2 A construction of automorphisms of vertex algebras

Let V be a vertex algebra. As usual, we set $a_{(n)}=\operatorname {Res}_{z=0}z^{n} a(z)$ for $a\in V$, where $a(z)$ is the quantum field corresponding to a. We have

(8)$$ \begin{align} [a_{(m)},b_{(n)}]=\sum_{j\geqslant 0}\begin{pmatrix} m\\ j\end{pmatrix}(a_{(j)}b)_{(m+n-j)} \end{align} $$

for $a,b\in V$, $m,n\in \mathbb {Z}$. In particular, $[a_{(0)},b_{(n)}]=(a_{(0)}b)_{(n)}$.

Let A be an element of V such that its zero mode $A_{(0)}$ acts semisimply on V so that $V=\bigoplus _{\lambda \in \mathbb {C} }V[\lambda ]$, where $V[\lambda ]=\{v\in V\mid A_{(0)}v=\lambda v\}$. We assume that $A\in V[0]$. Set $V[\geqslant \lambda ]=\bigoplus \limits _{\mu \in \lambda +\mathbb {R}_{\geqslant 0}} {V[\mu ]\supset V[> \lambda ]}=\bigoplus \limits _{\mu > \lambda } V[\mu ]$.

Suppose that there exists another element $\hat {A}\in V[\geqslant 0]$ such that $\hat {A}\equiv A \pmod {V[>0]}$ and $\hat {A}_{(0)}$ acts locally finitely on V. Then, $\hat {A}_{(0)}$ acts semisimply on V, and for $v\in V[\lambda ]$, there exists a unique eigenvector $\tilde {v}$ of $\hat {A}_{(0)}$ eigenvalue $\lambda $ such that $\tilde {v}\equiv v\pmod {V[>\lambda ]}$. By extending this correspondence linearly, we obtain a linear map

(9)$$ \begin{align} V\rightarrow V,\quad v\mapsto \tilde{v}. \end{align} $$

We also have the following.

Let M be a V-module on which both $A_{(0)}$ and $\hat {A}_{(0)}$ act semisimply. Set $M[\lambda ]=\{m\in M\mid A_{(0)}m=\lambda m\}$, $M[>\lambda ]=\sum _{\mu >\lambda }M[\mu ]$. We can define a linear isomorphism

(10)$$ \begin{align} M{\;\stackrel{_{\sim}}{\to}\;} M, \quad m\mapsto \tilde{m} \end{align} $$

that sends $m\in M[\lambda ]$ to a unique eigenvector $\tilde {m}$ of $\hat {A}_{(0)}$ of eigenvalue $\lambda $ such that $\tilde {m}\equiv m\pmod {M[>\lambda ]}$. The following assertion can be shown in the same manner as Lemma 2.1.

3 Preliminaries on Drinfeld–Sokolov reduction

Let $\mathfrak {g}$ be a simple Lie algebra over $\mathbb {C}$, and let $V^{k}(\mathfrak {g})$ be the universal affine vertex algebra associated with $\mathfrak {g}$ at level k as in the introduction. A $V^{k}(\mathfrak {g})$-module is the same as a smooth module M of level k over the affine Kac–Moody algebra $\widehat {\mathfrak {g}}$. Here, a $\widehat {\mathfrak {g}}$-module M is called smooth if $x(z)=\sum _{n\in \mathbb {Z}}(xt^{n})z^{-n-1}$ is a (quantum) field on M for all $x\in \mathfrak {g}$, that is, $(xt^{n})m=0$ for a sufficiently large n for any $m\in M$.

Let f be a nilpotent element of $\mathfrak {g}$. Let

(11)$$ \begin{align} \mathfrak{g}=\bigoplus_{j\in \frac{1}{2}\mathbb{Z}}\mathfrak{g}_{j} \end{align} $$

be a good grading ([Reference Kac, Roan and WakimotoKRW03]) of $\mathfrak {g}$ for f, that is, $f\in \mathfrak {g}_{-1}$, $\operatorname {\mathrm {ad}} f:\mathfrak {g}_{j}\rightarrow \mathfrak {g}_{j-1}$ is injective for $j\geqslant 1/2$ and surjective for $j\leqslant 1/2$. Denote by $x_{0}$ the semisimple element of $\mathfrak {g}$ that defines the grading, i.e.,

(12)$$ \begin{align} \mathfrak{g}_{j} =\{x\in \mathfrak{g}\mid [x_{0},x] = jx\}. \end{align} $$

We write $\deg x=d$ if $x\in \mathfrak {g}_{d}$.

Let $\{e,h,f\}$ be an $\mathfrak {sl}_{2}$-triple associated with f in $\mathfrak {g}$. Then the grading defined by $x_{0}=1/2h$ is good and is called a Dynkin grading.

Fix a Cartan subalgebra $\mathfrak {h}$ of $\mathfrak {g}$ that is contained in the Lie subalgebra $\mathfrak {g}_{0}$. Let $\Delta $ be the set of roots of $\mathfrak {g}$, $\mathfrak {g}=\mathfrak {h}\mathop {\oplus } \bigoplus _{\alpha \in \Delta }\mathfrak {g}_{\alpha }$ the root space decomposition. Set $\Delta _{j}=\{\alpha \in \Delta \mid \mathfrak {g}_{\alpha }\subset \mathfrak {g}_{j}\}$ so that $\Delta =\bigsqcup _{j\in \frac {1}{2}\mathbb {Z}}\Delta _{j}$. Put $\Delta _{>0}=\bigsqcup _{j>0}\Delta _{j}$. Let $I=\{1,2,\dots , \operatorname {rk}\mathfrak {g}\}$, and let $\{x_{a}\mid a\in I\sqcup \Delta \}$ be a basis of $\mathfrak {g}$ such that $x_{\alpha }\in \mathfrak {g}_{\alpha }$, $\alpha \in \Delta $, and $x_{i}\in \mathfrak {h}$, $i\in I$. Denote by $c_{a b}^{d}$ the corresponding structure constant.

Set $\mathfrak {g}_{\geqslant 1}=\bigoplus _{j\geqslant 1}\mathfrak {g}_{j}$, $\mathfrak {g}_{>0}=\bigoplus _{j\geqslant 1/2}\mathfrak {g}_{j}$, and let $\chi :\mathfrak {g}_{\geqslant 1}\rightarrow \mathbb {C}$ be the character defined by $\chi (x)=(f|x)$. We extend $\chi $ to the character $\hat {\chi }$ of $\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ by setting $\hat \chi (xt^{n})= \delta _{n,-1}\chi (x)$. Define

$$ \begin{align*}F_{\chi}=U(\mathfrak{g}_{>0}[t,t^{-1}])\otimes_{U(\mathfrak{g}_{>0}[t]+\mathfrak{g}_{\geqslant 1}[t,t^{-1}])}\mathbb{C}_{\hat\chi},\end{align*} $$

where $\mathbb {C}_{\hat \chi }$ is the one-dimensional representation of $\mathfrak {g}_{>0}[t]+\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ on which $\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ acts by the character $\hat \chi $ and $\mathfrak {g}_{>0}[t]$ acts trivially. Since it is a smooth $\mathfrak {g}_{>0}[t,t^{-1}]$-module, the space $F_{\chi }$ is a module over the vertex subalgebra $V(\mathfrak {g}_{>0 })\subset V^{k}(\mathfrak {g})$ generated by $x_{\alpha }(z)$ with $\alpha \in \Delta _{>0}$. For $\alpha \in \Delta _{> 0}$, let $\Phi _{\alpha }(z)$ denote the image of $x_{\alpha }(z)$ in $(\operatorname {\mathrm {End}} F_{\chi })[[z,z^{-1}]]$. Then

$$ \begin{align*} \Phi_{\alpha}(z)=\chi(x_{\alpha})\quad \text{for }\alpha\in \Delta_{\geqslant 1} \end{align*} $$

and

$$ \begin{align*} \Phi_{\alpha}(z)\Phi_{\beta}(w)\sim \frac{\chi([x_{\alpha},x_{\beta}])}{z-w}. \end{align*} $$

There is a unique vertex algebra structure on $F_{\chi }$ such that $|0{\rangle }=1{\otimes } 1$ is the vacuum vector and

$$ \begin{align*} Y((\Phi_{\alpha})_{(-1)}|0{\rangle} ,z)=\Phi_{\alpha}(z) \end{align*} $$

for $\alpha \in \Delta _{>0}$. (Note that $(\Phi _{\alpha })_{(-1)}|0{\rangle }=\chi (x_{\alpha })|0{\rangle } $ for $\alpha \in \Delta _{\geqslant 1}$.) In other words, $F_{\chi }$ has the structure of the $\beta \gamma $-system associated with the symplectic vector space $\mathfrak {g}_{1/2}$ with the symplectic form

(13)$$ \begin{align} \mathfrak{g}_{1/2}\times \mathfrak{g}_{1/2}\rightarrow \mathbb{C},\quad (x, y)\mapsto \chi([x,y]). \end{align} $$

Next, let $\bigwedge ^{\infty /2+\bullet }(\mathfrak {g}_{>0})$ be the vertex superalgebra generated by odd fields $\psi _{\alpha }(z)$, $\psi _{\alpha }^{*}(z)$, $\alpha \in \Delta _{>0}$, with the OPEs

$$ \begin{align*} \psi_{\alpha}(z)\psi_{\beta}^{*}(z)\sim \frac{\delta_{\alpha,\beta}}{z-w}, \quad \psi_{\alpha}(z)\psi_{\beta}(z)\sim \psi_{\alpha}^{*}(z)\psi_{\beta}^{*}(z)\sim 0 \end{align*} $$

Let $\bigwedge ^{\infty /2+\bullet }(\mathfrak {g}_{>0})=\bigoplus _{n\in \mathbb {Z}} \bigwedge ^{\infty /2+n}(\mathfrak {g}_{>0})$ be the $\mathbb {Z}$-gradation defined by $\deg |0{\rangle } =0$, $\deg (\psi _{\alpha })_{(n)}=-1$, $\deg (\psi _{\alpha }^{*})_{(n)}=1$.

For a smooth $\widehat {\mathfrak {g}}$-module M, set

(14)$$ \begin{align} C(M):=M{\otimes} F_{\chi}{\otimes} \bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0}). \end{align} $$

Then $C(M)=\bigoplus \limits _{i\in \mathbb {Z}}C^{i}(M)$, $C^{i}(M)=M{\otimes } F_{\chi }{\otimes } \bigwedge \nolimits ^{\infty /2+i}(\mathfrak {g}_{>0})$. Note that $C(V^{k}(\mathfrak {g}))$ is naturally a vertex superalgebra, and $C(M)$ is a module over the vertex superalgebra $C(V^{k}(\mathfrak {g}))$ for any smooth $\widehat {\mathfrak {g}}$-module M. Define

$$ \begin{align*} Q(z)=\sum_{\alpha\in \Delta_{>0}}x_{\alpha}(z)\psi_{\alpha}^{*}(z) +\sum_{\alpha\in \Delta_{>0}}\Phi_{\alpha}(z)\psi_{\alpha}^{*}(z) -\frac{1}{2} \sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^{\gamma} \psi_{\alpha}^{*}(z)\psi_{\beta}^{*}(z)\psi_{\gamma}(z), \end{align*} $$

where we have omitted the tensor product symbol. Then $Q(z)Q(w)\sim 0$, and we have $Q_{(0)}^{2}=0$ on any $C(V^{k}(\mathfrak {g}))$-module. The cohomology

$$ \begin{align*} H^{\bullet}_{DS,f}(M): =H^{\bullet}(C(M),Q_{(0)}) \end{align*} $$

is called the BRST cohomologyof the Drinfeld–Sokolov reduction associated with f with coefficients in M ([Reference Feigin and FrenkelFF90a, Reference Kac, Roan and WakimotoKRW03]; see also [Reference ArakawaAra05]). By definition [Reference FeĭginFeĭ84], $H^{\bullet }_{DS,f}(M)$ is the semi-infinite $\mathfrak {g}_{>0}[t,t^{-1}]$-cohomology $H^{\frac {\infty }{2}+\bullet }(\mathfrak {g}_{>0}[t,t^{-1}],M\otimes F_{\chi })$ with coefficients in the diagonal $\mathfrak {g}_{>0}[t,t^{-1}]$-module $M\otimes F_{\chi }$.

The vertex algebra

$$ \begin{align*}\mathscr{W}^{k}(\mathfrak{g},f):=H^{0}_{DS,f}(V^{k}(\mathfrak{g}))\end{align*} $$

is called the W-algebra associated with $(\mathfrak {g},f)$ at level k, which is conformal provided that $k\ne -h^{\vee }$. The vertex algebra structure of $\mathscr {W}^{k}(\mathfrak {g},f)$ does not depend on the choice of a good grading ([Reference Arakawa, Kuwabara and MalikovAKM15]); however, its conformal structure does. The central charge of $\mathscr {W}^{k}(\mathfrak {g},f)$ is given by

(15)$$ \begin{align} \dim \mathfrak{g}-\frac{1}{2}\dim \mathfrak{g}_{1/2}-12|\frac{\rho}{\sqrt{k+h^{\vee}}}-\sqrt{k+h^{\vee}}x_{0}|^{2}, \end{align} $$

where $\rho $ is the half sum of positive roots of $\mathfrak {g}$.

Let $\mathscr {W}_{k}(\mathfrak {g},f)$ be the unique simple graded quotient of $\mathscr {W}^{k}(\mathfrak {g},f)$.

Let $\operatorname {KL}$ be the full subcategory of $\widehat {\mathfrak {g}}$-modules consisting of objects on which $\mathfrak {g}$ acts semisimply and $t\mathfrak {g}[t]$ acts locally nilpotently, and let $\operatorname {KL}_{k}$ be the full subcategory of $\operatorname {KL}$ consisting of modules of level k.

Let Q be the root lattice of $\mathfrak {g}$, $\check {Q}$ the coroot lattice, P the weight lattice, $\check {P}$ the coweight lattice, $P_{+}$ the set of dominant integral weights and $\check {P}_{+}$ the set of dominant integral coweights of $\mathfrak {g}$. For $\lambda \in P_{+}$, set

$$ \begin{align*} \mathbb{V}^{k}_{\lambda} := U(\widehat{\mathfrak{g}}) \otimes_{U(\mathfrak{g}[t]\oplus \mathbb{C} K)} E_{\lambda}\in \operatorname{KL}_{k}, \end{align*} $$

where $E_{\lambda }$ is the irreducible finite-dimensional $\mathfrak {g}$-module of highest weight $\lambda $ lifted to a $\mathfrak {g}[t]$-module by letting $\mathfrak {g}[t]t$ act trivially and K by multiplication with the scalar k. Note that $V^{k}(\mathfrak {g})=\mathbb {V}^{k}_{0}$ as a $\widehat {\mathfrak {g}}$-module. We denote by $\mathbb {L}^{k}_{\lambda }$ the unique simple graded quotient of $\mathbb {V}^{k}_{\lambda }$. More generally, for any weight $\lambda $ of $\mathfrak {g}$, we denote by $\mathbb {L}^{k}_{\lambda }$ the irreducible highest weight representation of $\widehat {\mathfrak {g}}$ with highest weight $\lambda $ and level k.

