\end{equation}

Applying the Chern character to \eqref{eq: Ktheorysplitting}, we can write in the language of the Ext descendent algebra
$$\ch_k(\Ext)=\sum_{a+b=k}\sum_{i\in I} \ch^R_a(\alpha_i^L)\boxtimes \ch_b^L(\alpha_i^R)\,.
This is the analogue of equation (45) in \cite{blm}. The rest of the argument is a straightforward adaptation of the proof in loc. cit.
\end{proof}

\begin{remark}
We remark that it is proven along the way that the operators
$$\ch_n^L(\alpha)\cap -: H_\ast(\CN^{\,\bfT})\rightarrow H_\ast(\CN^{\,\bfT})
satisfy the following bracket with the creation/annihilation operators:
\begin{equation}\label{eq: pldagger}
\end{equation}
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm: natural iso}]
By Proposition \ref{prop: vertex iso}, it suffices to prove that the vector space isomorphism \eqref{eq: vector space iso} is natural in the sense of Remark \ref{rem: natural}. Since the isomorphism is constructed by the realization homomorphism $\xi_\alpha$ and a perfect pairing $\widetilde\chi_\bfT(-,-)$ in the diagram \eqref{eq: construction diagram}, it suffices to show the naturality of $\xi_\alpha$ and $\widetilde\chi_\bfT(-,-)$.

Let ${\bf f}:\bfT_1\rightarrow \bfT_2$ be a quasi-functor with the properties described in Remark \ref{rem: natural}. Naturality of the pairing $\widetilde\chi_\bfT(-,-)$ directly follows from the fact that it is induced from $\chi_{\bfT}$ and we have a lattice embedding $K(\bfT_1)\rightarrow K(\bfT_2)$. To show the naturality of $\xi_\alpha$, let $\alpha_1\in K(\bfT_1)$ and $\alpha_2\coloneqq f(\alpha_1)\in K(\bfT_2)$. Denote the induced morphism between the moduli stacks by $f:\CN^{\,\bfT_1}_{\alpha_1}\rightarrow\CN^{\,\bfT_2}_{\alpha_2}$. Note that the construction of the Ext descendent algebra is a contravariant functor in the sense that we have
$$f^*:\BD^{\bfT_2}_{\alpha_2}\rightarrow \BD^{\bfT_1}_{\alpha_1},\quad \ch^L_k(\beta)\mapsto \ch^L_k(g(\beta))
where $g$ is the left adjoint of $f$. Naturality of the realization homomorphism then refers to the commutativity of the diagram
\begin{center}
\begin{tikzcd}
H^*(\CN^{\,\bfT_1}_{\alpha_1})                 & H^*(\CN^{\,\bfT_2}_{\alpha_2}) \arrow[l, "f^*"']                  \\
\BD^{\bfT_1}_{\alpha_1} \arrow[u, "\xi_{\alpha_1}"] & \BD^{\bfT_2}_{\alpha_2} \arrow[u, "\xi_{\alpha_2}"'] \arrow[l, "f^*"']
\end{tikzcd}
\end{center}
which follows from the adjunction quasi-isomorphism
$$f^*\Ext_{\bfT_2}(\beta,-)\simeq \Ext_{\bfT_1}(g(\beta),-)\,.\qedhere

\subsubsection{Virasoro operators}
We finish the section by introducing Virasoro operators on the Ext descendent algebra and comparing them to the operators acting on the lattice vertex algebra.

\begin{definition}\label{def: K Virasoro}
For each $n\geq -1$, we define the Ext Virasoro operator $\bL_n$ acting on $\BD^\bfT$ as a summation $\bL_n=\bR_n+\bT_n$ of a derivation operator $\bR_n$ such that
\begin{align*}
\bR_n(\ch^L_k(\alpha))&=k\cdot(k+1)\cdots(k+n)\,\ch_{k+n}^L(\alpha)\,,\\
\bR_n(\ch^R_k(\alpha))&=k\cdot(k+1)\cdots(k+n)\,\ch_{k+n}^R(\alpha)\,,
\end{align*}
and a multiplication operator
$$\bT_n\coloneqq\sum_{a+b=n}a!b!(-1)^a\ch^R_a\ch^L_b(\Delta^K)=\sum_{a+b=n}\sum_{i\in I}a!b!(-1)^a \sum_{i\in I}\ch^R_a(\alpha_i^L)\ch_b^L(\alpha_i^R)\,.
\end{definition}

\begin{proposition}\label{prop: duality of operators}
Virasoro operators $\bL_n$ and $L_n$ acting on $\BD^\bfT_\alpha$ and $e^\alpha\otimes \BD_{K(\bfT)}$, respectively, are dual to each other with respect to the pairing $\widetilde\chi_\bfT(-,-)$ in \eqref{eq: construction diagram}.
\end{proposition}
This is the dg category version of \cite[Theorem 4.12]{blm}. We will give here a new proof that is cleaner and does not assume the existence of a conformal element. Given an operator $P$ we will denote by $P^\dagger$ its dual with respect to the perfect pairing $\widetilde\chi_\bfT(-,-)$.

\begin{proof}
We first note that when $n=-1$ the claim can be proven exactly in the same way as \cite[Lemma 4.9]{blm}. We focus on the case $n>0$; note that the statement for $n=0$ is easy to check directly, but it also follows from $n=-1$ and $n=1$ together with the Virasoro bracket.

We need to show that the operators $\bL_n^\dagger$, $n>0$ satisfy \eqref{eq: virlva1} and \eqref{eq: virlva2}.
\begin{claim}
We have $\bL_n^\dagger(e^\alpha)=0$ for $n>0$, i.e. equation \eqref{eq: virlva1} holds.
\end{claim}
\begin{proof}
The operator $\bL_n$ increases cohomological degree by $2n$, so $\bL^\dagger_n$ decreases homological degree by $2n$.\qedhere
\end{proof}
We denote by $p_l(\beta)\colon \BD^{\bfT}_\alpha\to \BD^{\bfT}_\alpha$ the operator of multiplication by $l!\ch_l^L(\beta)$. Its dual $p_l^\dagger(\beta)$ is the operator of capping with $l!\ch_l^L(\beta)$ and its bracket with the creation/annihilation operators is given by \eqref{eq: pldagger}.
\begin{claim}\label{claim: 2}
Let $\beta\in K(\bfT)$. We have
\[[\bR_n^\dagger, \beta_{(-k)}]=\begin{cases}
k\cdot \beta_{(-k+n)}&\textup{ if }k>n> 0\\
0 &\textup{ if  }n\geq k>0\,.
\end{cases}\]
\end{claim}
\begin{proof}
We prove the claim by showing that the duals of both sides agree when applied to $1\in \BD^{\bfT}_\alpha$ and have the same commutator with the operators $p_l(\gamma)$ of multiplication by descendents. Indeed we have $\bR_n(1)=\beta_{(-k)}^\dagger (1)=0$ and also $\beta_{(-k+n)}^\dagger (1)=0$ when $k>n$; this follows from the fact that $\beta_{(-k)}^\dagger$ has cohomological degree $-2k<0$. Hence the duals of both sides annihilate $1$.

We now use \eqref{eq: pldagger} to calculate
\begin{align*}[[\bR_n^\dagger, \beta_{(-k)}]^\dagger, p_{l}(\gamma)]&=[[\beta_{(-k)}^\dagger, \bR_n],p_l(\gamma)]=-[[\beta_{(-k)}^\dagger, p_l(\gamma)], \bR_n]+[\beta_{(-k)}^\dagger, [\bR_n, p_l(\gamma)]]\\
&=[\beta_{(-k)}^\dagger, l \cdot p_{l+n}(\gamma)]=l\cdot k\cdot\delta_{l+n, k}\cdot\chi_{\bfT}(\beta,\gamma)\cdot\id\,.
\end{align*}
If $n\geq k$ then $\delta_{l+n, k}=0$ for every $l> 0$. If $n<k$ then this agrees with
\[[k\cdot \beta_{(-k+n)}^\dagger, p_l(\gamma)]=k[p_l(\gamma)^\dagger, \beta_{(-k+n)}]^\dagger=l\cdot k\cdot\delta_{l+n, k}\cdot\chi_{\bfT}(\beta,\gamma)\cdot\id\,.\qedhere\]
\end{proof}

To deal with $T_n$ we prove the following auxiliary claim.

\begin{claim}\label{claim: 3}
For any $\beta\in K(\bfT)$ and $l\geq 0$ we have
\[p_l^\dagger(\beta)+\sum_{i\in I}\chi_\bfT(\alpha_i^R,\beta)\,p_l^\dagger(S_{\bfT}^{-1}(\alpha_i^L))=\beta_{(l)}\,.\]
\end{claim}
\begin{proof}
If $l>0$ then both sides annihilate $e^\alpha$. If $l=0$ then both sides applied to $e^\alpha$ give $\chi^\sym_\bfT(\beta, \alpha)\cdot e^\alpha$ since it is so defined on the right hand side and for the left hand side we have
\begin{align*}
\chi_\bfT(\beta,\alpha)+\sum_{i\in I}\chi_\bfT(\alpha_i^R,\beta)\chi(S_{\bfT}^{-1}(\alpha_i^L),\alpha)
&=\chi_\bfT(\beta,\alpha)+\sum_{i\in I}\chi_{\bfT}(\alpha,\alpha_i^L)\chi_\bfT(\alpha_i^R,\beta)\\
\end{align*}
Hence, it is enough to show that both sides have the same commutator with creation operators $\gamma_{(-s)}$ for $s>0$ and $\gamma\in K(\bfT)$. Indeed,
\begin{align*}
&\Big[ p_l^\dagger(\beta)+\sum_{i\in I}\chi_\bfT(\alpha_i^R,\beta)\,p_l^\dagger(S_{\bfT}^{-1}(\alpha_i^L)), \gamma_{(-s)}\Big]\\
=\,&\left(l\cdot\delta_{l,s}\cdot \chi_{\bfT}(\beta,\gamma)+l\cdot\delta_{l,s}\sum_{i\in I}\chi_\bfT(\alpha_i^R,\beta)\chi_{\bfT}(S_{\bfT}^{-1}(\alpha_i^L),\gamma)\right)\cdot \id\\
=\,&l\cdot\delta_{l,s}\cdot \chi^\sym_{\bfT}(\gamma,\beta)\cdot \id=[\beta_{(l)},\gamma_{(-s)}]\,.\qedhere
\end{align*}
\end{proof}

\begin{claim}\label{claim: 4}
Let $\beta\in K(\bfT)$. We have
\[[\bT_n^\dagger, \beta_{(-k)}]=\begin{cases}
k\cdot \beta_{(n-k)}&\textup{ if }n\geq k> 0\\
0 &\textup{ if  }k> n>0\,.
\end{cases}\]
\end{claim}
\begin{proof}
Recall that
$$\bT_n=\sum_{a+b=n}\sum_{i\in I}a!\ch_a^L(S^{-1}_{\bfT}(\alpha_i^L))\cdot b!\ch_b^L(\alpha_i^R)\,.
Therefore, we compute the bracket as
\begin{align*}
[\bT_n^\dagger, \beta_{(-k)}]
&=\sum_{a+b=n}\sum_{i\in I}\Big[p_b^\dagger(\alpha_i^R)p_a^\dagger(S^{-1}_{\bfT}(\alpha_i^L)),\beta_{(-k)}\Big]\\
&=\sum_{a+b=n}\sum_{i\in I}\Big(
\Big[p_b^\dagger(\alpha_i^R),\beta_{(-k)}\Big]p_a^\dagger(S^{-1}_{\bfT}(\alpha_i^L))+
\end{align*}
If $k>n\geq a,b$ then every commutator in the last line vanishes and we have $[\bT_n^\dagger, \beta_{(-k)}]=0$. Otherwise,
\begin{align*}[\bT_n^\dagger, \beta_{(-k)}]
&=k\sum_{i\in I}\Big(
\chi_\bfT(\alpha_i^R,\beta)p_{n-k}^\dagger(S^{-1}_{\bfT}(\alpha_i^L))+
&=k\sum_{i\in I}\Big(
\chi_\bfT(\alpha_i^R,\beta)p_{n-k}^\dagger(S^{-1}_{\bfT}(\alpha_i^L))+
&=k\sum_{i\in I}
\chi_\bfT(\alpha_i^R,\beta)p_{n-k}^\dagger(S^{-1}_{\bfT}(\alpha_i^L))+kp_{n-k}^\dagger(\beta)=k\cdot \beta_{(n-k)}
\end{align*}
by Claim \ref{claim: 3}.\qedhere
\end{proof}

Now Claims \ref{claim: 2} and \ref{claim: 4} together show that $\bL_n^\dagger$ satisfy equation \eqref{eq: virlva2}, so $\bL_n^\dagger=L_n$. \qedhere\end{proof}

\section{Virasoro constraints via wall-crossing}\label{sec: Virasoro and wall-crossing}

In this section, we prove the Virasoro constraints for quasi-smooth dg quivers as stated in Theorem \ref{thm: wt0virasoro}. In fact, we prove its generalization in terms Joyce's invariant classes as in Theorem \ref{thm: quivervirasoro}, allowing strictly semistable representations. Proof strategy is comparable to that of \cite{blm}. We first wall-cross to simpler moduli spaces where we can verify the Virasoro constraints directly and then show the compatibility between wall-crossing and Virasoro constraints using vertex algebras.

\subsection{Wall-crossing}\label{sec: wall-crossing}

Joyce's vertex algebra was introduced with the purpose of writing down and proving wall-crossing formulas for descendent integrals. In the case of quivers, this gives a complete algorithm to calculate descendent integrals in any moduli space of stable representations of an acyclic quiver with relations.

Let $\dgQ$ be a quasi-smooth dg quiver and denote the associated quiver with relations by $(Q,I)$. We denote by $V^\bfQ=H_\ast(\CN^\dgQ)$ the vertex algebra associated to $\dgQ$. As explained in the beginning of Section \ref{sec: vertexalgebra}, the quotient $\widecheck V^\bfQ=V^\bfQ/T(V^\bfQ)$ is naturally a Lie algebra.

If $d\in \BN^{\dgQ_0}$ and $\theta$ is a stability condition such that $ M_{d}^{\theta-\textup{ss}}= M_{d}^{\theta-\textup{st}}$, then the moduli space admits a virtual fundamental class
\[[ M_{d}^{\theta}]^\vir\in H_{2-2{\chi}_\dgQ(d,d)}( M_{d}^{\theta})\,.\]

This virtual fundamental class can be push-forwarded to $\widecheck V$. Indeed, it can be shown \cite[Proposition 3.24]{Jo17} that if $d\neq 0$ then
\[H_\ast(\CN^{\dgQ}_d)/T\big(H_\ast(\CN^{\dgQ}_d)\big) \cong H_\ast\Big(\big(\CN^\dgQ_d\big)^{\textup{pl}}\Big)\]
is isomorphic to the homology of the rigidication of the stack $\CN^\dgQ_d$. There is a map $\iota\colon M_{d}^\theta\to \big(\CN^\dgQ_d\big)^{\textup{pl}}$, so we can pushforward the virtual fundamental class and regard it as an element of the Lie algebra
\[[ M_{d}^{\theta}]^\vir\in H_\ast\Big(\big(\CN^\dgQ_d\big)^{\textup{pl}}\Big)\subseteq \widecheck V^\bfQ\,.\]
Alternatively, we can think of the class $[M_d^\theta]^\vir\in \widecheck V^{\bfQ}$ as follows: by Proposition \ref{prop: realization iso}, $V_d^{\bfQ}=H_\ast(\CN_d^\dgQ)$ is the space of functionals on $\BD^{\dgQ}_d$. Since $T$ is dual to $\bR_{-1}$, $\widecheck V_d^{\bfQ}$ is the space of functionals on $\BD^{\dgQ}_{d, \inv}=\ker\big(\bR_{-1}\colon \BD^{\dgQ}_{d}\to \BD^{\dgQ}_{d}\big)$. Recall that by Lemma \ref{lem: weignt zero realization} there is a realization morphism from $\BD^{\dgQ}_{d, \inv}$ to $H^\ast(M_d^\theta)$; composing this realization morphism with integration against the virtual fundamental class defines the required functional. Wall-crossing formulas are written using the Lie bracket on $\widecheck V^{\bfQ}$.

\begin{theorem}[{\cite[Theorems 5.7-5.9, 6.16]{Jo21}}]\label{thm: joycewc}
For any $d$ and $\theta$ there are uniquely defined classes $[M_d^{\theta}]^\inva\in \widecheck V^{\bfQ}$ that agree with the virtual fundamental classes when it is defined, and satisfy the wall-crossing formula. More precisely,
\[[M_d^{\theta}]^\inva=[M_d^{\theta}]^\vir\,,\quad\textup{when }M_{d}^{\theta-\textup{ss}}=M_{d}^{\theta-\textup{st}}\,,\]
and for any two stability conditions $\theta_1, \theta_2$ we have
\[[M_d^{\theta_2}]^\inva=\sum_{d_1+\ldots+d_\ell=d}U(d_1, \ldots, d_\ell; \theta_1, \theta_2)\Big[\Big[\ldots\Big[[M_{d_1}^{\theta_1}]^\inva,[M_{d_2}^{\theta_1}]^\inva\Big],\ldots \Big],[M_{d_\ell}^{\theta_1}]^\inva\Big]\]
in $\widecheck V^{\bfQ}$, where $U(d_1, \ldots, d_\ell; \theta_1, \theta_2)\in \BQ$ are explicitly computable coefficients.
\end{theorem}
\begin{remark}\label{rem: abelian and triangulated}
The previous theorem in \cite{Jo21} is proved using a vertex algebra whose underlying vector space is given by the homology group of the moduli stack $\CM^{Q,I}$ of representations of $(Q,I)$. Since any representation of $\BC[Q]/I$ induces a dg module of $\BC[\dgQ]$ by restriction of scalars, we have a morphism of (higher) stacks
$$i:\CM^{Q,I}\rightarrow \CN^{\dgQ}\,.
This morphism clearly intertwines the zero maps, the direct sum maps and the $B\BG_m$-actions which were part of the geometric input for defining the vertex algebra structure. Most importantly, the perfect complex $\Ext^{\textnormal{Joy}}$ used in \cite{Jo21} depends on a choice of generators $I=(r_1,\dots, r_n)$. Since we use a quasi-smooth dg quiver $\dgQ$ according to this choice of generators, it follows that the morphism $i$ intertwines the perfect complex, i.e.,
$$(i\times i)^*\Ext_{\dgQ} \simeq \Ext^{\textnormal{Joy}}\,.
Therefore, $i$ induces a vertex algebra homomorphism $V^{Q,I}\rightarrow V^\dgQ$ and a Lie algebra homomorphism $\widecheck V^{Q,I}\rightarrow \widecheck V^\dgQ$\,. This allows to transport the wall-crossing formula of \cite{Jo21} to the Lie algebra associated to $\dgQ$.

