)\lambda_\psi\\
% &= (\iota_v\iota_u d - d\iota_v \iota_u)\lambda_\psi\\
% &= \iota_v\iota_u d \lambda_\psi -0.
% \end{align}
% With this we have by integral by parts,
% \begin{align}
%     % -d\Th_\psi(\dot \psi,\ring \psi)
%     % &= \int \iota_u\iota_v d \lambda_\psi \ \mu
%     % = \int d \lambda_\psi \wedge \iota_v\iota_u\mu
%     % = - \int \lambda_\psi \wedge d\iota_v\iota_u\mu \\
%     % &=-\langle \Lambda(\psi), d\iota_v\iota_u \mu\rangle
%     % =-\langle \partial \Lambda(\psi), \iota_v\iota_u \mu\rangle.
%     d\Th_\psi(\dot \psi,\ring \psi)
%     &= \int \iota_v\iota_u d \lambda_\psi \ \mu
%     = \int d \lambda_\psi \wedge \iota_u\iota_v\mu
%     = - \int \lambda_\psi \wedge d\iota_u\iota_v\mu \\
%     &=-\langle \Lambda(\psi), d\iota_u\iota_v \mu\rangle
%     =-\langle \partial \Lambda(\psi), \iota_u\iota_v \mu\rangle.
% \end{align}
% Finally, applying \lemref{lem:lam_psi_in_fiber}, we obtain for $\gamma\coloneqq \Pi\psi$ that,
% \begin{align}
%     -\langle \partial \Lambda(\psi), \iota_u\iota_v \mu\rangle 
%     = -\langle \delta_{\gamma}, \iota_u\iota_v \mu\rangle 
%     = \omega_{\delta_\gamma} (-\cL_u \delta_\gamma, -\cL_v \delta_\gamma )
%     = \Pi^*\omega_{\delta_\gamma} (\dot\psi,\ring \psi)
%     = \tilde Z_\psi(\dot\psi,\ring \psi).
% \end{align}
% \end{proof}

% \begin{remark}[The case of $M=\RR^d$]\label{rmk:treatment_of_Rd}
%     When $M$ is not compact, \autoref{th:quantum_liouville_form} does not hold as is, since $\iota_u\mu$ may be globally supported. In that case, the last step of the proof using the Coarea formula (\autoref{prop:coarea_formula}) and namely \lemref{lem:lam_psi_in_fiber} does not work. 
%     We may fix this issue simply by modifying $\SemiD=\Diff_0(M)\ltimes C^\infty(M)$ to $\Diff_{0,c}(M)\ltimes C^\infty(M)$ and its Lie algebra $\frak{S}$ to $\diff_c(M)\ltimes C^\infty(M)$ by considering only the compactly supported vector fields.
% \end{remark}

\vspace{5pt}

In this section, we have shown that \(\Th\) is a Liouville form for $\omega$ in the prequantum sense, that is, \(d\Theta=\Pi^*\omega\). We conclude this section by noting that \(\Th\) does not directly induce a Liouville form for the symplectic structure $\omega$ on $\cO$ in the classical sense. This is because we cannot factor \(\Th\) onto \(\cO\), as the kernel of \(d\Pi\) for \(\Pi \colon \cU \to \cO \) is not contained within the kernel of \(\Th\). For example, for \(\dot \psi = e^{ic_1} \psi\), \(\ring \psi = e^{ic_2} \psi \in T_\psi \cU\) with some real constants \(c_1 \neq  c_2\), we have \(d\Pi(\dot \psi) = d\Pi(\ring \psi) = 0 \in T_{\Pi(\psi)} \cO\), but the values \(\Th(\dot \psi) = \frac{ \Vol(M) c_1}{2\pi}\) and \(\Th(\ring \psi) =  \frac{ \Vol(M) c_2}{2\pi}\) are different.

However, it is still possible to take a quotient space of \(\cU\) where the Liouville form can descend onto. This construction actually leads to a prequantum bundle structure, as we will explain in the next section.



% $\phi\coloneqq \arg \psi\colon M\setminus\{\psi=0\} \to \SS^1 $ and the circle differential $\lambda_\psi\in \Om^{1}(M\setminus \{\psi=0\})$ is given by $\lambda_\psi=\phi^* \sigma$ with the Haar measure $\sigma$ on $\SS^1=\RR/2\pi \ZZ$ of unit mass $\sigma(\SS^1)=1$. 
%  In the sequel, we also use the vector field $\nabla_{\SS^1}\phi\coloneqq (\lambda_\psi)^\flat$ where the flat operator is with respect to the specified Riemannian metric $g$ on $M$.



% In $\cD_{m-1}(M)$ and $\cD_{m-2}(M)$ which are the middle and the bottom layers of the above sequence, we define differential forms.  
% On each $\Diffp(M)$-orbit in $\cD_{m-1}(M)$ we define a 1-form $\vartheta$. At each $h\in \cV $, we define
% \begin{align}
%     \vartheta_{h}(\cL_v h)\coloneqq \on{Flux}(h,v)=\langle h, \iota_v \mu\rangle
% \end{align}
% using the flux for $m-1$ currents (\defref{def:flux}).
% In a similar way, we define a 2-form on each $\Diffp(M)$-orbit in $\cD_{m-2}(M)$ by
% \begin{align}
%     \omega_\beta(\cL_u \beta, \cL_v \beta)=\langle \beta, \iota_v \iota_u \mu\rangle.
% \end{align}


% \begin{proposition}\label{prop:prequantum_flux_relation}
% Let $\cV$ and $\cO$ be $\Diffp(M)$-orbits in $\cD_{m-1}(M)$ and $\cD_{m-2}(M)$ with $\partial \cV=\cO$ respectively. Then the fibration $\partial\colon \cV \to \cO$ defines a formal prequantization, \ie, for any $h\in \cV$ and $\beta \in \cO$ with $\partial h=\beta$, we have 
%     \begin{align}
%         d\vartheta_h=\partial^*\omega_\beta.
%     \end{align}
%     \begin{proof}
%     We have by direct computation that,
%         \begin{align}
%             d\vartheta(\cL_u h, \cL_v h)
%             &=\nabla_{\cL_u h}\vartheta(\cL_v h)-\nabla_{\cL_v h}\vartheta(\cL_u h)-\vartheta([\cL_u h, \cL_v h])\\
%             &=\nabla_{\cL_u h} \langle h, \iota_v \mu \rangle - \nabla_{\cL_v h} \langle h, \iota_u \mu \rangle - \langle h,\iota_{[u,v]}\mu \rangle\\
%             &= \langle h, \cL_u \iota_v \mu \rangle - \langle h, \cL_v \iota_u \mu \rangle - \langle h,\iota_{[u,v]}\mu \rangle.\\
%         \end{align}
%         Here we have,
%         \begin{align}
%             \cL_u \iota_v \mu - \cL_v \iota_u\mu - \iota_{[u,v]}\mu
%             &=\iota_{[u,v]}\mu + \iota_v \cL_u\mu  - \cL_v \iota_u\mu -\iota_{[u,v]}\mu\\
%             &=\iota_v \cL_u\mu -  \cL_v \iota_u\mu\\
%              &=  \io_v \io_u d \mu +  \io_v d\io_u\mu - \io_v d \iota_u\mu - d \io_v \iota_u\mu\\
%              &=0 - d \io_v \iota_u\mu.
%         \end{align}
%         Hence we have
%         \begin{align}
%             \langle h, \cL_u \iota_v \mu \rangle - \langle h, \cL_v \iota_u \mu \rangle - \langle h,\iota_{[u,v]}\mu \rangle
%             &= -\langle h, d\io_v\io_u \mu \rangle
%             % = -\langle \partial h, \io_v\io_u \mu \rangle
%             =- \langle \delta_\gamma,  \io_v\io_u \mu \rangle
%             =-\omega_\beta(\cL_u \beta, \cL_v\beta),
%         \end{align}
%         which concludes the proof.
%     \end{proof}
% \end{proposition} 


% \begin{lemma}\label{lem:theta_pullback_relation}
%     We have $\Lambda^*\vartheta_{h_\psi}=\Th_\psi$.

