# Virasoro constraints for K3 surfaces and monodromy operators

Weisheng Wang

###### Abstract.

The Virasoro constraints for moduli spaces of stable torsion free sheaves on a
surface with only $(p,p)$-cohomology were recently proved by Bojko-Moreira-
Lim. The rank 1 case, which is _not_ restricted to surfaces with only
$(p,p)$-cohomology, was established by Moreira. We formulate conjectural
Virasoro constraints in any positive rank without requiring only
$(p,p)$-cohomology. We prove our conjecture for K3 surfaces using Markman
monodromy operators, which allow us to reduce to the rank 1 case. We also
prove new Virasoro constraints in rank 0. Finally, for K3 surfaces, we
introduce new Virasoro operators in negative degree which, together with the
previous Virasoro operators, give a representation of Virasoro algebra with
central charge $24$.

##  1\. Introduction

The Virasoro operators in Gromov-Witten theory were first proposed in [3],
where the Virasoro operators are constructed for some Fano varieties. Here, I
will recall the form of the Virasoro operators following [9]. The Virasoro
constraints on the moduli of stable pairs side is obtained by the GW/PT
correspondence on $3$-folds [7], I will briefly review this. The Virasoro
constraints on moduli of stable sheaves were first obtained on Hilbert schemes
of points on a surface $S$ by restrict the stable pairs case to
$S\times\mathbb{P}^{1}$ [7][6]. It is then generalised to moduli of stable
sheaves of higher ranks [2][1].

###  1.1. Virasoro constraint in Gromov-Witten theory

Gromov-Witten theory is defined by integration over the moduli space of stable
maps. Let $X$ be a non-singular projective variety over $\mathbb{C}$. A map
from a connected pointed nodal curve to $X$ is a stable map if it has finite
automorphism group (for more detail see chapter 24 [4]). A stable map $f$
represents a homology class $\beta\in H_{2}(X,\mathbb{Z})$ if
$f_{\ast}[C]=\beta$. $\overline{\mathcal{M}}_{g,n}(X,\beta)$ denotes the
moduli space of stable maps from $n$-pointed genus $g$ nodal curves to $X$
representing the class $\beta$, it is a proper Deligne-Mumford stack. There
are $n$ evaluation maps
$\operatorname{ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$ given by:

$\operatorname{ev}_{i}(\Sigma,p_{1},\dots,p_{n},f)=f(p_{i})\quad(1\leq i\leq
n).$

At each point $[\Sigma,p_{1},\dots,p_{n},f]$ of
$\overline{\mathcal{M}}_{g,n}(X,\beta)$, the cotangent line to $\Sigma$ at
point $p_{i}$ is a one-dimensional vector space; those spaces glue together to
give a line bundle $\mathbb{L}_{i}$ called the $i$th tautological line bundle.
Define $\psi_{i}:=c_{1}(\mathbb{L}_{i})$.

Let $\\{\gamma_{a}\\}$ be a homogeneous basis of $H^{\ast}(X,\mathbb{C})$. The
descendent Gromov-Witten invariants of $X$ are:

(1)
$\displaystyle\langle\tau_{k_{1}}(\gamma_{a_{1}})\dots\tau_{k_{n}}(\gamma_{a_{n}})\rangle_{g,\beta}^{X}=\int_{\left[\overline{\mathcal{M}}_{g,n}(X,\beta)\right]^{\text{vir}}}\psi_{1}^{k_{1}}\operatorname{ev}_{1}^{\ast}(\gamma_{a_{1}})\dots\psi_{n}^{k_{n}}\operatorname{ev}_{n}^{\ast}(\gamma_{a_{n}}).$

Let $\\{t^{a}_{k}\\}$ be a set of variables, and define the generating
function $F^{X}(t,\lambda)$ as:

$F^{X}=\sum_{g\geq 0}\lambda^{2g-2}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}q^{\beta}\sum_{n\geq
0}\frac{1}{n!}\sum_{\begin{subarray}{c}a_{1}\dots a_{n}\\\ k_{1}\dots
k_{n}\end{subarray}}t^{a_{n}}_{k_{n}}\dots
t^{a_{1}}_{k_{1}}\langle\tau_{k_{1}}(\gamma_{a_{1}})\dots\tau_{k_{n}}(\gamma_{a_{n}})\rangle^{X}_{g,\beta}.$

