text:
\section{Introduction}

Coadjoint orbits of the canonical central extension
$$
1 \to S^1 \to \widehat{{\rm Diff}^+(S^1)} \to {\rm Diff}^+(S^1) \to 1
$$
of the group $\mathcal{G}={\rm Diff}^+(S^1)$ of orientation preserving diffeomorphisms of the circle (also called Virasoro coadjoint orbits) attracted attention both in mathematics and physics literature since long time, see {\em e.g.} \cite{AS1, Kirillov, LP, Segal, W1}. The coadjoint action on the hyperplane corresponding to the coordinate $c$ (dual to
${\rm Lie}(S^1) \cong \mathbb{R}$) is defined on the space of quadratic differentials on the circle $T(x) dx^2$, and it is given by formula
$$
\chi: T(x) dx^2 \mapsto T^\chi(x) dx^2 = \left(T(\chi(x)) \chi'(x)^2 + \frac{c}{12} \, {\rm Sch}(\chi)\right)  dx^2,
$$
where ${\rm Sch}(\chi)$ is the Schwarzian derivative
$$
{\rm Sch}(\chi) = \frac{\chi'''(x)}{\chi'(x)} - \frac{3}{2} \left( \frac{\chi''(x)}{\chi'(x)} \right)^2.
$$

For $c\neq 0$, one of the Virasoro coadjoint orbits is of special importance. It corresponds to $T(x)=\frac{c}{24} \, dx^2$, and it is the unique orbit with the stabiliser isomorphic to the group ${\rm PSL}(2, \mathbb{R})$. This orbit (also called the Teichm\"uller orbit) naturally embeds in the universal Teichm\"uller space $T(1)$\footnote{Teichm\"uller spaces for curves of all finite genera naturally embed in $T(1)$.}:
$$
O_{\rm Teich}  \cong {\rm Diff}^+(S^1)/{\rm PSL}(2, \mathbb{R}) \subset {\rm QS}(S^1)/{\rm PSL}(2, \mathbb{R}),
$$
where ${\rm QS}(S^1)$ is the group of quasi-conformal mappings of the circle.

Consider the space ${\rm Hyp}(\mathbb{D})$ of geodesically complete hyperbolic metrics on the unit disk $\mathbb{D} \subset \mathbb{C}$. A typical example in this class is the standard Poincar\'e metric.
The group of orientation preserving diffeomorphisms ${\rm Diff}^+(\mathbb{D})$ acts transitively on ${\rm Hyp}(\mathbb{D})$. Now consider the group ${\rm Diff}^+(\mathbb{D}, \partial \mathbb{D}) \subset {\rm Diff}^+(\mathbb{D})$ which fixes the boundary of the disk
$\partial \mathbb{D} \cong S^1$.
It was argued in the physics literature (see \cite{SSS}) that $O_{\rm Teich}$ is  symplectomorphic to the following moduli space:
$$
O_{\rm Teich} \cong {\rm Hyp}(\mathbb{D})/{\rm Diff}^+(\mathbb{D}, \partial \mathbb{D}).
$$
Formal Duistermaat-Heckman integrals over this space were defined and studied in \cite{Stanford_Witten, Alekseev_Shatashvili_char_orbits_DH}.

Recall that for $\chi \in {\rm QS}(S^1)$ there exist two univalent holomorphic functions
$f_+: \mathbb{D} \to \mathbb{C}, f_-: \mathbb{D}^* \to \mathbb{C}$ such that
$$
f_+(e^{i \chi(x)}) = f_-(e^{ix})
$$
for $x \in \mathbb{R}$. Here $\mathbb{D}^*$ is the unit disk centered at infinity. The functions $f_+(z)$ and $f_-(z)$ are called components of conformal welding of $\chi$.
The K\"ahler potential of the Weil-Petersson metric on $O_{\rm Teich}$ is given by the Takhtajan-Teo (TT) energy functional (see \cite{TT}\footnote{In fact, in \cite{TT} a new Hilbert manifold structure on $T(1)$ was introduced, and it was shown that $S(\chi)$ is a K\"ahler potential for the Weil-Petersson metric on the Hilbert submanifold $T_{0}(1) \subset T(1)$.}):
$$
S(\chi) = \int_\mathbb{D} \left| \frac{f''_+(z)}{f'_+(z)} \right|^2 \, d^2z +
4 \pi \log(|f'_+(0)|) +
\int_{\bar{\mathbb{D}}} \left| \frac{f''_-(z)}{f'_-(z)} \right|^2 \, d^2z -
4 \pi \log(|f'_-(\infty)|).
$$

In this paper, we focus our attention on the subgroup ${\rm Diff}^+_{\rm hol}(S^1) \subset
{\rm Diff}^+(S^1)$ which is characterized by the property that the map $z=e^{ix} \mapsto e^{i\chi(x)}$ extends to a holomorphic function on an annulus $\mathcal{A}_{r,R}=\{ z \in \mathbb{C}; r < |z| < R\}$ for some $r <1 <R$.
 For this subgroup, following Nag-Sullivan \cite{Sullivan} we define a group homomorphism to the group of restricted symplectic transformations acting on the Hilbert space $H=H_+ \oplus H_-$ of holomorphic functions (modulo constants):
$$
{\rm Diff}^+_{\rm hol}(S^1) \to {\rm Sp}^{\rm res}(H_+ \oplus H_-).
$$
Here $H_+$ is spanned by $z^n$ for $n\geq 1$, and $H_-$ by $z^n$ for $n \leq -1$, and $||z^n||=|n|$.
We then use the metaplectic representation of ${\rm Sp}^{\rm res}(H_+ \oplus H_-)$ defined by Berezin formalism of normal symbols (see \cite{Berezin}) to construct operators
\begin{equation}  \label{eq:quantum_welding}
N_\chi = N_{f_+^{-1}} * N_{f_-}.
\end{equation}
Here $*$ is the product of operators acting on the Fock space $\mathcal{F}$ defined by the polarization $H = H_+ \oplus H_-$.  In this sense, equation \eqref{eq:quantum_welding} defines a  quantization of conformal welding.

Our first main result is as follows:
\begin{thm}  \label{thm:intro1}
For $\chi, \phi \in {\rm Diff}^+_{\rm hol}(S^1)$, we have
$$
N_\chi N_\phi = C(\chi, \phi) N_{\chi \circ  \phi},
$$
where $C(\chi, \phi) \in \mathbb{C}^*$ is a multiplicative group 2-cocylce with
the property that
\begin{equation}  \label{eq:intro_surprize}
C(\chi, \phi) = C(f_-, g_+),
\end{equation}
and $f_\pm$ define a conformal welding of $\chi$, and $g_\pm$ define a conformal welding of $\phi$.
Furthermore, for $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$
$$
\log(|C(f_-^{-1}, f_+)|) = \frac{S(\chi)}{24\pi}.
$$
\end{thm}
Equation \eqref{eq:intro_surprize} is surprizing since the cocycle $C(\chi, \phi)$ depends only on the components $f_-$ and $g_+$! For a more precise statement of Theorem \ref{thm:intro1}, see Theorem \ref{thm:C_N}
and Theorem \ref{thm:C_N_welding}.

Our second main result is the following theorem (see also Theorem \ref{thm:main2}):
\begin{thm}  \label{thm:intro2}
For $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$, the operator
$$
U_\chi = e^{S(\chi)/48\pi} N_\chi
$$
is unitary, and the cocycle $C(\chi, \phi)$ satisfies the equality
\begin{equation}    \label{eq:mystery}
\log(|C(\chi, \phi)|) = \frac{S(\chi \circ \phi) - S(\chi) - S(\phi)}{48\pi}.
\end{equation}
\end{thm}
The left hand side and the right hand side of \eqref{eq:mystery} have rather different analytic forms.  Equation \eqref{eq:mystery} follows from Berezin formalism, but at this point we are not aware of its direct proof.

We believe that our findings admit extensions to other representations of the group of diffeomorphisms of the circle defined in terms of free fields. In particular, this applies to representations of affine Kac-Moody algebras on Wakimoto modules. We also believe that our results may find applications in Theoretical Physics. Our original motivation comes from the work \cite{Polyakov} which introduced conformal welding in the study of Fermions in a gravitational field in 2 dimensions (see also the analysis of the gravitational Wess-Zumino functionals in \cite{AS2}).

The structure of the paper is as follows: in Section 2, we recall the definition of the Bott-Virasoro 2-cocycle on ${\rm Diff}^+(S^1)$, and we extend it to the groupoid of conformal maps.
In Section 3, we explain how holomorphic maps define symplectic transformations on the space $H$, and we define their Grunsky coefficients. In Section 4, we set up the Berezin formalism for normal and unitary symbols of operators on the Fock space $\mathcal{F}$. Finally, in Section 5 we describe quantization of conformal welding, the cocycles for normal and unitary symbols and their relation to the TT functional.

\vskip 0.2cm

{\bf Acknowledgements:}
Research of AA was supported in part by the grants 182767, 208235 and 200400,
and by the NCCR SwissMAP of the Swiss National Science Foundation (SNSF).
Research of AA and SS was partly supported by the award of
the Simons Foundation to the Hamilton Mathematics Institute of the Trinity College Dublin
under the program ``Targeted Grants to Institutes''.

\section{Group cocycles}

In this Section, we recall the notion of a group 2-cocycle. We then focus our attention on the Bott-Virasoro cocycle on the group of orientation preserving diffeomorphisms of the circle ${\rm Diff}^+(S^1)$ and on its extension to holomorphic maps.

\subsection{Group 2-cocycles: definition and basic properties}

Let $G$ be a group and $\mathbb{K}$ be the basic field (in this article, $\mathbb{R}$ or $\mathbb{C}$) viewed as a trivial $G$-module. A map $c: G \times G  \to \mathbb{K}$ is an additive group 2-cocycle if
\begin{equation} \label{eq:2-cocycle}
c(f,g) + c(fg, h) = c(f, gh) + c(g,h)
\end{equation}
for all $f, g, h \in G$. The definition implies $c(e,g)=c(e,e)=c(g,e)$ for all $g \in G$. Also, for all $k \in \mathbb{K}$ the assignment $c(f,g)=k$ is a 2-cocycle.

For every map $b: G \to \mathbb{K}$ one defines a trivial 2-cocycle
$$
\delta b(f, g) = b(f) - b(fg) + b(g).
$$
Note that for $b(f) =k \in \mathbb{K}$ we obtain $c(f,g)=k$. Hence, for any 2-cocycle $c$ there is a cohomologous normalized cocycle
$$
\tilde{c}(f,g) = c(f,g) - c(e,e)
$$
which has the property $\tilde{c}(e,e)=0$.
We will use the following cyclic property of 2-cocycles:
\begin{prop}  \label{prop:cyclic}
Assume that a 2-cocycle $c$ has the property $c(f, f^{-1})=0$ for all $f \in G$. Then,
\begin{equation}  \label{eq:cyclic}
c(f, g) = c(g, h) = c(h, f),
\end{equation}
where $fgh=e$.
\end{prop}

\begin{proof}
In equation \eqref{eq:2-cocycle}, put $fgh=e$ to obtain
$$
c(f, g) + c(h^{-1}, h) = c(f, f^{-1}) + c(g, h).
$$
By assumption, $c(h^{-1}, h)=c(f, f^{-1})=0$. Hence, we get $c(f, g)=c(g, h)$. The last equality follows since
$fgh=e$ implies $hfg=e$.
\end{proof}

Assume that the group $G$ possesses a Lie algebra $\mathfrak{g} = {\rm Lie}(G)$, denote by $\exp: \mathfrak{g} \to G$ the exponential map, and assume that finite products of the type
$\exp(u_1) \dots \exp(u_m)$ cover $G$. We define a map $\beta: \mathfrak{g} \times G \to \mathbb{K}$ by formula
\begin{equation}
\beta(u, g)=\frac{d}{dt} \, c(\exp(tu), g)|_{t=0}.
\end{equation}
The condition $c(g, e)=c(e,e)$ implies that $\beta(u, e)=0$ for all $u \in \mathfrak{g}$.

\begin{prop} \label{prop:b=>c}
The map $\beta: \mathfrak{g} \times G \to \mathbb{K}$ uniquely determines a normalized group 2-cocycle $c$.
\end{prop}

\begin{proof}
Put $f=\exp(tu)$ in equation \eqref{eq:2-cocycle} and differentiate in $t$ at $t=0$. We obtain
$$
\frac{d}{dt} c(\exp(tu)g, h) = \beta(u, gh) - \beta(u,g).
$$
Hence, if $\beta(u,g)=0$ then the normalized cocycle $c(f,g)$ vanishes, as required.
\end{proof}

Define
$$
\alpha(u,v) = \frac{d}{ds} \, \beta(u, \exp(sv))|_{s=0}
=\frac{\partial^2}{\partial s \partial t} \, c(\exp(tu), \exp(sv))|_{s=t=0}
$$
and
$$
a(u,v) = \frac{1}{2}\left(\alpha(u,v) - \alpha(v,u)\right).
$$
Recall that $a \in \wedge^2 \mathfrak{g}^*$ is a Lie algebra 2-cocycle, and that it satisfies the equation
\begin{equation}
a(u, [v, w]) + a(w, [u, v]) + a(v, [w, u])=0.
\end{equation}
Note that there is no analog of Proposition \ref{prop:b=>c} which would allow to reconstruct maps $\beta$ and $c$ starting from the map $a$ (or the map $\alpha$). Indeed,
adding a trivial cocycle $\delta b$ such that $d/dt \, b(\exp(tu))|_{t=0} =0$ does not affect the maps $\alpha$ and $a$, but it changes $\beta$ and $c$, in general.