\end{remark}

Theorem \ref{thm: joycewc} allows us to determine the invariant classes $[M_d^\theta]^\inva$ completely by writing everything in terms of an increasing stability condition. We say that $\theta$ is increasing if $\theta_{t(e)}>\theta_{s(e)}$ for any $e\in Q_1$. The following proposition says that for an increasing stability condition, the invariant classes of moduli spaces  are essentially trivial.

\begin{proposition}[{\cite[Theorem 6.19]{Jo21}}]\label{prop: increasingstability}
Let $\theta$ be an increasing stability condition. We have
\[
[M_d^\theta]^\inva=\begin{cases}\overline{e^d\otimes 1}&\textup{ if }d=\delta_v \textup{ for some }v\in Q_0\\
0 & \textup{ otherwise.}
\end{cases}\]
\end{proposition}
Together with the wall-crossing formula, this determines the classes $[M_d^\theta]^\inva$ for every $\theta$.

\begin{example}\label{ex: wcgrassmannian}
We consider the Grassmannian $\Gr(N,k)$ and express its class using the wall-crossing formula. Recall the notations from Example \ref{ex: Grassmannian}. Let $\theta_2$ be a decreasing stability condition of the Kronecker quiver $K_N$ which corresponds to the Grassmannian and $\theta_1$ be an increasing stability condition. By Theorem \ref{thm: joycewc}, we have a wall-crossing formula
\[[M_{(1, k)}^{\theta_2}]=\sum_{d_1+\ldots+d_\ell=(1,k)}U(d_1, \ldots, d_\ell; \theta_1, \theta_2)\Big[\Big[\ldots\Big[[M_{d_1}^{\theta_1}]^\inva,[M_{d_2}^{\theta_1}]^\inva\Big],\ldots \Big],[M_{d_\ell}^{\theta_1}]^\inva\Big]\]
in the Lie algebra $\widecheck V^{K_N}$. By Proposition \ref{prop: increasingstability}, it suffices to consider the contributions where all $d_i$'s are either $\delta_1$ or $\delta_\infty$ and compute the wall-crossing coefficients. The computation of the coefficients can be done as in \cite[Proposition 13.10]{JS12} and we find
$$[\Gr(N,k)]=\frac{1}{k!}[e^{\delta_1},\ldots [e^{\delta_1},[e^{\delta_1},e^{\delta_\infty}]]\ldots ]\,.$$
In Section \ref{sec: sym Heke}, we further simplify this expression using the symmetrized Hecke operators and relate this formula to Schubert calculus.
\end{example}

\subsubsection{Vertex algebra and wall-crossing for framed moduli}

We briefly explain here the natural adaptation of these ideas to moduli of framed representations. For each $f\in \BN^{\dgQ_0}$ and $d\in \BZ^{\dgQ_0}$, we let $\CN^{\dgQ}_{f\rightarrow d}=\CN^{\dgQ}_d$ and define
$$\CN^{\dgQ, \textup{fr}}\coloneqq \bigsqcup_{(f,d)\in \BN^{\dgQ_0}\times \BZ^{\dgQ_0}}\CN^{\dgQ}_{f\rightarrow d}\,.
Define the framed version of the Ext $K$-theory class on $\CN^{\dgQ, \textup{fr}}\times \CN^{\dgQ, \textup{fr}}$ as
$$\Ext_\dgQ^{\textup{fr}}:=\Ext_\dgQ-\sum_{v\in \dgQ_0}\textup{R}\CH om(\CO^{(f_1)_v},\CV_v^{(2)})
where $f_1$ is the framing vector of the first factor and $\CV^{(2)}_v$ is the universal complex pulled back from the second factor. The rank of this $K$-theory class restricted to $\CN^{\dgQ}_{f_1\rightarrow d_2}\times \CN^{\dgQ}_{f_2\rightarrow d_2}$ is given by
\[\chi_{\dgQ, \textup{fr}}\big((f_1, d_1), (f_2, d_2)\big):=\chi_{\dgQ}(d_1, d_2)-f_1\cdot d_2\,.\]

The stack $\CN^{\dgQ, \textup{fr}}$ is equipped with the following geometric data:
\begin{enumerate}
\item A zero map $0:\pt\rightarrow \CN^{\dgQ, \textup{fr}}$.
\item A direct sum map $\Sigma:\CN^{\dgQ}_{f_1\rightarrow d_2}\times \CN^{\dgQ}_{f_2\rightarrow d_2}\rightarrow \CN^{\dgQ}_{(f_1+f_2)\rightarrow (d_1+d_2)}$.
\item A $B\BG_m$-action $\rho:B\BG_m\times \CN^{\dgQ, \textup{fr}}\rightarrow \CN^{\dgQ, \textup{fr}}$.
\item A $K$-theory class $\Theta^{\textup{fr}}=\sigma^*\Ext_\dgQ^{\textup{fr}}+(\Ext_\dgQ^{\textup{fr}})^\vee$.
\end{enumerate}
The first three items come directly from the corresponding maps of the unframed stack $\CN^{\dgQ}$. This data defines a framed vertex algebra
$$V^{\dgQ, \textup{fr}}\coloneqq \Big(H_\ast(\CN^{\dgQ, \textup{fr}})\,,\, \ket{0}\,,\, T\,,\, Y\Big)\,.
We note that this construction is similar to the one in \cite[Section 5.2]{Jo21}, where the author considers representations of a quiver for which some subset of the vertices $\Ddot{Q}_0\subseteq Q_0$ are framed.

Structure of the framed vertex algebra can be described precisely. The same proof of Theorem \ref{thm: natural iso} shows that we have a vertex algebra embedding
$$V^{\dgQ,\textup{fr}}\hookrightarrow \textup{VA}\Big(K(\dgQ)\times K(\dgQ), \chi_{\dgQ, \textup{fr}}^\sym\Big)\,.
On the level of underlying vector spaces, this embedding at the connected component $\CN^{\dgQ}_{f\rightarrow d}$ is given by the composition
$$H_*(\CN^{\dgQ}_{f\rightarrow d})=H_*(\CN^{\dgQ}_{d})=e^{(f,d)}\cdot\BD_{K(\dgQ)}\hookrightarrow e^{(f,d)}\cdot \BD_{K(\dgQ)\times K(\dgQ)}
where the last embedding is induced from $K(\dgQ)=\{0\}\times K(\dgQ)\subseteq K(\dgQ)\times K(\dgQ)$.

The framed vertex algebra is a natural place to write down the wall-crossing formulas for framed moduli spaces. Let $\dgQ$ be a quasi-smooth quiver and consider a framed moduli space $M^{\theta}_{f\rightarrow d}$. The universal framed representation of the moduli space induces a morphism $M^{\theta}_{f\rightarrow d}\rightarrow \CN^{\dgQ}_{f\rightarrow d}$, so we can pushforward its virtual class to the framed vertex algebra
$$[M^{\theta}_{f\rightarrow d}]^\vir\in V^{\dgQ, \textup{fr}}\,.
Note that this is different from the case of unframed representations, where moduli spaces only define a class on the homology of the rigidification or, equivalently, on the Lie algebra.

Finally, we explain how to obtain the wall-crossing formula in $V^{\dgQ,\textup{fr}}$ from the usual wall-crossing formula via framed/unframed correspondence. Given a fixed framing vector $f$, consider the auxiliary quiver $\bfQ^f$ as explained in Section \ref{sec: f/uf corr}. Then framed/unframed correspondence reads $M^{\theta}_{f\rightarrow d}=M^{\tilde{\theta}}_{(1,d)}$ for some $\tilde{\theta}$. On the other hand, if $\theta'$ is any stability condition of $\dgQ^f$, then wall-crossing formula reads
\[[M_{(1,d)}^{\tilde\theta}]^\inva=\sum_{\tilde d_1+\ldots+\tilde d_\ell=(1,d)}U(\tilde d_1, \ldots, \tilde d_\ell; \theta', \tilde \theta)\Big[\Big[\ldots\Big[[M_{\tilde d_1}^{\theta'}]^\inva,[M_{\tilde d_2}^{\theta'}]^\inva\Big],\ldots \Big],[M_{\tilde d_\ell}^{\theta'}]^\inva\Big]\]
where $\tilde d_i=(1, d_i)$ for exactly one $1\leq i \leq \ell$ and $\tilde d_j=(0, d_j)$ for $j\neq i$. This identity holds in the Lie algebra $\widecheck V^{\dgQ^f}$. Since we have
\begin{align*}
\chi_{\dgQ, \textup{fr}}((0, d_1), (0, d_2))&=\chi_{\bfQ^f}((0,d_1), (0,d_2))
\chi_{\dgQ, \textup{fr}}((f, d_1), (0, d_2))&=\chi_{\bfQ^f}((0,d_1), (1,d_2))
\chi_{\dgQ, \textup{fr}}((0, d_1), (f, d_2))&=\chi_{\bfQ^f}((1,d_1), (0,d_2))
\end{align*}
the above wall-crossing formula in $\widecheck V^{\dgQ^f}$ can instead be written as an identity in $\widecheck V^{\dgQ, \textup{fr}}$ after we replace $M_{(0,d)}^{\theta'}(Q^f, I^f)$ by $M_{0\to d}^{\theta'}(Q,I)$ and  $M_{(1,d)}^{\theta'}(Q^f, I^f)$ by $M_{f\to d}^{\theta'}(Q,I)$.\footnote{This requires a version of framed/unframed correspondence beyond limit stability condition which can be obtained by straightforward adaptation of Section \ref{sec: f/uf corr}.} Moreover, the same argument as explained in \cite[Lemma 5.11]{blm} shows that this formula can actually be lifted to the vertex algebra $V^{\dgQ, \textup{fr}}$.

\begin{example}
\label{ex: grassva}
We return to the example of the Grassmannian and the quiver with one vertex $Q=A_1$. We denote by $V^\Gr$ the vertex algebra
\[V^\Gr=V^{A_1, \textup{fr}}=H_\ast(\CN^{A_1, \textup{fr}})= H_\ast(BU)[Q, q^{\pm 1}]\]
where we write $Q^N q^k=e^{(N, k)}$ for $(N, k)\in \BN\times \BZ$ corresponding to the connected component of framed representations $N\to k$. This is a vertex subalgebra of
\[\textup{VA}(\BZ\times \BZ, \chi_{\Gr}^\sym)\]
\[\chi_{\Gr}((N_1, k_1), (N_2, k_2))=k_2(k_1-N_1)\,.\]
The wall-crossing formula from Example \ref{ex: wcgrassmannian} becomes
\[[\Gr(N, k)]=\frac{1}{k!}[q, \ldots [q, [q, Q^N]]\ldots ]\in V^\Gr\,,\]
where $[u, v]=u_0 v$ is the partial lift of the Borcherds' Lie bracket discussed in \cite[Lemma 3.12]{blm}.
\end{example}

\subsection{Vertex algebra comparison for dg quivers}\label{sec: assumptiondgquivers}

In this section, we show that Assumption \ref{ass: dg category} is satisfied for the dg category $\bfT$ associated to the dg quiver $\bfQ$. Thus, the general results from Section \ref{subsec: comparisonva} apply to $\bfQ$.

We first connect the language specific to dg quivers and the general language for any dg category. Recall that we had two types of descendent theory, one using cohomology $H(\bfQ)$ as in Section \ref{sec: descendent algebra} and the other using the perfect complex $\Ext_{\bfQ}$ as in Section \ref{sec: Ext descendent}. We remark that the realization homomorphism in \ref{sec: descendent algebra} can be lifted, using the universal complex $\CV_v$ for each $v\in \dgQ_0$, to the stack of a dg quiver
\begin{center}
\begin{tikzcd}
\BD^{\dgQ}_d \arrow[r] \arrow[rd, dashed] & H^*(\CM_{d}^{Q,I})                 \\
& H^*(\CN_{d}^{\dgQ}) \arrow[u, "i^*"']
\end{tikzcd}
\end{center}
where $i:\CM_d^{Q,I}\rightarrow \CN_d^\dgQ$ is the morphism explained in Remark \ref{rem: abelian and triangulated}.

As a first step toward the proof, we show that the two types of descendent theories, and respective Virasoro operators, are compatible.

\begin{proposition}\label{prop: comparison of two descendents}
Let $\dgQ$ be a quasi-smooth dg quiver and $\bfT$ be the natural dg enhancement of $D^b(\bfQ)$. Then we have a commutative diagram
\begin{center}
\begin{tikzcd}[row sep=small]
\BD^\bfT_d \arrow[rd, ""] \arrow[dd, "\sim"  {rotate=90, anchor=north}, "\phi"'] &   \\
& H^*(\CN_d^{\dgQ}) \\
\BD^\dgQ_d \arrow[ru, ""']                 &
\end{tikzcd}
\end{center}
where $\phi$ maps the Ext descendent $\ch^L_k([P(v)])$ to the cohomological descendent $\ch_k(v)$. Furthermore, $\phi$ intertwines the Virasoro operators on the two descendent algebras.
\end{proposition}
\begin{proof}
For each $v\in\dgQ_0$ and a dg module $M$ of $\dgQ$, we have a quasi-isomorphism $\Hom^\bullet_\dgQ(P(v),M)\simeq M_v$. This implies $\Ext_\dgQ(P(v),-)\simeq \CV_v$ which justifies the definition of $\phi$ and commutativity.

Intertwining property of the derivation parts is clear. To show that the multiplication parts intertwines through $\phi$, we use the formula for the $K$-theoretic diagonal of a dg quiver
$$\Delta^K=\sum_{v\in \dgQ_0}[S(v)]\otimes [P(v)]\in K(\dgQ)\otimes K(\dgQ)\,.
Therefore, the multiplication part of the Ext Virasoro operator is
\begin{align}\label{eq: Ext Tn}
\nonumber    \bT_n
&=\sum_{a+b=n}a!b!\,\sum_{v\in \dgQ_0} (-1)^a\ch_a^R(S(v))\ch_b^L(P(v))\\
&=\sum_{a+b=n}a!b!\,\sum_{v\in \dgQ_0} \ch_a^L(S_\bfT^{-1}(S(v)))\ch_b^L(P(v))\,.
\end{align}
For each $v$, we have rational numbers $a_{vw}$ such that
$$S_\bfT^{-1}(S(v))=\sum_{w\in \dgQ_0} a_{vw} [P(w)]\in K(\dgQ)\,.
Taking $\chi_\dgQ(-,S(u))$ on the both sides, we obtain
$$\chi_\dgQ(S(u),S(v))=\chi_\dgQ(S_\bfT^{-1}(S(v)),S(u))=\sum_{w\in \dgQ_0}a_{vw}\chi_\dgQ(P(w),S(u))= a_{vu}
In conclusion, $\phi$ maps the expression \eqref{eq: Ext Tn} to
\sum_{a+b=n}a!b!\sum_{v,w\in \dgQ_0} a_{vw}\,\ch_a(w)\ch_b(v)=\sum_{a+b=n}a!b!\sum_{v,w\in \dgQ_0} \chi_\dgQ(v,w)\ch_a(v)\ch_b(w)
which is precisely $\bT_n$ on $\BD^\bfQ$. \qedhere
\end{proof}

By the commutative diagram in the above proposition, Assumption \ref{ass: dg category} (2) is satisfied if and only if the cohomological realization homomorphism is an isomorphism. We prove this for any finite acyclic dg quivers.

\begin{proposition}\label{prop: realization iso}
Let $\dgQ$ be a finite and acyclic dg quiver and $\bfT$ be the natural dg enhancement of $D^b(\bfQ)$. Then the realization homomorphism \eqref{eq: assumption iso} is an isomorphism.
\end{proposition}
\begin{proof}
When $\dgQ$ is a usual quiver $Q$, this statement is essentially \cite[Proposition 5.13]{Jo17}. Joyce remarks in the last paragraph of Section 6.2 in \cite{Jo21} that the same result in the abelian category setting for $Q$ with relations is true, as long as the relations are homogeneous (i.e. $r(q)$ is a linear combination of paths of the same length). Motivated by this, we give a proof for any dg quiver.

