2.1 Conformal structure

A conformal structure with central charge $c$ on a vertex algebra ${\mathcal {A}}$ is a Virasoro vector $L(z) = \sum _{n\in \mathbb {Z}} L_n z^{-n-2} \in {\mathcal {A}}$ satisfying

(2.4)\begin{equation} L(z) L(w) \sim \frac{c}{2}(z-w)^{-4} + 2 L(w)(z-w)^{-2} + \partial L(w)(z-w)^{-1}, \end{equation}

such that in addition, $L_{-1} \alpha = \partial \alpha$ for all $\alpha \in {\mathcal {A}}$, and $L_0$ acts diagonalizably on ${\mathcal {A}}$. We say that $\alpha$ has conformal weight $d$ if $L_0(\alpha ) = d \alpha$, and we denote the conformal weight $d$ subspace by ${\mathcal {A}}[d]$. In all our examples, the conformal weight grading will be by $\mathbb {Z}_{\geq 0}$,

\[ {\mathcal{A}} = \bigoplus_{d\geq 0} {\mathcal{A}}[d]. \]

All vertex algebras in this paper will be purely even, that is, $|\alpha | = 0$ for all $\alpha \in {\mathcal {A}}$. As a matter of notation, we say that an even vertex algebra ${\mathcal {A}}$ is of type

\[ {\mathcal{W}}(d_1,d_2,\ldots) \]

if it has a minimal strong generating set consisting of one field in each conformal weight $d_1, d_2, \ldots$. If ${\mathcal {A}}$ is freely generated of type ${\mathcal {W}}(d_1,d_2,\ldots )$, it has graded character

(2.5)\begin{equation} \chi({\mathcal{A}}, q) = \sum_{n\geq 0} \text{dim}({\mathcal{A}}[n]) q^n = \prod_{i\geq 1} \prod_{k\geq 0} \frac{1}{1-q^{d_i +k}}. \end{equation}

2.2 Important identities

We recall some important identities that hold in any vertex algebra ${\mathcal {A}}$. For any fields $a,b,c \in {\mathcal {A}}$, we have

(2.6)\begin{gather} (\partial a)_{(n)} b = -na_{(n-1)} b, \quad \forall \, n\in \mathbb{Z}, \end{gather}

(2.7)\begin{gather}a_{(n)} b = (-1)^{|a||b|} \sum_{p \in \mathbb{Z}} (-1)^{p+1} (b_{(p)} a)_{(n-p-1)} 1,\quad \forall \, n\in \mathbb{Z}, \end{gather}

(2.8)\begin{gather}:(:ab:)c:\ - \ :abc:\ = \sum_{n\geq 0}\frac{1}{(n+1)!}\big( :(\partial^{n+1} a)(b_{(n)} c):\ + (-1)^{|a||b|} (\partial^{n+1} b)(a_{(n)} c):\big), \end{gather}

(2.9)\begin{gather}a_{(n)} (:bc:) -\ :(a_{(n)} b)c:\ - (-1)^{|a||b|}\ :b(a_{(n)} c): \ = \sum_{i=1}^n \binom{n}{i} (a_{(n-i)}b)_{(i-1)}c, \quad \forall \, n \geq 0. \end{gather}

Given fields $a,b,c$ and integers $m,n \geq 0$, the following identities are known as Jacobi relations of type $(a,b,c)$.

(2.10)\begin{equation} a_{(r)}(b_{(s)} c) = (-1)^{|a||b|} b_{(s)} (a_{(r)}c) + \sum_{i =0}^r \binom{r}{i} (a_{(i)}b)_{(r+s - i)} c. \end{equation}

For each triple of fields $a,b,c$, these identities are nontrivial only for finitely many choices of $r,s$.

2.3 $C_2$-cofiniteness and rationality

Given a vertex algebra ${\mathcal {V}}$, let $C({\mathcal {V}})$ denote the vector space quotient of ${\mathcal {V}}$ by the span of elements of the form $a_{(-2)} b$ for all $a,b \in {\mathcal {V}}$. By (2.7) and (2.8), the normally ordered product on ${\mathcal {V}}$ induces a commutative, associative product on $C({\mathcal {V}})$, which is known as Zhu's commutative algebra [Reference ZhuZhu96]. If $C({\mathcal {V}})$ is finite-dimensional as a vector space, ${\mathcal {V}}$ is called $C_2$-cofinite. This condition was introduced by Zhu in [Reference ZhuZhu96], and has the important consequence that ${\mathcal {V}}$ has finitely many simple $\mathbb {Z}_{\geq 0}$-graded modules. A vertex algebra ${\mathcal {V}}$ is called rational if every $\mathbb {Z}_{\geq 0}$-graded ${\mathcal {V}}$-module is completely reducible.

2.4 Affine vertex algebras

Given a simple, finite-dimensional Lie algebra ${\mathfrak g}$, the universal affine vertex algebra $V^k({\mathfrak g})$ is freely generated by fields $X^{\xi }$ which are linear in $\xi \in {\mathfrak g}$ and satisfy

(2.11)\begin{equation} X^{\xi}(z) X^{\eta}(w) \sim k ( \xi, \eta) (z-w)^{-2} + X^{[\xi,\eta]}(w)(z-w)^{-1}, \end{equation}

where $(\cdot ,\cdot )$ denotes the normalized Killing form $({1}/{2h^{\vee }}) \langle \cdot ,\cdot \rangle$. For all $k\neq -h^{\vee }$, $V^k({\mathfrak g})$ has a conformal vector

(2.12)\begin{equation} L^{{\mathfrak g}} = \frac{1}{2(k+h^{\vee})} \sum_{i=1}^n :X^{\xi_i} X^{\xi'_i}: \end{equation}

of central charge

\[ c = \frac{k\ \text{dim}({\mathfrak g})}{k+h^{\vee}}. \]

Here $\xi _i$ runs over a basis of ${\mathfrak g}$, and $\xi '_i$ is the dual basis with respect to $(\cdot ,\cdot )$.

As a module over the affine Lie algebra $\hat {{\mathfrak g}} = {\mathfrak g}[t,t^{-1}] \oplus \mathbb {C}$, $V^k({\mathfrak g})$ is isomorphic to the vacuum $\hat {{\mathfrak g}}$-module. For generic values of $k$, the vacuum module is irreducible and $V^k({\mathfrak g})$ is a simple vertex algebra. At certain special values of $k$, $V^k({\mathfrak g})$ is not simple. We denote by $L_k({\mathfrak g})$ the quotient of $V^k({\mathfrak g})$ by its maximal proper ideal graded by conformal weight.

2.5 Affine ${\mathcal {W}}$-algebras

Given a simple Lie algebra ${\mathfrak g}$ and a nilpotent element $f \in {\mathfrak g}$, there is a vertex algebra ${\mathcal {W}}^k({\mathfrak g}, f)$ known as an affine ${\mathcal {W}}$-algebra. The standard construction is via the quantum Drinfeld–Sokolov reduction and its generalizations, and the reader is referred to [Reference Feigin and FrenkelFF90, Reference Kac, Roan and WakimotoKRW03] for details. We briefly recall some basic features of the ${\mathcal {W}}$-algebras of ${\mathfrak s}{\mathfrak l}_n$ associated to the principal, subregular, and minimal nilpotent elements.