Theorem 3.1 [Reference Frenkel and GaitsgoryFG10, Reference ArakawaAra15a]

We have $H_{DS,f}^{i}(M)=0$ for $i\ne 0$ and $M\in \operatorname {KL}_{k}$. In particular, the functor

$$ \begin{align*} \operatorname{KL}_{k}\rightarrow \mathscr{W}^{k}(\mathfrak{g},f)\operatorname{-Mod},\quad M\mapsto H_{DS,f}^{0}(M) \end{align*} $$

is exact.

Note that for $M\in \operatorname {KL}_{k}, N\in \operatorname {KL}_{\ell }$, we have $M{\otimes }N\in \operatorname {KL}_{k+\ell }$. Therefore, $H^{i}_{DS,f}(M{\otimes } N)=0$ for $i\ne 0$. In particular, $H^{i}_{DS,f}(M{\otimes } L)=0$ for $i\ne 0$ if $M\in \operatorname {KL}$ and L is an integrable representation of $\widehat {\mathfrak {g}}$.

4 Proof of Theorem 1

Let $k\in \mathbb {C}$, $\ell \in \mathbb {Z}_{\geqslant 0}$, and set

$$ \begin{align*} C:=C(V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}))=V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}){\otimes} F_{\chi{\otimes}}\bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0}). \end{align*} $$

For $t\in \mathbb {C}$, define the element $Q_{t}\in C$ by

(16)$$ \begin{align} Q_{t}(z)=& \sum_{\alpha\in \Delta_{>0}}(\pi_{1}(x_{\alpha})(z)+t^{2\alpha(x_{0})}\pi_{2}(x_{\alpha})(z))\psi_{\alpha}^{*}(z)\\ & +\sum_{\alpha\in \Delta_{>0}}\Phi_{\alpha}(z)\psi_{\alpha}^{*}(z) -\frac{1}{2} \sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^{\gamma} \psi_{\alpha}^{*}(z)\psi_{\beta}^{*}(z)\psi_{\gamma}(z), \nonumber \end{align} $$

where $\pi _{1}(x_{a})(z)$ (resp. $\pi _{2}(x_{a})(z)$) denotes the action of $x_{a}(z)$, $a\in I\sqcup \Delta $, on $V^{k}(\mathfrak {g})$ (resp. on $\mathbb {L}_{\ell }(\mathfrak {g})$). Then $Q_{t}(z)Q_{t}(w)\sim 0$, and therefore, $(Q_{t})_{(0)}^{2}=0$. It follows that $(C,(Q_{t})_{(0)})$ is a differential graded vertex algebra, and the corresponding cohomology $H^{\bullet }(C,(Q_{t})_{(0)})$ is naturally a vertex algebra. Clearly,

(17)$$ \begin{align} &H^{i}(C,(Q_{t=0})_{(0)})\cong H_{DS,f}^{i}(V^{k}(\mathfrak{g})){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}) =\delta_{i,0}\mathscr{W}^{k}(\mathfrak{g},f){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}), \end{align} $$

(18)$$ \begin{align} &H^{i}(C,(Q_{t=1})_{(0)})\cong H_{DS,f}^{i}(V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})) =\delta_{i,0}H_{DS,f}^{0}(V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})); \end{align} $$

see Theorem 3.1.

By [Reference ArakawaAra05, 3.7], the differential $(Q_{t})_{(0)}$ decomposes as

(19)$$ \begin{align} (Q_{t})_{(0)}=d_{t}^{st}+d^{\chi}, \quad (d_{t}^{st})^{2}=(d^{\chi})^{2}=\{d_{t}^{st},d^{\chi}\}=0, \end{align} $$

where

(20)$$ \begin{align} \nonumber &d_{t}^{st}=\sum_{\alpha\in \Delta_{>0}}\sum_{n\in \mathbb{Z}}(\pi_{1}(x_{\alpha})_{(n)}+t^{2\alpha(x_{0})}\pi_{2}(x_{\alpha})_{(n)})\psi_{\alpha,(-n)}^{*}\\ &\qquad\qquad\quad+\sum_{\alpha\in \Delta_{1/2}}\sum_{n< 0}\Phi_{\alpha,(n)}\psi^{*}_{\alpha,(-n)} \end{align} $$

(21)$$ \begin{align} &\qquad \qquad \quad -\frac{1}{2} \sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^{\gamma} \sum_{m,n\in \mathbb{Z}} \psi_{\alpha,(m)}^{*}\psi_{\beta,(n)}^{*}\psi_{\gamma,(-m-n)}, \nonumber \\ &d^{\chi}=\sum_{\alpha\in \Delta_{1/2}}\sum_{n\geqslant 0}\Phi_{\alpha,(n)}\psi^{*}_{\alpha,(-n)} +\sum_{\alpha\in \Delta_{1}}\chi(x_{\alpha})\psi^{*}_{\alpha,(0)}, \end{align} $$

and $\{x,y\}=xy+yx$.

Define the Hamiltonian H on C by

$$ \begin{align*} H=H_{standard}^{V^{k}(\mathfrak{g})}+(\omega_{\mathbb{L}_{\ell}(\mathfrak{g})})_{(1)} +(\omega_{F_{\chi}})_{(1)}+(\omega_{\bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0})})_{(1)} -\pi_{1}(x_{0})_{(0)}-\pi_{2}(x_{0})_{(0)}, \end{align*} $$

where $H_{standard}^{V^{k}(\mathfrak {g})}$ is the standard Hamiltonian of $V^{k}(\mathfrak {g})$ that gives $\pi _{1}(x)$, $x \in \mathfrak {g}$, conformal weight one and $\omega _{\mathbb {L}_{\ell }(\mathfrak {g})}$ is the Sugawara conformal vector of $\mathbb {L}_{\ell }(\mathfrak {g})$,

$$ \begin{align*} &\omega_{F_{\chi}}(z)=\frac{1}{2}\sum\limits_{\alpha\in \Delta_{1/2}}:\partial_{z}\Phi^{\alpha}(z) \Phi_{\alpha}(z):,\\ &\omega_{\bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0})}(z)= \sum_{j>0}\sum_{\alpha\in \Delta_{j}}j:\psi_{\alpha}^{*}(z)\partial \psi_{\alpha}(z): +\sum_{j>0}\sum_{\alpha\in \Delta_{j}}(1-j):(\partial \psi_{\alpha}^{*}(z))\psi_{\alpha}(z):. \end{align*} $$

Here, $\{\Phi ^{\alpha }\}$ is a dual basis to $\{\Phi _{\alpha }\}$, that is, $\Phi ^{\alpha }(z)\Phi _{\beta }(w)\sim \delta _{\alpha \beta }/(z-w)$. Then $[H, d_{t}^{st}]=0=[H,d_{t}^{\chi }]$, and thus, H defines a Hamiltonian on the vertex algebra $H^{\bullet }(C,(Q_{t})_{(0)})$.

Following [Reference Frenkel and Ben-ZviFBZ04, Reference Kac, Roan and WakimotoKRW03], define

$$ \begin{align*} J^{a}(z)={\pi}_{1}(x_{a})(z)+\sum_{\beta,\gamma\in \Delta_{+}}c_{a,\beta}^{\gamma} :\psi_{\gamma}(z)\psi_{\beta}^{*}(z): \end{align*} $$

for $a\in I\sqcup \Delta $. We also denote by $J^{x}$ the linear combination of $J^{a}$, $a\in I\sqcup \Delta _{\leqslant 0}$, corresponding to $x\in \mathfrak {g}_{\leqslant 0}:=\bigoplus _{j\leqslant 0}\mathfrak {g}_{j}$. Let $C_{\leqslant 0}$ be the vertex subalgebra of C generated by $J^{x}(z$) ($x\in \mathfrak {g}_{\leqslant 0}$), $\pi _{2}(x)(z)$ ($x\in \mathfrak {g}$), $\psi ^{*}_{\alpha }(z)$ ($\alpha \in \Delta _{>0}$) and $\Phi _{\alpha }(z)$ ($\alpha \in \Delta _{1/2}$), and let $C_{> 0}$ be the vertex subalgebra of C generated by $\psi _{\alpha }(z)$ and

$$ \begin{align*}((Q_{t})_{(0)}\psi_{\alpha})(z)=J^{\alpha}(z)+t^{\alpha(h)}\pi_{2}(x_{\alpha})(z)+\Phi_{\alpha}(z)\end{align*} $$

($\alpha \in \Delta _{>0}$).

As in [Reference Frenkel and Ben-ZviFBZ04, Reference Kac, Roan and WakimotoKRW03], we find that both $C_{\leqslant 0}$ and $C_{> 0}$ are subcomplexes of $(C, (Q_{t})_{(0)})$ and that $C\cong C_{\leqslant 0}{\otimes } C_{>0}$ as complexes. Moreover, we have

$$ \begin{align*}H^{i}(C_{>0},(Q_{t})_{(0)})=\begin{cases}\mathbb{C}&\text{for }i= 0\\ 0&\text{for }i\ne 0,\end{cases}\end{align*} $$

and therefore,

(22)$$ \begin{align} H^{\bullet}(C, (Q_{t})_{(0)})\cong H^{\bullet}(C_{\leqslant 0},(Q_{t})_{(0)}) \end{align} $$

as vertex algebras. Since the cohomological gradation takes only nonnegative values on $C_{\leqslant 0}$, it follows that $H^{0}(C,(Q_{t})_{(0)})= H^{0}(C_{\leqslant 0},(Q_{t})_{(0)})$ is a vertex subalgebra of $C_{\leqslant 0}$.

Note that the vertex algebra $C_{\leqslant 0}$ does not depend on the parameter $t\in \mathbb {C}$. Also, $C_{\leqslant 0}$ is preserved by the action of both $d_{t}^{st}$ and $d^{\chi }$.

Let $C_{\leqslant 0,\Delta }=C_{\leqslant 0}\cap C_{\Delta }$ so that $C_{\leqslant 0}=\bigoplus _{\Delta }C_{\leqslant 0,\Delta }$.

Since both $d_{t}^{st}$ and $d^{\chi }$ preserve $C_{\leqslant 0,\Delta }$, we can consider the spectral sequence $E_{r}\Rightarrow H^{\bullet }(C_{\leqslant 0})$ such that $d_{0}=d^{\chi }$ and $d_{1}=d_{t}^{st}$, which is converging since each $C_{\leqslant 0,\Delta }$ is finite dimensional. As in [Reference Frenkel and Ben-ZviFBZ04, Reference Kac and WakimotoKW04, Reference Kac and WakimotoKW05], we find that

(23)$$ \begin{align} E_{1}^{\bullet,q} =H^{q}(C_{\leqslant 0},d^{\chi})=\delta_{q,0}V^{k^{\natural}}(\mathfrak{g}^{f}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}), \end{align} $$

where $V^{k^{\natural }}(\mathfrak {g}^{f})$ is the vertex subalgebra of $C_{\leqslant 0}$ generated by $J^{x}(z)$, $x\in \mathfrak {g}^{f}$. Thus, the spectral sequence collapses at $E_{1}=E_{\infty }$, and we get the vertex algebra isomorphism

(24)$$ \begin{align} \operatorname{\mathrm{gr}} H^{q}(C_{t,\leqslant 0})\cong \delta_{q,0}V^{k^{\natural}}(\mathfrak{g}^{f}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g}). \end{align} $$

Here, $\operatorname {\mathrm {gr}} H^{q}(C_{t,\leqslant 0})$ is the associated graded vertex algebra with respect to the filtration that defines the spectral sequence. By equation (24), for each $v\in V^{k^{\natural }}(\mathfrak {g}^{f}){\otimes } \mathbb {L}_{\ell }(\mathfrak {g})$, there exists a cocycle

$$ \begin{align*} \hat{v}=v_{0}+v_{1}+v_{2}+\dots \quad\text{(a finite sum)} \end{align*} $$

such that

$$ \begin{align*} v_{0}=v,\quad d^{st}v_{i}=-d^{\chi}v_{i+1}. \end{align*} $$

Set

(25)$$ \begin{align} A:=\pi_{2}(x_{0})\in \mathbb{L}_{\ell}(\mathfrak{g}), \end{align} $$

where we recall that $x_{0}$ is defined by equation (12). Then $C_{\leqslant 0}=\bigoplus _{\lambda \in \frac {1}{2}\mathbb {Z}}C_{\leqslant 0} [\lambda ]$, $C_{\leqslant 0}[\lambda ]=\{c\in C_{\leqslant 0}\mid A_{(0)}c=\lambda c\}$, and $A\in C_{\leqslant 0}[0]$. Consider the corresponding cocycle $\hat {A}$ that has cohomolological degree zero. Since $d^{st}_{t}A\subset C_{\leqslant 0}[>0]$ and $d^{\chi }C_{\leqslant 0}[\lambda ]\subset C_{\leqslant 0}[\lambda ]$, we may assume that $\hat {A}\equiv A\pmod {C_{\leqslant 0}[>0]}$. Moreover, we may assume that $\hat {A}$ is homogenous with respect to the Hamiltonian H so that $\hat {A}$ preserves each finite-dimensional subcomplex $C_{\leqslant 0,\Delta }$. In particular, $\hat {A}$ is locally finite on $C_{\leqslant 0}$. Therefore, we can apply the construction of Section 2 to obtain the automorphism

(26)$$ \begin{align} \varphi_{t}\colon C_{\leqslant 0}{\;\stackrel{_{\sim}}{\to}\;} C_{\leqslant 0},\quad c\mapsto \tilde{c} \end{align} $$

of vertex algebras.

Note that $\tilde {A}\equiv A\pmod {C[>0]}$ and the operator $A_{(0)}$ acts semisimply on the whole complex C so that $C=\bigoplus _{\lambda \in \frac {1}{2}\mathbb {Z}}C[\lambda ]$, $C[\lambda ]=\{c\in C\mid A_{(0)}c=\lambda c\}$. Since $\varphi _{t}$ is an automorphism of the vertex algebra, $\tilde {A}\in \varphi _{t}(\mathbb {L}_{\ell }(\mathfrak {g}))$ generates a Heisenberg vertex subalgebra. In particular, $\tilde {A}_{(0)}$ acts semisimply on C. Therefore, by replacing $\hat {A}$ by $\tilde {A}$ (see Lemma 2.2), we can extend the automorphism (26) to the automorphism

(27)$$ \begin{align} C{\;\stackrel{_{\sim}}{\to}\;} C\quad c\mapsto \tilde{c}, \end{align} $$

which is also denoted by $\varphi _{t}$.