We abuse the notation by writing $Q_0$ for a quiver with the set of vertices given by $\dgQ_0$ but no arrows. Recall that $S=\BC[Q_0]$. We have a sequence of maps of higher stacks
$$\CN^{Q_0}\xrightarrow{f}\CN^{\bfQ}\xrightarrow{g} \CN^{Q_0}
whose $B$-points for any commutative algebra $B$ is given as follows: $f$ sends a perfect $S\otimes B$-module $M$ to the $\BC[\bfQ]\otimes B$-module $\BC[\bfQ]\otimes_S M$  and $g$ sends a $\BC[\bfQ]\otimes B$-module $N$ perfect over $B$ to the $S\otimes B$-module $\BC[\bfQ]/J\otimes B$-module $\BC[\bfQ]/J\otimes_{\BC[\bfQ]} N$ . Since $\BC[\dgQ]/J\simeq S$, the composition $g\circ f$ is an identity morphism.

Assume for a moment that $f\circ g:\CN^{\bfQ}\rightarrow\CN^{\bfQ}$ is homotopic to the identity map. Since $\CN^{Q_0}\simeq \prod_{v\in Q_0}\Perf^{(v)}$ is a product of $|Q_0|$ copies of the stack of perfect $\BC$-complexes $\Perf$, this gives an isomorphism
$$g^*:H^*(\CN^{Q_0})=\bigotimes_{v\in Q_0}H^*(\Perf^{(v)})\xrightarrow{\sim} H^*(\CN^{\bfQ}).
The stack $\Perf_{d_v}$ is well-known to be homotopically equivalent to $BU$ (see \cite[Theorem 4.5]{B16} or \cite[Proposition 5.9]{Jo17}) and so
\[H^\ast(\Perf_{d_v})=\BQ[\ch_1(\CV_v), \ch_2(\CV_v), \ldots]\]
is the free polynomial ring in the Chern characters of the universal complex $\CV_v$. This proves that the realization homomorphism $\BD^{\dgQ}_d\rightarrow H^*(\CN_{d}^{\dgQ})$ is an isomorphism.

We are left to construct a homotopy between $f\circ g$ and $\id$. Since $\bfQ$ is acyclic, we can pick an arbitrary function $w\colon Q_0\to \BZ$ that is increasing along arrows, i.e. such that $w(e)\coloneq w(t(e))-w(s(e))\in \BZ_{>0}$ for every $e\in \bfQ_1$. The weight extends to any path in $\BC[\bfQ]$ multiplicatively, i.e., $w(e_\ell\cdots e_1)\coloneqq\sum_{i=1}^\ell w(e_i)$. We define a map of higher stacks
$$r\colon \BA^1\times \CN_{\bfQ}\rightarrow \CN_{\bfQ}
whose action on $B$-points for any commutative algebra $B$ is described as follows. If $t\in B$ and $M$ is a $\BC[\bfQ]\otimes B$-module which is $B$-perfect, then $r(t,M)$ is a $\BC[\bfQ]\otimes B$-module whose underlying $B$-module is the same as $M$ with $\BC[\bfQ]$-module structure twisted by $w$ and $t$, i.e.,
$$a\cdot_w m\coloneqq t^{w(a)}(a\cdot m),\quad a\in \BC[\bfQ],\ m\in M
when $a=e_\ell\cdots e_1$ and we extend the definition linearly.
Since $B$ is a commutative algebra, this indeed defines a $\BC[\bfQ]\otimes B$-module. The morphism $r$ restricted to $1\in \BA^1$ is clearly the identity map. On the other hand, if we restrict $r$ to $0\in \BA^1$, then the $w$-twisted multiplication map by $\BC[\bfQ]$ becomes trivial away from $S=\BC[Q_0]$ because path of positive length have positive weight. In other words, $r$ restricted to $0\in \BA^1$ becomes the map $f\circ g$. This proves that $f\circ g$ and $\id$ are homotopic to each other after taking the topological realizations.
\end{proof}

\begin{proposition}\label{prop: checking assumptions}
Let $\dgQ$ be a finite and acyclic dg quiver and $\bfT$ be the natural dg enhancement of $D^b(\bfQ)$. Then $\bfT$ satisfies Assumption \ref{ass: dg category}.
\end{proposition}
\begin{proof}
Recall that $K(\bfQ)\simeq \BZ^{\dgQ_0}$ since its $K$-theory is generated by projective dg modules $P(v)$ and they satisfy $\chi(P(v),S(w))=\delta_{vw}$. Also, the proof of Proposition \ref{prop: realization iso} shows, in particular, that $\pi_0(\CN^\dgQ)$ is $\BZ^{\dgQ_0}$. Since Proposition \ref{prop: realization iso} precisely covers the Assumption \ref{ass: dg category} (2), we are left with Part (3). This follows from the standard resolution; indeed, the diagonal bimodule corresponds to $\BC[\bfQ]$ regarded as a bimodule over itself, and by Theorem \ref{thm: dgstandardresolution} its class in $K$-theory is
\[\sum_{v\in \bfQ_0}[P(v)]\boxtimes [R(v)]-\sum_{e\in \bfQ_1}(-1)^{|e|}[P(t(e))]\boxtimes [R(s(e))]=\sum_{v\in \bfQ_0}[P(v)]\boxtimes [S(v)]\]
where $R(v)\coloneqq {\bf 1}_{v}\cdot \BC[\bfQ]$. The equality comes from the following exact sequence of $\BC[\bfQ]^{\textup{op}}$-modules for each $v\in \dgQ_0$:
\[0\rightarrow \hspace{-10pt}\bigoplus_{e\in \dgQ_1,\ t(e)=v}\hspace{-10pt} R(s(e))[-|e|]\rightarrow R(v)\rightarrow S(v)\rightarrow 0\,. \qedhere
\]
\end{proof}

\begin{corollary}\label{cor: grossisoquiver}
Let $\dgQ$ be a finite and acyclic dg quiver. Then there is a natural isomorphism of vertex algebras
$$V^\bfQ\rightarrow \textup{VA}(K(\bfQ),\chi_{\bfQ}^{\sym}).
\end{corollary}
\begin{proof}
By Proposition \ref{prop: checking assumptions}, Theorem \ref{thm: natural iso} applies to $\bfQ$.
\end{proof}

\subsection{Proof of Virasoro constraints for quivers}\label{sec: VA and Virasoro}

In this section, we apply the general theory developed up until here and show that the wall-crossing formula from Section \ref{sec: wall-crossing} implies the Virasoro constraints for quasi-smooth quivers, recoving the Virasoro constraints proven in \cite{bojko}. Recall Definition \ref{def: primarystates} of primary state.

\begin{corollary}
Suppose that $M_d^{\theta-\textup{ss}}=M_d^{\theta-\textup{st}}$. Then $M_d^{\theta}$ satisfies the Virasoro constraints (Theorem \ref{thm: wt0virasoro}) if and only if $[M_d^\theta]^\vir\in \widecheck V^{\bfQ}$ is a primary state.
\end{corollary}
\begin{proof}
By duality of Virasoro operators as in Proposition \ref{prop: duality of operators} and by Proposition \ref{prop: comparison of two descendents}, the Virasoro constraints of Section \ref{subsec: virasoroops} are equivalent to
\[0=\bL_\inv^\dagger([M_d^\theta]^\vir)=\sum_{n\geq -1}\frac{(-1)^n}{(n+1)!}L_{-1}^{n+1}\circ L_n([M_d^\theta]^\vir)\,.\]
The conclusion follows from Corollary \ref{cor: primarywt0}.
\end{proof}

In light of the previous corollary, it makes sense to talk about Virasoro constraints even when there are semistable representations, thanks to Joyce's classes from Theorem~\ref{thm: joycewc}. We say that $M_d^{\theta-\textup{ss}}$ satisfies Virasoro constraints if and only if $[M_d^{\theta}]^\inva \in \widecheck V^{\bfQ}$ is a primary state.

\begin{theorem}\label{thm: quivervirasoro}
Let $\dgQ$ be any quasi-smooth dg quiver. For any $d\in \BN^{Q_0}$ and stability condition $\theta$, the class $[M_d^{\theta}]^\inva$ is a primary state. In other words, every moduli space of quiver representations satisfies the Virasoro constraints.
\end{theorem}
\begin{proof}
When $\theta$ is an increasing stability condition, the claim follows trivially from Proposition \ref{prop: increasingstability} and the definition of the operators $L_n$, namely \eqref{eq: virlva1}. By Joyce's wall-crossing formula (Theorem \ref{thm: joycewc}) and the fact that $\widecheck P_0$ is a Lie subalgebra (Corollary \ref{cor: liesubalgebra}) the result follows for every $\theta$.\qedhere
\end{proof}

\section{Application to surfaces with exceptional collections}\label{sec: applicaton}

The goal of this section is to clarify how the quiver Virasoro constraints are related to the sheaf Virasoro constraints for surfaces admitting exceptional collections, especially for del Pezzo surfaces. In particular, we prove Theorem \ref{main thm: abch implies Virasoro} and \ref{main thm: three surfaces}.

\subsection{Exceptional collection and derived equivalence}
\label{sec: beilinson}

We start by recalling the notion of exceptional collection and the induced equivalence between a triangulated category with a full and strong exceptional collection and the derived category of a quiver with relations. From now on let $\CD$ be a triangulated category (e.g. the derived category of coherent sheaves on $X$ or representations of a quiver with relations $(Q,I)$).

\begin{definition}
A sequence $\FE=(E_1,  \ldots, E_n)$ of objects of $\CD$ is called an exceptional collection if
\[\Hom_\CD(E_i, E_j[\ell])=\begin{cases}
\BC& \textup{ if }i=j\textup{ and }\ell=0\\
0& \textup{ if }i>j\textup{ or }[i=j\textup{ and }\ell\neq 0]\,.
\end{cases}\]
An exceptional collection is said to be strong if moreover for any $i,j$ we have
\[\Hom_\CD(E_i, E_j[\ell])=0\quad\textup{ for }\ell \neq 0\,.\]
An exceptional collection is said to be full if the smallest triangulated subcategory of $\CD$ containing $\FE$ is $\CD$ itself.
\end{definition}

If $\CD$ admits a full exceptional collection $\FE$ then
\[K(\CD)=\bigoplus_{i=1}^n \BZ\cdot [E_i]\]
and $\chi_\CD\colon K(\CD)\times K(\CD)\to \BZ$ is given in the basis above by a triangular matrix with $1$ along the diagonal. In particular $\chi_\CD$ is non-degenerate.

A full exceptional collection $\FE$ always admits a unique left dual exceptional collection $\widetilde \FE=(\widetilde E_n, \ldots, \widetilde E_1)$ such that
\[\Hom_\CD(\widetilde E_i, E_j[\ell])=\begin{cases}\BC&\textup{ if }i=j\textup{ and }\ell=0\\
0&\textup{ otherwise}
\end{cases}\,.\]
In particular
\[\chi\big(\widetilde E_i, E_j\big)=\delta_{ij}\,.\]

\begin{example}
Let $(Q, I)$ be an acyclic quiver with relations; without loss of generality denote the nodes as $Q_0=\{1, 2, \ldots, n\}$ in a way that there are no arrows $i\to j$ for $i>j$.
Recall from Example \ref{ex: simpleprojective} the simple and projective modules $S(i)$ and $P(i)$. Then
\[E_i=S(i)\,,\quad \widetilde E_i=P(i)\]
are dual full exceptional collections on $D^b(Q, I)$. We call $(E_1,\dots, E_n)$ the standard exceptional collection of $(Q, I)$. The dual collection is strong by projectivity of $P(i)$. The same applies to a dg quiver, see Example \ref{ex: hprojectivequiver}.
\end{example}

\begin{example}\label{ex: exceptionalcollections}
The derived category of any smooth projective toric variety admits a full exceptional collection \cite{kawamata}. For $D^b(\BP^2)$ we can take
\[\FE=\big(\CO_{\BP^2}(-1)[2], \CO_{\BP^2}[1], \CO_{\BP^2}(1)\big)\,,\quad \widetilde\FE=\big(\CO_{\BP^2}(1), T_{\BP^2}, \CO_{\BP^2}(2)\big)\,.\]
Both $\FE$ and $\widetilde \FE$ are strong. For $D^b(\BP^1\times \BP^1)$ we can take
\begin{align*}\FE&=\big(\CO_{\BP^1\times \BP^1}(0,-1)[2], \CO_{\BP^1\times \BP^1}[1], \CO_{\BP^1\times \BP^1}(1,-1)[1], \CO_{\BP^1\times \BP^1}(1,0)\big)\,,\quad\\ \widetilde
\FE&=\big(\CO_{\BP^1\times \BP^1}(1,0), \CO_{\BP^1\times \BP^1}(1,1), \CO_{\BP^1\times \BP^1}(2,0), \CO_{\BP^1\times \BP^1}(2,1)\big)\,.
\end{align*}
In this case, $\FE$ is not strong, but $\widetilde \FE$ is strong.
\end{example}
Given a triangulated category $\CD$ and an exceptional collection $\FE$ whose left dual $\widetilde \FE$ is full and strong, we associate a quiver $Q$ together with an ideal of relations $I\subseteq \BC [Q]$. The set of vertices of $Q$ is $\{1, \ldots, n\}$ and the number of arrows $i\to j$ is equal to $\dim \Ext^1(E_i, E_j)$ for $i\neq j$; since there are no arrows $i\to j$ if $i\geq j$, the quiver $Q$ is acyclic. Then there is an ideal of relations $I$ such that
\[\BC [Q]/I=\textup{End}_{\CD}\Big(\bigoplus_{i=1}^n \widetilde E_i\Big)\,.\]

In the next theorem we let $\CD$ be a sufficiently nice triangulated category such as $D^b(X)$ for some smooth projective variety $X$ or $D^b(Q, I)$.

\begin{theorem}[{\cite[Theorem 6.2]{bondal}}]\label{thm: beilinson}
Suppose that $\CD$ has an exceptional collection $\FE$ such that its left dual exceptional collection $\widetilde \FE$ is full and strong. Then there is an isomorphism of derived categories
\[B\colon \CD\to D^b(Q, I)\]
sending $\FE$ to the canonical exceptional collection of $(Q, I)$. The isomorphism sends an object $F$ of $\CD$ to the complex of representations of $(Q, I)$ which at vertex $i$ has complex $\RHom_\CD\big(\widetilde E_i, F\big)$.
\end{theorem}

By Theorem \ref{thm: koszuldgquiver}, there is a dg quiver $\bfQ$ with $\dim \Ext_{\CD}^{k+1}(E_i, E_j)$ arrows from $i$ to $j$ of degree $-k$ such that
\[\CD\simeq D^b(Q, I)\simeq D^b(\bfQ)\,.\]
Up to choices of basis, this dg quiver is canonically associated to the pair $(\CD, \FE)$.

\begin{example}The theorem applied to the derived category of $D^b(\BP^2)$ with the full exceptional collection of Example \ref{ex: exceptionalcollections} establishes an isomorphism between $D^b(\BP^2)$ and the derived category of the Beilinson quiver with relations $B_3$, see Example \ref{ex: beilinson}. See also Example \ref{ex: beilinsondgquiver} where the associated dg quiver is discussed.

