Unordered WZW top fiber with label assignments

Fix $n$ WZW labels ${\lambda }_1,\dots ,{\lambda }_n$ and write $\mathrm{\Lambda }$ for the finite set of distinct labels, with multiplicity $m_{\lambda }$ for $\lambda \in \mathrm{\Lambda }$. On the ordered configuration space $F_n(\mathbb{C})=\{(z_1,\dots ,z_n):z_i\mathrm{\ne }{z}_j\text{ for }i\mathrm{\ne }j\},$ an ordered label assignment is a map $a:\{1,\dots ,n\}\to \mathrm{\Lambda }$ with $\mathrm{\mid }{a}^{-1}(\lambda )\mathrm{\mid }=m_{\lambda }$ for every $\lambda$. Its tensor fiber is $W_a={⨂}_{r=1}^n{V}_{a(r)}^{\ast }$, ordered by coordinate index. Let $W={⨁}_a{W}_a$. The symmetric group $S_n$ acts on the base by $(\pi z{)}_r=z_{{\pi }^{-1}(r)}$, sends $a$ to $\pi a$, where $(\pi a)(r)=a({\pi }^{-1}r)$, and reorders tensor factors by the same permutation.

The quotient of $F_n(\mathbb{C})\times W$ by this action is the unordered WZW top bundle with label assignments over $C_n(\mathbb{C})=F_n(\mathbb{C})\mathrm{/}{S}_n$. Equivalently, its fiber at an unordered configuration $Q$ is the direct sum over all distinct assignments $a:Q\to \mathrm{\Lambda }$ with the specified multiplicities of ${⨂}_{q\in Q}{V}_{a(q)}^{\ast }$; the finite tensor product is identified under its canonical permutation maps. When all labels agree there is one assignment, and this construction reduces to the usual simultaneous permutation quotient with fiber $(V_{\lambda }^{\ast }{)}^{\otimes n}$.

This definition records the finite KZ top-space bundle only. It does not identify it with the full affine conformal-block sheaf.