2.1. Borel–Moore homology

For an introduction to Borel–Moore homology, see, for example, [Reference Chriss and Ginzburg7], and for an introduction to equivariant Borel–Moore homology, see, for example, [Reference Edidin and Graham13]. Given an algebraic variety X over ${\mathbb {C}}$ , we define its Borel–Moore homology $H^{\mathrm {BM}}(X)$ as

$$ \begin{align*}H^{\mathrm{BM}}(X)^{-n}=H^{\mathrm{BM}}_n(X)=H^n_c(X){^{\vee}},\end{align*} $$

where $H^n_c(X)$ denotes the cohomology with compact support and coefficients in ${\mathbb {Q}}$ . If X is smooth and has dimension $d_X$ , then the Poincaré duality implies that

$$ \begin{align*}H^{\mathrm{BM}}(X)\simeq H^*(X)[2d_X].\end{align*} $$

Given an algebraic group G acting on X, we define the equivariant Borel–Moore homology $H^{\mathrm {BM}}_G(X)=H^{\mathrm {BM}}([X/G])$ as

$$ \begin{align*}H^{\mathrm{BM}}_G(X)^{-n}=H^{\mathrm{BM}}_{G,n}(X)=H^n_{c,G}(X){^{\vee}},\end{align*} $$

where $H^n_{c,G}(X)$ denotes G-equivariant cohomology with compact support and coefficients in ${\mathbb {Q}}$ . If X is smooth and $X,G$ have dimensions $d_X,d_G$ , respectively, then

$$ \begin{align*}H^{\mathrm{BM}}_G(X)\simeq H^*_G(X)[2d_X-2d_G].\end{align*} $$

2.2. Graded vector spaces

Let us consider the category $\operatorname {\mathrm {Vect}}^L$ of L-graded vector spaces $V=\bigoplus _{\alpha \in L}V_\alpha $ , with morphisms of degree zero. It has a structure of a closed monoidal category with tensor products and internal $\operatorname {\mathrm {Hom}}$ -objects defined by

$$ \begin{align*}V\otimes W=\bigoplus_{\alpha\in L}(V\otimes W)_\alpha,\qquad (V\otimes W)_\alpha=\coprod_{\beta\in L}V_{\beta}\otimes W_{\alpha-\beta},\end{align*} $$

$$ \begin{align*}{\underline{\textrm{Hom}}}(V,W)=\bigoplus_{\alpha\in L}\operatorname{\mathrm{Hom}}_\alpha(V,W),\qquad \operatorname{\mathrm{Hom}}_\alpha(V,W)=\prod_{\beta\in L}\operatorname{\mathrm{Hom}}(V_\beta,W_{\alpha+\beta}).\end{align*} $$

We equip $\operatorname {\mathrm {Vect}}^L$ with a symmetric monoidal category structure, where the braiding morphism is defined using the symmetric bilinear form $(\cdot ,\cdot )$ on L:

(1) $$ \begin{align} \sigma\colon V\otimes W\to W\otimes V,\qquad a\otimes b\mapsto(-1)^{(\alpha,\beta)}b\otimes a,\qquad a\in V_\alpha,\, b\in W_\beta. \end{align} $$

In what follows, we shall often use ${\mathcal {C}}$ to denote $\operatorname {\mathrm {Vect}}^L$ equipped with the thus defined symmetric closed monoidal category structure.

We define associative algebras in ${\mathcal {C}}$ as monoid objects in this category; in particular, for each $V\in {\mathcal {C}}$ , the object $\underline {\textrm {End}}(V)={\underline {\textrm {Hom}}}(V,V)\in {\mathcal {C}}$ is an associative algebra. Using the braiding $\sigma $ , one may also define commutative algebras and Lie algebras, and their modules. (Alternatively, one may note that the category ${\mathcal {C}}$ contains the category $\operatorname {\mathrm {Vect}}$ as a full symmetric monoidal subcategory of objects of degree zero, and so one may consider objects in ${\mathcal {C}}$ , which are algebras over the classical operads $\mathsf {Ass}$ , $\mathsf {Com}$ and $\mathsf {Lie}$ in $\operatorname {\mathrm {Vect}}$ .) In particular, as in the case of $\operatorname {\mathrm {Vect}}$ , the free associative algebra generated by an object X of ${\mathcal {C}}$ is the tensor algebra $\mathsf {T}(X)=\coprod _{n\ge 0} X^{\otimes n}$ , and the free commutative algebra generated by an object X of ${\mathcal {C}}$ is the symmetric algebra $S(X)=\coprod _{n\ge 0} (X^{\otimes n})_{\Sigma _n}$ .

Given an associative algebra $A\in {\mathcal {C}}$ , we may equip it with the bracket

$$ \begin{align*}[-,-]\colon A\otimes A\to A,\qquad [a,b]=\mu(a\otimes b)-\mu\sigma(a\otimes b) ;\end{align*} $$

this defines a functor from the category of associative algebras in ${\mathcal {C}}$ to the category of Lie algebras in ${\mathcal {C}}$ . This functor has a left adjoint functor, the functor of the universal enveloping algebra $U({\mathfrak {g}})$ of a Lie algebra ${\mathfrak {g}}$ . We shall use two different versions of the Poincaré–Birkhoff–Witt theorem for universal enveloping algebras. The first of them asserts that if ${\mathfrak {g}}$ is a Lie algebra and ${\mathfrak {h}}\subset {\mathfrak {g}}$ is a Lie subalgebra, the universal enveloping algebra $U({\mathfrak {g}})$ is a free $U({\mathfrak {h}})$ -module, and the vector space of generators of this module is isomorphic to $S({\mathfrak {g}}/{\mathfrak {h}})$ ; moreover, if there exists a Lie subalgebra ${\mathfrak {h}}'$ such that ${\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {h}}'$ , we may take $U({\mathfrak {h}}')\subset U({\mathfrak {g}})$ as the space of generators. The second version uses the fact that $U({\mathfrak {g}})$ has a canonical coproduct $\Delta $ for which elements of ${\mathfrak {g}}$ are primitive, and this coproduct makes $U({\mathfrak {g}})$ a cocommutative coassociative coalgebra. The theorem asserts that as a coalgebra, $U({\mathfrak {g}})$ is isomorphic to $S^c({\mathfrak {g}})$ , the cofree cocommutative coassociative conilpotent coalgebra generated by ${\mathfrak {g}}$ (note that since we work over ${\mathbb {Q}}$ , the underlying object of $S^c(X)$ is the same as that of $S(X)$ for all X in ${\mathcal {C}}$ ). To prove the Poincaré–Birkhoff–Witt theorem in ${\mathcal {C}}$ , one may use the methods of [Reference Bergman2] or [Reference Dotsenko and Tamaroff12] for the first version and the methods of [Reference Quillen52, Appendix B] or [Reference Loday44] for the second version.

We note that for an abelian group L equipped with a homomorphism $p\colon L\to \mathbb {Z}_2$ (for example, the parity $\mathbb {Z}\to \mathbb {Z}_2$ ), one normally thinks of the category of L-graded vector spaces with the braiding morphism (the Koszul sign rule)

(2) $$ \begin{align} \sigma\colon V\otimes W\to W\otimes V,\qquad a\otimes b\mapsto(-1)^{p(\alpha)p(\beta)}b\otimes a,\qquad a\in V_\alpha,\, b\in W_\beta \end{align} $$

and refers to the corresponding commutative (or Lie) algebras as super-commutative algebras (or Lie superalgebras). In this paper, we mostly encounter the more general setting discussed above and suppress the qualifier ‘super’.

2.3. Laurent series

Let V be a vector space. Let $V{[\![{z^{\pm 1}}]\!]}$ be the vector space of doubly infinite Laurent series with coefficients in V and $V{(\;\!\!\!({z})\!\!\!\;)}\subset V{[\![{z^{\pm 1}}]\!]}$ be the subspace of formal Laurent series (that is, series for which only finitely many negative components are non-zero). For a series $v=\sum _{n\in \mathbb {Z}} v_nz^n \in V{[\![{z^{\pm 1}}]\!]}$ , we denote the coefficient $v_{-1}$ by $\operatorname {\mathrm {Res}}_z v$ .

Formal Laurent series with coefficients in an algebra do themselves form an algebra. It is important to note that doubly infinite Laurent series in several variables contain several different subspaces of formal Laurent series in those variables: for instance, the subspaces ${\mathbb {Q}}{(\;\!\!\!({z})\!\!\!\;)}{(\;\!\!\!({w})\!\!\!\;)}$ and $\mathbb {Q}{(\;\!\!\!({w})\!\!\!\;)}{(\;\!\!\!({z})\!\!\!\;)}$ of ${\mathbb {Q}}{[\![{z^{\pm 1},w^{\pm 1}}]\!]}$ are different. A lot of formulas in the theory of vertex algebras use the fact that rational functions like $\frac 1{z-w}$ can be expanded as elements of both of these rings. To avoid unnecessarily heavy formulas, we shall frequently use the binomial expansion convention for such expansion; see for example, [Reference Lepowsky and Li40, §2.2]: for $n\in \mathbb {Z}$ , we define the formal Laurent series $(z+w)^n$ by the formula

$$ \begin{align*}(z+w)^n =\sum_{k\ge 0}\binom{n}{k}z^{n-k}w^k.\end{align*} $$

In plain words, we expand powers of binomials as power series in the second summand. Thus, for example, $(z-w)^n=(-1)^n(-z+w)^n$ for all $n\in \mathbb {Z}$ , but it is equal to $(-1)^n(w-z)^n$ only for $n\ge 0$ . An important doubly infinite Laurent series in ${\mathbb {Q}}{[\![{z^{\pm 1}}]\!]}$ is the ‘delta function’

$$\begin{align*}\delta(z)=\sum_{n\in\mathbb{Z}}z^n. \end{align*}$$

Doubly infinite Laurent series cannot be multiplied, but in the instances where we use the above series, we shall use the following version, which can be multiplied by any formal Laurent series in w:

$$ \begin{align*}\delta{\left({\frac{z-w}u}\right)}=\sum_{n\in\mathbb{Z}}\sum_{k\ge0}(-1)^k\binom nk u^{-n}z^{n-k}w^k.\end{align*} $$

A useful formula involving this expression is $\operatorname {\mathrm {Res}}_z \delta {\left ({\frac {z-w}u}\right )}=u$ .

2.4. Characters and Poincaré series

Given an object $M\in \operatorname {\mathrm {Vect}}^{\mathbb {Z}}$ with finite-dimensional components, we define its character $\operatorname {\mathrm {ch}}(M)$ by the formula

(3) $$ \begin{align} \operatorname{\mathrm{ch}}(M)=\sum_{k\in\mathbb{Z}}\dim M^kq^{-{\frac12} k}. \end{align} $$

The number $-{\frac 12} k$ will be called the weight of the component $M^k$ (it may be convenient to view $-k$ as a homological degree). Note that

$$ \begin{align*}\operatorname{\mathrm{ch}}(M[n])=q^{{\frac12} n}\operatorname{\mathrm{ch}}(M).\end{align*} $$

In particular, for an algebraic variety X, we have

$$ \begin{align*}\operatorname{\mathrm{ch}}(H^{\mathrm{BM}}(X)) =\sum_{k\in\mathbb{Z}} \dim H^{\mathrm{BM}}(X)^{-k}q^{{\frac12} k} =\sum_{k\in\mathbb{Z}} \dim H^k_c(X)q^{{\frac12} k} =P_c(X,q),\end{align*} $$

the Poincaré polynomial (with compact support) of X.

More generally, given an object $M\in (\operatorname {\mathrm {Vect}}^{L})^{\mathbb {Z}}\simeq \operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ with finite-dimensional components $M_{\mathbf {d}}^k$ , for ${\mathbf {d}}\in L$ and $k\in \mathbb {Z}$ , we define its Poincaré series $Z(M,x,q)$ by the formula

(4) $$ \begin{align} Z(M,x,q)=\sum_{{\mathbf{d}}\in L}(-1)^{({\mathbf{d}},{\mathbf{d}})}\operatorname{\mathrm{ch}}(M_{\mathbf{d}})x^{\mathbf{d}} =\sum_{{\mathbf{d}}\in L}\sum_{k\in\mathbb{Z}}(-1)^{({\mathbf{d}},{\mathbf{d}})} \dim M_{\mathbf{d}}^k \cdot q^{-{\frac12} k}x^{\mathbf{d}}. \end{align} $$

2.5. Plethystic exponential

For more information on $\lambda $ -rings and plethystic exponentials, see, for example, [Reference Getzler24, Reference Mozgovoy48]. Consider the ring

$$ \begin{align*}R={\mathbb{Q}}{(\;\!\!\!({q^{\frac12}})\!\!\!\;)}{[\![{x_i{\colon} i\in I}]\!]},\qquad I={ {\left\{{1,\dots,r}\right\}}}\end{align*} $$

and its unique maximal ideal ${\mathfrak {m}}$ . We equip R with the ${\mathfrak {m}}$ -adic topology. We define plethystic exponential to be the continuous group isomorphism

$$ \begin{align*}\operatorname{\mathrm{Exp}}\colon ({\mathfrak{m}},+)\to(1+{\mathfrak{m}},*)\end{align*} $$

defined on monomials by

$$ \begin{align*}\operatorname{\mathrm{Exp}}(q^k x^{\mathbf{d}})=\sum_{n\ge0}q^{nk}x^{n{\mathbf{d}}},\qquad k\in\tfrac12\mathbb{Z},\,{\mathbf{d}}\in{\mathbb{N}}^I\backslash{ {\left\{0\right\}}}.\end{align*} $$

Let us assume now that $L=\mathbb {Z}^I$ is equipped with a symmetric bilinear form $(\cdot ,\cdot )$ and let $\operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ be the corresponding symmetric monoidal category (with the signs in the braiding depending just on L-degrees). We consider a subcategory ${\mathcal {A}}\subset \operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ consisting of objects

$$ \begin{align*}M=\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^I}M_{\mathbf{d}} =\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^I}\bigoplus_{k\in\mathbb{Z}}M_{\mathbf{d}}^k\end{align*} $$

such that $M_{\mathbf {d}}^k$ are finite-dimensional and $M_{\mathbf {d}}^k=0$ for $k\gg 0$ . Then the Poincaré series (4) induces a ring homomorphism

$$ \begin{align*}Z\colon K_0({\mathcal{A}})\to R.\end{align*} $$

This homomorphism is actually a $\lambda $ -ring homomorphism (with the $\lambda $ -ring structure on $K_0({\mathcal {A}})$ induced by the symmetric monoidal category structure on ${\mathcal {A}}$ that we defined earlier see, for example, [Reference Getzler24, Reference Heinloth31]). We will formulate this fact in the following way.

Theorem 2.1. For any graded space $M\in {\mathcal {A}}$ with $M_0=0$ , we have

$$ \begin{align*}Z(S(M))=\operatorname{\mathrm{Exp}}(Z(M)).\end{align*} $$

Proof. It is enough to prove the statement for a one-dimensional space $M=M_{\mathbf {d}}^k$ . If $({\mathbf {d}},{\mathbf {d}})$ is odd, then $Z(M)=-q^{-{\frac 12} k}x^{\mathbf {d}}$ . However, $S(M)={\mathbb {Q}}\oplus M$ ; hence,

$$ \begin{align*}Z(S(M))=1-q^{-{\frac12} k}x^{\mathbf{d}} =\operatorname{\mathrm{Exp}}{\left({-q^{-{\frac12} k}x^{\mathbf{d}}}\right)} =\operatorname{\mathrm{Exp}}(Z(M)). \end{align*} $$

If $({\mathbf {d}},{\mathbf {d}})$ is even, then $Z(M)=q^{-{\frac 12} k}x^{\mathbf {d}}$ . On the other hand, $S(M) =\coprod _{n\ge 0}(M^{\otimes n})_{\Sigma _n} =\coprod _{n\ge 0}M^{\otimes n}$ , hence

$$ \begin{align*}Z(S(M))=\sum_{n\ge0}q^{-n{\frac12} k}x^{n{\mathbf{d}}} =\operatorname{\mathrm{Exp}}{\left({q^{-{\frac12} k}x^{\mathbf{d}}}\right)} =\operatorname{\mathrm{Exp}}(Z(M)).\\[-37pt] \end{align*} $$

3.1. Definition of CoHA

For more details on the results of this section, see [Reference Kontsevich and Soibelman36]. For any ${\mathbf {d}}\in {\mathbb {N}}^{I}$ , we define the space of representations

$$ \begin{align*}R_{\mathbf{d}}=R(Q,{\mathbf{d}})=\bigoplus_{a\colon i\to j}\operatorname{\mathrm{Hom}}({\mathbb{C}}^{{\mathbf{d}}_i},{\mathbb{C}}^{{\mathbf{d}}_j}),\end{align*} $$

which has the standard action of $G_{\mathbf {d}}=\prod _{i\in I}\operatorname {\mathrm {GL}}_{{\mathbf {d}}_i}({\mathbb {C}})$ . We define the cohomological Hall algebra (CoHA) with the underlying L-graded object in $D(\operatorname {\mathrm {Vect}})\simeq \operatorname {\mathrm {Vect}}^{\mathbb {Z}}$ equal