Let $\rho: G \to {\rm End}(V)$ be a  projective representation, and assume that
$$
\rho(f) \rho(g) = e^{i c(f, g)} \rho(fg),
$$
where $c: G \times G \to \mathbb{C}$ is a complex valued function. Then, $c$ verifies the identity \eqref{eq:2-cocycle} modulo $2\pi \mathbb{Z}$. This is a direct consequence of associativity of the product in ${\rm End}(V)$.
If the group $G$ is connected, and the function $c$ is a continuous function, then it is actually a 2-cocycle. Indeed, in this case the defect in equation \eqref{eq:2-cocycle} is also a continuous function of $f, g, h$ which vanishes for $f=g=h=e$. Hence, it vanishes for all $f, g, h \in G$.
Furthermore, assume that $V$ is a Hilbert space and that $\rho: G \to U(V)$ is a unitary representation. Then, $c(f,g) \in \mathbb{R}$ is a real valued 2-cocycle.

Every 2-cocycle $c: G \times G \to \mathbb{C}$ defines a group law on $\hat{G} = G \times \mathbb{C}^*$ defined by formula
$$
(f, z) \cdot (g, w) = (fg, zw \exp(ic(f,g))).
$$
This group fits into a short exact sequence
$$
1 \to \mathbb{C}^* \to \hat{G} \to G \to 1
$$
and defines a central extension of $G$. If the cocycle $c$ is real valued, this central extension is by the circle $S^1$ (instead of $\mathbb{C}^*$).

\subsection{The Bott-Virasoro cocycle}

Consider the group $G={\rm Diff}^+(S^1)$ of orientation preserving diffeomorphisms of the circle. We recall the following basic fact:
\begin{thm}[Bott-Virasoro cocycle]  \label{thm:Bott-Virasoro}
The map $c_{\rm BV}: G \times G \to \mathbb{R}$ defined by formula
$$
c_{\rm BV}(\chi,\phi) = \int_0^{2\pi} \log(\chi'(\phi(x))) \log(\phi'(x))'\,  dx
$$
is a normalized real valued group 2-cocycle.
Furthermore, it satisfies the cyclic property \eqref{eq:cyclic}.
\end{thm}

\begin{proof}
For convenience of the reader, we give a proof of this statement.
The left hand side of equation \eqref{eq:2-cocycle} is as follows:
$$
\begin{array}{lll}
 c_{\rm BV}(\chi,\phi)+c_{\rm BV}(\chi \circ \phi,\psi) & =    &
 \int_0^{2\pi}\left( \log(\chi'(\phi(x))(\log(\phi'(x)))' + \log((\chi\circ \phi)'(\psi(x))) (\log(\psi'(x)))' \right) \, dx \\
  & = & \int_0^{2\pi} \left( \log(\chi'(\phi(\psi(x)))) \log(\psi'(x))' + \log(\phi'(\psi(x))) \log(\psi'(x))'\right) \, dx \\
  & +  & \int_0^{2\pi} \log(\chi'(\phi(x))(\log(\phi'(x)))'\, dx, \\
\end{array}
$$
and the right hand has the following form:
$$
\begin{array}{lll}
c_{\rm BV}(\chi, \phi \circ \psi) + c_{\rm BV}(\phi,\psi)  & = &
\int_0^{2\pi}\left( \log(\chi'(\phi(\psi(x)))) \log((\phi\circ \psi)'(x))' +
\log(\phi'(\psi(x))) \log(\psi'(x))'\right) \, dx\\
& = & \int_0^{2\pi}\left( \log(\chi'(\phi(\psi(x)))) \log(\psi'(x))' + \log(\phi'(\psi(x))) \log(\psi'(x))'\right) \, dx \\
& + & \int_0^{2\pi} \log(\chi'(\phi(\psi(x))) \log(\phi'(\psi(x)))' \, dx.
\end{array}
$$
Note that the second lines in the two expressions coincide term by term, and the third line of the right hand side of \eqref{eq:2-cocycle} is obtained from the third line of the left hand side by the change of variable $x \mapsto \psi(x)$.

For the cyclic property, put $\chi = \phi^{-1}$. Then, $\log(\chi'(\phi(x))) = - \log(\phi'(x))$ and
$$
c_{\rm BV}(\phi^{-1}, \phi) = - \int_0^{2\pi} \log(\phi'(x)) \log(\phi'(x))' \, dx = - \frac{1}{2} \, \log(\phi'(x))^2|_0^{2\pi} = 0.
$$
Here we have used the fact that $\phi'(x)$ is periodic. The cyclic property follows by Proposition \ref{prop:cyclic}.
\end{proof}

It is instructive to compute the map $\beta_{\rm BV}$:
$$
\beta_{\rm BV}(u,\phi) = \int_0^{2\pi} u'(\phi(x)) \log(\phi'(x))' \, dx =
- \int_0^{2\pi} u'(y) \log((\phi^{-1})'(y))' \, dy.
$$
Here we made a change of variables $y=\phi(x)$ in the integral.
The map $\alpha_{\rm BV}$ is given by formula,
$$
\alpha_{\rm BV}(u,v) = \int_0^{2\pi} u'(x) v''(x) dx,
$$
and it is skew-symmetric in $u$ and $v$.

\subsection{Extension to holomorphic maps}
In this Section, we extend the Bott-Virasoro cocycle to a certain class of holomorphic maps. In more detail, let $\mathcal{S}$ be a connected open subset of the complex plane with $\pi_1(\mathcal{S}) \cong \mathbb{Z}$. By the Uniformization Theorem, such a domain is holomorphically isomorphic to an annulus
$$
\mathcal{A}_{r, R} = \{ z \in \mathbb{C}; r < |z| <R \} .
$$
We will consider triples $(\mathcal{S}, f, \mathcal{T})$, where $\mathcal{S}$ and $\mathcal{T}$ are two such domains, and $f: \mathcal{S} \to \mathcal{T}$ is a holomorphic isomorphism between $\mathcal{S}$ and $\mathcal{T}$. Sometimes it is convenient to label the domain and the range of $f$ by $\mathcal{S}_f$ and $\mathcal{T}_f$, respectively. Note that the domain $\mathcal{S}_f$ contains a closed curve $C_f$ which represents the generator of $\pi_{1}(\mathcal{S}_f)$. Its image $f(C_f)$ represents the generator of $\pi_1(\mathcal{T}_f) \cong \mathbb{Z}$. By the analytic continuation principle, the holomorphic function $f$ is uniquely determined by its restriction to $C_f$.

Triples $(\mathcal{S}_f, f, \mathcal{T}_f)$ form a groupoid with composition law
$$
(\mathcal{S}_f, f, \mathcal{T}_f) \circ (\mathcal{S}_g, g, \mathcal{T}_g) =
(\mathcal{S}_g, f\circ g, \mathcal{T}_f).
$$
Two triples are composable if $\mathcal{S}_f = \mathcal{T}_g$. The curves $g(C_g)$ and $C_f$ are homotopic to each other in $\mathcal{S}_f = \mathcal{T}_g$.

We define a subset ${\rm Diff}^+_{\rm hol}(S^1) \subset {\rm Diff}^+(S^1)$ by the following property: $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$ if the map
$$
z = e^{ix} \mapsto e^{i \chi(x)}
$$
extends to a univalent holomorphic map $f$ on an annulus $\mathcal{A}_{r, R}$ with $r < 1 < R$. It is easy to see that ${\rm Diff}^+_{\rm hol}(S^1)$ is a subgroup of ${\rm Diff}^+(S^1)$.

Consider two diffeomorphisms $\chi, \phi \in {\rm Diff}^+_{\rm hol}(S^1)$ and the corresponding univalent holomorphic functions $f$ and $g$ such that
$$
f(e^{ix})=e^{i\chi(x)}, \hskip 0.3cm g(e^{ix}) = e^{i \phi(x)}.
$$
By making the annulus $\mathcal{S}_g$ smaller if needed, one can always achieve $\mathcal{T}_g \subset \mathcal{S}_f$. By restricting $f$ to $\mathcal{T}_g$, one obtains a pair of composable holomorphic maps, and
$$
f(g(e^{ix})) = f(e^{i\phi(x)}) = e^{i\chi(\phi(x))}.
$$
Since the analytic function $f \circ g$ is uniquely determined by its values on the unit circle, we conclude that it corresponds to the diffeomorphism $\chi \circ \phi$.

\begin{prop}  \label{prop:cBVextend}
Let $\chi, \phi \in {\rm Diff}^+_{\rm hol}(S^1)$ and $f,g$ the corresponding composable holomorphic functions. Then,
\begin{equation}  \label{eq:cBVextend}
c_{\rm BV}(\chi, \phi) = \int_{C_1} \log\left( \frac{g(z)f'(g(z))}{f(g(z))} \right) \,
\log\left( \frac{z g'(z)}{g(z)}\right)' \, dz.
\end{equation}
\end{prop}

\begin{proof}
The proof is by a direct calculation. In particular, for $z=e^{ix}, g(z)=e^{i\phi(x)}$ we have $zg'(z)/g(z)=\phi'(x)$, and  $g(z)f'(g(z))/f(g(z)) = \chi'(\phi(x))$.
\end{proof}

For a pair of composable univalent holomorphic maps $f$ and $g$, one can use the right hand side of equation \eqref{eq:cBVextend} as a definition of a  functional of a pair $(f,g)$:
\begin{equation}  \label{eq:CBV}
C_{\rm BV}(f,g)=\int_{C_g} \log\left( \frac{g(z)f'(g(z))}{f(g(z))} \right) \,
\log\left( \frac{z g'(z)}{g(z)}\right)' \, dz.
\end{equation}
Here the integration is over the curve $C_g$ on which both homorphic functions $g$ and $f \circ g$ are well defined.
Note that $C_{\rm BV}$ is complex valued, in general. This is in contrast to the cocycle $c_{\rm BV}$ which takes values in $\mathbb{R}$.

\begin{prop}  \label{prop:C_BV}
The map $C_{\rm BV}$ is a groupoid 2-cocylce. That is, for all composable triples $(f,g,h)$, it satisfies the equation
$$
C_{\rm BV}(f,g)+C_{\rm BV}(fg, h) = C_{\rm BV}(f, gh) + C_{\rm BV}(g,h).
$$
Furthermore, it satisfies the cyclic property \eqref{eq:cyclic}.
\end{prop}

\begin{proof}
The proof of the cocycle condition is analogous to the one of Theorem \ref{thm:Bott-Virasoro}. The only non-trivial step in the proof is as follows: one needs to check that
$$
\int_{C_h} \log\left( \frac{g(h(z))f'(g(h(z)))}{f(g(h(z)))} \right) \,
\log\left(\frac{h(z)g'(h(z))}{g(h(z))}\right)' \, dz =
\int_{C_g} \log\left( \frac{g(w)f'(g(w))}{f(g(w))}\right) \, \log\left( \frac{wg'(w)}{g(w)} \right)' dw.
$$
Two integrals are related by the change of variable $w=h(z)$. After this change of variables, the integration contour on the left hand side is $C_h$, and on the right hand side it is $h^{-1}(C_g)$. Both these curves represent the generator of $\pi_1(\mathcal{S}_h)$, and therefore they are homotopic to each other.

For the cyclic property, put $f=g^{-1}$. Then,
$$
\log\left( \frac{g(z)f'(g(z))}{f(g(z))} \right) = - \log\left( \frac{z g'(z)}{g(z)}\right)
$$
and
$$
C_{\rm BV}(g^{-1}, g) = - \frac{1}{2} \int_{C_g} \frac{d}{dz} \, \left(\log\left( \frac{z g'(z)}{g(z)}\right)\right)^2 \, dz =0.
$$
The proof of Proposition \ref{prop:cyclic} applies verbatim to the case of groupoids. This completes the proof.
\end{proof}

Note that the expression \eqref{eq:CBV} can be re-written using  the change of variables $z=g^{-1}(w)$.  We get
$$
C_{\rm BV}(f, h^{-1}) =
- \int_{C_f} \log\left( \frac{w f'(w)}{f(w)}\right) \, \log\left(
\frac{w (g^{-1})'(w)}{g^{-1}(w)}\right)' \, dw.
$$
Holomorphic functions $f: \mathcal{S}_f \to \mathcal{T}_f$ and $h: \mathcal{T}_g \to \mathcal{S}_g$ are actually defined on the same domain
$\mathcal{S}_f = \mathcal{T}_g$ which contains the curve $C_f$. We compute the expression $\beta$ for $C_{\rm BV}$.
By putting $f(w)=w + t u(w) + O(t^2)$, we obtain
$$
\beta_{\rm BV}(u, h^{-1}) =
- \int_{C_1} \left(u'(w) - \frac{u(w)}{w}\right) \, \log\left(
\frac{w h'(w)}{h(w)}\right)' \, dw.
$$

\begin{rem}
Yet another groupoid cocycle which has the cyclic property is given by formula
\begin{equation}  \label{eq:tildeC}
\tilde{C}(f,g) = \int_{C_g} \log(f'(g(z))) \log(g'(z))' dz.
\end{equation}
The proof is similar to those of Theorem \ref{thm:Bott-Virasoro} and of Proposition \ref{prop:C_BV}.
\end{rem}

\section{Symplectic transformations}

In this Section, we recall the notion of symplectic transformations associated to holomorphic maps.

\subsection{Symplectic transformations in finite dimensions}

Recall the following standard setup: let $U$ be a complex vector space, and $\omega \in \wedge^2 U$ be a non-degenerate (symplectic) 2-form. A linear map $A \in {\rm End}(U)$ is called symplectic if it preserves $\omega$:
$$
\omega(A(u_1), A(u_2)) = \omega(u_1, u_2)
$$
for all $u_1, u_2 \in U$.

The following set of examples is of special interest for us: let $V$ be a finite dimensional complex vector space. Then, one can equip the direct sum
$U= V \oplus V^*$ with a  natural symplectic form
$$
\omega(a+a^*, b+b^*) = \langle b^* , a\rangle - \langle a^*, b\rangle,
$$
where $a,b \in V, a^*,b^* \in V^*$. A splitting $U= V \oplus V^*$ is also called a polarization of the symplectic space $U$.

