Uniform nilpotence for root currents of positive-integral-level simple affine VOAs

Fix $\alpha \in \mathrm{\Delta }$ and set $\gamma =\delta -\alpha$. By [F1]–[F2], $\gamma$ is a positive real root with coroot pairing $m_{\alpha }$. In its ${\mathfrak{s}\mathfrak{l}}_2(\gamma )$ triple, the highest affine vacuum $\mathbf{1}$ is killed by the positive-root vector $f_{\alpha }\otimes t$ and has $h_{\gamma }$-weight $m_{\alpha }$; the lowering vector is $e_{\alpha }\otimes t^{-1}=x_{\alpha }(-1)$ up to nonzero scale. This lowering operator is locally nilpotent on the integrable module. Indeed, each simple-root generator is locally nilpotent by [F1]. For a Weyl translate, define $T_i=\exp (F_i)\exp (-E_i)\exp (F_i)$ on the module, where $E_i,F_i$ are the simple Chevalley operators. These exponentials are finite on every vector and invertible. The finite commutator identity $\exp (F_i)X\exp (-F_i)=\exp (\mathrm{ad}{F}_i)X$ (and its $E_i$ version) follows by expanding the conjugation polynomial; the adjoint series terminates on each affine root vector by [F2]. Therefore $T_i$ implements the Lie automorphism in [F2], and conjugates a simple-root operator to the corresponding real-root operator. Local nilpotence is preserved by this invertible conjugation. Hence the ${\mathfrak{s}\mathfrak{l}}_2(\gamma )$ lowering operator is locally nilpotent, its string on $\mathbf{1}$ is finite-dimensional, and [F3] gives $x_{\alpha }(-1{)}^{m_{\alpha }+1}\mathbf{1}=0$. Since $N_{\mathrm{\ell }}=1+{\max }_{\beta \in \mathrm{\Delta }}{m}_{\beta }$, also $x_{\alpha }(-1{)}^{N_{\mathrm{\ell }}}\mathbf{1}=0$; the maximum is finite because $\mathrm{\Delta }$ is finite.

For fields $A(z)={\sum }_n{A}_n{z}^{-n-1}$ and $B(z)$, the state-field map preserves the $(-1)$-product [F5], whose formal-residue definition [F5] is ${\mathrm{Res}}_x({\iota }_{x,z}(x-z{)}^{-1}A(x)B(z)-{\iota }_{z,x}(x-z{)}^{-1}B(z)A(x))$. Expanding the two binomials gives $A_{-}(z)B(z)+B(z)A_{+}(z)$, the stated normal product; module Jacobi gives the same identity on every weak module. For a root vector, its modes commute by [F4], so this normal product agrees coefficientwise with the ordinary product whenever the coefficient sums are finite on the vector. The state $x_{\alpha }(-1{)}^{N_{\mathrm{\ell }}}\mathbf{1}=0$ from step 1.1 therefore has zero field on $V$ and on every weak module.

Fix a weak module $M$, a root vector $x_{\alpha }$, and $v\in M$; abbreviate its commuting modes by $X_j$. Choose an integer $k$ with $X_{j}v=0$ for all $j>k$. Let $C$ be the span of all finite products of the $X_j$ applied to $v$, including the empty product. Commutativity makes $C$ invariant and gives $X_{j}C=0$ for $j>k$. In any coefficient of $J_M^{x_{\alpha }}(z{)}^{N_{\mathrm{\ell }}}$ on $C$, discard tuples containing an index above $k$; a fixed total index then bounds each remaining index below, so only finitely many tuples remain. The coefficient with total index $N_{\mathrm{\ell }}k$ is $X_k^{N_{\mathrm{\ell }}}$, hence it vanishes on $C$ by step 2.1. Descend through the integers $n\le k$. Suppose for $n<k$ that positive integers $r_j$ satisfy $X_j^{r_j}C=0$ for $n<j\le k$. In the coefficient with total index $N_{\mathrm{\ell }}n$, the only tuple with all indices at most $n$ is $(n,\dots ,n)$. Thus the zero-field identity gives $X_n^{N_{\mathrm{\ell }}}=-Q$ on $C$, where $Q$ is a finite sum of commuting mode monomials, each containing some $X_j$ with $n<j\le k$. Put $R=1+{\sum }_{j=n+1}^k(r_j-1)$. Every monomial in $Q^R$ contains at least $R$ such higher-index factors, so some $X_j$ occurs at least $r_j$ times; hence $Q^{R}C=0$ and $X_n^{N_{\mathrm{\ell }}R}C=0$. Finite downward induction reaches every specified integer $n$; modes above $k$ already vanish on $v$. This proves local nilpotence and the exponent $N_{\mathrm{\ell }}$ at a truncation index.

For each finite simple root choose $a_i=\exp (\mathrm{ad}{e}_i)f_i=f_i+h_i-e_i$. Its current modes are conjugate to those of $f_i$ by the invertible finite exponential $\exp (e_i(0))$: step 3.1 makes $e_i(0)$ locally nilpotent on the VOA and every weak module; the zero-mode Jacobi identity gives $[e_i(0),x_n]=[e_i,x{]}_n$, and the rank-one commutators make the Baker–Campbell–Hausdorff expansion finite. Thus the iterated normal product of $N_{\mathrm{\ell }}$ copies of $J^{a_i}(z)$ vanishes on every weak module, as the same conjugation carries the zero product for $f_i$. The element $a_i$ has $[a_i,a_i]=0$ and $B(a_i,a_i)=0$ because it is an inner conjugate of the isotropic root vector $f_i$ and the automorphism preserves the invariant form; hence its current modes commute. The root vectors together with the $a_i$ span $\mathfrak{g}$, since $a_i-(f_i-e_i)=h_i$ and the simple coroots span the Cartan. Choose a basis from this spanning set containing both $e_i$ and $f_i$ for each finite simple root and both $e_{\theta }$ and $f_{\theta }$ for the highest finite root, counting any rank-one duplicates once.

For each conjugate basis vector its modes commute and its iterated normal product vanishes by step 4.1. The cyclic-space coefficient argument of step 3.1 therefore applies to it, giving local nilpotence of every mode and the exponent $N_{\mathrm{\ell }}$ at a truncation index. The basis contains the finite simple-root generators and both highest-root vectors, so every affine simple raising and lowering generator, including $f_{\theta }(1)$ and $e_{\theta }(-1)$, acts locally nilpotently.