(5) $$ \begin{align} {\mathcal{H}}_Q=\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^I}{\mathcal{H}}_{Q,{\mathbf{d}}}, \end{align} $$

where

(6) $$ \begin{align} {\mathcal{H}}_{Q,{\mathbf{d}}} =H^{\mathrm{BM}}_{G_{\mathbf{d}}}(R_{\mathbf{d}})[\chi({\mathbf{d}},{\mathbf{d}})] \simeq H^*_{G_{\mathbf{d}}}(R_{\mathbf{d}})[-\chi({\mathbf{d}},{\mathbf{d}})] \end{align} $$

is considered as an object of $D(\operatorname {\mathrm {Vect}})$ (note that the stack $[R_{\mathbf {d}}/G_{\mathbf {d}}]$ has dimension $-\chi ({\mathbf {d}},{\mathbf {d}})$ ). Multiplication in this algebra is constructed as follows. Given ${\mathbf {d}},{\mathbf {e}}\in {\mathbb {N}}^I$ , consider

$$ \begin{align*}V_{\mathbf{d}}=\bigoplus_{i\in I}{\mathbb{C}}^{{\mathbf{d}}_i}\subset V_{{\mathbf{d}}+{\mathbf{e}}}=\bigoplus_{i\in I}{\mathbb{C}}^{{\mathbf{d}}_i+{\mathbf{e}}_i}\end{align*} $$

and let $R_{{\mathbf {d}},{\mathbf {e}}}\subset R_{{\mathbf {d}}+{\mathbf {e}}}$ be the subspace of representations that preserve $V_{\mathbf {d}}$ . It is equipped with an action of the parabolic subgroup $G_{{\mathbf {d}},{\mathbf {e}}}\subset G_{{\mathbf {d}}+{\mathbf {e}}}$ consisting of maps that preserve $V_{\mathbf {d}}$ . We have morphisms of stacks

$$ \begin{align*}[R_{\mathbf{d}}/G_{\mathbf{d}}]\times[R_{\mathbf{e}}/G_{\mathbf{e}}] \xleftarrow{q}[R_{{\mathbf{d}},{\mathbf{e}}}/G_{{\mathbf{d}},{\mathbf{e}}}] \xrightarrow{p}[R_{{\mathbf{d}}+{\mathbf{e}}}/G_{{\mathbf{d}}+{\mathbf{e}}}],\end{align*} $$

where q has dimension $-\chi ({\mathbf {e}},{\mathbf {d}})$ . These maps induce morphisms in $D(\operatorname {\mathrm {Vect}})$

$$ \begin{align*}H^{\mathrm{BM}}_{G_{\mathbf{d}}}(R_{\mathbf{d}})\otimes H^{\mathrm{BM}}_{G_{\mathbf{e}}}(R_{\mathbf{e}}) \xrightarrow{q^*} H^{\mathrm{BM}}_{G_{{\mathbf{d}},{\mathbf{e}}}}(R_{{\mathbf{d}},{\mathbf{e}}})[2\chi({\mathbf{e}},{\mathbf{d}})] \xrightarrow{p_*} H^{\mathrm{BM}}_{G_{{\mathbf{d}}+{\mathbf{e}}}}(R_{{\mathbf{d}}+{\mathbf{e}}})[2\chi({\mathbf{e}},{\mathbf{d}})]. \end{align*} $$

Taking the composition, we obtain the multiplication map

$$ \begin{align*}{\mathcal{H}}_{Q,{\mathbf{d}}}\otimes{\mathcal{H}}_{Q,{\mathbf{e}}}\to{\mathcal{H}}_{Q,{\mathbf{d}}+{\mathbf{e}}}\end{align*} $$

in $D(\operatorname {\mathrm {Vect}})$ . It was proved in [Reference Kontsevich and Soibelman36] that this multiplication is associative.

3.2. Shuffle algebra description

For any $n\ge 0$ , define the graded algebra

$$ \begin{align*}\Lambda_n={\mathbb{Q}}[x_1,\dots,x_n]^{\Sigma_n},\end{align*} $$

where $x_i$ has degree $2$ and $\Sigma _n$ is the symmetric group on n elements. Similarly, for any ${\mathbf {d}}\in {\mathbb {N}}^{I}$ , define

$$ \begin{align*}\Lambda_{\mathbf{d}}=\bigotimes_{i\in I}\Lambda_{{\mathbf{d}}_i} ={\mathbb{Q}}[x_{i,k}{\colon} i\in I,\, 1\le k\le {\mathbf{d}}_i]^{\Sigma_{\mathbf{d}}},\qquad \Sigma_{\mathbf{d}}=\prod_{i\in I}\Sigma_{{\mathbf{d}}_i}.\end{align*} $$

Then

$$ \begin{align*}H^*_{\operatorname{\mathrm{GL}}_n}({\mathbf{pt}})\simeq \Lambda_n,\qquad H^*_{G_{\mathbf{d}}}({\mathbf{pt}})\simeq \Lambda_{\mathbf{d}}.\end{align*} $$

This implies that

(7) $$ \begin{align} {\mathcal{H}}_{Q,{\mathbf{d}}} =H^*_{G_{\mathbf{d}}}(R_{\mathbf{d}})[-\chi({\mathbf{d}},{\mathbf{d}})] =\Lambda_{\mathbf{d}}[-\chi({\mathbf{d}},{\mathbf{d}})]. \end{align} $$

It was proved in [Reference Kontsevich and Soibelman36] that the product ${\mathcal {H}}_{Q,{\mathbf {d}}}\otimes {\mathcal {H}}_{Q,{\mathbf {e}}}\to {\mathcal {H}}_{Q,{\mathbf {d}}+{\mathbf {e}}}$ is given by the shuffle product

(8) $$ \begin{align} f*g=\sum_{\sigma\in\operatorname{\mathrm{Sh}}({\mathbf{d}},{\mathbf{e}})}\sigma(fgK), \end{align} $$

where the sum runs over all $({\mathbf {d}},{\mathbf {e}})$ -shuffles, meaning $\sigma \in \Sigma _{{\mathbf {d}}+{\mathbf {e}}}$ satisfying

$$ \begin{align*}\sigma_i(1)<\dots<\sigma_i({\mathbf{d}}_i),\qquad \sigma_i({\mathbf{d}}_i+1)<\dots<\sigma_i({\mathbf{d}}_i+{\mathbf{e}}_i) \qquad \forall i\in I,\end{align*} $$

and the kernel K is a function in the localisation of $\Lambda _{\mathbf {d}}\otimes \Lambda _{\mathbf {e}}$ defined by

(9) $$ \begin{align} K(x,y) =\frac {\prod_{a:i\to j}\prod_{k=1}^{{\mathbf{d}}_i}\prod_{\ell=1}^{{\mathbf{e}}_j} (y_{j,\ell}-x_{i,k})} {\prod_{i\in I}\prod_{k=1}^{{\mathbf{d}}_i}\prod_{\ell=1}^{{\mathbf{e}}_i} (y_{i,\ell}-x_{i,k})} =\prod_{i,j\in I} \prod_{k=1}^{{\mathbf{d}}_i}\prod_{\ell=1}^{{\mathbf{e}}_j} (y_{j,\ell}-x_{i,k})^{-\chi(i,j)}. \end{align} $$

The above formula for the shuffle product implies that

(10) $$ \begin{align} f*g=(-1)^{\chi({\mathbf{d}},{\mathbf{e}})}g*f,\qquad f\in{\mathcal{H}}_{Q,{\mathbf{d}}},\,g\in{\mathcal{H}}_{Q,{\mathbf{e}}}. \end{align} $$

This formula implies that ${\mathcal {H}}_Q$ is a commutative algebra in the symmetric monoidal category $\operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}\simeq (\operatorname {\mathrm {Vect}}^{\mathbb {Z}})^L$ with the braiding arising from the Euler form $\chi $ .

Remark 3.1. It was observed in [Reference Kontsevich and Soibelman36, §2.6] that it is possible to modify multiplication in ${\mathcal {H}}_Q$ (non-canonically) to make it super-commutative. For that, one defines the parity map $p\colon L\to \mathbb {Z}_2$ , ${\mathbf {d}}\mapsto \chi ({\mathbf {d}},{\mathbf {d}})\pmod 2$ and chooses a group homomorphism $\varepsilon \colon L\times L\to \mu _2={{\left \{{\pm 1}\right \}}}$ such that

(11) $$ \begin{align} \varepsilon({\mathbf{d}},{\mathbf{e}}) =(-1)^{p({\mathbf{d}})p({\mathbf{e}})+\chi({\mathbf{d}},{\mathbf{e}})} \varepsilon({\mathbf{e}},{\mathbf{d}}),\qquad {\mathbf{d}},{\mathbf{e}}\in L. \end{align} $$

Then the product on ${\mathcal {H}}_Q$ defined by

(12) $$ \begin{align} f\star g=\varepsilon({\mathbf{d}},{\mathbf{e}})f*g,\qquad f\in{\mathcal{H}}_{Q,{\mathbf{d}}},\,g\in{\mathcal{H}}_{Q,{\mathbf{e}}} \end{align} $$

is super-commutative: $f\star g=(-1)^{p({\mathbf {d}})p({\mathbf {e}})}g\star f$ for all $f\in {\mathcal {H}}_{Q,{\mathbf {d}}}$ , $g\in {\mathcal {H}}_{Q,{\mathbf {e}}}$ . We shall avoid non-canonical choices and not use this sign twist. Instead, we shall interpret ${\mathcal {H}}_Q$ as a commutative algebra in $\operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ or $\operatorname {\mathrm {Vect}}^L$ with the symmetric monoidal category structure arising from $\chi $ .

3.3. Modules over CoHA

For more details about modules over CoHAs, see for example, [Reference Franzen18, Reference Soibelman62]. Let ${\mathbf {w}}\in {\mathbb {N}}^{I}$ be a vector, called a framing vector. We may consider a new (framed) quiver $Q^{{\mathbf {w}}}$ by adding a new vertex $\infty $ and ${\mathbf {w}}_i$ arrows $\infty \to i$ for all $i\in I$ . For any ${\mathbf {d}}\in {\mathbb {N}}^{I}$ , let $\bar {\mathbf {d}}=({\mathbf {d}},1)\in {\mathbb {N}}^{Q_0^{\mathbf {w}}}$ and

$$ \begin{align*}R_{{\mathbf{d}},{\mathbf{w}}}^{\mathsf{f}} =R(Q^{\mathbf{w}},\bar{\mathbf{d}}) =R_{\mathbf{d}}\oplus F_{{\mathbf{d}},{\mathbf{w}}} =R_{\mathbf{d}}\oplus\bigoplus_{i\in I}\operatorname{\mathrm{Hom}}({\mathbb{C}}^{{\mathbf{w}}_i},{\mathbb{C}}^{{\mathbf{d}}_i}).\end{align*} $$

Let $R^{{\mathsf {f}},\mathrm {st}}_{{\mathbf {d}},{\mathbf {w}}}\subset R^{\mathsf {f}}_{{\mathbf {d}},{\mathbf {w}}}$ be the open subset of stable representations, consisting of representations M generated by $M_\infty $ . Then $G_{\mathbf {d}}$ acts freely on $R^{{\mathsf {f}},\mathrm {st}}_{{\mathbf {d}},{\mathbf {w}}}$ , and we consider the moduli space

$$ \begin{align*}\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}}=R^{{\mathsf{f}},\mathrm{st}}_{{\mathbf{d}},{\mathbf{w}}}/G_{\mathbf{d}},\end{align*} $$

called the non-commutative Hilbert scheme. Using this moduli space, one can define a module over CoHA, denoted by ${\mathcal {M}}_{\mathbf {w}}$ . Its underlying L-graded object in $D(\operatorname {\mathrm {Vect}})\simeq \operatorname {\mathrm {Vect}}^{\mathbb {Z}}$ is

$$ \begin{align*}{\mathcal{M}}_{\mathbf{w}} =\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^I}{\mathcal{M}}_{{\mathbf{w}},{\mathbf{d}}},\qquad {\mathcal{M}}_{{\mathbf{w}},{\mathbf{d}}}=H^{\mathrm{BM}}(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}})[\chi({\mathbf{d}},{\mathbf{d}})-2{\mathbf{w}}\cdot{\mathbf{d}}].\end{align*} $$

The CoHA-action is defined as follows. Let $R^{\mathsf {f}}_{{\mathbf {d}},{\mathbf {e}},{\mathbf {w}}}\subset R^{\mathsf {f}}_{{\mathbf {d}}+{\mathbf {e}},{\mathbf {w}}}$ be the subspace of framed representations that preserve $V_{\mathbf {d}}=\bigoplus _{i\in I}{\mathbb {C}}^{{\mathbf {d}}_i}$ , and let $R^{{\mathsf {f}},\mathrm {st}}_{{\mathbf {d}},{\mathbf {e}},{\mathbf {w}}}\subset R^{\mathsf {f}}_{{\mathbf {d}},{\mathbf {e}},{\mathbf {w}}}$ be the open subset of stable representations. It is equipped with a free action of the group $G_{{\mathbf {d}},{\mathbf {e}}}$ , and we define $\operatorname {\mathrm {Hilb}}_{{\mathbf {d}},{\mathbf {e}},{\mathbf {w}}}=R^{{\mathsf {f}},\mathrm {st}}_{{\mathbf {d}},{\mathbf {e}},{\mathbf {w}}}/G_{{\mathbf {d}},{\mathbf {e}}}$ . We have morphisms of stacks and algebraic varieties

$$ \begin{align*}[R_{\mathbf{d}}/G_{\mathbf{d}}]\times\operatorname{\mathrm{Hilb}}_{{\mathbf{e}},{\mathbf{w}}} \xleftarrow{q}\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{e}},{\mathbf{w}}} \xrightarrow{p}\operatorname{\mathrm{Hilb}}_{{\mathbf{d}}+{\mathbf{e}},{\mathbf{w}}},\end{align*} $$

where q has dimension $n=-\chi ({\mathbf {e}},{\mathbf {d}})+{\mathbf {w}}\cdot {\mathbf {d}}$ . These maps induce morphisms in $D(\operatorname {\mathrm {Vect}})$

$$ \begin{align*}H^{\mathrm{BM}}_{G_{\mathbf{d}}}(R_{\mathbf{d}})\otimes H^{\mathrm{BM}}(\operatorname{\mathrm{Hilb}}_{{\mathbf{e}},{\mathbf{w}}}) \xrightarrow{q^*} H^{\mathrm{BM}}_{}(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{e}},{\mathbf{w}}})[-2n] \xrightarrow{p_*} H^{\mathrm{BM}}_{}(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}}+{\mathbf{e}},{\mathbf{w}}})[-2n]. \end{align*} $$

Taking the composition, we obtain the action map

$$ \begin{align*}{\mathcal{H}}_{Q,{\mathbf{d}}}\otimes{\mathcal{M}}_{{\mathbf{w}},{\mathbf{e}}}\to{\mathcal{M}}_{{\mathbf{w}},{\mathbf{d}}+{\mathbf{e}}}\end{align*} $$

in $D(\operatorname {\mathrm {Vect}}).$ The compatibility with the product of CoHA is proved in the same way as the associativity of that product.

The module ${\mathcal {M}}_{\mathbf {w}}$ can be also described using shuffle algebras [Reference Franzen17, Reference Franzen18]. Consider the forgetful map $j\colon R^{{\mathsf {f}},\mathrm {st}}_{{\mathbf {d}},{\mathbf {w}}}\to R_{\mathbf {d}}$ and the corresponding map of stacks

$$ \begin{align*}j\colon \operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}}\to[R_{\mathbf{d}}/G_{\mathbf{d}}]\end{align*} $$

having dimension ${\mathbf {w}}\cdot {\mathbf {d}}$ . It induces a map

$$ \begin{align*}j^*\colon H^{\mathrm{BM}}_{G_d}(R_{\mathbf{d}})\to H^{\mathrm{BM}}(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}})[-2{\mathbf{w}}\cdot{\mathbf{d}}],\end{align*} $$

which induces a map of L-graded objects in $D(\operatorname {\mathrm {Vect}})$

(13) $$ \begin{align} j^*\colon {\mathcal{H}}_Q=\bigoplus_{{\mathbf{d}}}{\mathcal{H}}_{Q,{\mathbf{d}}}\to{\mathcal{M}}_{\mathbf{w}}=\bigoplus_{\mathbf{d}}{\mathcal{M}}_{{\mathbf{w}},{\mathbf{d}}}. \end{align} $$

It was proved in [Reference Franzen17, Theorem 5.2.1] that this map is an epimorphism of ${\mathcal {H}}_Q$ -modules and that its kernel is equal to

(14) $$ \begin{align} \ker (j^*)= \sum_{{\mathbf{d}}'\ge0,{\mathbf{d}}>0}{\mathcal{H}}_{Q,{\mathbf{d}}'}* e_{\mathbf{d}}^{\mathbf{w}}{\mathcal{H}}_{Q,{\mathbf{d}}} ={\left({e_{\mathbf{d}}^{\mathbf{w}}{\mathcal{H}}_{Q,{\mathbf{d}}}{\colon} {\mathbf{d}}>0}\right)}, \end{align} $$

where the last expression means the ideal with respect to the product in ${\mathcal {H}}_Q$

$$ \begin{align*}e_{{\mathbf{d}}}^{{\mathbf{w}}}=\prod_{i\in I}\prod_{k=1}^{{\mathbf{d}}_i}x_{i,k}^{{\mathbf{w}}_i}\in\Lambda_{\mathbf{d}}\end{align*} $$

is the product of appropriate powers of elementary symmetric functions and $e_{\mathbf {d}}^{\mathbf {w}}{\mathcal {H}}_{Q,{\mathbf {d}}}$ means the product in $\Lambda _{\mathbf {d}}$ (corresponding to the cup product in cohomology).

