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\begin{document}

\title[The graded Turaev cobracket as divergence]{Diagrammatic Objects, Expansions, and the Graded Turaev Cobracket as Divergence}
\author{}
\maketitle

Let $\Sigma$ be a compact oriented surface with first homology
$H=H_1(\Sigma;\mathbb Q)$ carrying its intersection form
$\omega\in\Lambda^2H$.  
The vector space spanned by free homotopy classes of loops on~$\Sigma$
carries the \emph{Goldman--Turaev Lie bialgebra} structure:
the bracket records signed intersections, and the cobracket records self--intersections.
After completing and grading by powers of the augmentation ideal of~$\pi_1(\Sigma)$,
a \emph{symplectic expansion}
\[
Z:\ \widehat{\mathbb Q[\pi_1(\Sigma)]}\longrightarrow \widehat{T}(H)
\]
identifies the associated graded with cyclic words
\(
\mathrm{Cyc}(H)=T(H)/[T(H),T(H)].
\)

\vskip.4em
\noindent{\bf 1. The graded cobracket.}
Under~$Z$, the graded Turaev cobracket is the operator
\[
\delta_T(a_1\cdots a_n)=
\sum_{i<j}\omega^{ij}\,(a_{i+1}\cdots a_{j})\otimes(a_{j+1}\cdots a_{i}),
\]
where $\omega^{ij}$ are the components of~$\omega^{-1}$ in a symplectic basis.
Diagrammatically, $\delta_T$ ``cuts'' a beaded circle and inserts~$\omega$ at the cut.

\begin{center}
\begin{tikzpicture}[scale=1.0,baseline={(current bounding box.center)}]
  % main circle
  \draw[thick] (0,0) circle (1);
  % beads
  \foreach \ang in {15,75,135,195,255,315} {
    \filldraw[fill=white] (\ang:1) circle (2pt);
  }
  % cut marks
  \draw[red,very thick,decorate,decoration={markings,
      mark=at position 0.5 with {\arrow{stealth}}}] (30:1.05) -- (30:1.35);
  \draw[red,very thick,decorate,decoration={markings,
      mark=at position 0.5 with {\arrow{stealth}}}] (210:1.05) -- (210:1.35);
  % omega label
  \node[red] at (1.3,0.1) {$\omega$};
  % two arcs
  \draw[dashed] (30:1.1) arc (30:210:1.1);
  \draw[dashed] (210:0.9) arc (210:390:0.9);
\end{tikzpicture}

\vskip.2em
{\small A beaded circle split into two by inserting the symplectic form $\omega$.}
\end{center}

\vskip.4em
\noindent{\bf 2. The interpretation map.}
For a quadratic Lie algebra $(\mathfrak g,\langle\ ,\ \rangle)$,
each $h\in H$ acts by contraction with the corresponding infinitesimal holonomy variable.
This yields an \emph{interpretation map}
\[
I:\mathrm{Cyc}(H)\longrightarrow \operatorname{Inv}\bigl(S(\mathfrak g)\bigr),
\]
sending a cyclic word $a_1\cdots a_n$ to the invariant tensor
obtained by replacing each $a_i$ with the corresponding element of~$\mathfrak g$
and contracting indices using $\langle\ ,\ \rangle$ and~$\omega$.

\vskip.4em
\noindent{\bf 3. The differential--geometric avatar.}
A theorem of Alekseev--Kawazumi--Kuno--Naef (\emph{Invent.~Math.}~2023) asserts that,
for a symplectic expansion~$Z$, the diagram
\[
\begin{tikzcd}
\mathrm{Cyc}(H) \arrow[r,"\delta_T"] \arrow[d,"I"'] &
\mathrm{Cyc}(H)\otimes\mathrm{Cyc}(H) \arrow[d,"I\otimes I"] \\
\operatorname{Inv}\bigl(S(\mathfrak g)\bigr)
\arrow[r,"\mathrm{div}"] &
\operatorname{Inv}\bigl(S(\mathfrak g)\bigr)\otimes
\operatorname{Inv}\bigl(S(\mathfrak g)\bigr)
\end{tikzcd}
\]
commutes, where $\mathrm{div}$ is the \emph{divergence operator}—the trace of the covariant derivative
with respect to the canonical symplectic connection on the moduli of flat $\mathfrak g$--connections.

\vskip.4em
\noindent{\bf 4. Aphorism.}
\emph{Diagrammatic objects pine for specializations:
under the interpretation map, the beads of the Turaev cobracket become tensor indices,
and the cobracket itself becomes the divergence operator.}

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