If we apply it to $D^b(\BP^1\times \BP^1)$ with the exceptional collection from Example \ref{ex: exceptionalcollections} we obtain an isomorphism with the derived category of the quiver
\begin{center}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoAmAXVJADcBDAGwFcYkQBGEAX1PU1z5CKcqWLU6TVu3I8+IDNjwEiogCwSGLNohABmOfyVCiaipqk6QanhJhQA5vCKgAZgCcIAWyRkQOCCQOGgAjGDAoJABaPT8taV16AH1ZGkZ6MMYABQFlYRAsMGxYQxAPb18aAKDQ8MjEWLSMmGzck11C4rYaeKtkrl43Tx9EYP9AhtqI6MaQdMyc4xUOoqwSnst2AGMU0vKRserJkDDphr95lsXBZYLV9cltbaSB+X3K8aRRE7qZi+bWkt8p01t1HgkTrtBmVhh8jt9TvVZpdATdgfcwb12CEXntYYhvkc9FN6jF-gs2rcQQ8sbooFC3vjCRNiT8zsiAdc8uxqZjNnTcdxKNwgA
\begin{tikzcd}  &  & \tikz \node[draw, circle]{2}; \arrow[rrdd, "c_2" description, bend right] \arrow[rrdd, "c_1" description, bend left] &  &   \\
&  &                              &  &   \\
\tikz \node[draw, circle]{1};\arrow[rruu, "a_2" description, bend right] \arrow[rruu, "a_1" description, bend left] \arrow[rrdd, "b_2" description, bend right] \arrow[rrdd, "b_1" description, bend left] &  &                   &  & \tikz \node[draw, circle]{4}; \\ &  &   &   \\
&  & \tikz \node[draw, circle]{3};\arrow[rruu, "d_2" description, bend right] \arrow[rruu, "d_1" description, bend left] &  &
\end{tikzcd}
\end{center}
subject to relations
\[c_1a_1+ d_1b_1=c_2a_2+ d_2b_2=c_2 a_1+d_1 b_2=c_1 a_2+d_2 b_2=0\,.\]
\end{example}

\subsubsection{Comparing vertex algebras}
Let $X$ be a smooth projective variety and suppose that $\CD=D^b(X)$ satisfies the hypothesis of Theorem \ref{thm: beilinson}. The isomorphism of derived categories $D^b(X)\simeq D^b(Q, I)\simeq D^b(\bfQ)$ can be upgraded to an equivalence between the natural dg enhancements, so it induces a quasi-equivalence between the derived stacks $\mathbf{B}\colon \bfCN^X\to \bfCN^\bfQ$. As in Remark \ref{rem: natural}, the pushforward on homology
\[B_\ast\colon H_\ast(\CN^{X})\to H_\ast(\CN^{\bfQ})\]
is an isomorphism between the vertex algebras canonically associated to (the derived enhancements of) $D^b(X)$ and $D^b(\bfQ)$.

These vertex algebras are naturally isomorphic to the lattice vertex algebras $\textup{VA}(K(X), \chi_X^{\sym})$ and $\textup{VA}(K(Q), \chi_{\dgQ}^{\sym})$, see Corollary \ref{cor: grossisoquiver}. The isomorphism of derived categories induces an isomorphism $\phi\colon K(X)\to K(Q)$ that preserves the respective Euler pairings $\chi_X$ and $\chi_\bfQ$. The naturality of this isomorphism amounts to commutativity of the following diagram:
\begin{center}
\begin{tikzcd}
H_\ast(\CN^{X})\arrow[rr]\arrow[d, "B_\ast"]& &\textup{VA}(K(X), \chi_X^{\sym})\arrow[d, "\phi"]\\
H_\ast(\CN^{\dgQ})\arrow[rr]& &\textup{VA}(K(Q), \chi_{\dgQ}^{\sym})
\end{tikzcd}
\end{center}

\subsection{Bridgeland stability and the ABCH program}

Given a (sufficiently nice, for instance $D^b(X)$, $D^b(Q, I)$ or $D^b(\bfQ)$) triangulated category $\CD$, Bridgeland defined in \cite{Bri} a notion of (numerical) stability conditions on $\CD$ which we refer to as Bridgeland stability conditions. A Bridgeland stability consists in a pair $\sigma=(\CA, Z)$ where $\CA$ is the heart of a t-structure on $\CD$ and $Z\colon K_{\textnormal{num}}(\CD)\to \BC$ is a group homomorphism, called the central charge.\footnote{Here $K_{\textnormal{num}}(\CD)$ denotes the quotient of $K(\CD)$ by the kernel of the Euler paring. The numerical $K$-group is equal to the original $K$-group in all cases of our applications. } Among other things, it is required that $Z(F)\in \BH$ for $F\in \CA\backslash\{0\}$ where $\BH$ is the upper half plane with a negative real line
\[\BH=\{z\in \BC\colon \Im(z)>0\}\cup (-\infty, 0)\,.\]
A Bridgeland stability defines a slope function
\[\mu^\sigma(F)=-\frac{\Re(Z(F))}{\Im(Z(F))}\in \BR\cup\{+\infty\}\]
on $\CA\backslash\{0\}$. An object of $\CA$ is said to be $\sigma$-(semi)stable if every nonzero proper subobject has smaller (or equal) slope. Given $\v\in K(\CD)$ we denote by $\CM^{\sigma-\textup{st}}_\v\subseteq \CM^{\sigma-\textup{ss}}_\v$ the moduli stacks of $\sigma$-stable and $\sigma$-semistable objects in class $\v$. A stability condition defines a slicing of $\CD$, i.e. a collection of full aditive subcategories $\CP(\phi)\subseteq \CD$ for each $\phi\in\BR$: if $0<\phi\leq 1$ then $\CP(\phi)$ consists of the zero object and $\sigma$-semistable objects of $\CA\backslash\{0\}$ such that $Z(F)=m(F)e^{\pi \sqrt{-1} \phi}$ for some $m(F)\in \BR^+$; for the remaining $\phi$, we extend this notion by setting $\CP(\phi+1)=\CP(\phi)[1]$.

Bridgeland shows that the space of stability conditions $\Stab(\CD)$ is a complex manifold; more precisely, the map that forgets $\CA$ and only remembers $Z$ defines a local homomorphism from $\Stab(\CD)$ to $\Hom_\BZ(K_{\textup{num}}(\CD), \BC)$. There is an $\BR$-action
\[\BR\times \Stab(\CD)\ni(\phi, \sigma)\mapsto \sigma[\phi]\in \Stab(\CD)\] on $\Stab(\CD)$ which translates the associated slicing of $\sigma$ by $\phi$. In particular, for $0<\phi<1$ we have
\[\CM^{\sigma[\phi]-\textup{ss}}_\v=\begin{cases}
\CM^{\sigma-\textup{ss}}_\v & \textup{if }\pi\geq \arg(Z(\v))>\phi \pi\\
\CM^{\sigma-\textup{ss}}_\v[1] & \textup{if }0<\arg(Z(\v))\leq \phi \pi\,.\\
\end{cases}\]
By the notation $\CM^{\sigma-\textup{ss}}_\v[1]$, we mean the same moduli stack $\CM^{\sigma-\textup{ss}}_\v$ with a universal object shifted by $[1]$.

\subsubsection{Quiver stability conditions}

Suppose that $\FE$ is an exceptional collection of $\CD$ such that its dual $\widetilde \FE$ is full and strong. Then the extension closure of $E_1, \ldots, E_n$ is an abelian category $\CA_\FE$, which under the derived equivalence in Theorem \ref{thm: beilinson} corresponds to the abelian category $\Rep_{(Q,I)}$ of representations of a quiver with relations $(Q, I)$. For any choice of $\zeta=(\zeta_1, \ldots, \zeta_n)$ with $\zeta_i \in \BH\,\setminus (-\infty,0)$, we define a central charge $Z_{\zeta}$ by setting $Z_\zeta(E_i)=\zeta_i$ and extending it additively to $K(\CD)$. We say that a stability condition $\sigma=(\CA_\FE, Z_\zeta)$ constructed in this way is a quiver stability condition.

If $\zeta_i=-\theta_i+\sqrt{-1}\in \BH$, the slope associated to $\sigma=(\CA_\FE, Z_\zeta)$ matches precisely the slope defined for representations of quivers in Section \ref{subsubsec: stabilityquivers}, so the derived equivalence $B:\CD\xrightarrow{\sim}D^b(Q,I)$ identifies $\CM^{\sigma-\textup{ss}}_\v$ with $\CM_{d}^{\theta-\textup{ss}}(Q, I)$ where $d$ is the image of $\v$ under the isomorphism $K(\CD)\simeq K(Q, I)$. It was observed in \cite[Proposition 8.1]{abch} that, more generally, for arbitrary $\zeta_i\in \BH\,\backslash(-\infty,0)$  the equivalence $B$ still identifies $\CM^{\sigma-\textup{ss}}_\v$ with $\CM_{d}^{\theta-\textup{ss}}(Q, I)$ where $\theta$ is defined by
\[\theta_i=-\Re(\zeta_i)+\Im(\zeta_i)\frac{\sum_{j=1}^n \Re(\zeta_j)d_j}{\sum_{j=1}^n \Im(\zeta_j)d_j}\,.\]
Note that $\theta(d)=0$ for this choice of $\theta$.

\subsubsection{Geometric stability conditions} Suppose now that $S$ is a smooth projective surface and $\CD=D^b(S)$. Recall that given an ample divisor $H$ we may define the slope of a torsion-free sheaf $F$ by
\[\mu^H(F)=\frac{c_1(F)\cdot H}{\rk(F)}\,.\]
This defines the notion of $\mu^H$-(semi)stable torsion-free sheaves. The slope of pure one-dimensional sheaf $G$ is defined as
\[\mu^H(G)=\frac{\ch_2(G)}{c_1(G)\cdot H}\,.\]
This defines the notion of $\mu^H$-(semi)stability for pure one-dimensional sheaves.

In \cite{ab} the authors construct stability conditions $\sigma_{E, H}=(\CA_{E, H}, Z_{E, H})$ where $H, E$ are $\BR$-divisors with $H$ being ample. We call these geometric stability conditions. The heart of the $t$-structure is obtained as a tilt
\[\CA_{E, H}=\langle \Coh_{H, \leq E\cdot H}[1], \Coh_{H, > E\cdot H}\rangle\]
of $\Coh(S)$ with respect to $\mu^H$-stability. Here we write $\Coh_{H, >s}$ for the extension closure of torsion sheaves and $\mu^H$-semistable torsion-free sheaves $T$ with $\mu^H(T)>s$; similarly, $\Coh_{H, \leq s}$ is the extension closure of $\mu^H$-semistable torsion-free sheaves $F$ with $\mu^H(F)\leq s$. The central charge is given by
\[Z_{E, H}(F)=-\int_S e^{-E-\sqrt{-1}\, H}\ch(F)\in \BC\,. \]
Note that these stability conditions have the property that structure sheaves of points $\CO_x$ are stable and $Z(\CO_x)=-1$. By result of \cite{toda}, see also \cite{macri_schmidt}, geometric stability condition defines an Artin stack of finite type $\CM^{\sigma_{E,H}-\textup{ss}}_\v$. Furthermore, loc. cit. proves that $\CM^{\sigma_{E,H}-\textup{ss}}_\v$ is an open substack of all perfect complexes with trivial negative Ext groups constructed in \cite{lieblich}. Therefore $\CM^{\sigma_{E,H}-\textup{ss}}_\v$ is equipped with an obstruction theory whose associated virtual tangent complex at a point $F$ is given by $\RHom_S(F,F)[1]$.

Geometric stability is closely related to so called twisted Gieseker stability which we explain now. Recall from \cite{matsukiwentworth} that the $(H,E-\frac{K}{2})$-twisted Hilbert polynomial of $F\in \Coh(S)$ is defined as
$$P_{(H,E-\frac{K}{2})}(F,t)\coloneqq\int_X \ch(F)\cdot e^{-(E-\frac{K}{2})+tH}\cdot \td(S)\in \BR[t]\,.
The choice of $-K/2$, where $K=K_S$ denotes the canonical divisor, is made so that
$$e^{\frac{K}{2}}\cdot \td(S)=(1,0,c\,[\pt]),\quad c=\chi(\CO_S)-\textstyle\frac{K^2}{8}\in \BQ
has no codimension one part. Writing $\ch^E(-)\coloneqq\ch(-)\cdot e^{-E}$, we have
$$P_{(H,E-\frac{K}{2})}(F,t)=t^2\cdot \frac{\rk(F)H^2}{2}+t\cdot (H\cdot \ch^E_1(F))+(\ch^E_2(F)+c\cdot\rk(F))\,.
If $F$ is torsion-free, we define the $(H,E-\frac{K}{2})$-twisted reduced Hilbert polynomial is
$$p_{(H,E-\frac{K}{2})}(F,t)=t^2+t\cdot \frac{H\cdot \ch^E_1(F)}{\rk(F)H^2/2}
We say that a torsion-free sheaf $F$ is $(H,E-\frac{K}{2})$-twisted Gieseker (semi)stable if for all $0\subsetneq F'\subsetneq F$ we have
$$p_{(H,E-\frac{K}{2})}(F',t)(\leq) p_{(H,E-\frac{K}{2})}(F,t)\quad \textnormal{for all}\quad t\gg0\,.
In other words, $F$ is $(H,E-\frac{K}{2})$-twisted Gieseker (semi)stable if for all $0\subsetneq F'\subsetneq F$ we have one of the followings:
\begin{enumerate}
\item [(i)]$\displaystyle \frac{H\cdot \ch^E_1(F')}{\rk(F')}
<\frac{H\cdot \ch^E_1(F)}{\rk(F)}$\vspace{3pt}
\item [(ii)]$\displaystyle \frac{H\cdot \ch^E_1(F')}{\rk(F')}
=\frac{H\cdot \ch^E_1(F)}{\rk(F)}$ \ and \ $\displaystyle \frac{\ch_2^E(F')}{\rk(F')}
\end{enumerate}
For 1-dimensional sheaves, $(H,E-\frac{K}{2})$-twisted Gieseker stability is equivalent to $(H,E-\frac{K}{2})$-twisted slope stability, where the twisted slope of $F$ is given by
\[\frac{\ch_2^E(F)}{H\cdot \ch_1(F)}\,.\]

The geometric stability recovers $(H,E-\frac{K}{2})$-twisted Gieseker stability
in the large volume limit. Note that $\CA_{E,H}=\CA_{E,tH}$ for any $t>0$ in the following statement.

\begin{lemma}[\cite{BriK3}]\label{lem: large volume}
Let $H,E$ be $\BR$-divisors with $H$ being ample and $\v\in K(S)$. Let $t\gg 0$ be large enough according to $H$, $E$ and $\v$. Let $F\in \CA_{E,H}$ of type $\v$.
\begin{enumerate}
\item If $\dim(\v)=1$, then
\[F\textup{ is }\sigma_{E, tH}\textup{-(semi)stable}\Leftrightarrow F\textup{ is }(H,E-{\textstyle\frac{K}{2}})\textup{-twisted slope (semi)stable sheaf.}\]
\item If $\dim(\v)=2$ and we assume further that $E\cdot H<\mu^H(\v)$, then
\[F\textup{ is }\sigma_{E, tH}\textup{-(semi)stable}\Leftrightarrow F\textup{ is }(H,E-{\textstyle\frac{K}{2}})\textup{-twisted Gieseker (semi)stable sheaf.}\]
\end{enumerate}
\end{lemma}
\begin{proof}

The $\dim(\v)=2$ case is essentially Proposition 14.2 of loc. cit. Even though the cited reference considers only $K3$ surfaces, the same proof applies to arbitrary surfaces once we observe the characterization of the $(H,E-{\textstyle\frac{K}{2}})$-twisted Gieseker (semi)stability in the preceding paragraphs. See also \cite[Lemma 3.5]{rekuski} where this was used.

Even though the $\dim(\v)=1$ case is also standard, for completeness we explain its proof by adapting the one for the $\dim(\v)=2$ case. For fixed $H, E$ and $\v$, there are only finitely many walls for the parameter $t>0$ according to \cite[Lemma 6.24]{macri_schmidt}. Therefore, we can choose $t$ large enough so that there are no higher walls. Suppose that $F$ is $\sigma_{E,tH}$-(semi)stable. By the choice of $t$, $F$ is $\sigma_{E,t'H}$-(semi)stable for all $t'\gg 0$. Thus $F$ satisfies the assumption of \cite[Lemma 6.18]{macri_schmidt}, so we conclude that $F$ is a torsion sheaf supported in degree zero\footnote{This is because the other two conclusions in \cite[Lemma 6.18]{macri_schmidt} occur only if $\dim(\v)=2$.}. If $F$ is not $(H, E-\frac{K}{2})$-twisted slope (semi)stable, then we get a destabilizing subsheaf $0\subsetneq G\subsetneq F$. In other words,
$$\frac{\ch^E_2(G)}{H\cdot \ch_1(G)}(\nleq)\frac{\ch^E_2(F)}{H\cdot \ch_1(F)}\,.
Since $G$ is a torsion sheaf, it is inside the tilted heart $\CA_{E,tH}=\CA_{E,H}$. This contradicts the $\sigma_{E,t'}$-(semi)stability of $F$ because for torsion sheaves we have
\begin{equation}\label{eq: torsion comparison}
\mu^{\sigma_{E,t'H}}(G)=\frac{1}{t'}\cdot\frac{\ch_2^E(G)}{H\cdot \ch_1(G)},\quad \mu^{\sigma_{E,t'H}}(F)=\frac{1}{t'}\cdot\frac{\ch_2^E(F)}{H\cdot \ch_1(F)}\,.
\end{equation}
Thus $F$ is $(H,E-\frac{K}{2})$-twisted slope (semi)stable.