Proposition 4.2. We have $\varphi _{t}(Q_{t=0})_{(0)}=( Q_{t})_{(0)}$ on C. In particular, $\varphi _{t}$ defines an isomorphism

$$ \begin{align*} (C,(Q_{t=0})_{(0)})\cong (C,(Q_{t})_{(0)}) \end{align*} $$

of differential graded vertex algebras.

By Proposition 4.2, $\varphi :=\varphi _{t=1}$ defines an isomorphism

(28)$$ \begin{align} (C,(Q_{t=0})_{(0)})\cong (C,(Q_{t=1})_{(0)}) \end{align} $$

of differential graded vertex algebras. We have shown the following assertion.

Theorem 4.3. The automorphism $\varphi $ induces an isomorphism $H^{\bullet }(C,(Q_{t=0})_{(0)})\cong H^{\bullet }(C,(Q_{t=1})_{(0)})$. In particular,

$$ \begin{align*} \mathscr{W}^{k}(\mathfrak{g},f){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})\cong H_{DS,f}^{0}(V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})) \end{align*} $$

as vertex algebras.

Example 4.4. Let $\mathfrak {g}=\mathfrak {sl}_{2}=\operatorname {span}_{\mathbb {C}}\{e,h,f\}$ and $\ell =1$, and consider the Drinfeld–Sokolov reduction associated with f. Then $C=V^{k}(\mathfrak {g}){\otimes } \mathbb {L}_{1}(\mathfrak {g}){\otimes } \bigwedge \nolimits ^{\infty /2+\bullet }(\mathfrak {g}_{>0})$, and $\bigwedge \nolimits ^{\infty /2+\bullet }(\mathfrak {g}_{>0})$ is generated by odd fields $\psi (z)=\psi _{\alpha }(z)$, $\psi ^{*}(z) =\psi _{\alpha }^{*}(z)$. We have

$$ \begin{align*}Q_{t}(z)=(e_{1}(z)+t^{2} e_{2}(z))\psi^{*}(z)+\psi^{*}(z),\end{align*} $$

where we have set $x_{i}=\pi _{i}(x)$ for $x\in \mathfrak {g}$, $i=1,2$. The vertex subalgebra $C_{\leqslant 0}$ is generated by $J^{f_{1}}(z)=f_{1}(z)$, $J^{h_{1}}(z)=h_{1}(z)+2:\!\psi (z)\psi ^{*}(z)\!:$, $e_{2}(z)$, $h_{2}(z)$, $f_{2}(z)$ and $\psi ^{*}(z)$. We have $A=h_{2}/2$, and

$$ \begin{align*} \hat{A}(z)=h_{2}(z) + t^{2} J^{h_{1}}(z)e_{2}(z). \end{align*} $$

We find that the isomorphism

$$ \begin{align*} \varphi_{t}\colon V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{1}(\mathfrak{g}){\otimes} \bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0}){\;\stackrel{_{\sim}}{\to}\;} V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{1}(\mathfrak{g}) {\otimes} \bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0}) \end{align*} $$

is given by

$$ \begin{align*} &e_{1}(z)\mapsto e_{1}(z)(1-t^{2} e_{2}(z)), \\ &h_{1}(z)\mapsto h_{1}(z)-t^{2} k\partial_{z} e_{2}(z),\\ &f_{1}(z)\mapsto f_{1}(z)(1+t^{2} e_{2}(z)),\\ &e_{2}(z)\mapsto e_{2}(z),\\ &h_{2}(z)\mapsto h_{2}(z)+t^{2}J^{h_{1}}(z)e_{2}(z),\\ &f_{2}(z)\mapsto f_{2}(z) -\frac{t^{2}}{2}J^{h_{1}}(z)h_{2}(z) + \partial_{z}J {}^{h_{1}}(z)-\frac{t^{4}}{4} :\!e_{2}(z) J^{h_{1}}(z)^{2}\!:, \\ &\psi(z)\mapsto \psi(z)(1-t^{2} e_{2}(z)),\\ &\psi^{*}(z)\mapsto \psi^{*}(z)(1+t^{2}e_{2}(z)). \end{align*} $$

Note that we have $(1-t^{2} e_{2}(z))(1+t^{2} e_{2}(z))=1$ on $\mathbb {L}_{1}(\mathfrak {g})$.

5 Remarks on superalgebras

We restricted our attention to vertex algebras, and here we remark that the results also hold for vertex superalgebras, i.e.,

First, if we take $\mathfrak {g} = \mathfrak {osp}_{1|2n}$ and a positive integer $\ell $, then $\mathbb {L}_{\ell }(\mathfrak {g})$ is rational [Reference Creutzig and LinshawCL21, Thm. 7.1]. Hence, Theorem 1 holds for $\mathfrak {g} = \mathfrak {osp}_{1|2n}$.

Unfortunately, except for the $\mathfrak {g} = \mathfrak {osp}_{1|2n}$ cases, there are only few integrable representations of $\widehat {\mathfrak {g}}$ [Reference Kac and WakimotoKW94, Theorem 8.1]. For this reason, the notion of principal integrable representations and subprincipal integrable representations of $\widehat {\mathfrak {g}}$ was introduced in [Reference Kac and WakimotoKW01], which are integrable modules with respect to a certain affine vertex subalgebra, call it $\mathbb {L}_{\ell }(\mathfrak a)$, with $\mathfrak {a}\subset \mathfrak {g}_{\bar 0}$, where $\mathfrak {g}_{\bar 0}$ is the even part of $\mathfrak {g}$.

Theorem 5.3. Let $\mathfrak {g}$ be a basic classical Lie superalgebra with $\mathfrak {g}_{\bar 0}=\mathfrak {a}\mathop {\oplus } \mathfrak {a^{\prime }}$, $\mathfrak {a}$ semisimple, $\mathfrak {a}^{\prime }$ reductive. Suppose that $\mathbb {L}_{\ell }(\mathfrak {g})$ is integrable as a representation of $\widehat {\mathfrak {a}}\subset \widehat {\mathfrak {g}}$, and f is a nilpotent element in $\mathfrak {a}$. For a quotient V of $V^{k}(\mathfrak {g})$, we have a vertex algebra isomorphism

(29)$$ \begin{align} H_{DS,f}^{0}(V\otimes \mathbb{L}_{\ell}(\mathfrak{g}))\cong H_{DS,f}^{0}(V)\otimes \mathbb{L}_{\ell}(\mathfrak{g}), \end{align} $$

where in the left-hand side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } \mathbb {L}_{\ell }(\mathfrak {g})$. More generally, let V be a vertex algebra equipped with a vertex algebra homomorphism $V^{k}(\mathfrak {g})\rightarrow V$. Then we have an isomorphism

(30)$$ \begin{align} H_{DS,f}^{\bullet}(V\otimes \mathbb{L}_{\ell}(\mathfrak{g}))\cong H_{DS,f}^{\bullet}(V)\otimes \mathbb{L}_{\ell}(\mathfrak{g}) \end{align} $$

of graded vertex algebras, and for any V-module M, $ \mathbb {L}_{\ell }(\mathfrak {g})$-module N, there is an isomorphism

$$ \begin{align*} H_{DS,f}^{\bullet}(M{\otimes}N)\cong H_{DS,f}^{\bullet}(M){\otimes} N, \end{align*} $$

as modules over $H_{DS,f}^{\bullet }(V\otimes \mathbb {L}_{\ell }(\mathfrak {g}))\cong H_{DS,f}^{\bullet }(V)\otimes \mathbb {L}_{\ell }(\mathfrak {g})$.

6 Urod conformal vector

In this section, we assume that k is not critical, and discuss how the conformal structure match under the isomorphism

$$ \begin{align*} \varphi\colon \mathscr{W}^{k}(\mathfrak{g},f){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})\cong H_{DS,f}^{0}(V^{k}(\mathfrak{g}){\otimes} \mathbb{L}_{\ell}(\mathfrak{g})) \end{align*} $$

of vertex algebras.

On the right-hand side, the conformal vector of $H_{DS,f}^{0}(V^{k}(\mathfrak {g}){\otimes } \mathbb {L}_{\ell }(\mathfrak {g}))$ is given by

$$ \begin{align*} \omega_{total}=\omega_{V^{k}(\mathfrak{g})}+\omega_{\mathbb{L}_{\ell}(\mathfrak{g})}+ \omega_{F_{\chi}}+\omega_{\bigwedge\nolimits^{\infty/2+\bullet}(\mathfrak{g}_{>0})}+T \pi_{1}(x_{0})+T\pi_{2}(x_{0}), \end{align*} $$

where $\omega _{V^{k}(\mathfrak {g})}$ is the Sugawara conformal vector of $V^{k}(\mathfrak {g})$. It has the central charge

(31)$$ \begin{align} \frac{k\dim \mathfrak{g}}{k+h^{\vee}}& +\frac{\ell\dim \mathfrak{g}}{\ell+h^{\vee}} -\frac{(k+\ell)\dim \mathfrak{g}}{k+\ell+h^{\vee}}\\ &+ \dim \mathfrak{g}_{0}-\frac{1}{2}\dim \mathfrak{g}_{1/2}-12|\frac{\rho}{\sqrt{k +\ell+h^{\vee}}}-\sqrt{k+\ell+h^{\vee}}x_{0}|^{2}. \nonumber \end{align} $$

Clearly, $\varphi ^{-1}(\omega _{total})$ is a conformal vector of $\mathscr {W}^{k}(\mathfrak {g},f){\otimes } \mathbb {L}_{\ell }(\mathfrak {g})$. Set

(32)$$ \begin{align} \omega_{Urod}=\varphi^{-1}(\omega^{total})-\omega_{\mathscr{W}^{k}(\mathfrak{g},f)}, \end{align} $$

where $\omega _{\mathscr {W}^{k}(\mathfrak {g},f)}$ is the conformal vector of $\mathscr {W}^{k}(\mathfrak {g},f)$. Since it commutes with $\omega _{\mathscr {W}^{k}(\mathfrak {g},f)}$, $\omega _{Urod}$ defines a conformal vector of $\mathbb {L}_{\ell }(\mathfrak {g})$ ([Reference Lepowsky and LiLL04, 3.11]), which is called the Urod conformal vector of $\mathbb {L}_{\ell }(\mathfrak {g})$. It depends on the choice of $x_{0}$ and k and has the central charge

(33)$$ \begin{align} \frac{k\dim \mathfrak{g}}{k+h^{\vee}}& +\frac{\ell\dim \mathfrak{g}}{\ell+h^{\vee}} -\frac{(k+\ell)\dim \mathfrak{g}}{k+\ell+h^{\vee}}\\ + 12&\left(|\frac{\rho}{\sqrt{k +h^{\vee}}}-\sqrt{k+h^{\vee}}x_{0}|^{2} -|\frac{\rho}{\sqrt{k +\ell+h^{\vee}}}-\sqrt{k+\ell+h^{\vee}}x_{0}|^{2}\right). \nonumber \end{align} $$

In the case that $\mathfrak {g}$ is simply laced and $f=f_{prin}$, the central charge (33) becomes

(34)$$ \begin{align} -\frac{\ell(\ell h+h^{2}-1)\dim \mathfrak{g}}{\ell+h}, \end{align} $$

where h is the Coxeter number, and we have used the strange formula $|\rho |^{2}/2h^{\vee }=\dim \mathfrak {g}/24$, and so it does not depend on the parameter k.

By definition, the conformal vertex algebra $\mathscr {W}^{k}(\mathfrak {g},f){\otimes } \mathbb {L}_{\ell }(\mathfrak {g})$ with the conformal vector $\omega _{\mathscr {W}^{k}(\mathfrak {g},f)}+\omega _{Urod}$ is isomorphic to $H_{DS,f}^{0}(V^{k}(\mathfrak {g}){\otimes } \mathbb {L}_{\ell }(\mathfrak {g})) $ as conformal vertex algebras.

Lemma 6.1. The Hamiltonian of $\mathbb {L}_{\ell }(\mathfrak {g})$ defined by $\omega _{Urod}$ coincides with

$$ \begin{align*}H_{Urod}:=(\omega_{\mathbb{L}_{\ell}(\mathfrak{g})})_{(1)}-(x_{0})_{(0)}.\end{align*} $$

Example 6.2 Continued from Example 4.4

The Urod conformal vector of $\mathbb {L}_{1}(\mathfrak {g})$ is given by

$$ \begin{align*} \omega_{Urod}=\omega_{\mathbb{L}_{1}(\mathfrak{g})}+T h/2-(k+1)T^{2} e/2, \end{align*} $$

which agrees with [Reference Bershtein, Feigin and LitvinovBFL16].

7 Compatibility with twists

In this section, we show that the various twisted Drinfeld–Sokolov reduction functors commute with tensoring integrable representations as well.

For ${\check \mu } \in \check {P}$, we define a character $\hat {\chi }_{\check \mu }$ of $\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ by the formula

(35)$$ \begin{align} \hat{\chi}_{\check\mu}(x_{\alpha} f(t)) = \chi(x_{\alpha}) \cdot \operatorname{Res}_{t=0} f(t) t^{{\langle} {\check\mu},\alpha {\rangle}} dt, \qquad f(t) \in \mathbb{C}[t,t^{-1}]. \end{align} $$

Define

$$ \begin{align*}F_{\chi,{\check\mu}}=U(\mathfrak{g}_{>0}[t,t^{-1}])\otimes_{U(\mathfrak{g}_{>0}[t]+\mathfrak{g}_{\geqslant 1}[t,t^{-1}])}\mathbb{C}_{\hat\chi_{{\check\mu}}},\end{align*} $$

where $\mathbb {C}_{\hat \chi _{{\check \mu }}}$ is the one-dimensional representation of $\mathfrak {g}_{>0}[t]+\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ on which $\mathfrak {g}_{\geqslant 1}[t,t^{-1}]$ acts by the character $\hat \chi _{{\check \mu }}$ and $\mathfrak {g}_{>0}[t]$ acts trivially. Then $F_{\chi ,{\check \mu }}$ is naturally a $V(\mathfrak {g}_{>0})$-module, and we denote by $\Phi _{\alpha }^{{\check \mu }}(z)$ the image of $x_{\alpha }(z)$ in $(\operatorname {\mathrm {End}} F_{\chi ,{\check \mu }}))[z,z^{-1}]$. We have

$$ \begin{align*} &\Phi_{\alpha}^{{\check\mu}}(z)=z^{{\langle} {\check\mu},\alpha{\rangle}}\chi(x_{\alpha}) \quad \text{for }\alpha\in \Delta_{>0},\\ &\Phi_{\alpha}^{{\check\mu}}(z)\Phi_{\beta}^{{\check\mu}}(w)\sim \frac{w^{{\langle} {\check\mu},\alpha+\beta{\rangle}} \chi([x_{\alpha},x_{\beta}])}{z-w.}\end{align*} $$

For a smooth $\widehat {\mathfrak {g}}$-module M of level k, we define

(36)$$ \begin{align} H_{DS,f,{\check\mu}}^{\bullet}(M)=H^{\infty/2+\bullet}(\mathfrak{g}_{>0}[t,t^{-1}], M{\otimes} F_{\chi,{\check\mu}}), \end{align} $$

where $\mathfrak {g}_{>0}[t,t^{-1}]$ acts diagonally on $M{\otimes } F_{\chi ,{\check \mu }}$. By definition, $H_{DS,f,{\check \mu }}^{\bullet }(M)$ is the cohomology of the complex $(C_{{\check \mu }}(M), (Q_{{\check \mu }})_{(0)})$, where $C_{{\check \mu }}(M)=M{\otimes } F_{\chi ,{\check \mu }}{\otimes }\bigwedge \nolimits ^{\infty /2+\bullet }(\mathfrak {g}_{>0}) $ and $(Q_{{\check \mu }})_{(0)}$ is the zero mode of the field

$$ \begin{align*} Q_{{\check\mu}}(z)=\sum_{\alpha\in \Delta_{>0}}x_{\alpha}(z)\psi_{\alpha}^{*}(z) +\sum_{\alpha\in \Delta_{>0}}\Phi_{\alpha}^{{\check\mu}}(z)\psi_{\alpha}^{*}(z) -\frac{1}{2} \sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^{\gamma} \psi_{\alpha}^{*}(z)\psi_{\beta}^{*}(z)\psi_{\gamma}(z), \end{align*} $$

on $C_{{\check \mu }}(M)$.