1. (i) For $n\geq 3$, the principal ${\mathcal {W}}$-algebra ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ is freely generated of type

   \[ {\mathcal{W}}(2,3,\ldots, n). \]
   It is generated by the weights $2$ and $3$ fields [Reference Arakawa, Lam and YamadaALY19], and the Virasoro element $L$ has central charge
   \[ c = -\frac{(n-1) (n^2 + n k - n-1) (n^2 + k + n k)}{n + k}. \]

2. (ii) For $n\geq 3$, the subregular ${\mathcal {W}}$-algebra ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {subreg}})$ is freely generated of type

   \[ {\mathcal{W}}(1,2,3,\ldots, n-1, n/2, n/2). \]
   The weight $1$ field generates a Heisenberg subalgebra ${\mathcal {H}}$, and the Virasoro field has central charge
   \[ c = -\frac{((k +n)(n-1)-n)((k +n)(n-2)n - n^2+1)}{k+n}. \]

3. (iii) For $n\geq 4$, the minimal ${\mathcal {W}}$-algebra ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {min}})$ is freely generated of type

   \[ {\mathcal{W}} ( 1^{(n-2)^2}, 2, (3/2)^{2(n-2)}). \]
   The $(n-2)^2$ fields in weight $1$ generate an affine vertex algebra
   \[ V^{k+1}({\mathfrak g}{\mathfrak l}_{n-2}) = {\mathcal{H}} \otimes V^{k+1}({\mathfrak s}{\mathfrak l}_{n-2}), \]
   and the fields in weight $3/2$ are primary and transform under ${\mathfrak g}{\mathfrak l}_{n-2}$ as the sum of the standard representation $\mathbb {C}^{n-2}$ and its dual. The Virasoro element has central charge
   \[ c = -\frac{5 k + 6 k^2 + 4 n + 5 k n - n^2 - k n^2}{k + n}. \]

The complete OPE algebra of ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ is not known in general, although for $n\leq 5$ it has been worked out explicitly in the physics literature [Reference ZhuZhu93]. On the other hand, the OPE algebra of all minimal ${\mathcal {W}}$-algebras, including ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {min}})$, is simpler and appears in [Reference Kac and WakimotoKacW04]. By a theorem of Genra [Reference GenraGen17], ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {subreg}})$ coincides with the Feigin–Semikhatov algebra ${\mathcal {W}}^{(2)}_n$ at level $k$ [Reference Feigin and SemikhatovFS04], and part of the OPE algebra appears in [Reference Feigin and SemikhatovFS04].

2.6 Coset construction

Given a vertex algebra ${\mathcal {V}}$ and a subalgebra ${\mathcal {A}} \subseteq {\mathcal {V}}$, the coset or commutant of ${\mathcal {A}}$ in ${\mathcal {V}}$, denoted by $\text {Com}({\mathcal {A}},{\mathcal {V}})$, is the subalgebra of elements $v\in {\mathcal {V}}$ such that

\[ [a(z),v(w)] = 0,\quad\forall \, a\in{\mathcal{A}}. \]

This was introduced by Frenkel and Zhu in [Reference Frenkel and ZhuFZ92], generalizing earlier constructions in [Reference Goddard, Kent and OliveGKO85, Reference Kac and PetersonKP81]. Equivalently, $v\in \text {Com}({\mathcal {A}},{\mathcal {V}})$ if and only if $a_{(n)} v = 0$ for all $a\in {\mathcal {A}}$ and $n\geq 0$. Note that if ${\mathcal {V}}$ and ${\mathcal {A}}$ have Virasoro elements $L^{{\mathcal {V}}}$ and $L^{{\mathcal {A}}}$, $\text {Com}({\mathcal {A}},{\mathcal {V}})$ has Virasoro element $L = L^{{\mathcal {V}}} - L^{{\mathcal {A}}}$ as long as $L^{{\mathcal {V}}} \neq L^{{\mathcal {A}}}$.

2.7 Principal ${\mathcal {W}}$-algebras as cosets

Let ${\mathfrak g}$ be simply laced. There exists a diagonal homomorphism

\[ V^{k+1}({\mathfrak g}) \rightarrow V^k({\mathfrak g})\otimes L_1({\mathfrak g}),\quad X^{\xi}(z) \mapsto X^{\xi}(z) \otimes 1 + 1 \otimes X^{\xi}(z),\quad \xi \in {\mathfrak g}, \]

which descends to a homomorphism $L_{k+1}({\mathfrak g}) \rightarrow L_k({\mathfrak g})\otimes L_1({\mathfrak g})$ for all admissible values of $k$. The following result was recently proven in [Reference Arakawa, Creutzig and LinshawACL19].

Theorem 2.1 [Reference Arakawa, Creutzig and LinshawACL19, Main Theorems 1 and 2]

Let ${\mathfrak g}$ be simply laced and let $k, k'$ be complex numbers related by

\[ k' +h^{\vee}=\frac{k+h^{\vee}}{k+h^{\vee}+1}. \]

1. (i) For generic values of $k$, we have a vertex algebra isomorphism

   \[ {\mathcal{W}}^{k'}({\mathfrak g})\cong {\rm Com}(V^{k+1}({\mathfrak g}),V^k({\mathfrak g})\otimes L_1({\mathfrak g})). \]
   In particular, this isomorphism holds for all real numbers $k > -h^{\vee } -1$.

2. (ii) Suppose that $k$ is an admissible level for $\hat {{\mathfrak g}}$. Then $k'$ is a nondegenerate admissible level for $\hat {{\mathfrak g}}$ so that ${\mathcal {W}}_{k'}({\mathfrak g})$ is a minimal series ${\mathcal {W}}$-algebra, and we have a vertex algebra isomorphism

   \[ {\mathcal{W}}_{k'}({\mathfrak g})\cong {\rm Com}(L_{k+1}({\mathfrak g}),L_k({\mathfrak g})\otimes L_1({\mathfrak g})). \]

This was conjectured in [Reference Bais, Bouwknegt, Surridge and SchoutensBBSS88b] in the case of discrete series ${\mathcal {W}}$-algebras, which correspond to the case $k \in \mathbb {N}$, in [Reference Kac and WakimotoKacW89, Reference Kac and WakimotoKacW90] for arbitrary minimal series ${\mathcal {W}}$-algebras, and in the case of generic level in [Reference Feigin, Jimbo, Miwa and MukhinFJMM16]. In the case ${\mathfrak g} = {\mathfrak s}{\mathfrak l}_n$, the coset realization of ${\mathcal {W}}^{k'}({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ is very useful for our purposes because the weight $3$ field appears explicitly in [Reference Bais, Bouwknegt, Surridge and SchoutensBBSS88b].

2.8 Coincidences

For later use, we mention some recently established coincidences between certain cosets of affine or ${\mathcal {W}}$-algebras, and principal, rational ${\mathcal {W}}$-algebras of type $A$. First, recall the universal parafermion algebra $N^k({\mathfrak s}{\mathfrak l}_2) = \text {Com}({\mathcal {H}}, V^k({\mathfrak s}{\mathfrak l}_2))$, as well as its simple quotient $N_k({\mathfrak s}{\mathfrak l}_2) = \text {Com}({\mathcal {H}}, L_k({\mathfrak s}{\mathfrak l}_2))$ [Reference Dong, Lam and YamadaDLY09].