3.4. Characters of CoHA and CoHA modules

Let us collect some well-known formulas for the Poincaré series (4) of our objects of interest. We begin with the Poincaré series of CoHA.

Proposition 3.2. We have

$$ \begin{align*}Z({\mathcal{H}}_Q,x,q) =A_Q(x,q) :=\sum_{{\mathbf{d}}\in{\mathbb{N}}^I} \frac{(-q^{\frac12})^{-\chi({\mathbf{d}},{\mathbf{d}})}}{(q{^{-1}})_{\mathbf{d}}}x^{\mathbf{d}}, \end{align*} $$

where $(q)_{\mathbf {d}}=\prod _{i\in I} (q)_{{\mathbf {d}}_i}$ and $(q)_n=\prod _{k=1}^n(1-q^k)$ .

Proof. Consider $\Lambda _n={\mathbb {Q}}[x_1,\dots ,x_n]^{\Sigma _n}\simeq {\mathbb {Q}}[e_1,\dots ,e_n]$ , where $\deg x_i=2$ and $\deg e_i=2i$ . Then

$$ \begin{align*}\operatorname{\mathrm{ch}}(\Lambda_n)=\prod_{k=1}^n\frac1{1-q^{-k}}=\frac1{(q{^{-1}})_n}.\end{align*} $$

This implies that $\operatorname {\mathrm {ch}}(\Lambda _{\mathbf {d}})=\frac 1{(q{^{-1}})_{\mathbf {d}}}$ , and therefore, ${\mathcal {H}}_{Q,{\mathbf {d}}}=\Lambda _{\mathbf {d}}[-\chi ({\mathbf {d}},{\mathbf {d}})]$ has the character

$$ \begin{align*}\operatorname{\mathrm{ch}}({\mathcal{H}}_{Q,{\mathbf{d}}}) =\sum_k\dim\Lambda_{\mathbf{d}}^{k-\chi({\mathbf{d}},{\mathbf{d}})}q^{-{\frac12} k} =q^{-{\frac12}\chi({\mathbf{d}},{\mathbf{d}})}\operatorname{\mathrm{ch}}(\Lambda_{\mathbf{d}}) =\frac{q^{-{\frac12}\chi({\mathbf{d}},{\mathbf{d}})}}{(q{^{-1}})_{\mathbf{d}}}.\\[-37pt] \end{align*} $$

Remark 3.3. We also have

$$ \begin{align*}A_Q(x,q) =\sum_{\mathbf{d}} \frac{(-q^{\frac12})^{-\chi({\mathbf{d}},{\mathbf{d}})}}{(q{^{-1}})_{\mathbf{d}}}x^{\mathbf{d}} =\sum_{\mathbf{d}} (-q^{\frac12})^{\chi({\mathbf{d}},{\mathbf{d}})} \frac{P_c(R_{\mathbf{d}},q)}{P_c(G_{\mathbf{d}},q)}x^{\mathbf{d}}, \end{align*} $$

where we used the fact that $P_c({\mathbb {C}}^n,q)=q^n$ and $P_c(\operatorname {\mathrm {GL}}_n({\mathbb {C}}),q)=q^{n^2}(q{^{-1}})_n$ .

Let us now determine the Poincaré series of the CoHA module ${\mathcal {M}}_{\mathbf {w}}$ .

Proposition 3.4. We have

$$ \begin{align*}Z(\mathcal M_{{\mathbf{w}}},x,q) =A_Q(x,q)\cdot S_{-2{\mathbf{w}}}A_Q(x,q){^{-1}}, \end{align*} $$

where $S_{{\mathbf {w}}}(x^{\mathbf {d}})=q^{{\frac 12}{\mathbf {w}}\cdot {\mathbf {d}}}x^{\mathbf {d}}$ .

Proof. It follows from [Reference Engel and Reineke15, Theorem 5.2] that

$$ \begin{align*}\sum_{\mathbf{d}} (-q^{\frac12})^{\chi({\mathbf{d}},{\mathbf{d}})}P_c(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}},q)x^{\mathbf{d}} =S_{2{\mathbf{w}}}A_Q(x,q)\cdot A_Q(x,q){^{-1}}.\end{align*} $$

As $\mathcal M_{{\mathbf {w}},{\mathbf {d}}}=H^{\mathrm {BM}}(\operatorname {\mathrm {Hilb}}_{{\mathbf {d}},{\mathbf {w}}})[\chi ({\mathbf {d}},{\mathbf {d}})-2{\mathbf {w}}\cdot {\mathbf {d}}]$ , we obtain

$$ \begin{align*}Z(\mathcal M_{{\mathbf{w}}}) =\sum_{\mathbf{d}} (-q^{{\frac12}})^{\chi({\mathbf{d}},{\mathbf{d}})-2{\mathbf{w}}\cdot {\mathbf{d}}} P_c(\operatorname{\mathrm{Hilb}}_{{\mathbf{d}},{\mathbf{w}}},q)x^{\mathbf{d}} =A_Q(x,q)\cdot S_{-2{\mathbf{w}}}A_Q(x,q){^{-1}}.\\[-40pt] \end{align*} $$

We define (refined) DT invariants $\Omega _{\mathbf {d}}(q)$ of the quiver Q by the formula

(15) $$ \begin{align} Z({\mathcal{H}}_Q,x,q{^{-1}})=A_Q(x,q{^{-1}})=\operatorname{\mathrm{Exp}}{\left({\frac{\sum_{\mathbf{d}} (-1)^{\chi({\mathbf{d}},{\mathbf{d}})}\Omega_{\mathbf{d}}(q{^{-1}}) x^{\mathbf{d}}}{1-q}}\right)}. \end{align} $$

By a theorem of Efimov [Reference Efimov14], we have $\Omega _{\mathbf {d}}(q)\in {\mathbb {N}}[q^{\pm {\frac 12}}]$ . We shall give a new proof of this result in Theorem 5.7. The above formula for the Poincaré series of the CoHA module can be written in the form

(16) $$ \begin{align} Z(\mathcal M_{{\mathbf{w}}},x,q{^{-1}})= \operatorname{\mathrm{Exp}}{\left({\sum_{\mathbf{d}}\frac{1-q^{{\mathbf{w}}\cdot {\mathbf{d}}}}{1-q}(-1)^{\chi({\mathbf{d}},{\mathbf{d}})}\Omega_{\mathbf{d}}(q{^{-1}})x^{\mathbf{d}}}\right)}. \end{align} $$

4.1. Graded vertex algebras

In this section, we will introduce L-graded vertex algebras. We start with recalling the classical definition of a vertex superalgebra and then explain its generalisation to the L-graded case.

4.1.1. Classical vertex (super)algebras

Let $\operatorname {\mathrm {Vect}}^{\mathbb {Z}_2}$ be the category of $\mathbb {Z}_2$ -graded vector spaces (also called super vector spaces), equipped with a symmetric monoidal category structure using the Koszul sign rule (2). Given a $\mathbb {Z}_2$ -graded vector space $V=V_0\oplus V_1$ and $a\in V_p$ , we call $p(a)=p\in \mathbb {Z}_2$ the parity of a. The $\mathbb {Z}_2$ -graded vector space $\underline {\textrm {End}}(V)=\underline {\textrm {Hom}}(V,V)$ is equipped with the Lie bracket

(17) $$ \begin{align} [a,b]=ab-(-1)^{p(a)p(b)}ba,\qquad a,b\in\underline{\textrm{End}}(V). \end{align} $$

Define the space of fields

$$ \begin{align*}{\mathcal{F}}(V)={\left\{\left.{\sum_{n\in\mathbb{Z}} a(n)z^{-n-1}\in\underline{\textrm{End}}(V){[\![{z^{\pm1}}]\!]}}\ \right\vert{\forall v\in V,\, a(n)v=0\text{ for }n\gg0}\right\}}. \end{align*} $$

This means that, for every $\mathfrak {a}(z)\in {\mathcal {F}}(V)$ and $v\in V$ , we have $\mathfrak {a}(z)v\in V{(\;\!\!\!(z)\!\!\!\;)} $ . We equip ${\mathcal {F}}(V)$ with the $\mathbb {Z}_2$ -grading, where $\mathfrak {a}(z)=\sum _n a(n)z^{-n-1}$ has parity p if $a(n)$ has parity p for all $n\in \mathbb {Z}$ . We say that two fields $\mathfrak {a}(z),\mathfrak {b}(z)\in {\mathcal {F}}(V)$ are (mutually) local if

(18) $$ \begin{align} (z-w)^n[\mathfrak{a}(z),\mathfrak{b}(w)]=0,\qquad n\gg0. \end{align} $$

A vertex (super)algebra is a triple , where V is a $\mathbb {Z}_2$ -graded vector space, $Y\colon V\to {\mathcal {F}}(V)$ is a linear map, and is an element called vacuum, such that, for all $a,b\in V$ ,

1. (1) $Y(a,z)=\sum _{n}a(n)z^{-n-1}$ has the same parity as a.

2. (2) , the identity operator.

3. (3) for $n\ge 0$ and .

4. (4) $[T,Y(a,z)]=\partial _z Y(a,z)$ , where $T\in \operatorname {\mathrm {End}}(V)$ is defined by and $\partial _z=\frac {\partial }{\partial z}$ .

5. (5) $Y(a,z)$ and $Y(b,z)$ are local.

Remark 4.1. One can show that, for any $a\in V$ ,

Therefore,

$$ \begin{align*}T(a(n)b)=(Ta)(n)b+a(n)(Tb),\qquad (Ta)(n)=-n a(n-1).\end{align*} $$

4.1.2. Graded vertex algebras

In the remaining part of section of §4, we shall mostly work in the category ${\mathcal {C}}^{{\frac 12}\mathbb {Z}}\simeq \operatorname {\mathrm {Vect}}^{L\times {\frac 12}\mathbb {Z}}$ (the corresponding ${\frac 12}\mathbb {Z}$ -degrees will be called weights). It has a symmetric monoidal category structure induced from that of ${\mathcal {C}}$ (without any additional signs coming from weights). Consider an object $V=\bigoplus _{n\in {\frac 12}\mathbb {Z}}V_n\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ bounded below, meaning that $V_n=0$ for $n\ll 0$ . We define the space of fields

$$ \begin{align*}{\mathcal{F}}(V)=\underline{\textrm{End}}(V){[\![{z^{\pm1}}]\!]}\in{\mathcal{C}}^{{\frac12}\mathbb{Z}},\end{align*} $$

where z has L-degree zero and weight $-1$ . Note that if $\sum _n a(n)z^{-n-1}\in {\mathcal {F}}(V)$ has weight k, then $a(n)$ has weight $k-n-1$ . This implies that, for any $b\in V$ , we have

$$ \begin{align*}a(n)b=0,\qquad n\gg0,\end{align*} $$

by the assumption on V. Locality of fields is defined in the same way as in (18), using the bracket on $\underline {\textrm {End}}(V)$ . We define an L-graded vertex algebra to be a triple , where $V\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ is bounded below, $Y\colon V\to {\mathcal {F}}(V)$ is a linear map, and , such that, for all $a,b\in V$ ,

1. (1) Y preserves degrees, meaning that, for $Y(a,z)=\sum _na(n)z^{-n-1}$ , the operator $a(n)$ has the same L-degree as a and the weight $\operatorname {\mathrm {wt}}(a)-n-1$ .

2. (2) .

3. (3) for $n\ge 0$ and .

4. (4) $[T,Y(a,z)]=\partial _z Y(a,z)$ , where $T\in \underline {\textrm {End}}(V)$ is defined by .

5. (5) $Y(a,z)$ and $Y(b,z)$ are local.

Note that the map $T\colon V\to V$ has L-degree zero and weight $1$ . Note also that the map $Y\colon V\to {\mathcal {F}}(V)$ is encoded by the products

(19) $$ \begin{align} (n)\colon V\otimes V\to V,\qquad a\otimes b\mapsto a(n)b,\qquad n\in\mathbb{Z}. \end{align} $$

By definition, a morphism $f\colon V\to W$ between two graded vertex algebras is a degree zero linear map that preserves the vacuum and all these products. A module over a graded vertex algebra V is a graded vector space M equipped with a degree zero linear map $Y_M\colon V\to {\mathcal {F}}(M)$ that preserves products, maps to $\operatorname {\mathrm {id}}_M$ and maps V to a subspace of mutually local fields.

Remark 4.2. For a lattice L equipped with a symmetric bilinear form $(\cdot ,\cdot )$ , it is conventional to use the braiding on $\operatorname {\mathrm {Vect}}^{L}$ given by the Koszul signs $(-1)^{p(\alpha )p(\beta )}$ , where $p(\alpha ) = (\alpha ,\alpha )\pmod 2$ . This is justified by the desire to work in the familiar setting of vertex (super)algebras. However, this convention requires one to introduce certain non-canonical sign twists [Reference Kac34, §5.4] (needed to ensure mutual locality of vertex operators in lattice vertex algebras), very similar to the sign twists from Remark 3.1. Our convention for the braiding in $\operatorname {\mathrm {Vect}}^L$ (see §2.2) allows one to avoid any non-canonical choices both in the case of CoHAs and in the case of vertex algebras.

4.1.3. External products

Let $V\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ be bounded below, and let $\mathfrak {a}\in {\mathcal {F}}(V)_\alpha $ , $\mathfrak {b}\in {\mathcal {F}}(V)_\beta $ be two fields (here $\alpha ,\beta \in L$ ). For any $n\in \mathbb {Z}$ , we define

(20) $$ \begin{align} [\mathfrak{a}(z)\times_n\mathfrak{b}(w)] =(z-w)^n\mathfrak{a}(z)\mathfrak{b}(w) -(-1)^{(\alpha,\beta)}(-w+z)^n \mathfrak{b}(w)\mathfrak{a}(z) \end{align} $$

and consider the binary operation on ${\mathcal {F}}(V)$ defined by

(21)

We have by [Reference Li41, Lemma 3.1.4]. By Dong’s lemma [Reference Li41, Proposition 3.2.7], if $\mathfrak {a},\mathfrak {b},\mathfrak {c}\in {\mathcal {F}}(V)$ and $\mathfrak {a},\mathfrak {b}$ are local with $\mathfrak {c}$ , then is local with $\mathfrak {c}$ , for all $n\in \mathbb {Z}$ .

The coefficients of the field can be written explicitly as

(22)

In particular, for $m\ge 0$ , we have

(23)

For $m=-1$ , we obtain

Remark 4.3. The field

(24)

is the so-called normally ordered product $:{\mathfrak {a}}(z){\mathfrak {b}}(z)):$ of the fields $\mathfrak {a}$ and $\mathfrak {b}$ . Here, $f_+(z)=\sum _{n\ge 0}f_nz^n$ and $f_-(z)=\sum _{n<0}f_nz^n$ , for $f(z)=\sum _n f_nz^n$ .

Lemma 4.4. If V is a vertex algebra, then $Y\colon V\to {\mathcal {F}}(V)$ preserves the products (19), meaning that $a(n)b$ is mapped to for all $a,b\in V$ and $n\in \mathbb {Z}$ .

Proof. By the Jacobi identity [Reference Frenkel, Lepowsky and Meurman23], for any $a,b\in V$ , we have

(25) $$ \begin{align} [Y(a,z)\times_n Y(b,w)]=\operatorname{\mathrm{Res}}_u u^nw{^{-1}} Y(Y(a,u)b,w) \delta{\left({\frac{z-u}{w}}\right)}. \end{align} $$

Applying $\operatorname {\mathrm {Res}}_z\delta {\left ({\frac {z-u}{w}}\right )}=w$ , we obtain

$$ \begin{align*}\operatorname{\mathrm{Res}}_z[Y(a,z)\times_n Y(b,w)] =\operatorname{\mathrm{Res}}_u u^n Y(Y(a,u)b,w)=Y(a(n)b,w).\end{align*} $$

This implies the statement.

For any $\mathfrak {a}\in {\mathcal {F}}(V)$ and $n\in \mathbb {Z}$ , we obtain , and we define

Let $\operatorname {\mathrm {id}}_V\in \underline {\textrm {End}}(V)\subset {\mathcal {F}}(V)$ be the identity operator and $T=\partial _z\in \underline {\textrm {End}}({\mathcal {F}}(V))$ . Then, for any $\mathfrak {a}\in {\mathcal {F}}(V)$ , we have [Reference Li41, Lemmas 3.1.6-3.1.7]

1. (1) $Y(\operatorname {\mathrm {id}}_V,u)\mathfrak {a}=\mathfrak {a}$ .

2. (2) $Y(\mathfrak {a},u)\operatorname {\mathrm {id}}_V=e^{uT}\mathfrak {a}$ .

3. (3) $Y(T\mathfrak {a},u)=[T,Y(\mathfrak {a},u)]=\partial _u Y(\mathfrak {a},u)$ .

The last property implies . In particular,

A subspace $W\subset {\mathcal {F}}(V)$ is called a field algebra if it is closed under products and contains $\operatorname {\mathrm {id}}_V$ . It is called a local field algebra if it consists of mutually local fields. In this case, consider the restriction

By the locality assumption, we have $[\mathfrak {a}(z)\times _n\mathfrak {b}(w)]=0$ ; hence, , for $\mathfrak {a},\mathfrak {b}\in W$ and $n\gg 0$ . Therefore, $Y(\mathfrak {a},u)$ is a field, and we obtain a linear map $Y:W\to {\mathcal {F}}(W)$ . The triple $(W,Y,\operatorname {\mathrm {id}}_V)$ is a vertex algebra by [Reference Li41, Theorem 3.2.10]. For any subset $S\subset {\mathcal {F}}(V)$ , consisting of mutually local (homogeneous) fields, let ${{\left \langle S\right \rangle }} $ be the smallest subspace of ${\mathcal {F}}(V)$ containing $S\cup {{\left \{{\operatorname {\mathrm {id}}_V}\right \}}}$ and closed under all products . Then by Dong’s lemma, ${{\left \langle {S}\right \rangle }}$ is a local field algebra, and we conclude that it is a vertex algebra.

