Two-insertion KZ monodromy is a Casimir exponential

Let $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, let $V={\mathbb{C}}^2$ be its standard representation with its standard Hermitian inner product, and use the Killing-form Casimir tensor ${\mathrm{\Omega }}_{12}$ on $V\otimes V$. For two insertions set $w=z_1-z_2\mathrm{\ne }0$. At the basepoint $w_{\ast }=1$, take $z_2=0$ and $z_1=w$; along a chosen logarithm branch, a horizontal section with value $v_{\ast }$ at $w_{\ast }$ is $s(w)=\exp (({\mathrm{\Omega }}_{12}\mathrm{/}\kappa )\log (w\mathrm{/}{w}_{\ast }))v_{\ast }$. The positive loop $w(t)=w_{\ast }{e}^{2\pi it}$, equivalently $z_1(t)=e^{2\pi it}$ and $z_2(t)=0$, has horizontal transport

$P_{\gamma }=\exp (2\pi i{\mathrm{\Omega }}_{12}\mathrm{/}\kappa )\mathrm{.}$

For the library's first-then-second fundamental-group product, the associated left monodromy representation uses $P_{\gamma }^{-1}$; equivalently, $P_{\gamma }$ is the right monodromy action by horizontal transport. This transport need not be unitary: with $\kappa =i$, it has eigenvalue $e^{\pi \mathrm{/}4}$.

Infinitesimal braid identities imply KZ flatness and braid monodromy

Casimir KZ connection on ordered configurations

The Killing form of a semisimple Lie algebra

The diagram uses $w=z_1-z_2$ as the relative coordinate; the arrow indicates increasing $t$ in $w(t)=w_{\ast }{e}^{2\pi it}$, and the removed origin is the collision $z_1=z_2$. It records the positive loop only; the proof below derives transport from the differential equation and does not rely on the picture.

diagram failed to render:
\begin{tikzpicture}[>=stealth,thick]
  \draw[->] (-1.35,0) -- (1.45,0) node[right] {$\operatorname{Re}w$};
  \draw[->] (0,-1.35) -- (0,1.45) node[above] {$\operatorname{Im}w$};
  \draw (0,0) circle (1);
  \fill (0,0) circle (0.045) node[below right] {$0$};
  \fill (1,0) circle (0.045) node[below right] {$w_*$};
  \draw[->] (0.707,0.707) arc[start angle=45,end angle=90,radius=1];
  \node at (-0.25,1.12) {$w(t)$};
\end{tikzpicture}