Theorem 2.2 [Reference Arakawa, Lam and YamadaALY19, Theorem 6.1]

For all positive integers $k \geq 3$,

\[ N_k({\mathfrak s}{\mathfrak l}_2) \cong {\mathcal{W}}_{k'}({\mathfrak s}{\mathfrak l}_k, f_{{\rm prin}}),\quad k' = -k + \frac{2 + k}{1 + k},\ -k + \frac{1 + k}{2 + k}, \]

which has central charge $c = {2 (k-1)}/({k+2})$, and is rational and $C_2$-cofinite.

Next, recall the Bershadsky–Polyakov algebra, which is just the ${\mathcal {W}}$-algebra ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_3, f_{\text {min}})$. As in [Reference Arakawa, Creutzig and LinshawACL17], it is convenient to introduce a shift of level; we define

\[ {\mathcal{W}}^{\ell} = {\mathcal{W}}^{\ell - 3/2}({\mathfrak s}{\mathfrak l}_3, f_{\text{min}}), \]

and we denote its simple quotient by ${\mathcal {W}}_{\ell }$. The weight $1$ field generates a Heisenberg subalgebra ${\mathcal {H}}$. We define ${\mathcal {C}}^{\ell } = \text {Com}({\mathcal {H}}, {\mathcal {W}}^{\ell })$ and ${\mathcal {C}}_{\ell } = \text {Com}({\mathcal {H}}, {\mathcal {W}}_{\ell })$, which is the simple quotient of ${\mathcal {C}}^{\ell }$.

Theorem 2.3 [Reference Arakawa, Creutzig and LinshawACL17, Theorem 8.3]

For all positive integers $\ell$,

\[ {\mathcal{C}}_{\ell} \cong {\mathcal{W}}_{\ell'}({\mathfrak s}{\mathfrak l}_{2\ell}, f_{{\rm prin}}),\quad \ell' = -2 l + \frac{3 + 2 l}{1 + 2 l},\ -2 l + \frac{1 + 2 l}{3 + 2 l}, \]

which has central charge $c = -{3(2\ell -1)^2}/({2\ell +3})$ and is $C_2$-cofinite and rational.

Finally, recall the ${\mathcal {W}}$-algebra ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_4, f_{\text {subreg}})$, and its simple quotient ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_4, f_{\text {subreg}})$. Let ${\mathcal {C}}^k = \text {Com}({\mathcal {H}}, {\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_4, f_{\text {subreg}}))$ and ${\mathcal {C}}_k = \text {Com}({\mathcal {H}}, {\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_4, f_{\text {subreg}}))$ which is the simple quotient of ${\mathcal {C}}^k$.

Theorem 2.4 [Reference Creutzig and LinshawCL18, Theorem 6.2]

Let $n\geq 3$ and assume that $n-1$ is coprime to at least one of $n+1$ and $n+4$. Then

\[ {\mathcal{C}}_k \cong {\mathcal{W}}_{k'}({\mathfrak s}{\mathfrak l}_n, f_{{\rm prin}}), \quad k = -4 + \frac{4 + n}{3}, \quad k' = -n + \frac{4 + n}{1 + n},\ -n + \frac{1+ n}{4 + n}, \]

which has central charge $c = -{4 (n-1) (2n -1)}/({4 + n})$ and is $C_2$-cofinite and rational.

These theorems were proven by exhibiting $V_k({\mathfrak s}{\mathfrak l}_2)$, ${\mathcal {W}}_{\ell }$, and ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_4, f_{\text {subreg}})$, respectively, as simple current extensions of $V_L \otimes {\mathcal {W}}$. Here $V_L$ is a certain rank one lattice vertex algebra that extends ${\mathcal {H}}$, and ${\mathcal {W}}$ is the appropriate principal ${\mathcal {W}}$-algebra of type $A$. This approach does not easily generalize if the vertex algebras involved are not rational, or if the larger vertex algebra is not a simple current extension of a tensor product of two nice subalgebras. One of the goals of this paper is to establish many more coincidences of this kind. For example, we shall classify all cases where the above cosets $N_k({\mathfrak s}{\mathfrak l}_2)$, ${\mathcal {C}}_{\ell }$, and ${\mathcal {C}}_k$, are isomorphic to a principal ${\mathcal {W}}$-algebra of type $A$. There are several families of coincidences which do not involve rational or $C_2$-cofinite vertex algebras.

3.1 Degenerate nonlinear conformal algebras

As pointed out on the last page of [Reference De Sole and KacDK05], it is interesting to relax the Jacobi identity (3.14). A nonlinear conformal algebra $A$ satisfying (3.13) but not all of the identities (3.14) is called a degenerate nonlinear conformal algebra. The universal enveloping vertex algebra $U(A)$ can still be defined as a quotient of the tensor algebra ${\mathcal {T}}(A)$. For simplicity, we restrict to case $\Gamma = \mathbb {Z}_{\geq 0}$. For $a,b,c\in {\mathcal {T}}(A)$ with $\Delta (a) + \Delta (b) + \Delta (c) -2 = \Delta$, note that

\[ L_{\lambda}(a, L_{\mu}(b,c)) - (-1)^{|a||b|} L_{\mu}(b, L_{\lambda}(a,c)) - L_{\lambda + \mu}(L_{\lambda}(a,b),c) \in \mathbb{C}[\lambda,\mu] \otimes {\mathcal{T}}(A)_{\Delta}. \]

Let ${\mathcal {J}}_{\Delta }(A)$ be the span of the coefficients of $\lambda ^i \mu ^j$ in all such expressions, for all $i,j\geq 0$. Then we define ${\mathcal {M}}'_{\Delta }(A) \subseteq {\mathcal {T}}_{\Delta }(A)$ to be the span of ${\mathcal {M}}_{\Delta }(A)$ together with the following elements:

1. (i) $N(a,b)$ and $N(b,a)$ for all $a\in {\mathcal {T}}_{\Delta _1}(A)$ and $b \in {\mathcal {J}}_{\Delta _2}(A)$ with $\Delta _1 + \Delta _2 = \Delta$;

2. (ii) the coefficient of $\lambda ^i$ for all $i\geq 0$ in $L_{\lambda }(a,b)$ and $L_{\lambda }(b,a)$ for all $a\in {\mathcal {T}}_{\Delta _1}(A)$ and $b \in {\mathcal {J}}_{\Delta _2}(A)$, with $\Delta _1 + \Delta _2-1 = \Delta$.

Setting

(3.18)\begin{equation} {\mathcal{M}}'(A) = \bigcup_{\Delta \in \Gamma} {\mathcal{M}}'_{\Delta}(A), \end{equation}

it follows that $\mathbb {C}[\lambda ] \otimes {\mathcal {M}}'(A)$ is a two-sided ideal of $\mathbb {C}[\lambda ] \otimes {\mathcal {T}}(A)$ under the operations $N$ and $L_{\lambda }$. Setting

(3.19)\begin{equation} U(A) = {\mathcal{T}}(A) /{\mathcal{M}}'(A), \end{equation}

by a similar argument as in the case of nonlinear Lie conformal algebras, $U(A)$ has a vertex algebra structure. Note that if $A$ is a nonlinear Lie conformal algebra, ${\mathcal {M}}(A) = {\mathcal {M}}'(A)$, so this definition of $U(A)$ agrees with (3.15). The assignment $A \mapsto U(A)$ is functorial, and $U(A)$ satisfies a universal property analogous to Theorem 3.3. This functor defines an equivalence between the category of degenerate nonlinear conformal algebras and an appropriate category of vertex algebras, which we continue to call the De Sole–Kac correspondence.