We define the structure of a $\mathscr {W}^{k}(\mathfrak {g},f)$-module on $H_{DS,f,\check {\mu }}^{i}(M), i \in \mathbb {Z}$, as follows. Let $\Delta (J^{\{{\check \mu }\}},z)$ be Li’s delta operator corresponding to the field

(37)$$ \begin{align} J^{\{{\check\mu}\}}(z):={\check\mu}(z) + \sum_{\alpha \in \Delta_{>0}} {\langle} \alpha,{\check\mu} {\rangle} :\! \psi_{\alpha}(z) \psi^{*}_{\alpha}(z) \! : -\frac{1}{2}\sum_{\alpha \in \Delta_{1/2}}{\langle} \alpha,{\check\mu}{\rangle} :\! \Phi_{\alpha}(z) \Phi^{\alpha}(z) \! : \end{align} $$

in $C(V^{k}(\mathfrak {g}))$, where $\Phi ^{\alpha }(z)$ is a linear sum of $\Phi ^{\beta }(z)$ corresponding to the vector $x^{\alpha }$ dual to $x_{\alpha }$ with respect to the symplectic form (13), that is, $(f|[x_{\alpha },x^{\beta }])=\delta _{\alpha ,\beta }$. We have

$$ \begin{align*} J^{\{{\check\mu}\}}(z)x_{\alpha}(w)\sim \frac{{\langle} \alpha,{\check\mu} {\rangle} }{z-w}x_{\alpha}(w), \quad J^{\{{\check\mu}\}}(z)\Phi_{\alpha}(w)\sim \frac{{\langle} \alpha,{\check\mu} {\rangle} }{z-w}\Phi_{\alpha}(w),\\ J^{\{{\check\mu}\}}(z)\psi_{\alpha}(w)\sim \frac{{\langle} \alpha,{\check\mu} {\rangle} }{z-w}\psi_{\alpha}(w), \quad J^{\{{\check\mu}\}}(w)\psi^{*}_{\alpha}(z)\sim -\frac{{\langle} \alpha,{\check\mu} {\rangle} }{z-w}\psi^{*}_{\alpha}(w); \end{align*} $$

see [Reference Kac, Roan and WakimotoKRW03]. For any smooth $\widehat {\mathfrak {g}}$-module M, we can twist the action of $C(V^{k}(\mathfrak {g}))$ on $C(M)$ by the correspondence

(38)$$ \begin{align} \sigma_{\check\mu} : Y(A,z) \mapsto Y_{C(M)}(\Delta(J^{\{{\check\mu}\}},z)A,z). \end{align} $$

for any $A\in C(V^{\kappa }(\mathfrak {g}))$. Since we have

$$ \begin{align*} & \sigma_{\check\mu}(x_{\alpha}(z))=z^{- {\langle} \alpha,{\check\mu} {\rangle} }x_{\alpha}(z),\quad \sigma_{\check\mu}(\Phi_{\alpha}(z))=z^{- {\langle} \alpha,{\check\mu} {\rangle} }\Phi_{\alpha}(z),\\ & \sigma_{\check\mu}(\psi_{\alpha}(z))=z^{- {\langle} \alpha,{\check\mu} {\rangle} }\psi_{\alpha}(z), \quad \sigma_{\check\mu}(\psi^{*}_{\alpha}(z))=z^{{\langle} \alpha,{\check\mu} {\rangle} }\psi^{*}_{\alpha}(z), \end{align*} $$

it follows that the resulting complex $(C(M),\sigma _{\check \mu }(Q_{(0)}))$ is naturally identified with $(C_{{\check \mu }}(M), (Q_{{\check \mu }})_{(0)})$. Therefore, $(C_{{\check \mu }}(M), (Q_{{\check \mu }})_{(0)})$ has the structure of a differential graded module over the differential vertex algebra $(C(V^{k}(\mathfrak {g})),Q_{(0)}))$, and hence, each cohomology $H_{DS,f,\check {\mu }}^{i}(M), i \in \mathbb {Z}$, is a module over $\mathscr {W}^{k}(\mathfrak {g},f)=H_{DS,f}^{0}(M)$.

The functor $H_{DS,f,\check {\mu }}^{0}(?)$ was introduced in [Reference Arakawa and FrenkelAF19] in the case that f is a principal nilpotent element.

Let V be vertex algebra equipped with a vertex algebra homomorphism $V^{k}(\mathfrak {g})\rightarrow V$, and let M be a V-module. Then the same construction as above gives $H_{DS,f,\check {\mu }}^{i}(M)$ a structure of a module over the vertex algebra $H_{DS,f}^{0}(V)$.

Theorem 7.1. Let V be a quotient of the universal affine vertex algebra $V^{k}(\mathfrak {g})$, $\ell \in \mathbb {Z}_{\geqslant 0}$, and let ${\check \mu } \in \check {P}$. For any V-module M, $\mathbb {L}_{\ell }(\mathfrak {g})$-module N and $i\in \mathbb {Z}$, there is an isomorphism

$$ \begin{align*} H_{DS,f,{\check\mu}}^{i}(M{\otimes}N)\cong H_{DS,f,{\check\mu}}^{i}(M){\otimes} \sigma_{\check\mu}^{*} N \end{align*} $$

as modules over $H_{DS,f}^{0}(V\otimes \mathbb {L}_{\ell }(\mathfrak {g}))\cong H_{DS,f}^{0}(V)\otimes \mathbb {L}_{\ell }(\mathfrak {g})$, where $\sigma _{\check \mu }^{*} N$ is the twist of N on which $A\in \mathbb {L}_{\ell }(\mathfrak {g})$ acts as $\sigma _{{\check \mu }}(A(z))$.

More generally, let

$$ \begin{align*}w=(y,{\check\mu})\end{align*} $$

be an element of the extented affine Weyl group $\tilde {W}=W\ltimes P^{\vee }$, where W is the Weyl group of $\mathfrak {g}$, and let $\tilde {y}$ be a Tits lifting of y to and automorphism of $\mathfrak {g}$ so that $\tilde {y}(x_{\alpha })=c_{\alpha }x_{y(\alpha )}$ for some $c_{\alpha }\in \mathbb {C}^{*}$. Then

$$ \begin{align*} \sigma_{w}\colon x_{\alpha}t^{n}\mapsto \tilde{y}(x_{\alpha})t^{n-{\langle} \alpha,{\check\mu}{\rangle}} \end{align*} $$

defines a Tits lifting of w to an automorphism of $\widehat {\mathfrak {g}}$. Set

$$ \begin{align*}F_{\chi,w}=U(\tilde{w}(\mathfrak{g}_{>0}[t,t^{-1}]))\otimes_{U(\tilde{w}(\mathfrak{g}_{>0}[t]+\mathfrak{g}_{\geqslant 1}[t,t^{-1}]))}\mathbb{C}_{\hat\chi_{{\check\mu}}},\end{align*} $$

where $\mathbb {C}_{\hat \chi _{{\check \mu }}}$ is the one-dimensional representation of $\tilde {w}(\mathfrak {g}_{>0}[t]+\mathfrak {g}_{\geqslant 1}[t,t^{-1}])$ on which $\tilde {w}(\mathfrak {g}_{\geqslant 1}[t,t^{-1}])$ acts by the character $\hat \chi _{w}: x_{\alpha }t^{n}\mapsto ( x_{\alpha }t^{n}|\tilde {w}(ft^{-1})) $ and $ \tilde {w}(\mathfrak {g}_{>0}[t])$ acts trivially. Then

(39)$$ \begin{align} H_{DS,f,w}^{i}(M)=H^{\infty/2+i}(y(\mathfrak{g}_{>0})[t,t^{-1}], M{\otimes} F_{\chi,w}) \end{align} $$

is equipped with a $\mathscr {W}^{k}(\mathfrak {g},f)$-module structure.

The proof of the following assertion is the same as that of Theorem 7.1.

Theorem 7.2. Let V be a quotient of the universal affine vertex algebra $V^{k}(\mathfrak {g})$, $\ell \in \mathbb {Z}_{\geqslant 0}$, and let $w\in \tilde {W}$. For any V-module M, $\mathbb {L}_{\ell }(\mathfrak {g})$-module N, and $i\in \mathbb {Z}$, there is an isomorphism

$$ \begin{align*} H_{DS,f,w}^{i}(M{\otimes}N)\cong H_{DS,f,w}^{i}(M){\otimes} \sigma_{w}^{*} N \end{align*} $$

as modules over $H_{DS,f}^{0}(V\otimes \mathbb {L}_{\ell }(\mathfrak {g}))\cong H_{DS,f}^{0}(V)\otimes \mathbb {L}_{\ell }(\mathfrak {g})$.

In the above, $\sigma _{w}^{*} N$ is the twist of N on which $A\in \mathbb {L}_{\ell }(\mathfrak {g})$ acts as $\sigma _{w}(A(z))$, which is again an integrable representation of $\widehat {\mathfrak {g}}$ of level $\ell $.

Suppose that the grading (11) is even, that is, $\mathfrak {g}_{j}=0$ unless $j\in \mathbb {Z}$. Then $x_{0}\in P^{\vee }$. For a smooth $\widehat {\mathfrak {g}}$-module M of level k, “$-$”-Drinfeld–Sokolov reduction $H_{DS,f,-}^{\bullet }(M)$ ([Reference Frenkel, Kac and WakimotoFKW92, Reference ArakawaAra11]) is nothing but the twisted reduction for $w=(w_{0},-x_{0})$, where $w_{0}$ is the longest element of W:

(40)$$ \begin{align} H_{DS,f,-}^{\bullet}(M)=H_{DS,f,(w_{0},-x_{0})}^{\bullet}(M) \end{align} $$

Corollary 7.3. Let V be a quotient of the universal affine vertex algebra $V^{k}(\mathfrak {g})$, $\ell \in \mathbb {Z}_{\geqslant 0}$. Suppose that the grading (11) is even. For any V-module M, $\mathbb {L}_{\ell }(\mathfrak {g})$-module N and $i\in \mathbb {Z}$, there is an isomorphism

$$ \begin{align*} H_{DS,f,-}^{i}(M{\otimes}N)\cong H_{DS,f,-}^{i}(M){\otimes} \sigma_{(w_{0},-x_{0})}^{*} N \end{align*} $$

as modules over $H_{DS,f}^{0}(V\otimes \mathbb {L}_{\ell }(\mathfrak {g}))\cong H_{DS,f}^{0}(V)\otimes \mathbb {L}_{\ell }(\mathfrak {g})$.

8 Higher-rank Urod algebras

In the case that f is a principal nilpotent element $f_{prin}$, we denote by $\mathscr {W}^{k}(\mathfrak {g})$ the principal W-algebra $\mathscr {W}^{k}(\mathfrak {g},f_{prin})$ and by $\mathscr {W}_{k}(\mathfrak {g})$ the unique simple quotient of $\mathscr {W}^{k}(\mathfrak {g})$. We have the Feigin–Frenkel duality ([Reference Feigin and FrenkelFF91, Reference Aganagic, Frenkel and OkounkovAFO18]; see also [Reference Arakawa, Creutzig and LinshawACL19]) which states that

(41)$$ \begin{align} \mathscr{W}^{k}(\mathfrak{g})\cong \mathscr{W}^{\check{k}}({}^{L}\mathfrak{g}), \end{align} $$

where ${}^{L}\mathfrak {g}$ is the Langlands dual Lie algebra and $\check {k}$ is defined by the formula $r^{\vee }(k+h^{\vee })(\check {k}+{}^{L}h^{\vee })=1$. Here, $r^{\vee }$ is the lacety of $\mathfrak {g}$, and ${}^{L}h^{\vee }$ is the dual Coxeter number of ${}^{L}\mathfrak {g}$.

Suppose that $\mathfrak {g}$ is simply laced and $k+h^{\vee }-1\not \in \mathbb {Q}_{\leqslant 0}$. By [Reference Arakawa, Creutzig and LinshawACL19], we have

$$ \begin{align*} \mathscr{W}^{\ell}(\mathfrak{g})\cong (V^{k-1}(\mathfrak{g}){\otimes} \mathbb{L}_{1}(\mathfrak{g}))^{\mathfrak{g}[t]}; \end{align*} $$

that is, $\mathscr {W}^{\ell }(\mathfrak {g})$ is isomorphic to the commutant of $V^{k}(\mathfrak {g})$ in $V^{k-1}(\mathfrak {g}){\otimes } \mathbb {L}_{1}(\mathfrak {g})$, where $V^{k}(\mathfrak {g})$ is considered as a vertex subalgebra of $V^{k-1}(\mathfrak {g}){\otimes } \mathbb {L}_{1}(\mathfrak {g})$ by the diagonal embedding and $\ell $ is defined by equation (6). In other words, we have a conformal vertex algebra embedding

(42)$$ \begin{align} V^{k}(\mathfrak{g}){\otimes} \mathscr{W}^{\ell}(\mathfrak{g})\hookrightarrow V^{k-1}(\mathfrak{g}){\otimes} \mathbb{L}_{1}(\mathfrak{g}). \end{align} $$

By Main Theorem 1, (42) induces the conformal vertex algebra homomorphism

(43)$$ \begin{align} \mathscr{W}^{k}(\mathfrak{g},f){\otimes}\mathscr{W}^{\ell}(\mathfrak{g})\rightarrow H_{DS,f}^{0}(V^{k-1}(\mathfrak{g}){\otimes} \mathbb{L}_{1}(\mathfrak{g})) \overset{\varphi^{-1}}{{\;\stackrel{_{\sim}}{\to}\;}} \mathscr{W}^{k-1}(\mathfrak{g},f){\otimes} \mathcal{U}(\mathfrak{g},f), \end{align} $$

which is again embedding by Theorem 3.1. Here, $\mathcal {U}(\mathfrak {g},f)$ is the vertex algebra $\mathbb {L}_{1}(\mathfrak {g})$ equipped with the Urod conformal vector $\omega _{Urod}$, which we call the Urod algebra associated with $(\mathfrak {g},f)$. We set

$$ \begin{align*}\mathcal{U}(\mathfrak{g})=\mathcal{U}(\mathfrak{g},f_{prin}).\end{align*} $$

8.1 Generic decompositions

In this subsection, we assume that k is irrational.