Also define the full partition function $Z^{X}=\exp(F^{X})$. $Z^{X}$ is the
partition function corresponds to the standard disconnected Gromov-Witten
bracket $\langle,\rangle^{X,\bullet}_{g,\beta}$, where the domain nodal curve
could be disconnected:

(2) $\displaystyle Z^{X}=\sum_{g\geq 0}\lambda^{2g-2}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}q^{\beta}\sum_{n\geq
0}\frac{1}{n!}\sum_{\begin{subarray}{c}a_{1}\dots a_{n}\\\ k_{1}\dots
k_{n}\end{subarray}}t^{a_{n}}_{k_{n}}\dots
t^{a_{1}}_{k_{1}}\langle\tau_{k_{1}}(\gamma_{a_{1}})\dots\tau_{k_{n}}(\gamma_{a_{n}})\rangle^{X,\bullet}_{g,\beta}.$

A set of formal differential operators $\\{L_{k}\\}_{k\geq-1}$ are defined in
[9], They are defined in variables $t_{k}^{a}$ and only depend upon the
intersection pairing $g_{ab}=\int_{X}\gamma_{a}\cup\gamma_{b}$, with
$\gamma_{a},\gamma_{b}\in H^{\ast}(X,\mathbb{C})$, the Hodge decomposition
$\gamma_{a}\in H^{p_{a},q_{a}}(X,\mathbb{C})$ and the action of $c_{1}(X)$ on
$\\{\gamma_{a}\\}$. For the precise form of operators $\\{L_{k}\\}_{k\geq-1}$,
one can refer section 4 of [9]. Those operators satisfy the Virasoro bracket,

$[L_{k},L_{l}]=(k-l)L_{k+l}.$

The Virasoro conjecture in Gromov-Witten theory states as follows:

######  Conjecture 1.1.

For all non-singular projective varieties $X$, $L_{k}(Z^{X})=0$.

This conjecture has been proven for curves $C_{g}$ of genus $g$ and
nonsingular projective toric varieties.

###  1.2. GW/Pairs correspondence for 3-folds and Virasoro constraints for
stable pairs

The moduli space of stable maps $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is
essentially based upon the geometry of curves in $X$, there is another way to
approach the moduli of curves in $X$, which is the moduli of stable pairs.

######  Definition 1.2.

A stable pair $(F,s)$ on $X$ is a coherent sheaf $F$ on $X$ and a section
$s\in H^{0}(X,F)$ satisfying the following two stability conditions: (1) $F$
is pure of dimension 1, (2) the section $s:\mathcal{O}_{X}\to F$ has cokernel
of dimensional 0.

Given a stable pair $\mathcal{O}_{X}\to F$, the kernel of $s$ defines a Cohen-
Macaulay subcurve $C\subset X$, i.e. $C$ has no embedded points; the support
of the cokernel of $s$ defines a $0$-dimensional subscheme of $C$. To a stable
pair, the Euler characteristic and the class of the support $C$ of $F$ is
associated:

$\chi(F)=n\in\mathbb{Z}\quad\text{and}\quad[C]=\beta\in H_{2}(X,\mathbb{Z}).$

For a fixed $n$ and $\beta$, there is a projective moduli space of stable
pairs $P_{n}(X,\beta)$, it is non empty only if $\beta$ is an effective curve
class. This moduli space is studied in [10], it has been shown that
$P_{n}(X,\beta)$ is a fine moduli space with a universal stable pair
$(\mathbb{F},s)$ over $X\times P_{n}(X,\beta)$. Let $\pi_{X}$ and $\pi_{P}$ be
the projection to the first and second factor, then one can define the
descendent class by:

$\displaystyle\operatorname{ch}_{k}(\gamma)=\pi_{P\ast}\left(\operatorname{ch}_{k}(\mathbb{F}-\mathcal{O}_{X\times
P_{n}(X,\beta)})\cdot\pi_{X}^{\ast}\gamma\right).$

The invariant in the stable pair theory has the following form:

$\displaystyle\langle\operatorname{ch}_{k_{1}}(\gamma_{1})\dots\operatorname{ch}_{k_{m}}(\gamma_{m})\rangle^{X,\text{PT}}_{n,\beta}:=\int_{\left[P_{n}(X,\beta)\right]^{\text{vir}}}\prod_{i=1}^{m}\operatorname{ch}_{k_{i}}(\gamma_{i}).$

The moduli of stable maps and stable pair are both based upon the geometry of
curves in $X$, therefore one may hope there exists some links between those
two descendent theories. Indeed, for non-singular projective 3-folds, this
correspondence conjecturally holds: the Gromov-Witten and stable pairs
descendent series are related after a change of variables.