Consider a transformation $A \in {\rm End}(V \oplus V^*)$ defined by formula
\begin{equation}      \label{eq:AinSp}
\left(
\begin{array}{ll}
a \\
a^*
\end{array}
\right) \mapsto
\left(
\begin{array}{ll}
\tilde{a} \\
\tilde{a}^*
\end{array}
\right) =
\left(
\begin{array}{ll}
\alpha & \beta \\
\gamma & \delta
\end{array}
\right) \cdot
\left(
\begin{array}{ll}
a \\
a^*
\end{array}
\right).
\end{equation}
Here $\alpha: V \to V, \beta: V^*\to V, \gamma: V \to V^*, \delta: V^* \to V^*$.
This transformation is symplectic if and only if the following conditions are verified:
$$
\begin{array}{lll}
\beta \alpha^t = (\beta \alpha^t)^t,
 &  \alpha^t \gamma = (\alpha^t \gamma)^t,
& \alpha \delta^t - \beta \gamma^t = 1, \\
\delta^t \beta = (\delta^t \beta)^t, &
\gamma \delta^t = (\gamma \delta^t)^t, &
\alpha^t \delta - \gamma^t \beta = 1.
\end{array}
$$
Note that if $\alpha$ and $\delta$ are  invertible, the following operators are symmetric: $(\alpha^{-1} \beta)$, $(\beta \delta^{-1})$, $(\gamma \alpha^{-1})$, $(\delta^{-1} \gamma)$. In this case, one can also express $\alpha$ and  $\delta$ in terms of three other operators:
$$
\delta = (\alpha^t)^{-1} + \gamma \alpha^{-1} \beta, \hskip 0.3cm
\alpha = (\delta^t)^{-1} + \beta \delta^{-1} \gamma.
$$

\subsection{Symplectic transformations and holomorphic maps}
We now pass to the infinite dimensional context and apply the theory of symplectic transformations to holomorphic functions. As in the previous Section, let $\mathcal{S} \subset \mathbb{C}$ be a connected domain with $\pi_1(\mathcal{S})=\mathbb{Z}$, and let $C \subset \mathcal{S}$ be a closed oriented curve which realizes the generator of $\pi_1(\mathcal{S})$. We consider the space $\mathcal{H}_\mathcal{S}$ of holomorphic functions on $\mathcal{S}$ and define the following 2-form
$$
\omega_\mathcal{S} = \frac{1}{4\pi} \int_C \, (\delta \phi(z)) \partial_z(\delta \phi(z)) \, dz.
$$
Here $\delta$ is the de Rham differential on $\mathcal{H}_\mathcal{S}$ and $\partial_z$ is the $z$-derivative. It is clear that the definition of $\omega_\mathcal{S}$ is independent of the choice of the curve $C$.

Let $f: \mathcal{S} \to \mathcal{T}$ be a holomorphic isomorphism. It induces an isomorphism $f^*: \mathcal{H}_\mathcal{T} \to \mathcal{H}_\mathcal{S}$ by composition: $\phi \mapsto f^*\phi(z) = \phi(f(z))$. In turn, the map $f^*$ induces a pull-back map of differential forms that we denote by $(f^*)^*$.

\begin{prop}
 $(f^*)^*\omega_\mathcal{S}=\omega_\mathcal{T}$.
\end{prop}

\begin{proof}
We compute,
$$
\begin{array}{lll}
4 \pi (f^*)^* \omega_\mathcal{S} & = &
\int_{C_\mathcal{S}} (\delta \phi(f(z))) \partial_z (\delta \phi(f(z))) \, dz \\
& = &
\int_{f(C_\mathcal{S})} (\delta \phi(w)) \partial_w (\delta \phi(w)) \, dw \\
& = &
\int_{C_\mathcal{T}} (\delta \phi(w)) \partial_w (\delta \phi(w)) \, dw \\
& = &
4 \pi \omega_\mathcal{T}.
\end{array}
$$
Here we made a change of variables $z=f^{-1}(w)$, and then used the fact that
$f(C_\mathcal{S})$ is homotopic of $C_\mathcal{T}$ in $\mathcal{T}$.
\end{proof}

For an annulus
$$
\mathcal{A}_{r, R} = \{ z \in \mathbb{C}; r < |z| < R\}
$$
with $r < 1 < R$, one can choose $C_\mathcal{S}$ to be the unit circle.
We will consider the space $H = \mathcal{H}_\mathcal{A}/\mathbb{C}$ of holomorphic functions modulo constants.
Using the Fourier transform,
$$
\phi(z) = \sum_{n=1}^\infty \frac{a_n}{\sqrt{n}} z^n + \sum_{n=1}^\infty \frac{a_n^*}{\sqrt{n}}  z^{-n} + a_0,
$$
we obtain a formula for $\omega_\mathcal{A}$:
$$
\omega_\mathcal{A} =\frac{i}{2} \, \sum_{n =1}^\infty \delta a_n  \wedge \delta a_n^*.
$$
This form is symplectic on $H$. In what follows, it will be more convenient to work with functions
$$
\psi(z) = \phi'(z) =  \sum_{n=1}^\infty \sqrt{n} \, a_n z^{n-1} - \sum_{n=1}^\infty \sqrt{n} \, a^*_n z^{-n-1}
$$
which do not contain the superfluous constant $a_0$. One can view Fourier components $\{ a_n, a^*_n\}$ as coordinates on the infinite dimensional symplectic space of holomorphic functions.

The space $H$ admits a polarization
$$
H = H_+ \oplus H_-,
$$
where $H_+$ is spanned by monomials $z^n$ with $n \geq 1$ and $H_-$ by monomials $z^n$ with $n\leq -1$. Holomorphic maps induce symplectic transformations
$$
f: \psi \mapsto \tilde{\psi}(z)=\psi(f(z))f'(z).
$$
In more detail,
\begin{equation}  \label{eq:f_acts_psi}
\sum_{n=1}^\infty \sqrt{n} \, (\tilde{a}_n z^{n-1} -
 \tilde{a}^*_n z^{-n-1})
 = f'(z) \sum_{n=1}^\infty \sqrt{n} ({a}_n (f(z))^{n-1} - {a}^*_n (f(z))^{-n-1}).
\end{equation}
This equation implies
\begin{equation} \label{eq:alpha}
\alpha_{m,n} = \frac{1}{2\pi i} \int_C \sqrt{\frac{n}{m}}
\int_C \frac{f(z)^{n-1} f'(z)}{z^m} \, dz,
\end{equation}
\begin{equation} \label{eq:beta}
\beta_{m,n} = - \frac{1}{2\pi i} \int_C \sqrt{\frac{n}{m}}
\int_C \frac{f(z)^{-n-1} f'(z)}{z^m} \, dz,
\end{equation}
\begin{equation} \label{eq:gamma}
\gamma_{m,n} = -\frac{1}{2\pi i} \int_C \sqrt{\frac{n}{m}}
\int_C f(z)^{n-1} f'(z) z^m \, dz,
\end{equation}
\begin{equation} \label{eq:delta}
\delta_{m,n} =  \frac{1}{2\pi i} \int_C \sqrt{\frac{n}{m}}
\int_C f(z)^{-n-1} f'(z) z^m \, dz,
\end{equation}
where $\alpha_{m,n}, \beta_{m,n}, \gamma_{m,n}, \delta_{m,n}$ are infinite dimensional matrices representing operators $\alpha, \beta, \gamma, \delta$.

The map from holomorphic maps to symplectic transformations is a group anti-homomorphism:
\begin{prop}  \label{prop:A_fg}
Let $f, g$ be two composable holomorphic maps and $A_f, A_g$ be the corresponding symplectic transformations. Then,
$$
A_{f \circ g} = A_g A_f.
$$
\end{prop}

\begin{proof}
The proof is by a direct computation.
\end{proof}

\subsection{Grunsky coefficients and symplectic transformations}

We will need the following simple properties of holomorphic functions.

Let $f(z)$ be a univalent holomorphic function on neighborhood of zero with $f(0)=0$. Then,  $f(z)=\sum_{n=1}^\infty f_n z^n$ with $f_1 \neq 0$. Recall that the function
\begin{equation}  \label{eq:Grunsky1}
\log\left(\frac{f(z)-f(w)}{z-w}\right) = \sum_{m,n=0}^\infty F_{m,n} z^m w^n
\end{equation}
is regular in $z$ and $w$. Here $F_{m,n}$ are the Grunsky coefficients of $f(z)$
(see \cite{Pom} for details).

In a similar fashion, let $f(z)$ be a univalent holomorphic function on a neighborhood of infinity with $f(\infty) = \infty$. Then, $f(z) = \sum_{n = - \infty}^1 f_n z^n$ with $f_1 \neq 0$, and
\begin{equation}   \label{eq:Grunsky2}
\log\left(\frac{f(z)-f(w)}{z-w}\right) = \log(f_1) + \sum_{m,n=1}^\infty F_{-m,-n} z^{-m} w^{-n}.
\end{equation}

The following proposition will be important for the rest of the paper:
\begin{prop}   \label{prop:f_h}
Let $f=\sum_{n=1}^\infty f_n z^n$ with $f_1 \neq 0$ be a univalent holomorphic map on a neighborhood of zero. Then, the corresponding symplectic transformation is upper-triangular, the operator $\gamma$ vanishes, and the symmetric operator $(\alpha^{-1} \beta)$ is of the following form:
\begin{equation}   \label{eq:TT_formula1}
\sum_{m,n=1}^\infty \sqrt{mn} \, (\alpha^{-1} \beta)_{m,n} u^{m-1} w^{-1}n = \frac{(f^{-1})'(u)(f^{-1})'(w)}{(f^{-1}(z)-f^{-1}(w))^2} - \frac{1}{(u-w)^2}.
\end{equation}

Let $f=\sum_{n=-\infty}^1 f_n z^n$ with $f_1 \neq 0$ be a univalent holomorphic map on a neighborhood of infinity. Then, the corresponding symplectic transformation is lower triangular, the operator $\beta$ vanishes, and the symmetric operator $(\gamma \alpha^{-1})$ is of the following form:
\begin{equation}   \label{eq:TT_formula2}
- \sum_{m,n=1}^\infty \sqrt{mn} \, (\gamma \alpha^{-1})_{m,n} z^{-m-1} w^{-n-1} =  \frac{f'(z)f'(w)}{(f(z)-f(w))^2} - \frac{1}{(z-w)^2}
\end{equation}
\end{prop}

\begin{proof}
Let $f=\sum_{n=1}^\infty f_n z^n$ with $f_1 \neq 0$. Observe that
the expression  $f(z)^{n-1} f'(z)z^m$ in equation \eqref{eq:gamma} is regular at zero and its integral over $C$ vanishes. Hence, the operator $\gamma$ vanishes. Furthermore, for $n>m$ the function $f(z)^{n-1} f'(z)/z^m$ in equation \eqref{eq:alpha} is also regular at zero which implies $\alpha_{m,n}=0$. Therefore, $\alpha$ is an upper-triangular (infinite) matrix. A similar argument shows that $\delta$ is also upper-triangular.

Let $h$ be the inverse function of $f$. Equation \eqref{eq:alpha} implies
$$
\alpha^{-1}_{m,n} = \frac{1}{2\pi i} \, \sqrt{\frac{{n}}{{m}}} \, \int_{C'} \frac{h(w)^{n-1} h'(w)}{w^m} dw,
$$
where $C'$ is some (possibly different from $C$) circle around zero.
Combining with equation \eqref{eq:beta}, we obtain
$$
(\alpha^{-1} \beta)_{m,n} = \sum_{k=1}^\infty \alpha^{-1}_{m,k} \beta_{k,n} =
- \frac{1}{(2\pi i)^2} \, \sqrt{\frac{n}{m}} \, \int_{C \times C'} \frac{h'(w) f'(z)}{w^m f(z)^{n+1}} \,
\sum_{k=1}^\infty \frac{h(w)^{k-1}}{z^k} dw dz.
$$
Summing up a geometric series and making a substitution $z = h(u)$ yields
$$
(\alpha^{-1} \beta)_{m,n} = - \frac{1}{(2\pi i)^2} \,\sqrt{\frac{n}{m}} \, \int_{C' \times C'} \frac{h'(w) }{w^m u^{n+1}} \, \frac{1}{h(u)-h(w)} \, du dw
$$
Finally, integration by parts over $u$ gives rise to
$$
(\alpha^{-1} \beta)_{m,n} = \frac{1}{(2\pi i)^2} \, \frac{1}{\sqrt{mn}} \,
\int_{C' \times C'} \frac{1}{w^m u^{n}} \, \frac{h'(u)h'(w)}{(h(u)-h(w))^2} \, du dw.
$$
The function
$$
\frac{h'(u)h'(w)}{(h(u)-h(w))^2} - \frac{1}{(u-w)^2} = \frac{\partial^2}{\partial u \partial w} \, \log\left( \frac{f(u)-f(w)}{u-w}\right)
$$
is regular in $u,w$. Hence, it is given by the Taylor series \eqref{eq:TT_formula1}.

Proof of equation \eqref{eq:TT_formula2} is similar.
\end{proof}

\section{Metaplectic representation and Berezin formalism}

In this Section, we recall the metaplectic representation of the symplectic group and Berezin formalism in finite and infinite dimensions.

\subsection{Heisenberg Lie algebra and normal symbols}
To a symplectic vector space $V \oplus V^*$ one can naturally associate a
 Heisenberg Lie algebra with generators
$\hat{a}, \hat{a}^*$ for $a\in V, a^* \in V^*$ defined by canonical commutation relations
$$
[\hat{a}, \hat{b}] = [\hat{a}^*, \hat{b}^*]=0, \hskip 0.3cm
[ \hat{a}, \hat{b}^*] = \omega(b^*, a)= \langle b, a\rangle.
$$
Choose a Hermitian scalar product $(\cdot, \cdot)$ on $V$. Then, the symmetric algebra $SV^*$ also carries a Hermitian product, and it can be completed to a Fock space
$$
\mathcal{F} = \overline{SV^*}.
$$
The Fock space $\mathcal{F}$ carries a natural action (by unbounded operators) of the Heisenberg algebra, where $\hat{a} \cdot 1 = 0$ for all $a \in V$ and
$$
(\hat{a} + \hat{a}^*) \cdot f = a^* f + \partial_a f,
$$
where $\partial_a$ is a constant vector field acting on $SV^*$.