%     \begin{proof}
%         Let $\dot \psi=\cL_v \psi+\varphi \psi$ with some $v\in \diffp(M)$ and $\varphi\in C^\infty(M,\CC)$. We have on $\{\psi \neq 0\}$ that
%         \begin{align}
%             \Im \frac{\dot \psi \bar \psi}{|\psi|^2}
%             = \Im\left(\frac{ d\psi[v] }{\psi} +\frac{\div v}{2}+\varphi\right)=\Im\left(\frac{ d\psi[v] }{\psi} +\varphi\right).
%         \end{align}
%         Here we have
%         \begin{align}
%             d\psi=d|\psi| \cdot e^{i\phi}+ |\psi| e^{i\phi} i \lambda_\psi=\psi \left( \frac{d|\psi|}{|\psi|}+ i \lambda_\psi  \right)
%         \end{align}
%         where $\phi=\arg\psi$. Hence we have
%         \begin{align}
%             \Im\left(\frac{ d\psi[v] }{\psi} +\varphi\right)
%             = \Im \left(\frac{d|\psi|[v]}{|\psi|} + i \lambda_\psi[v]+ \varphi\right)
%             = \lambda_\psi[v]+ \Im \varphi. 
%         \end{align}
%             Thus we have
%             \begin{align}
%             \Th_\psi(\dot \psi)
%             &=\lim_{\epsilon\to 0}\int_{M\setminus B_\epsilon}  \Im \frac{\dot \psi \bar \psi}{|\psi|^2}\mu
%             = \lim_{\epsilon\to 0}\int_{M\setminus B_\epsilon} \lambda_\psi \left[v+ \Im \varphi\frac{\nabla_{\SS^1}\phi}{|\nabla_{\SS^1}\phi|^2}\right]\mu 
%             \\&= \lim_{\epsilon\to 0}\int_{M\setminus B_\epsilon}  \lambda_\psi \wedge \iota_{\tilde v} \mu 
%             =\langle h_\psi,\iota_{\tilde v}\mu\rangle=\vartheta_{h_\psi}(\cL_{\tilde v} h_\psi)
%             \end{align}
%             where $\tilde v=v+ \Im \varphi\frac{\nabla_{\SS^1}\phi}{|\nabla_{\SS^1}\phi|^2}$.

%             Using the coarea formula (\autoref{prop:coarea_formula}) again, it is easy to show $d \Lambda |_\psi (\dot \psi)=\cL_{\tilde v}h_\psi$, which concludes the proof.
%     \end{proof}
% \end{lemma}


% Now \autoref{th:quantum_liouville_form}  is a direct consequence of the results we have given.
% \begin{proof}[Proof of \autoref{th:quantum_liouville_form}]
%     Combining \autoref{prop:prequantum_flux_relation}, \lemref{lem:lam_psi_in_fiber}, and \lemref{lem:theta_pullback_relation} we have
%     \begin{align}
%           \Lambda^* \partial^* \omega_{\delta_\gamma}
%           = \Lambda^*d\vartheta_{h_\psi}
%           = d \Th_\psi.
%     \end{align}
% \end{proof}

\section{Prequantum structure}\label{sec:prequantum_bundle}

We have shown that each $\SemiD$-orbit $\cU$ in the fiber bundle $\cF_\cO=\Pi^{-1}\cO$ admits a Liouville form in the prequantum sense for the MW structure on $\cO$. The full fiber $\cF_\cO$ inherits these symplectic and Liouville structures as $\cF_\cO$ is a  $\Diff_0(M) \ltimes C^\infty(M, \CC^\times)$-orbit, foliated by $\SemiD$-orbits, as discussed in \autoref{sec:geometry_of_fibers}.

Building on this, we now construct a prequantum bundle as a quotient bundle of $\cF_\cO$, equipped with a connection form whose curvature form recovers the MW symplectic form on $\cO$. This framework enables us to define a unique horizontal lift in $\cF_\cO$  over a path in $\cO$.

% \Albert{[Last sentence: I don't see a direct implication that prequantum circle bundle suddenly allows us to do the horizontal lift.  Maybe just add a few more words in between. We could say ``In particular this prequantum bundle has a connection whose curvature equals the symplectic structure of the base \(\cX\). This allows us to consider horizontal lifts...'' or something like that.]}

% \begin{definition}[Pseudo principal circle bundle]\label{def:pseudo_principal_circle_bundle}
% Let $\pi\colon E\to B$ a fiber bundle. If $E$ is equipped with a fiber-preserving $\SS^1$ action which is free and transitive in each connected component of each fiber, we say $\pi$ is a pseudo principal circle bundle. 
% \end{definition}

% \begin{definition}[Prequantum G-bundle]\label{def:prequantum_bundle}
%     Let $\pi\colon (E,\alpha)\to (B,\beta)$ be a formal prequantization (\autoref{def:formal_prequantization}), we say If $\pi$ is a pseudo principal circle bundle and the 1-form $\alpha$ is a connection form, we say $\pi$ is \emph{a prequantum bundle}.
% \end{definition}

\begin{definition}[Prequantum $G$-bundle]\label{def:prequantum_bundle}
    \emph{A prequantum $G$-bundle} over a symplectic manifold $(B,\beta)$ is a principal $G$-bundle $\pi\colon E\to B$ equipped with a connection form $\alpha\in \Omega^1(E;\mathfrak{g})$ satisfying 
\begin{align}\label{eq:curvature_relation}
        \pi^*\beta=d\alpha+\frac{1}{2}[\alpha\wedge\alpha].
    \end{align}
\end{definition}
% If $G$ is abelian, the relation \eqref{eq:curvature_relation} reduces to $ \pi^*\beta=d\alpha$, which is always the case in this thesis.

\rev{
\begin{remark}
    \autoref{def:prequantum_bundle} relaxes the standard definition of a prequantum bundle, which often requires the structure group to be a circle or a (complex) line. In fact, the prequantum bundle we construct in this section, the structure group $G$ is not a circle but copies of a circle unless the ambient space $M$ has trivial first cohomology. In any either, $G$ is one-dimensional and hence we have $\pi^*\beta =d\alpha$ without the term $\frac{1}{2}[\alpha\wedge\alpha]$, as in the case of a circle bundle.
\end{remark}
% \footnote{\autoref{def:prequantum_bundle} relaxes the standard definition of a prequantum bundle, which often requires the structure group to be a circle or a line. In fact, the prequantum bundle we construct in this section, the structure group $G$ is copies of a circle unless the ambient space $M$ has trivial first cohomology. In any either, $G$ is one-dimensional and hence abelian. So it satisfies $\pi^*\beta =d\alpha$ without extra term $\frac{1}{2}[\alpha\wedge\alpha]$, as in the case of a circle bundle.}
}


% The pseudo-principal circle bundle (\autoref{def:pseudo_principal_circle_bundle}) and the prequantum bundle (\autoref{def:prequantum_bundle}) are relaxed versions of the standard definitions of principal circle bundle and prequantum circle bundle that require each fiber to be a topological circle, and the circle action to be transitive. In contrast, \autoref{def:pseudo_principal_circle_bundle} and \autoref{def:prequantum_bundle} only require transitivity connected components-wise.

% These notions are necessary because each fiber $\Pi^{-1}\gamma$ in the space of implicit representations may have multiple connected components corresponding to $H^1(M, \ZZ)$, as discussed in \autoref{sec...}. Consequently, the prequantum bundles introduced in this section inherit this disconnectedness.
% However, this does not prevent us from defining the horizontal lift, as follows:

For a prequantum $G$-bundle, the vertical and horizontal distributions are defined in the standard way for principal bundles:
\begin{definition}[Vertical and horizontal distributions, and horizontal lift]\label{def:unique_horizontal_lift}
    Let $\pi\colon (E,\alpha)\to (B,\beta)$ be a prequantum $G$-bundle. At each point $x \in E$, the tangent space splits as \( T_x E = V_x E \oplus H_x E \), where the vertical distribution is defined by \( V_x E = \ker d\pi|_x \), and the horizontal distribution is given by \( H_x E = \ker \alpha|_x \). A \emph{horizontal lift} of a path \( \{ \gamma_t \}_t \subset B \) is a path \( \{ \ell_t \}_t \subset E \) satisfying \( \pi \circ \ell_t = \gamma_t \) and \( \partial_t \ell_t \in H_{\ell_t} E \) for all \( t \). Such a lift is uniquely determined by the choice of the initial point \( \ell_0 \).
\end{definition}




\autoref{th:quantum_liouville_form} states that the fibration \( \Pi\colon (\cF_\cO, \Th) \to (\cO, \omega) \) is a formal prequantization \ie, \( \Pi^* \omega = d\Th \). However, it is not a prequantum $G$-bundle since \( \Th \) is not a connection form. Consequently, it does not define a unique horizontal lift: for a path \( \{ \gamma_t \}_t \subset \cO \), there exist infinitely many paths \( \{ \ell_t \}_t \subset \cF_\cO \) with fixed initial point \( \ell_0 \) satisfying \( d\Pi (\partial_t \ell_t) = \partial_t \gamma_t \) and \( \partial_t \ell_t \in \ker \Th_{\ell_t} \) for all \( t \).
% More concretely, given any tangent vector \( \dot{\gamma} \in T_\gamma \cO \) and any choice of \( (v, \varphi) \in \AlgSemiD_\gamma \subset \AlgSemiD \), the vector \( \dot{\psi} = -\cL_v \psi + \varphi \psi \in T_\psi \cF_\cO \) lies in both \( \ker d\Pi \) and \( \ker \Th_\psi \). 
To define a bundle on which \( \Th \) becomes a genuine connection 1-form, we take the quotient of the tangent space at each \( \psi \in  \cF_\cO \) by the intersection \( \ker \Th \cap \ker d\Pi \). This quotient process is characterized by the following equivalence relation.

\begin{definition}[Volume class and volume bundle]
     For $\psi_0,\psi_1\in \cF_\cO$, we say $ \psi_0\sim_\cP \psi_1$ if there is a path $\{\psi_t\}_{t\in[0,1]} $ in a $\SemiD$-orbit joining $\psi_0$ and $\psi_1$ such that $\partial_t\psi_t\in \ker d\Pi_{\psi_t} \cap \ker\Theta_{\psi_t}$ for any $t$. We call each equivalence class $[\psi]_\cP$ \emph{a volume class} and the resulting quotient space $\cP\coloneqq \cF_\cO / \sim_\cP$ \emph{the volume bundle}. The tangent space at each $[\psi]_\cP$ is $T_{[\psi]_\cP}\cP=T_\psi \cF_\cO/\sim_\cP$ where $\dot\psi\sim_\cP \ring\psi$ if $\dot\psi-\ring\psi\in \ker d\Pi|_\psi\cap \ker \Th|_\psi$.
     