In particular, assume that is a vertex algebra. The map $Y\colon V\to {\mathcal {F}}(V)$ is injective as . Let $W=Y(V)\subset {\mathcal {F}}(V)$ . Then W consists of mutually local fields by the definition of a vertex algebra. It is closed with respect to products by Lemma 4.4, and it contains . Therefore, W is a vertex algebra and $Y\colon V\to W$ is an isomorphism of vertex algebras.

4.2. Conformal algebras

In this section, we will introduce the notion of conformal algebras, closely related to the notion of vertex algebras. For more details on this subject, see, for example, [Reference Primc51, Reference Roitman57].

4.2.1. Definition of conformal algebras

We define a conformal algebra to be an object $C\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ equipped with a linear map $\partial \colon {\mathfrak {C}}\to {\mathfrak {C}}$ (of L-degree zero and weight $1$ ) and bilinear operations (of L-degree zero and weight $-n-1$ ), for $n\in {\mathbb {N}}$ , such that for all $a,b\in {\mathfrak {C}}$ , we have

1. (1) for $n\gg 0$ .

2. (2) .

3. (3) .

Let us recall a fundamental class of examples of conformal algebras obtained via a procedure reminiscent of the construction of external products in §4.1.3. Suppose that ${\mathfrak {L}}\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ is an algebra, by which we mean a graded vector space equipped with a bilinear operation

$$ \begin{align*}[-,-]\colon{\mathfrak{L}}\otimes{\mathfrak{L}}\to{\mathfrak{L}} \end{align*} $$

(despite the suggestive notation, we do not assume this operation to satisfy any of the properties of a Lie bracket). We define external products on the space ${\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ (where z has L-degree zero and weight $-1$ ) by the formula (cf. (21))

(26)

for $\mathfrak {a}(z)=\sum _{n\in \mathbb {Z}}a(n)z^{-n-1}$ and $\mathfrak {b}(z)=\sum _{n\in \mathbb {Z}}b(n)z^{-n-1}$ in ${\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ . The coefficient of $z^{-m-1}$ in this series equals (cf. (23))

(27)

We also define $\partial =\frac {\partial }{\partial z}\colon {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}\to {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ , so that the map $\partial $ and the products satisfy the axioms (2) and (3) above. Note that the axiom (1) is satisfied by the fields $\mathfrak {a},\mathfrak {b}\in {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ if they are (mutually) local, meaning that

$$ \begin{align*}(z-w)^n[\mathfrak{a}(z),\mathfrak{b}(w)]=(z-w)^n[\mathfrak{b}(w),\mathfrak{a}(z)]=0,\qquad n\gg0.\end{align*} $$

In this case, we have [Reference Kac34, §2.2]

(28)

We see that if ${\mathfrak {C}}\subset {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ is a subspace consisting of pairwise mutually local elements which is closed under all external products and the map $\partial $ , then ${\mathfrak {C}}$ is a conformal algebra.

4.2.2. Coefficient algebras of conformal algebras

For a conformal algebra , we shall now define the coefficient algebra $\operatorname {\mathrm {Coeff}}( )$ . Note that we have

hence, $\partial C\subset C$ is an ideal with respect to the product , and we can equip $C/\partial C$ with the bracket

(29)

In particular, we can apply this construction to the affinization of C [Reference Kac34, §2.7], which is defined to be $\tilde C=C[t^{\pm 1}]$ equipped with the derivation $\tilde \partial =\partial \otimes 1+1\otimes \partial _t$ and the products

(30)

where ${\partial _t^{(k)}} =\frac 1{k!}\partial _t^k$ . Now we define the algebra of coefficients

$$ \begin{align*}{\mathfrak{L}}=\operatorname{\mathrm{Coeff}}(C)=\tilde C/\tilde\partial\tilde C\end{align*} $$

with the bracket given by . More explicitly, let us write $C(n)=Ct^n\subset \tilde C$ and $a(n)=at^n\in C(n)$ , for $a\in C$ , $n\in \mathbb {Z}$ . Then ${\mathfrak {L}}$ is equal to the quotient of $\tilde C=\bigoplus _{n\in \mathbb {Z}}C(n)$ by the subspace generated by

(31) $$ \begin{align} (\partial a)(n)+na(n-1),\qquad a\in {\mathfrak{C}},\, n\in\mathbb{Z}. \end{align} $$

The operation $[-,-]$ given by (cf. (28))

(32)

descends from $\tilde C$ to ${\mathfrak {L}}$ . We will usually denote the image of $a(n)\in \tilde C$ in ${\mathfrak {L}}$ by $a(n)$ . The map

$$ \begin{align*}\partial:{\mathfrak{L}}\to{\mathfrak{L}},\qquad \partial(a(n))=(\partial a)(n)=-na(n-1)\end{align*} $$

is a derivation of the algebra ${\mathfrak {L}}$ . Let

$$ \begin{align*}{\mathfrak{L}}(n)=\operatorname{\mathrm{im}}\Big({\mathfrak{C}}(n)\hookrightarrow \tilde C\to {\mathfrak{L}}\Big).\end{align*} $$

From (31), we obtain that ${\mathfrak {L}}(n-1)$ embeds into ${\mathfrak {L}}(n)$ for $n\ne 0$ . Therefore, we have increasing filtrations

$$ \begin{align*}{\mathfrak{L}}(0)\subset {\mathfrak{L}}(1)\subset\dots\subset {\mathfrak{L}}_+:=\sum_{n\ge0}{\mathfrak{L}}(n),\qquad \dots\subset {\mathfrak{L}}(-2)\subset {\mathfrak{L}}(-1)=: {\mathfrak{L}}_-.\end{align*} $$

All of these subspaces are preserved by the derivation $\partial $ . It is clear from (32) that

$$ \begin{align*}[{\mathfrak{L}}(m),{\mathfrak{L}}(n)]\subset{\mathfrak{L}}(m+n)\end{align*} $$

for $m,n\ge 0$ or $m,n<0$ . In particular, ${\mathfrak {L}}_\pm $ are subalgebras of ${\mathfrak {L}}$ . Moreover, we have an internal direct sum decomposition [Reference Primc51, Reference Roitman57]

(33) $$ \begin{align} {\mathfrak{L}}={\mathfrak{L}}_+\oplus {\mathfrak{L}}_- \end{align} $$

of the vector space ${\mathfrak {L}}$ , and an isomorphism of vector spaces

$$ \begin{align*}{\mathfrak{C}}\to {\mathfrak{L}}_-={\mathfrak{L}}(-1),\qquad a\mapsto a(-1).\end{align*} $$

Under the isomorphism $C\simeq {\mathfrak {L}}_-$ , the bracket on C is given by [Reference Primc51, p. 4.14]

(34)

Note that the map $C\to \tilde C$ , $a\mapsto a(0)$ , is a homomorphism of conformal algebras; hence, it induces a morphism of algebras

(35) $$ \begin{align} C/\partial C\to {\mathfrak{L}}(0)\subset{\mathfrak{L}}=\tilde C/\tilde\partial\tilde C,\qquad a+\partial C\mapsto a(0). \end{align} $$

By [Reference Roitman57, Prop. P 1.3], the map $C\to {\mathfrak {L}}(0),\,a\mapsto a(0)$ has the kernel $\partial C+\sum _{k\ge 1}{\ker \partial ^k}$ ; hence, $C/\partial C\to {\mathfrak {L}}(0)$ is an isomorphism if $\partial $ is injective. It is important to note that the brackets on C and $C/\partial C$ are not related.

The map

$$ \begin{align*}\phi\colon {\mathfrak{C}}\to {\mathfrak{L}}{[\![{z^{\pm1}}]\!]},\qquad a\mapsto \sum_{n\in\mathbb{Z}}a(n)z^{-n-1}\end{align*} $$

is a homomorphism of conformal algebras (meaning that it preserves the products and the derivation). Since $a\mapsto a(-1)$ is an isomorphism between ${\mathfrak {C}}$ and ${\mathfrak {L}}(-1)$ , this map is injective.

The coefficient algebra ${\mathfrak {L}}=\operatorname {\mathrm {Coeff}}({\mathfrak {C}})$ satisfies the following universal property: if ${\mathfrak {A}}$ is an algebra and $\psi \colon {\mathfrak {C}}\to {\mathfrak {A}}{[\![{z^{\pm 1}}]\!]}$ is a morphism of conformal algebras, then there exists a unique algebra morphism $\varrho \colon {\mathfrak {L}}\to {\mathfrak {A}}$ (given by $a(n)\mapsto \operatorname {\mathrm {Res}}_z z^n\psi (a)$ ) such that the diagram

is commutative.

4.2.3. Vertex Lie algebras and their enveloping vertex algebras

Using coefficient algebras, one can define particular types of conformal algebras. In particular, a conformal algebra ${\mathfrak {C}}$ is called a vertex Lie algebra (or Lie conformal algebra) if the algebra $\operatorname {\mathrm {Coeff}}({\mathfrak {C}})$ is a Lie algebra. Equivalently, this means that, for $a,b\in {\mathfrak {C}}$ and $m,n\in {\mathbb {N}}$ , we have

1. (1) (Jacobi identity), or equivalently,

   .

2. (2) , for $a{\kern-1pt}\in{\kern-1pt} {\mathfrak {C}}_\alpha ,b{\kern-1pt}\in{\kern-1pt} {\mathfrak {C}}_\beta $ (anti-commutativity).

Under these conditions, the bracket (29) defined on $C/\partial C$ is a Lie bracket; see, for example, [Reference Kac34, Rem. 2.7a]. Let ${\mathfrak {C}}$ be a vertex Lie algebra and ${\mathfrak {L}}=\operatorname {\mathrm {Coeff}}({\mathfrak {C}})$ be its coefficient algebra. We define the Verma module

$$ \begin{align*}V:=U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}\simeq U({\mathfrak{L}}_-),\end{align*} $$

where ${\mathbb {Q}}$ is equipped with the trivial ${\mathfrak {L}}_+$ -module structure. Let be the cyclic vector of this module. One can show that, for any $a\in {\mathfrak {C}}$ and $v\in V$ , we have $a(n)v=0$ for $n\gg 0$ . This implies that $\phi (a)=\sum _n a(n)z^{-n-1}\in {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ induces a field $\psi (a)\in {\mathcal {F}}(V)$ . The map $\psi :{\mathfrak {C}}\to {\mathcal {F}}(V)$ is injective as the map ${\mathfrak {C}}\simeq {\mathfrak {L}}_-\to \underline {\textrm {End}}(V)$ is injective.

The image of $\psi \colon {\mathfrak {C}}\to {\mathcal {F}}(V)$ consists of mutually local fields and generates a vertex algebra $W\subset {\mathcal {F}}(V)$ . By [Reference Roitman57, §2.4], the vertex algebra W has a structure of an ${\mathfrak {L}}$ -module, and the map $W\to V$ , is an isomorphism of ${\mathfrak {L}}$ -modules. We shall call the vertex algebra $W\simeq V$ the universal enveloping vertex algebra of ${\mathfrak {C}}$ and denote it by ${\mathcal {U}}(C)$ . Note that we have a commutative diagram of graded vector spaces

If V is a vertex algebra, then V equipped with the products $(n)\colon V\otimes V\to V$ , $n\ge 0$ , and the map $\partial =T\colon V\to V$ is a vertex Lie algebra. By [Reference Primc51, Theorem 5.5], the corresponding forgetful functor from the category of vertex algebras to the category of vertex Lie algebras has a left adjoint functor which is precisely the universal enveloping algebra functor $C\mapsto {\mathcal {U}}(C)$ .

4.3. Vertex bialgebras

Let ${\mathcal {C}}$ be a symmetric monoidal category and $(V,\Delta ,\varepsilon )$ be a (counital) coalgebra in ${\mathcal {C}}$ . We define the set of group-like elements

(36)

where is the unit object equipped with the canonical structure of a counital coalgebra. We say that V is connected if $G(V)$ contains exactly one element.

For example, let $L=\mathbb {Z}^I$ be equipped with a symmetric bilinear form and let ${\mathcal {C}}=\operatorname {\mathrm {Vect}}^L$ be the corresponding symmetric monoidal category. Let $V\in {\mathcal {C}}$ be a coalgebra with degrees concentrated in ${\mathbb {N}}^I\subset L$ and $\dim V_0=1$ . Then V is connected.

Following [Reference Li42, §4], we define a (local) vertex bialgebra to be a vertex algebra V equipped with a coalgebra structure $(V,\Delta ,\varepsilon )$ such that $\Delta :V\to V\otimes V$ and $\varepsilon :V\to {\mathbb {Q}}$ are homomorphisms of vertex algebras. We say that a vertex bialgebra V is cocommutative if its coproduct is cocommutative. We define the subspace of primitive elements

(37) $$ \begin{align} P(V)={\left\{\left.{x\in V}\ \right\vert{\Delta(x)=x\otimes 1+1\otimes x}\right\}}. \end{align} $$

An important family of cocommutative connected vertex bialgebras arises from vertex Lie algebras [Reference Han, Li and Xiao29, Reference Li42].

Proposition 4.5. [Reference Li42, §4] Let C be a vertex Lie algebra and $V={\mathcal {U}}(C)$ be its universal enveloping vertex algebra. Then there exists vertex algebra homomorphisms

$$ \begin{align*}\Delta:V\to V\otimes V,\qquad\varepsilon:V\to{\mathbb{Q}},\end{align*} $$

uniquely determined by

The vertex algebra V equipped with $\Delta $ and $\varepsilon $ is a cocommutative vertex bialgebra.

Conversely, we can associate a vertex Lie algebra with a vertex bialgebra.

The next result is an analogue of the Milnor-Moore theorem for vertex bialgebras.

4.4. Free vertex algebras

As was observed already in [Reference Borcherds4, §4], the locality axiom contains a quantifier that does not allow one to define the free vertex algebra on a given set of generators, but if one restricts to some specific orders of locality, it is possible to define such a universal object; this has been done in [Reference Roitman57]. In this section, we recall the definition of the locality order and explain how free vertex algebras of given non-negative locality are constructed.

4.4.1. Locality order

Definition 4.9. Let $V\in {\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ be a bounded below object. We say that two fields $\mathfrak {a}\in {\mathcal {F}}(V)_\alpha $ and $\mathfrak {b}\in {\mathcal {F}}(V)_\beta $ are (mutually) local of order $N\in \mathbb {Z}$ if

(38) $$ \begin{align} (z-w)^N\mathfrak{a}(z)\mathfrak{b}(w) -(-1)^{(\alpha,\beta)}(-w+z)^N \mathfrak{b}(w)\mathfrak{a}(z) =0, \end{align} $$

where the binomial expansion convention is used. Note that this condition means that $[\mathfrak {a}(z)\times _N\mathfrak {b}(w)]=0$ ; see (20).

We note that for $N\ge 0$ , the binomial expansion is finite and the locality condition becomes simply

$$ \begin{align*}(z-w)^N[\mathfrak{a}(z),\mathfrak{b}(w)]=0,\end{align*} $$

being in agreement with the definition of locality (of unspecified order) in §4.1.1.

Lemma 4.10. Let V be a vertex algebra and $a,b\in V$ . Then $Y(a,z),Y(b,z)$ are local of order N if and only if $a(n)b=0$ for all $n\ge N$ .

Proof. Let $\mathfrak {a}(z)=Y(a,z)$ and $\mathfrak {b}(z)=Y(b,z)$ . If $[\mathfrak {a}(z)\times _N \mathfrak {b}(w)]=0$ , then $[\mathfrak {a}(z)\times _n \mathfrak {b}(w)]=0$ for all $n\ge N$ . This implies that . As $Y\colon V\to {\mathcal {F}}(V)$ is injective and preserves products, we conclude that $a(n)b=0$ for all $n\ge N$ . Conversely, assume that $a(n)b=0$ for all $n\ge N$ . By the Jacobi identity (25),

$$ \begin{align*}[\mathfrak{a}(z)\times_N \mathfrak{b}(w)] =\operatorname{\mathrm{Res}}_u u^Nw{^{-1}} Y(\mathfrak{a}(u)b,w)\delta{\left({\frac{z-u}{w}}\right)} .\end{align*} $$

Under our assumption, all powers of u on the right are non-negative, and we conclude that $[\mathfrak {a}(z)\times _N\mathfrak {b}(w)]=0$ .

