Flatness and braid monodromy of the WZW KZ connection

Each ${\eta }_{ij}$ is closed on $F_n(\mathbb{C})$, since it is $d\log (z_i-z_j)$ away from $z_i=z_j$. Write $A={\kappa }^{-1}{\sum }_p{\mathrm{\Omega }}_p^{\vee }{\eta }_p$, with $p$ ranging over unordered pairs. Then $dA=0$ and the curvature of $d-A$ is $A\wedge A$. Terms for disjoint pairs vanish by [F1]. For each triple $i<j<k$, put $a={\eta }_{ij}$, $b={\eta }_{jk}$, $c={\eta }_{ik}$, $P={\mathrm{\Omega }}_{ij}^{\vee }$, $Q={\mathrm{\Omega }}_{jk}^{\vee }$, $R={\mathrm{\Omega }}_{ik}^{\vee }$. The contribution is ${\kappa }^{-2}([P,Q]a\wedge b+[P,R]a\wedge c+[R,Q]c\wedge b)\mathrm{.}$ By [F1], $[P,R]=-[P,Q]$ and $[R,Q]=-[P,Q]$. Also, with $u=z_i-z_j$, $v=z_j-z_k$, direct differentiation gives $a\wedge b-a\wedge c+b\wedge c=\frac{du\wedge dv}{uv}-\frac{du\wedge dv}{u(u+v)}-\frac{du\wedge dv}{v(u+v)}=0.$ Thus every triple contribution is zero and the curvature vanishes.

The flat connection on the ordered cover is permutation-equivariant: a permutation reorders the tensor factors and sends each pair $(i,j)$ and form ${\eta }_{ij}$ to the corresponding relabeled pair and form. Hence the action in [F2] preserves $A$ and $\mathrm{\nabla }$, and the connection descends to the quotient bundle.

By [F3] the horizontal sections form a local system. Parallel transport along a path is defined by solving the finite-dimensional linear ODE in local trivializations. To see its homotopy invariance, subdivide a compact path homotopy into small rectangles lying in coordinate balls from [F4]; transport around each rectangle is the identity by endpoint dependence, and transports along shared interior edges cancel in inverse pairs. Thus based-loop transport depends only on the loop class, giving the right monodromy action; with the first-loop-then-second convention, its inverse is the associated left representation.

Put $h(z)=z\mathrm{/}(1+\mathrm{\mid }z\mathrm{\mid })$ and $q_0={\bar{h}}^{-1}([Q_n])$, with $Q_n$ as in [F6]. The induced map $\bar{h}$ gives a based homeomorphism from $C_n(\mathbb{C})$ at $q_0$ to $C_n(\mathrm{int}{D}^2)$ at $[Q_n]$. By [F5], inclusion into $C_n(D^2)$ induces a fundamental-group isomorphism at that same base configuration; [F6] then identifies this group with the boundary-fixed mapping-class group based at $Q_n$. Thus the local-system monodromy at $q_0$ is a braid-group action with exactly the basepoint used by the mapping-class theorem. At a different base configuration $q$, choose a path to $q_0$; [F9] gives the resulting basepoint isomorphism, and parallel transport identifies the fibers, so the actions are conjugate. The identification depends on the chosen path, and no path-independent isomorphism is asserted. The direct sum in [F2] retains the label-reordering rule: identical labels give one summand, while a general label list requires all assignments.

Under [A2], parallel transport is intertwined by $\iota$, so a loop $\gamma$ acts on the block fiber by ${\iota }^{-1}{M}_{\gamma }\iota$, where $M_{\gamma }$ is its KZ holonomy from step 4.1. Conjugating the KZ monodromy law by the same fiberwise isomorphism preserves the group law; hence the resulting braid action is well-defined on $\mathcal{B}$. The missing construction of [A2] remains an explicit prerequisite.