Let $\mathbb{D}_{\text{PT}}^{X}$ be the commutative $\mathbb{Q}$-algebra with
generators

$\\{\operatorname{ch}_{i}(\gamma)|i\geq 0,\gamma\in H^{\ast}(X,\mathbb{Q})\\}$

subject to the nature relations

$\displaystyle\operatorname{ch}_{i}(\lambda\cdot\gamma)$
$\displaystyle=\lambda\operatorname{ch}_{i}(\gamma)$
$\displaystyle\operatorname{ch}_{i}(\hat{\gamma}+\gamma)$
$\displaystyle=\operatorname{ch}_{i}(\hat{\gamma})+\operatorname{ch}_{i}(\gamma),$

For $\lambda\in\mathbb{Q}$ and $\gamma,\hat{\gamma}\in H^{\ast}(X)$. Define
$\mathbb{D}^{X}_{\text{GW}}$ similarly as $\mathbb{D}_{\text{PT}}^{X}$ using
generators $\tau_{i}(\gamma)$’s. The GW/PT correspondence is a linear map
$\mathfrak{C}^{\bullet}:\mathbb{D}_{\text{PT}}^{X\bigstar}\to\mathbb{D}^{X}_{\text{GW}}$,
where $\mathbb{D}_{\text{PT}}^{X\bigstar}$ is a subalgebra of
$\mathbb{D}_{\text{PT}}^{X}$ called essential descendants and the map
$\mathfrak{C}^{\bullet}$ is defined on monomials, for the precise definition
of $\mathfrak{C}^{\bullet}$ and $\mathbb{D}_{\text{PT}}^{X\bigstar}$ one can
refer [7]. In toric 3-fold case, the precise correspondence statement is:

######  Theorem 1.3 (Theorem 6 of [7]).

Let $X$ be a nonsingular projective toric 3-fold. Let
$D\in\mathbb{D}_{\mathrm{PT}}^{X\bigstar}$, $\beta\in H_{2}(X,\mathbb{Z})$
with $d_{\beta}=\int_{\beta}c_{1}(X)$. Then the $\mathrm{GW/PT}$
correspondence holds:

$(-q)^{-d_{\beta}/2}\left(\sum_{n\in\mathbb{Z}}q^{n}\left\langle
D\right\rangle_{n,\beta}^{X,\mathrm{PT}}\right)=(-iu)^{d_{\beta}}\left(\sum_{g\in\mathbb{Z}}u^{2g-2}\left\langle\mathfrak{C}^{\bullet}\left(D\right)\right\rangle_{g,\beta}^{X,\mathrm{GW}}\right)$

after the change of variable $-q=e^{iu}$.

In [7], Virasoro constraints on moduli of stable pairs are proven using the
Virasoro constraints for the Gromov-Witten theory of toric 3-folds and the
above GW/PT correspondence. The Virasoro operators
$\mathcal{L}^{\text{PT}}_{k}$ with $k\geq-1$ are defined as a operators on
$\mathbb{D}_{\text{PT}}^{X}$. For the precise definition one can refer [7].
The Virasoro constraints states as follows:

######  Conjecture 1.4 ([7]).

Let $X$ be a nonsingular projective 3-fold with only $(p,p)$-cohomology, and
let $\beta\in H_{2}(X,\mathbb{Z})$. For all $k\geqslant-1$ and
$D\in\mathbb{D}_{\mathrm{PT}}^{X}$, we have

$\left\langle\mathcal{L}_{k}^{\mathrm{PT}}(D)\right\rangle_{\beta}^{X,\mathrm{PT}}=0,$

where

$\left\langle
D\right\rangle_{\beta}^{X,\mathrm{PT}}:=\sum_{n\in\mathbb{Z}}q^{n}\int_{\left[P_{n}(X,\beta)\right]^{vir}}D.$