\end{definition}

We note that the projection $\Pi \colon \cF_\cO \to \cO$ decomposes into two projections: $\rev{\pi_\cP} \colon \cF_\cO \to \cP$ and $\rev{\Pi_\cP} \colon \cP \to \cO$.
By construction, the Liouville form $\Th$ descends to a 1-form $\Th_\cP \in \Omega^1(\cP)$ by $\pi_\cP^* \Th_\cP = \Th$. Since $d\Th = \Pi^* \omega$, we have $d\Th_\cP = \Pi_\cP^* \omega$, and therefore the fibration $\Pi_\cP \colon (\cP, \Th_\cP) \to (\cO, \omega)$ is a formal prequantization.

In addition, the volume bundle $\cP$ forms a principal $G$-bundle over $\cO$, where the structure group is $G = \SS^1 \times H^1_{dR}(M, \ZZ)$, acting on $\cP$ as follows. We first define a circle action by constant phase shift:
\begin{align}
\label{eq:S1ActionOnF}
    e^{ic} \rhd [\psi]_\cP \coloneqq [\psi e^{ic}]_\cP, \quad \text{for } \psi \in \cF, \; e^{ic} \in \SS^1.
\end{align}
As shown in \autoref{prop:cardinality_of_fiber_components}, each fiber $\Pi^{-1}\gamma$ may contain multiple connected components indexed by the discrete group $H^1_{dR}(M, \ZZ)$. 
The $\SS^1$-action \eqref{eq:S1ActionOnF} is free and transverse within each connected component.

We may further specify a group action of \( H^1_{dR}(M, \ZZ) \) on \( \cP \) as a deck transformation between the connected components of each fiber.  That is, it is a fiber-preserving action that maps points from one connected component of a fiber to another component.
We fix a lattice basis $[\eta_1],\ldots,[\eta_n]$ for \(H_{dR}^1(M,\ZZ)\), where $n$ is the first Betti number of $M$. Recall that there is an isomorphism
\begin{eqnarray}
    \frak{i} \colon C^\infty(M,\CC^\times)/\on{Exp}(C^\infty(M,\CC^\times)) &\to  H^1_{dR}(M,\ZZ)\\
     \left[\kappa\right]\qquad \qquad\qquad&\mapsto\left[\Im \oneOverTwoPi \frac{d\kappa}{\kappa}  \right].
\end{eqnarray}
Choose arbitrary representatives $\tau_i\in C^\infty(M,\CC^\times)$ of $\frak{i}^{-1}[\eta_i]$ for $i=1,\ldots,n$.
Using these functions \(\tau_i\), we define the action of $H^1_{dR}(M,\ZZ)$ by
\begin{align}
    [\eta] \rhd [\psi]_\cP \coloneqq [\tau_1^{c_1}\cdots\tau_n^{c_n} \cdot \psi]_\cP, \quad \psi \in \cF, \ [\eta]= [c_1 \eta_1+\ldots + c_n \eta_n]\in H^1_{dR}(M,\ZZ).
\end{align}

These $\SS^1$- and $H^1_{dR}(M,\ZZ)$-actions together define a $G=\SS^1\times H^1_{dR}(M,\ZZ)$-action
\begin{align}\label{eq:G_action}
    (e^{ic}, [\eta])\rhd [\psi]_\cP = [e^{ic} \tau_1^{c_1}\cdots\tau_n^{c_n}\psi]_\cP, \quad e^{ic}\in \SS^1, [\eta]\in H^1_{dR}(M,\ZZ),
\end{align}
which is free, transverse, and fiber preserving. We thus obtain the following result.
\begin{proposition}\label{prop:principal_G_bundle}
The fibration $\pi_\cP\colon \cF\to\cO$ equipped with the above action of $G=\SS^1\times H^1_{dR}(M,\ZZ)$ is a principal $G$-bundle. 
\end{proposition}



% Furthermore, $\Th_\cP$\rev{, when scaled by a normalization factor,} 

\rev{We now define scaled versions of the MW form $\omega\in \Omega^2(\cO)$ and the Liouville form $\Th_\cP\in \Omega^1(\cP)$ and show that the latter defines a connection form on $\cP$. Let us define the normalized MW form and Liouville form  $\omega^\normalized\coloneqq \normconst\omega$ and $\Th_\cP^\normalized\coloneqq \normconst\Th_\cP$ with the normalization factor $\normconst=2\pi/\Vol(M)$. }

\begin{proposition}\label{prop:connection_form}
    On the principal $G$-bundle $\Pi_\cP\colon\cP\to \cO$ defined above, the 1-form \rev{$\Th_\cP^\normalized$} is a connection form. That is, \rev{$\Th_\cP^\normalized$} satisfies the following two properties:
\begin{enumerate}
    \item Equivariance under the $G$-action $\Phi_g\colon \cP\to \cP$; i.e., $\Phi_g^*\Th_\cP=\Th_\cP$ for every $g\in G$.
    \item Vertical reproducibility; i.e., there exists a nonzero constant $C\neq 0$ such that $\Th_\cP(\hat\xi)=C \xi$ for any $\xi\in \frak{g}$ and its fundamental vector field $\hat \xi\in \Gamma(T\cP)$.%
    % \footnote{A standard definition requires $C = 1$ or $C\in 2\pi\ZZ$, which can be achieved by simply rescaling $\Theta_\cP$ or $\mu$. In this thesis, however, we retain the constant $C$ in the definition for expositional simplicity.}

\end{enumerate}
\end{proposition}

\begin{proof}
We first verify equivariance. Let $g=(e^{ic},[\sum_i c_i\eta_i])\in G=\SS^1\times H^1_{dR}(M,\ZZ)$, and denote by $\Phi_g$ the $g$-action \eqref{eq:G_action}. By direct computation, we have
\begin{align}
    \rev{\Th_\cP^\normalized}|_{\Phi_g([\psi]_\cP)}([\Phi_{g*}\dot\psi]_\cP)
    &=\rev{\frac{1}{\Vol(M)}}\int \Im \frac{\dot \psi e^{ic}\tau_1^{c_1}\cdots \tau_n^{c_n} \cdot \overline{\psi e^{ic}\tau_1^{c_1}\cdots \tau_n^{c_n}}}{|\psi e^{ic}\tau_1^{c_1}\cdots \tau_n^{c_n}|^2} \ \mu 
    \\&= \rev{\frac{1}{\Vol(M)}} \int \Im \frac{\dot \psi \cdot \bar\psi }{|\psi |^2} \ \mu 
    =\Th_\cP|_{[\psi]_\cP}([\dot \psi]_\cP),
\end{align}
where $\psi\in \cF_\cO$ and $\dot \psi\in T_\psi \cF_\cO$ are any representatives of $[\psi]_\cP$ and $[\dot\psi]_\cP$ respectively.

Next, we verify vertical reproducibility. Let $\xi\in \frak{g}=T_{(1,[0])}G=T_1\SS^1=\RR$, and let $\hat\xi \in \Gamma(T\cP)$ be the corresponding fundamental vector field, defined by
\begin{align}
    \hat\xi([\psi]_\cP) = \left[\frac{d}{dt}\Big|_{t=0} e^{i\xi t}\psi \right]_\cP = [i\xi \psi]_\cP.
\end{align}
Then, we have
\begin{align} 
\rev{\Th_\cP^\normalized}
|_{[\psi]_\cP}(\hat\xi)
= \rev{\frac{1}{\Vol(M)}}\int \Im \frac{i\xi \psi \bar\psi}{|\psi|^2} \ \mu
=\rev{\xi}
% = \frac{\xi \on{Vol}(M)}{2\pi}.
\end{align}
\end{proof}

\rev{Since $\Th_\cP^\normalized$ and $\omega^\normalized$ are constant multiples of $\Th_\cP$ and $\omega$,  all the results established so far applies except \autoref{prop:connection_form}. Therefore, combining the results that
$\Pi_\cP\colon(\cP, \Th_\cP)\to(\cO,\omega)$ is a formal prequantization (\autoref{th:quantum_liouville_form}) and a principal $G$-bundle with $G=\SS^1 \times H^1_{dR}(M,\ZZ)$ (\autoref{prop:principal_G_bundle}), and that $\Th_\cP^\normalized$ is a connection form (\autoref{prop:connection_form}), we obtain the main theorem.}


% Combining the results that $\Pi_\cP\colon(\cP, \Th_\cP)\to(\cO,\omega)$ is a formal prequantization (\autoref{th:quantum_liouville_form}) and a principal $G$-bundle with $G=\SS^1 \times H^1_{dR}(M,\ZZ)$ (\autoref{prop:principal_G_bundle}),
% \rev{ and that $\Th_\cP^\normalized$ is a connection form on }

% equipped with a connection form $\Th_\cP$ (\autoref{prop:connection_form}), we obtain our main theorem:

\begin{theorem}\label{th:prequantum_circle_bundle}
The fibration 
% $\Pi_\cP\colon (\cP,\Th_\cP)\to (\cO, \omega)$
\rev{$\Pi_\cP\colon (\cP,\Th_\cP^\normalized)\to (\cO, \omega^\normalized)$}
is a prequantum $G$-bundle with $G = \SS^1 \times H^1_{dR}(M,\ZZ)$.
\end{theorem}

\rev{
\begin{remark}[Role of normalization]
We normalized the MW form $\omega$ and the Liouville form $\Theta_\cP$ for two reasons, while noting that our results are essentially unaffected by the presence or absence of normalization. 