Suppose that $A$ has basis $\{\alpha ^1, \alpha ^2,\ldots \}$ as a $\mathbb {C}[T]$-module, where $\Delta (\alpha ^i) = d_i >0$. Then $U(A)$ is strongly generated by the corresponding fields $\{\alpha ^1, \alpha ^2,\ldots \}$, but is not freely generated unless $A$ is a nonlinear Lie conformal algebra. If we declare that $T^k(\alpha ^i)$ has conformal weight $d_i+k$, then $U(A)$ will be graded by conformal weight. We can still consider the category $C_A$ consisting of vertex algebras graded by conformal weight with strong generators $\{\alpha ^1, \alpha ^2,\ldots \}$, and OPE relations determined by $A$. Then $U(A)$ is a universal object in $C_A$ which admits a homomorphism to any other object in the category, and $C_A$ contains another universal object $S(A)$ which is the simple quotient of $U(A)$ by its maximal proper graded ideal.

Unlike the category of commutative rings, not every vertex algebra is the quotient of a freely generated vertex algebra with the same strong generating set. This is because Jacobi relations that fail to hold will give rise to null fields, which are elements of ${\mathcal {M}}'(A)$ that correspond to nontrivial normally ordered polynomial relation among the generators and their derivatives. A typical example is the parafermion algebra $N^k({\mathfrak s}{\mathfrak l}_2) = \text {Com}({\mathcal {H}}, V^k({\mathfrak s}{\mathfrak l}_2))$, which is of type ${\mathcal {W}}(2,3,4,5)$. These fields close nonlinearly and the OPE algebra can be found in the paper [Reference Dong, Lam and YamadaDLY09]. Starting in weight $8$, there are Jacobi relations that do not hold as consequences of (2.6)–(2.9) alone, which give rise to nontrivial relations among the generators and their derivatives.

4.1 Conformal structure and graded character

Let ${\mathcal {V}}$ be a vertex algebra over $R$ and let $c\in R$. Suppose that ${\mathcal {V}}$ contains a field $L$ satisfying (2.4), such that $L_0$ acts on ${\mathcal {V}}$ by $\partial$ and $L_1$ acts diagonalizably, and we have an $R$-module decomposition

\[ {\mathcal{V}} = \bigoplus_{d\in R} {\mathcal{V}}[d], \]

where ${\mathcal {V}}[d]$ is the $L_0$-eigenspace with eigenvalue $d$. If each ${\mathcal {V}}[d]$ is in addition a free $R$-module of finite rank, we have a well-defined graded character

\[ \chi({\mathcal{V}} ,q) = \sum_{d \in R} \text{rank}_R({\mathcal{V}}[d]) q^d. \]

In all our examples, the grading will be by $\mathbb {Z}_{\geq 0}$ regarded as a subsemigroup of $R$, and ${\mathcal {V}}[0] \cong R$. A typical example is $V^k({\mathfrak g})$ where $k$ is regarded as a formal variable, so $V^k({\mathfrak g})$ is a vertex algebra over the polynomial ring $\mathbb {C}[k]$. As such, it has no conformal vector, but if we define $R$ to be the localization $D^{-1} \mathbb {C}[k]$ where $D$ is the multiplicatively closed set generated by $(k + h^{\vee })$, then $V^k({\mathfrak g})$ has Virasoro vector $L^{{\mathfrak g}}$ given by (2.12).

In [Reference Creutzig and LinshawCL15, Reference Creutzig and LinshawCL19], a class of examples called deformable families were considered. These are vertex algebras over certain rings of rational functions of degree at most zero in a parameter $\kappa$. This notion was used to study orbifolds and cosets of affine vertex algebras and ${\mathcal {W}}$-algebras by passing to the limit as $\kappa \rightarrow \infty$, which has a simpler structure.

Here we allow $R$ to be the ring of functions on a variety $X \subseteq \mathbb {C}^n$, so that $X = \text {Specm}(R)$. We can think of the vertex algebra as being defined on $X$ in the sense that for each point $p\in X$, the evaluation at $p$ yields a vertex algebra over $\mathbb {C}$. There is also the notion of specialization along a subvariety. Let $I\subseteq R$ be an ideal corresponding to a subvariety $Y \subseteq X$, and let $I \cdot {\mathcal {V}}$ denote the set of finite sums of the form $\sum _i f_i v_i$ where $f_i \in I$ and $v_i \in {\mathcal {V}}$. Clearly $I \cdot {\mathcal {V}}$ is the vertex algebra ideal generated by $I$, and the quotient

\[ {\mathcal{V}}^I = {\mathcal{V}} / (I \cdot {\mathcal{V}}) \]

is a vertex algebra over the ring $R/I$.

4.2 Simplicity

Let ${\mathcal {V}}$ be a vertex algebra over a ring $R$ with weight grading

(4.1)\begin{equation} {\mathcal{V}} = \bigoplus_{n\geq 0} {\mathcal{V}}[n],\quad {\mathcal{V}}[0] \cong R. \end{equation}

A vertex algebra ideal ${\mathcal {I}} \subseteq {\mathcal {V}}$ is called graded if

\[ {\mathcal{I}} = \bigoplus_{n\geq 0} {\mathcal{I}}[n],\quad {\mathcal{I}}[n] = {\mathcal{I}} \cap {\mathcal{V}}[n]. \]

We say that ${\mathcal {V}}$ is simple if for every proper graded ideal $\mathcal {I}\subseteq \mathcal {V}$, $\mathcal {I}[0] \neq \{0\}$. This is different from the usual notion of simplicity, namely, that ${\mathcal {V}}$ has no nontrivial proper graded ideals, but it coincides with the usual notion if $R$ is a field. If $I \subseteq R$ is a nontrivial proper ideal, it will generate a nontrivial proper graded vertex algebra ideal $I\cdot {\mathcal {V}}$. Then

\[ \mathcal{V}^I = \mathcal{V} / (I\cdot \mathcal{V}) \]

is a vertex algebra over the ring $R/I$. Even if $\mathcal {V}$ is simple over $R$, $\mathcal {V}^I$ need not be simple over $R/I$.

Let $R$ be the ring of functions on a variety $X$, and let $\mathcal {V}$ be a simple vertex algebra over $R$ with weight grading (4.1). Let $I\subseteq R$ be an ideal such that ${\mathcal {V}}^I$ is not simple. This means that ${\mathcal {V}}^I$ possesses a maximal proper graded ideal $\mathcal {I}$ such that $\mathcal {I}[0] = \{0\}$. Then the quotient

\[ {\mathcal{V}}_{I} = {\mathcal{V}}^I / {\mathcal{I}} \]

is a simple vertex algebra over $R/I$. Letting $Y\subseteq X$ be the closed subvariety corresponding to $I$, we can regard ${\mathcal {V}}_I$ as a simple vertex algebra defined over $Y$. Continuing this process, we define the degeneration poset of all ideals $I\subseteq R$ for which ${\mathcal {V}}^I$ is not simple over $R/I$. It corresponds to the poset of closed subvarieties of $X$ where degeneration occurs.