The statement of this conjecture about stationary descendants for non-singular
projective toric 3-folds is proven in [7]. A special case is when
$X=S\times\mathbb{P}^{1}$ where $S$ is a smooth projective toric surface, then
the Virasoro constraints for this toric 3-fold $X$ are:

$\forall
k\geqslant-1,\quad\left\langle\mathcal{L}_{k}^{\mathrm{PT}}\prod_{i=1}^{r}\mathrm{ch}_{m_{i}}\left(\gamma_{i}\times\boldsymbol{\operatorname{p}}\right)\right\rangle_{n\left[\mathbb{P}^{1}\right]}^{X,\mathrm{PT}}=0,$

where $\gamma_{i}\in H^{\ast}(S)$, $\boldsymbol{\operatorname{p}}\in
H^{2}(\mathbb{P}^{1})$ is the point class and $[\mathbb{P}^{1}]\in H_{2}(X)$
is the fiber class. Specializing to the space
$P_{n}(S\times\mathbb{P}^{1},n[\mathbb{P}^{1}])\cong\operatorname{Hilb}^{n}(S)$,
one gets a new set of Virasoro constraints for tautological classes on
$\operatorname{Hilb}^{n}(S)$ for toric surfaces. In [6], this constraint is
proven for simply connected surfaces, I will recall the full form of the
Virasoro operators in this Hilbert scheme of points setting in the next
section.

The Hilbert scheme of points can be viewed as a moduli of stable sheaves of
rank $1$, therefore one may expect that there is a Virasoro constraint for the
moduli of stable sheaves. In [2], D. van Bree has proposed a Virasoro
conjecture for the moduli space of Gieseker semistable sheaves for surfaces
with only $(p,p)$-cohomology. In [1], Bojko, Lim and Moreira have proved the
Virasoro constraint for moduli spaces of stable torsion-free sheaves on any
curve and on surfaces with only $(p,p)$-cohomology classes, in particular the
conjecture of D. van Bree is proven.

In this paper, I will combine Moreira’s Virasoro constraints on Hilbert
schemes of points on simply connected surfaces and van Bree’s Virasoro
constraints on moduli of sheaves on surfaces with only (p,p)-cohomology to
propose the following conjecture of Virasoro constraints on moduli of stable
sheaves on simply connected surfaces and provide a proof for this Virasoro
constraint in the case of K3 surface.

######  Conjecture 1.5.

Let $S$ be a smooth projective simply connected surface over $\mathbb{C}$ and
let $H$ be a fixed polarisation. Choose numbers $r>0$ and $c_{2}$ and a line
bundle $L$. Let $M=M_{S}^{H}(r,L,c_{2})$ be the moduli space of Gieseker
semistable sheaves of rank $r$, with determinant $L$ and second Chern class
$c_{2}$. Assume that all semistable sheaves are also stable and $M$ has a
(twisted) universal sheaf $\mathcal{F}$. Let $\mathbb{D}^{S}$ be the
holomorphic descendents defined in the section 2. Let $\xi:\mathbb{D}^{S}\to
H^{\ast}(M,\mathbb{C})$ be the geometric realization defined in (4) and let
$\mathcal{L}_{k},k\geq-1$ be the Virasoro operators defined in (6). Then
$\forall k\geq-1$ and $\forall D\in\mathbb{D}^{S}$, we have

$\int_{[M]^{\mathrm{vir}}}\xi_{\mathcal{F}\otimes(\operatorname{det}\mathcal{F})^{-1/r}}\left(\mathcal{L}_{k}D\right)=0$

In Theorem 2.2, the above conjecture is proven for $S$ being a K3 surface. In
section 5, the rank zero case is considered and a modified version of the
Virasoro constraints is proven in proposition 5.6. The idea of the proof is to
use the Markman operator to transform the descendent integrals on moduli of
stable sheaves to descendent integrals on Hilbert schemes of points where the
conjecture is proven.

In section 2, I will recall the Virasoro constraints of Moreira and van Bree
and formulate the Virasoro constraints on K3 surfaces. In section 3, I will
recall the Markman operator following G.Oberdieck’s expositions [8]. In
section 4, I will prove the Virasoro constraint on moduli of positive rank
stable sheaves on K3 surfaces. In section 5, I will consider the rank 0 case.
In section 6, I will propose a set of negative Virasoro operators on K3
surface. Those negative Virasoro operators combined with the existing
$L_{k\geq-1}$ operators will satisfy the Virasoro algebra with central charge
$e(S)$.