The first reason is to align our setting with the standard definition of vertical reproducibility of a connection form, $\Th(\hat \xi) = \xi$ for $\xi \in \frak{g}$. Without normalization we have   $\Th(\hat \xi) = \tfrac{1}{\normconst}\xi$ instead, which is harmless: one may relax the definition to allow a constant factor, and all of our results remain valid.

Second, normalization makes $\omega$ physically natural in terms of physical units. Note that the value $\omega(-\cL_u \delgam, -\cL_v \delgam)=\int_\gamma \iota_v\iota_u\mu$ carries the dimension of $\on{length}^m/\on{time}^2$ as $u, v$ have $\on{length}/\on{time}$, so $\int_\Sigma \omega$ for a topological disc $\Sigma \subset \cO$ has unit $\on{length}^m$.
% as \begin{align}
% \int_{\Sigma(D)} \omega = \int_D  \Sigma^* \omega = \int_D \omega(\partial_{t_1}, \partial_{t_2}) dt_1 dt_2 
% \end{align}
% with interpretation of the unit of $dt_1, dt_2$ being $\on{time}$. 
Thus, multiplying $\omega$ by the factor $2\pi/\Vol(M)$ makes its unit $\on{radius}$, canceling $\on{length}^m$. By relaxing the definition of vertical reproducibility as above, one could even use $1/\Vol(M)$ as the normalization factor to make $\omega$ dimension-less.

Additionally the \emph{prequantizability} of a symplectic form on $\cO$ is often expressed in the literature \cite{HallerVizman2003NonlinearGA} : ``$\omega$ is prequantizable if $\Vol(M) = 2\pi$ or if it is of integral class times $2\pi$''. However, the specific value of $\Vol(M)$ depends on the unit of measurement \eg, cubic meter $m^3$ and cubic yard $yd^3$ in 3D. We believe that the choice of unit for $\Vol(M)$ should not affect the prequantizability of the mechanics on the space. This motivates our normalization to remove this artificial dependence. 

% That being said, the normalization is optional: the essential results remain unchanged with or without it.
\end{remark}
}

\vspace{5pt}

Using \rev{either the normalized or the unnormalized} connection form $\Theta_\cP$, each tangent space $T_{[\psi]_\cP}\cP$  splits into the vertical distribution $V_{[\psi]_\cP}\cP=\ker d\Pi_\cP|_{[\psi]_\cP}$ and the horizontal distribution $H_{[\psi]_\cP}\cP=\ker\Th_\cP|_{[\psi]_\cP}$. Since $\ker d\Pi_\cP|_{[\psi]_\cP}=\ker d\Pi/ \ker \Th|_\psi$ for any representative $\psi\in \pi^{-1}_\cP[\psi]_\cP$, and $\Th\in \Omega^1(\cF_\cO)$ decomposes the tangent space $T_\psi \cF_\cO$ into codimension-1 hyperplanes as the level sets of $\Th_\psi$, the vertical distribution $V_{[\psi]_\cP}\cP$ has precisely one dimension. 

Hence we can now define a unique horizontal lift on $\cP$ (\autoref{def:unique_horizontal_lift}) over a path in $\cO$. In light of \autoref{rmk:meaning_of_Th}, this horizontal lift can be interpreted as the evolution of the implicit representation $[\psi]_\cP$ such that the average swept volume of phase level hypersurfaces remains zero at all times.
\rev{
That is, for any representative $\psi_t$ of a horizontal lift $[\psi_t]_\cP$, we have for the family of phase hypersurfaces \(\{\sigma_t^s\}_{s\in \SS^1}\) defined by $\sigma_t^s = \phi_t^{-1}(s)$,
\begin{align}
    \oneOverTwoPi\int_{\SS^1}\int_{\sigma_t^s}\iota_{\partial_t{\sigma}_t^s} \mu \ ds = 0
\end{align}
at each $t$ where $\partial_t \sigma_t^s$ is the velocity defined on each hypersurface $\sigma_t^s$. 
}


% 
This also reveals a geometric interpretation of the MW form as the curvature form of the Liouville form $\Th_\cP$, measuring the holonomy induced by parallel transport on $\cP$ over a closed path in $\cO$:
% 
% \begin{figure}[htbp]
%     \centering
% \includegraphics[width=0.7\textwidth]{images/horizontal_lift.png}
%     \caption{\Sadashige{this figure is a place holder}}
%     \label{fig:MW_as_curvature}
% \end{figure}
% 
% Since $\Th_\cP$ measures the flux of the vector field and the phase shift due to the perturbation, the horizontal lift gives a perturbation causing no global phase shift.  Then, via the prequantization $\Pi\colon \cP\to \cO$, the Marsden-Weinstein form can be interpreted as the curvature form and the holonomy caused by the parallel transport on $\cP$ over a closed path on $\cO$ is measured by $\Th_\cP$. 
% 


\begin{corollary}[Average swept volume]\label{cor:average_swept_volume}
Consider a closed path \(\partial\Sigma\) in \(\cO\) that bounds a 2-dimensional disk \(\Sigma\), representing a cyclic motion of a codimension-2 submanifold \(\gamma_t \subset M\) for \(0 \leq t \leq 1\), with \(\gamma_0 = \gamma_1\) and let $[\psi_t]_\cP$ be a horizontal lift over $\gamma_t$ with respect to $\Theta_\cP$.
% such that the phase map $\phi_t\coloneqq \psi_t/|\psi_t|\colon M\setminus \psi^{-1}(0) \to \SS^1$ is a submersion for each \(t\).
\rev{Suppose $[\psi_0]_\cP$ and $[\psi_1]_\cP$ are on the same connected component of the fiber $\Pi_\cP^{-1}\gamma$. }
Then, \rev{for representatives $\psi_0$ and $\psi_1$ of $[\psi_0]_\cP$ and $[\psi_1]_\cP$,} the volume enclosed between phase hypersurface \(\sigma_0^s\) and \(\sigma_1^s\), averaged over \(s\in \SS^1\), equals to \(\iint_{\Sigma} \omega\), where \(\omega\) is the MW form on \(\cO\).
\end{corollary}

\rev{
\begin{proof}   
    % First notice that $\psi_0$ and $\psi_1$ are in the same connected component of the fiber $\Pi^{-1}\gamma_0$ due to the assumption.
    The integral of the curvature is given by the monodromy of the parallel transport in $\cP$ over a closed path bounding $\Sigma$ in $\cO$, which is measured by the connection form $\Th_\cP^\normalized$ (and unnormalized form $\Th_\cP$ corresponding to $\omega$). That is, 
     \begin{align}
        \int_\Sigma \omega^\normalized = \int^1_0 \Th_\cP^\normalized \left(\frac{d}{dt} \left( e^{i\Arg([\psi_1]_\cP, [\psi_0]_\cP)t}\rhd [\psi_1]_\cP\right)\right) dt = \Arg([\psi_1]_\cP, [\psi_0]_\cP) \quad \on{mod} 2\pi, 
    \end{align}
    where $\Arg([\psi_1]_\cP, [\psi_0]_\cP)\in\RR/2\pi\ZZ$ is defined by $e^{i\Arg([\psi_1]_\cP, [\psi_0]_\cP)}\rhd[\psi_1]_\cP=[\psi_0]_\cP$ when $[\psi_0]_\cP$ and $[\psi_1]_\cP$ are in the same connected component of a fiber. Explicitly, 
    \begin{align}
        \Arg([\psi_1]_\cP, [\psi_0]_\cP) = \arg \int \bar \psi_1\psi_0 \,\mu
    \end{align}
    with any representatives $\psi_0,\psi_1$.
    
    For unnormalized forms $\omega$ and $\Th_\cP$, the result is multiplied by $1/\normconst=\Vol(M)/2\pi$, 
    \begin{align}
        \int_\Sigma \omega = \int^1_0 \Th_\cP \left(\frac{d}{dt} \left(e^{i\Arg([\psi_1]_\cP, [\psi_0]_\cP) t}\rhd [\psi_1]_\cP\right)\right) dt = \frac{\Vol(M)}{2\pi} \Arg([\psi_1]_\cP^, [\psi_0]_\cP) \quad \on{mod} \Vol(M), 
    \end{align}
    
    % due to the assumption that $\psi_0$ and $\psi_1$ are in the same connected component of the fiber. 
    