Using the binomial expansion convention in Equation (38) and extracting the coefficient of $z^{N-m-1}w^{-n-1}$ , one can obtain an equivalent collection of equations

(39) $$ \begin{align} \sum_{k\ge0}(-1)^k\binom Nk & a(m-k)b(n+k)-\\ \nonumber & (-1)^{(\alpha,\beta)}\sum_{k\le N}(-1)^{k}\binom N{N-k} b(n+k)a(m-k)=0 \end{align} $$

for all $m,n\in \mathbb {Z}$ . If $N\ge 0$ , then the locality conditions can be written in the form

(40) $$ \begin{align} \sum_{k=0}^N(-1)^k\binom Nk[a(m-k),b(n+k)]=0. \end{align} $$

4.4.3. Free vertex algebras of non-negative locality

Suppose that the locality function ${\mathcal {N}}$ is non-negative. We set

$$ \begin{align*}X={ {\left\{{i(n){\colon} i\in I,\, n\in\mathbb{Z}}\right\}}}.\end{align*} $$

The vector space ${\mathbb {Q}} X$ spanned by X can be viewed as an object of ${\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ , where we set

(41) $$ \begin{align} \deg_L i(n)=\deg_L(i)\in L, \qquad \operatorname{\mathrm{wt}} i(n)=\operatorname{\mathrm{wt}}(i)-n-1\in{\frac12}\mathbb{Z}. \end{align} $$

In the free associative algebra $\mathsf {T}({\mathbb {Q}} X)$ , we may consider the locality relations (40) with $a,b\in I$ and $N={\mathcal {N}}(a,b)$ . These relations are manifestly homogeneous, so they generate a homogeneous ideal ${\mathcal {I}}\subset \mathsf {T}({\mathbb {Q}} X)$ . The quotient ${\mathcal {B}} =\mathsf {T}({\mathbb {Q}} X)/{\mathcal {I}}$ by that ideal is an associative algebra in ${\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ . Let us consider the subset

$$ \begin{align*}X_+={\left\{\left.{i(n)\in X}\ \right\vert{n\ge0}\right\}}\end{align*} $$

and the left ${\mathcal {B}}$ -module ${\mathcal {P}}={\mathcal {B}} /{\mathcal {B}} X_+$ ; we denote by the cyclic element of that module.

Lemma 4.11. The module ${\mathcal {P}}$ is restricted, meaning that, for all $i\in I$ and $x\in {\mathcal {P}}$ , we have $i(n)x=0$ for $n\gg 0$ .

Proof. Let $w=i_1(n_1)\dots i_p(n_p)$ be a monomial in $\mathsf {T}({\mathbb {Q}} X)$ , and suppose that there exists $i\in I$ such that $i(n)w\notin {\mathcal {B}} X_+$ for arbitrary large n. We can assume that p is minimal possible. Then

$$ \begin{align*}i_1(k)i_2(n_2)\dots i_p(n_p)\in {\mathcal{B}} X_+\end{align*} $$

for $k\gg 0$ . Therefore, we can assume that for any $k>n_1$ , we have

$$ \begin{align*}i(n)i_1(k)i_2(n_2)\dots i_p(n_p)\in {\mathcal{B}} X_+\end{align*} $$

for $n\gg 0$ . Let $N={\mathcal {N}}(i,i_1)$ . Then we have a locality relation

$$ \begin{align*} \sum_{k=0}^N(-1)^k\binom Nk[i(n-k),i_1(n_1+k)]=0 \end{align*} $$

in the algebra ${\mathcal {B}}$ . This relation implies that $i(n)w$ is contained in the left ideal generated by elements

$$ \begin{align*}i(n')i_1(k)i_2(n_2)\dots i_p(n_p),\qquad n'\ge n-N,\,k>n_1,\end{align*} $$

$$ \begin{align*}i(n')i_2(n_2)\dots i_p(n_p),\qquad n'\ge n-N.\end{align*} $$

By our assumption, these elements are contained in ${\mathcal {B}} X_+$ for $n\gg 0$ .

This lemma implies that for every $i\in I$ , we have a field

$$ \begin{align*}{\mathbf i}(z)=\sum_{n\in\mathbb{Z}}i(n)z^{-n-1}\in {\mathcal{F}}({\mathcal{P}}).\end{align*} $$

These fields are mutually local by construction. Let us define a derivation T of $\mathsf {T}({\mathbb {Q}} X)$ by the formula $T i(n)=-ni(n-1)$ .

Lemma 4.12. The derivation T preserves the ideal ${\mathcal {I}}$ . In particular, T induces endomorphisms of ${\mathcal {B}}$ and ${\mathcal {P}}$ .

Proof. Let us consider ${\mathbf i}(z)=\sum _{n\in \mathbb {Z}}i(n)z^{-n-1}$ as an element of $\mathsf {T}({\mathbb {Q}} X){[\![{z^{\pm 1}}]\!]}$ . Using this notation, we see that the relations that generate ${\mathcal {I}}$ are coefficients of $(z-w)^N[{\mathbf i}(z),\mathfrak {j}(w)]$ with $N={\mathcal {N}}(i,j)$ . Since $T{\mathbf i}(z)=\partial _z{\mathbf i}(z)$ , we have

$$ \begin{align*}\partial_z\Big((z-w)^N[{\mathbf i}(z),\mathfrak{j}(w)]\Big) =N(z-w)^{N-1}[{\mathbf i}(z),\mathfrak{j}(w)]+(z-w)^N[T{\mathbf i}(z),\mathfrak{j}(w)]\end{align*} $$

and

$$ \begin{align*}\partial_w\Big((z-w)^N[{\mathbf i}(z),\mathfrak{j}(w)]\Big) =-N(z-w)^{N-1}[{\mathbf i}(z),\mathfrak{j}(w)]+(z-w)^N[{\mathbf i}(z),T\mathfrak{j}(w)].\end{align*} $$

Adding these equations, we obtain

$$ \begin{align*}T(z-w)^N[{\mathbf i}(z),\mathfrak{j}(w)]= \partial_z\Big((z-w)^N[{\mathbf i}(z),\mathfrak{j}(w)]\Big)+\partial_w\Big((z-w)^N[{\mathbf i}(z),\mathfrak{j}(w)]\Big), \end{align*} $$

which means that locality relations are preserved by T.

Applying the reconstruction theorem (see [Reference Kac34, Theorem 4.5] or [Reference Frenkel and Ben-Zvi20, §4.4]) to the fields ${\mathbf i}(z)\in {\mathcal {F}}({\mathcal {P}})$ and the endomorphism $T\in \operatorname {\mathrm {End}}({\mathcal {P}})$ , we obtain a vertex algebra structure on ${\mathcal {P}}$ with

The vertex algebra ${\mathcal {P}}$ is the free vertex algebra with respect to locality ${\mathcal {N}}$ and degree function $\deg $ . We will sometimes denote the element by i.

4.4.4. Free vertex algebras as universal enveloping algebras

Next, we will show that the free vertex algebra ${\mathcal {P}}$ of non-negative locality can be identified with the universal enveloping algebra of a vertex Lie algebra. Let ${\mathfrak {L}}$ be the Lie algebra with generators $i(n)\in X$ subject to the locality relations (40) with $a,b\in I$ and $N={\mathcal {N}}(a,b)$ , and let ${\mathfrak {L}}_+\subset {\mathfrak {L}}$ be the subalgebra generated by $X_+\subset X$ . Then we have

$$ \begin{align*}{\mathcal{B}}\simeq U({\mathfrak{L}}),\qquad {\mathcal{P}}={\mathcal{B}}/{\mathcal{B}} X_+\simeq U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}.\end{align*} $$

For every $i\in I$ , we consider ${\mathbf i}(z)=\sum _n i(n)z^{-n-1}\in {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ . These elements are mutually local by definition; therefore, by Dong’s Lemma [Reference Li41, Prop. 3.2.7], they generate, under non-negative external products and the derivation $\partial _z$ , a conformal algebra

$$ \begin{align*}{\mathfrak{C}}\subset {\mathfrak{L}}{[\![{z^{\pm1}}]\!]},\end{align*} $$

which is a vertex Lie algebra as conditions from §4.2.3 are automatically satisfied.

Proposition 4.13. The free vertex algebra ${\mathcal {P}}$ of non-negative locality is isomorphic to the universal enveloping vertex algebra of the vertex Lie algebra C.

Proof. First, let us show that the algebra morphism

$$ \begin{align*}\varrho:\operatorname{\mathrm{Coeff}}({\mathfrak{C}})\to {\mathfrak{L}},\qquad \mathfrak{a}(z)(n)\mapsto\operatorname{\mathrm{Res}}_z z^n\mathfrak{a}(z)\end{align*} $$

is an isomorphism (cf. [Reference Roitman57, Prop. 3.1]). (In fact, this vertex Lie algebra is the free vertex Lie algebra with respect to locality ${\mathcal {N}}$ and degree function $\deg $ .) To construct the inverse map, we need to show that elements ${\mathbf i}(z)(n)$ satisfy the locality relations. This means that for $N={\mathcal {N}}(i,j)$ , we have

$$ \begin{align*}\sum_{k=0}^N(-1)^k\binom Nk [{\mathbf i}(z)(m-k),\mathfrak{j}(z)(n+k)]=0\end{align*} $$

in $\operatorname {\mathrm {Coeff}}(C)$ . Using the formula for the bracket in $\operatorname {\mathrm {Coeff}}(C)$ , this equation can be written in the form

Note that for $l\ge N$ by the locality relations. We claim that, however, for any $0\le l<N$ , we have

$$ \begin{align*}\sum_{k=0}^N(-1)^k\binom Nk \binom{m-k}{l}=0. \end{align*} $$

This claim would imply the required statement. We have

$$ \begin{align*} \sum_{k,l\ge0}\binom Nk \binom{m-k}{l}x^ky^l =\sum_{k\ge0}\binom Nk x^k(1+y)^{m-k} =(1+y)^m{\left({1+\frac x{1+y}}\right)}^N\\ =(1+x+y)^N(1+y)^{m-N}. \end{align*} $$

In particular, for $x=-1$ , we obtain

$$ \begin{align*}\sum_{k,l\ge0}(-1)^k\binom Nk \binom{m-k}{l}y^l =y^N(1+y)^{m-N}, \end{align*} $$

which has only powers of y that are $\ge N$ . This proves the claim.

The derivation $\partial $ on $\operatorname {\mathrm {Coeff}}(C)$ corresponds to the derivation T on ${\mathfrak {L}}\subset {\mathcal {B}}$ from Lemma 4.12. Using the isomorphism $\varrho $ , we obtain the direct sum decomposition

$$ \begin{align*}{\mathfrak{L}}={\mathfrak{L}}_+\oplus{\mathfrak{L}}_-,\end{align*} $$

where both subalgebras are preserved by T. By [Reference Roitman57, §3.2], the algebra ${\mathfrak {L}}_+\subset {\mathfrak {L}}$ is exactly the algebra generated by $X_+={\left \{\left .{i(n)\in X}\ \right \vert {n\ge 0}\right \}}$ , which we introduced earlier.

This implies that ${\mathcal {P}}={\mathcal {B}}/{\mathcal {B}} X_+$ is isomorphic to the Verma module $U({\mathfrak {L}})\otimes _{U({\mathfrak {L}}_+)}{\mathbb {Q}}$ as a ${\mathcal {B}}$ -module, and therefore, the free vertex algebra ${\mathcal {P}}$ is isomorphic to the universal enveloping vertex algebra ${\mathcal {U}}({\mathfrak {C}})$ .

4.5. Lattice vertex algebras

In this section, we recall an important class of vertex algebras called lattice vertex algebras. They play a prominent role in our work, for they can be used to construct convenient realisations of certain free vertex algebras.

4.5.1. Fock spaces

We consider the ${\mathbb {Q}}$ -vector space ${\mathfrak {h}}=L\otimes _{\mathbb {Z}}{\mathbb {Q}}$ associated to L and define the Heisenberg Lie algebra $\hat {\mathfrak {h}}$ as a one-dimensional extension

$$ \begin{align*}0\to{\mathbb{Q}} K\to\hat{\mathfrak{h}}\to{\mathfrak{h}}\otimes{\mathbb{Q}}[t^{\pm1}]\to0\end{align*} $$

with central K and

$$ \begin{align*}[h_m,h^{\prime}_n]=m\delta_{m,-n}\cdot (h,h')K.\end{align*} $$

Here and below, we define $h_n:=h\otimes t^n$ , for $h\in {\mathfrak {h}}$ and $n\in \mathbb {Z}$ . If we set weight degrees

$$ \begin{align*}\operatorname{\mathrm{wt}}(h_n)=-n,\qquad\operatorname{\mathrm{wt}}(K)=0,\end{align*} $$

then $\hat {\mathfrak {h}}$ becomes a Lie algebra in $\operatorname {\mathrm {Vect}}^{{\frac 12}\mathbb {Z}}$ .

Let ${\mathsf {H}}$ be the associative Heisenberg algebra which is the quotient of the universal enveloping algebra of $\hat {\mathfrak {h}}$ modulo the ideal generated by $K-1$ . Consider the commutative subalgebra

$$ \begin{align*}{\mathsf{H}}_+=U({\mathfrak{h}}\otimes{\mathbb{Q}}[t])\simeq S({\mathfrak{h}}\otimes{\mathbb{Q}}[t])\subset{\mathsf{H}}.\end{align*} $$

For any $\lambda \in L{^{\vee }}=\operatorname {\mathrm {Hom}}(L,\mathbb {Z})$ , let ${\mathbb {Q}}_\lambda $ be the $1$ -dimensional module over ${\mathsf {H}}_+$ , where, for each $h\in {\mathfrak {h}}$ , $h_0$ acts by multiplication with $\lambda (h)$ and $h_n$ with $n>0$ acts trivially. We define the Fock space $V_\lambda $ to be the ${\mathsf {H}}$ -module

$$ \begin{align*}V_\lambda={\mathsf{H}}\otimes_{{\mathsf{H}}_+}{\mathbb{Q}}_\lambda .\end{align*} $$

As a vector space, it is isomorphic to $S({\mathfrak {h}}\otimes t{^{-1}}{\mathbb {C}}[t{^{-1}}])$ . We denote the cyclic vector of $V_\lambda $ by ${|\lambda \rangle }$ . In particular, for any $\alpha \in L$ , we may consider the linear function $(\alpha ,\cdot )\in L{^{\vee }}$ , and by abuse of notation, we shall denote the corresponding Fock space by $V_\alpha $ and its generator by ${|\alpha \rangle }$ .

For any $\alpha \in L$ , we have a field $\alpha (z)=\sum _{n\in \mathbb {Z}}\alpha _nz^{-n-1}\in {\mathcal {F}}(V_0)$ . One knows that $V_0$ has a vertex algebra structure with the vacuum ${| 0\rangle }$ and with

In particular, $Y(\alpha _{-1}{|0\rangle },z)=\alpha (z)$ . Similarly, using $\operatorname {\mathrm {id}}_{V_\lambda }$ in the above formula, we can equip $V_\lambda $ with a module structure over the vertex algebra $V_0$ .

4.5.2. Vertex operators and lattice vertex algebras

Let us consider the total Fock space

$$ \begin{align*}V_L=\bigoplus_{\alpha\in L}V_\alpha .\end{align*} $$

This space is automatically L-graded. We equip it with a weight grading for which $\operatorname {\mathrm {wt}}({|\beta \rangle })={\frac 12}(\beta ,\beta )$ , so that

(42) $$ \begin{align} \operatorname{\mathrm{wt}}(\alpha^{(1)}_{n_1}\dots \alpha^{(p)}_{n_p}{|\beta\rangle}) ={\frac12}(\beta,\beta)-\sum_{k=1}^p n_k\in{\frac12}\mathbb{Z}. \end{align} $$

Thus, $V_L$ may be considered as an object in ${\mathcal {C}}^{{\frac 12}\mathbb {Z}}$ .

Let $S_\alpha \colon V_L\to V_L$ be the unique linear map satisfying

1. (1) $S_\alpha {|\beta \rangle }={|{\alpha +\beta }\rangle }$ .

2. (2) $[h_n,S_\alpha ]=0$ for $n\ne 0$ and $h\in {\mathfrak {h}}$ .

Let $z^{(\alpha ,-)}\colon V_L\to V_L{[\![{z^{\pm 1}}]\!]}$ be the map that acts on $V_\beta $ by multiplication with $z^{(\alpha ,\beta )}$ . Note that

$$ \begin{align*}z^{(\alpha,-)} S_\beta=z^{(\alpha,\beta)}S_\beta z^{(\alpha,-)}.\end{align*} $$

For any $\alpha \in L$ , we let

$$ \begin{align*}\Gamma^-_\alpha(z)= \exp{\left({-\sum_{n<0}\frac{\alpha_n}nz^{-n}}\right)} ,\qquad \Gamma^+_\alpha(z)=\exp{\left({-\sum_{n>0}\frac{\alpha_n}nz^{-n}}\right)} .\end{align*} $$

Using these series, we now define the vertex operator (a field on $V_L$ )

(43) $$ \begin{align} \Gamma_\alpha(z) =\sum_n \Gamma_\alpha(n)z^{-n-1} =S_\alpha \Gamma^-_\alpha(z)\Gamma^+_\alpha(z)z^{(\alpha,-)}. \end{align} $$

Let us check that the operators $\Gamma _\alpha (z)$ are pairwise local.

Lemma 4.14. For any $\alpha ,\beta \in L$ , we have

$$ \begin{align*}(z-w)^{-(\alpha,\beta)}\Gamma_\alpha(z)\Gamma_\beta(w) =(w-z)^{-(\alpha,\beta)}\Gamma_\beta(w)\Gamma_\alpha(z).\end{align*} $$

Equivalently, $\Gamma _\alpha (z)$ , $\Gamma _\beta (z)$ are local of order $-(\alpha ,\beta )$ .