For $(\lambda ,{\check \mu })\in P_{+}\times \check {P}_{+}$, define

$$ \begin{align*} T_{\lambda,{\check\mu}}^{k}=H_{DS,f_{prin},{\check\mu}}^{0}(\mathbb{V}^{k}_{\lambda})\in \mathscr{W}^{k}(\mathfrak{g})\operatorname{-Mod}. \end{align*} $$

It was shown in [Reference Arakawa and FrenkelAF19] that the $\mathscr {W}^{k}(\mathfrak {g})$-modules $T_{\lambda ,{\check \mu }}^{k}$ are simple, and the isomorphism

(44)$$ \begin{align} T_{\lambda,{\check\mu}}^{k}\cong T_{{\check\mu},\lambda}^{\check{k}} \end{align} $$

holds under the Feigin–Frenkel duality (41).

Let $\ell $ be nonnegative integer. Then $\{\mathbb {L}^{\ell }_{\lambda }\mid \lambda \in P^{\ell }_{+}\}$ gives the complete set of isomorphism classes of simple $\mathbb {L}_{\ell }(\mathfrak {g})$-modules, where

$$ \begin{align*}P^{\ell}_{+}=\{\lambda\in P_{+}\mid {\langle} \lambda,\theta^{\vee}{\rangle} \leqslant \ell\}\subset P_{+}\end{align*} $$

is the set of integrable dominant weight of $\mathfrak {g}$ of level $\ell $. In particular, $\{\mathbb {L}^{1}_{\lambda }\mid \lambda \in P^{1}_{+}\}$ gives the complete set of isomorphism classes of simple $\mathcal {U}(\mathfrak {g},f)$-modules.

Let

(45)$$ \begin{align} \mathbb{V}^{k}_{\mu,f}=H_{DS,f}^{0}(\mathbb{V}^{k}_{\mu})\in \mathscr{W}^{k}(\mathfrak{g},f)\operatorname{-Mod}. \end{align} $$

Theorem 8.1. Let $\mathfrak {g}$ be simply laced, and let f be any nilpotent element of $\mathfrak {g}$. For $\lambda ,\mu \in P_{+}$ and $\nu \in P_{+}^{1}$, we have

$$ \begin{align*} \mathbb{V}^{k-1}_{\mu,f}{\otimes} \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\substack{\lambda\in P_{+}\\ \lambda-\mu-\nu\in Q}}\mathbb{V}^{k}_{\lambda,f}{\otimes} T_{\lambda, \mu}^{\ell} \end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g},f){\otimes }\mathscr {W}^{\ell }(\mathfrak {g})$-modules (see equation (43)).

In the case $f=f_{prin}$, we have the following more general statement.

Theorem 8.2. Let $\mathfrak {g}$ is simply laced. For $\lambda ,\mu ,\mu ^{\prime }\in P_{+}$ and $\nu \in P_{+}^{1}$, we have

(46)$$ \begin{align} T^{k-1}_{\mu,\mu^{\prime}}{\otimes} \mathbb{L}^{1}_{\nu} =\bigoplus_{\substack{\lambda\in P_{+}\\ \lambda-\mu-\mu^{\prime}-\nu\in Q}}T^{k}_{\lambda,\mu^{\prime}}{\otimes} T_{\lambda, \mu}^{\ell} \end{align} $$

as $\mathscr {W}^{k}(\mathfrak {g}){\otimes }\mathscr {W}^{\ell }(\mathfrak {g})$-modules.

Note that Theorem 8.2 is compatible with equation (44) since $\check {(k-1)}=\ell -1$.

Proof of Theorem 8.1 and Theorem 8.2.

Let $\lambda ,\mu \in P_{+}$ and $\nu \in P_{+}^{1}$. By [Reference Arakawa, Creutzig and LinshawACL19, Main Theorem 3], we have

(47)$$ \begin{align} \mathbb{V}^{k-1}_{\mu}{\otimes} \mathbb{L}^{1}_{\nu}=\bigoplus_{\substack{\lambda\in P_{+}\\ \lambda-\mu-\nu\in Q}} \mathbb{V}^{k}_{\lambda}{\otimes} T^{\ell}_{\lambda,\mu} \end{align} $$

as $V^{k+1}(\mathfrak {g}){\otimes }\mathscr {W}^{\ell }(\mathfrak {g})$-modules. Applying Main Theorem 1 to $\mathbb {V}^{k-1}_{\mu }{\otimes } \mathbb {L}^{1}_{\nu }$, we obtain Theorem 8.1. Next, under the identification $P/Q\cong P^{1}_{+}$, we have $\sigma _{\mu ^{\prime }}^{*}\mathbb {L}^{1}_{\nu +\mu ^{\prime }+Q}\cong \mathbb {L}^{1}_{\nu +Q}$. Hence, we obtain Theorem 8.2 by applying Theorem 7.1.

Let $\pi ^{k}$ be the Heisenberg vertex subalgebra of $V^{k}(\mathfrak {g})$ generated by $h(z)$, $h\in \mathfrak {h}$, and let $\pi ^{k}_{\lambda }$ be the irreducible highest weight representation of $\pi ^{k}$ with highest weight $\lambda $. There is a vertex algebra embedding

$$ \begin{align*} \mathscr{W}^{k}(\mathfrak{g})\hookrightarrow \pi^{k+h^{\vee}} \end{align*} $$

called the Miura map ([Reference Feĭgin and FrenkelFF90b]; see also [Reference ArakawaAra17]), and hence, each $\pi ^{k+h^{\vee }}_{\lambda }$ is a $\mathscr {W}^{k}(\mathfrak {g})$-module.

Theorem 8.3. Let $\mathfrak {g}$ is simply laced, $\mu \in \mathfrak {h}^{*}$ be generic, $\nu \in P^{1}_{+}$. We have the isomorphism

$$ \begin{align*} \pi_{\mu}^{k-1+h^{\vee}}{\otimes} \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda \in \mathfrak{h}^{*} \atop \lambda-\mu-\nu\in Q} \pi^{k+h^{\vee}}_{\lambda}{\otimes} \pi^{\ell+h^{\vee}}_{\lambda-(\ell+h^{\vee})\mu} \end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g}){\otimes } \mathscr {W}^{\ell }(\mathfrak {g})$-modules.

For a generic $\lambda \in \mathfrak {h}^{*}$, the $\mathscr {W}^{k}(\mathfrak {g})$-module $\pi ^{k+h^{\vee }}_{\lambda }$ is irreducible and isomorphic to a Verma module $\mathbb {M}^{k}(\chi _{\lambda })$ with highest weight ([Reference Feigin and FrenkelFF90a, Reference FrenkelFre92, Reference ArakawaAra07]). Here, $\chi _{\lambda }:\operatorname {Zhu}(\mathscr {W}^{k}(\mathfrak {g}))\rightarrow \mathbb {C}$ is described in [Reference Arakawa, Creutzig and LinshawACL19, (27)], where $\operatorname {Zhu}(\mathscr {W}^{k}(\mathfrak {g}))$ is the Zhu algebra of $\mathscr {W}^{k}(\mathfrak {g})$. Hence, the following assertion immediately follows from Theorem 8.3.

Corollary 8.4. Let $\mathfrak {g}$ is simply laced, $\mu \in \mathfrak {h}^{*}$ be generic, $\nu \in P^{1}_{+}$. We have the isomorphism

$$ \begin{align*} \mathbb{M}^{k-1}(\chi_{\mu}){\otimes} \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda \in \mathfrak{h}^{*} \atop \lambda-\mu-\nu\in Q} \mathbb{M}^{k}(\chi_{\lambda}){\otimes}\mathbb{M}^{\ell}(\chi_{\lambda-(\ell+h^{\vee})\mu}) \end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g}){\otimes } \mathscr {W}^{\ell }(\mathfrak {g})$-modules.

Let $\mathbb {W}^{k}_{{\lambda }}$ be the Wakimoto module [Reference Feĭgin and FrenkelFF90b] of $\widehat {\mathfrak {g}}$ at level k with highest weight $\lambda \in \mathfrak {h}^{*}$. For $\lambda =0$, $\mathbb {W}^{k}:=\mathbb {W}^{k}_{0}$ is a vertex algebra, and we have an embedding $V^{k}(\mathfrak {g})\hookrightarrow \mathbb {W}^{k}$ of vertex algebras ([Reference Feĭgin and FrenkelFF90b, Reference FrenkelFre05]). We have $H_{DS,f_{prin}}^{i}( \mathbb {W}^{k}_{\lambda })\cong \delta _{i,0}\pi ^{k+h^{\vee }}_{\lambda }$, and the Miura map is by definition [Reference Feĭgin and FrenkelFF90b] the map $\mathscr {W}^{k}(\mathfrak {g})\rightarrow H_{DS,f_{prin}}^{0}( \mathbb {W}^{k})\cong \pi ^{k+h^{\vee }}$ induced by the embedding $V^{k}(\mathfrak {g})\hookrightarrow \mathbb {W}^{k}$.

Proof of Theorem 8.3.

Since $\nu $ is generic, $\mathbb {W}^{k-1}_{\mu }{\otimes } \mathbb {L}^{1}_{\nu }$ is a direct sum of irreducible Verma modules as a diagonal $\widehat {\mathfrak {g}}$-module or, equivalently, a direct sum of irreducible Wakimoto modules $\mathbb {W}^{k}_{\lambda }$. So we can write $\mathbb {W}^{k-1}_{\mu }{\otimes } \mathbb {L}^{1}_{\nu }=\bigoplus _{\lambda \in \mathfrak {h}^{*}}\mathbb {W}^{k}_{\lambda {\otimes }} m_{\mu ,\nu }^{\lambda }$, where $m_{\mu ,\nu }^{\lambda }$ be the multiplicity of $\mathbb {W}^{k}_{\lambda }$ in $\mathbb {W}^{k-1}_{\mu }{\otimes } \mathbb {L}^{1}_{\nu }$. Note that $m_{\mu ,\nu }^{\lambda }$ is a $\mathscr {W}^{\ell }(\mathfrak {g})$-module by equation (42). We have

$$ \begin{align*} m_{\mu,\nu}^{\lambda}\cong \operatorname{\mathrm{Hom}}_{\pi^{k+1+h^{\vee}}}(\pi^{k+1+h^{\vee}}_{\lambda}, H^{\infty/2+0}({\mathfrak n}[t,t^{-1}],\mathbb{W}^{k-1}_{\mu}{\otimes} \mathbb{L}^{1}_{\nu})\\ \cong \begin{cases} \pi^{\ell+h^{\vee}}_{\mu-(\ell+h^{\vee})\lambda}&\text{if }\lambda-\mu-\nu\in Q,\\ 0&\text{otherwise } \end{cases} \end{align*} $$

as $\mathscr {W}^{\ell }(\mathfrak {g})$-modules by [Reference Arakawa, Creutzig and LinshawACL19, Proposition 8.3]. The assertion follows by applying Main Theorem 1 to

(48)$$ \begin{align} \mathbb{W}^{k-1}_{\mu}{\otimes} \mathbb{L}^{1}_{\nu}=\bigoplus\limits_{\lambda\in \mathfrak{h}^{*} \atop \lambda-\mu-\nu\in Q}\mathbb{W}^{k}_{\lambda{\otimes}} \pi_{\mu-(\ell+h^{\vee})}. \end{align} $$

8.2 Decomposition at admissible levels

Let k be an admissible number for $\widehat {\mathfrak {g}}$, that is, $\mathbb {L}_{k}(\mathfrak {g})$ is admissible ([Reference Kac and WakimotoKW89]) as a $\widehat {\mathfrak {g}}$-module. In the case that $\mathfrak {g}$ is simply laced, this condition is equivalent to

(49)$$ \begin{align} k+h^{\vee}=\frac{p}{q},\quad p,q\in \mathbb{Z}_{\geqslant 1}, \ (p,q)=1,\ p\geqslant h^{\vee}. \end{align} $$

For an admissible number k, a simple module over $\mathbb {L}_{k}(\mathfrak {g})$ need not be ordinary, that is, in $\operatorname {KL}$, unless k is a nonnegative integer. The classification of simple highest weight representations of $\mathbb {L}_{k}(\mathfrak {g})$ was given in [Reference ArakawaAra16a]. For our purpose, we need only the ordinary representations of $\mathbb {L}_{k}(\mathfrak {g})$. By [Reference ArakawaAra16a], the complete set of isomorphism classes of ordinary simple $\mathbb {L}_{k}(\mathfrak {g})$-modules is given by

$$ \begin{align*} \{\mathbb{L}^{k}_{\lambda}\mid \lambda \in Adm_{\mathbb{Z}}^{k}\}, \end{align*} $$

where $Adm_{\mathbb {Z}}^{k}=\{\lambda \in P_{+}\mid \mathbb {L}^{k}_{\lambda }\text { is admissible}\}$. We have

(50)$$ \begin{align} Adm_{\mathbb{Z}}^{k}=P^{p-h^{\vee}}_{+} \end{align} $$

if $\mathfrak {g}$ is simply laced and k is of the form (49).

