\section{The proof of Theorem \ref{thm:main}}
\label{sec:theproof}

In this section we will prove the main Theorem \ref{thm:main}. There are two things left to be shown:
\begin{enumerate}
\item The morphism $\morphU^{cyc}$ is a quasi-isomorphism. This will follow from Shoikhet's work and the fact that on divergence free vector fields our morphism agrees with Shoikhet's.

\item It is an $L_\infty$-morphism of dglas. The proof is a typical ``Kontsevich-Stokes-proof''. The only unusual thing is that we have a non-vanishing ``bulk'' term due to the non-closedness of $\eta_z$, which will provide the differential $u \dv$ on the polyvector field side. This trick has probably been invented by Cattaneo and Felder \cite{cafe}.
\end{enumerate}

\subsection{It is a Quasi-Isomorphism}
This is a direct consequence of the following Proposition and Theorem 2.3 of \cite{shoikhet}:
\begin{prop}
On the sub-Lie algebra $(T_{poly}(M))_{div}[u]$ of divergence free poly vector fields, the morphism $\morphU^{cyc}$ agrees with Shoikhet's morphism from \cite{shoikhet}, up to signs due to different conventions.
\end{prop}
\begin{proof}
Shoikhet's morphism contains a sum of graphs with dashed pairs. Using the notation of the proof of Lemma \ref{lem:angleforms}, a dashed pair at $i$-th position introduces a factor $du_i\wedge du_{i+1}$ into the weight form. Summing over all $i$ we get just the form $\varpi_z$. 
\end{proof}


\input qweights.tex

\subsection{It is an $L_\infty$-morphism}
We want to show that \eref{equ:ourLinfty} holds for $\mathcal{F}=\morphU^{cyc}$.
Each of the three terms occuring has a representation in terms of graphs. The first term on the left can be identified with the sum
\begin{multline*}
\sum_\Gamma \sum_{i=1}^m (-1)^{\sum_{r=1}^{i-1}k_r} w_{\Gamma}(j_1,..,j_i+1,..,j_m) D_{\Gamma} (\gamma_1 \otimes \cdots \otimes\dv \gamma_i \otimes \cdots \otimes \gamma_m)
=\\=
\sum_\Gamma \sum_{i=1}^m (-1)^{\sum_{r=1}^{i-1}k_r} w_{\Gamma}(j_1,..,j_i+1,..,j_m) \sum_{s(i,i)} (-1)^{s(i,i)} D_{\Gamma+(i,i)} (\gamma_1 \otimes \cdots \otimes \gamma_m)
=\\=
\sum_\Gamma \sum_{i\in Tp(\Gamma)} (-1)^{\sum_{r=1}^{i-1}k_r+s(i,i)} w_{\Gamma-(i,i)}(j_1,..,j_i+1,..,j_m) D_\Gamma(\gamma_1 \otimes \cdots \otimes \gamma_m).
\end{multline*}
Here we substituted $x_i=u^{j_i}\gamma_i$ into \eref{equ:ourLinfty}. In the second line an edge $(i,i)$ is added to $\Gamma$ and put at the $s(i,i)+1$st position in the ordering on $Star(i)$. The sum on the right is over all such possible positions, $s(i,i)=0,..,|Star(i)|-1$. For the last line, we changed the summation variable $\Gamma$.

The second term on the left of \eref{equ:ourLinfty} can be identified with
\begin{multline*}
\sum_\Gamma 
\sum_{i\neq j}^n \epsilon (i,j,..,\hat{i},..,\hat{j},.., m)
w_\Gamma(j_i+j_j,j_1,..,\hat{j}_i,..,\hat{j}_j,..,j_m) \\
D_\Gamma((\gamma_i\bullet \gamma_j)\otimes \gamma_1\otimes\cdots\otimes \hat{\gamma}_i \otimes \cdots \otimes\hat{\gamma}_j \otimes \cdots \gamma_n)
=\\=
\sum_\Gamma