Conversely, assume that $F$ is $(H,E-\frac{K}{2})$-twisted slope (semi)stable. Let $t$ be chosen as before so that there are no higher walls. Suppose that $F$ is not $\sigma_{E,tH}$-(semi)stable. Let $G$ be a $\sigma_{H,tE}$-destabilizing subobject, i.e.,
$$\mu^{\sigma_{E,tH}}(G)(\nleq) \mu^{\sigma_{E,tH}}(F)\,.
Note that the cohomology sheaf $H^{-1}(G)$ is zero because $H^{-1}(F)=0$. We divide into two cases; $\rk(G)>0$ and $\rk(G)=0$. If $\rk(G)>0$, then we have
$$\lim_{t'\rightarrow \infty} \mu^{\sigma_{E,t'H}}(G)=-\infty < 0 =\lim_{t'\rightarrow \infty}\mu^{\sigma_{E,t'H}}(F)\,.
By continuity of the slope formula, there exists some $t'>t$ such that $\mu^{\sigma_{E,t'H}}(G)=\mu^{\sigma_{E,t'H}}(F)$, contradicting the fact that there are no walls above $t$. Therefore $\rk(G)=0$, i.e., $G$ is a torsion sheaf. Since torsion sheaves on $S$ form an abelian subcategory of both $\CA_{E,H}$ and $\Coh(S)$, $G$ being a subobject of $F$ in $\CA_{E, H}$ implies that $0\subsetneq G\subsetneq F$ is a subobject in $\Coh(S)$. By the computation in \eqref{eq: torsion comparison}, $G$ destabilizes $F$ with respect to the $(H,E-\frac{K}{2})$-twisted slope (semi)stability, contradicting the assumption.
\end{proof}

\begin{remark}\label{rem: Gieseker is geometric}
Consider the moduli stack $\CM^{H-\textup{ss}}_\v$ of Gieseker $H$-semistable sheaves of type $\v$. By Lemma \ref{lem: large volume}, we can identify this moduli stack as
for some geometric stability condition $\sigma$. When $\dim(\v)=1$, $\mu^H$-(semi)stability is equivalent to $\sigma_{(\frac{K}{2},tH)}$-(semi)stability for $t\gg0$. Now let $\dim(\v)=2$. Pick some small enough $n$ such that $(nH+\frac{K}{2})\cdot H<\mu^H(\v)$. Since $H$ and $nH$ are linearly dependent, $(H,nH)$-twisted Gieseker (semi)stability is equivalent to the usual Gieseker (semi)stability with respect to $H$. On the other hand, the lemma says that $(H,nH)$-twisted Gieseker (semi)stability is equivalent to $\sigma_{nH+\frac{K}{2},tH}$~-~(semi)stability for some $t\gg 0$.
\end{remark}

The following Lemma will be useful later.

\begin{lemma} \label{lem: dimension 2}
Let $S$ be a surface with nef anticanonical surface. Let $\sigma_{E, H}$ be a geometric stability condition and $\phi\in \BR$. Then the abelian category $\CA_{E, H}[\phi]$ has homological dimension 2.
\end{lemma}
\begin{proof}
Suppose that $F_1, F_2$ are objects in $\CA_{E, H}[\phi]=\CP((\phi, \phi+1])$. It was proven in \cite[Theorem 7.7]{mozgovoy} that $\Hom(\CP(\phi_1), \CP(\phi_2))=0$ if $\phi_2-\phi_1>2$. Since $F_2[k]$ is an object in $\CP((\phi+k, \phi+k+1])$ it follows that $\Ext^k(F_1, F_2)=\Hom(F_1, F_2[k])=0$ for $k\geq 3$, showing that the homological dimension is at most 2. Since $\CO_x[m]$ belongs to $\CA_{E, H}[\phi]$ for some $m\in \BZ$ and $\Ext^2(\CO_x[m],\CO_x[m])\neq 0$, homological dimension is precisely $2$.
\end{proof}

\subsubsection{The ABCH program} Let $S$ be a surface which admits (possibly many) full and strong exceptional collections, for instance any del Pezzo surfaces \cite[Example 8.6]{Bri-St}.\footnote{Having geometric helix implies that we have a full and strong exceptional collection.} It was proposed in \cite{abch, am} that one could try to use the exceptional collections on $S$ and corresponding quiver stability conditions to study moduli spaces with respect to geometric stability conditions. Note that the quiver stability conditions that we will consider are typically not geometric; for example $\CO_{\BP^2}(-1)[2]$ appears in the exceptional collection in Example \ref{ex: exceptionalcollections} but it is not in the heart of any geometric stability condition. However, in some cases the $\BR$-action on the space of stability conditions can be used to relate geometric and quiver stability conditions.

\begin{definition}\label{def: quiver description}
Let $\sigma$ be a geometric stability condition and $\v\in K(S)$. We say that $\CM_\v^{\sigma-\textup{ss}}$ admits a quiver description if there is another geometric stability condition $\sigma'$ and $\phi\in \BR$ such that $\sigma'[\phi]$ is a quiver stability condition and, for $F$ in class $\v$,
\[F\textup{ is }\sigma\textup{-(semi)stable}\,\Leftrightarrow\, F\textup{ is }\sigma'\textup{-(semi)stable}\,.\]
\end{definition}
In particular, when this happens the moduli stack $\CM_\v^{\sigma-\textup{ss}}$ can be identified with a moduli stack of representations $\CM^{\theta-\textup{ss}}_d(Q, I)$ of some quiver with relations via the equivalence $D^b(S)\simeq D^b(Q,I)$ composed with a twist $[k]$.

A careful study of the wall and chamber structure on the space of stability conditions and an identification of ``quiver regions'' associated to certain choices of exceptional collections allowed the authors to prove the following:

\begin{theorem}[\cite{abch, am}]\label{thm: ABCH for three surfaces}
Let $S=\BP^2$, $S=\BP^1\times \BP^1$ or $S=\textup{Bl}_\pt(\BP^2)$. Then, for any geometric stability condition $\sigma$ and $\v\in K(S)$, $\CM_\v^{\sigma-\textup{ss}}$ admits a quiver description.
\end{theorem}

The authors in \cite[Conjecture 1]{am} conjecture that the same is true for any del Pezzo surfaces. We remark that in the case of $S=\BP^2$ the only exceptional collections used are the ones obtained from tensoring by a line bundle Example \ref{ex: exceptionalcollections}.

\subsection{Application to sheaf Virasoro constraints}

Let $X$ be a smooth projective variety admitting a full exceptional collection. Virasoro constraints for moduli spaces of sheaves on $X$ are formulated in \cite{blm} using a descendent algebra $\BD^X$ generated by symbols of the form $\ch_k(\gamma)$ for a cohomology class $\gamma\in H^\ast(X)$ and Virasoro operators $\bL_n\colon \BD^X\to \BD^X$; the descendent algebra admits a realization map to $H^\ast(\CN^X)$, see Definition 2.5 in \cite{blm}. On the other hand, we defined in Definition~\ref{def: ext descendent} the Ext descendent algebra $\BD^\bfT$ to be generated by symbols of the form $\ch_k^L(\alpha)$ for $\alpha\in K(X)$, and a realization map to $H^\ast(\CN^X)$. We have an isomorphism between the two descendent algebras
$$\psi:\BD^{\bfT}\rightarrow \BD^X,\quad \ch^L_k(\alpha)\mapsto \ch_k\big(\ch(\alpha^\vee)\td(X)\big).

As we did for quivers in Proposition \ref{prop: comparison of two descendents}, we can show that this isomorphism commutes with realization homomorphisms and intertwines the Virasoro operators on $\BD^\bfT$ (cf. Definition \ref{def: K Virasoro}) and on $\BD^X$ (cf. \cite[Section 2.3]{blm}).\footnote{Since $X$ admits a full exceptional collections, we have $H^{p,q}(X)=0$ for all $p\neq q$ by \cite[Proposition 1.9]{marcollitabuada}. In the language of \cite{blm} this means that $\ch_k(\gamma)=\ch_k^H(\gamma)$.} Indeed, the operators both on $\BD^{\bfT}$ and $\BD^X$ are shown to be dual to the canonical Virasoro operators on the lattice vertex algebra $V^X\coloneqq H_\ast(\CN^X)=H_\ast(\CN^{\,\bfT})$, see Proposition \ref{prop: duality of operators} and \cite[Theorem 4.12]{blm}.

Suppose we are given a coarse moduli space $M_\v^\sigma$ parametrizing sheaves or complexes on $X$ with respect to some stability condition $\sigma$ such that there are no strictly $\sigma$-semistable objects of type $\v$. Suppose further that the natural perfect obstruction theory on $M_\v^\sigma$ is 2-term, so we have a natural virtual fundamental class $[M_\v^\sigma]^\vir$. The sheaf Virasoro constraints conjectures in \cite{blm} are the statement that the class $[M_\v^\sigma]^\vir$ in $\widecheck V^X$ is a primary state.

\begin{theorem}\label{thm: ABCH implies Virasoro constraints}
Let $S$ be a del Pezzo surface, $\sigma$ be a geometric stability condition and $\v\in K(S)$. Suppose that $\CM_\v^{\sigma-\textup{ss}}$ admits a quiver description.
\begin{enumerate}
\item [(i)]  If there are no strictly $\sigma$-semistable objects of type $\v$, then the coarse moduli space $M^\sigma_\v$ together with its natural virtual class satisfies the Virasoro constraints.
\item [(ii)]  If $\Ext^2_S(F,F')=0$ for all $\sigma$-semistable objects $F$ and $F'$ of type $\v$, then the moduli stack $\CM_\v^{\sigma-\textup{ss}}$ is smooth and tautologically generated. Furthermore, the Virasoro operators are geometric, i.e. they descend to the cohomology ring via the realization homomorphism $\BD^S\to H^\ast(\CM_\v^{\sigma-\textup{ss}})$.
\end{enumerate}
\end{theorem}

\begin{proof}
Let $\CM_\v^{\sigma-\textup{ss}}$ be a moduli stack admitting a quiver description. By definition, this means that we have an exceptional collection $\FE=(E_1,\dots, E_n)$ of $D^b(S)$ and a stability condition $\theta$ for a corresponding $(Q,I)$ such that
This isomorphism is obtained by the equivalence $B:D^b(S)\rightarrow D^b(Q,I)$, possibly composed by a shift $[k]$. By definition of $B$, it lifts to a quasi-equivalence between the natural dg enhancements which in turn defines an quasi-isomorphism between derived stacks ${\bf B}:\bfCN^S\rightarrow \bfCN^\dgQ$. Here $\dgQ$ is the canonical dg quiver associated to $(Q,I)$ which is quasi-smooth by Lemma \ref{lem: dimension 2}.  Then we have a homotopy commutative diagram
\begin{equation}\label{eq: comm 1}
\begin{tikzcd}
\CM_\v^{\sigma-\textup{ss}} \arrow[r, "B"] \arrow[d, hook] & {\CM}_d^{\theta-\textup{ss}} \arrow[d, hook] \\
\CN^S \arrow[r, "B"]                 & \CN^\dgQ
\end{tikzcd}
\end{equation}
where the vertical morphisms are open embeddings. Note that this induces an obstruction theory for both $\CM_\v^{\sigma-\textup{ss}}$ and $\CM_d^{\theta-\textup{ss}}$ whose virtual tangent complexes at points $F\in \CM_\v^{\sigma-\textup{ss}}$ and $V\in \CM_d^{\theta-\textup{ss}}$ are given by
$$\RHom_S(F, F)[1],\quad \RHom_{\dgQ}(V,V)[1]
respectively. By commutativity of \eqref{eq: comm 1}, $B:\CM_\v^{\sigma-\textup{ss}}\rightarrow \CM_d^{\theta-\textup{ss}}$ also identifies the obstruction theories. By Lemma \ref{lem: dimension 2}, the complex $\RHom_S(F, F)[1]$ lies in the perfect amplitude $[-1,1]$, hence so is $\RHom_{\dgQ}(V,V)[1]$.

To show the statement (i), assume that there are no strictly $\sigma$-semistable objects of type $\v$. Then the good moduli spaces are simply the $\BG_m$-rigidification of the moduli stacks and we have a homotopy commutative diagram
\begin{equation}\label{eq: comm 2}
\begin{tikzcd}
M_\v^{\sigma} \arrow[r, "B"] \arrow[d, hook] & {M}_d^{\theta} \arrow[d, hook] \\
(\CN^S)^{\textup{pl}} \arrow[r, "B"]                 & (\CN^\dgQ)^{\textup{pl}}
\end{tikzcd}
\end{equation}
with the vertical morphisms being open embeddings. The good moduli spaces are equipped with the induced truncated obstruction theory whose virtual tangent complex lies in the amplitude $[0,1]$. Pushforward of the virtual classes define the elements in the Lie algebras associated to vertex algebras
$$[M^\sigma_\v]^\vir\in \widecheck V^S,\quad [{M}^\theta_d]^\vir \in \widecheck V^\dgQ\,.
By Corollary \ref{cor: intertwining Virasoro}, $B_*:V^S\rightarrow V^\dgQ$ is a vertex algebra isomorphism intertwining the Virasoro operators; hence the induced isomorphism of Lie algebras $B_\ast\colon \widecheck V^S\to \widecheck V^\bfQ$ preserves the subspaces of primary states. Since $B_*([M^\sigma_\v]^\vir)=[{M}^\theta_d]^\vir$ by commutativity of \eqref{eq: comm 2}, $M^\sigma_\v$ satisfies the Virasoro constraints with its natural virtual class if and only if ${M}^\theta_d$ does. By Theorem \ref{thm: quivervirasoro}, we know that ${M}^\theta_d$ satisfies the Virasoro constraints with respect to the virtual class defined in Section \ref{sec: explicit virtual class}. To show that this virtual class is equal to the one coming from the derived enhancement of $\CN^\dgQ$, it suffices to compare the $K$-theory class of the virtual tangent complex by the result of \cite{Sie}. By Proposition \ref{prop: extquasismooth}, the $K$-theory class coming from the derived enhancement of $\CN^\dgQ$ is
$$\CO-\sum_{i\in Q_0}\CH\textup{om}(\CV_i, \CV_i)
+\sum_{e\in Q_1}\CH\textup{om}(\CV_{s(e)}, \CV_{t(e)})
-\sum_{\tilde r\in Q_2}\CH\textup{om}(\CV_{s(\tilde r)}, \CV_{t(\tilde r)})\,,
which matches precisely the $K$-theory class of the virtual tangent complex given by the explicit description in Section \ref{sec: explicit virtual class}, proving the statement (i).

\iffalse
Recall that the $K$-theory class of the virtual tangent complex coming from $\CN^\dgQ$ is
\begin{equation}\label{eq: rigidified Tvir}
\CO-\RHom_\dgQ(\CV,\CV)
\end{equation}
where $\CV$ is the universal representation of $(Q,I)$. The standard dg resolution for the universal representation $\CV$ implies
$$\CV=\sum_{i\in \dgQ_0} P(i)\otimes \CV_i-\sum_{e\in \dgQ_1^{(0)}}P({t(e)})\otimes \CV_{s(e)}+\sum_{\tilde r\in \dgQ_1^{(-1)}}P({t(\tilde r)})\otimes \CV_{s(\tilde r)}
in $K$-theory. Therefore \eqref{eq: rigidified Tvir} can be rewritten as
\begin{equation}\label{eq: rigidified Tvir 2}
\CO-\sum_{i\in \dgQ_0}\RHom_\dgQ(P(i)\otimes \CV_i, \CV)
+\sum_{e\in \dgQ_1^{(0)}}\RHom_\dgQ(P({t(e)})\otimes \CV_{s(e)}, \CV)
-\sum_{\tilde r\in \dgQ_1^{(-1)}}\RHom_\dgQ(P({t(\tilde r)})\otimes \CV_{s(\tilde r)}, \CV)\,.
\end{equation}
On the other hand, the adjuncton formula and property of the standard projective module $P(i)$, we have \wcomment{One way of looking at this formula is to consider $\CN^\dgQ \times \dgQ$ just like we do for $\CN^X\times X$. Then $P_i$ is pulled back from $\dgQ$ and $\CV$ is the universal object on the product and $\CV_i=\RHom_p(q^*P_i, \CV)$ is pulled back from $\CN^{\dgQ}$. I am not sure how to justify this discussion and terms involved.}
$$\RHom_\dgQ(P_i\otimes \CV_j, \CV)=\CH\textup{om}(\CV_j,\CV_i)\,.
Therefore, we can further simplify \eqref{eq: rigidified Tvir 2} as
$$\CO-\sum_{i\in \dgQ_0}\CH\textup{om}(\CV_i, \CV_i)
+\sum_{e\in \dgQ_1^{(0)}}\CH\textup{om}(\CV_{s(e)}, \CV_{t(e)})
-\sum_{\tilde r\in \dgQ_1^{(-1)}}\CH\textup{om}(\CV_{s(\tilde r)}, \CV_{t(\tilde r)})\,.
\mcomment{This computation is already in Proposition 3.7 (although it's in the dga setting)}\wcomment{Write one more Proposition 3.9 that we need here exactly and refer to it.}
Note that this precisely matches the $K$-theory class of the virtual tangent complex given by the explicit description in Section \ref{sec: explicit virtual class}, proving hte statement (i).
\fi