Let k be an admissible number and $\lambda \in Adm^{k}_{\mathbb {Z}}$. By [Reference ArakawaAra15a], the associated variety $X_{\mathbb {L}^{k}_{\lambda }}$ [Reference ArakawaAra12] of $\mathbb {L}^{k}_{\lambda }$ is the closure of some nilpotent orbit $\mathbb {O}_{k}$ which depends only on the denominator q of k. More explicitly, we have

(51)$$ \begin{align} X_{\mathbb{L}^{k}_{\lambda}} =\{x\in \mathfrak{g}\mid (\operatorname{\mathrm{ad}} x)^{2q}=0\} \end{align} $$

in the case that $\mathfrak {g}$ is simply laced. By [Reference ArakawaAra15a], we have

(52)$$ \begin{align} H_{DS,f}^{0}(\mathbb{L}^{k}_{\lambda})\ne 0 \quad \iff \quad f \in X_{\mathbb{L}^{k}_{\lambda}}=\overline{\mathbb{O}}_{k}. \end{align} $$

An admissible number k is called nondegenerate if $ X_{\mathbb {L}^{k}_{\lambda }}$ equals to the nilpotent cone $\mathcal {N}$ of $\mathfrak {g}$ for some $\lambda \in Adm^{k}_{\mathbb {Z}}$ or, equivalently, for all $Adm^{k}_{\mathbb {Z}}$. In the case that $\mathfrak {g}$ is simply laced, this happens if and only if the denominator q of k is equal or greater than $h^{\vee }$. If this is the case, the simple principal W-algebra $\mathscr {W}_{k}(\mathfrak {g})=\mathscr {W}_{k}(\mathfrak {g},f_{prin})$ is rational and lisse ([Reference ArakawaAra15a, Reference ArakawaAra15b]), and the complete set of the isomorphism classes of $\mathscr {W}_{k}(\mathfrak {g})$-module is given by

(53)$$ \begin{align} \{\mathbf{L}^{k}_{[\lambda,{\check\mu}]}\mid [\lambda,{\check\mu}]\in (Adm_{\mathbb{Z}}^{k}\times Adm_{\mathbb{Z}}^{\check{k}})/\tilde{W}_{+}\}, \end{align} $$

where

(54)$$ \begin{align} \mathbf{L}^{k}_{[\lambda,{\check\mu}]}:=H_{DS,f_{prin}}^{0}(\mathbb{L}^{k}_{\lambda-(k+h^{\vee}){\check\mu}}), \end{align} $$

and $\tilde {W}_{+}$ is the subgroup of the extended affine Weyl group of $\mathfrak {g}$ consisting of elements of length zero that acts on the set $Adm_{\mathbb {Z}}^{k}\times Adm_{\mathbb {Z}}^{\check {k}}$ diagonally. We have

(55)$$ \begin{align} \mathbf{L}^{k}_{[\lambda,{\check\mu}]}\cong \mathbf{L}^{\check{k}}_{[{\check\mu},\lambda]} \end{align} $$

under the Feigin–Frenkel duality.

The following assertion is new for nonzero ${\check \mu }$.

Theorem 8.5. Let k be a nondegenerate admissible number, and let $\lambda \in Adm_{\mathbb {Z}}^{k}$, ${\check \mu }\in Adm_{\mathbb {Z}}^{\check {k}}$. We have

$$ \begin{align*}H_{DS,f_{prin},{\check\mu}}^{i}(\mathbb{L}^{k}_{\lambda})\cong \begin{cases} \mathbf{L}^{k}_{[\lambda,{\check\mu}]}&\text{for }i=0,\\ 0&\text{for }i\ne 0 \end{cases}\end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g})$-modules.

Proof. The case ${\check \mu }=0$ has been proved in [Reference ArakawaAra04, Reference ArakawaAra07]. In particular,

(56)$$ \begin{align} H_{DS,f_{prin}}^{0}(\mathbb{L}_{k}(\mathfrak{g}))\cong \mathscr{W}_{k}(\mathfrak{g}). \end{align} $$

By [Reference Arakawa and FrenkelAF19, Theorem 2.1], we have $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {V}^{k}_{\lambda })=0$ for all $i\ne 0$, $\lambda \in P_{+}$. It follows that $H_{DS,f_{prin},{\check \mu }}^{i\ne 0}(M)=0$, $i\ne 0$, for any object M in $\operatorname {KL}$ that admits a Weyl flag. This implies that

(57)$$ \begin{align} H_{DS,f_{prin},{\check\mu}}^{i}(M)=0\quad \text{for }i>0,\ M\in \operatorname{KL}; \end{align} $$

see the proof of [Reference ArakawaAra04, Theorem 8.3].

By [Reference ArakawaAra14], the admissible representation $\mathbb {L}^{k}_{\lambda }$ admits a two-sided resolution

(58)$$ \begin{align} \dots C^{-1}\rightarrow C^{0}\rightarrow C^{1}\rightarrow \dots \end{align} $$

of the form $C^{i}=\bigoplus \limits _{w\in \widehat {W}(\lambda +k\Lambda _{0})\atop \ell ^{\infty /2}(w)=i}\mathbb {W}^{k}_{w\circ {\lambda }}$, where $\widehat {W}(\lambda +k\Lambda _{0})$ is the integral Weyl group of $\lambda +k\Lambda _{0}$ and $\ell ^{\infty /2}(w)$ is the semi-infinite length of w. By [Reference Arakawa and FrenkelAF19, Lemma 3.2], we have

$$ \begin{align*}H_{DS,f_{prin},{\check\mu}}^{0}(\mathbb{W}^{k}_{{\lambda}})\cong \begin{cases} \pi_{\lambda-(k+h^{\vee}){\check\mu}}^{k}&\text{for }i=0,\\ 0&\text{for }i\ne 0 \end{cases}\end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g})$-modules. It follows that $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ is the i-th cohomology of the complex obtained by applying the functor $H_{DS,f_{prin},{\check \mu }}^{0}(?)$ to the resolution (58). In particular, $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ is a subquotient of the $\mathscr {W}^{k}(\mathfrak {g})$-module

$$ \begin{align*}H_{DS,f_{prin},{\check\mu}}^{0}(C^{i})\cong \bigoplus\limits_{w\in \widehat{W}(\lambda+k\Lambda_{0})\atop \ell^{\infty/2}(w)=i}\pi^{k}_{w\circ \lambda-(k+h^{\vee}){\check\mu}}.\end{align*} $$

On the other hand, since $\mathbb {L}^{k}_{\lambda }$ is a $\mathbb {L}_{k}(\mathfrak {g})$-module, each $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ is a module over the simple W-algebra $\mathscr {W}_{k}(\mathfrak {g})=H_{DS,f_{prin}}^{0}(\mathbb {L}_{k}(\mathfrak {g}))$. Since $\mathscr {W}_{k}(\mathfrak {g})$ is rational, $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ is a direct sum of simple modules of the form (53). However, such a module appears in the local composition factor of $\pi ^{k}_{w\circ \lambda -(k+h^{\vee }){\check \mu }}$ if and only if $w\in W$ ([Reference ArakawaAra07]), where $W\subset \widehat {W}(k\Lambda _{0})$ is the Weyl group of $\mathfrak {g}$. As

$$ \begin{align*} \ell^{\infty/2}(w)\geqslant 0\text{ for }w\in W\text{ and the equality holds if and only if }w=1\text{,} \end{align*} $$

$H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ must vanish for $i< 0$. Together with equation (57), we get $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })=0$ for $i\ne 0$. Finally, since $\mathbf {L}^{k}_{[\lambda ,{\check \mu }]}$ is the unique simple $\mathscr {W}_{k}(\mathfrak {g})$-module that appears in the local composition factor of $\pi ^{k}_{\lambda -(k+h^{\vee }){\check \mu }}$ and it appears with multiplicity one ([Reference ArakawaAra07]), $H_{DS,f_{prin},{\check \mu }}^{i}(\mathbb {L}^{k}_{\lambda })$ is either zero or isomorphic to $\mathbf {L}^{k}_{[\lambda ,{\check \mu }]}$. The assertion follows since the Euler character of $H_{DS,f_{prin}}^{\bullet }(\mathbb {L}_{k}(\mathfrak {g}))$ is equal to the character of $\mathbf {L}^{k}_{[\lambda ,{\check \mu }]}$.

For an admissible number k and $\lambda \in Adm^{k}_{\mathbb {Z}}$, define

(59)$$ \begin{align} \mathbb{L}^{k}_{\lambda,f}=H_{DS,f}^{0}(\mathbb{L}^{k}_{\lambda})\in \mathscr{W}^{k}(\mathfrak{g},f)\operatorname{-Mod}. \end{align} $$

Note that $\mathbb {L}^{k}_{\lambda ,f_{prin}}=\mathbf {L}_{[\lambda ,0]}^{k}\cong \mathbf {L}_{[0,\lambda ]}^{\check {k}}$.

Let $\mathfrak {g}$ be simply laced. Observe that if $k-1$ is an admissible number, then so is k, and $\ell $ is a nondegenerate admissible number. Moreover, we have

$$ \begin{align*} Adm_{\mathbb{Z}}^{k-1}=Adm_{\mathbb{Z}}^{\check{\ell}} =P^{p-h^{\vee}}_{+},\quad Adm_{\mathbb{Z}}^{k}=Adm_{\mathbb{Z}}^{\ell}=P^{p+q-h^{\vee}}_{+} \end{align*} $$

if $k-h^{\vee }+1=p/q$ with $p\geqslant h^{\vee }$, $q\geqslant 1$, $(p,q)=1$.

Theorem 8.6. Let $\mathfrak {g}$ be simply laced, and let $k-1 $ be admissible. Suppose that $f\in X_{\mathbb {L}_{k}(\mathfrak {g})}=X_{\mathbb {L}_{k-1}(\mathfrak {g})}$. For $\mu \in Adm^{k-1}_{\mathbb {Z}}$, $\nu \in P_{+}^{1}$, we have

$$ \begin{align*} \mathbb{L}^{k-1}_{\mu,f}\otimes \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda\in Adm^{k}_{\mathbb{Z}}\atop \lambda-\mu-\nu\in Q} \mathbb{L}^{k}_{\lambda,f}\otimes \mathbf{L}_{[\lambda,\mu]}^{\ell} \end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g},f){\otimes } \mathscr {W}^{\ell }(\mathfrak {g})$-modules.

In the case $f=f_{prin}$, we have the following more general statement.

Theorem 8.7. Let $\mathfrak {g}$ be simply laced, and let $k-1 $ be nondegenerate admissible. For $\mu \in Adm^{k-1}_{\mathbb {Z}}$, $\mu ^{\prime }\in Adm^{\check {k-1}}_{\mathbb {Z}}=Adm^{\check {k}}_{\mathbb {Z}}$, $\nu \in P_{+}^{1}$, we have

$$ \begin{align*} \mathbf{L}_{[\mu,\mu^{\prime}]}^{k-1}\otimes \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda\in Adm^{k}_{\mathbb{Z}}\atop \lambda-\mu-\mu^{\prime}-\nu\in Q} \mathbf{L}_{[\lambda,\mu^{\prime}]}^{k}\otimes \mathbf{L}_{[\lambda,\mu]}^{\ell} \end{align*} $$

as $\mathscr {W}_{k}(\mathfrak {g}){\otimes } \mathscr {W}_{\ell }(\mathfrak {g})$-modules.

Corollary 8.8. Let $\mathfrak {g}$ be simply laced. Let $k+h^{\vee }=(2h^{\vee }+1)/h^{\vee }$ so that $\ell +h^{\vee }=(2h^{\vee }+1)/(h^{\vee }+1)$.

1. 1. There is an embedding of vertex algebras

   $$ \begin{align*} \mathscr{W}_{k}(\mathfrak{g}){\otimes} \mathscr{W}_{\ell}(\mathfrak{g})\hookrightarrow \mathbb{L}_{1}(\mathfrak{g}), \end{align*} $$
   and $\mathscr {W}_{k}(\mathfrak {g})$ and $ \mathscr {W}_{\ell }(\mathfrak {g})$ form a dual pair in $ \mathbb {L}_{1}(\mathfrak {g})$.

2. 2. For $\nu \in P^{1}_{+}$, we have

   $$ \begin{align*} \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda\in Adm_{\mathbb{Z}}^{k}\cap (\nu+Q)} \mathbf{L}_{[\lambda,0]}^{k}\otimes \mathbf{L}_{[\lambda,0]}^{\ell} \end{align*} $$
   as $\mathscr {W}_{k}(\mathfrak {g}){\otimes } \mathscr {W}_{\ell }(\mathfrak {g})$-modules.

Proof. The assertion follows from Theorem 8.7 noting that $\mathscr {W}_{k-1}(\mathfrak {g})=\mathbf {L}_{[0,0]}^{k-1}=\mathbb {C}$ if $k+h^{\vee }-1=(h^{\vee }+1)/h^{\vee }$ or $h^{\vee }/(h^{\vee }+1)$.

Proof of Theorem 8.6 and Theorem 8.7.

By [Reference Arakawa, Creutzig and LinshawACL19, Main Theorem 3], we have

$$ \begin{align*} \mathbb{L}^{k-1}_{\mu}\otimes \mathbb{L}^{1}_{\nu}\cong \bigoplus_{\lambda\in Adm^{k}_{\mathbb{Z}}\atop \lambda-\mu-\nu\in Q} \mathbb{L}^{k}_{\lambda}\otimes \mathbf{L}_{[\lambda,\mu]}^{\ell} \end{align*} $$

as $\mathbb {L}_{k}(\mathfrak {g}){\otimes } \mathscr {W}_{\ell }(\mathfrak {g})$-modules. Hence, the assertion follows from Main Theorem 1 and Theorem 7.1.

9 Application to the extension problem of vertex algebras

In this section, we apply Theorem 8.2 to prove the existence of the extensions of vertex algebras that are expected by four-dimensional supersymmetric gauge theories ([Reference Creutzig and GaiottoCG17, Reference Creutzig, Gaiotto and LinshawCGL18]).

For $\lambda \in P_{+}$, let $\lambda ^{*}=-w_{0}(\lambda )$ so that $E_{\lambda }^{*}\cong E_{\lambda ^{*}}$.

Theorem 9.1. Let $\mathfrak {g}$ be simply laced, and let $k, k^{\prime }$ be irrational complex numbers satisfying

$$ \begin{align*} \frac{1}{k+h^{\vee}} + \frac{1}{k^{\prime}+ h^{\vee}} = n \end{align*} $$

for $n\in \mathbb {Z}_{\geqslant 1}$. Then

$$ \begin{align*} {A}^{n}[\mathfrak{g}] :=\bigoplus_{\lambda\in P_{+}\cap Q}T^{k}_{\lambda,0}{\otimes} T^{k^{\prime}}_{\lambda^{*},0} \end{align*} $$

can be equipped with a structure of simple vertex operator algebra of central charge

$$ \begin{align*} 2\operatorname{rk}\mathfrak{g}+4h \dim \mathfrak{g}- n h \dim \mathfrak{g} \left(1 + \frac{\psi^{2}}{n \psi-1} \right), \end{align*} $$

where $\psi = k+h^{\vee }$ and h is the Coxeter number (which equals to $h^{\vee }$). The vertex operator algebra $A^{n}[\mathfrak {g}]$ is of CFT type if $n\geqslant 2$.

Proof. We shall prove the assertion on induction on n using the fact that

(60)$$ \begin{align} T^{k^{\prime}}_{\mu,\mu^{\prime}}{\otimes}\mathcal{U}(\mathfrak{g})\cong \bigoplus_{\lambda\in P_{+}\atop \lambda-\mu-\mu^{\prime}\in Q} T^{k^{\prime}+1}_{\lambda,\mu^{\prime}}{\otimes} T^{\ell}_{\lambda,\mu} \cong \bigoplus_{\lambda\in P_{+}\atop \lambda-\mu-\mu^{\prime}\in Q} T^{k^{\prime}+1}_{\lambda,\mu^{\prime}}{\otimes} T^{\check{\ell}}_{\mu,\lambda}, \end{align} $$

with $\check {\ell }$ satisfying the relation

$$ \begin{align*} \frac{1}{\check{\ell}+ h^{\vee}} = \frac{1}{k^{\prime}+ h^{\vee}}+1 \end{align*} $$

which follows from Theorem 8.2 and equation (44).