Proof. First of all, we have

$$ \begin{align*}\Gamma^+_\alpha(z)\Gamma^-_\beta(w) ={\left({1-\frac wz}\right)}^{(\alpha,\beta)} \Gamma^-_\beta(w)\Gamma^+_\alpha(z),\end{align*} $$

which follows from the fact that

$$ \begin{align*} &\exp{\left({ {\left[{\sum_{n>0}\frac{\alpha_n}nz^{-n}, \sum_{n<0}\frac{\beta_n}nw^{-n}}\right]} }\right)} =\\& \quad\exp{\left({(\alpha,\beta)\sum_{n>0}\frac{(w/z)^n}{-n}}\right)} ={\left({1-\frac wz}\right)}^{(\alpha,\beta)}. \end{align*} $$

Now we obtain

(44) $$ \begin{align} \Gamma_\alpha(z)\Gamma_\beta(w)& =S_\alpha \Gamma^-_\alpha(z)\Gamma^+_\alpha(z)z^{(\alpha,-)}\cdot S_\beta \Gamma^-_\beta(w)\Gamma^+_\beta(w)w^{(\beta,-)}\\\nonumber&=z^{(\alpha,\beta)}S_{\alpha+\beta} \Gamma^-_\alpha(z)\Gamma^+_\alpha(z)\Gamma^-_\beta(w)\Gamma^+_\beta(w) z^{(\alpha,-)} w^{(\beta,-)} \\\nonumber&=(z-w)^{(\alpha,\beta)}S_{\alpha+\beta} \Gamma^-_\alpha(z)\Gamma^-_\beta(w)\Gamma^+_\alpha(z)\Gamma^+_\beta(w) z^{(\alpha,-)} w^{(\beta,-)}, \end{align} $$

and similarly for $\Gamma _\beta (w)\Gamma _\alpha (z)$ . This implies the first claim of the lemma. For the second claim, it is enough to note that $(w-z)^{-(\alpha ,\beta )}=(-1)^{(\alpha ,\beta )}(-w+z)^{-(\alpha ,\beta )}$ and recall Definition 4.9.

Note that the weight of z is $-1$ and the weight of $\alpha _n$ is $-n$ . Therefore, $\alpha _n z^{-n}$ has weight zero, and the same is true for $\Gamma _\alpha ^{\pm }(z)$ . The element $S_\alpha z^{(\alpha ,-)}{|\beta \rangle }=z^{(\alpha ,\beta )}{|{\alpha +\beta }\rangle }$ has weight ${\frac 12}(\alpha ,\alpha )+{\frac 12}(\beta ,\beta )$ . We conclude that the operator $\Gamma _\alpha (z)$ has weight ${\frac 12}(\alpha ,\alpha )$ , which is also the weight of ${|\alpha \rangle }$ . The total Fock space $V_L$ has a structure of an L-graded vertex algebra, called the lattice vertex algebra, that extends the vertex algebra structure on $V_0$ and satisfies

$$ \begin{align*}Y({|\alpha\rangle},z)=\Gamma_\alpha(z).\end{align*} $$

4.5.3. Principal free vertex algebras

Let $(e_i)_{i\in I}$ be a basis of the lattice L. Recall that, according to Lemma 4.14, the fields $\Gamma _i(z):=\Gamma _{e_i}(z)$ and $\Gamma _j(z):=\Gamma _{e_j}(z)$ on $V_L$ are local of order $-(e_i,e_j)$ . We define the locality function

$$ \begin{align*}{\mathcal{N}}:I\times I\to\mathbb{Z},\qquad {\mathcal{N}}(i,j)=-(e_i,e_j)\end{align*} $$

and the degree map

$$ \begin{align*}\deg:I\to L\times{\frac12}\mathbb{Z},\qquad \deg(i)={\left({e_i,{\frac12}(e_i,e_i)}\right)}.\end{align*} $$

Let ${\mathcal {P}}$ be the free vertex algebra with respect to locality ${\mathcal {N}}$ and degree $\deg $ , which we shall also call the principal free vertex algebra. By the universality property of the free vertex algebra, there is a unique morphism of graded vertex algebras

$$ \begin{align*}{\mathcal{P}}\to V_L,\qquad i\mapsto {|{e_i}\rangle}.\end{align*} $$

By [Reference Roitman58, Theorem 2], this morphism is injective. Therefore, the free vertex algebra ${\mathcal {P}}$ can be identified with the vertex subalgebra of $V_L$ generated by elements ${|{e_i}\rangle }$ , $i\in I$ . This subalgebra was studied in [Reference Milas and Penn47] under the name principal subalgebra.

5.1. Three incarnations

In this section, we shall discuss three different incarnations of the same vertex algebra ${\mathcal {P}}_Q$ : as a principal free algebra, as a free algebra of non-negative locality, and as the universal enveloping algebra of a vertex Lie algebra. From (40), we know that for ${\mathcal {N}}(i,j)\ge 0$ , the locality relation between $Y(i,z)$ and $Y(j,z)$ is a collection of ‘honest’ Lie algebra relations (finite combinations of Lie brackets). In our situation, the only possible negative value of the locality function is ${\mathcal {N}}(i,j)=-1$ , in which case, $i=j$ and $\chi (e_i,e_i)=1$ .

Proof. We already know that locality of order $-1$ implies locality of order $0$ , so we only need to prove the reverse implication. The locality of order $-1$ is equivalent to the relations

(45) $$ \begin{align} \sum_{k\ge0}a(m-k)a(n+k)- \sum_{k<0}a(n+k)a(m-k)=0 \end{align} $$

for all $m,n\in \mathbb {Z}$ . The locality of order $0$ is equivalent to

(46) $$ \begin{align} [a(m),a(n)]=a(m)a(n)+a(n)a(m)=0,\qquad m,n\in\mathbb{Z}. \end{align} $$

It remains to note that, assuming that $m\ge n$ , Condition (45) can be rewritten as

$$ \begin{align*}\sum_{k=0}^{m-n}a(m-k)a(n+k)=0, \end{align*} $$

which vanishes whenever (46) holds.

It follows from Lemma 5.1 that if ${\mathcal {N}}(i,i)=-1$ , then the field $Y(i,z)$ is local of order $0$ with itself, so the principal free vertex algebra ${\mathcal {P}}_Q$ coincides with the free vertex algebra for the non-negative locality function ${\mathcal {N}}_+:=\max ({\mathcal {N}},0)$ . In particular, even though some values of the locality function for ${\mathcal {P}}_Q$ can be negative, Proposition 4.13 implies that ${\mathcal {P}}_Q$ is isomorphic to the universal enveloping vertex algebra ${\mathcal {U}}({\mathfrak {C}})$ of a vertex Lie algebra ${\mathfrak {C}}$ .

Let us recall some notation for future reference. We consider the sets

$$ \begin{align*}X={\left\{\left.{i(n)}\ \right\vert{i\in I,\,n\in\mathbb{Z}}\right\}},\qquad X_+={\left\{\left.{i(n)}\ \right\vert{i\in I,\,n\ge0}\right\}}\end{align*} $$

and define ${\mathfrak {L}}$ to be the Lie algebra generated by X subject to locality relations (40) with $a,b\in I$ and $N={\mathcal {N}}_+(a,b)$ , and ${\mathfrak {L}}_+\subset {\mathfrak {L}}$ to be the subalgebra generated by $X_+\subset X$ .

For every $i\in I$ , we consider ${\mathbf i}(z)=\sum _n i(n)z^{-n-1}\in {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ , and we define ${\mathfrak {C}}\subset {\mathfrak {L}}{[\![{z^{\pm 1}}]\!]}$ to be the vertex Lie algebra generated by these series. We know from Proposition 4.13 that ${\mathfrak {L}}\simeq \operatorname {\mathrm {Coeff}}(C)$ and that we have a direct sum decomposition

$$ \begin{align*}{\mathfrak{L}}={\mathfrak{L}}_+\oplus{\mathfrak{L}}_-,\end{align*} $$

where ${\mathfrak {L}}_{\pm }\subset {\mathfrak {L}}$ are Lie subalgebras closed under the derivation

(47) $$ \begin{align} \partial:{\mathfrak{L}}\to{\mathfrak{L}},\qquad \partial(i(n))=-n i(n-1). \end{align} $$

We have an isomorphism of graded vector spaces (compatible with derivations)

(48) $$ \begin{align} C\xrightarrow{\ \sim\ }\operatorname{\mathrm{Coeff}}(C)_-\xrightarrow{\ \sim\ }{\mathfrak{L}}_-,\qquad \mathfrak{a}(z)\mapsto \mathfrak{a}(z)(-1)\mapsto\operatorname{\mathrm{Res}}_z z{^{-1}}\mathfrak{a}(z). \end{align} $$

Moreover, the principal free vertex algebra ${\mathcal {P}}_Q$ is isomorphic to the universal enveloping vertex algebra of the vertex Lie algebra C

(49) $$ \begin{align} {\mathcal{P}}_Q\simeq {\mathcal{U}}(C)=U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}. \end{align} $$

We also have an isomorphism of graded vector spaces

(50) $$ \begin{align} {\mathcal{P}}_Q\simeq U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}\simeq U({\mathfrak{L}}_-). \end{align} $$

5.2. The graded dual of the principal free vertex algebra

Our first goal is to exhibit a relationship between the underlying objects of ${\mathcal {P}}_Q$ and ${\mathcal {H}}_Q$ . Recall from (41) that ${\mathcal {P}}_Q$ is equipped with the L-degree as well as the weight

As we are aiming to relate vertex algebras to cohomological Hall algebras, we shall implement a relationship between half-integer weights and integer cohomological degrees, defining, for $u\in {\mathcal {P}}_Q$ , its (cohomological) degree to be

$$ \begin{align*}\deg(u)=-2\operatorname{\mathrm{wt}}(u)\in\mathbb{Z}.\end{align*} $$

If we superpose the L-degree and the degree we just defined, ${\mathcal {P}}_Q$ becomes an object of $\operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ with finite-dimensional components ${\mathcal {P}}_{Q,{\mathbf {d}}}^n$ , for ${\mathbf {d}}\in {\mathbb {N}}^I$ and $n\in \mathbb {Z}$ ; this immediately follows from the fact that the lattice realisation of free vertex algebras allows us to view ${\mathcal {P}}_Q$ as a subspace of the total Fock space $V_L$ . We define the graded dual space of ${\mathcal {P}}_Q$ by the formula

$$ \begin{align*}{\mathcal{P}}{_Q^{\vee}} =\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^{I}}{\mathcal{P}}{_{Q,{\mathbf{d}}}^{\vee}},\qquad {\mathcal{P}}{_{Q,{\mathbf{d}}}^{\vee}}={\underline{\textrm{Hom}}}({\mathcal{P}}_{Q,{\mathbf{d}}},{\mathbb{Q}}) =\bigoplus_{n\in\mathbb{Z}}({\mathcal{P}}_{Q,{\mathbf{d}}}^{-n}){^{\vee}}. \end{align*} $$

Let us describe this graded dual directly. For a dimension vector ${\mathbf {d}}\in {\mathbb {N}}^I$ , let $p={\left \lvert {\mathbf {d}}\right \rvert }$ and $\operatorname {\mathrm {Hom}} j=(j_1,\dots ,j_p)$ be any sequence of vertices with $e_{j_1}+\cdots +e_{j_p}={\mathbf {d}}$ . We shall associate to such sequence $\operatorname {\mathrm {Hom}} j$ and $\xi \in {\mathcal {P}}{_{0,{\mathbf {d}}}^{\vee }}$ an element $F_{\xi ,\operatorname {\mathrm {Hom}} j}(z_1,\ldots ,z_p) \in {\mathbb {Q}}{(\;\!\!\!({z_1})\!\!\!\;)}\dots {(\;\!\!\!({z_p})\!\!\!\;)}$ given by the formula

Proof. By the locality properties of the free vertex algebra ${\mathcal {P}}_Q$ , we have

$$ \begin{align*}(z-w)^{{\mathcal{N}}(i,j)}Y(i,z)Y(j,w) =(w-z)^{{\mathcal{N}}(i,j)}Y(j,w)Y(i,z),\qquad i,j\in I.\end{align*} $$

This shows that the Laurent series

is fully symmetric under the action of $\Sigma _p$ : the product

$$ \begin{align*}Y(j_1,z_1)Y(j_2,z_2)\cdots Y(j_p,z_p)\end{align*} $$

is multiplied by the exactly correct factors to make it completely symmetric.

To prove that $F_{\xi ,\operatorname {\mathrm {Hom}} j}(z_1,\ldots ,z_p)$ is a polynomial, we shall argue as follows. Note that

(51)

Examining this formula, we immediately conclude that the Laurent series $F_{\xi ,\operatorname {\mathrm {Hom}} j} (z_1,\ldots ,z_p)$ does not contain any negative power of the last variable $z_p$ : the vacuum vector of ${\mathcal {P}}_Q$ is annihilated by all generators $j_p(n)$ with $n\ge 0$ , and multiplication by the product $\prod _{k<l}(z_k-z_l)^{{\mathcal {N}}(j_k,j_l)}$ does not create negative powers of $z_p$ because of the convention for the geometric series expansion of $(z+w)^n$ . Since we already know that the result is symmetric, we conclude that (51) has no negative powers of any variable, and thus is a formal power series in $z_1$ , …, $z_p$ . Finally, since we are dealing with the graded dual vector space, the linear functional $\xi $ is supported at finitely many degrees, and so in

one ends up ignoring all coefficients of our formal power series but finitely many, and we obtain a polynomial.

We shall now state and prove the main result of this section. For a dimension vector ${\mathbf {d}}\in {\mathbb {N}}^I$ , we now make a choice of a concrete sequence of vertices $\operatorname {\mathrm {Hom}} i=(i_1,\dots ,i_p)$ with $e_{i_1}+\cdots +e_{i_p}={\mathbf {d}}$ . For that, we choose an order of I, use this order to identify I with ${{\left \{{1,\dots ,r}\right \}}}$ , where $r=|I|$ , and put

$$ \begin{align*}i_{{\mathbf{d}}_1+\dots+{\mathbf{d}}_{j-1}+k}=j,\qquad j\in I,\,1\le k\le {\mathbf{d}}_j.\end{align*} $$

We shall use formal variables $z_{{\mathbf {d}}_1+\dots +{\mathbf {d}}_{j-1}+k}=x_{j,k}$ associated to this sequence of vertices. Recall from §3.2 that

$$ \begin{align*}\Lambda_{\mathbf{d}} =\bigotimes_{j\in I}\Lambda_{{\mathbf{d}}_j} ={\mathbb{Q}}[x_{j,k}{\colon} j\in I,\, 1\le k\le {\mathbf{d}}_j]^{\Sigma_{\mathbf{d}}}\end{align*} $$

is a graded algebra with $\deg (x_{j,k})=2$ (corresponding to weight $-1$ , cf. §4.1.2).

Proposition 5.3. We have an isomorphism of graded vector spaces

(52) $$ \begin{align} F\colon {\mathcal{P}}{_Q^{\vee}}\to {\mathcal{H}}_Q. \end{align} $$

Specifically, for any dimension vector ${\mathbf {d}}$ , the isomorphism ${{\mathcal {P}}}{_{Q,\mathbf {d}}^{\vee }}\to {\mathcal {H}}_{Q,{\mathbf {d}}}=\Lambda _{\mathbf {d}}[-\chi ({\mathbf {d}},{\mathbf {d}})]$ is given by the formula

Proof. We begin with noticing that, according to Lemma 5.2, the polynomial $F_\xi $ in variables $x_{j,k}$ is symmetric with respect to the action of the group $\Sigma _{\mathbf {d}}=\prod _{i\in I}\Sigma _{{\mathbf {d}}_i}$ (the action of a permutation from $\Sigma _{\mathbf {d}}$ simultaneously on the vertices $i_k$ and the variables $z_k$ is the same as the action on the variables only). Thus, F is a well-defined map into $\Lambda _{\mathbf {d}}[-\chi ({\mathbf {d}},{\mathbf {d}})]$ .

Let us first check that F is a map of degree zero. Recall that $Y(i,z)$ has weight ${\frac 12}\chi (e_i,e_i)$ , so by our convention $\deg (u)=-2\operatorname {\mathrm {wt}}(u)$ , the degree of $Y(i,z)$ is equal to $-\chi (e_i,e_i)$ . Therefore, $F_\xi $ has degree

$$ \begin{align*} \deg(\xi)+2\sum_{k<l}{\mathcal{N}}(i_k,i_l)-&\sum_k\chi(e_{i_k},e_{i_k}) =\\ &\deg(\xi)-\sum_{k,l}\chi(e_{i_k},e_{i_l}) =\deg(\xi)-\chi({\mathbf{d}},{\mathbf{d}}). \end{align*} $$

This constant shift of degrees by $-\chi ({\mathbf {d}},{\mathbf {d}})$ means exactly that the map

$$ \begin{align*}F\colon {\mathcal{P}}{_{Q,{\mathbf{d}}}^{\vee}}\to\Lambda_{\mathbf{d}}[-\chi({\mathbf{d}},{\mathbf{d}})]\end{align*} $$

has degree zero.

Let us show that F is injective. Suppose that for some $\xi \in {\mathcal {P}}{_{Q,{\mathbf {d}}}^{\vee }}$ , we have $F_\xi =0$ . According to Lemma 5.2, this implies that

for any sequence of vertices with $e_{i_1}+\cdots +e_{i_p}={\mathbf {d}}$ . Since $F_\xi $ is a polynomial, its product with $\prod _{k<l}(z_k-z_l)^{-{\mathcal {N}}(j_k,j_l)}$ (expanded according to the binomial expansion convention) is well defined, so we may conclude that

Extracting coefficients of individual monomials $z_1^{m_1}\cdots z_p^{m_p}$ , we see that $\xi $ vanishes on all vectors , so $\xi =0$ .