Now we consider the statement (ii). Assume that $\Ext^2_S(F,F')=0$ for all $F,F'$ in $\CM_\v^{\sigma-\textup{ss}}$. Via the isomorphism $B:\CM_\v^{\sigma-\textup{ss}}\xrightarrow{\sim}{\CM}_d^{\theta-\textup{ss}}$ this implies the analogous statement for $(Q,I)$. Therefore $(Q,I)$, $d$ and $\theta$ satisfy Assumption \ref{ass: smoothness}. By Theorem~\ref{thm: virasoro rep}, we have a surjective realization homomorphism
$$\xi_d:\BD^{\dgQ}_d\rightarrow H^*({\CM}_d^{\theta-\textup{ss}})
through which the operators $\bR_n^{\dgQ}$ descend. On the other hand, by taking cohomology of the diagram \eqref{eq: comm 1}, we obtain a commutative diagram
\begin{center}
\begin{tikzcd}
H^*(\CM_\v^{\sigma-\textup{ss}}) & H^*({\CM}_d^{\theta-\textup{ss}}) \arrow[l, "B^*"]  \\
\BD_\v^{S}      \arrow[u]      & \BD_d^{\dgQ} \arrow[l, "B^*"]    \arrow[u]
\end{tikzcd}
\end{center}
where the isomorphism $B^*:\BD^{\dgQ}_d\to \BD^S_\v$ is defined as in the proof of Theorem \ref{thm: natural iso}; explicitly, it is given by
\[B^\ast \ch_k(i)=\ch_k\big(\ch(\widetilde E_i^\vee)\td(X)\big)\,,\quad i\in Q_0.\] Note that this isomorphism $B^*$ intertwines the Virasoro operators $\bR_n^S$ and $\bR_n^\dgQ$ which can be checked directly from the definition of $B^*$.\footnote{More conceptually, this stems from the fact that the descendent algebras $\BD^S, \BD^\bfQ$ and respective Virasoro operators are identified with the naturally defined Ext descendent algebra and Ext Virasoro operators, see Definitions \ref{def: ext descendent} and \ref{def: K Virasoro}.} Therefore, statements about surjectivity of the realization homomorphism and descent of $\bR_n^S$ follows from the corresponding statements for $(Q,I)$. \end{proof}

By applying the above theorem to del Pezzo surfaces where the ABCH program is proven, we obtain the following result concerning sheaf moduli spaces.

\begin{corollary}\label{cor: three cases}
Let $\CM_\v^{H-\textup{ss}}$ be a moduli stack of nonzero-dimensional $H$-semistable sheaves on $\BP^2$, $\BP^1\times \BP^1$ or $\textup{Bl}_\pt(\BP^2)$.
\begin{enumerate}
\item [(i)] If there are no strictly $H$-semistable sheaves of type $\v$, then the coarse moduli space $M_\v^H$ satisfies the Virasoro constraints with smooth fundamental class.
\item [(ii)] The moduli stack $\CM^{H-\textup{ss}}_\v$ is a smooth and tautologically generated. Furthermore, the Virasoro operators descends to the cohomology ring via the realization homomorphism.
\end{enumerate}

\end{corollary}

\begin{proof}
Let $\CM_\v^{H-\textup{ss}}$ be a moduli stack as in the statement. By Remark \ref{rem: Gieseker is geometric} and Theorem \ref{thm: ABCH for three surfaces}, it admits a quiver description. For any positive dimensional $H$-semistable sheaves $F$ and $F'$ of type $\v$, we have
$$\Ext^2_S(F,F')\simeq \Hom_S(F',F\otimes K_S)^\vee=0
since $F'$ and $F\otimes K_S$ are $\mu^H$-semistable sheaves with $\mu^H(F')>\mu(F\otimes K_S)$ by negativity of $K_S$. This implies that the moduli stack is smooth by deformation theory. Therefore (i) and (ii) follows from the corresponding statements from Theorem~\ref{thm: ABCH implies Virasoro constraints}.
\end{proof}
Note that part (i) of the above Theorem is also shown in \cite{blm, bojko} when $\dim(\v)=2$. The case $\dim(\v)=1$ was previously only known conditionally on some technical conditions necessary for the application of a wall-crossing formula \cite[Assumption 5.8]{blm}.

The statement about the Virasoro operators descending for the moduli spaces of one-dimensional sheaves on $\BP^2$ is applied to the study of the cohomology ring of such moduli spaces in terms of generators and relations in \cite{KLMP}.

\section{Grassmannian and symmetric polynomials}\label{sec: Grass}

In this section, we study the Virasoro constraints of Grassmannians from the perspective of symmetric functions. Previously, notation $\Gr(N,k)$ was used to denote the Grassmannian parametrizing $k$-dimensional {\it quotients} of a fixed vector space $\BC^N$. In this section, we change the notation and use $\Gr(k,N)$ to denote the Grassmannian parametrizing $k$-dimensional {\it subspaces} of $\BC^N$. We shift to this convention because Grassmannians and symmetric functions are discussed often in this way.

\subsection{Algebra of symmetric functions}

We will now recall some basic definitions regarding the algebra of symmetric functions, following \cite{macdonald}. Let
\[\Lambda=\varprojlim_{n} \BQ[x_1, \ldots, x_n]^{S_n}\]
be the ring of symmetric functions in infinitely many variables, where the inverse limit is understood in the category of graded rings. The ring $\Lambda$ can be described as the polynomial ring in infinitely many variables in a few different ways:
\begin{enumerate}
\item $\Lambda=\BQ[e_1, e_2, \ldots]$ where $e_j$ is the $j$-th elementary symmetric function.
\item $\Lambda=\BQ[p_1, p_2, \ldots]$ where $p_j$ is the $j$-th power sum.
\item $\Lambda=\BQ[h_1, h_2, \ldots]$ where $h_j$ is the $j$-th complete symmetric function.
\end{enumerate}
The different sets of generators can be written in terms of each other by Newton's identities:
\begin{align*}\sum_{j\geq 0} e_j &=\exp\left(\sum_{k\geq 1}\frac{(-1)^{k+1}}{k}p_k\right)\\
\sum_{j\geq 0} h_j &=\exp\left(\sum_{k\geq 1}\frac{1}{k}p_k\right)=\left(\sum_{j\geq 0} (-1)^j e_j\right)^{-1}\,.\end{align*}

Above, we set $e_0=h_0=1$; the identities are understood in the completion of $\Lambda$ with respect to the degree. Each set of generators defines a basis of $\Lambda$ indexed by partitions $\lambda=(\lambda_1\geq \lambda_2\geq \cdots\geq \lambda_{\ell(\lambda)})$\,:
\[e_\lambda=\prod_{i=1}^{\ell(\lambda)}e_{\lambda_i}\,, \quad p_\lambda=\prod_{i=1}^{\ell(\lambda)}p_{\lambda_i}\,, \quad h_\lambda=\prod_{i=1}^{\ell(\lambda)}h_{\lambda_i}\,.\]
Another natural basis indexed by partitions is $\{m_\lambda\}$ where
\[m_\lambda(x_1, \ldots, x_n)=\sum_{\sigma} \prod_{i=1}^n x_i^{\sigma_i}\]
where the sum is over all the distinct permutations of $\lambda$.

The ring $\Lambda$ admits a non-degenerate bilinear pairing $\langle -,-\rangle$, called the Hall inner product, that plays a particularly important role. This pairing is defined by setting
\[\langle p_\lambda, p_\mu\rangle=\delta_{\lambda \mu}z_\lambda\,.\]
In the formula above
\[z_\lambda=\prod_{i\geq 1}i^{m_i}m_i!\]
where $m_i$ is the number of times that $i$ appears in $\lambda$. It can be checked that for every $n>0$ the multiplication operator $p_n$ and the annihilation operator $p_{-n}:=n\frac{\partial}{\partial p_n}$ are adjoint with respect to the Hall pairing, i.e.,
\[\langle p_n f, g\rangle=\langle f, p_{-n}\,g\rangle\,.\]

Schur polynomials $\{s_\lambda\}$ form yet another natural basis of $\Lambda$ indexed by partitions. They can be defined by the following property:
\begin{definition}
The Schur polynomials $\{s_\lambda\}$ are the unique orthonormal basis of $\Lambda$ with respect to $\langle -,- \rangle$  such that
\[s_\lambda=m_\lambda+\sum_{\mu<\lambda}a_{\lambda\mu}m_\mu\]
for some $a_{\lambda\mu}\in \BQ$ where the sum runs over partitions $\mu$ which are smaller than $\lambda$ in the lexicographic order.
\end{definition}
There are several different ways to define Schur polynomials more explicitly. For example they can be written in terms of the complete symmetric functions $h_i$ by a determinant formula:
\[s_\lambda=\det\big(h_{\lambda_i-i+j}\big)_{1\leq i,j\leq \ell(\lambda)}\,.\]
Another possible definition of Schur polynomials is presented in Proposition \ref{prop: heckeproperties}.

The ring $\Lambda$ admits an involution $\sigma\colon \Lambda\to \Lambda$ defined by $\sigma(p_j)=(-1)^{j-1}p_j$. This involution sends $s_\lambda$ to $s_{\lambda^t}$ where $\lambda^t$ is the conjugate partition.

\begin{example}\label{ex: schur}
Elementary symmetric functions and complete symmetric functions are both easily written as Schur functions:
\[e_j=s_{(1)^j}\,,\quad h_j=s_{(j)}\,.\]
We also have
\[s_{(2,2)}=\frac{1}{12}p_1^4 + \frac{1}{4}p_2^2 - \frac 13 p_1 p_3\,.\]
\end{example}

\subsection{Vertex algebra from symmetric functions}\label{sec: vertex, symmetric}

Let $\Lambda[Q,q^{\pm 1}]$ be the extension of $\Lambda$ by the algebra $\BQ[Q,q^{\pm 1}]$ associated to the monoid $\BN\times \BZ$. The Hall pairing extends to this vector space
[ Q,q^{\pm 1}]\otimes \Lambda_*[Q,q^{\pm 1}]\xrightarrow{\langle-,-\rangle}\BQ\,.
The additional subscript and superscript of $*$ are just to indicate their relation to the cohomology and homology as we discuss next.

Recall the Grassmannian vertex algebra $V^\Gr=H_*(\CN^{A_1,\textup{fr}})$ from Example \ref{ex: grassva}.\footnote{We need to use the dual convention for the framed vertex algebra since we changed the convention of Grassmannian in this section.} Then, we have a  diagram similar to \eqref{eq: construction diagram}
\begin{equation}\label{eq: grassva and symmetric functions}
\begin{tikzcd}
H^*(\CN_{k\rightarrow N})   \arrow[r,phantom,"\otimes" description]       &[-2em] H_*(\CN_{k\rightarrow N}) \arrow[rr,""] \arrow[d, "\xi^\dagger"] & \ & \BQ \arrow[d, equal] \\
Q^Nq^k\cdot \Lambda^*  \arrow[u,"\xi"]\arrow[r,phantom,"\otimes" description]    &[-2em]   Q^Nq^k\cdot \Lambda_* \arrow[rr,""]   &    \   & \BQ
\end{tikzcd}
\end{equation}
where $\xi$ is a ring isomorphism sending $p_n$ to $n!\ch_n(\CV)$ and $\xi^\dagger$ is induced from the perfect pairings defined by topological pairing on the top and Hall pairing on the bottom. The resulting isomorphism $\xi^\dagger:V^{\Gr}\xrightarrow{\sim}\Lambda_*[Q,q^{\pm 1}]$ endows $\Lambda_*[Q,q^{\pm 1}]$ with a vertex algebra structure such that it becomes a vertex subalgebra of $\textup{VA}(\BZ\times\BZ,\chi^\sym_{\Gr})$ where
$$\chi^\sym_{\Gr}\big((k_1,N_1),(k_2,N_2)\big)=2k_1k_2 - k_1N_2-k_2N_1\,.

Using the ambient lattice vertex algebra $\textup{VA}(\BZ\times\BZ,\chi^\sym_{\Gr})$, we can explicitly write down fields of $\Lambda_*[Q,q^{\pm 1}]$. Here, we record two of the fields acting on the component $Q^Nq^k\cdot \Lambda_*$ that are relevant to us:
\begin{align*}
Y(p_1, z)&\coloneqq\sum_{n>0}p_{n}z^{-1+n}+2kz^{-1}+\sum_{n>0}2 p_{-n}z^{-1-n}\,,\\
Y(q,z)&\coloneqq e^q\cdot \exp\left(\sum_{j>0}\frac{p_{j}}{j}z^j\right)\exp\left(-\sum_{j>0}\frac{2 p_{-j}}{j}z^{-j}\right)\,.
\end{align*}
We warn the reader that the annihilation operators in this vertex algebra are $2p_{-n}$ rather than $p_{-n}$ for $n>0$ since $\chi^\sym_\Gr((1,0),(1,0))=2$.

\subsection{Schubert calculus}

The descendent algebra $\BD^{A_1}_k$ of the quiver with only one node $A_1$ can be naturally identified with the algebra of symmetric functions $\Lambda^\ast$:
\begin{equation}\label{eq: symmetric functions and descendent}
\begin{tikzcd}[row sep=0cm]
Q^Nq^k\cdot \Lambda^\ast \arrow[r] & \BD^{A_1}_k \arrow[r] & H^\ast(\CN_{k\rightarrow N})\\
p_n \arrow[r, mapsto]& n! \ch_n \arrow[r, mapsto]& n!\ch_n(\CV)
\end{tikzcd}
\end{equation}
where $\CV$ is the universal rank $k$ complex over $\CN_{k\rightarrow N}=\CN_k$. Note that the composition of these maps is precisely $\xi$ in \eqref{eq: grassva and symmetric functions}. Similarly, the elementary symmetric functions $e_j$ are sent to the Chern classes of $\CV$ and the signed complete symmetric functions $(-1)^j h_j$ are sent to the Segre classes of $\CV$.

Let $\BF$ be the rank $k$ universal subbundle of $\Gr(k,N)$. The geometric realization map for the Grassmannian
\[\xi_\BF\colon \Lambda^\ast\to H^\ast(\Gr(k,N))\]
sends by definition $e_j$ to $c_j(\BF)$. We recall that the cohomology ring of the Grassmannian is
\[H^\ast(\Gr(k,N))\cong \BQ[c_1, \ldots, c_k]/I\]
where $I$ is the ideal generated by
\[\big[(1+c_1+c_2+\ldots+c_k)^{-1}\big]_j\,,\quad \textup{for }j>N-k\]
where $[-]_j$ denotes the degree $j$ part. The (Poincaré duals of) Schubert cycles
\[S_\lambda\in H^{\ast}(\Gr(k,N))\,,\quad \lambda\subseteq (N-k)^k\]
form a linear basis of the cohomology $H^\ast(\Gr(k,N))$. By Pieri's formulas, the map $\Lambda^\ast\to H^\ast(\Gr(k,N))$ sends the Schur polynomial $s_\lambda$ to the corresponding Schubert cycle $(-1)^{|\lambda|}S_\lambda$. In particular, $s_\lambda$ is mapped to 0 whenever $\lambda$ is not contained in the rectangular shape $(N-k)^k$. As an easy consequence of this fact, we can identify the class of the Grassmannian in the Grassmannian vertex algebra:

\begin{proposition}\label{prop: grassschur}
The class of the Grassmannian $\Gr(k,N)$ is given by
\[[\Gr(k,N)]=Q^N q^k \otimes (-1)^{k(N-k)}s_{(N-k)^k}\in \Lambda_\ast[Q, q^{\pm 1}]\,.\]
\end{proposition}
\begin{proof}
Write $[\Gr(k,N)]=Q^N q^k \otimes g_{N, k}$ for some $g_{N, k}\in \Lambda_\ast$. Let $\lambda$ be a partition and consider the associated Schur polynomial $s_\lambda \in \Lambda^\ast$. By the Schubert calculus considerations above we have
\[\langle g_{N, k}, s_\lambda\rangle=\int_{\Gr(k,N)} (-1)^{|\lambda|}S_\lambda=\begin{cases}(-1)^{k(N-k)}&\textup{if }\lambda=(N-k)^k\\
since $S_\lambda=0$ for any other partition of size at least $k(N-k)$. Since $\{s_\lambda\}_\lambda$ form an orthonormal basis for the Hall inner product it follows that
\[g_{N, k}=(-1)^{k(N-k)}s_{(N-k)^k}\,. \qedhere\]
\end{proof}

\subsection{Virasoro constraints for the Grassmannian}

In this section we will write down the Virasoro constraints for the Grassmannian using the language of symmetric functions explained in the previous section.

