Fixed-coordinate WZW sphere current coinvariants

Every rational function has a Laurent expansion with only finitely many negative powers at each marked point. Each finite insertion is restricted by [F1]. For the infinity factor, [F3] reduces the claim to a basis monomial $y_{-r_1}^1\cdots y_{-r_s}^s\mathbf{1}$ with $r_i\ge 1$. If $m>{\sum }_i{r}_i$, commute $x_m$ successively to the right using $[x_q,y_{-r_i}]=[x,y{]}_{q-r_i}+q{\delta }_{q,r_i}B(x,y)K$. At every commutation the current index $q$ exceeds the sum of the remaining $r_i$, so it stays positive and no central term occurs; the final positive mode kills $\mathbf{1}$ by [F2]. Thus $x_m$ kills that monomial for all sufficiently large $m$, and by linearity the vacuum module is restricted. Consequently, on each fixed algebraic tensor the negative tail is finite and the sufficiently high nonnegative modes kill every factor. Every local sum, and hence the sum over the finitely many marked points, is finite.

On each algebraic tensor, step 1.1 makes the local mode sums finite. For local Laurent expansions $f={\sum }_m{f}_m{t}^m$ and $g={\sum }_n{g}_n{t}^n$, the loop-bracket coefficients in [F4] are those of $[x,y]\otimes fg$. The central coefficient is $kB(x,y){\sum }_{m}m{f}_m{g}_{-m}=kB(x,y)\mathrm{Res}((df)g)$, a finite sum because each Laurent expansion has only finitely many negative powers. The central generator acts by the same level $k$ on every tensor factor. Thus the commutator of the summed actions of rational currents $x\otimes f$ and $y\otimes g$ is the summed action of $[x,y]\otimes fg$, plus $kB(x,y){\sum }_{p\in \{z_1,\dots ,z_n,\mathrm{\infty }\}}{\mathrm{Res}}_p((df)g)\mathrm{.}$ The differential $(df)g$ is meromorphic on ${\mathbb{P}}^1$, and its residues at all its poles sum to zero. Hence the central term vanishes and the global current action is a Lie representation. Its image span is invariant because $a(bv)=b(av)+[a,b]v$, and both terms lie in that span. The quotient is therefore defined. A functional on the tensor product descends uniquely to the quotient exactly when it annihilates this image span, by the universal property of a vector-space quotient.