If $I_1, I_2$ are ideals in the degeneration poset, let $\mathcal {V}_{I_1} = \mathcal {V}^{I_1} / \mathcal {I}_1$ and $\mathcal {V}_{I_2} = \mathcal {V}^{I_2} / \mathcal {I}_2$ be the corresponding simple vertex algebras over the rings $R/I_1$ and $R / I_2$, respectively. Let $Y_1, Y_2 \subseteq X$ be the closed subvarieties corresponding to $I_1, I_2$, and let $p \in Y_1 \cap Y_2$ be a point in the intersection. Let $\mathcal {V}^p_{I_1}$ and $\mathcal {V}^p_{I_2}$ be the vertex algebras over $\mathbb {C}$ obtained by evaluating at $p$. Then $\mathcal {V}^p_{I_1}$ and $\mathcal {V}^p_{I_2}$ need not be simple, and we let $\mathcal {V}_{I_1,p}$ and $\mathcal {V}_{I_2,p}$ denote their simple quotients. Clearly $p$ corresponds to a maximal ideal $M \subseteq R$ containing both $I_1$ and $I_2$, and we have isomorphisms

\[ \mathcal{V}_{I_1,p} \cong \mathcal{V}_M \cong \mathcal{V}_{I_2,p}. \]

In general, ${\mathcal {V}}_{I_1}$ and ${\mathcal {V}}_{I_2}$ can be very different, and the above isomorphism which arises from the intersection of the varieties $Y_1$ and $Y_2$, is nontrivial.

4.3 Shapovalov determinant and spectrum

Given a conformal vertex algebra ${\mathcal {V}} = \bigoplus _{n\geq 0} {\mathcal {V}}[n]$ over $R$ with ${\mathcal {V}}[0] \cong R$, ${\mathcal {V}}[n]$ has a symmetric bilinear form

(4.2)\begin{equation} \langle\,{,}\,\rangle_n: {\mathcal{V}}[n]\otimes_{R} {\mathcal{V}}[n] \rightarrow R,\quad \langle \omega,\nu \rangle_n = \omega_{(2n-1)}\nu. \end{equation}

If each ${\mathcal {V}}[n]$ is a free $R$-module of finite rank, we define the level $n$ Shapovalov determinant $\text {det}_n \in R$ to be the determinant of the matrix of $\langle \,{,}\,\rangle _n$. Under some mild hypotheses, namely, ${\mathcal {V}}[0] \cong R$ and ${\mathcal {V}}[1]$ is annihilated by $L_1$, an element $\omega \in {\mathcal {V}}[n]$ lies in the radical of the form $\langle \,{,}\,\rangle _n$ if and only of $\omega$ lies in the maximal proper graded ideal of ${\mathcal {V}}$ [Reference LinshawLi94]. Then ${\mathcal {V}}$ is simple if and only if $\text {det}_n \neq 0$ for all $n$. If ${\mathcal {V}}$ is simple and $R$ is a unique factorization ring, each irreducible factor $a$ of $\text {det}_n$ give rise to a prime ideal $(a) \subseteq R$. Note that if $a|\text {det}_n$, then $a| \text {det}_m$ for all $m>n$. We define the Shapovalov spectrum $\text {Shap}({\mathcal {V}})$ to be the set of distinct prime ideals of the form $(a) \subseteq R$ such that $a$ is a divisor of $\text {det}_n$ for some $n$. We say that $(a) \in \text {Shap}({\mathcal {V}})$ has level $n$ if $a$ divides $\text {det}_n$ but does not divide $\text {det}_m$ for $m < n$.

Note that the varieties $V(a)$ for $a\in \text {Shap}({\mathcal {V}})$ are precisely the irreducible subvarieties of $\text {Specm}(R)$ where degeneration of ${\mathcal {V}}$ occurs. For example, if ${\mathcal {V}} = V^k({\mathfrak g})$ and $R = D^{-1} \mathbb {C}[k]$ where $D$ is the multiplicative set generated by $k + h^{\vee }$ as above, $\text {Shap}(V^k({\mathfrak g}))$ is the set of all ideals $(k -k_0)$ where the vacuum module $V^{k_0}({\mathfrak g})$ is reducible. This was described explicitly by Gorelik and Kac in [Reference Gorelik and KacGK07].

4.4 De Sole–Kac correspondence over rings

In a similar way to [Reference De Sole and KacDK05], one can define the notions of nonlinear conformal algebra and nonlinear Lie conformal algebra over a ring $R$. A nonlinear conformal algebra over $R$ will be a free $R[T]$-module $A$ with a grading

\[ A = \bigoplus_{\Delta \in \Gamma \setminus \{0\}} A[\Delta] \]

for some semigroup $\Gamma$, and an $R$-module homomorphism

\[ [-_{\lambda} -]: A\otimes_R A \rightarrow R[\lambda] \otimes_R {\mathcal{T}}(A) \]

satisfying (3.1) and (3.9). Here ${\mathcal {T}}(A)$ is the tensor algebra of $A$ in the category of $R$-modules, which inherits the $\Gamma$-grading as before. Then the normally ordered product

\[ N: {\mathcal{T}}(A) \otimes_R {\mathcal{T}}(A) \rightarrow {\mathcal{T}}(A), \]

and the lambda-bracket

\[ L_{\lambda}: {\mathcal{T}}(A) \otimes_R {\mathcal{T}}(A) \rightarrow R[\lambda]\otimes_R {\mathcal{T}}(A), \]

can be defined on ${\mathcal {T}}(A)$ as in § 3, and the identities (3.10) and (3.11) hold for all $a,b,c \in A$ and $B,C,D\in {\mathcal {T}}(A)$.

Here we only need the case $\Gamma = \mathbb {Z}_{\geq 0}$. The subspaces ${\mathcal {M}}_{\Delta }(A)$ and ${\mathcal {M}}(A)$ are defined as in § 3. We call $A$ a nonlinear Lie conformal algebra over $R$ if it also satisfies (3.13) and (3.14). As in § 3, we can define the universal enveloping vertex algebra $U(A) = {\mathcal {T}}(A) / {\mathcal {M}}(A)$, which is a vertex algebra over $R$. It is graded by conformal weight if we declare $T^k(a)$ to have weight $\Delta (a) + k$. We only consider examples where $U(A)$ contains a Virasoro element $L$ whose $L_0$-eigenspace grading coincides with the weight grading. If $\{\alpha ^1,\alpha ^2,\ldots \}$ is an ordered basis of $A$ as a free $R[T]$-module, $U(A)$ is freely generated by the corresponding fields as a vertex algebra over $R$. In particular, each weighted component $U(A)[n]$ is a free $R$-module. If $\Delta (\alpha ^i) = d_i > 0$, the graded character of $U(A)$ is given by (2.5) with $\text {dim}(U(A)[n])$ replaced by $\text {rank}_R(U(A)[n])$.

Finally, suppose that $A$ is a degenerate nonlinear conformal algebra over $R$, meaning that it satisfies (3.13) but not (3.14). Then the universal enveloping vertex algebra $U(A) = {\mathcal {T}}(A) / {\mathcal {M}}'(A)$ is still defined, where ${\mathcal {M}}'(A)$ is defined as in § 3. If $\{\alpha ^1,\alpha ^2,\ldots \}$ is an ordered basis for $A$ as a $R[T]$-module, $U(A)$ is strongly generated by the corresponding fields, but is not freely generated. It is still the direct sum of its weight graded pieces $U(A)[n]$, but since nontrivial normally ordered polynomial relations among the monomials hold, $U(A)[n]$ need not be a free $R$-module.