     Let us now consider any path $\{\psi_{t'}\}_{t'\in [0,1]}\subset \Pi^{-1}\gamma_0$ joining representatives $\psi_1$ and $\psi_0$ within the fiber. 
     As observed in \autoref{rmk:meaning_of_Th}, the Liouville form $\Th$ measures the infinitesimal swept volume of the phase level hypersurfaces $\sigma_t^s$ averaged in $\SS^1$.
     Therefore, the integral of $\Th$ along $\psi_{t'}$ is the average signed volume enclosed by pairs $\{(\sigma_1^s, \sigma_0^s)\}_s$, which is formally
    \begin{align}
       \frac{\Vol(M)}{2\pi}\Arg([\psi_1]_\cP, [\psi_0]_\cP)
       =\int^1_0 \Theta(\dot \psi_{t'}) dt' = \int_{\SS^1} \Vol(\partial^{-1}(\sigma^0_s - \sigma^1_s) ) ds \quad \mod \Vol(M).
    \end{align}
\end{proof}
}
\rev{In the proof, we used a known fact that the integral of curvature form is the monodromy of parallel transport measured by the connection form. In \autoref{sec:curvature_monodromy}, we gave a precise statement and a proof. }

\rev{As illustrated in \autoref{fig:psi_interpolation}, we may compress the phase field so that it is non-constant only within narrow bands.}
By considering a limiting case of \autoref{cor:average_swept_volume}, where  $\arg \psi_t\coloneqq M\setminus\psi^{-1}(0)\to \TT^1$ becomes constant except a \(2\pi\) jump at on a single hypersurface \(\sigma_t\) bounding \(\gamma_t\), we obtain the following:

\begin{corollary}[Swept volume by a hypersurface] \label{cor:single_surface_swept_volume}
Let \(\Sigma \subset \cO\) and \(\{\gamma_t\}_{t \in [0,1]} \subset M\) be as in \autoref{cor:average_swept_volume}. Suppose that each \(\gamma_t\) bounds a hypersurface, i.e., \(\gamma_t = \partial \sigma_t\), and that the  volume swept out by \(\sigma_t\) remains zero at each \(t\), meaning
\(\int_{\sigma_t}\iota_{\partial_t {\sigma}_t} \mu = 0\).
Then, the volume enclosed between \(\sigma_0\) and \(\sigma_1\) is given by \(\iint_{\Sigma} \omega\).
\end{corollary}
Note that the interpretation of the MW form in \autoref{cor:single_surface_swept_volume} reduces to the swept volume of a single surface, no longer explicitly involving the complex function.
\begin{remark}
   \autoref{cor:single_surface_swept_volume} can also be shown directly in the framework for the explicit shape space (\autoref{sec:Liouville_classical}). Let us take an \(m{-}1\) dimensional submanifold \(\sigma_0\) of \(M\) bounding some \(\gamma_0 \in \cO\), and consider the orbit \(\cS \coloneqq \Diff_0(M) \rhd \sigma_0\), where the action is defined by \(f \rhd \sigma = f \circ \sigma\) for \(f \in \Diff_0(M)\) and \(\sigma \in \cS\). Then \(\pi \colon \cS \to \cO\) is a fiber bundle where the fibration is given by the boundary operator, i.e., \(\pi(\sigma) = \partial \sigma\), and the tangent space at each \(\sigma\) is \(T_\sigma \cS = \{u \circ \sigma \mid u \in \diff(M)\}\).

   Define a 1-form \(\eta\) on \(\cS\) by \(\eta_\sigma(u \circ \sigma) = \fluxMone(\delta_\sigma, u)\) where $\delta_\sigma$ is a de Rham $m-1$ current of $\sigma$. Then \(\eta\) serves as a formal prequantization, i.e., \(d\eta = \pi^*\omega\), which can be shown in a manner similar to the proof of \autoref{th:liouville_classical}. For a path \(\{\gamma_t\} \subset \cO\), there exist infinitely many lifts $\{\ell_t\}\subset \cS$ such that $\partial_t\ell_t\in\ker \eta$ for all $t$, but the notion of no swept volume still makes sense, and we recover \autoref{cor:single_surface_swept_volume}.
\end{remark}

\section{Additional structures}\label{sec:additional_structures}
Our implicit representations of shapes and the resulting prequantum bundle structure reveal some additional structures. 

\subsection{Hamiltonian systems on the space of implicit representations}\label{sec:implicit_Hamiltonian_vector_field}

In \autoref{sec:explicit_Hamiltonian_field}, we reviewed Hamiltonian vector fields on each \(\Diff_0(M)\)-orbit \(\cO\) in the explicit shape space with respect to the MW form \(\omega\). We now consider Hamiltonian vector fields in the implicit shape space \(\cF_\cO = \Pi^{-1} \cO\).

Define a closed 2-form \(\tilde Z\) on \(\cF_\cO\) by \(\tilde Z \coloneqq \Pi^* \omega\). The closedness of \(\tilde Z\) follows from that of \(\omega\). Note that \(\tilde Z\) is degenerate with \(\ker \tilde Z = \ker d\Pi\), which is infinite-dimensional. Hence, \(\tilde Z\) is not symplectic but merely a \emph{presymplectic} form, in contrast to \(\omega\) being symplectic.

Consequently, Hamiltonian vector fields are defined only up to \(\ker \tilde Z\). Let \(\cH \colon \cO \to \RR\) be a function with \(d\cH \in \operatorname{im} \flat^\omega\), and define \(H \colon \cF_\cO \to \RR\) by \(H = \cH \circ \Pi\). We say that any vector field \(X_H \in \Gamma(T\cF_\cO)\) satisfying \(dH = \iota_{X_H} \tilde{Z}\) is a Hamiltonian vector field of \(H\), despite the non-uniqueness.


\subsubsection{Horizontal Hamiltonian vector fields}
The prequantum structure we have built allows a canonical choice among these Hamiltonian vector fields.
We can define a unique \emph{horizontal Hamiltonian} vector field on the prequantum bundle \(\cP\) over a Hamiltonian vector field on \(\cO\), using \(\Th_{\cP}\) as the connection form.

Since \(H = \cH\circ\Pi\) is by construction constant along each fiber $\Pi^{-1}\gamma$ over $\gamma\in \cO$, it descends onto $\cP$ as a function $H_\cP=\cH\circ \Pi_\cP$ defined using $\Pi_\cP\colon \cP\to \cO$. 
Then there is one degree of freedom within the vector fields \(X_{H_\cP} \in \Gamma(T\cP)\) satisfying
\(dH_{\cP} = \iota_{X_{H_\cP}} Z_\cP\)
where \(Z_\cP \coloneqq \Pi_\cP^*\omega\) is a presymplectic structure on \(\cP\).
Among these, the horizontal Hamiltonian vector field is the unique one additionally satisfying the horizontality condition \(X_{H_\cP} \in H_{[\psi]_\cP}\cP=\ker \Th_\cP|_{[\psi]_\cP}\) at each $[\psi]_\cP\in\cP$.

\subsubsection{Momentum maps}
Momentum maps can also be defined for presymplectic structures \(\tilde Z=\Pi^*\omega \in \Omega^2(\cF_\cO)\) in much the same way as for the MW symplectic structure \(\omega\) explained in \autoref{sec:explicit_momentum_maps}. Given an action of a Lie group \(G\) on \(\cF_\cO\), we say that \(J\colon \cF_\cO \to \frak{g}^*\) is a momentum map if 
\(d\langle J(\cdot),\xi\rangle = \iota_{\hat\xi}\tilde Z\) for any \(\xi \in \frak{g}\) and its corresponding fundamental vector field \(\hat \xi\in \Gamma(T\cF_\cO)\). When the Hamiltonian \(H\) is invariant under the $G$-action, \(J(\psi_t) \in \frak{g}^*\) is conserved along the Hamiltonian flow \(\psi_t\).


Similar to the case of the Liouville form $\eta$ for the MW symplectic form $\omega$ on the explicit shape space $\cO$, the Liouville form \(\Theta\) of the presymplectic form \(\tilde Z\) in the implicit shape space $\cF_\cO$ can describe conserved quantities of Hamiltonian flows. When $\Th$ is invariant under the $G$-action, the corresponding conserved quantity $J(\psi)$ satisfies
\(\langle J(\psi),\xi\rangle = -\Theta(\hat \xi(\psi))\), up to an additive constant. 

As observed in \autoref{sec:explicit_momentum_maps}, the Liouville form \(\eta\) for the MW form $\omega$ on $\UEmb(S,\RR^m)$ is not invariant under the action of the entire \(\SDiffp(\RR^m)\), but only under the volume-preserving linear transformations in \(\on{SL}(\RR^m)\). In contrast, the Liouville form \(\Th\) in the implicit shape space $\cF_\cO$ is invariant under the entire group \(\SDiffp(M)\). We, however, acknowledge that their settings differ: $\eta$ on $\cO$ is the Liouville form in the classical sense \ie, $d\eta=\omega$ and the ambient manifold $M$ has an exact volume form, whereas $\Th$ on $\cF_\cO$ is defined in the prequantum sense \ie, $d\Theta=\pi^*\omega$ and the ambient manifold is a  closed manifold \(M\). 