Introduce an orthonormal basis $\{ a_i\}$ of $V$ and the dual basis $\{ a_i^* \}$ of $V^*$. In this basis, $\omega$ takes the canonical form
$$
\omega(a_i, a_j)=\omega(a^*_i, a^*_j)=0, \hskip 0.3cm
\omega(a_i, a^*_j) = \delta_{ij}.
$$
Then, operators $\hat{a}_i, \hat{a}^*_i$ on $\mathcal{F}$ are conjugate to each other under the Hermitian structure on $\mathcal{F}$.
To multi-indices $I=(i_1, \dots, i_m), J=(j_i, \dots, j_n)$ we associate
monomials
$$
{a}_I = {a}_{i_1} \dots {a}_{i_m}, \hskip 0.3cm
{a}_J={a}^*_{j_1} \dots {a}^*_{j_n}.
$$
To a power series in formal variables $a_i, a^*_i$
$$
N_q(a, a^*) = \sum_{I, J} q_{I,J} a_I a^*_J
$$
one associates an operator
$$
\hat{q} = \sum_{I, J} q_{I, J} \hat{a}^*_J \hat{a}_I.
$$
If the sum is finite, this operator is well defined, and $N_q(a, a^*)$ is called its normal symbol. Sometimes, $\hat{q}$ is well defined even for infinite series $N_q(a, a^*)$. The operator product $q \cdot r$ is represented by a formal Gaussian integral in terms of normal symbols:
\begin{equation} \label{eq:f*g}
N_q * N_r = N_{q \cdot r}(a, a^*) = \int  N_q(a+b, a^*) \, e^{-\langle b^*, b\rangle} \,
N_r(a, a^* + b^*) \, dbdb^*.
\end{equation}
Note that this formal integral in defined modulo sign since in general it involves a square root of the determinant (see below for a more detailed discussion).

\subsection{Berezin formalism}
In the finite dimensional context, the group of symplectic transformations
${\rm Sp}(V\oplus V^*)$ has a double cover
$$
1 \to \mathbb{Z}_2 \to {\rm Mp}(V \oplus V^*) \to {\rm Sp}(V \oplus V^*) \to 1
$$
called the metaplectic group. We will also need the associated central extension of ${\rm Sp}(V \oplus V^*)$ by $\mathbb{C}^*$:
$$
1 \to \mathbb{C}^* \to \widehat{\rm Sp}(V \oplus V^*)=
{\rm Mp}(V \oplus V^*) \times_{\mathbb{Z}_2} \mathbb{C}^*
\to {\rm Sp}(V \oplus V^*) \to 1.
$$
It comes with a natural representation  on the Fock space $\mathcal{F}$ which can also be viewed as a projective representation of ${\rm Sp}(V\oplus V^*)$.

For a given symplectic transformation $A \in {\rm Sp}(V \otimes V^*)$
one says that an invertible operator $\hat{A}$ on $\mathcal{F}$ implements it if it represents a lift of $A$ in
$\widehat{\rm Sp}(V\oplus V^*)$. In more detail, it means that
\begin{equation}

    \hat{A}
    \left(
    \begin{array}{l}
    \hat{a} \\
    \hat{a}^*
    \end{array}
    \right)
    \hat{A}^{-1}
    =
    \left(
    \begin{array}{ll}
    \alpha & \beta \\
    \gamma & \delta
    \end{array}
    \right)
    \left(
    \begin{array}{l}
    \hat{a} \\
    \hat{a}^*
    \end{array}
    \right)
    \end{equation}
    for all $a \in V, a^* \in V^*$.

We will call a symplectic transformation $A \in {\rm Sp}(V \oplus V^*)$ admissible if its components $\alpha: V \to V$ and $\delta: V^* \to V^*$ are invertible.
The following theorem summarizes a result of Berezin (see \cite{Berezin}):
\begin{thm}  \label{thm:berezin}
Let $A \in {\rm Sp}(V \oplus V^*)$ be an admissible symplectic transformation. Then, $A$ is implemented by a unique operators $\hat{A}$ with a normal symbol $N_A$ whose constant term is equal to 1. This normal symbol is given by formula
\begin{equation}   \label{eq:normal_symbols}
N_A(a, a^*)  =
\exp\left( \langle a^*, (\alpha^{-1} -1) a\rangle - \frac{1}{2} (a^*, (\alpha^{-1} \beta) a^*) + \frac{1}{2} (a, (\gamma \alpha^{-1}) a) \right)
\end{equation}
\end{thm}

We will call a pair of admissible symplectic transformations $A_1, A_2$ composable if $A_1A_2$ is also an admissible transformation. We use a similar terminology for triples.
The following proposition gives a product rule in terms of normal symbols:

\begin{prop}  \label{prop:berezin_product}
Let $A_1, A_2 \in {\rm Sp}(V \oplus V^*)$ be a composable pair.
Then,
\begin{equation}   \label{eq:berezin_product}
N_{A_1} * N_{A_2} = \frac{1}{{\rm det}^{1/2}(1 + (\alpha_2^{-1} \beta_2) (\gamma_1 \alpha_1^{-1}))} \, N_{A_2A_1}.
\end{equation}
\end{prop}

\begin{proof}
The proof is by a direct calculation of the Gaussian integral \eqref{eq:f*g}.
\end{proof}

Note that the product rule for normal symbols \eqref{eq:berezin_product} is not quite well defined because of the square root of the determinant. In fact, the subset of admissible elements in the metaplectic group admits the following description:
$$
{\rm Mp}_{\rm adm}(V \oplus V^*) = \{ (A, z) \in {\rm Sp}_{\rm adm}(V \oplus V^*) \times \mathbb{C}^*; {\rm det}(\alpha) = z^2\}.
$$
Observe that
$$
{\rm det}(1 + (\alpha_2^{-1} \beta_2)( \gamma_1 \alpha_1^{-1}))
= \frac{{\rm det}(\alpha)}{{\rm det(\alpha_1)} {\rm det}(\alpha_2)}.
$$
Hence, we can re-write the product rule of normal symbols in terms of the metaplectic group as follows:
$$
N_{(A_1, z_1)} * N_{(A_2, z_2)} = \frac{z_1z_2}{z} N_{(A, z)},
$$
where $A=A_2A_1$.
One can summarize the properties of the product rule as follows:
\begin{prop}
The expression
\begin{equation}   \label{eq:berezin_cocycle}
C_N(A_2, z_2; A_1, z_1) = \frac{1}{{\rm det}^{1/2}(1 + (\alpha_2^{-1} \beta_2) (\gamma_1 \alpha_1^{-1}))} = \frac{z_1z_2}{z}
\end{equation}
is a multiplicative 2-cocylce. That is, for all composable triples $A_1, A_2, A_3$ we have
$$
C_N(A_2, A_1) C_N(A_3, A_2A_1) = C_N(A_3A_2, A_1)C_N(A_3, A_2)
$$
\end{prop}

\begin{proof}
The statement follows from the fact that $C_N=z_1z_2/z$ is a trivial 2-cocylce. This fact reflects  associativity of the operator product $*$.
\end{proof}

For operators with normal symbols $G_{(A,z)}=z^{-1} N_A$, we obtain a group anti-homomorphism:
\begin{equation}
    G_{(A_2, z_2)} * G_{(A_1, z_1)} = G_{(A_1A_2, z_1z_2)}.
\end{equation}

The group ${\rm Sp}(V \oplus V^*)$ contains a subgroup ${\rm USp}(V \oplus V^*)$ which has the following property: $\tilde{a}_i$ is the conjugate of $\tilde{a}^*_i$ for all $i$. This condition imposes an extra requirement on the components $\alpha, \beta, \gamma, \delta$ of $A$:
$$
\gamma = \overline{\beta}, \hskip 0.3cm
\delta=\overline{\alpha}.
$$
Here $\overline{\alpha}, \overline{\beta}$ are complex conjugate of $\alpha$ and $\beta$, respectively. In particular, we obtain
$$
\alpha \alpha^* = \alpha \delta^t = 1 + \beta \gamma^t = 1 + \beta \beta^*.
$$
This implies that $\alpha$ is invertible, and that so is $\delta = \overline{\alpha}$. Furthermore, we have the following useful identity:
and
\begin{equation}   \label{eq:alpha_alpha_-1}
\alpha^{-1}(\alpha^*)^{-1} = \alpha^{-1}(\alpha \alpha^* - \beta \beta^*)(\alpha^*)^{-1} = 1 - (\alpha^{-1} \beta)(\alpha^{-1}\beta)^*.
\end{equation}
This implies that all transformations $A \in {\rm USp}(V \oplus V^*)$ are admissible, all pairs $A_1, A_2$ are composable, and Theorem \ref{thm:berezin} and Proposition \ref{prop:berezin_product} apply without further assumptions. Also, the map $(A, z) \mapsto G_{(A,z)}$ defines a group anti-homomorphism from the corresponding subgroup of the metaplectic group ${\rm MUSp}(V \oplus V^*)$ to unitary operators on the Fock space $\mathcal{F}$.

\subsection{The infinite dimensional case}
Most of the facts reviewed in the previous Section generalize to the infinite dimensional setup. Let $V$ be a Hilbert space. This allows to identify $V^* \cong V$. In what follows, we list special features which distinguish the infinite dimensional situation from the finite dimensional one.

Instead of the symplectic group ${\rm Sp}(V \oplus V^*)$, one considers the restricted symplectic group
$$
{\rm Sp}^{\rm res}(V \oplus V^*) = \{ A \in {\rm Sp}(V \oplus V^*); \alpha, \delta \, {\rm are} \, {\rm Fredholm},
\beta, \gamma \, {\rm are}\, {\rm Hilbert-Schmidt}\}.
$$
For admissible elements of this subgroup, Theorem \ref{thm:berezin} and Proposition \ref{prop:berezin_product} hold true verbatim. In particular, the determinant
$$
{\rm det}(1 + (\alpha_2^{-1} \beta_2) (\gamma_1 \alpha_1^{-1}))
$$
is well defined since the operator
$$
1 + (\alpha_2^{-1} \beta_2) (\gamma_1 \alpha_1^{-1}) = \alpha_2^{-1} \alpha \alpha_1^{-1}
$$
is invertible, the operators $(\alpha_2^{-1} \beta_2)$ and $(\gamma_1 \alpha_1^{-1})$ are Hilbert-Schmidt, and hence the operator $(\alpha_2^{-1} \beta_2) (\gamma_1 \alpha_1^{-1})$ is of trace class. The product formula \eqref{eq:berezin_product} still makes sense on the metaplectic double cover.

However, the determinant ${\rm det}(\alpha)$ is not well defined, in general. Therefore, the admissible part of the metaplectic group does not allow for a simple description using the equation $z^2={\rm det}(\alpha)$, and the cocycle $C_N(A_1, A_2)$ is {\em a priori} nontrivial. It is instructive to write the corresponding Lie algebra cocycle
$a_N(x_1, x_2)$ on a pair of elements of the symplectic Lie algebra:
$$
x_i =
\left(
\begin{array}{ll}
a_i & b_i \\
c_i & d_i
\end{array}
\right), \, i=1,2,
$$
where $a_i$ and $d_i$ are bounded operators, and $b_i$ and $c_i$ are Hilbert-Schmidt operators.
An easy calculation shows that
$$
a_N(x_1, x_2) = {\rm Tr}(b_1c_2 -c_1b_2).
$$
The right hand side is well defined because both terms $b_1c_2$ and $c_1b_2$ are of trace class.

The restricted group ${\rm USp}^{\rm res}(V \oplus V^*)$ is defined as before:
$$
{\rm USp}^{\rm res}(V \oplus V^*) = \{ A \in {\rm Sp}^{\rm res}(V \oplus V^*); \delta=\overline{\alpha}, \gamma=\overline{\beta}\} .
$$
Again, all elements $A \in {\rm USp}^{\rm res}(V \oplus V^*) $ are admissible and all pairs $A_1, A_2$ are composable.

Another important result of \cite{Berezin} is as follows:
\begin{thm}  \label{prop:berezin_unitary}
Let $A \in {\rm USp}^{\rm res}(V \oplus V^*)$. Then, the normal symbol
$$
U_A = \pm \frac{1}{{\rm det}(\alpha \alpha^*)^{1/4}} \, N_A
= \pm {\rm det}^{1/4}(1 - (\alpha^{-1}\beta)(\alpha^{-1}\beta)^*) \, N_A
$$
defines a unitary operator on $\mathcal{F}$.
\end{thm}
Here we have used equation \eqref{eq:alpha_alpha_-1}. Note that the
resulting Fredholm determinant is well defined since the operator
$(\alpha^{-1}\beta)(\alpha^{-1}\beta)^*$ is of trace class.
Operators $U_A$ satisfy the product rule
$$
U_{A_1} * U_{A_2} = C_U(A_1, A_2) U_{A_2 A_1},
$$
where the cocycle $C_U(A_1, A_2)$ is given by formula
$$
C_U(A_1, A_2) = \left(\frac{{\rm det}(\alpha \alpha^*)}{
{\rm det}(\alpha_1 \alpha_1^*){\rm det}(\alpha_2 \alpha_2^*)}\right)^{\frac{1}{4}} \, \cdot \, \frac{1}{{\rm det}^{1/2}(1 + (\alpha_2^{-1} \beta_2)( \gamma_1 \alpha_1^{-1}))}.
$$
It is defined on the metaplectic double cover, and it
takes values in $S^1 \cong \{ z\in \mathbb{C}; |z|=1\}$ instead of $\mathbb{C}^*$. This cocycle is non trivial, in general. The corresponding Lie algebra cocycle is the same as for $C_N(A_1, A_2)$ (up to second order, the normalization factor is symmetric in $A_1, A_2$):
$$
a(x_1, x_2) = {\rm Tr}(b_1 \overline{b}_2 - \overline{b}_1 b_2).
$$

\section{Quantization of conformal welding}

In this Section, we apply Berezin quantization to triangular decomposition of symplectic transformations induced by holomorphic maps, and in particular to conformal welding.