Let us show the assertion for $n=1$. Let $\mathbf {I}_{G}^{k}$ be the chiral universal centralizer on G at level k ([Reference ArakawaAra18]), which was introduced earlier in [Reference Frenkel and StyrkasFS06] for the $\mathfrak {g}=\mathfrak {sl}_{2}$ case as the modified regular representation of the Virasoro algebra. By definition, $\mathbf {I}_{G}^{k}$ is obtained by taking the principal Drinfeld–Sokolov reduction with respect to two commuting actions of $\widehat {\mathfrak {g}}$ on the algebra of the chiral differential operators [Reference Malikov, Schechtman and VaintrobMSV99, Reference Beilinson and DrinfeldBD04] $\mathcal {D}_{G,k}^{ch}$ on G at level k. The $\mathbf {I}_{G}^{k}$ is a conformal vertex algebra of central charge $2 \operatorname {rk}\mathfrak {g} + 48(\rho |\rho ^{\vee }) =2 \operatorname {rk}\mathfrak {g}+4 h^{\vee }\dim \mathfrak {g}$, equipped with a conformal vertex algebra homomorphism $\mathscr {W}^{k}(\mathfrak {g}){\otimes } \mathscr {W}^{k^{*}}(\mathfrak {g})\rightarrow \mathbf {I}_{G}^{k}$, where $k^{*}$ the dual level of k defined by the equation

$$ \begin{align*}\frac{1}{k+h^{\vee}}+\frac{1}{k^{*}+h^{\vee}}=0.\end{align*} $$

As explained in [Reference ArakawaAra18], $\mathbf {I}_{G}^{k}$ is a strict chiral quantization ([Reference ArakawaAra18]) of the universal centralizer $\mathcal {S}\times _{\mathfrak {g}^{*}}(G\times \mathcal {S})$, where $ \mathcal {S}$ is the Kostant–Slodowy slice. Since $\mathcal {S}\times _{\mathfrak {g}^{*}}(G\times \mathcal {S})$ is a smooth symplectic variety, $\mathbf {I}_{G}^{k}$ is simple ([Reference Arakawa and MoreauAM]). For an irrational k, we have the decomposition

$$ \begin{align*}\mathbf{I}_{G}^{k}\cong \bigoplus_{\lambda\in P_{+}} T_{\lambda,0}^{k}{\otimes} T_{\lambda^{*},0}^{k^{*}}\end{align*} $$

as $\mathscr {W}^{k}(\mathfrak {g}){\otimes } \mathscr {W}^{k^{*}}(\mathfrak {g})$-modules, which follows from the decomposition [Reference Arkhipov and GaitsgoryAG02, Reference ZhuZhu08] of $\mathcal {D}_{G,k}^{ch}$ as $\widehat {\mathfrak {g}}\times \widehat {\mathfrak {g}}$-modules. Hence, by equation (60),

$$ \begin{align*} \mathbf{I}_{G}^{k}{\otimes} \mathcal{U}(\mathfrak{g}) \cong \bigoplus_{\lambda\in P_{+}}T_{\lambda,0}^{k}{\otimes} T_{\lambda^{*},0}^{k^{*}}{\otimes} \mathcal{U}(\mathfrak{g}) \cong \bigoplus_{\lambda\in P_{+}, \ \mu\in P_{+} \atop \mu-\lambda^{*}\in Q}T_{\lambda,0}^{k} {\otimes} T_{\mu,0}^{k^{*}+1}{\otimes} T_{\lambda^{*},\mu}^{\check{\ell}}, \end{align*} $$

and $\check {\ell }$ satisfies the relation

$$ \begin{align*}\frac{1}{k+h^{\vee}}+\frac{1}{\check{\ell}+h^{\vee}}=1.\end{align*} $$

It follows that

$$ \begin{align*} {A}^{1}[\mathfrak{g}]:=\operatorname{Com}(\mathscr{W}^{k^{*}+1}(\mathfrak{g}), \mathbf{I}_{G}^{k}{\otimes}\mathcal{U}(\mathfrak{g})) \cong \bigoplus_{\lambda\in P_{+}\cap Q}T^{k}_{\lambda,0}{\otimes} T^{\check{\ell}}_{\lambda^{*},0}. \end{align*} $$

Moreover, since $\mathbf {I}_{G}^{k}{\otimes } \mathcal {U}(\mathfrak {g})$ is simple ${A}^{1}[\mathfrak {g}]$ is simple as well by [Reference Creutzig, Genra and NakatsukaCGN, Proposition 5.4].

Assuming that the statement is true for $n\in \mathbb {Z}_{\geqslant 1}$, we find similarly that

$$ \begin{align*}{A}^{n+1}[\mathfrak{g}]:=\operatorname{Com}(\mathscr{W}^{k^{\prime}+1}(\mathfrak{g}), {A}^{n}[\mathfrak{g}]{\otimes} \mathcal{U}(\mathfrak{g}))\end{align*} $$

has the required decomposition.

For the central charge computation, it is useful to introduce

$$ \begin{align*} \psi:= k+ h^{\vee}\qquad \text{and} \qquad \phi_{n} := \frac{\psi}{n\psi -1 } \end{align*} $$

so that

$$ \begin{align*} \frac{1}{\psi} + \frac{1}{\phi_{n}} = n. \end{align*} $$

By equation (34) and the fact that the central charge of $\mathscr {W}^{k+1}(\mathfrak {g})$ is

$$ \begin{align*} (1-h(h+1)(k+h)^{2}/(k+h+1))\operatorname{rk}\mathfrak{g} = \operatorname{rk}\mathfrak{g} - \dim \mathfrak{g} h \frac{\psi^{2}}{\psi +1} \end{align*} $$

(here we used that $ (h+1) \operatorname {rk} \mathfrak {g} = \dim \mathfrak {g}$), we have

$$ \begin{align*} \begin{aligned} c_{A^{n+1}[\mathfrak{g}]} &= c_{A^{n}[\mathfrak{g}]}-\frac{h^{2}+h-1}{h+1}\dim \mathfrak{g} - \operatorname{rk}\mathfrak{g} + h\frac{\psi^{2}}{(n\psi-1)((n+1)\psi-1)})\dim\mathfrak{g} \\ &= - \dim \mathfrak{g} h + h \phi_{n} \phi_{n+1}\dim\mathfrak{g}, \end{aligned} \end{align*} $$

where $c_{V}$ is the central charge of V, and we have put $A^{0}[\mathfrak {g}]=\mathbf {I}_{G}^{k}$. Note that

$$ \begin{align*} \phi_{n} - \phi_{n+1} = \phi_{n}\phi_{n+1} = \psi ( n\phi_{n} - (n+1) \phi_{n+1} ), \end{align*} $$

and so by induction for n, we have

$$ \begin{align*} c_{A^{n}[\mathfrak{g}]}&=2\operatorname{rk}\mathfrak{g}+4h \dim \mathfrak{g}- n h \dim \mathfrak{g} (1 + \psi \phi_{n}). \end{align*} $$

The conformal dimension of $T_{\lambda ,0}^{k}{\otimes } T_{\lambda ^{*},0}^{k^{\prime }}$ is

$$ \begin{align*} \frac{n}{2}(|\lambda+\rho|^{2}-|\rho|^{2})-2(\lambda|\rho) =\frac{n}{2}|\lambda|^{2}+(n-2)(\lambda|\rho), \end{align*} $$

which is an integer for $\lambda \in Q$. If $n\geqslant 2$, this is clearly nonnegative and is equal to zero if and only if $\lambda =0$, whence the last assertion.

Remark 9.2. More generally, it is expected [Reference Creutzig and GaiottoCG17, Reference Creutzig, Gaiotto and LinshawCGL18] that if k and $k^{\prime }$ are irrational numbers related by

$$ \begin{align*} \frac{1}{k+h^{\vee}} + \frac{1}{k^{\prime}+ h^{\vee}} = n\in \mathbb{Z}, \end{align*} $$

then

$$ \begin{align*} \bigoplus_{\lambda \in Q \cap P^{+} } \mathbb{V}^{k}_{\lambda, f} \otimes \mathbb{V}^{k^{\prime}}_{\lambda^{*}, f^{\prime}} \end{align*} $$

can be given the structure of a simple vertex operator algebra for any nilpotent elements f, $f^{\prime }$.

10 Fusion categories of modules over quasi-lisse W-algebras

For a vertex operator algebra V, let $\mathcal {C}_{V}^{ord}$ be the full subcategory of the category of finitely generated V-modules consisting of modules M on which $L_{0}$ acts locally finitely, the $L_{0}$-eigenvalues of M are bounded from below and all the generalized $L_{0}$-eigenspaces are finite dimensional. A simple object in $\mathcal {C}_{V}^{ord}$ is called an ordinary representation of V.

Recall that a finitely strongly generated vertex algebra V is called quasi-lisse [Reference Arakawa and KawasetsuAK18] if the associate variety $X_{V}$ has finitely many symplectic leaves. By [Reference Arakawa and KawasetsuAK18, Theorem 4.1], if V is quasi-lisse, then $\mathcal {C}_{V}^{ord}$ has only finitely many simple objects.

In the case that V is lisse, that is, $X_{V}$ is zero dimensional, Conjecture 1 has been proved in [Reference HuangHua09]. Conjecture 1 is true if one can show that every object in $\mathcal {C}_{V}^{ord}$ is $C_{1}$-cofinite and if grading-restricted generalized Verma modules for V are of finite length [Reference Creutzig and YangCY20]. If V is a vertex algebra that has an affine vertex subalgebra at admissible level, then another possibility of proving Conjecture 1 is the following. First, show that the affine vertex subalgebra is simple; second, prove that a suitable category of modules of the coset by the affine subalgebra in V has vertex tensor category structure. Then use the theory of vertex algebra extensions [Reference Creutzig, Kanade and McRaeCKM17] to deduce vertex tensor category structure on $\mathcal {C}_{V}^{ord}$. This is a promising direction as, for example, many simple quotients of cosets of $\mathcal W$-algebras of type A are rational and lisse by [Reference Creutzig and LinshawCL20, Cor. 6.13], and similar results for $\mathcal W$-algebras of type $B, C$ and D are work in progress.

In the case that Conjecture 1 is true, we denote by $M\boxtimes _{V} N$ the tensor product of V-modules M and N in $\mathcal {C}_{V}^{ord}$, or simply by $M\boxtimes N$ if no confusion should occur.

Now let k be an admissible number for $\widehat {\mathfrak {g}}$. As explained in Subsection 8.2,

$$ \begin{align*}X_{\mathbb{L}_{k}(\mathfrak{g})}=\overline{\mathbb{O}_{k}}\end{align*} $$

for some nilpotent orbit $\mathbb {O}_{k}$, and hence, $\mathbb {L}_{k}(\mathfrak {g})$ is quasi-lisse. By the conjecture of Adamović and Milas [Reference Adamović and MilasAM95] that was proved in [Reference ArakawaAra16a], the category $C_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$ is semisimple and $\{ \mathbb {L}^{k}_{\lambda } | \lambda \in Adm^{k}_{\mathbb {Z}}\}$ gives a complete set of isomorphism classes in $\mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$. Note that, in the case k is nonnegative integer, this is a well-known fact [Reference Frenkel and ZhuFZ92], and if this is the case, $\mathbb {L}_{k}(\mathfrak {g})$ is rational and lisse, and hence, Conjecture 1 holds by [Reference Huang and LepowskyHL95]. In the case k is admissible but not an integer, $\mathbb {L}_{k}(\mathfrak {g})$ is neither rational nor lisse anymore. Nevertheless, Conjecture 1 has been proved for $V=\mathbb {L}_{k}(\mathfrak {g})$ in [Reference Creutzig, Huang and YangCHY18] provided that $\mathfrak {g}$ is simply laced. Moreover, it was shown in [Reference CreutzigCre19] that $\mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$ is a fusion category, i.e., any object is rigid, and we have an isomorphism

(61)$$ \begin{align} K[C_{\mathbb{L}_{p-h^{\vee}}(\mathfrak{g})}^{ord}]\cong K[C_{\mathbb{L}_{k}(\mathfrak{g})}^{ord}], \quad [\mathbb{L}^{p-h^{\vee}}_{\lambda}]\mapsto [\mathbb{L}^{k}_{\lambda}] \end{align} $$

of Grothendieck rings of our fusion categories, where p is the numerator of $k+h^{\vee }$.

Let f be a nilpotent element of $\mathfrak {g}$. By [Reference ArakawaAra15a],

$$ \begin{align*}X_{H_{DS,f}^{0}(\mathbb{L}_{k}(\mathfrak{g}))}=X_{\mathbb{L}_{k}(\mathfrak{g})}\cap \mathcal{S}_{f},\end{align*} $$

where $\mathcal {S}_{f}$ is the Slodowy slice at f in $\mathfrak {g}$. Therefore, in the case that k is admissible, $X_{H_{DS,f}^{0}(\mathbb {L}_{k}(\mathfrak {g}))}=\overline {\mathbb {O}_{k}}\cap \mathcal {S}_{f}$ is a nilpotent Slodowy slice, that is, the intersection of the Slodowy slice with a nilpotent orbit closure, provided that $H_{DS,f}^{0}(\mathbb {L}_{k}(\mathfrak {g}))\ne 0$ or, equivalently, $\overline {\mathbb {O}_{k}}\cap \mathcal {S}_{f}\ne \emptyset $, that is, $f\in \overline {\mathbb {O}_{k}}$. In particular, $H_{DS,f}^{0}(\mathbb {L}_{k}(\mathfrak {g}))$ is quasi-lisse if $f\in \overline {\mathbb {O}_{k}}$ and so is the simple W-algebra $\mathscr {W}_{k}(\mathfrak {g},f)$. If $f\in \mathbb {O}_{k}$, then $\overline {\mathbb {O}_{k}}\cap \mathcal {S}_{f}$ is a point by the transversality of the Slodowy slices, and therefore, $\mathscr {W}_{k}(\mathfrak {g},f)$ is lisse.

The good grading (11) is called even of $\mathfrak {g}_{j}=0$ unless $j\in \mathbb {Z}$. If this is the case, $\mathscr {W}^{k}(\mathfrak {g},f)$ is $\mathbb {Z}_{\geqslant 0}$-graded and thus of CFT type.

The following assertion was stated in the case that $f\in \mathbb {O}_{k}$ in [Reference Arakawa and van EkerenAvE], but the same proof applies.

Theorem 10.4. Let $\mathfrak {g}$ be simply laced, k admissible, and let f be an even nilpotent element in $\overline {\mathbb {O}}_{k}$. Suppose Conjecture 1 is true for $\mathscr {W}_{k+1}(\mathfrak {g},f)$ and also that $\mathscr {W}_{k+1}(\mathfrak {g},f)$ is self-dual. Then the functor

$$ \begin{align*}\mathcal{C}_{\mathbb{L}_{k}(\mathfrak{g})}^{ord}\rightarrow \mathcal{C}_{\mathscr{W}_{k}(\mathfrak{g},f)}^{ord}, \quad M\mapsto H_{DS,f}^{0}(M),\end{align*} $$

is a unital braided tensor functor. In particular, the modules $\mathbb {L}^{k}_{\lambda ,f}$, $\lambda \in Adm^{k}_{\mathbb {Z}}$, are rigid.