Let us finally show that F is surjective. For that, we shall use the lattice vertex operator realisation, replacing $Y(i_k,z_k)$ by vertex operators $\Gamma _{{i_k}}(z_k)=\Gamma _{e_{i_k}}(z_k)$ and by ${|0\rangle }$ . Recall that we established the equality (44)

$$\begin{align*}\Gamma_\alpha(z)\Gamma_\beta(w) =(z-w)^{\chi(\alpha,\beta)}S_{\alpha+\beta} \Gamma^-_\alpha(z)\Gamma^-_\beta(w)\Gamma^+_\alpha(z)\Gamma^+_\beta(w) z^{(\alpha,-)} w^{(\beta,-)}. \end{align*}$$

This implies, more generally, that

$$ \begin{align*} & \Gamma_{i_1}(z_1)\dots\Gamma_{i_p}(z_p)=\\ & \prod_{k<l}(z_k-z_l)^{\chi(e_{i_k},e_{i_l})}\Gamma^-_{i_1}(z_1)\dots\Gamma^-_{i_p}(z_p)\Gamma^+_{i_1}(z_1)\dots\Gamma^+_{i_p}(z_p) z_1^{(e_{i_1},-)}\cdots z_p^{(e_{i_p},-)}, \end{align*} $$

and therefore,

where the last equality uses the definition $\chi (e_{i_k},e_{i_l})=-{\mathcal {N}}(i_k,i_l)$ of the Euler form and the obvious fact that $\Gamma ^+_{i}(z)|0\rangle =|0\rangle $ for all i. Thus, we may write

If we now expand the exponentials, we obtain the formal infinite sum of all standard basis vectors $\prod _{j_s,n_s} e_{j_s,-n_s}^{a_s}$ of the Fock space $V_{\mathbf {d}}$ with coefficients being scalar multiples of monomials in the power sum $\Sigma _{\mathbf {d}}$ -symmetric functions [Reference Macdonald46]. Since every $\Sigma _{\mathbf {d}}$ -symmetric function is a polynomial in the power sum symmetric functions, we may obtain each $\Sigma _{\mathbf {d}}$ -symmetric function for a suitable choice of a linear functional on $V_{\mathbf {d}}$ , and the surjectivity claim follows.

Using our result, we can immediately recover the formula for the Poincaré series $Z({\mathcal {P}}_Q,x,q)$ proved in [Reference Milas and Penn47, Theorem 5.3] using combinatorial bases for free vertex algebras.

Corollary 5.4. We have

$$ \begin{align*}Z({\mathcal{P}}_Q,x,q) =\sum_{{\mathbf{d}}\in{\mathbb{N}}^I} \frac{(-q^{{\frac12}})^{\chi({\mathbf{d}},{\mathbf{d}})}}{(q)_{\mathbf{d}}}x^{\mathbf{d}}. \end{align*} $$

Proof. By Proposition 3.2, we have

$$ \begin{align*}Z({\mathcal{H}}_Q,x,q) =\sum_{{\mathbf{d}}\in{\mathbb{N}}^I} \frac{(-q^{\frac12})^{-\chi({\mathbf{d}},{\mathbf{d}})}}{(q{^{-1}})_{\mathbf{d}}}x^{\mathbf{d}}.\end{align*} $$

However, by the established isomorphism, we have $Z({\mathcal {P}}_Q,x,q)=Z({\mathcal {P}}{_Q^{\vee }},x,q{^{-1}})=Z({\mathcal {H}}_Q,x,q{^{-1}})$ .

5.3. The coalgebra structure of the principal free vertex algebra

Our next goal is to unravel a canonical coproduct on ${\mathcal {P}}_Q$ (cf. Proposition 4.5) and to relate it to the product on ${\mathcal {H}}_Q$ . The universal enveloping algebra $U({\mathfrak {L}})$ is a bialgebra, with the canonical coproduct $\Delta $ . Because of this coproduct, the tensor product ${\mathcal {P}}_Q\otimes {\mathcal {P}}_Q$ acquires a $U({\mathfrak {L}})$ -module structure; note that the action of $U({\mathfrak {L}})$ has obvious signs arising from the braiding.

Proof. Recall that ${\mathcal {P}}_Q$ may also be described as $U({\mathfrak {L}})/U({\mathfrak {L}})X_+$ , where

$$ \begin{align*}X_+={\left\{\left.{i(n)\in X}\ \right\vert{n\ge0}\right\}}.\end{align*} $$

The $U({\mathfrak {L}})$ -module morphism condition implies that

$$\begin{align*}\delta(i(n)w)=(i(n)\otimes\operatorname{\mathrm{id}}+\operatorname{\mathrm{id}}\otimes i(n))\delta(w) \end{align*}$$

for all $w\in {\mathcal {P}}_Q$ , and so it is clear that if the morphism $\delta $ exists, then it is unique, and is induced by $\Delta $ . Thus, all we have to show is that $\Delta $ descends to the quotient by the left ideal generated by $X_+$ , which follows from the property

$$\begin{align*}\Delta(X_+)\subset X_+\otimes {\mathbb{Q}} + {\mathbb{Q}}\otimes X_+.\\[-36pt] \end{align*}$$

Recall that we have an isomorphism of graded vector spaces (50)

$$ \begin{align*}{\mathcal{P}}_Q\simeq U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}\simeq U({\mathfrak{L}}_-).\end{align*} $$

The above coproduct on ${\mathcal {P}}_Q$ can be identified with the canonical coproduct on $U({\mathfrak {L}}_-)$ (cf. Proposition 4.6). We are now ready to state and prove the main result of this section.

Proof. Suppose that $\zeta \in {\mathcal {P}}_{Q,{\mathbf {d}}}^\vee $ , $\xi \in {\mathcal {P}}_{Q,{\mathbf {e}}}^\vee $ and $\operatorname {\mathrm {Hom}} i=(i_1,\dots ,i_p)$ is a sequence of vertices such that $\sum _{k}e_{i_k}={\mathbf {d}}+{\mathbf {e}}$ . According to the proof of Proposition 5.3,

(53)

The map $\delta ^\vee $ is completely defined by

$$ \begin{align*}{ {\left\langle{\delta^\vee(\zeta\otimes\xi), u}\right\rangle}}={ {\left\langle{\zeta\otimes\xi, \delta(u)}\right\rangle}}\end{align*} $$

for all $u\in {\mathcal {P}}_Q$ . Note that

where $\kappa (A,B)=\sum _{a>b}\chi ({i_a},{i_b})$ , for $A\sqcup B=[1,p]={{\left \{{1,\dots ,p}\right \}}}$ , and the products over A and B are taken in the increasing order. For the calculation of the symmetric polynomial (53), we need to consider only $A,B$ with ${\mathbf {dim}}(A)=\sum _{a\in A}e_{i_a}={\mathbf {d}}$ and ${\mathbf {dim}}(B)={\mathbf {e}}$ . Recalling that the pairing of ${\mathcal {P}}_{Q,{\mathbf {d}}}^\vee \otimes {\mathcal {P}}_{Q,{\mathbf {e}}}^\vee $ with ${\mathcal {P}}_{Q,{\mathbf {d}}}\otimes {\mathcal {P}}_{Q,{\mathbf {e}}}$ produces an extra sign $(-1)^{\chi ({\mathbf {d}},{\mathbf {e}})}$ according to the braiding, we see that $F_{\delta ^\vee (\zeta \otimes \xi )}$ is equal to

The product $\prod _{k<l}(z_k-z_l)^{{\mathcal {N}}(i_k,i_l)}$ is equal to the product of four terms, according to whether each of the two elements k and l belongs to A or to B. Note that

is precisely $F_\zeta $ , and

is precisely $F_\xi $ , so we just need to investigate the factor by which their product is multiplied; that is,

$$ \begin{align*} (-1)^{\kappa(A,B)+\chi({\mathbf{d}},{\mathbf{e}})} \prod_{{{\substack{{a<b}\\{a\in A,b\in B}}}}} (z_{a}-z_{b})^{{\mathcal{N}}(i_{a},i_{b})} \prod_{{{\substack{{b<a}\\{a\in A,b\in B}}}}} (z_{b}-z_{a})^{{\mathcal{N}}(i_{b},i_{a})} \\ =(-1)^{\kappa(A,B)+\chi({\mathbf{d}},{\mathbf{e}})} \prod_{{{\substack{{b<a}\\{a\in A,b\in B}}}}}(-1)^{{\mathcal{N}}(i_{a},i_{b})} \prod_{a\in A, b\in B} (z_{a}-z_{b})^{{\mathcal{N}}(i_a,i_b)}, \\ =(-1)^{\chi({\mathbf{d}},{\mathbf{e}})} \prod_{a\in A, b\in B}(z_{a}-z_{b})^{{\mathcal{N}}(i_a,i_b)} =\prod_{a\in A, b\in B} (z_{b}-z_{a})^{{\mathcal{N}}(i_a,i_b)}. \end{align*} $$

Recalling the shuffle product formula (8), we see that

$$\begin{align*}F_{\delta^\vee(\zeta\otimes\xi)} = F_\zeta * F_\xi, \end{align*}$$

so the isomorphism F recovers precisely the shuffle product of CoHA.

5.4. A new proof of positivity of Donaldson–Thomas invariants

The above results together with the isomorphism of graded vector spaces (50)

$$ \begin{align*}{\mathcal{P}}_Q\simeq U({\mathfrak{L}})\otimes_{U({\mathfrak{L}}_+)}{\mathbb{Q}}\simeq U({\mathfrak{L}}_-)\end{align*} $$

have an interesting consequence: another proof of Efimov’s positivity theorem for refined Donaldson–Thomas invariants [Reference Efimov14, Theorem 1.1]. To see that, we shall study the Lie algebra ${\mathfrak {L}}_-$ in more detail. Recall from §5.1 that ${\mathfrak {L}}_-$ is stable under the derivation $\partial \colon {\mathfrak {L}}\to {\mathfrak {L}}$ .

Theorem 5.7. Let Q be a symmetric quiver. Then the coalgebra ${\mathcal {H}}{_Q^{\vee }}$ has a canonical structure of a cocommutative connected vertex bialgebra. The space of primitive elements

$$ \begin{align*}C=P({\mathcal{H}}{_Q^{\vee}})\in\operatorname{\mathrm{Vect}}^{L\times\mathbb{Z}}\end{align*} $$

is a vertex Lie algebra (also having a structure of a Lie algebra) such that ${\mathcal {H}}{_Q^{\vee }}\simeq {\mathcal {U}}(C)$ as vertex bialgebras. The derivation $\partial $ on C has L-degree zero and cohomological degree $-2$ , and C is a free ${\mathbb {Q}}[\partial ]$ -module such that the space of generators

$$ \begin{align*}C/\partial C=\bigoplus_{{\mathbf{d}}\in L}W_{\mathbf{d}}=\bigoplus_{{\mathbf{d}}\in L,k\in\mathbb{Z}}W_{\mathbf{d}}^k\end{align*} $$

has finite-dimensional components $W_{\mathbf {d}}$ and $k\equiv \chi ({\mathbf {d}},{\mathbf {d}})\pmod 2$ whenever $W_{\mathbf {d}}^k\ne 0$ .

Proof. We have ${\mathcal {H}}{_Q^{\vee }}\simeq {\mathcal {P}}_Q\simeq {\mathcal {U}}(C)$ for the vertex Lie algebra C; see §5.1. Therefore, ${\mathcal {H}}{_Q^{\vee }}$ has a structure of a cocommutative connected vertex bialgebra and $P({\mathcal {H}}{_Q^{\vee }})\simeq C$ by Propositions 4.5 and 4.6. We can identify C with ${\mathfrak {L}}_-$ , and we will show that ${\mathfrak {L}}_-$ satisfies the required properties. Theorem 5.6 implies that we have an isomorphism of coalgebras

$$ \begin{align*}{\mathcal{H}}{_Q^{\vee}}\simeq {\mathcal{P}}_Q\simeq U({\mathfrak{L}}_-).\end{align*} $$

Let us define another derivation of ${\mathfrak {L}}$ , which we shall denote t; it acts on the generators of ${\mathfrak {L}}$ by the formula $t(i(n))=-i(n+1)$ . By direct inspection, t preserves all relations of ${\mathfrak {L}}$ , and thus acts on this algebra. Moreover, this derivation preserves the subalgebra ${\mathfrak {L}}_+$ , and so acts on the vector space ${\mathfrak {L}}/{\mathfrak {L}}_+\simeq {\mathfrak {L}}_-$ . We note that $[\partial ,t]=\operatorname {\mathrm {id}}$ on the space of generators; it follows that $[\partial ,t]={\left \lvert {\mathbf {d}}\right \rvert }\cdot \operatorname {\mathrm {id}}$ on each graded component $({\mathfrak {L}}_-)_{\mathbf {d}}$ . Thus, each such component is a $\mathbb {Z}$ -graded module over the Weyl algebra $A_1={\mathbb {Q}}[t,\partial ]$ of polynomial differential operators on the line (on which $\partial $ acts by an endomorphism of weight $1$ and t acts by an endomorphism of weight $-1$ ). Since we can embed $({\mathfrak {L}}_-)_{\mathbf {d}}\subset {\mathcal {P}}_{0,{\mathbf {d}}}\subset V_{\mathbf {d}}$ (the Fock space), weights of $({\mathfrak {L}}_-)_{\mathbf {d}}$ are bounded from below. Therefore, by Lemma 5.8, we obtain ${\mathfrak {L}}_-\simeq W\otimes {\mathbb {Q}}[\partial ]$ , for $W\simeq {\mathfrak {L}}_-/\partial {\mathfrak {L}}_-$ .

The Fock space $V_{\mathbf {d}}$ has finite-dimensional weight components, and its weights are bounded below. Moreover, each vector of $V_{\mathbf {d}}$ is obtained from ${|{\mathbf {d}}\rangle }$ by action of elements of integer weights; hence, all weights of $V_{\mathbf {d}}$ are of the form ${\frac 12}\chi ({\mathbf {d}},{\mathbf {d}})+n$ , for some $n\in \mathbb {Z}$ . The corresponding degree is congruent to $\chi ({\mathbf {d}},{\mathbf {d}})\pmod 2$ . The same applies to $W_{\mathbf {d}}\subset V_{\mathbf {d}}$ , and we conclude that $\operatorname {\mathrm {ch}}(W_{\mathbf {d}})\in {\mathbb {N}}{(\;\!\!\!({q^{\frac 12}})\!\!\!\;)}$ . It remains to establish that $W_{\mathbf {d}}$ is finite-dimensional, and for this, we will show that $\operatorname {\mathrm {ch}}(W_{\mathbf {d}})\in {\mathbb {N}}[q^{\pm {\frac 12}}]$ .

Since ${\mathcal {H}}{_Q^{\vee }}\simeq U({\mathcal {L}}_-)$ and ${\mathfrak {L}}_-\simeq W\otimes {\mathbb {Q}}[\partial ]$ , we obtain

$$ \begin{align*}Z({\mathcal{H}}_Q,x,q{^{-1}})=\operatorname{\mathrm{Exp}}{\left({Z({\mathfrak{L}}_-,x,q)}\right)} =\operatorname{\mathrm{Exp}}{\left({\frac{Z(W,x,q)}{1-q}}\right)}. \end{align*} $$

Applying formula (15) for DT invariants, we obtain

$$ \begin{align*}Z(W,x,q)= \sum_{\mathbf{d}}(-1)^{\chi({\mathbf{d}},{\mathbf{d}})}\Omega_{\mathbf{d}}(q{^{-1}})x^{\mathbf{d}},\end{align*} $$

meaning that $\operatorname {\mathrm {ch}}(W_{\mathbf {d}})=\Omega _{\mathbf {d}}(q{^{-1}})$ . Let us now use characters of CoHA-modules. According to (16), for any ${\mathbf {w}}\in {\mathbb {N}}^I$ , we have

$$ \begin{align*}Z(\mathcal M_{{\mathbf{w}}},x,q{^{-1}})= \operatorname{\mathrm{Exp}}{\left({\sum_{\mathbf{d}}\frac{q^{{\mathbf{w}}\cdot {\mathbf{d}}}-1}{q-1}(-1)^{\chi({\mathbf{d}},{\mathbf{d}})}\Omega_{\mathbf{d}}(q{^{-1}})x^{\mathbf{d}}}\right)}. \end{align*} $$

All components $\mathcal M_{{\mathbf {w}},{\mathbf {d}}}$ of the CoHA module $\mathcal M_{\mathbf {w}}$ are finite-dimensional (as the cohomology of an algebraic variety); hence, $Z(\mathcal M{_{{\mathbf {w}}},x,q^{-1}})\in \mathbb {Z}[q^{\pm {\frac 12}}]{[\![{x_i{\colon } i\in I}]\!]}$ . This implies that for all ${\mathbf {d}}$ ,

$$ \begin{align*}\frac{q^{{\mathbf{w}}\cdot {\mathbf{d}}}-1}{q-1}\operatorname{\mathrm{ch}}(W_{\mathbf{d}}) =\frac{q^{{\mathbf{w}}\cdot {\mathbf{d}}}-1}{q-1}\Omega_{\mathbf{d}}(q{^{-1}})\end{align*} $$

is an element of ${\mathbb {Q}}[q^{\pm {\frac 12}}]$ (actually $\mathbb {Z}[q^{\pm {\frac 12}}]$ , but we do not need this). We can choose ${\mathbf {w}}\in {\mathbb {N}}^I$ such that ${\mathbf {w}}\cdot {\mathbf {d}}>0$ . We have just shown that the product of the polynomial $\frac {q^{{\mathbf {w}}\cdot {\mathbf {d}}}-1}{q-1}\in {\mathbb {N}}[q]$ and the series $\operatorname {\mathrm {ch}}(W_{\mathbf {d}})\in {\mathbb {N}}{(\;\!\!\!({q^{{\frac 12}}})\!\!\!\;)}$ is a Laurent polynomial. This implies that $\operatorname {\mathrm {ch}}(W_{\mathbf {d}})\in {\mathbb {N}}[q^{\pm {\frac 12}}]$ , as required.