\subsection{Triangular decomposition and conformal welding}
In this Section, we discuss an analogue of triangular decomposition for holomorphic maps.

Let $f: \mathcal{A}_{r, R} \to \mathbb{C}$ be a univalent holomorphic map defined on the annulus $\mathcal{A}_{r, R}$ such that its range is contained in another annulus: $f(\mathcal{A}_{r, R}) \subset \mathcal{A}_{r', R'}$. We say that $f$ admits a triangular decomposition if there exist univalent holomorphic maps $f_+: \mathbb{D}_{R'} \to \mathbb{C}, f_-: \mathbb{D}^*_{r'}$ such that $f_+(0)=0, f_-(\infty)=\infty$ and
\begin{equation}  \label{eq:triangular}
f_- = f_+ \circ f
\end{equation}
on $\mathcal{A}_{r, R}$. Here
$$
\mathbb{D}_R=\{ z \in \mathbb{C}; |z|<R\}, \hskip 0.3cm
\mathbb{D}^*_r = \{ z \in \mathbb{C}; |z| >r\} \cup \{ \infty\}
$$
are the discs centered at $0$ and $\infty$, respectively.
Holomorphic functions $f_+$ and $f_-$ admit Taylor expansions at $0$ and $\infty$:
\begin{equation}   \label{eq:f_pm}
f_+(z) = \sum_{k=1}^\infty (f_+)_k z^k, \hskip 0.3cm
f_-(z) = \sum_{k=-\infty}^1 (f_-)_k z^k,
\end{equation}
where $(f_+)_1 \neq 0, (f_-)_1 \neq 0$.

A special case of triangular decomposition is given by conformal welding of diffeomorphisms of the circle. Let $f$ be a holomorphic function which corresponds to $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$. That is,
$$
f(e^{ix})= e^{i \chi(x)}.
$$
In this case, one can choose the domain of $f$ to be an annulus $\mathcal{A}_{r, R}$ with $r < 1 < R$. The following theorem follows from results of \cite{Kirillov_welding} on conformal welding for elements of
${\rm Diff}^+(S^1)$. We will only be interested in the subgroup ${\rm Diff}^+_{\rm hol}(S^1) \subset{\rm Diff}^+(S^1)$.
\begin{thm}
Let $f: \mathcal{A}_{r, R} \to \mathbb{C}$ be a univalent holomorphic map which corresponds to a diffeomorphism of the circle $\chi$. Then, it admits a unique triangular decomposition $f =f_+^{-1} \circ f_-$ with $(f_+)'(0)=1$.
\end{thm}

\begin{rem}
One says that the univalent holomorphic functions $f_\pm$ provide a conformal welding of the diffeomorphism $\chi$.
Note that the normalization $f'_+(0)=(f_+)_1=1$ (here $f_+(z)=\sum_{n=1}^\infty (f_+)_n z^n$)
can be replaced by the normalization $f'_-(\infty)=(f_-)_1 =1$ (here $f_-(z) =
\sum_{n=-\infty}^1 (f_-)_n z^n$. This is achieved by dividing both $f_+$ and $f_-$ by $f'_-(\infty)$.
\end{rem}

Equation \eqref{eq:triangular} can also be re-written in the form
$$
f = f_+^{-1} \circ f_-
$$
which resembles of the Gauss decomposition of matrices. It turns out that it induces a Gauss decomposition on the corresponding elements of the infinite dimensional symplectic group. Indeed, by Proposition \ref{prop:f_h},  symplectic transformations $A_{f_+}$ and $A_{f_-}$ are upper- and lower-triangular, respectively. And by Proposition \ref{prop:A_fg}, we have
a Gauss type decomposition (recall that the map $f \mapsto A_f$ is a group anti-homomorphism):
$$
A_f = A_{f_-} A_{f_+^{-1}}.
$$
It is convenient to introduce the notation
$$
A_f =
\left(
\begin{array}{ll}
\alpha_f & \beta_f \\
\gamma_f & \delta_f
\end{array}
\right)
$$
for components of the symplectic transformation $A_f$. Then, we have
$$
A_{f} =
\left(
\begin{array}{ll}
\alpha_{f_-} &  0 \\
\gamma_{f_-} & \delta_{f_-}
\end{array}
\right)
\left(
\begin{array}{ll}
\alpha_{f_+^{-1}} & \beta_{f_+^{-1}} \\
0 & \delta_{f_+^{-1}}
\end{array}
\right) =
\left(
\begin{array}{cc}
\alpha_{f_-} \alpha_{f_+^{-1}}& \alpha_{f_-} \beta_{f_+^{-1}} \\
\gamma_{f_-} \alpha_{f_+^{-1}} & \gamma_{f_-} \beta_{f_+^{-1}} + \delta_{f_-} \delta_{f_+^{-1}}
\end{array}
\right).
$$
Assuming that $\alpha_{f_-}$ and $\alpha_{f_+}^{-1}$ are invertible, this implies
\begin{equation}  \label{eq:Gauss}
    \alpha_f^{-1} \beta_f = \alpha_{f_+^{-1}}^{-1}\beta_{f_+^{-1}}, \hskip 0.3cm
    \gamma_f \alpha_f^{-1} = \gamma_{f_-} \alpha_{f_-}^{-1}.
\end{equation}

\subsection{Normal symbols of holomorphic maps}
In this Section, we apply Berezin theory of normal symbols to holomorphic maps and the corresponding symplectic transformations.

In order to do that, we equip the space of holomorphic functions (modulo constants) $H$ with a structure of a Hilbert space (following \cite{Sullivan}) by declaring $|| z^n || = |n|$ for all $n \neq 0$. Then, symplectic transformations induced by holomorphic maps admit a metaplectic projective representation on the corresponding Fock space.

In more detail, let $f$ be a holomorphic map, and assume that the corresponding symplectic transformation $A_f$ is admissible. Then, it is convenient to denote by $N_f$ (instead of $N_{A_f}$) its normal symbol.
\begin{prop}  \label{prop:N*N}
Let $f$ and $g$ be a pair of composable holomorphic maps, and assume that the corresponding symplectic transformations $A_f$ and $A_g$ are admissible. Then,
$$
N_f * N_g = \frac{1}{{\rm det}^{1/2}(1 + (\alpha_g^{-1}\beta_g)(\gamma_f \alpha_f^{-1}))} \, N_{f\circ g}
$$
Furthermore, if $f=\sum_{n=1}^\infty f_n z^n$ with $f_1 \neq 0$, or if
$g=\sum_{-\infty}^1 g_n z^n$ with $g_1 \neq 0$, then
$$
N_f * N_g = N_{f\circ g}.
$$
\end{prop}

\begin{proof}
The first statement follows from Proposition \ref{prop:berezin_product}.
For the second statement, note that if $f=\sum_{n=1}^\infty f_n z^n$, then
by Proposition \ref{prop:f_h} $\gamma_f=0$. Similarly, if $g=\sum_{-\infty}^1 g_n z^n$, then $\beta_g=0$. This completes the proof.
\end{proof}

We are now ready to state one of our main results:
\begin{thm}   \label{thm:C_N}
Let $f$ and $g$ be a pair of composable holomorphic maps, and assume that they admit triangular decompositions. Then,
\begin{equation} \label{eq:product_N}
N_{f} * N_{g} = C_N(f, g) N_{f \circ g},
\end{equation}
where the 2-cocycle $C_N(f,g)$ is of the form
\begin{equation}  \label{eq:C_N}
    C_N(f,g)=\frac{1}{ {\rm det}^{1/2}(1 + (\alpha_{g_+^{-1}}^{-1}\beta_{g_+^{-1}})(\gamma_{f_-}\alpha_{f_-^{-1}}))}.
\end{equation}
The corresponding map $\beta_N$ is given by formula
\begin{equation}  \label{eq:beta_N}
    \beta_N(u, g) = - \frac{1}{(2\pi i)^2}
    \int_{C \times C} \frac{u_-(z)-u_-(w)}{z-w} \, \frac{\partial^2}{\partial z \partial w} \, \log\left( \frac{g_+(z)-g_+(w)}{z-w} \right) \, dzdw.
\end{equation}

\end{thm}

\begin{proof}
For the first statement, we use Proposition \ref{prop:N*N}, and we observe that by formula \eqref{eq:Gauss}
$$
\alpha_g^{-1}\beta_g = \alpha_{g_+^{-1}} \beta_{g_+^{-1}}, \hskip 0.3cm
\gamma_f \alpha_f^{-1} = \gamma_{f_-} \alpha_{f_-}^{-1}.
$$
In order to prove the formula for $\beta_N$, put $f(t)=\exp(tu)$, where
$u=u(z)\frac{\partial}{\partial z}$ is a vector field with
$$
u(z) = u_+(z) + u_-(z) = \sum_{n=1}^\infty u_n z^n + \sum_{n=-\infty}^0 u_n z^n.
$$
Note that $f(t)=f_+(t)^{-1} f_-(t)$ and $\log(f_-(t)) = tu_- + O(t^2)$. Hence,
$$
\beta_N(u, g) = \frac{d}{dt} \, C(\exp(tu), g)|_{t=0} =
{\rm Tr} (\alpha_{g+^{-1}}^{-1} \beta_{g_+^{-1}}) \left(\frac{d}{dt} \, (\gamma_{\exp(tu_-)} \alpha_{\exp(tu_-)}^{-1})\right)_{t=0}.
$$
Recall that
$$
\sum_{m,n} \sqrt{mn} (\gamma_{\exp(tu_-)} \alpha_{\exp(t u_-)}^{-1})_{m,n} z^{-m-1}w^{-n-1} =
- \frac{\partial^2}{\partial z \partial w} \, \log\left(\frac{\exp(tu_-)(z)-\exp(tu_-)(w)}{z-w}\right),
$$
where $\exp(tu_-)(z)$ is the image of $z$ under the holomorphic map $\exp(tu_-)$. The derivative in $t$ at $t=0$ yields (after integrating over $z$ and $w$)
$$
\sum_{m,n} \frac{1}{\sqrt{mn}} \, \frac{d}{dt}(\gamma_{f_-} \alpha_{f_-}^{-1})_{m,n}|_{t=0} z^{-m}w^{-n} =
- \frac{u_-(z)-u_-(w)}{z-w}.
$$
Also recall that
$$
\sum_{m,n} \sqrt{mn} (\alpha_{g_+^{-1}} \beta_{g_+})_{m,n} z^{m-1}w^{n-1}=
\frac{\partial^2}{\partial z \partial w} \, \log\left( \frac{g_+(z)-g_+(w)}{z-w}\right).
$$
Next, we convert the trace in $m,n$ into a double contour integral.
Since the factors $\sqrt{mn}$ and $1/\sqrt{mn}$ cancel out, we obtain the desired result.
\end{proof}

Surprisingly, the cocycle
$$
C_N(f,g)=C_N(f_+^{-1} f_-, g_+^{-1} g_-)
$$
has the following polarization property: it is independent of the components $f_+$ and $g_-$ in triangular decompositions of $f$ and $g$.

The following result of \cite{TT} (see Corollary 2.9 in Chapter 2) establishes important properties of symplectic transformations associated to conformal welding:
\begin{thm}  \label{thm:TT1}
Let $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$, $f$ the corresponding holomorphic function, and
$f_\pm$ the components of conformal welding of $\chi$.
Then, the maps $A_{f_+^{-1}}, A_{f_-}$ and $A_f = A_{f_-} A_{f_+^{-1}}$ belong to the restricted symplectic group ${\rm Sp}^{\rm res}(H_+ \oplus H_-)$. In particular, symmetric operators
$(\alpha_{f_+^{-1}}^{-1} \beta_{f_+^{-1}})$ and $(\gamma_{f_-} \alpha_{f_-}^{-1})$ are Hilbert-Schmidt.
\end{thm}

Theorem \ref{thm:TT1} implies that conditions of Proposition \ref{prop:N*N} are verified for conformal welding, and we have
$$
N_f = N_{f_+^{-1}} * N_{f_-}.
$$

Furthermore, let  $\chi, \phi \in {\rm Diff}^+_{\rm hol}(S^{1})$. Recall that the correpsonding
conformal maps $f$ and $g$ are always composable. By Theorem \ref{thm:TT1}, $A_f$ and $A_g$ are admissible. Hence, Theorem \ref{thm:C_N} applies and we obtain the formula
\eqref{eq:product_N} for the product of normal symbols $N_f * N_g$.

\subsection{The Takhtajan-Teo energy functional}

In this Section, we  recall the definition and main properties of the Takhtajan-Teo (TT) energy functional which will be important in the discussion of unitary symbols for conformal welding.