5.1 Step 1: $S_{n,k}$ for $n\leq 9$

Since $W^4 = W^3_{(1)} W^3$ by definition, the most general OPE of $W^3$ with itself that is compatible with the $\mathbb {Z}_2$-symmetry is

\begin{align*} W^3(z) W^3(w) &\sim \frac{c}{3}(z-w)^{-6} + a_0 L(w)(z-w)^{-4} + a_1 \partial L(w)(z-w)^{-3}\\ & \quad + W^4(w)(z-w)^{-2} + \big( a_2 \partial W^4 + a_3 \partial^3 L+a_4 :(\partial L)L: \big)(w) (z-w)^{-1}. \end{align*}

It is not difficult to check that imposing the Jacobi relations of type $(L,W^3,W^3)$ forces

(5.4)\begin{align} W^3(z) W^3(w) &\sim \frac{c}{3}(z-w)^{-6} + 2 L(w)(z-w)^{-4} + \partial L(w)(z-w)^{-3} \nonumber\\ & \quad + W^4(w)(z-w)^{-2} + \bigg(\frac{1}{2} \partial W^4 -\frac{1}{12} \partial^3 L\bigg)(w)(z-w)^{-1}. \end{align}

The most general OPE of $L$ and $W^4$ compatible with the $\mathbb {Z}_2$-symmetry is

\begin{align*} L(z) W^4(w) &\sim a_0 (z-w)^{-6} + a_1 L(w)(z-w)^{-4} + a_2 \partial L(w)(z-w)^{-3} \\ & \quad + 4 W^4(w)(z-w)^{-2}+ \partial W^4(w)(z-w)^{-1}. \end{align*}

By imposing the Jacobi relations of type $(L,L,W^4)$ and $(L,W^3,W^3)$ this forces

(5.5)\begin{align} L(z) W^4(w) &\sim 3 c (z-w)^{-6} + 10 L(w)(z-w)^{-4} + 3 \partial L(w)(z-w)^{-3} \nonumber\\ & \quad + 4W^4(w)(z-w)^{-2} + \partial W^4(w)(z-w)^{-1}. \end{align}

Next, the most general OPE of $L$ and $W^5$ compatible with the $\mathbb {Z}_2$-symmetry is

\begin{align*} L(z) W^5(w) &\sim b_0 W^3(z)(z-w)^{-4} + b_1 \partial W^3(z)(z-w)^{-3}\\ & \quad +5 W^5(w)(z-w)^{-2} + \partial W^5(w)(z-w)^{-1}. \end{align*}

Similarly, the most general OPE of $W^3$ and $W^4$ compatible with the $\mathbb {Z}_2$-symmetry is

\begin{align*} W^3(z) W^4(w) &\sim a_0 W^3(w)(z-w)^{-4} +a_1 \partial W^3(w)(z-w)^{-3}+ W^5(w)(z-w)^{-2} \\ & \quad +(a_2 :L\partial W^3: + a_3 :(\partial L) W^3: +a_4\partial W^5 + a_5 \partial^3 W^3) (w)(z-w)^{-1}, \end{align*}

It turns out that unlike (5.4) and (5.5), if we impose all Jacobi identities of type $(L,W^3, W^4)$, $(L, L, W^5)$, and $(W^3, W^3, W^3)$ there is an additional free parameter $\lambda$ in the above OPEs. They have the form

(5.6)\begin{align} L(z) W^5(w) &\sim \big(185 - 80 \lambda (2 + c)\big) W^3(z)(z-w)^{-4} + \big(55 - 16\lambda (2 + c) \big) \partial W^3(z)(z-w)^{-3}\nonumber\\ & \quad + 5 W^5(w)(z-w)^{-2} + \partial W^5(w)(z-w)^{-1}, \end{align}

(5.7)\begin{align} W^3(z) W^4(w) &\sim (31-16 \lambda (2+c)) W^3(w)(z-w)^{-4} + \frac{8}{3} (5 - 2\lambda (2 + c) ) \partial W^3(w)(z-w)^{-3} \nonumber\\ & \quad + W^5(w)(z-w)^{-2} +\bigg(\frac{2}{5} \partial W^5 + \frac{32}{5} \lambda :L\partial W^3: -\frac{48}{5} \lambda :(\partial L) W^3: \nonumber\\ & \quad + \frac{2}{15} (-5 + 2 \lambda (-1 + c))\partial^3 W^3\bigg) (w)(z-w)^{-1}. \end{align}

Similarly, letting $L = W^2$ and imposing all Jacobi identities of type $(W^i, W^j, W^k)$ for $i+j+k \leq 11$, all terms in the OPE of $W^i(z) W^j(w)$ for $2\leq i \leq j$ and $i+j \leq 9$ are uniquely determined as $\mathbb {C}[c, \lambda ]$-linear combinations of normally ordered monomials in the generators $W^i$. These formulas are given in the appendix; see (A.1)–(A.6). By imposing the identities (2.6) and (2.7), this uniquely determines the OPEs

\[ \partial^a W^i(z) \partial^b W^j(w),\quad a,b\geq 0,\quad i>j \geq 2,\quad i+j \leq 9. \]

5.2 Step 2: Induction

We make the following inductive hypothesis.

1. (i) For $2 \leq i\leq j$ and $i+j \leq n$, all terms in the OPE of $W^i(z) W^j(w)$ have been expressed as $\mathbb {C}[c, \lambda ]$-linear combinations of normally monomials in $L, W^3,\ldots , W^{n-2}$ and their derivatives. The OPEs are weight homogeneous and compatible with $\mathbb {Z}_2$-symmetry, i.e. all terms appearing in $W^i_{(k)} W^j$ have weight $i+j-k-1$ and eigenvalue $(-1)^{i+j}$.

2. (ii) We impose (2.6) and (2.7), which then determines all OPEs of the form

   \[ \partial^a W^i(z) \partial^b W^j(w), \quad a,b \geq 0,\quad i+j \leq n. \]

3. (iii) The coefficients $a_{i,j}$ and $b_{i,j}$ are independent of $c, \lambda$ for $i+j \leq n$. Here $a_{i,j}$ and $b_{i,j}$ are given by (5.2) and (5.3), respectively.

By inductive data, we mean the collection of all OPE relations

(5.8)\begin{equation} \partial^a W^i(z) \partial^b W^j(w),\quad a,b \geq 0,\quad i+j \leq n. \end{equation}

If we impose (2.8) and (2.9), it is a consequence of our inductive hypothesis that if $\alpha$ is a normally ordered polynomial in the generators $L, W^3, \ldots , W^{n-i}$ and their derivatives of weight $w \leq n+1-i$, the OPE $W^i(z) \alpha (w)$ is uniquely determined by the inductive data.

Lemma 5.1 If we impose the Jacobi identity

(5.9)\begin{align} L_{(2)} (W^3_{(0)} W^{n-2}) & = (L_{(2)} W^3)_{(0)} W^{n-2} + W^3 _{(0)}( L_{(2)} W^{n-2}) \nonumber\\ &\quad + 2(L_{(1)} W^3)_{(1)} W^{n-2} + (L_{(0)} W^3)_{(2)} W^{n-2}, \end{align}

we must have

\[ b_{3,n-2} = \frac{2}{n-1}. \]

In particular, $b_{3,n-2}$ is independent of $c$ and $\lambda$.