###  1.3. Acknowledgement

The author would like to thank his supervisor Martijn Kool for many useful
suggestions. He is also grateful to Woonam Lim for useful conversations. The
author is supported by the ERC Consolidator Grant FourSurf 101087365.

##  2\. Virasoro constraints

Let $S$ be a non-singular and projective K3 surface. There is a bilinear form,
called Mukai pairing, on $\Lambda:=H^{\ast}(S,\mathbb{Z})$ defined as:

$\displaystyle(x,y):=-\int_{S}x^{\vee}y,$

where $\bullet^{\vee}$ is defined as follows: if one writes $x=(r,D,n)$ as the
decomposition of degree then $x^{\vee}=(r,-D,n)$. This pairing is symmetric,
unimodular, of signature $(4,20)$ and the resulting lattice is called the
Mukai lattice. For $x=(r,D,n)\in\Lambda$, I will also write

(3) $\displaystyle\operatorname{rk}(x)=r,\quad c_{1}(x)=D,\quad v_{2}(x)=n.$

For a coherent sheaf $\mathcal{F}$ on $S$, its Mukai vector is defined to be
$v=\operatorname{ch}(\mathcal{F})\cdot\sqrt{\operatorname{td}_{S}}$. Let
$v\in\Lambda$ and an ample line bundle $H$ be chosen such that $M:=M_{H}(v)$,
the moduli space of $H$-semistable sheaves on $S$ with Mukai vector $v$, does
not contain any strictly semistable sheaves and is a smooth projective and
admits a (twisted) universal sheaf. For example, one could choose $v\in
H^{1,1}(S)$, $v\neq 0$ and $(v,v)\geq-2$ and $v$ is not a multiple of a class
by an integer larger than $1$, those choices of $v$ are called effective and
primitive in the sense of [5]. Also, for such $v$, there always exists an
ample line bundle $H$ on $S$ such that $M_{H}(v)$ has the above mentioned
properties [5].

Let $\mathbb{D}^{S}$ be the commutative algebra generated by symbols called
holomorphic descendents of the form:

$\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)\quad\text{ for }i\geq 0,\gamma\in
H^{\ast}(S,\mathbb{C})$

subject to the linearity relations

$\operatorname{ch}^{\mathrm{H}}_{i}(\lambda_{1}\gamma_{1}+\lambda_{2}\gamma_{2})=\lambda_{1}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma_{1})+\lambda_{2}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma_{2})$

for $\lambda_{1},\lambda_{2}\in\mathbb{C}$. I also write
$\operatorname{ch}^{\mathrm{H}}_{\bullet}(\gamma)$ for the element

$\operatorname{ch}^{\mathrm{H}}_{\bullet}(\gamma)=\sum_{i\geq
0}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)\in\mathbb{D}^{S}.$

Consider the moduli space $M:=M_{H}(v)$ with $r:=\operatorname{rk}(v)\geq 1$
and the product $M\times S$, let $\pi_{M}$ and $\pi_{S}$ be the projection
from $M\times S$ to $M$ and $S$. Let $\mathcal{F}$ be a universal sheaf or a
twisted universal sheaf. Let me recall the definition in [1] of the geometric
realization with respect to $\mathcal{F}$ on $M\times S$ as the algebra
homomorphism

(4) $\displaystyle\xi_{\mathcal{F}}:\mathbb{D}^{S}\to H^{\ast}(M),$

which acts on generators $\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$ with
$\gamma\in H^{p,q}(S)$ as

$\xi_{\mathcal{F}}\left(\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)\right)=\pi_{M\ast}\left(\operatorname{ch}_{i+\operatorname{dim}(S)-p}(\mathcal{F})\pi_{S}^{\ast}\gamma\right)$

Next, I will define the Virasoro operators for K3 surfaces. I combined the
form of Virasoro operators in [6] and in [2]. For $k\geq-1$, I define
operators $R_{k}$, $T_{k}$, $S_{k}$ on $\mathbb{D}^{S}$ as follows:

$\bullet$ The operator $R_{k}:\mathbb{D}^{S}\to\mathbb{D}^{S}$ is defined as a
derivation by fixing its action on the generators: given $\gamma\in
H^{p,q}(S)$,

$\displaystyle R_{k}(\operatorname{ch}_{i}^{\mathrm{H}}(\gamma))$
$\displaystyle=\left(\prod_{j=0}^{k}(i+j)\right)\operatorname{ch}_{i+k}^{\mathrm{H}}(\gamma)$

I take the following conventions: the above product is $1$ if $k=-1$ and
$\operatorname{ch}^{\mathrm{H}}_{i+k}(\gamma)=0$ if $i+k<0$.