Let $\mathcal{D} \subset \mathbb{C}$ be a simply connected domain on the complex plane, $C \subset \mathcal{D}$ be a simple closed curve and $
\mathcal{C} \subset \mathcal{D}$ be a compact domain bounded by $C$.
We consider $u: \mathcal{D} \to \mathbb{C}$ a univalent holomorphic function.
To a pair $(\mathcal{C}, u)$ we associate a functional
\begin{equation}
    E(\mathcal{C}, u) = \int_\mathcal{C} \, \left| \frac{u''(z)}{u'(z)} \right|^2 d^2z.
\end{equation}

Two pairs $(u, \mathcal{C}_u)$ and $(v, \mathcal{C}_v)$ are called compatible if $v(C_v)=C_u$ and  $v(\mathcal{C}_v) = \mathcal{C}_u$.

\begin{prop}   \label{prop:TT_tranform}
For compatible pairs $(u, \mathcal{C}_u)$ and $(v, \mathcal{C}_v)$, we have
\begin{equation}
    E(u \circ v) = E(u) + E(v) -
    2\,  {\rm Re} \, \int_{\mathcal{C}_u} \, \frac{u''(w)}{u'(w)} \cdot \frac{\overline{(v^{-1})''(w)}}{\overline{(v^{-1})'(w)}} \, d^2w.
\end{equation}
\end{prop}

\begin{proof}
By direct substitution, we obtain
$$
\frac{(u \circ v)''(z)}{(u \circ v)'(z)} =
\frac{u''(v(z))}{u'(v(z))} \, u'(z) + \frac{v''(z)}{v'(z)}.
$$
This implies
$$
E(u \circ v) = E(u) + E(v) + 2 \, {\rm Re} \,
\int_{\mathcal{C}_v} \, \frac{u''(v(z))}{u'(v(z))} \, u'(z) \cdot
\frac{\overline{v''(z)}}{\overline{v'(z)}} \, d^2z.
$$
We now make a change of variables $z=v^{-1}(w)$ in the last term, and use that
$(v^{-1})'(w)=u'(z)^{-1}$ and $(v^{-1})''(w)=-u''(z)/u'(z)^3$ to complete the proof.
\end{proof}

As an example, consider the case where $\mathcal{C}_u = \mathcal{C}_v=\mathbb{D}$ is the unit disk. In this case, we denote
$$
E_+(v) = E(\mathbb{D}, v).
$$
Let $v$ be a M\"obius transformation:
$$
v(z)= \frac{az+b}{\bar{b} z + \bar{a}},
$$
with $|a|^2 - |b|^2=1$. Then,
$$
E_+(v) = 4 |b|^2 \int_\mathbb{D} \, \frac{d^2z}{|\bar{b} z + \bar{a}|^2} =
- 4 \pi \, \log\left(1 - \left| \frac{b}{a} \right|^2 \right) = 4\pi \log(|a|^2).
$$
The inverse M\"obius transformation $v^{-1}(w)$ is of the form
$$
v^{-1}(w)=\frac{\bar{a} w - b}{a - \bar{b}w},
$$
and we have
$$
\begin{array}{lll}
\int_{\mathbb{D}} \, \frac{u''(w)}{u'(w)} \cdot \frac{\overline{(v^{-1})''(w)}}{\overline{(v^{-1})'(w)}} \, d^2w & = &
2\pi\left( \log(u'(b/\bar{a})) - \log(u'(0)) \right) \\
& = &
2\pi(\log((u\circ v)'(0)) - \log(u'(0)) + \log(\bar{a}^2)).
\end{array}
$$
Here we have used that $v(0)=b/\bar{a}$ and that
$(u \circ v)'(0) = u'(v(0))v'(0) = u'(v(0))/\bar{a}$.
Putting things together, we obtain the following interesting equality for the functional $E_+$:
$$
E_+(u \circ v) + 4\pi \log|(u \circ v)'(0)| = E_+(u) + 4\pi \log|u'(0)|.
$$
Hence, on the unit disk the expression
$$
S_+(u) = E_+(u) +4\pi \log(|u'(0)|)= \int_\mathbb{D} \left| \frac{u''(z)}{u'(z)} \right|^2 \, d^2z
+ 4\pi \log(|u'(0)|)
$$
is invariant under M\"obius transformations. Similarly, for a function $u$ defined on the unit disk $\mathbb{D}^*$ centered at infinity, we define
$$
S_-(u) = E_-(u) - 4\pi \log(|u'(\infty)|)= \int_{\mathbb{D}^*} \left| \frac{u''(z)}{u'(z)} \right|^2 \, d^2z
- 4\pi \log(|u'(\infty)|).
$$

Let $\chi \in {\rm Diff}^+_{\rm hol}$, $f$ the corresponding holomorphic map and $f=f_+^{-1} \circ f_-$ the  conformal welding of $\chi$. The TT energy functional is defined as
\begin{equation}
    S(\chi) = S_+(f_+) + S_-(f_-).
\end{equation}
By the invariance properties of $S_\pm$,
$S(\chi)$ is left- and right-invariant under M\"obius transformations:
$$
S(m_1 \circ \chi \circ m_2) = S(\chi).
$$
In particular, it descends to the quotient space
$$
{\rm Diff}^+_{\rm hol}(S^1)/{\rm PSL}(2, \mathbb{R}) \subset {\rm Diff}^+(S^1)/{\rm PSL}(2, \mathbb{R}).
$$

We recall the following highly nontrivial property of the TT energy functional (see Theorem 3.8 in Chapter 2 of \cite{TT}):
\begin{thm} \label{thm:S_symmetry}
For all $\chi \in{\rm Diff}^+(S^1)$, we have
$$ S(\chi^{-1}) = S(\chi).$$
\end{thm}

We observe the following interesting property of the functional $S(\chi)$:
\begin{prop}  \label{prop:TT=cocycle+}
For $\chi \in {\rm Diff}^+_{\rm hol}$, we have
$$
S(\chi) =  {\rm Im} \, \tilde{C}(f_-^{-1}, f_+) - \int_{C} (\chi'(x) + 1) \log(\chi'(x)) dx,
$$
where $\tilde{C}$ is the groupoid 2-cocycle defined by equation \eqref{eq:tildeC}.
\end{prop}

\begin{proof}
First, observe that
$$
E_+(f_+) = \frac{i}{2} \, \int_{C} \,  \log(f'_+) d \log(\overline{f'_+}), \hskip 0.3cm
E_-(f_-) = - \frac{i}{2} \, \int_{C} \, \log(f'_-) d \log(\overline{f'_-}).
$$
Also, note that
$$
{\rm Im}\, \tilde{C}(f_-^{-1}, f_+) = {\rm Im} \,  \tilde{C}(f_+, f) = - \frac{i}{2} \int_C (\log(f'_+(f)) d \log(f') - \log(\overline{f'_+(f)}) d \log(\overline{f'})).
$$
We consider
$$
\begin{array}{lll}
A_1(\chi)  & = &
E_+(f_+) + E_-(f_-) - {\rm Im} \,  \tilde{C}(f_-^{-1}, f_+) \\
& = &  \frac{i}{2} \, \int_{C} \,  \log(f'_+) d \log(\overline{f'_+}) \\
& - & \frac{i}{2} \, \int_{C} \, (\log(f'_+(f) + \log(f')) d (\log(\overline{f'_+(f)}) + \log(\overline{f'})) \\
& + & \frac{i}{2} \int_C (\log(f'_+(f)) d \log(f') - \log(\overline{f'_+(f)}) d \log(\overline{f'}))
\end{array}
$$
Using that for $z=e^{ix}$
$$
\log(f'(z))=\log(\chi'(x)) + i (\chi(x)-x),
$$
we obtain
$$
A_1(\chi) = - \int_C (\log(\chi'(x)) + \log(|f'_+(f(z))|^2)(\chi'(x)-1) dx.
$$
Next, we consider
$$
\begin{array}{lll}
A_2(\chi) & = &
4\pi(\log(|f'_+(0)|) - \log(|f'_-(\infty)|) \\
& = & \int_C (\log(|f'_+(z)|^2) - \log(|f'_-(z)|^2)) dx \\
& = & \int_C (\log(|f'_+(z)|^2) - \log(|f'_+(f(z))|)^2 - \log(|f'(z)|^2)) dx \\
& = & \int_C (\log(|f'_+(f(z))|)^2(\chi'(x) -1) - 2 \log(\chi'(x))) dx.
\end{array}
$$
Adding up the expressions $A_1(\chi)$ and $A_2(\chi)$, we conclude
$$
S_+(f_+) + S_-(f_-) - {\rm Im} \, \tilde{C}(f_-^{-1}, f_+) = - \int_C (\chi'(x) + 1)\log(\chi'(x)) dx,
$$
as required.
\end{proof}

Note that the statement of Proposition \ref{prop:TT=cocycle+} can be re-written as
\begin{equation}  \label{eq:tildeC=}
{\rm Im} \, \tilde{C}(f_-^{-1}, f_+)=S(\chi)  + \int_{C}  (\log(\chi'(x)) - \log((\chi^{-1})'(x))) dx.
\end{equation}
By Theorem \ref{thm:S_symmetry}, the first term on the right hand side is invariant under the involution $\chi \mapsto \chi^{-1}$
while the second term is anti-invariant.

\subsection{Unitary symbols and quantization of welding}
In this Section, we study properties of unitary symbols $U_f$ corresponding to diffeomorphisms of the circle using conformal welding.

Let $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$ and $f$ the corrsponding holomorphic map. Recall that the symplectic transformation $A_f \in {\rm USp}^{\rm res}(H_+ \oplus H_-)$ belongs to the unitary symplectic group. This implies that $\alpha_f$ is invertible, $\gamma_f = \bar{\beta}_f$, and %
$$
\alpha_f^{-1} (\alpha_f^*)^{-1} = \alpha_f^{-1}(\alpha_f\alpha_f^* - \beta_f \beta_f^*) (\alpha_f^*)^{-1} = 1 - (\alpha_f^{-1} \beta_f)(\alpha_f^{-1} \beta_f)^*.
$$
Since the symmetric operator $\alpha_f^{-1} \beta_f = \alpha_{f_+^{-1}}^{-1} \beta_{f_+^{-1}}$ is Hilbert-Schmidt, the operator $(\alpha_{f}^{-1} \beta_f)(\alpha_f^{-1} \beta_f)^*$ is trace class and $\alpha_f^{-1} (\alpha_f^*)^{-1}$ possesses a Fredholm determinant. The following result is an adaptation of Theorem 3.8 in Chapter 2 of \cite{TT}:
\begin{thm}  \label{thm:TT2}
The operator $\alpha_f$ is bounded and invertible. The Fredholm determinant of the self-adjoint operator $\alpha_f^{-1} (\alpha_f^*)^{-1}$ is given by
\begin{equation}
{\rm det}(\alpha_f^{-1} (\alpha_f^*)^{-1}) = e^{-S(\chi)/12\pi},
\end{equation}
where $S(f)$ is the TT energy functional.
\end{thm}

By Proposition \ref{prop:berezin_unitary}, the unitary operator representing the symplectic transformation  $A_f$ is given by
\begin{equation}  \label{eq:U_f}
U_f = \frac{1}{{\rm det}^{1/4}(\alpha_f \alpha_f^*)} \, N_f = e^{S(\chi)/48 \pi} \, N_f = e^{S(\chi)/48\pi } \, N_{f_+^{-1}} * N_{f_-}.
\end{equation}
Using this construction, we obtain the following interesting relation between the cocycle $C_N$ and the TT energy functional:
\begin{thm}  \label{thm:C_N_welding}
Let $\chi \in {\rm Diff}^+_{\rm hol}(S^1)$, $f$ the corresponding holomorphic function, and $f_\pm$ the components of the conformal welding of $\chi$.
Then,
$$
\log(|C_N(f_-^{-1}, f_+)|) = \frac{S(\chi)}{24\pi}.
$$
\end{thm}

\begin{proof}
On the one hand,  equation \eqref{eq:U_f} implies:
$$
U_f^{-1} = e^{- S(\chi)/48\pi } \, N_{f_-}^{-1} * N_{f_+^{-1}}^{-1} = e^{- S(\chi)/48\pi } \, N_{f_-^{-1} } * N_{f_+} = e^{- S(\chi)/48 \pi} \, C_N(f_-^{-1}, f_+)  N_{f^{-1}}.
$$
Here we have used the facts that $N_{f_-^{-1}}$ is the operator inverse of $N_{f_-}$ and $N_{f_+^{-1}}$ is the operator inverse of $N_{f_+}$.
On the other hand, we have
$$
U_{f^{-1}} = e^{S(\chi^{-1})/48 \pi} \, N_{f^{-1}}.
$$
Since both $U_f$ and $U_{f^{-1}}$ are unitary operators, their product $U_fU_{f^{-1}} = z \cdot  {\rm Id}$ which implements $f \circ f^{-1} = e$ is a multiple of the identity with $|z|=1$.
Hence,
$$
U_{f^{-1}} = z \, U_f^{-1} = z \, e^{- S(\chi)/48 \pi} \, C_N(f_-^{-1}, f_+)  N_{f^{-1}}.
$$
By comparing the two expressions for $U_{f^{-1}}$, we obtain the following equality:
$$
\log(|C_N(f_-^{-1}, f_+)|) = \frac{S(\chi) + S(\chi^{-1})}{48\pi} = \frac{S(\chi)}{24\pi},
$$
as required.
\end{proof}

\begin{rem}
The statement of Theorem \ref{thm:C_N_welding} should be compared to Proposition \ref{prop:TT=cocycle+} and equation
\eqref{eq:tildeC=}.
In particular, it would be interesting to find a more explicit relation between the cocycles
$\log(|C_N|)$ and $\tilde{C}$.
\end{rem}

The second main result of this article is the following theorem:
\begin{thm}  \label{thm:main2}
Let $\chi, \phi \in{\rm Diff}^+_{\rm hol}(S^1)$, and  $f$ and $g$ be the corresponding holomorphic maps. Then,
$$
U_f * U_g = C_U(f,g) \, U_{f \circ g},
$$
where
\begin{equation}  \label{thm:C_U}
    C_U(f,g) = e^{(S(\chi) + S(\phi) - S(\chi \circ \phi))/48 \pi} \, C_N(f,g).
\end{equation}
Furthermore,
\begin{equation}  \label{eq:C_N=TT}
\log(|C_N(f,g)|) = \frac{S(\chi \circ \phi)-S(\chi)-S(\phi)}{48 \pi}.
\end{equation}
\end{thm}

\begin{proof}
Formula for $C_U(f,g)$ is a direct consequence of Theorem \ref{thm:C_N} and Theorem \ref{thm:TT2}. Since $U_f, U_g$ and $U_{f \circ g}$ correspond to unitary operators, $C_U(f,g)$ takes values in $S^1 \cong \{ z \in \mathbb{C}; |z|=1\}$ and the real part of its  logarithm vanishes.
\end{proof}

\begin{rem}
Explicit expressions for the left and the right hand sides of equation \eqref{eq:C_N=TT} are of very different nature. The determinant on the left hand side uses Grunsky coefficients of
the welding components $f_-$ and $g_-$. On the right hand side, the TT energy functional  is
an integral of local expressions in terms of all welding components $f_\pm, g_\pm, (f \circ g)_\pm$. It would be interesting to find a direct proof of the surprising equality \eqref{eq:C_N=TT} between these expressions.
\end{rem}

\begin{rem}
The imaginary part ${\rm Im} \,\log(C_U(f,g))$  of the cocycle $C_U(f,g)$ is an additive real valued group 2-cocycle on ${\rm Diff}^+(S^1)$. Assuming that
$$
H^2({\rm Diff}^+_{\rm hol}(S^1), \mathbb{R}) =H^2({\rm Diff}^+(S^1), \mathbb{R}) \cong \mathbb{R},
$$
this cocycle must be cohomologous to the Bott-Virasoro cocycle. It would be interesting to find an explicit expression for the coboundary which represents their difference.
\end{rem}

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Spiralling into financial debt? Collection agencies harassing you at home and at work? We can help! Stop worrying and live debt free. Control your debt before it controls you. Talk to a Qualified Insolvency Counsellor today.
It is time to take action! It is never too late to control your debt again. There are ways to manage your debt without declaring bankruptcy. Financial & Debt Solutions is your partner in helping you get your life back.
You may not know that you are about to go into a downward spiral in managing your debt. Take our free, easy 9-point self-assessment tool and find out if you are in danger of losing control of your debt.
You may be noticing some warning signs that you are struggling with your finances. Talk to one of our qualified insolvency counsellors today and find out how you can again be in control of your finances.
Hope is only a call away – talk to one of our qualified insolvency counsellors & secure your life.
I felt horrible and depressed with all the holiday debt I had, but talking to FDS helped me not only control my debt but also have a better handle of my finances.
I was being harassed every night - coming home from work, only to get calls from creditors. FDS helped me stop these calls, and pay off my debt.
Thank God I didn't have to declare bankruptcy. FDS counsellors saved my credit ratings.
I gave a sigh of relief after talking to Financial & Debt Solutions. I don't know why I didn't talk to them sooner but I am glad I did!
Financial and Debt Solutions (FDS) has a knowledgable team of insolvency counsellors who can help you get your debt under control, so you can go back to enjoying your life without financial worries weighing you down.
Address: 700 Upper James St., Hamilton, ON.