To see the invariance of $\Th$ under the $\SDiffp(M)$-action, denote by \(R_f \colon \psi \mapsto \psi \circ f^{-1}\) the action of \(f \in \SDiffp(M)\) on \(\psi \in \cF\). A direct computation shows that 
\({R_f}_* \dot \psi = \dot \psi \circ f^{-1}\) for \(\dot \psi \in T_\psi \cF_\cO\).
Then, using \(f^* \mu = \mu\), we have
\begin{align}
   \Th_{R_f (\psi)}({R_f}_* \dot \psi)
   &= \Th_{\psi \circ f^{-1}}(\dot \psi \circ f^{-1}) \\
   &= \oneOverTwoPi \int \Im \frac{\dot \psi \circ f^{-1} \cdot \overline{\psi} \circ f^{-1}}{|\psi \circ f^{-1}|^2} \, \mu \\
   &= \oneOverTwoPi \int \Im \frac{\dot \psi \cdot \overline{\psi}}{|\psi|^2} \, f^* \mu 
   = \Th_\psi(\dot \psi).
\end{align}
Note that the fundamental vector field \(\hat u \in \Gamma(T\cF_\cO)\) of \(u \in \sdiff(M)\) with respect to the \(\SDiffp(M)\)-action on \(\cF_\cO\) is given by \(\hat u(\psi) = -\cL_u \psi\).
Therefore, the corresponding momentum map \(J \colon \cF_\cO \to \sdiff(M)^*\) is computed as
\begin{align}
    \langle J(\psi), u \rangle = -\Th_\psi(-\cL_u \psi)
    = \oneOverTwoPi \fluxMone(\Lambda(\psi), u),
\end{align}
where \(\Lambda(\psi) \in \cD_{m-1}(M)\) is the circle differential current (\autoref{lem:lam_psi_in_fiber}) and \(\fluxMone\) is the flux functional (\autoref{def:flux}).
In the case where \(u\) is exact divergence-free \ie, $\iota_u\mu$ is exact, this further reduces to 
\(\langle J(\psi), u \rangle = \fluxMtwo(\delta_{\Pi \psi}, u)\). This refers only to the explicit shape \(\Pi \psi \in \cO\) and coincides with the momentum map on \(\cO\) given in \autoref{eg:momentum_map_SO3}.


\subsection{Marsden--Weinstein  structure in terms of implicit representations}\label{sec:MW_structure_implicit_representation}
Just as \(\Emb(S, M)\) provides parametrizations of unparametrized shapes \(\UEmb(S, M)\), we may regard \(\cF_\cO = \Pi^{-1} \cO\) as another parametrization space for an orbit \(\cO\) in $\UEmb(S,M)$. This perspective allows us to express the MW form \(\omega\) in terms of implicit representations.

Let \(\tilde Z = \Pi^* \omega\) be the presymplectic form on \(\cF_\cO\) defined in the previous subsection, and let us now suppose that \((M, \mu)\) is equipped with a Riemannian metric inducing the volume form \(\mu\). Then \(\tilde Z\) admits an explicit expression, as we describe below.

At each zero $p \in \psi^{-1}(0)$, the tangent space decomposes as $T_pM = T_p\gamma(S) \oplus N_p\gamma(S)$ with respect to the given Riemannian metric $g$. We can choose a unit frame field $(n_1, n_2)$ over the normal bundle $N\gamma(S)$ such that $\mu(\partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma, n_1, n_2) > 0$ for any local coordinates $(s_1, \ldots, s_n)$ consistent with the orientation of $S$.

Let $d\psi_N \colon N_p\gamma(S) \to \RR^2$ be the restriction of the differential $d\psi \colon T_p M \to \CC \simeq \RR^2$ to the normal space $N_p\gamma(S)$. This is a linear map between real 2-dimensional vector spaces, and is invertible since $d\psi$ is surjective on $\psi^{-1}(0)$ by the definition of $\cF$. We define the value $\det(d\psi_N) \in \RR^\times$ by
\begin{align}
    \det(d\psi_N)\, \nu_N(\cdot, \cdot) = \nu_\CC(d\psi_N\, \cdot, d\psi_N\, \cdot)
\end{align}
using the area form $\nu_N$ on $N\gamma(S)$ induced by $g$, and the standard area form $\nu_\CC$ on $\CC$, defined by $\nu_\CC(h, k) = \Re(h i \bar{k})$ for $h, k \in \CC$.
% 
% On $N_p \gamma(S)$, we can define a volume form $dn_1\wedge dn_2$using $n_1$ and $n_2$ as local coordinates and a value $\det d\psi_N\in \RR^\times$ by the relation
% \Albert{[\(n_1,n_2\) are basis vectors.  Not sure what type \(dn_1,dn_2\) are.  Maybe write \(n_1^\flat\wedge n_2^\flat\)?  In fact, you can just write \(\det(d\psi_N)\) without explanation further than noting that both the domain and target spaces are equipped with (induced) volumes.]}\Sadashige{Good idea! Modified the text. }
% \begin{align}
%     \det d\psi_N  dn_1 \wedge d n_2 = dx\wedge dy (d\psi_N \cdot, d\psi_N \cdot)
% \end{align}
% where $x$ and $y$ are the standard Cartesian coordinates on $\RR^2\simeq\CC$.
% 
% 
% 
% \Albert{[Define \(d\psi_N\) when there is a metric.  When there is a metric on \(M\), for each \(p\in\gamma\) (\(\psi(p) = 0\)), we have  \(d\psi\colon T_pM\to\CC\).  Here \(T_pM\) is orthogonally decomposed into \(T_pM = T_p\gamma \oplus N_p\gamma\).  Let \(d\psi_N\) be the restriction of \(d\psi\) on the normal plane.  Note that \(d\psi_N\) maps a real 2D metric vector space to a real 2D metric vector space and is invertible.]}
Note that $\det d\psi_N$ depends on the choice of Riemannian metric $g$, but is independent of the choice of the frame field $(n_1, n_2)$. The following proposition gives an explicit expression for $\tilde Z_\psi$.

\begin{proposition}\label{prop:expression_of_Z}
    Given a Riemannian metric $g$ on $M$, the presymplectic form $\tilde Z_\psi$ is explicitly written as
    \begin{align}\label{eq:expression_of_Z}
        \tilde Z_\psi(\dot \psi, \ring \psi)
        = \int_{\{\psi = 0\}} \det(d\psi_N^{-1}) \Re(\dot\psi\, i\, \bar{\ring \psi})\, d\cH^n,
    \end{align}
    where $d\cH^n$ is the $n$-dimensional Hausdorff measure induced by $g$.
\end{proposition}
We note that in \eqref{eq:expression_of_Z}, both $\det(d\psi_N^{-1})$ and $d\cH^n$ depend on the metric $g$, but their product is independent of $g$.


\begin{proof}
Let \(\gamma = \Pi\psi\) and \(\dot \psi = -\mathcal{L}_u \psi + a \psi\), \(\ring \psi = -\mathcal{L}_v \psi + b \psi\) with \(u, v \in \mathrm{Diff}(M)\) and \(a, b \in C^\infty(M; \mathbb{C})\). From \(d\Pi(\dot\psi) = -\mathcal{L}_u \delgam \in T_{\delgam}\mathcal{D}_\mathcal{O}\) and \(d\Pi(\ring\psi) = -\mathcal{L}_v \delgam\), we have
\begin{align}
    \Pi^*\omega(\dot \psi, \ring \psi)
    &=\omega(-\cL_u\delgam, -\cL_v\delgam)
    = \int_\gamma \iota_v \iota_u \mu 
    = \int_\gamma \iota_{v^\perp} \iota_{u^\perp} \mu
\end{align}
where \(u^\perp, v^\perp\) denote the orthogonal projections of \(u, v\) onto the normal space with respect to \(g\) on the zeros, which are defined on $\psi^{-1}(0)$.