Recall from Section \ref{sec: vertex, symmetric} that $\Lambda_*[Q,q^{\pm 1}]$ is a vertex subalgebra of $\textup{VA}(\BZ\times\BZ,\chi^\sym_\Gr)$. Since the latter is a lattice vertex algebra associated to the nondegenerate pairing $\chi^\sym_\Gr$, it is equipped with the natural conformal element and the corresponding Virasoro operators. It is easy to check that the half of the Virasoro operators $\{L_n\}_{\geq -1}$ preserve the vertex subalgebra $\Lambda_*[Q,q^{\pm 1}]$ since $\BZ\times\{0\}\subset \BZ\times \BZ$ is the sublattice. Explicitly, the Virasoro operators on each component $Q^Nq^k\cdot \Lambda_*$ are given by\footnote{If we set $N=0$ in the formula, we recover precisely half of the Virasoro operators on $\textup{VA}(\BZ,2)$. }
\begin{equation}\label{eq: operatorsgrva}
\displaystyle\sum_{j\geq 1}p_{j}p_{-n-j}+\sum_{\substack{a+b=n\\a,b>0}}p_{-a} p_{-b}+(2k-N)p_{-n}& n>0\\
\displaystyle\sum_{j>0}p_{j}p_{-j}+k(k-N)\cdot\textup{id} & n=0
\end{cases}
\end{equation}
with $L_{-1}$ being the translation operator $T$. By using the Hall pairing, we obtain the dual Virasoro operators acting on $Q^Nq^k\cdot \Lambda^*$:
\begin{equation}\label{eq: operatorsgrva dual}
\bL_n:=(L_n)^\dagger=\begin{cases}
\displaystyle\sum_{j\geq 1}p_{n+j}p_{-j}+\sum_{\substack{a+b=n\\a,b>0}}p_{a} p_{b}+(2k-N)p_{n}& n>0\\
\displaystyle\sum_{j>0}p_{j}p_{-j}+k(k-N)\cdot\textup{id} & n=0
\end{cases}
\end{equation}
Under the identification $Q^Nq^k\cdot \Lambda^*\simeq \BD^{A_1}_k$ in \eqref{eq: symmetric functions and descendent}, these dual Virasoro operators agree with the framed Virasoro operators $\bL_n^{*\rightarrow N}$ defined in Section \ref{subsubsec: framedvirasoro}.\footnote{Here, we are again using a different convention for the framing as we do in Section \ref{sec: Grass}. } Therefore, the Virasoro constraints for the Grassmannian $\Gr(k,N)$ are equivalent to its class in $\Lambda_*[Q,q^{\pm 1}]$ being annihilated by $L_n$ operators for $n\geq 0$.

By Proposition \ref{prop: grassschur} and Hall pairing duality between $\bL_n^{*\rightarrow N}$ and  $L_n$, the Virasoro constraints of $\Gr(k,N)$ amount to the following identity: for $n>0$, we have\footnote{The $n=0$ case is a simple consequence of the dimension formula $\dim\Gr(k,N)=k(N-k)$. }
\begin{equation}
\label{eq: virasorograssexplicit}
\left(\sum_{j\geq 1}(n+j)p_{j}\frac{\partial}{\partial p_{n+j}}+\sum_{\substack{a+b=n\\a,b>0}}ab\frac{\partial }{\partial p_a}\frac{\partial }{\partial p_b}+(2k-N)n\frac{\partial }{\partial p_n}\right)s_{(N-k)^k}=0\,.
\end{equation}
This equation is a consequence of Theorem \ref{thm: quivervirasoro}, but it is also a well-known fact about the representation theory of the Virasoro Lie algebra \cite{mimachiyamada}, see Theorem \ref{thm: singularvectorsjack}. For completeness we will also give a self-contained proof in Proposition \ref{prop: virasorogr} using Hecke correspondences for Grassmannians.

For the Grassmannian, it happens that the Virasoro constraints fully determine the descendent integrals (up to scalar). This can be thought of as the analog that the Virasoro constraints for the Gromov--Witten of a point (equivalent to Witten's conjecture together with the string equation) determine all the integrals of $\psi$ classes. This is again well-known in the representation theory of the Virasoro Lie algebra, see the uniqueness part of Theorem \ref{thm: singularvectorsjack}.

\begin{proposition}\label{prop: uniquenessgrass}
Suppose that $Q^N q^k \otimes f\in \Lambda_\ast[Q, q^{\pm 1}]$ is a primary state in the Grassmannian vertex algebra, i.e. $L_n(Q^N q^k \otimes f)=0$ for $n\geq 0$ for the operators defined in~\eqref{eq: operatorsgrva}. Then $f$ is proportional to $s_{(N-k)^k}$.
\end{proposition}
\begin{proof}
We start by noting that the $L_0$ equation is equivalent to $f$ being homogeneous of degree $d\coloneqq k(N-k)$. Since we have already showed that $s_{(N-k)^k}$ satisfies the Virasoro constraints $L_n(Q^N q^k \otimes f)=0$ it is enough to show that the Virasoro constraints determine $f$ after one fixes $\langle p_1^d, f\rangle$. Consider the partial order of the set of partitions defined by
\[\lambda'\prec \lambda\textup{ if and only if }\ell(\lambda')>\ell(\lambda)\textup{, or }\ell(\lambda')=\ell(\lambda)\textup{ and }m_1(\lambda')>m_1(\lambda)\]
where $m_1(\lambda)$ denotes the number of times that 1 appears in $\lambda$. Note that the partition $1^d$ is smaller than any other partition of $d$ with respect to $\prec$. We claim that if $f$ satisfies the Virasoro constraints then for any partition $\lambda\neq 1^d$ we can write $\langle p_\lambda, f\rangle$ as a linear combination of $\langle p_{\lambda'}, f\rangle$ for $\lambda'\prec \lambda$.

Let $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_\ell)$ with $\lambda_i$ non-decreasing. Let $m=m_1(\lambda)\geq 0$. Assuming that $\lambda\neq 1^d$ we have $m<\ell$ so let $t=\lambda_{m+1}>1$. Define the partition $\widetilde \lambda$ of $d-t+1$ by
\[\widetilde \lambda=(\underbrace{1, \ldots, 1, 1}_{m+1}, \lambda_{m+2}, \ldots, \lambda_\ell)\,.\]
By the Virasoro constraints for $f$ we have
\begin{equation}\label{eq: schubertfromvirasoro}0=\langle \bL_{t-1}(p_{\widetilde \lambda}),f\rangle=(m+1)\langle p_\lambda, f\rangle+\sum_{\lambda'\prec \lambda}C_{\lambda'}\langle p_{\lambda'}, f\rangle\,
\end{equation}
for explicit constants $C_{\lambda'}\in \BZ$. It is straightforward to see that only partitions $\lambda'\prec \lambda$ appear: all the partitions coming from the linear part of $\bL_{t-1}$ have bigger length and all the remaining partitions coming from the derivation part have $m_1(\lambda')=m+1>m=m_1(\lambda)$. It follows by induction with respect to the order $\prec$ that $\langle p_\lambda, f\rangle$ can be determined from $\langle p_1^d, f\rangle$, proving the proposition.
\end{proof}

\begin{example}
We illustrate the algorithm used in the proof of Proposition \ref{prop: uniquenessgrass} by computing the integrals of descendents in $\Gr(4,2)$. We have
\begin{align*}
0&=\int_{\Gr(4,2)}\bL_1(p_1^3)=3\int_{\Gr(4,2)}p_1^2 p_2\\
0&=\int_{\Gr(4,2)}\bL_2(p_1^2)=2\int_{\Gr(4,2)}p_1p_3 +\int_{\Gr(4,2)} p_1^4\\
0&=\int_{\Gr(4,2)}\bL_1(p_1p_2)=\int_{\Gr(4,2)} p_2^2+2\int_{\Gr(4,2)}p_1p_3\\
0&=\int_{\Gr(4,2)}\bL_3(p_1)=\int_{\Gr(4,2)}p_4+2\int_{\Gr(4,2)} p_1^2 p_2\,.
\end{align*}
In each equality, the first term of the right hand side is the leading term in the sense of the proof of the proposition and the equalities can be used to recursively determine these leading terms. By further using that $\int_{\Gr(4,2)}c_2^2=1$ \cite[Corollary 4.2]{EH} or $\int_{\Gr(4,2)}p_1^4=2$ \cite[Exercise 4.38]{EH}  we determine all the integrals of descendents:
\[\int_{\Gr(4,2)}p_1^4=2=\int_{\Gr(4,2)}p_2^2\,,\quad \int_{\Gr(4,2)}p_1p_3=-1\,,\quad \int_{\Gr(4,2)}p_4=0=\int_{\Gr(4,2)}p_1^2 p_2\,.\]
Note that these can be obtained from the expression for $s_{(2,2)}$ in Example \ref{ex: schur}.
\end{example}

\subsection{Hecke operators}
We define Hecke operators $H_n\colon \Lambda\to \Lambda$ as the following vertex operators:
\[H(z)=\sum_{n\in \BZ}H_n z^n= \exp\left(\sum_{j>0}\frac{p_{j}}{j}z^j\right)\exp\left(-\sum_{j>0}\frac{p_{-j}}{j}z^{-j}\right)\,.\]
This corresponds to the field $Y(q,z)$ of the lattice vertex algebra associated to $(\BZ,1)$ via isomorphism $\textup{VA}(\BZ,1)\simeq \Lambda_*[q^{\pm 1}]$. By Newton's identities we can rewrite the Hecke operators as
\[H_n=\sum_{j\geq 0}(-1)^j h_{j+n}\circ e_j^\perp\]
where $(-)^\perp$ denotes the adjoint operator with respect to the Hall pairing. Such operators have been studied in depth from the point of view of the combinatorics of $\Lambda$, see for instance \cite{jing}. In the literature they are often called Bernstein operators or simply vertex operators; we have opted to call them Hecke operators due to their geometric meaning that we explain below. We summarize some of the well-known properties of these operators in the following proposition:
\begin{proposition}[{\cite{jing}}]\label{prop: heckeproperties}
The Hecke operators satisfy the following properties:
\begin{enumerate}
\item We have
\[[H_n, p_{m}]=-H_{n+m} \,,\quad\textup{for every }n\in \BZ,\, m\in \BZ\setminus \{0\}\,.\]
\item The adjoint to a Hecke operator is given by
\[H_n^\perp=(-1)^n \sigma \circ H_{-n}\circ \sigma\,.\]
\item The Hecke operators satisfy the commuting relations
\[H_nH_m=-H_{m-1}H_{n+1}\,.\]
In particular $H_{n}H_{n+1}=0$.
\item The Schur polynomial $s_\lambda$ can be given in terms of Hecke operators by
\[s_\lambda=H_{\lambda_1}H_{\lambda_2}\ldots H_{\lambda_{\ell}}1.\]
\end{enumerate}
\end{proposition}
\begin{proof}
Parts (2), (3) and (4) are (2.5), (2.12) and (3.9) in \cite{jing} in the limit $t\rightarrow 0$. Part (1) is the defining property of vertex operators, see for example (5.4.3b) in \cite{Ka98}.\qedhere
\end{proof}
Note that the expression for Schur polynomials given by $(4)$ makes sense for any tuple $\lambda=(\lambda_1, \ldots, \lambda_\ell)$ of (not necessarily positive) integers. The commuting relations (3) can be used to reduce the Schur polynomial associated to any tuple to the Schur polynomial of a partitions.

These Hecke operators play a role in the following geometric setup. Consider the Grassmannians $\Gr(k,N)$ and $\Gr(k+1,N)$ and their respective universal bundles $\BF_k\subseteq \CO^N$, $\BF_{k+1}\subseteq \CO^N$. The flag variety $\Flag(k, k+1; N)$ is a projective bundle over both Grassmannians:
\begin{equation}\label{eq: heckeprojectivebundles}
\begin{tikzcd}
\BP(\CO^N/\BF_k)\arrow[d]& \Flag(k, k+1; N)\arrow[l, equal]\arrow[ld, "\pi_k"]\arrow[r, equal]\arrow[rd,"\pi_{k+1}"'] &
\BP(\BF_{k+1}^\vee)\arrow[d]\\
\Gr(k,N) & & \Gr(k+1,N)
\end{tikzcd}
\end{equation}
The universal flag on $\Flag(k, k+1; N)$ is
\[\BF_k\subseteq \BF_{k+1}\subseteq \CO^N\]
where we omit the pullbacks via $\pi_k, \pi_{k+1}$. The line bundle $\BF_{k+1}/\BF_k$ can be interpreted in each projective bundle description as
\[\BF_{k+1}/\BF_k=\CO_{\BP(\CO^N/\BF_k)}(-1)=\CO_{\BP(\BF_{k+1}^\vee)}(1)\,.\]
Let $\zeta=c_1(\BF_{k+1}/\BF_k)\in H^2(\Flag(k, k+1; N))$. We define \textit{geometric} Hecke operators by
\begin{align*}\widetilde H_\ell\colon H^\ast(\Gr(k,N))&\longrightarrow H^\ast(\Gr(k+1,N))\\
\gamma&\longmapsto (\pi_{k+1})_\ast\big( \zeta^\ell\cdot \pi_k^\ast \gamma\big)\,.
\end{align*}
The connection between these geometric Hecke operators and the previously defined Hecke operators on $\Lambda$ is explained in the next proposition:
\begin{proposition}\label{prop: heckegeom}
For every $\ell\geq 0$ the following diagram commutes:
\begin{center}
\begin{tikzcd}
\Lambda^\ast \arrow[d, two heads]\arrow[r, "H_{\ell-k}"]& \Lambda^\ast \arrow[d, two heads]\\
H^\ast(\Gr(k,N))\arrow[r, "\widetilde H_\ell"] & H^\ast(\Gr(k+1,N))\,.
\end{tikzcd}
\end{center}
\end{proposition}
\begin{proof}
The proposition is easily proved by comparing the two paths applied to $1\in \Lambda^\ast$ and by showing that $\widetilde H_\ell$ satisfies commutation relations with $p_a$ similar to the ones in Proposition \ref{prop: heckeproperties}(1). Indeed, we have by the projective bundle formula
\[\widetilde H_\ell(1)=(\pi_{k+1})_\ast(\zeta^\ell)=s_{\ell-k}(\BF_{k+1}^\vee)=h_{\ell-k}(\BF_{k+1})\,.\]
On the other hand, $H_{\ell-k}(1)=h_{\ell-k}$. By the push-pull formula we have
\[\widetilde H_\ell\big(p_a(\BF_k)\gamma\big)=(\pi_{k+1})_\ast \big((p_a(\BF_{k+1})-\zeta^a)\cdot \zeta^\ell\cdot \pi_{k}^\ast \gamma)=p_a(\BF_{k+1})\widetilde H_\ell(\gamma)-\widetilde H_{\ell+a}(\gamma)\,.\]
Comparing with Proposition \ref{prop: heckeproperties}(1) and using induction the result follows.\qedhere
\end{proof}

The geometric picture with the Hecke operators allows us to give a second proof of Proposition \ref{prop: grassschur}.

\begin{proof}[Proof of Proposition \ref{prop: grassschur} via Hecke operators]
Denote by $g_{N, k}\in \Lambda_\ast$ the symmetric function such that $[\Gr(k,N)]=Q^N q^k\otimes g_{N,k}$. Let $f\in \Lambda^\ast$. Using the projective bundles in \eqref{eq: heckeprojectivebundles} and Proposition \ref{prop: heckegeom} we find that
\begin{align*}
\langle g_{N, k}, f\rangle&=\int_{\Gr(k,N)}f=\int_{\Flag(k, k+1; N)}(-\zeta)^{N-k-1} \pi_k^\ast f\\
&=(-1)^{N-k-1}\int_{\Gr(k+1,N)}H_{N-2k-1}(f)=(-1)^{N-k-1}\langle g_{N, k+1}, H_{N-2k-1}(f)\rangle\\
&=(-1)^{k}\langle \sigma H_{2k+1-N}\sigma (g_{N, k+1}), f\rangle\,.
\end{align*}
In the last step we used \ref{prop: heckeproperties}(2). Thus we conclude that \[g_{N, k}=(-1)^{k}\sigma H_{2k+1-N}\sigma (g_{N, k+1})\,.\]
Using $g_{N, N}=1$ we can recursively obtain
\begin{equation}\label{eq: sign1}g_{N, k}=(-1)^{\sum_{j=k}^{N-1} j}\sigma H_{2k+1-N}H_{2k+3-N}\ldots H_{N-3}H_{N-1}(1)\,.
\end{equation}
Using Proposition \ref{prop: heckeproperties} (3) iteratively we find that
\begin{equation}\label{eq: sign2}H_{2k+1-N}H_{2k+3-N}\ldots H_{N-3}H_{N-1}=(-1)^{\sum_{i=0}^{N-k-1} i}H_{k}H_{k}\ldots H_{k}
\end{equation}
Note that the signs of \eqref{eq: sign1} and \eqref{eq: sign2} combine to
\[\sum_{j=k}^{N-1} j+\sum_{i=0}^{N-k} i=\sum_{i=0}^{N-k-1}(2i+k)\equiv k(N-k) \mod 2\,.\]
Thus, putting  \eqref{eq: sign1} and \eqref{eq: sign2} together gives
\[g_{N, k}=(-1)^{k(N-k)}\sigma H_{k}H_{k}\ldots H_{k}(1)=(-1)^{k(N-k)}\sigma(s_{k^{N-k}})=(-1)^{k(N-k)}s_{(N-k)^k}\,.\qedhere\]
\end{proof}

We take the opportunity to give a direct and elementary proof of the Virasoro constraints for the Grassmannian based on these Hecke operators. For that, we need the commutator between the Virasoro and Hecke operators. In the following two propositions, we use the operators $\bL_n$ and $L_n$ from \eqref{eq: operatorsgrva} and \eqref{eq: operatorsgrva dual} with $k=N=0$, i.e.,
\begin{equation*}
\displaystyle\sum_{j\geq 1}p_{j}p_{-n-j}+\sum_{\substack{a+b=n\\a,b>0}}p_{-a} p_{-b}& n>0\\
\displaystyle\sum_{j>0}p_{j}p_{-j} & n=0
\end{cases}
\end{equation*}
and $\bL_n=(L_n)^\dagger$.