Proof. Recall that

\[ W^3_{(0)} W^{n-2} = b_{3,n-2} \partial W^{n-1} + D_{3,n-2} \]

where $D_{3,n-2}$ is a normally ordered polynomial in $L, W^3, \ldots , W^{n-3}$ and their derivatives. Then

\[ L_{(2)} (W^3_{(0)} W^{n-2}) = b_{3,n-2} L_{(2)} \partial W^{n-1} + L_{(2)} D_{3,n-2}. \]

Also, $L_{(2)}D_{3,n-2}$ has no terms depending on $W^{n-1}$ by inductive assumption, so it does not contribute to the coefficient of $\partial W^{n-1}$. We have

\[ L_{(2)} \partial W^{n-1} = -(\partial L)_{(2)} W^{n-1} + \partial (L_{(2)} W^{n-1}). \]

Note that $L_{(2)} W^{n-1}\in S_{n+1,2}$ and is therefore not yet known, but by weight and $\mathbb {Z}_2$-symmetry considerations it only depends on $L, W^3, \ldots , W^{n-3}$ and their derivatives. Then $\partial (L_{(2)} W^{n-1})$ only depends on $L, W^3, \ldots , W^{n-3}$. Modulo terms which only depend on $L, W^3, \ldots , W^{n-3}$ and their derivatives, we have

\[ L_{(2)} \partial W^{n-1} \equiv - (\partial L)_{(2)} W^{n-1} = 2 L_{(1)} W^{n-1} = 2(n-1) W^{n-1}. \]

So the left-hand side of (5.9) is $2(n-1) b_{3,n-2} W^{n-1}$ up to terms which do not depend on $W^{n-1}$.

Next, the term $(L_{(2)} W^3)_{(0)} W^{n-2}$ from (5.9) vanishes because $W^3$ is assumed primary. The term

\[ W^3 _{(0)}( L_{(2)} W^{n-2}) \]

from (5.9) has no contribution to the coefficient of $W^{n-1}$, since $L_{(2)} W^{n-2}$ only depends on $L, W^3, \ldots , W^{n-4}$ and their derivatives. The term

\[ 2(L_{(1)} W^3)_{(1)} W^{n-2} \]

from (5.9) contributes $6 W^3_{(1)}W^{n-2} = 6 W^{n-1}$. The term

\[ (L_{(0)} W^3)_{(2)} W^{n-2} = \partial W^3_{(2)} W^{n-2} = -2 W^3_{(1)} W^{n-2} = -2 W^{n-1}. \]

We conclude that

\[ 2(n-1) b_{3,n-2} = 4, \]

and hence the lemma follows.

Proof. We first impose the Jacobi relation

(5.10)\begin{equation} W^3_{(1)} (W^3_{(0)} W^{n-3}) = (W^3_{(1)} W^3)_{(0)} W^{n-3} + W^3_{(0)} (W^3_{(1)} W^{n-3}) + (W^3_{(0)} W^3)_{(1)} W^{n-3}. \end{equation}

Since $W^3_{(0)} W^{n-3} = ({2}/({n-2})) \partial W^{n-2} + D_{3,n-3}$, the left-hand side of (5.10) is

(5.11)\begin{align} &W^3_{(1)} \bigg(\frac{2}{n-2} \partial W^{n\!-\!2} + D_{3,n-3} \bigg)\nonumber\\ &\quad = -\frac{2}{n-2}(\partial W^3)_{(1)} W^{n-2} +\frac{2}{n-2} \partial \big(W^3_{(1)} W^{n-2}\big) + W^3_{(1)} D_{3,n-3}\nonumber\\ &\quad = \frac{2}{n-2} \bigg(\frac{2}{n-1} \partial W^{n-1} +D_{3,n-2}\bigg) + \frac{2}{n-2} \partial W^{n-1} + W^3_{(1)} D_{3,n-3}\nonumber\\ &\quad = \frac{2 (1 + n)}{(n-2) (n-1)} \partial W^{n-1} + \frac{2}{n-2} D_{3,n-2} + W^3_{(1)} D_{3,n-3}. \end{align}

Next,

(5.12)\begin{equation} \begin{aligned} (W^3_{(1)} W^3)_{(0)} W^{n-3} & = W^4_{(0)} W^{n-3} = b_{4,n-3} \partial W^{n-1} + D_{4,n-3},\\ W^3_{(0)} (W^3_{(1)} W^{n-3}) & = W^3_{(0)} W^{n-2} = \frac{2}{n-1} \partial W^{n-1} + D_{3,n-2},\\ (W^3_{(0)} W^3)_{(1)} W^{n-3} & = \bigg(\frac{1}{2} \partial W^4 - \frac{1}{12} \partial^3 L\bigg)_{(1)} W^{n-3}\\ & = -\frac{1}{2} W^4_{(0)} W^{n-3} = -\frac{1}{2} \big(b_{4,n-3} \partial W^{n-1} + D_{4,n-3}\big). \end{aligned} \end{equation}

Collecting terms, we conclude that

\[ b_{4,n-3} = \frac{12}{(n-1)(n-2)}. \]

Inductively, we impose the Jacobi relation

(5.13)\begin{equation} W^3_{(1)} (W^{i-1}_{(0)} W^{n+1-i}) = (W^3_{(1)} W^{i-1})_{(0)} W^{n+1-i} + W^{i-1}_{(0)} (W^3_{(1)} W^{n+1-i}) + (W^3_{(0)} W^{i-1})_{(1)} W^{n+1-i}. \end{equation}

The left side of (5.13) is

(5.14)\begin{align} W^3_{(1)} \big( b_{i-1,n+1-i} \partial W^{n-2} + D_{i-1,n+1-i}\big) &= b_{i-1,n+1-i} W^3_{(1)} \partial W^{n-2} + W^3_{(1)}D_{i-1,n+1-i}\nonumber\\ &= b_{i-1,n+1-i} \bigg(\frac{2}{n-1} \partial W^{n-1}+ D_{3,n-2} + \partial W^{n-1}\bigg)\nonumber\\ &\quad + W^3_{(1)}D_{i-1,n+1-i}. \end{align}

The right side of (5.13) is

(5.15)\begin{equation} b_{i,n+1-i} \partial W^{n-1} + D_{i,n+1-i} + b_{i-1, n+2-i} \partial W^{n-1} + D_{i-1,n+2-i} -\frac{2}{i} \big(b_{i,n+1-i} \partial W^{n-1} + D_{i,n+1-i}\big). \end{equation}

Recall that $b_{i-1,n+1-i}$ and $D_{i-1,n+1-i}$ are part of our inductive data, and we are assuming inductively that $b_{i-1,n+1-i}$ is independent of $c, \lambda$. Since $D_{i-1,n+1-i}$ is a normally ordered polynomial of weight $n-1$ in $L, W^3, \ldots , W^{n-4}$ and their derivatives, it follows that $W^3_{(1)} D_{n-1,n+1-i}$ is uniquely determined by inductive data. This shows that $b_{i,n+1-i}$ is uniquely determined from inductive data together with $b_{i-1,n+2-i}$ for $4\leq i < {n}/{2}$. Finally, since $b_{3,n-2}$ is independent of $c,\lambda$, it is clear that each $b_{i,n+1-i}$ is independent of $c,\lambda$.

Similarly, we will show that $a_{4,n-3} ={4}/({n-2})$ and that $a_{i,n+1-i}$ is determined from the Jacobi identities for all $i$; see Lemma 5.4 below. Combining these observations, we obtain the following lemma.