From \(u^\perp = d\psi_N^{-1} \dot\psi\) and \(v^\perp = d\psi_N^{-1} \ring\psi\), we have
\begin{align}
    \int_\gamma \iota_{v^\perp} \iota_{u^\perp} \mu
    &= \int_\gamma \mu(d\psi_N^{-1} \dot\psi, d\psi_N^{-1} \ring\psi, \cdots) \\
    &= \int_\gamma \nu_N\left(d\psi_N^{-1} \dot\psi, d\psi_N^{-1} \ring\psi\right) \iota_{n_2} \iota_{n_1} \mu \\
    &= \int_\gamma \det(d\psi_N^{-1}) \nu_\mathbb{C}(\dot\psi, \ring\psi) \iota_{n_2} \iota_{n_1} \mu,
\end{align}
which reads the stated expression.
        % where $v$ and $u$ are vector fields on $M$ such that $\dot \gamma=u\circ \gamma, \ring \gamma=v\circ \gamma$.
        % Note that
        
        % \Sadashige{Below is the version with matrix representation. }
        % At each $p\in M$ on the zeros of $\psi$, let  $D\Psi$ be the $2\times m$ matrix representation $D\Psi$ of $d\psi$ with respect to the basis $(\partial_{s_1}\gamma,\ldots,\partial_{s_n}\gamma,n_1,n_2)\in T_p\gamma(N) \oplus  N_p\gamma(N)$ using any fixed local charts $(s_1,\ldots, s_{n})$ consistent with the orientation of $S$ and $(1,i)\in T_0\CC\simeq \RR^2$. 
        % We have the singular value decomposition,
        % \begin{align}
        %     D\Psi=D\Psi_N (n_1, n_2)^T
        % \end{align}
        %  with $D\Psi_N$ the $2\times 2$ real-matrix representation of $d\psi_N$ with respect to the basis $(n_1,n_2)$ and $(1,i)$. 
        %  With a pseudo-inverse of $D\Psi$
        %  \begin{align}
        %       D\Psi^\dag=(n_1, n_2) D\Psi_N^{-1},
        %  \end{align}
        %  we have
        % \begin{align}
        %     u^\perp
        %     = D\Psi^\dag \dot \Psi 
        %     =(n_1, n_2) D\Psi_N^{-1} \dot \Psi,\quad v^\perp=(n_1, n_2) D\Psi_N^{-1} \ring \Psi.
        % \end{align}
        % on $\{\psi=0\}$ where $\dot \Psi, \ring \Psi$ are $\dot \psi, \ring \psi$ represented by 2 dimensional real vectors with basis $(1,i)$.  Then we get
        % \begin{align}
        %      \int_\gamma \iota_{v^\perp} \iota_{u^\perp} \mu
        %      &= \int_\gamma  \mu\left( (n_1, n_2)D\Psi_N^{-1}\dot \Psi,  (n_1, n_2)D\Psi_N^{-1}\ring \Psi,\cdots \right)\\
        %      &=\int_\gamma \det(D\Psi_N^{-1}\dot \Psi, D\Psi_N^{-1}\ring \Psi) \iota_{n_2} \iota_{n_1} \mu\\
        %      &=\int_\gamma \det D\Psi_N^{-1} \det (\dot \Psi, \ring \Psi)\iota_{n_2} \iota_{n_1} \mu.
        % \end{align}
        % From $\det (\dot \Psi, \ring \Psi)= \Re \dot \psi \Im \ring \psi - \Im \dot \psi \Re \ring \psi$, the stated expression follows.
        
        % \Sadashige{Below is the version with lesser use of matrix representation. }\Albert{I like this better.}
        % \Sadashige{Using the modified definition of  $\det d\psi_N$ above, I completely removed matrix representations.}
 
        % where $ \left( d\psi_N^{-1}\dot\psi, d\psi_N^{-1}\ring\psi \right)$ is a real $2\times 2$ matrix comprising the coefficients of $d\psi_N^{-1}\dot\psi,d\psi_N^{-1}\ring\psi\in N_p \gamma(S)$ with respect to the basis $(n_1,n_2)$. We have $\det \left( d\psi_N^{-1}\dot\psi, d\psi_N^{-1}\ring\psi \right)
            % =\det d\psi_N^{-1} \det ( \dot\psi, \ring\psi)$ where $( \dot\psi, \ring\psi)$ is again a real $2\times 2$ matrix with the real and imaginary parts of $\dot\psi$ and $\ring\psi$ as its entries. Now the stated expression follows from $\det ( \dot\psi, \ring\psi)=\Re(\dot \psi i \bar {\ring \psi})$.
        \end{proof} 

\subsubsection{Riemannian and K\"ahler structures}
We can also describe Riemmanian and formal K\"ahler structures we defined in \autoref{sec:kaehler_explicit} in terms of implicit representations.

We first observe a relation between the presymplectic form $\tilde Z$ and the standard symplectic structure on the space of complex valued functions. The expression \eqref{eq:expression_of_Z} of $\tilde Z$ in \autoref{prop:expression_of_Z} appears similar to the symplectic structure on  $C^\infty(M,\CC)$ given by 
\begin{align}
    \frak{Z}(\dot \psi, \ring \psi)=\int_M \Re (\dot\psi i \bar {\ring \psi} )\mu
\end{align}
for $\dot \psi, \ring \psi\in T_\psi C^\infty(M,\CC)=C^\infty(M,\CC)$, which forms a K\"ahler structure, together with a Riemannian metric 
\begin{align}
    \frak{G}(\dot \psi, \ring \psi)=\int_M \Re ( \dot\psi \bar {\ring \psi} ) \mu
\end{align}
and the almost complex structure $\tilde \cJ_i\colon \dot \psi \mapsto i\dot \psi$ on $C^\infty(M,\CC)$. In light of this, it is tempting to define a degenerate Riemannian metric $\tilde B$ on $\cF_\cO$ by $\tilde B(\cdot, \cdot)=\tilde Z(\cdot, \tilde \cJ_i\cdot)$. Explicitly,
\begin{align}
    \tilde B_\psi(\dot \psi, \ring \psi)
        =\int_{\{\psi=0\}} \det{d\psi_N^{-1} } \Re (\dot\psi \bar{ \ring \psi})\ d\cH^{n}.
\end{align}
Then $(\tilde B, \tilde Z, \tilde \cJ_i)$ is formally a degenerate K\"ahler structure.  At a glance, this setting seems natural and intrinsic as it does not require any Riemannian metric of the ambient manifold unlike the formal K\"ahler structure \((\cG,\omega,\cJ)\) on the explicit shape space, explained in \autoref{sec:kaehler_explicit}.

% This structure, however, does not descend onto the explicit shape space $\cO$, as follows 
However, we emphasize that the triple \((\tilde B,\tilde Z, \tilde\cJ_i)\) on $\cF_\cO$ does not descend onto the base space $\cO$.
% \Albert{[This sentence is broken by too many commas.  In general, there are too many ``however''s in these paragraphs that the readers might not get what the take away is.  Just make one clear statement and the rest are explanations for the statement.  ``However, we emphasize that this K\"ahler structure \((\tilde B,\tilde Z,\cJ_i)\) is a different one from the formal K\"ahler structure \((g,\omega,\cJ)\) of \autoref{sec:kaehler_explicit}.  A simple way to tell the difference is to observe that the \(\cJ_i\) action is different from \(\cJ\) as demonstrated in the subsequent example.  Moreover, while \(\tilde Z\) descends onto \(\cO\) as \(\tilde Z = \Pi^*\omega\), the metric \(\tilde B\) and the almost complex structure \(\cJ_i\) do not.'']}.
To see this, first note that the presymplectic form $\tilde Z$ is by design invariant under the action of $\SemiD_\gamma=\Diff_\gamma(M)\ltimes C^\infty(M,\CC)$ on each fiber $\Pi^{-1}\gamma$ over $\gamma\in \cO$ and $\ker \tilde Z_\psi$ agrees with 
\begin{align}
     \ker d\Pi_\psi=\{-\cL_u\psi +\varphi \psi \mid (u,\varphi)\in \diff_0(M)\ltimes C^\infty(M,\CC) \text{ s.t. } -\cL_u\delgam=0\}.
\end{align}
Hence $\tilde Z$ descends onto $\cO$ and defines the MW symplectic structure $\omega$ by $\tilde Z=\Pi^*\tilde \omega$. 

However, both $\tilde B$ and $\tilde \cJ_i$ do not descend onto $\cO$, and hence fail to define a Riemannian metric and an almost complex structure on $\cO$, as shown in the following example.  
In particular, 
% $\tilde \cJ_i d\Pi\neq d\Pi \cJ$.
the diagram
\[
\begin{tikzcd}
T\cF_\cO \arrow[r,"\tilde \cJ_i"] \arrow[d,"d\Pi"] & T\cF_\cO \arrow[d,"d\Pi"] \\
T\cO \arrow[r,"\cJ"] & T\cO
\end{tikzcd}
\]
does not commute.

\begin{example}
The almost complex structure $\tilde \cJ_i$ behaves differently at $\psi_0$ and $\psi_1$ in the same fiber $\Pi^{-1}\gamma$ if they are in different conformal classes (\autoref{def:conformal_and_twist}). 

Consider the setting in \autoref{eg:non_transitivity_conformal} where $\psi_0(x,y)=x+iy$ and $\psi_1(x,y)=x+y+iy$ on $\RR^2$ so $\Pi \psi_j=\delta_0$ for $j=0,1$. With a fixed vector field $u(x,y)=(1,0)$ on $\RR^2$, we have $\cL_u\psi_j=1$ and $\tilde \cJ_i (\cL_u \psi_j)=i\cdot 1=i$ for both $j=1,2$. 
However, observe that $\tilde \cJ_i (\cL_u \psi_j)=i=\cL_{u_j}\psi_j$ with different fixed vector fields $u_0(x,y)=(0,1)$ and $u_1(x,y)=(1,1)$. Namely $d\Pi (\tilde \cJ_i (\cL_u \psi_j))=d\Pi |_{\psi_j}(i)=-\cL_{u_j}\delta_0$ for $j=1,2$ yield different motions of the zero \ie, $-\cL_{u_0}\delta_0\neq -\cL_{u_1}\delta_0$. 