Remark 5.9. In the proof of [Reference Efimov14, Theorem 1.1], a free action of the polynomial ring in one variable on ${\mathcal {H}}_{Q,{\mathbf {d}}}$ is used to find a space of free generators of ${\mathcal {H}}_Q$ . This action on ${\mathcal {H}}_{Q,{\mathbf {d}}}$ is given by multiplication with

(54) $$ \begin{align} \sigma_{\mathbf{d}}=\sum_{j\in I, 1\le k\le {\mathbf{d}}_j} x_{j,k}, \end{align} $$

and it can also be obtained as a by-product of our argument. Recall the derivation t from the proof of Theorem 5.7 that acts on the generators by the formula $t(i(n))=-i(n+1)$ . This means that

$$ \begin{align*} t(Y(i,z))&=t\left(\sum_ni(n)z^{-n-1}\right)=\\&\quad -\sum_ni(n+1)z^{-n-1}=-\sum_ni(n)z^{-n-2}z=-zY(i,z). \end{align*} $$

Since t is a derivation, this implies that

$$ \begin{align*} &\left.t(Y(i_1,z_1)Y(i_2,z_2)\cdots Y(i_p,z_p))\right|{}_{z_{{\mathbf{d}}_1+\dots+{\mathbf{d}}_{j-1}+k}=x_{j,k}}=\\ &-\left(\sum_{j\in I, 1\le k\le {\mathbf{d}}_j} x_{j,k}\right)\cdot \left.Y(i_1,z_1)Y(i_2,z_2)\cdots Y(i_p,z_p)\right|{}_{z_{{\mathbf{d}}_1+\dots+{\mathbf{d}}_{j-1}+k}=x_{j,k}}. \end{align*} $$

We see that up to a sign the multiplication by (54) is precisely the action of the endomorphism $t{^{\vee }}$ on ${\mathcal {P}}{_{Q,\mathbf {d}}^{\vee }}$ . We saw that ${\mathcal {P}}_Q$ is isomorphic to $U({\mathfrak {L}}_-)\simeq S^c(W\otimes {\mathbb {Q}}[\partial ])$ as a coalgebra; after taking graded duals (which can be implemented by doing the Fourier transform for differential operators), the modules become ${\mathbb {Q}}[t{^{\vee }}]$ -free.

Let us record a simple consequence of Theorem 5.7.

Corollary 5.10. For any symmetric quiver Q, consider the graded vertex bialgebra ${\mathcal {H}}{_Q^{\vee }}$ and its space of primitive elements $C=P({\mathcal {H}}{_Q^{\vee }})$ , which is a vertex Lie algebra. Then the corresponding Donaldson–Thomas invariants satisfy

$$ \begin{align*}\Omega_{\mathbf{d}}(q{^{-1}})=\operatorname{\mathrm{ch}}(C_{\mathbf{d}}/\partial C_{\mathbf{d}})\in{\mathbb{N}}[q^{\pm{\frac12}}].\end{align*} $$

Proof. We established that ${\mathcal {H}}{_Q^{\vee }}\simeq {\mathcal {P}}_Q$ and that the principal free vertex algebra ${\mathcal {P}}_Q$ is isomorphic to universal enveloping vertex algebra of a vertex Lie algebra C. We have seen in the proof of Theorem 5.7 that $\Omega _{\mathbf {d}}(q{^{-1}})=\operatorname {\mathrm {ch}}(W_{\mathbf {d}})$ , where ${\mathfrak {L}}_-=W\otimes {\mathbb {Q}}[\partial ]$ , so that $W\simeq {\mathfrak {L}}_-/\partial {\mathfrak {L}}_-$ . Recall that for any vertex Lie algebra C, its coefficient algebra ${\mathfrak {L}}$ has a decomposition ${\mathfrak {L}}={\mathfrak {L}}_-\oplus {\mathfrak {L}}_+$ such that $C\simeq {\mathfrak {L}}_-$ (as a graded vector space). Moreover, we have

(55) $$ \begin{align} {\mathfrak{L}}/\partial{\mathfrak{L}}\simeq {\mathfrak{L}}_-/\partial{\mathfrak{L}}_-\simeq C/\partial C, \end{align} $$

which completes the proof.

This statement is easily generalisable to the following appealing conjecture suggesting a relationship between vertex algebras and more general DT invariants.

5.5. Relationship to the vertex algebras of Joyce

In this section, we will briefly explain the relationship of our results to the geometric construction of vertex algebras proposed in [Reference Joyce33] (see also [Reference Bojko3, Reference Gross27, Reference Gross, Joyce and Tanaka28, Reference Latyntsev37]). In order to do this, we will need to formulate a minor generalisation of that construction. Let us assume that we have the following data

1. (1) A lattice L equipped with a symmetric bilinear form $\chi $ .

2. (2) An abelian category ${\mathcal {A}}$ and a linear map $\operatorname {\mathrm {cl}}:K_0({\mathcal {A}})\to L$ .

3. (3) A moduli stack ${\mathcal {M}}$ of object in ${\mathcal {A}}$ such that the substack ${\mathcal {M}}_{\mathbf {d}}$ of objects $E\in {\mathcal {M}}$ with $\operatorname {\mathrm {cl}}(E)={\mathbf {d}}$ is open and closed and there exist natural morphisms of stacks

   1. (a) $\Phi :{\mathcal {M}}\times {\mathcal {M}}\to {\mathcal {M}}$ that maps $(E,F)\mapsto E\oplus F$ .

   2. (b) $\Psi :B{\mathbb {G}_{\mathrm {m}}}\times {\mathcal {M}}\to {\mathcal {M}}$ that maps ${\mathbb {G}_{\mathrm {m}}}\times \operatorname {\mathrm {Aut}} (E)\ni (t,f)\mapsto tf\in \operatorname {\mathrm {Aut}}(E)$ , for $E\in {\mathcal {A}}$ .

4. (4) A perfect complex ${\Theta }$ on ${\mathcal {M}}\times {\mathcal {M}}$ such that

   1. (a) ${\Theta }$ is weakly-symmetric, meaning that

      (56) $$ \begin{align} [\sigma^*{\Theta}{^{\vee}}]=[{\Theta}] \end{align} $$
      in the Grothendieck group of ${\mathcal {M}}\times {\mathcal {M}}$ , where $\sigma :{\mathcal {M}}\times {\mathcal {M}}\to {\mathcal {M}}\times {\mathcal {M}}$ is the permutation of factors.

   2. (b) The restriction ${\Theta }_{{\mathbf {d}},{\mathbf {e}}}={\Theta }|_{{\mathcal {M}}_{\mathbf {d}}\times {\mathcal {M}}_{\mathbf {e}}}$ has constant rank $\chi ({\mathbf {d}},{\mathbf {e}})$ for all ${\mathbf {d}},{\mathbf {e}}\in L$ .

   3. (c) We have

      $$ \begin{align*}(\Phi\times\operatorname{\mathrm{id}}_{\mathcal{M}})^*{\Theta}\simeq \pi_{13}^*{\Theta}\oplus\pi_{23}^*{\Theta},\qquad (\operatorname{\mathrm{id}}_{\mathcal{M}}\times\Phi)^*{\Theta}\simeq \pi_{12}^*{\Theta}\oplus\pi_{13}^*{\Theta}, \end{align*} $$
      $$ \begin{align*}(\Psi\times\operatorname{\mathrm{id}}_{\mathcal{M}})^*{\Theta}\simeq {\mathcal{U}}\boxtimes{\Theta},\qquad (\operatorname{\mathrm{id}}_{\mathcal{M}}\times\Psi)^*{\Theta}\simeq {\mathcal{U}}{^{\vee}}\boxtimes{\Theta}, \end{align*} $$
      where $\pi _{ij}:{\mathcal {M}}^3\to {\mathcal {M}}^2$ is the projection to the corresponding factors and ${\mathcal {U}}$ is the universal line bundle over $B{\mathbb {G}_{\mathrm {m}}}$ .

The proof of the following result goes through the same lines as in [Reference Joyce33].

Theorem 5.12. Consider the $L\times \mathbb {Z}$ -graded vector space

$$ \begin{align*}V=\bigoplus_{{\mathbf{d}}\in L} V_{\mathbf{d}} =\bigoplus_{{\mathbf{d}}\in L} H_*({\mathcal{M}}_{\mathbf{d}})[\chi({\mathbf{d}},{\mathbf{d}})]\end{align*} $$

so that the component of $V_{\mathbf {d}}$ of homological degree k (and weight ${\frac 12} k$ ) is $H_{k-\chi ({\mathbf {d}},{\mathbf {d}})}({\mathcal {M}}_{\mathbf {d}})$ . Then V has a structure of a graded vertex algebra (in the symmetric monoidal category $\operatorname {\mathrm {Vect}}^{L\times \mathbb {Z}}$ with the braiding induced by $\chi $ ) defined by

1. (1) , where $\eta :{\mathbf {pt}}\to {\mathcal {M}}$ is the inclusion of the zero object.

2. (2) The operator $T:V\to V$ of homological degree $2$ (and weight $1$ ) is defined by $T(v)=\Psi _*(t\boxtimes v)$ , where $t\in H_2(B{\mathbb {G}_{\mathrm {m}}})=H_2({\mathbb {P}}^\infty )$ is the canonical generator.

3. (3) For $u\in H_k({\mathcal {M}}_{\mathbf {d}})$ , $v\in H_*({\mathcal {M}}_{\mathbf {e}})$ , we define

   $$ \begin{align*}Y(u,z)v= (-1)^{k\chi({\mathbf{d}},{\mathbf{d}})}z^{\chi({\mathbf{d}},{\mathbf{e}})} \Phi_*{\left({(e^{zT}\otimes\operatorname{\mathrm{id}})\left({(u\boxtimes v)\cap c_{z{^{-1}}}({\Theta}_{{\mathbf{d}},{\mathbf{e}}})}\right) }\right)}.\end{align*} $$

Let us show that the above vertex algebra construction can be applied in the case of the moduli stack of representations of a symmetric quiver. Let Q be a symmetric quiver with the set of vertices I. Let $L=\mathbb {Z}^I$ and $\chi $ be the Euler form of Q. Let ${\mathcal {A}}$ be the category of representations of Q and ${\mathcal {M}}$ be the stack of representations of Q. We have a linear map ${\mathbf {dim}}:K_0({\mathcal {A}})\to L$ . We define the perfect complex ${\Theta }=\operatorname {\mathrm {RHom}}$ over ${\mathcal {M}}\times {\mathcal {M}}$ such that its fiber over $(M,N)\in {\mathcal {M}}\times {\mathcal {M}}$ is isomorphic to $\operatorname {\mathrm {RHom}}(M,N)\in D^b(\operatorname {\mathrm {Vect}})$ . The rank of ${\Theta }$ over ${\mathcal {M}}_{\mathbf {d}}\times {\mathcal {M}}_{\mathbf {e}}$ is equal to $\chi ({\mathbf {d}},{\mathbf {e}})$ . In order to apply Theorem 5.12, we need to prove that ${\Theta }$ is weakly-symmetric (56).

Lemma 5.13. We have

$$ \begin{align*}[\sigma^*{\Theta}{^{\vee}}]=[{\Theta}]\end{align*} $$

in the Grothendieck group of ${\mathcal {M}}\times {\mathcal {M}}$ .

Proof. Let $A={\mathbb {C}} Q$ be the path algebra of Q. For every $i\in I$ , let $e_i\in A$ be the corresponding idempotent and $P=Ae_i$ be the corresponding projective A-module. For any representation M, we have the standard projective resolution

$$ \begin{align*}0\to\bigoplus_{a:i\to j}P_j\otimes M_i\to\bigoplus_{i}P_i\otimes M_i\to M\to0.\end{align*} $$

Therefore, $\operatorname {\mathrm {RHom}}(M,N)$ can be written as a complex

$$ \begin{align*}\dots\to0\to\bigoplus_{i}\operatorname{\mathrm{Hom}}(M_i,N_i)\to\bigoplus_{a:i\to j}\operatorname{\mathrm{Hom}}(M_i,N_j)\to0\to\dots\end{align*} $$

Similarly, $\operatorname {\mathrm {RHom}}(N,M)$ can be written as a complex

$$ \begin{align*}\dots\to0\to\bigoplus_{i}\operatorname{\mathrm{Hom}}(N_i,M_i)\to\bigoplus_{a:i\to j}\operatorname{\mathrm{Hom}}(N_i,M_j)\to0\to\dots\end{align*} $$

Using the fact that Q is symmetric and that $\operatorname {\mathrm {Hom}}(N_i,M_j)\simeq \operatorname {\mathrm {Hom}}(M_j,N_i){^{\vee }}$ , we can rewrite this complex in the form

$$ \begin{align*}\dots\to0\to\bigoplus_{i}\operatorname{\mathrm{Hom}}(M_i,N_i){^{\vee}}\to \bigoplus_{a:i\to j}\operatorname{\mathrm{Hom}}(M_i,N_j){^{\vee}}\to0\to\dots\end{align*} $$

This implies that $[\operatorname {\mathrm {RHom}}(M,N){^{\vee }}]=[\operatorname {\mathrm {RHom}}(N,M)]$ . The statement of the lemma is a global version of this observation.

Applying the previous lemma and the construction of Theorem 5.12, we obtain a vertex algebra structure on $H_*({\mathcal {M}})\simeq {\mathcal {H}}{_Q^{\vee }}$ .

However, in [Reference Latyntsev37], one constructed a quantum vertex bialgebra structure on $H_*({\mathcal {M}})$ associated with a complex ${\Theta }$ which is not necessarily weakly-symmetric. In the case of symmetric quivers and ${\Theta }=\operatorname {\mathrm {RHom}}$ , the resulting quantum vertex algebra structure on $H_*({\mathcal {M}})\simeq {\mathcal {H}}{_Q^{\vee }}$ is actually a vertex algebra (in an appropriate symmetric monoidal category) because of Lemma 5.13. Therefore, by the results of [Reference Latyntsev37], one has a vertex bialgebra structure on $H_*({\mathcal {M}})\simeq {\mathcal {H}}{_Q^{\vee }}$ . We expect that Conjecture 5.14 can be lifted to the level of vertex bialgebras, where ${\mathcal {P}}_Q\simeq {\mathcal {H}}{_Q^{\vee }}$ is equipped with a vertex bialgebra structure by Theorem 5.7.

6.1. Modified coproduct

For each ${\mathbf {w}}\in {\mathbb {N}}^I$ , let us consider the subset

$$ \begin{align*}X_{{\mathbf{w}}}={\left\{\left.{i(n)\in X}\ \right\vert{n\ge-{\mathbf{w}}_i}\right\}}\end{align*} $$

of the set of generators of the Lie algebra ${\mathfrak {L}}$ , and the corresponding $U({\mathfrak {L}})$ -module ${\mathcal {P}}_{\mathbf {w}}=U({\mathfrak {L}})/U({\mathfrak {L}})X_{\mathbf {w}}$ with the cyclic vector . For each $i\in I$ , the series $\mathbf {i}(z)=\sum _{n\in \mathbb {Z}}i(n)z^{-n-1}$ defines a field on ${\mathcal {P}}_{\mathbf {w}}$ . These series can be used to equip ${\mathcal {P}}_{\mathbf {w}}$ with a structure of a module over the vertex algebra ${\mathcal {P}}_Q$ ; clearly, ${\mathcal {P}}_Q$ itself is a particular case of this construction for ${\mathbf {w}}=0$ .

Note that since ${\mathbf {w}}\in {\mathbb {N}}^I$ , we have $X_+\subset X_{\mathbf {w}}$ , and therefore, there is a canonical surjection of $U({\mathfrak {L}})$ -modules $\pi \colon {\mathcal {P}}_Q\to {\mathcal {P}}_{\mathbf {w}}$ . We shall now see how the graded dual

$$ \begin{align*}{\mathcal{P}}_{\mathbf{w}}^\vee=\bigoplus_{{\mathbf{d}}\in{\mathbb{N}}^{I}}{\mathcal{P}}{_{{\mathbf{w}},{\mathbf{d}}}^{\vee}} \end{align*} $$

of ${\mathcal {P}}_{\mathbf {w}}$ is included in ${\mathcal {P}}{_Q^{\vee }}$ . As before, we fix a dimension vector ${\mathbf {d}}\in {\mathbb {N}}^I$ and let $p={\left \lvert {\mathbf {d}}\right \rvert }$ and $\operatorname {\mathrm {Hom}} j=(j_1,\dots ,j_p)$ be any sequence of vertices with $e_{j_1}+\cdots +e_{j_p}={\mathbf {d}}$ .

Lemma 6.1. For any $\xi \in {\mathcal {P}}{_{{\mathbf {w}},{\mathbf {d}}}^{\vee }}$ , the Laurent series

is completely symmetric under the action of $\Sigma _p$ permuting simultaneously the vertices $j_p$ and the variables $z_p$ . Moreover, it is a polynomial divisible by the product $z_1^{{\mathbf {w}}_{j_1}}\cdots z_p^{{\mathbf {w}}_{j_p}}$ .