Proof. Since $b_{i,n+1-i}$ is determined from this data, it suffices to show that $D_{i,n+1-i}$ is also determined for $3\leq i \leq {n}/{2}$. It follows from (5.11) and (5.12) that, modulo terms which are determined from inductive data,

(5.16)\begin{equation} D_{4,n-3} \equiv -\frac{2 (n-4)}{n-2} D_{3,n-2}. \end{equation}

Using (5.14) and (5.15) in the case $i=5$, we get

(5.17)\begin{equation} \frac{12}{(n-2)(n-3)} D_{3,n-2} \equiv D_{5,n-4} + D_{4,n-3} - \frac{2}{5} D_{5,n-4}, \end{equation}

modulo inductive data. Similarly, (5.14) and (5.15) show that there are nontrivial relations $D_{i,n+1-i} \equiv p_i(n) D_{3,n-2}$ for rational functions $p(n)$ for all $i$, modulo inductive data. Hence it is enough to find three linearly independent relations between $D_{3,n-2}$, $D_{4,n-3}$, and $D_{5,n-4}$. From the Jacobi relation

\[ W^3_{(0)} (W^4_{(1)} W^{n-4}) = (W^3_{(0)} W^4)_{(1)} W^{n-4} + W^4_{(1)} (W^3_{(0)} W^{n-4}), \]

we get

\[ \frac{4}{n-3} D_{3,n-2} \equiv -\frac{2}{5} D_{5,n-4} + \frac{2}{n-3} D_{4,n-3} + \frac{2}{n-3} \partial C_{4,n-3}, \]

modulo inductive data. Since $C_{4,n-3}$ is determined by $S_{n+1,1}$, we get the relation

(5.18)\begin{equation} \frac{4}{n-3} D_{3,n-2} \equiv -\frac{2}{5} D_{5,n-4} + \frac{2}{n-3} D_{4,n-3} \end{equation}

modulo inductive data together with data determined by $S_{n+1,1}$. Then (5.16)–(5.18) are the desired linearly independent relations.

Proof. By assumption $L_{(1)} W^{n-1} = (n-1) W^{n-1}$ and $W^3_{(1)} W^{n-2} = W^{n-1}$, so we begin with the case $i\geq 4$. We impose the Jacobi relation

(5.19)\begin{equation} W^{n-3} _{(1)} (W^3_{(1)} W^{3}) = W^3 _{(1)} (W^{n-3}_{(1)} W^{3}) + (W^{n-3} _{(1)} W^{3})_{(1)}W^3 + (W^{n-3} _{(0)} W^3)_{(2)} W^{3}. \end{equation}

By (2.7), the left-hand side of (5.19) is

\[ W^{n-3}_{(1)} W^4 = \sum_{i\geq 0} (-1)^{i} \frac{1}{i!} \partial^{i}(W^4_{(i+1)} W^{n-3}). \]

Therefore modulo derivatives of elements of $S_{n+1,\ell }$ for $\ell \geq 2$,

\[ W^{n-3}_{(1)} W^4 \equiv W^4_{(1)} W^{n-3} = a_{4,n-3} W^{n-1} + C_{4,n-3}. \]

As for the right-hand side, modulo terms which are either derivatives of elements of $S_{n+1,\ell }$ for $\ell \geq 2$ or determined by inductive data, we have

\begin{gather*} W^3 _{(1)} (W^{n-3}_{(1)} W^{3}) \equiv W^3 _{(1)} (W^{3}_{(1)} W^{n-3}) =W^3 _{(1)} W^{n-2} = W^{n-1},\\ (W^{n-3} _{(1)} W^{3})_{(1)}W^3 \equiv (W^{3} _{(1)} W^{n-3})_{(1)}W^3 \equiv W^{n-2}_{(1)} W^3 \equiv W^{n-1}, \end{gather*}

\begin{align*} (W^{n-3} _{(0)} W^3)_{(2)} W^{3} &= \bigg(\sum_{i\geq 0} (-1)^{i+1} \frac{1}{i!} \partial^i (W^3_{(i)} W^{n-3}) \bigg)_{(2)} W^3\\ &= \bigg(- \frac{2}{n-2} \partial W^{n-2} + \partial W^{n-2}\bigg)_{(2)} W^3 = \bigg(\frac{4}{n-2} -2 \bigg)W^{n-2}_{(1)} W^3\\ &\equiv \bigg(\frac{4}{n-2} -2 \bigg)W^{3}_{(1)} W^{n-2} = \bigg(\frac{4}{n-2} -2 \bigg) W^{n-1}. \end{align*}

Collecting terms, we see that

\[ a_{4,n-3} =\frac{4}{n-2}, \]

and that $C_{4,n-3}$ is uniquely determined modulo inductive data and derivatives of elements of $S_{n+1,\ell }$ for $\ell \geq 2$.

Next, we impose the Jacobi relation

(5.20)\begin{equation} W^3 _{(1)} (W^4_{(1)} W^{n-4}) = W^4 _{(1)} (W^3_{(1)} W^{n-4}) + (W^3 _{(1)} W^4)_{(1)} W^{n-4} + (W^3 _{(0)} W^4)_{(2)} W^{n-4}. \end{equation}

Since $W^4_{(1)} W^{n-4} = ({4}/({n-3})) W^{n-2} + C_{4,n-4}$ where $C_{4,n-4}$ depends only on $L,W^3,\ldots , W^{n-4}$ and their derivatives and is determined by inductive data, the left-hand side of (5.20) is

\[ W^3_{(1)} \bigg(\frac{4}{n-3} W^{n-2} + C_{4,n-4}\bigg) \equiv \frac{4}{n-3} W^{n-1}. \]

For the right-hand side, we have

\begin{gather*} W^4 _{(1)} (W^3_{(1)} W^{n-4}) = W^4 _{(1)} W^{n-3} = \frac{4}{n-2} W^{n-1} + C_{4,n-3},\\ (W^3 _{(1)} W^4)_{(1)} W^{n-4} = W^5_{(1)} W^{n-4} = a_{5,n-4} W^{n-1} + C_{5,n-4}, \end{gather*}

\begin{align*} (W^3 _{(0)} W^4)_{(2)} W^{n-4} &= \bigg(\frac{2}{5} \partial W^5 + \frac{32}{5} \lambda :L\partial W^3: -\frac{48}{5} \lambda :(\partial L) W^3:\\ &\quad + \frac{2}{15} \big(-5 + 2 \lambda (-1 + c)\big)\partial^3 W^3\bigg)_{(2)} W^{n-4}\\ & \equiv -\frac{4}{5} \big(a_{5,n-4} W^{n-1} + C_{5,n-4}\big). \end{align*}

It is immediate that

\[ a_{5,n-4} = \frac{20}{(n-2) (n-3)}, \]

and that $C_{5,n-4}$ is determined uniquely modulo inductive data and derivatives of elements of $S_{n+1,\ell }$ for $\ell \geq 2$. Similarly, for $i>4$, by imposing the Jacobi relation

(5.21)\begin{equation} W^3 _{(1)} (W^i_{(1)} W^{n-i}) = W^i _{(1)} (W^3_{(1)} W^{n-i}) + (W^3 _{(1)} W^i)_{(1)} W^{n-i} + (W^3 _{(0)} W^i)_{(2)} W^{n-i},\end{equation}

the same argument shows that both $a_{i+1,n-i}$ and $C_{i+1,n-i}$ are uniquely determined modulo inductive data and derivatives of elements of $S_{n+1,\ell }$ for $\ell \geq 2$. This shows that $S_{n+1,1}$ is uniquely determined modulo this data.