Note that if $f\in \mathbb {O}_{k}$, then Conjecture 1 holds since $\mathscr {W}_{k}(\mathfrak {g},f)$ is lisse.

In the case that $f=f_{prin}$, $f\in \overline {\mathbb {O}}_{k}$ if and only if $\overline {\mathbb {O}}_{k}=\mathcal {N}$, the nilpotent cone of $\mathfrak {g}$. An admissible number k such that $\overline {\mathbb {O}}_{k}=\mathcal {N}$ is called nondegenerate. If this is the case, $\mathscr {W}^{k}(\mathfrak {g})$ is rational, and the complete fusion rule $\mathscr {W}_{k}(\mathfrak {g},f)$ has been determined previously in [Reference Frenkel, Kac and WakimotoFKW92, Reference Arakawa and van EkerenAvE19, Reference CreutzigCre19].

In the case that $\mathbb {O}_{k}$ is a subregular nilpotent orbit and $f\in \mathbb {O}_{k}$, then $\mathscr {W}_{k}(\mathfrak {g},f)$ is rational [Reference Arakawa and van EkerenAvE], and the fusion rules of $\mathscr {W}_{k}(\mathfrak {g},f)$ has been determined in [Reference Arakawa and van EkerenAvE].

The following assertion, which follows immediately from Theorem 10.4, is new except for type A ([Reference Arakawa and van EkerenAvE]) and the above cases since the conjectural rationality [Reference Kac and WakimotoKW08, Reference ArakawaAra15a] of $\mathscr {W}_{k}(\mathfrak {g},f)$ with $f\in \mathbb {O}_{k}$ is open otherwise.

Remark 10.7. Let $ \mathcal {C}_{\mathscr {W}_{k}(\mathfrak {g},f)}^{KL}$ denote the fusion category consisting of objects $H_{DS,f}^{0}(M)$, $M\in \mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$. By Theorem 10.4,

$$ \begin{align*}\mathcal{C}_{\mathbb{L}_{k}(\mathfrak{g})}^{ord}\rightarrow \mathcal{C}_{\mathscr{W}_{k}(\mathfrak{g},f)}^{KL},\quad M\mapsto H_{DS,f}^{0}(M)\end{align*} $$

is a quotient functor between fusion categories. It gives an equivalence if and only if the modules $\mathbb {L}^{k}_{\lambda ,f}$, $ \lambda \in Adm^{k}_{\mathbb {Z}}$ are distinct.

The rest of this section is devoted to the proof of Theorem 10.4.

Suppose Conjecture 1 holds for V, and let $\mathcal C$ be a monoidal full subcategory of $\mathcal {C}_{V}^{ord}$. Recall that $M, N\in \mathcal {C}$ are said to centralize each other [Reference MügerMü03] if the monodromy of M and N is trivial, i.e., is equal to the identity on $M\boxtimes _{V} N$, where the monodromy is the double braiding

$$ \begin{align*}M\boxtimes_{V} N \xrightarrow{b_{M, N}} N \boxtimes_{V} M \xrightarrow{b_{N, M}} M \boxtimes_{V} N.\end{align*} $$

Suppose that V is a vertex operator subalgebra of another quasi-lisse vertex operator algebra W and that W as a V-module is an object of $\mathcal {C}_{V}^{ord}$. We assume that Conjecture 1 holds for W as well. Let $\mathcal {D}$ be a monoidal full subcategory of $\mathcal C$ such that W and any object in $\mathcal {D}$ centralize each other. Then

$$ \begin{align*} \mathcal{F}(M):=W\boxtimes_{V} M \end{align*} $$

can be equipped with a structure of a module for the vertex operator algebra W, giving rise to the induction functor [Reference Creutzig, Kanade and McRaeCKM17]

$$ \begin{align*} \mathcal{D}\rightarrow C_{W}^{ord},\quad M\mapsto \mathcal{F}(M). \end{align*} $$

Let f be admissible, and let f be an even nilpotent element in $\overline {\mathbb {O}_{k}}$. Then $f\in \overline {\mathbb {O}_{k+1}}$ as well since $\overline {\mathbb {O}_{k}}$ depends only on the denominator of k. Let $\ell $ be the number defined by

$$ \begin{align*}\frac{1}{k+h^{\vee}+1}+\frac{1}{\ell+h^{\vee}}=1,\end{align*} $$

that is,

$$ \begin{align*} \ell+h^{\vee}=\frac{k+h^{\vee}+1}{k+h^{\vee}}. \end{align*} $$

Then $\ell $ is a nondegenerate admissible number. Note that

(62)$$ \begin{align} Adm^{\ell}_{\mathbb{Z}}=Adm^{k+1}_{\mathbb{Z}}=P_{+}^{p+q-h^{\vee}}, \quad Adm^{\check{\ell}}_{\mathbb{Z}}=Adm^{k}_{\mathbb{Z}}=P_{+}^{p-h^{\vee}}, \end{align} $$

where we have put $k+h^{\vee }=p/q$.

Consider the $\mathscr {W}_{\ell }(\mathfrak {g})$-modules

$$ \begin{align*}\mathbf{L}^{\ell}_{[0,\mu]}\cong \mathbf{L}^{\check{\ell}}_{[\mu,0]} =H_{DS,f_{prin}}^{0}(\mathbb{L}^{\check{\ell}}_{\mu}),\quad \mu\in Adm_{\mathbb{Z}}^{\ell}.\end{align*} $$

Since the stabilizer of $0\in Adm^{\ell }_{\mathbb {Z}}$ of the $\tilde {W}_{+}$-action is trivial, the simple $\mathscr {W}_{\ell }(\mathfrak {g})$-modules $\mathbf {L}^{\ell }_{[0,\mu ]}$, $\mu \in Adm_{\mathbb {Z}}^{\ell }$, are distinct. Therefore, by Theorem 10.4 (that is proved for $f=f_{prin}$ in [Reference CreutzigCre19]), the modules $\mathbf {L}^{\ell }_{[0,\mu ]}$, $\mu \in Adm_{\mathbb {Z}}^{\ell }$ form a fusion full subcategory of $\mathcal {C}^{ord}_{\mathscr {W}_{\ell }(\mathfrak {g})}$ that is equivalent to a category that can be called a simple current twist of $\mathcal {C}_{\mathbb {L}_{\check {\ell }}(\mathfrak {g})}^{ord}\cong \mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$ (see [Reference CreutzigCre19, Thm.7.1] for the details). This simple current twist of $\mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$ is the fusion subcategory of $\mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord} \boxtimes \mathcal {C}_{\mathbb {L}_{1}(\mathfrak {g})}^{ord}$ whose simple objects are the $\mathbb {L}^{k}_{\mu ,f}{\otimes } \mathbb {L}^{1}_{-\mu +Q}$. Call this category $\mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord, tw}$.

Now set

$$ \begin{align*} W:=\mathscr{W}_{k}(\mathfrak{g},f){\otimes} \mathbb{L}_{1}(\mathfrak{g})=\mathbb{L}^{k}_{0,f}{\otimes} \mathbb{L}^{1}_{\nu}, \quad V:=\mathscr{W}_{k+1}(\mathfrak{g},f){\otimes} \mathscr{W}_{\ell}(\mathfrak{g})=\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,0]}. \end{align*} $$

By Theorem 8.6,

(63)$$ \begin{align} W \cong \bigoplus_{\substack{\lambda \in Adm^{k+1}_{\mathbb{Z}}\cap Q }} \mathbb{L}^{k+1}_{\lambda,f} \otimes \mathbf{L}^{\ell}_{[\lambda,0]}. \end{align} $$

Hence, V is a vertex subalgebra of W. Moreover, each direct summand of W is an ordinary V-module and the sum is finite, and so W is an object of $\mathcal {C}_{V}^{ord}$.

Clearly, the V-modules

$$ \begin{align*}\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell} =\mathscr{W}_{k+1}(\mathfrak{g},f){\otimes} \mathbf{L}_{[0,\mu]}^{\ell},\quad \mu \in Adm^{\check{\ell}}_{\mathbb{Z}}\end{align*} $$

form a monoidal full subcategory of $\mathcal {C}^{ord}_{V}$ that is equivalent to $ \mathcal {C}_{\mathbb {L}_{k}(\mathfrak {g})}^{ord}$. We denote by $\mathcal {D}$ this fusion category.

By Theorem 10.8 and equation (63), we find that W centralizes any object of $\mathcal {D}$. Hence, we have the induction functor

$$ \begin{align*} \mathcal{F}\colon \mathcal{D}\rightarrow \mathcal{C}^{ord}_{W},\quad M\mapsto W\boxtimes_{V}M. \end{align*} $$

Proof. Thanks to [Reference CreutzigCre19, Theorem 3.5], it is sufficient to show that $\mathcal {F}(\mathbb {L}^{k+1}_{0,f}{\otimes } \mathbf {L}_{[0,\mu ]}^{\ell })$ is a simple W-module that is isomorphic to $\mathbb {L}^{k}_{\mu ,f}{\otimes } \mathbb {L}^{1}_{-\mu +Q}$ for all $\mu \in Adm^{\check {\ell }}_{\mathbb {Z}}$. As V-modules we have

$$ \begin{align*} \mathcal{F}(\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell}) = \bigoplus_{\substack{\lambda \in Adm^{k+1}_{\mathbb{Z}}\cap Q }} (\mathbb{L}^{k+1}_{\lambda,f} \otimes \mathbf{L}^{\ell}_{[\lambda,0]}) \boxtimes_{V} (\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell})\\ \cong \bigoplus_{\substack{\lambda \in Adm^{k+1}_{\mathbb{Z}}\cap Q }}\mathbb{L}^{k+1}_{\lambda,f} {\otimes} \mathbf{L}_{[\lambda,\mu]}^{\ell}\cong \mathbb{L}^{k}_{\mu,f}{\otimes} \mathbb{L}^{1}_{-\mu+Q} \end{align*} $$

by Theorem 8.6 and Theorem 10.8. We claim this is indeed an isomorphism of W-modules. To see this, it is sufficient to show that there is a nontrivial W-module homomorphism $\mathcal {F}(\mathbb {L}^{k+1}_{0,f}{\otimes } \mathbf {L}_{[0,\mu ]}^{\ell }) \rightarrow \mathbb {L}^{k}_{\mu ,f}{\otimes } \mathbb {L}^{1}_{-\mu +Q}$ since $\mathbb {L}^{k}_{\mu ,f}{\otimes } \mathbb {L}^{1}_{-\mu +Q}$ is simple. By the Frobenius reciprocity [Reference Kirillov and OstrikKO02, Reference Creutzig, Kanade and McRaeCKM17], we have

$$ \begin{align*} &\operatorname{\mathrm{Hom}}_{W\operatorname{-Mod}}(\mathcal{F}(\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell}), \mathbb{L}^{k}_{\mu,f}{\otimes} \mathbb{L}^{1}_{-\mu+Q})\\ &\cong \operatorname{\mathrm{Hom}}_{V\operatorname{-Mod}}(\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell},\mathbb{L}^{k}_{\mu,f}{\otimes} \mathbb{L}^{1}_{-\mu+Q}) \\ &\cong \bigoplus_{\substack{\lambda \in Adm^{k+1}_{\mathbb{Z}}\cap Q }} \operatorname{\mathrm{Hom}}_{V\operatorname{-Mod}}(\mathbb{L}^{k+1}_{0,f}{\otimes} \mathbf{L}_{[0,\mu]}^{\ell},\mathbb{L}^{k+1}_{\lambda,f} {\otimes} \mathbf{L}_{[\lambda,\mu]}^{\ell}). \end{align*} $$

It follows that there is a nontrivial homomorphism corresponding to the identity map $\mathbb {L}^{k+1}_{0,f}{\otimes } \mathbf {L}_{[0,\mu ]}^{\ell }\rightarrow \mathbb {L}^{k+1}_{0,f}{\otimes } \mathbf {L}_{[0,\mu ]}^{\ell }$.

Proof of Theorem 10.4.

By Theorem 10.9, the correspondence

$$ \begin{align*} \mathbb{L}^{k}_{\mu}{\otimes} \mathbb{L}^{1}_{-\mu+Q}\mapsto \mathbf{L}^{\ell}_{[0,\mu]}\mapsto \mathbb{L}^{k+1}_{0,f}{\otimes}\mathbf{L}^{\ell}_{[0,\mu]} \mapsto \mathbb{L}^{k}_{\mu,f}{\otimes} \mathbb{L}^{1}_{-\mu+Q}, \quad \mu\in Adm^{k}_{\mathbb{Z}}=Adm^{\check{\ell}}_{\mathbb{Z}} \end{align*} $$

gives a tensor functor

$$ \begin{align*} \mathcal{C}_{\mathbb{L}_{k}(\mathfrak{g})}^{ord, tw}\rightarrow C_{\mathscr{W}_{k}(\mathfrak{g},f){\otimes} \mathbb{L}_{1}(\mathfrak{g})}^{ord} =C_{\mathscr{W}_{k}(\mathfrak{g},f)}\boxtimes \mathcal{C}_{\mathbb{L}_{1}(\mathfrak{g})}^{ord}, \end{align*} $$

where the last $\boxtimes $ denotes the Deligne product. This functor is fully faithful if and only if all the $\mathbb {L}^{k}_{\mu ,f}$ are nonisomorphic. Since $\mathcal {C}_{\mathbb {L}_{1}(\mathfrak {g})}^{ord}$ is semisimple and its simple objects are all invertible, i.e., simple currents, this corresponds extends to a surjective tensor functor

$$ \begin{align*} \mathcal{C}_{\mathbb{L}_{k}(\mathfrak{g})}^{ord} \boxtimes \mathcal{C}_{\mathbb{L}_{1}(\mathfrak{g})}^{ord} \rightarrow C_{\mathscr{W}_{k}(\mathfrak{g},f)}\boxtimes \mathcal{C}_{\mathbb{L}_{1}(\mathfrak{g})}^{ord}, \end{align*} $$

which restricts to a surjective tensor functor

$$ \begin{align*} \mathcal{C}_{\mathbb{L}_{k}(\mathfrak{g})}^{ord} \rightarrow C_{\mathscr{W}_{k}(\mathfrak{g},f)},\qquad \mathbb{L}^{k}_{\mu}\rightarrow \mathbb{L}^{k}_{\mu,f}=H_{DS,f}^{0}(\mathbb{L}^{k}_{\mu}) \end{align*} $$

that again is fully faithful if and only if all the $\mathbb {L}^{k}_{\mu ,f}$ are nonisomorphic.

Let us conclude with a remark on the general case.