\begin{proposition}\label{prop: commvirasorohecke}
We have the following identities in $\End(\Lambda)$:
\begin{align*}
[\bL_n, H_m]&=(m+n-1)H_{m+n}+\sum_{j=1}^{n-1} H_{n+m-j}\circ p_j-H_m\circ p_n\\
&=(m+1)H_{n+m}+\sum_{j=1}^{n-1} p_j\circ H_{n+m-j}-p_n\circ H_m\,,\\
[L_n, H_m]&=(m-1)H_{m-n}+\sum_{j=1}^{n-1} H_{m-n+j}\circ p_{-j}-H_m\circ p_{-n}\\
&=(m-n+1)H_{m-n}+\sum_{j=1}^{n-1} p_{-j}\circ H_{m-n+j}-p_{-n}\circ H_m\,.
\end{align*}
\end{proposition}
\begin{proof}
Proofs can be obtained directly from Proposition \ref{prop: heckeproperties}(1), but see also the Appendix in \cite{liuyang} (note that the operators in loc. cit. differ from ours by a factor of $1/2$ in $\bT_n$, but since they analyse $\bR_n$ and $\bT_n$ separately we can deduce the identities here from their work).\qedhere
\end{proof}

We use this to give a direct proof of \eqref{eq: virasorograssexplicit}.

\begin{proposition}\label{prop: virasorogr}
Equation \eqref{eq: virasorograssexplicit} holds, i.e. for every $m, k, n>0$ we have
\[L_n(s_{m^k})=(m-k)p_{-n}s_{m^k}\,.\]
\end{proposition}
\begin{proof}
We prove the statement by induction on $k$. We leave $k=1$ for the reader to check. Suppose that the equation holds for some $k$. From Proposition \ref{prop: commvirasorohecke} we have
\[[L_n, H_m]-[L_{n-1}, H_{m-1}]=2p_{-n+1}\circ H_{m-1}-p_{-n}\circ H_m\,.\]
Note that we have $H_{m}s_{m^k}=s_{m^{k+1}}$ and $H_{m-1}s_{m^k}=0$; the latter holds due to Proposition \ref{prop: heckeproperties}(3).
\end{proof}

\subsection{Symmetrized Hecke operators and wall-crossing}\label{sec: sym Heke}

Note that the Hecke operators used in the previous section appear naturally as fields associated to the state $q\in \Lambda[q^{\pm 1}]$ when we give $\Lambda_\ast[q^{\pm 1}]$ the vertex algebra structure associated to the lattice $(\BZ, 1)$. However, the Grassmannian vertex algebra $\Lambda_\ast[q^{\pm  1}, Q]$ is a sub-algebra of the lattice vertex algebra associated to $(\BZ\times \BZ, \chi^{\sym}_\Gr)$ where
\[\chi^{\sym}_\Gr((N, k), (N', k'))=2kk'-(Nk'+N'k)\,.\]
In particular, the subalgebra $\Lambda[q^{\pm 1}]\subseteq V^\Gr$ is the lattice vertex algebra associated to $(\BZ, 2)$. Thus wall-crossing formulas on $V^\Gr$ are in principle not written using the previous Hecke operators, but rather
\[H^\sym(z)=\sum_{n\in \BZ}H_n^\sym z^n= \exp\left(\sum_{j>0}\frac{p_j}{j}z^j\right)\exp\left(-\sum_{j>0}\frac{2p_{-j}}{j}z^{-j}\right).\]
More precisely, we have
\[Y(q\otimes 1, z)Q^N q^k\otimes f=(-1)^{N-k}z^{2k-N}H^\sym(z)f\,.\]

Denoting by $[u, v]=u_{(0)}v$ the 0-th product in $V^\Gr=\Lambda_\ast[Q, q^{\pm 1}]$ we have in particular
\begin{equation}\label{eq: heckesymwc}
[q\otimes 1, Q^N q^k\otimes f]=(-1)^{N-k} Q^N q^{k+1}\otimes H_{N-2k-1}^\sym f
\end{equation}

In \cite[Theorem 3.10]{CJ}, Cai and Jing prove a formula for Schur polynomials of rectangular shapes in terms of the vertex operators $H^\sym_n$.\footnote{Their theorem is more general, it applies to Jack polynomials and to almost rectangular shapes, i.e. $\lambda=(m,m,\ldots, m, m-1)$.} Their proof is entirely combinatorial. We now use the wall-crossing formula from Example \ref{ex: wcgrassmannian} to give a new proof of this formula using the Grassmannian.

\begin{proposition}[{\cite[Theorem 3.10]{CJ}}]Let $m, k>0$. We have the following identity in $\Lambda$:
\begin{equation}\label{eq: sign4}H^\sym_{m-k+1}H^\sym_{m-k+3}\ldots H^\sym_{m+k-3}H^\sym_{m+k-1}(1)=(-1)^{\binom{k}{2}} k!s_{(m)^k}\,.\end{equation}
\end{proposition}
\begin{proof}Let $N=m+k$. We show this by comparing Proposition \ref{prop: grassschur} with the wall-crossing formula in Example \ref{ex: grassva}. On one hand, Proposition \ref{prop: grassschur} gives the formula for the class of the Grassmannian
\[[\Gr(k,N)]=Q^N q^k\otimes (-1)^{k(N-k)}s_{(N-k)^k}\]
We recall the wall-crossing formula from Example \ref{ex: grassva}:
\[[\Gr(k,N)]=\frac{1}{k!}[q,\ldots [q,[q,Q^N]]\ldots ]\,.\]
Applying \eqref{eq: heckesymwc} repeatedly gives us
\begin{equation}\label{eq: sign3}[\Gr(k,N)]=Q^N q^k\otimes \frac{(-1)^{\sum_{j=N-k+1}^{N}j}}{k!}H^\sym_{N-2k+1}H^\sym_{N-2k+3}\ldots H^\sym_{N-3}H^\sym_{N-1}1\,.\end{equation}
The conclusion follows from comparing the two expressions and a sign analysis similar to the one in the previous section.\qedhere
\end{proof}

\begin{remark}
The identity \eqref{eq: sign4} seems quite remarkable to us. In general there is no way to write Schur functions $s_\lambda$ using the vertex operators $H^\sym$, and \eqref{eq: sign4} is a special phenomenon of rectangular shapes $\lambda$. On the other hand, from the point of view of Joyce's wall-crossing, it is surprising that the class of the Grassmannian can be expressed using the Hecke operators $H_m$. We speculate that this phenomenon might be an indication that it is possible to write new wall-crossing formulas using a version of Joyce's vertex algebra in which the we do not consider a symmetrized complex $\Theta=\Ext^\vee+\sigma^\ast \Ext$, but rather take $\Theta=\Ext$. This variation is studied in \cite{latyntsev}: it is no longer a vertex algebra, but rather a braided vertex algebra (in particular it is a field algebra, also known as non-local vertex algebra), and it has connections to Cohomological Hall algebras.
\end{remark}

The Virasoro operators $L_n$ have simpler commutators with $H_m^\sym$ than with $H_m$. This is expected due to the fact that both $L_n$ and $H_m^\sym$ are obtained from fields in the vertex algebra associated to the lattice $(\BZ, 2)$.

\begin{proposition}
We have the identity in $\End(\Lambda_\ast)$:
\[[L_n, H_m^\sym]=(m+1)H^\sym_{m-n}-2  p_{-n}\circ H_m^\sym \,.\]
\end{proposition}
\begin{proof}
A proof can be given as in \cite{liuyang}. Alternatively, we can use the identity \cite[(2.7.1)]{Ka98} for the vertex algebra associated to the lattice $(\BZ, 2)$.
\end{proof}

\subsection{Geometricity of Virasoro and Calogero-Sutherland operator}\phantom{.}\\
We showed in Theorem \ref{thm: virasoro rep} that the Virasoro constraints imply that the operators $\bR_n$ descend to cohomology or, equivalently, they preserve the ideal of relations. We now illustrate this phenomena in the case of the Grassmannian; it turns out that showing geometricity directly in examples is often much easier than showing the full constraints.

The derivation $\bR_n$ is a priori defined in the descendent algebra $\Lambda^\ast$. The ideal of relations of the Grassmannian is the ideal $I$ generated by $e_{k+1}, e_{k+2}, \ldots$ and $h_{N-k+1}, h_{N-k+2}, \ldots$. Equivalently, $I$ is the linear span of the Schur polynomials $s_\lambda$ with $\lambda$ not contained in $(N-k)^k$. We now compute the action of $\bR_n$ on the generator $e_j$, for $j>k$, of the ideal. Denote by $e_\bullet$ the formal sum $\sum_{j\geq 0}e_j$. We have
\[\bR_n e_\bullet=\bR_n\exp\left(\sum_{\ell\geq 1}(-1)^{\ell-1}\frac{p_\ell}{\ell}\right)=\left(\sum_{\ell\geq 1}(-1)^{\ell-1} p_{\ell+n}\right)e_\bullet\,.\]
\begin{align*}
\bR_n e_j=\sum_{\ell=1}^j (-1)^{\ell-1}p_{n+\ell} e_{j-\ell}=(-1)^{n}(j+n)e_{j+n}+\sum_{s=0}^{n-1}(-1)^s e_{j+s}p_{n-s}
\end{align*}
by Newton's identity, so we conclude that $\bR_n(e_j)\in I$ for $j>k$. A similar computation shows the same for the remaining generators.

\subsubsection{Calogero-Sutherland operator}

The geometricity of the Virasoro operators can be explained at once by the geometricity of the Calogero-Sutherland operator (also Laplace-Beltrami operator in the literature). Define the operator $\Delta\colon \Lambda^\ast\to \Lambda^\ast$ by
\begin{equation}\label{eq: csoperator}
\Delta=\frac{1}{6}\sum_{a+b+c=0}\colon p_ap_bp_c\colon=\frac{1}{2}\left(\sum_{a,b>0}p_a p_b p_{-a-b}+p_{a+b}p_{-a} p_{-b}\right)\,.\end{equation}
It is straightforward to show that
\[[p_n,\Delta]=n\left(\sum_{a>0}p_{a+n}p_{-a}+\frac{1}{2}\sum_{\substack{a,b>0\\a+b=n}}p_ap_b\right)\colon\,.\]
The right hand side is $n$ times $\bR_n+\frac{1}{2}\bT_n$, which are the Virasoro operators associated to the vertex algebra $\textup{VA}(\BZ, 1)$. Since $p_n$ descends to cohomology for $n\geq 1$ (as multiplication by $n!\ch_n(\BF)$), the operator $\Delta$ descending to cohomology implies that $\bR_n$ also does. Indeed, this is true:

\begin{proposition}
The operator $\Delta$ on $\Lambda^\ast$ descends to an operator on $H^\ast(\Gr(k,N))$.
\end{proposition}
\begin{proof}
It is well-known that Schur polynomials are a basis of eigenvectors of $\Delta$, see e.g. \cite[Proposition 2]{frenkelwang} or \cite[Remark 1.11]{Sch-Vass}. For instance,
\[\Delta e_j=-\frac{j(j-1)}{2}e_j\textup{ and }\Delta h_j=\frac{j(j-1)}{2}h_j\,.\]
It follows immediately that $\Delta$ preserves the ideal of relations $I$.\qedhere
\end{proof}

\begin{remark}
In \cite{Sch-Vass} the authors study a certain algebra of operators, related to the $W_{1+\infty}$ Lie algebra, and show that it acts on $\bigoplus_{n\geq 0} H^\ast(\textup{Hilb}^n(\BQ^2))$ and, more generally, on the cohomology of moduli spaces of instantons. They consider the operators $\mathsf D_{0,l}$ on $\Lambda^\ast$ (which for simplicity we specialize at $\kappa=1$) defined to have Schur polynomials as eigenvectors with specified eigenvalues:
\[\mathsf D_{0,l}(s_\lambda)=\left(\sum_{\Box\in \lambda}\big(y(\Box)-x(\Box)\big)^{l-1}\right)s_\lambda\,.\]
In particular, $\mathsf D_{0,2}=\Delta$, see \cite[Remark 1.11]{Sch-Vass}. By the same reasoning as above, the operators $\mathsf D_{0,l}$ descend to the cohomology of the Grassmannian. We suspect that it should be possible to interpret $\mathsf D_{0,l}$ as vertex operators defined by $\textup{VA}(\BZ, 1)$, similarly to the formula \eqref{eq: csoperator} which exhibits $\Delta$ as an operator coming from the state $p_1^3$. For instance, it seems that $\mathsf D_{0,3}$ can be obtained from the state $p_1^4-p_2^2$.
\end{remark}

\subsection{Central charge $c\neq 1$ and Jack functions}
In this last section, we work with the coefficients in $\BC$ instead of $\BQ$. Recall that the Virasoro Lie algebra is defined by
\[\Vir=\bigoplus_{n\in \BZ}\BC L_n\oplus \BC C\]
with Lie bracket given by
\begin{align*}
[L_n, L_m]&=(n-m)L_{m+n}+\delta_{m+n}\frac{n^3-n}{12}C\\
[L_n, C]&=0
\end{align*}
We recall some basic notions of representation theory. Let $\Vir^{+}$ (respectively $\Vir^{-}$) be the sub Lie algebras spanned by $L_n$ for $n>0$ (respectively $n<0$) and let $\Vir^{0}$ be spanned by $L_0, C$. Given a representation $M$ of $\textup{Vir}$ we say that $w\in M$ is a singular vector of weight $(c, h)$ if
\[L_0(w)=hw\,,\quad C(w)=cw\,,\quad L_n(w)=0\,\textup{ for all }n>0\,. \]
We say that $w\in M$ is a highest weight vector if it is a singular vector and moreover $U(\Vir^-)\cdot w=M$.

Fix parameter $\alpha, \beta\in \BC$ and define
\[\beta_0=\frac{\beta}{2}-\frac{1}{\beta}\,,\quad c=1-12\beta_0^2\,,\quad h=\frac{1}{2}\alpha^2-\alpha \beta_0\,.\]
Then there is a representation of $\Vir$ on $\Lambda$ such that $1\in \Lambda$ is a highest weight vector of weight $(c,h)$; this is called the Fock space representation of highest weight $(c,h)$ and we denote it by $\Lambda_{\alpha, \beta}$. The positive part of the Virasoro algebra acts as
\[L_n=\frac{\beta^2}{2}\sum_{\substack{s+t=n\\s,t>0}}p_{-s}p_{-t}+\sum_{s>0}p_sp_{-s-n}+(\alpha+\beta_0(n+1))\beta p_{-n}\]
\[L_0(f)=(h+\deg(f))f\quad \textup{ for }f \textup{ homogeneous}\,.\]
The representation $\Lambda_{\alpha, \beta}$ is irreducible if and only if there are no singular vectors other than scalars if and only if the Verma module of highest weight $(c,h)$ is irreducible. Note that when $\beta=\sqrt{2}$ and $\alpha=(2k-N)/\sqrt{2}$ the operators $L_n$, $n>0$ specialize to \eqref{eq: operatorsgrva}; thus, the Virasoro constraints for the Grassmannian say that $s_{(N-k)^k}$ is a singular vector of $\Lambda_{(2k-N)/\sqrt{2}, \sqrt{2}}$. In this case the central charge is $c=1$.

The singular vectors of $\Lambda_{\alpha, \beta}$ are completely classified by \cite{mimachiyamada, wakimotoyamada}. It turns out that the singular vectors are given in terms of Jack functions of rectangular partitions. Jack functions $J_\lambda^t$ depend on a partition $\lambda$ and a parameter $t\in \BC$ and they are deformations of Schur polynomials: when $t=1$ the Jack function $J_\lambda^{t=1}$ is proportional to $s_\lambda$; we refer to \cite[Section VI.10]{macdonald} for an introduction to Jack functions.

\begin{theorem}\label{thm: singularvectorsjack}
The representation $\Lambda_{\alpha, \beta}$ has a singular vector $f\in \Lambda_{\alpha, \beta}$ of degree $d>0$ if and only if there are two integers $r, s>0$ such that \[d=rs\,,\quad \alpha=(1+r)\frac{\beta}{2}-(1+s)\frac{1}{\beta}\,.\]
When that is the case, the only singular vector of degree $d$ (up to a scalar) is given by
\[\sigma J^{\beta^2/2}_{(r)^s}\,\]
where $\sigma$ is the involution on $\Lambda$ sending $p_j\mapsto (-1)^{j-1}p_j$.
\end{theorem}
It would be very interesting to connect the singular vector $\sigma J^{\beta^2/2}_{(r)^s}$ to some moduli space of quiver representations/sheaves in the same way that $s_{(N-k)^k}$ is associated to the Grassmannian, and thus interpreting as Theorem \ref{thm: singularvectorsjack} as Virasoro constraints for such moduli space. Note that the Calogero-Sutherland operator \eqref{eq: csoperator} also admits a deformation for which the Jack polynomials are eigenvectors, see \cite[Remark 1.1]{Sch-Vass}.

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