% For another vector field $u_0=(0,1)$, we have $\cL_{u_0}\psi_0=i=\tilde \cJ_i (\cL_u \psi_0)$, and hence $d\Pi(\tilde \cJ_i (\cL_u \psi_0))=-\cL_{u_0}\delta_0$.
% We also have $\cL_u \psi_1=1$ and $\tilde \cJ_i(\cL_u d\psi_1)=i\cdot 1=i$. On the other hand, we have $\cL_{u_1}\psi_1=i=\cJ_i (\cL_u \psi_1)$ for a fixed vector field $u_1=(-1,1)$, namely $d\Pi(\tilde \cJ_i (\cL_u \psi_1))=-\cL_{u_1}\delta_0$. Since $u_0\neq u_1$, they give different velocities of the zero \ie, $-\cL_{u_0}\delta_0\neq -\cL_{u_1}\delta_0$. 

The ill-definedness of $\tilde\cJ_i$ for different conformal classes also affects the degenerate Riemannian structure $\tilde B$. For the above $\psi_0$ and $\psi_1$ we have
\begin{align}
    \det (d\psi_{j})=
    \det \begin{pmatrix}
        1 & 0 \\
        0 & 1
    \end{pmatrix}
    =1
\end{align}
for both $j=0,1$ with respect to the standard Euclidean metric, and hence $\tilde B_{\psi_j} (\dot \psi_j,\ring \psi_j)=\Re(\dot \psi_j\bar{\ring \psi}_j)$. With a vector field $v=(0,1)$, we have $\cL_v \psi_0=i$ and  $\cL_v \psi_1=1+i$, namely
\begin{align}
\tilde B_{\psi_0}(\cL_v \psi_0,\cL_v \psi_0)&=\Re \left(i \cdot (-i)\right)=1,\\
\tilde B_{\psi_1}(\cL_v \psi_1,\cL_v \psi_1)&=\Re\left( (1+i) \cdot (1-i)\right)=2.
\end{align}
Thus neither of $\tilde \cJ_i$ nor $\tilde B$ descends to $\cO$. 
\end{example}

\vspace{5pt}
We may still construct a degenerate Riemannian metric on $\cF_\cO$ that descends onto $\cO$. Let us define a degenerate metric $\tilde G$ on $\cF_\cO$ by pulling back the metric $\cG$ on $\cO$ defined in \autoref{sec:kaehler_explicit}:
\begin{align}
    \tilde G = \Pi^* \cG.
\end{align}
Explicitly, we have
\begin{align}
    \tilde G_{\psi}(-\cL_u \psi+ a \psi, -\cL_v \psi+ b \psi)
    = \cG_\delgam(-\cL_u \delgam, -\cL_v \delgam)
    = \int_\gamma \iota_{\hat v} \iota_u \mu,
\end{align}
for  $-\cL_u \psi + a \psi, -\cL_v \psi + b \psi\in T_\psi \cF_\cO$
where $\gamma = \Pi\psi$, and $\hat v$ is any vector field satisfying $\hat v \circ \gamma = \cJ (v \circ \gamma)$ for the almost complex structure $\cJ \colon T\cO \to T\cO$ defined on $\cO$.

The degenerate metric $\tilde G$ is compatible with the presymplectic form $\tilde Z$ through the homomorphism $\tilde J \colon T\cF_\cO \to T\cF_\cO$ defined by
\begin{align}
    \tilde J_\psi(-\cL_v \psi + \varphi \psi)
    = -\cL_{\hat v} \psi + i \varphi \psi,
\end{align}
where, again, $\hat v$ is chosen so that $\hat v \circ \gamma = \cJ (v \circ \gamma)$. In general, $\tilde J$ may not satisfy $\tilde J^2 = -\id$ pointwise, so it may fail to be an almost complex structure on $\cF_\cO$. However, the failure is controlled: the difference $\tilde J \circ \tilde J(\dot \psi) - (-\id) (\dot \psi)=(\tilde J \circ \tilde J + \id)(\dot \psi)$ lies in $\ker d\Pi|_\psi$ for any $\dot \psi \in T_\psi \cF_\cO$. Thus, the following diagram commutes:
\[
\begin{tikzcd}
T\cF_\cO \arrow[r,"\tilde J"]\arrow[d,"d\Pi"] &  T\cF_\cO\arrow[d,"d\Pi"]\\
T\cO \arrow[r, "\cJ"]&  T\cO
\end{tikzcd}
\]

By construction, both $\tilde G$ and $\tilde J$ descend to the shape space $\cO$, defining the Riemmanian and the almost complex structure $\cG$  and $\cJ$, hence  forming a formal K\"ahler structure $(\cG, \omega, \cJ)$ on $\cO$.

\rev{
\section{Appendix: Curvature as monodromy}\label{sec:curvature_monodromy}

In the proof of \autoref{cor:average_swept_volume} in this chapter and \autoref{cor:horizontal_lift_area_formula} in \autoref{ch:spherical_polygon}, we use the fact that the integral of curvature form is given as the monodromy of a horizontal lift, measured by a connection form. This is formalized as the following proposition.
\begin{proposition}\label{prop:curvature_monodromy}
    Let $\pi\colon (E,\alpha)\to(B,\beta)$ be a prequantum circle bundle
    % with a scaling factor $\normconst$ for the vertical reproducibility of $\alpha$ \ie, $\alpha(\hat \xi)=\normconst \xi$ for any $\xi\in T_1\SS^1$ and its fundamental vector field $\hat \xi\in \Gamma(TE)$. 
    and $\Sigma$ be a topological disc in $B$.
    Consider any smooth path $x\colon [0,1]\to E$ such that it is horizontal with respect to $\alpha$ and $\pi \circ x$ parametrizes the closed curve $\partial\Sigma\subset B$.
    Then we have 
    \begin{align}
        \int_\Sigma \beta = \Arg(x(1),x(0)) \mod 2\pi
    \end{align}
    % \begin{align}
    %     \int_\Sigma \omega = \frac{c}{\normconst} \mod \frac{2\pi}{\normconst}
    % \end{align}
    where $\Arg(x(1),x(0))\in [0,2\pi)$ is defined by $e^{i\Arg(x(1),x(0))}\rhd x(1) = x(0)$.
   
\end{proposition}

\begin{proof}
First consider any smooth lift $\ell$ of $\Sigma$ to $E$ which is diffeomorphic in its image and contains $x(0)$. Let $y\colon [0,1]\to E$ be the parametrization of $\partial (\ell(\Sigma))$ such that $\pi(x(t))=\pi(y(t))$ for any $t$. Since $y$ is a closed curve bounding a topological disc $\ell(\Sigma)$, we have by the Stokes theorem that 
\begin{align}
    \int_\Sigma \beta 
    &= \int_{\pi (\ell(\Sigma))} \beta 
     = \int_{\ell(\Sigma)}\pi^*\beta
     = \int_{\ell(\Sigma)} d\alpha
    = \int_{\partial (\ell(\Sigma))} \alpha
    =\int_0^{1} y^* \alpha.
\end{align}
   We now define a non-closed path $\tilde y \colon [0,1]\to E$ by 
\begin{align}
    \tilde y(t) = e^{-i \int^t_0 y^*\alpha}\rhd y(t).
\end{align}
The path $\tilde y$ is horizontal with respect to $\alpha$, which can be verified by direct computation. 

Since $y(1)=y(0)=\tilde y(0)$ by definition, we have 
\begin{align}
    \tilde y(1) =  e^{-i \int^{1}_0 y^*\alpha}\rhd y(0) = e^{-i \int^{1}_0 y^*\alpha}\rhd \tilde y(0) 
\end{align}
and hence
\begin{align}
\int_\Sigma \beta 
= \int^{1}_0 y^*\alpha
= \Arg(\tilde y(1), \tilde y(0)).
\end{align}
Finally, since $ \tilde x(0)=\tilde y(0)$ by design, we have $x=\tilde y$ due to the uniqueness of the horizontal lift for a given initial point, which concludes the proof.
\end{proof}
As in \autoref{cor:average_swept_volume}, when the connection form a has scaling factor in vertical reproducibility, the result of \autoref{prop:curvature_monodromy} is modified. For a connection $\alpha$ with a scaling factor $1/\cN$ in  vertical reproducibility \ie, $\alpha(\hat\xi)=\frac{1}{\cN}\xi$ for $\xi\in\frak{g}$ and its fundamental vector field $\hat \xi$, we have
\begin{align}
    \int_\Sigma \beta = \frac{1}{\cN}\Arg(x(1),x(0)) \mod \frac{2\pi}{\cN} 
\end{align}
since the horizontal vector field $\tilde y$ in terms of a path $y$ in the proof becomes  $\tilde y(t)=e^{-i\cN \int^t_0 y^* \alpha}\rhd y(t)$. Consequently, we have $\tilde y(1)=e^{-i\cN \int^t_0 y^* \alpha}\rhd \tilde y(0) $ and hence $\frac{1}{\cN} \Arg(\tilde y (1),\tilde y(0))=\int ^1_0 y^*\alpha=\int_\Sigma \beta$.

}