\sum_{i\neq j}^n \epsilon (i,j,..,\hat{i},..,\hat{j},.., m) (-1)^{k_i}
w_{\Gamma}(j_i+j_j,j_1,..,..,\hat{j}_j,..,j_m)
\sum_{\Gamma'} (-1)^{s(i,j)} D_{\Gamma'}(\gamma_1\otimes \cdots \otimes \gamma_m)
=\\=
\sum_\Gamma \sum_{(i,j)\in E(\Gamma) } 
(-1)^{s(i,j)} \epsilon (i,j,..,\hat{i},..,\hat{j},.., m)
w_{\tilde{\Gamma}}(j_i+j_j,j_1,..,\hat{j}_i,..,\hat{j}_j,\dots,j_m)
D_\Gamma(\gamma_1\otimes \cdots \otimes \gamma_m).
=\\=
\sum_\Gamma \sum_{\substack{(i,j)\in E(\Gamma) \\ (j,i)\notin E(\Gamma)}} 
(-1)^{s(i,j)} \epsilon (i,j,..,\hat{i},..,\hat{j},.., m)
w_{\tilde{\Gamma}}(j_i+j_j,j_1,..,\hat{j}_i,..,\hat{j}_j,\dots,j_m)
D_\Gamma(\gamma_1\otimes \cdots \otimes \gamma_m).
\end{multline*}
The sum over $\Gamma'$ is over graphs obtained from $\Gamma$ by the following procedure:
\begin{enumerate}
\item Insert an additional type I vertex, and renumber the vertices such that (i) the new vertex is vertex $j$ and (ii) the vertex of $\Gamma$ in which $\gamma_k$ was inserted is the new vertex $1$ in $\Gamma'$. Since the numbering of vertices is irrelevant for the definition of $D_{\Gamma'}$, there does not occur an additional sign.
\item Reconnect zero or more edges ending at vertex $i$ to vertex $j$, i.e., apply Leibniz' rule.  There does not occur a sign either. 
\item Reconnect the $k_j$ last (in the ordering on $Star(i)$) edges starting at $i$ such that they start at $j$, maintaining their ordering. 
\item Add an extra edge $(i,j)$. Make it the $s(i,j)+1$st in the ordering on $Star(i)$, where $s(i,j)=0,1,..,|Star(i)|-1$.
\end{enumerate}
To see the first equality, note that by Remark \ref{rem:wedges} (ii) we are allowed to replace the antisymmetric tensor $\gamma_i\bullet \gamma_j$ by its non-antisymmetrized version $\iota_{\p_r} \gamma_i \otimes \p_r\cdot\gamma_j$. The resulting term
\[
D_\Gamma((\iota_{\p_r} \gamma_i \otimes \p_r\cdot\gamma_j)\otimes \gamma_1\otimes\cdots\otimes \hat{\gamma}_i \otimes \cdots \otimes\hat{\gamma}_j \otimes \cdots \gamma_n)
\]
is equal to the sum
\[
\sum_{\Gamma'} (-1)^{s(i,j)} D_{\Gamma'}(\gamma_1\otimes \cdots \otimes \gamma_m).
\]

For the second equality, we changed the summation variables. Here $\tilde{\Gamma}$ is the graph obtained from $\Gamma$ by renumbering the vertices such that vertices $i$ and $j$ become vertices 1 and 2, maintaining the order of the other vertices. Note that we dropped here graphs containing a double edge $(i,j)$ since they do not contribute due to antisymmetry of the $\gamma$'s. Note that in this sum, there may be graphs containing an edge $(i,j)$ as well as an edge $(j,i)$. More precisely, these graphs come from applying the Leibniz rule to a tadpole edge. For the third equality, we used that all those graphs cancel. Concretely, if there is such a pair of edges, the $(i,j)$-term in the sum over edges cancels with the $(i',j')=(j,i)$-term. The relative sign of the two terms is a $(-1)^{k_ik_j}$ from the $\epsilon(\dots)$, times a $(-1)^{(k_j-1)k_i+k_i-1}$ from permuting the weight forms of the edges of vertices $i$ and $j$ appropriately. Hence the total relative sign is $-1$. This computation is reminiscent of the calculation showing that $\dv$ intertwines with the Schouten bracket.

The term on the right of \eref{equ:ourLinfty} is 
\begin{multline*}
\sum_{I\sqcup J=[n]}\epsilon(I,J) (-1)^{|k_I|} \sum_{\Gamma_1,\Gamma_2} w_{\Gamma_1}(j_I) w_{\Gamma_2}(j_J) D_{\Gamma_1}(\gamma_I) \circ D_{\Gamma_2}(\gamma_J)
=\\=
\sum_{I\sqcup J=[n]}\epsilon(I,J) (-1)^{|k_I|} \sum_{\Gamma_1,\Gamma_2} w_{\Gamma_1}(j_I) w_{\Gamma_2}(j_J) \sum_{l=1}^{n_1} (-1)^{(l+1)(n_2+1)} D_{\Gamma_1}(\gamma_I) \circ_l D_{\Gamma_2}(\gamma_J)
=\\=
\sum_{I\sqcup J=[n]}\epsilon(I,J) (-1)^{|k_I|}
\sum_{\Gamma_1,\Gamma_2} w_{\Gamma_1} w_{\Gamma_2} \sum_{\Gamma_{I,J}}(-1)^{(l+1)(n_2+1)} D_{\Gamma_{I,J}}(\gamma_1\otimes\cdots\otimes\gamma_n)
=\\=
\sum_\Gamma \sum_{\Gamma'\subset \Gamma}\epsilon(I,J) (-1)^{|k_I|+(l+1)(n_2+1)}
w_{\Gamma/\Gamma'}(j_I) w_{\Gamma'} (j_J)
D_\Gamma(\gamma_1\otimes \cdots \otimes \gamma_m).
\end{multline*}

Here the sum over $\Gamma_{I,J}$ in the second line is over all graphs that can be obtained by inserting the graph $\Gamma_2$ into one of the type II vertices $\bar{1},\dots,\overline{n_1}$ of $\Gamma_1$ and reconnecting the edges ending at that type two vertex in any possible way to vertices of $\Gamma_2$. The number $l$ in the second line is such the type II vertex $\Gamma_2$ is substituted for is $\bar{l}$.
In the last line, we again switched summation variables. The numbers $l$ and $n_2$ here are as in Section \ref{sec:qweights}.
To proceed further, note that 
\[
(-1)^{|k_I|+(l+1)(n_2+1)}=(-1)^{n-n_2+1+(l+1)(n_2+1)} = -(-1)^{(l+1)(n_2+1)+n-n_2}
\]
Comparing this to the quadratic weight relations of Proposition \ref{prop:weightrel}, one sees that the three summands sum up to 0. 
Hence Theorem \ref{thm:main} is proven. \qed
{"config": "arxiv", "file": "0806.4095/theproof.tex"}