The chart represents all Groups in the EP by color. Name and Members pop up when your mouse cursor is over any of them.
By clicking on a Group, all its Members will be displayed in a grid in alphabetical order.
You can get an overview of the Group based on Twitter metrics by selecting one of the available metrics in the unrollable list above the grid.
This column will contain the Tweet Search Box which allows you to scrutinize all tweets of the Group issued during the present mandate. After a filtering is applied, MEPs matching your filter will display in color and the rest will be grayed.
At this point you could read all tweets matching your filtering, just click on the MEP who issued them.
Select a Group and have fun.
All figures correct as at April 20th, 2019.


We're so excited to have Fawn Gehweiler, our amazing illustrator, visiting us from Portland, Oregon. She has been super busy this week giving the store a makeover inside and out. Check out how cute she is, plus samples of her work and stay tuned for big changes in the Fall.


FitsUWeddings agrees to perform the services specified in the "Client Information/Description of Services" attached to this Agreement and incorporated in this Agreement by reference.
FitsUWeddings may, at FitsUWeddings' own expense, employ any assistants that FitsUWeddings deems necessary to perform the services required of FitsUWeddings by this Agreement. Client may not control, direct, or supervise FitsUWeddings' assistants or employees in the performance of those services.
For services to be rendered under this Agreement, Client agrees to pay FitsUWeddings the sum set forth in "Client Information/Description of Services". Client agrees to pay to FitsUWeddings a good faith non-refundable deposit equaling 20% of the amount entered in the "Client Information/Description of Services" upon execution of this Agreement. Client agrees to pay the remaining balance in full no later than two (2) week prior to the wedding date.
Client agrees that reservation for requested services can only be made upon delivery of this agreement and payment of the nonrefundable deposit. If making payments online, Client understands that reservation for services requested will be made only upon receipt of the payment and this Agreement. If any payment due is not made in accordance with this agreement, Client agrees that Client may forfeit the services of FitsUWeddings and will forfeit the deposit, if any.
In addition to the retainer, Client agrees to pay any and all reasonable and necessary expenses incurred by FitsUWeddings on behalf of Client in connection with the rendering of services under this Agreement on a monthly basis. Client understands that no services will be rendered until the remaining balance, including any additional charges, have been paid in full. Any returned check is subject to a $35.00 fee in addition to any outstanding balance.
[registered mail or private courier or properly designated agent of FitsUWeddings].
However, in the event of equipment failure or uncontrollable/unforeseen events, the delivery period may be longer. FitsUWeddings will not be liable in any way if such an event occurs. On completion, FitsUWeddings will deliver to Client a master [on videotape or DVD] and will make extra sub-master copies available for Client at agreed price.
5(a) Weather Disruption: If it is not possible for a session to take place on a scheduled day due to weather, FitsUWeddings and Client will reschedule on an agreed date at the earliest date possible with no additional charge.
5(b) Cooperation/No Show: FitsUWeddings is not responsible if Client and/or key individuals fail to appear or cooperate during sessions or for missed images due to details not revealed to Videographer. If Client and/or key individuals fail to appear to a scheduled session, the 50% deposit will be forfeited. The session can be rescheduled at the original session full fee with a new 50% nonrefundable deposit. In the event that FitsUWeddings' photographer and/or videographer is injured or sick or otherwise unable to attend the sessions, FitsUWeddings will make every effort to reschedule to the earliest possible date. If the session must take place on the scheduled date, FitsUWeddings will make every effort to secure a replacement photographer and/or videographer.
5(c) Reshoots: In the unlikely event that Client is completely dissatisfied with the final results based solely on Client's personal preference; FitsUWeddings will reshoot at a 25% discount from the original session fee. In the unlikely event that the finished product from a session is unusable due to technical problems such as camera failure or exposure problems, FitsUWeddings will reshoot at no additional charge.
5(d) Cancellation/Postponement: Client understands that by entering into this Agreement, FitsUWeddings is foregoing other photographic and/or video jobs. In the event of a cancellation or postponement of the session by Client, the deposits paid will be forfeited. Deposits may be transferable if a date change is necessary and Client gives FitsUWeddings forty-eight (48) hour(s) notice of such cancellation or postponement.
6(a) Ownership: After completion and delivery of the package(s) as specified above, Client hereby agrees that the original and finished production becomes the property of FitsUWeddings. FitsUWeddings will retain ownership of production materials including, but not limited to, set designs, unused footage, costumes, and props and may copyright same. FitsUWeddings will have the right to use produced materials for demonstrations, publication, marketing, advertising, broadcast purposes or otherwise.
6(b) Model Release: Client hereby grants FitsUWeddings and its legal representatives and assigns the irrevocable and unrestricted right to use and publish photographs and/or video of Client(s) in which Client may be included for editorial, trade, advertising and other purposes, and in any other manner or medium; to alter the same without restriction; and to copyright the same. Client hereby releases FitsUWeddings and its legal representatives and assigns from all claims and liability relating to said images.
6(c) Exhibition: Client hereby grants FitsUWeddings permission to display photography and/or video resulting from this agreement as an example of photography and/or video work and for entrance into photographic and/or videographic competitions or to enter the produced materials in any other competitions for awards anywhere in the world. Client hereby releases all claims to profits that may arise from use of the images.
There may be additional charges if changes are made to this Agreement by Client. Client may increase the period of the photographer's and/or videographer's service on the session's date, provided photographer and/or videographer is available during the time requested. Added expenses resulting from such change, including additional service cost, digital capture cost, and change-related fees will be added onto the final balance and will be paid in full by Client immediately. Client may not decrease the period of FitsUWeddings service on the session date. If the period of photographer's and/or videographer's services are cut short by reason of fire, casualty, death, act of God, nature, or other causes beyond the control of the parties, FitsUWeddings shall still receive the deposit fees outlined in the "Client Information/Description of Services" attached to this Agreement.
Production Control FitsUWeddings is granted full production and editorial control by the client regarding all aspects of production and post-production. FitsUWeddings has the right to assign any videographer for the assignment set within.
FitsUWeddings takes utmost care with respect to the exposure, retouching, and transportation of digital captures, proofs and prints. However, in the unlikely event that all the originals are lost, stolen, damaged or destroyed while in FitsUWeddings' control or if a session cannot go forward as scheduled due to inability of FitsUWeddings photographer and/or videographer to attend session, and no suitable replacement can be found, it is agreed that in such event of a breach of this Agreement by FitsUWeddings, it would be impracticable or extremely difficult to determine the actual damage to Client. Therefore, FitsUWeddings is limited and will pay to Client, as liquidated damages and not as a penalty, the sum equal to the amount of all payments received, or a reshoot can be rescheduled at no additional cost. The limit of liability for partial loss of damage of originals shall be a prorated amount of the total amount based on the percentage of originals lost or damaged. In no event will FitsUWeddings liability or responsibility exceed the return of all payments received. The parties agree that this limited liability/liquidated damages represents a reasonable compensation for the loss incurred because of the breach.
FitsUWeddings will not be liable to Client, or to anyone who may claim any right due to a relationship with Client, for any acts or omissions in the performance of services under this Agreement or on the part of the employees or agents of FitsUWeddings. Client will indemnify and hold FitsUWeddings free and harmless from any obligations, costs, claims, judgments, and attorney's'' fees, and attachments arising from, growing out of, or in any way connected with the services rendered to Client under this Agreement.
9(a) Termination by Client for Default of FitsUWeddings: If FitsUWeddings defaults in the performance of this Agreement or materially breaches any of its provisions, Client, at Client's option, may terminate this Agreement by giving written notification to FitsUWeddings.
9(b) Termination by FitsUWeddings for Default of Client: If Client defaults in the performance of this Agreement or materially breaches any of its provisions, FitsUWeddings, at FitsUWeddings' option, may terminate this Agreement by giving written notification to Client. FitsUWeddings shall be entitled to keep all payments received in addition to any other damages it may be entitled to.
Client Obligations It is the client's responsibility to notify the ceremony location, reception location, rehearsal location, photographer, disc jockey, and those others directly associated, the presence of FitsUWeddings. Client further agrees they will provide dinner and refreshments of the same quality and in the same location as any attending guest for all FitsUWeddings videographers and/or photographers. Client agrees to cooperate fully with FitsUWeddings and understands that without full cooperation by client, satisfaction cannot be guaranteed.
FitsUWeddings makes no warranties, express or implied, with respect to the work of property that is the subject matter of this Agreement.
Any disputes that may arise between Client and FitsUWeddings regarding the terms of this Agreement will be submitted to final and binding arbitration in accordance with the Arbitration process agreeable to the parties. The arbitrator's award will be final and binding on the parties, and judgment may be entered on it by a court of competent jurisdiction.
Compatibility FitsUWeddings produces standard media that is compatible with virtually all players in North America. FitsUWeddings is not responsible for incompatibility with any media format created and the client's player. If it is determined that the media is defective, FitsUWeddings will replace the media at no additional charge if notified within (7) seven days after pickup or delivery of the finished production, less shipping and handling.
If any action at law or in equity, including an action for declaratory relief, is brought to enforce or interpret the provisions of this Agreement the prevailing party will be entitled to reasonable attorneys' fees in addition to any other relief to which that party may be entitled. The attorneys' fees may be set by the court in the same action or in a separate action brought for that purpose.
This Agreement supersedes any and all agreements, both oral and written, between the parties with respect to the rendering of services by FitsUWeddings for Client and contains all of the covenants and agreements between the parties with respect to the rendering of these services in any manner whatsoever. Each party acknowledges that no representations, inducements, promises, or agreements, written or oral, have been made by either party, or by anyone acting on behalf of either party, that are not embodied in this Agreement. Any modification of this Agreement will be effective only if it is in a writing signed by the party to be charged.
Any notices to be given under this Agreement by either party to the other may be effected either by personal delivery in writing or by registered or certified mail, with postage prepaid and return receipt requested. Mailed notices shall be addressed to the address indicated by the parties or at their last known addresses. Notices delivered personally will be deemed communicated at the time of delivery. Mailed notices will be deemed communicated one day after mailing.
If any provision of this Agreement is deemed unenforceable, that provision will be omitted only to the extent necessary to make this Agreement valid and enforceable and the remaining provisions will remain in full force and effect.


U+2625 was added to Unicode in version 1.1 (1993). It belongs to the block Miscellaneous Symbols in the Basic Multilingual Plane.
The glyph is not a composition. It has a Neutral East Asian Width. In bidirectional context it acts as Other Neutral and is not mirrored. The glyph can, under circumstances, be confused with 4 other glyphs. In text U+2625 behaves as Alphabetic regarding line breaks. It has type Other for sentence and Other for word breaks. The Grapheme Cluster Break is Any.
The ankh (/ˈæŋk/ or /ˈɑːŋk/; Egyptian: IPA: [ʕaːnax]; U+2625 ☥ or U+132F9 ㋹), also known as breath of life, the key of the Nile or crux ansata (Latin meaning "cross with a handle"), was the ancient Egyptian hieroglyphic character that read "life", a triliteral sign for the consonants ꜥ-n-ḫ.
It represents the concept of eternal life, which is the general meaning of the symbol. The Egyptian gods are often portrayed carrying it by its loop, or bearing one in each hand, arms crossed over their chest. The ankh appears in hand or in proximity of almost every deity in the Egyptian pantheon (including Pharaohs). Thus it is fairly and widely understood as a symbol of early religious pluralism: all sects believed in a common story of eternal life, and this is the literal meaning of the symbol. This rationale contributed to the adoption of the ankh by New Age mysticism in the 1960s, to mean essentially the same tolerance of diversity of belief and common ethics as in Ancient Egypt.


Housewarming parties are a fantastic and sweet sentiment you can do for a family member, neighbor, or friend. They really don't take much planning, just ask the home-owner if they'd be ok with a little shower for and in their new house, and be generous about the time frame because they may get self-conscious about how the house looks right after moving. Have a back plan for where to host the party in case you sense their discomfort.
If you're throwing your own housewarming party, that's cool too. Be ready with booze. Don't expect gifts, but it is ok to ask if folks would like to bring a dish or dessert to share. This is all easiest done via housewarming invites that you can customize.
Get your Housewarming Invitations from PurpleTrail, and we'll promise to give those cards as much love as you will your guests of honor.


HINGHAM — Police are searching for a burglar who they say stole cash, jewelry and electronic devices from a Gardner Street home.
Police said that a resident of the home arrived about 2:30 p.m. Tuesday and saw the back door had been kicked in. The resident, a student, didn't go inside and went to a neighbor's house to call the police and family members.
When officers and a police dog arrived on scene, the burglar wasn't in the home. Police said it appeared that the break-in happened between 10:45 a.m. and 2:30 p.m.


text:
У́водь — река в Ивановской и Владимирской областях; левый приток Клязьмы.

Длина реки составляет 185 км, площадь водосборного бассейна — 3770 км², средний расход в 30 километрах от устья — 19 м³/сек. Крупные притоки — Урожка, Талка (левые); Востра, Ухтохма, Вязьма, Тальша, Наромша (правые).

На реке расположены города Иваново и Кохма. На берегу реки в 0,75 км к северо-западу от эпонимного села и в 2,5 км к юго-востоку от села Писцово обнаружено селище Кожевники XVIII века.

Описание

Уводь вытекает из Андреевских болот к северо-востоку от города Комсомольска.

В верхнем течении — небольшая речка со слабоизвилистым руслом и редкими островами и перекатами. Ширина не превышает 10 м, берега лесистые, всхолмлённые.

К северу от Иванова на Уводи в 1937 году создано Уводьское водохранилище. Его ширина достигает 1,5 км, длина — 37 км, глубина — 15 м.

Целью водохранилища является снабжение водой Иванова. Так как маловодная Уводь не способна обеспечить нужное количество воды, водохранилище пополняется волжской водой по каналу Волга — Уводь. Сток из водохранилища в город регулируется плотиной у деревни Уводь. В черте самого Иванова на реке ещё две плотины.

Сразу за Ивановом Уводь протекает через город Кохму, от него до устья Вязьмы река течёт по сильно пересечённой лесистой равнине. Берега песчаные, скорость течения небольшая.

Приняв справа два крупных притока — Ухтохму и Вязьму, Уводь расширяется и начинает образовывать многочисленные старицы, протоки и острова, среди которых зачастую теряется крайне извилистое основное русло. В некоторых местах Уводь образует озёровидные расширения. Такой характер река сохраняет вплоть до устья.

В 2 км к северу от Коврова Уводь и Клязьма соединяются протокой. Основное русло Уводи впадает в Клязьму десятью километрами ниже.

В черте города Иваново через реку Уводь перекинуты мосты:
 Фабричный (Фабричный проезд),
 Нардомовский (улица Громобоя),
 Соковский (Шереметевский проспект),
 Театральный (проспект Ленина),
 Энергетиков (улица Суворова),
 Инженерный (улица Смирнова),
 Домостроителей (улица Павла Большевикова), а также четыре железнодорожных и три малых пешеходных моста.

Притоки
(км от устья)

 23 км: Луйка (левый)
 Сингорька (левый)
 29 км: Наромша (правый)
 42 км: река Тальша (правый)
 Колдомка (левый)
 Чешиха (левый)
 Мелена (левый)
 Варжевка (левый)
 Объедовка (правый)
 62 км: Вязьма (правый)
 Табачная (левый)
 Никулиха (левый)
 Никулина (левый)
 71 км: Ухтохма (правый)
 Жуковка (левый)
 Шепелевка (левый)
 Соседка (?) (правый)
 Поповка (правый)
 90 км: Востра (правый)
 Чёрная (левый)
 Страданка (правый)
 104 км: Сиверка (левый)
 Харинка (левый)
 121 км: Талка (левый)
 Кокуй (правый)
 Варгин ручей (правый).
 130 км: Чернавка (правый)
 Красавица (левый)
 Лагиш (правый)
 175 км: Урожка (левый)
 Чёрная, у с. Писцово (левый)

Примечания

Литература

Притоки Клязьмы
Реки Ковровского района
Реки Камешковского района
Реки Савинского района
Реки Лежневского района
Реки Ивановского района Ивановской области
Реки Комсомольского района Ивановской области

redpajama_set_name: RedPajamaWikipedia


text:
Fairhaven Homecoming Day Fair
Fairhaven Center
Fairhaven, MA 02719
Fairhaven's largest annual event features about 175 booths of handmade crafts and delicious foods in addition to live entertainment.,
Days/Hours Open: Sat 10am‑4pm
Address: 40 Center Street, Fairhaven, MA 02719
Entertainment: 1 stage - Local. Music: Variety ??
Other Activities: children's activities, art exhibit
# Food Booths: tba
# of Exhibitors: 170
Prize Money: na
Music: until full
Food: 04/05/2021
Promoter: Fairhaven Improvement Association
Show Dir.: Event Office
Email / contact Web Event Phone Food Fee Exhib. Fee
Sowa Open Market - May
Harvard Apple Blossom Festival
Andover Crafts in the Park
Massachusetts COVID Info
@MassDPH
Massachusetts Department of Public Health - Information on the Outbreakof Coronavirus Disease 2019 (COVID-19)
1/7/21: Order #60 continues 12/22 restrictions. 12/22: Beginning at 12:01 a.m. on December 26, 2020, Massachusetts will implement temporary capacity limits to stop the spread of COVID-19 as cases and hospitalizations rise. Businesses must adhere to the following capacity limitations. The revised limits below supersede existing limits in the Phase 3, Step 1 Sector-Specific Protocols. Except for the specified adjustments to capacity limitations, all other terms of COVID-19 gatherings order (including exceptions) and all sector-specific workplace safety rules remain unchanged. These limits are in effect until January 10, 2021 unless further extended by the Governor. 12/8: Required return to Phase III, Step 1 reopening for all Massachusetts communities. Made adjustments to Phase III, Step 1 protocols. Effective October 5, a limited number of sectors will be eligible to reopen, with restrictions, in Step II of Phase III for lower risk communities only: Indoor performance venues will be permitted to open with 50% capacity with a maximum of 250 people. Outdoor performance venue capacity will increase to 50% with a max of 250 people. For arcades and indoor and outdoor recreation businesses, additional Step II activities like trampolines, obstacle courses, roller rinks and laser tag will also be permitted to open and capacity will increase to 50%. Fitting rooms will be permitted to open in all types of retail stores. Gyms, museums, libraries and driving and flight schools will also be permitted to increase their capacity to 50%. Gov. Baker pasued Phase 3 on 8/7/20 after case numbers rise. The following businesses will be eligible to reopen in Step One of Phase III, subject to industry-specific rules concerning capacity and operations: Movie theaters and outdoor performance venues; Museums, cultural and historical sites; Fitness centers and health clubs;Certain indoor recreational activities with low potential for contact;Professional sports teams, under the authority of league-wide rules, may hold games without spectators. Rescinds CO 44. On July 2, Governor Baker issued an order initiating Phase III of the Commonwealth's reopening process beginning July 6.
11/2: Governor Baker also signed an updated order related to face-coverings. The revised order requires all persons to wear face-coverings in all public places, even where they are able to maintain 6 feet of distance from others. The revised order still allows for an exception for residents who cannot wear a face-covering due to a medical or disabling condition, but it allows employers to require employees to provide proof of such a condition. It also allows schools to require that students participating in in-person learning provide proof of such a medical or disabling condition. 5/6: Requirement. Governor Baker issued an Order effective Wednesday, May 6 requiring face masks or cloth face coverings in public places where social distancing is not possible. This applies to both indoor and outdoor spaces.
Currently, under COVID-19 Order #59 (effective Dec. 22, 2020), indoor gatherings of are limited to 10 people and outdoor gatherings are limited to 25 people.
This event is registered to Fairhaven Improvement Association and the User ID for this account is 5089940797. The email address that has control of this account is an∗∗∗ @ gmail.com. You must login with User ID 5089940797 to update this event.
If you do not have the password for this account and don't have access to an∗∗∗ @ gmail.com (to have the password reset for you), please contact support@festivalnet.com to gain access to this event.

redpajama_set_name: RedPajamaCommonCrawl


text:
La Orquesta Sinfónica de San Luis (sigla: SLSO) es una orquesta sinfónica estadounidense con sede en San Luis (Misuri). Fundada en 1880 por Joseph Otten como la Sociedad Coral de San Luis, es una de las orquestas más antiguas del país, solo precedida por la Filarmónica de Nueva York.

Directores

Enlaces externos
 Página oficial de la SLSO

San Luis
Cultura de San Luis
Organizaciones fundadas en 1880
Música en 1880
Estados Unidos en 1880


Google and Amazon are both running sales on smart speakers today. While these aren't necessarily the best prices ever for Amazon Echo or Google Home devices, you could certainly do worse.
For example, you can save as much as $25 when buying a Google Home + Google Home Mini bundle. Or you can save $15 on a new Amazon Echo or $10 on an Echo Dot.


LE's proprietary additives are used exclusively in LE lubricants. Almasol High Temperature Lubricant contains Almasol.
Almasol® solid wear reducing additive is able to withstand extremely heavy loads, chemical attack and temperatures up to 1,900°F(1,038°C). It is attracted to metal surfaces, forming a microscopic layer but not building on itself or affecting clearances. Almasol minimises metal-to-metal contact and the resulting friction, heat and wear.


The 21st Annual Komen Lowcountry Race for the Cure is on Saturday, October 18, 2014. Survivor Cafe opens at 7 a.m. and Survivors need to be at the stage area by 8:00 a.m. for the Survivor Ceremony and the race begins at 9:00 a.m.. This event will occur rain or shine. We reserve the right to cancel in extreme circumstances. In that event, there will be no refunds; rather, your entry fee will be used as a donation to the Komen Lowcountry Race for the Cure.
Yes! You need to register by 8:30 a.m.
How about the Kids Dash?
The Kids Dash will start at the corner of River Landing Dr. and Seven Farms and head toward the Family Circle Stadium. Parents are encouraged to run with their children or have someone able to meet them at the finish line by the stadium. We encourage families to enjoy the expo and Kids Area – or head back to the start line for the 1-mile fun run at 8:45 a.m.
Individual packet pickup is Fri. Oct. 17th from 9 a.m. – 6 p.m. and again on Sat. morning beginning at 7 p.m.
Participants can register online through Wednesday, October 15th at 12:00 noon. Or on-site on October 17th from 9-6:00 p.m. or Saturday morning.
If you do not take the Shuttles there will be Police and Volunteers directing traffic and showing you where to park. Please be patient and follow their directions. Check out the Parking & Shuttles information page.
The shuttle bus will continuously run from the parking lot at Mt. Pleasant Towne Centre and N. Charleston Coliseum (Lot E) beginning at 6:30 a.m. The last Shuttle to depart Daniel Island will be at 11 a.m. Participants are encouraged to use the free shuttle and to arrive early. Shuttle riders will be given a colored wrist band which will help them find their shuttle stops.
Please try to register and donate online first! Please do not send cash; instead send checks or money orders. Our mailing address is as follows: Komen Lowcountry, 9300 Medical Plaza Dr., Ste. F, N. Charleston, SC 29406.
For the safety of all participants, inline skates, bikes, scooters, skateboards and pets are not allowed at the Race. It's safer for your pets as well.
During your registration please select "Timed" within your participant type.
Please make all checks payable to: Komen Lowcountry. On the check memo line, please include your name or your team name.
Kids for the Cure, ages 6-12, registration is $15 (late registration $20). Children 5 and under are free although a parent must sign a waiver.
Yes! We will be selling Komen merchandise.


This book/lecture is meant for a faculty freshman point type in challenge fixing, the place the actual difficulties care for electric and digital circuits. it could even be utilized in a junior/senior point type in highschool to coach circuit research. the fundamental problem-solving paradigm utilized in this publication is that of answer of an issue into its part components. The reader learns the right way to take circuits of various degrees of complexity utilizing this paradigm. The problem-solving workouts additionally familiarize the reader with a couple of varied circuit parts together with resistors, capacitors, diodes, transistors, and operational amplifiers and their use in useful circuits. The reader should still come away with either an realizing of the way to strategy advanced difficulties and a "feel" for electric and digital circuits.
The publication addresses a variety of gains of programming languages from a mathematical standpoint. It discusses semantics of items from basic crucial language to failure to concurrency matters utilizing channel established and shared reminiscence innovations. it's a reliable ebook to get an concept approximately so much of this techniques in case you dispose of the maths concerned.
A useful book to all concerned about instructor schooling, providing the benefits and disadvantages of distance schooling or open studying.
Mashups have emerged as an leading edge software program pattern that re-interprets latest internet development blocks and leverages the composition of person elements in novel, value-adding methods. extra charm additionally derives from their capability to show non-programmers into builders. Daniel and Matera have written the 1st entire reference paintings for mashups.
4. The amount of time taken by the sinusoid to complete one cycle is called the period of the sinusoid, and is usually denoted by T. We can see that the period is simply the reciprocal of the frequency: T = 1/ f 1 Heinrich Rudolf Hertz was born on February 22, 1857, in Hamburg, Germany, and studied engineering and physics at Munich and Berlin. While he was a professor at the technical school in Karlsruhe, he conducted a set of experiments showing the propagation of electromagnetic waves (he called them electric rays).
We would like to have a framework in which to represent all these signals. J. Fourier came up with such a framework. He showed that all periodic signals could be written in terms of sines and cosines. Many signals that we are interested in, such as audio and video signals, are not periodic. However, as these signals are nonzero only over a finite period of time, we can convert them into a periodic signal by repeating the signal. For example, consider the signal f (t) shown in Fig. 2. By repeating it as shown in Fig.
3 we can generate a periodic signal that is identical to the original signal in the interval [0, To ]. Thus the method of Fourier allows us to represent a variety of signals in a unified manner. 2) where f is the frequency of the sinusoid and θ is the phase. The frequency of a sinusoid is measured in cycles per second or Hertz1 (Hz). A frequency of n Hz means that it takes 1 s for the sinusoid to complete n cycles. One cycle of the sinusoid V (t) = 10 cos(2π × 10t) is shown in Fig. 4. The amount of time taken by the sinusoid to complete one cycle is called the period of the sinusoid, and is usually denoted by T.