$\bullet$ The operator $T_{k}:\mathbb{D}^{S}\to\mathbb{D}^{S}$ is the operator
of multiplication by a fixed element of $\mathbb{D}^{S}$:

$\displaystyle T_{k}$
$\displaystyle=\sum_{i+j=k}(-1)^{\operatorname{dim}(S)-p^{L}}i!j!\operatorname{ch}_{i}^{\mathrm{H}}\operatorname{ch}_{j}^{\mathrm{H}}(\text{td}_{S})$

where
$(-1)^{\operatorname{dim}(S)-p^{L}}i!j!\operatorname{ch}_{i}^{\mathrm{H}}\operatorname{ch}_{j}^{\mathrm{H}}(\text{td}_{S})$
222In [1], $T_{k}$ operators are also defined for curves, in this case
$\operatorname{dim}(S)$ should be replaced by the dimension of the curve. is
defined as follows: let $\Delta:S\to S\times S$ be the diagonal map and let

$\sum_{t}\gamma_{t}^{L}\otimes\gamma_{t}^{R}=\Delta_{\ast}\operatorname{td}_{S}$

be the Künneth decomposition of $\Delta_{\ast}\operatorname{td}_{S}$ such that
$\gamma^{L}_{t}\in H^{p^{L}_{t},q^{L}_{t}}(S)$ for some $p^{L}_{t},q^{L}_{t}$,
Then

$(-1)^{\operatorname{dim}(S)-p^{L}}i!j!\operatorname{ch}_{i}^{\mathrm{H}}\operatorname{ch}_{j}^{\mathrm{H}}(\text{td}_{S})=\sum_{t}(-1)^{\operatorname{dim}(S)-p^{L}_{t}}i!j!\operatorname{ch}_{i}^{\mathrm{H}}(\gamma_{t}^{L})\operatorname{ch}_{j}^{\mathrm{H}}(\gamma_{t}^{R}).$

$\bullet$ For $\alpha\in H^{\ast}(S)$, define the derivation $R_{-1}[\alpha]$
by
$R_{-1}[\alpha](\operatorname{ch}^{\mathrm{H}}_{i}(\gamma))=\operatorname{ch}^{\mathrm{H}}_{i-1}(\alpha\gamma)$.
For $k\geq-1$, $S_{k}:\mathbb{D}^{S}\to\mathbb{D}^{S}$ is defined

(5) $\displaystyle
S_{k}(D)=-\frac{(k+1)!}{r}\sum_{\\{i|p^{L}_{i}=0\\}}R_{-1}[\delta^{L}_{i}]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\delta^{R}_{i})D\right),\quad\forall
D\in\mathbb{D}^{S},$

where $r$ is the rank of the sheaves that the moduli $M$ parameterize and the
sum runs over the terms $\delta_{i}^{L}\otimes\delta_{i}^{R}$ of the Künneth
decomposition of $\Delta_{\ast}\mathbbm{1}\in H^{\ast}(S\times S)$ such that
$p_{i}^{L}=0$.

Finally define operators $L_{k}$ and $\mathcal{L}_{k}$ for $k\geq-1$ as:

(6) $\displaystyle\begin{split}L_{k}&=R_{k}+T_{k}\\\
\mathcal{L}_{k}&=R_{k}+T_{k}+S_{k}\end{split}$

######  Remark 2.1.

Denote by $S^{[n]}$ the Hilbert scheme of points on $S$ parameterizing $0$
dimensional subschemes of $S$ with length $n$. Let $\mathcal{I_{\mathcal{Z}}}$
be the ideal sheaf of the universal subscheme; equivalently one can write
$\mathcal{I_{\mathcal{Z}}}=\mathcal{O}_{S^{[N]}\times
S}-\mathcal{O}_{{\mathcal{Z}}}$. When viewing $S^{[n]}$ as a moduli of rank
one sheaves, the universal sheaf $\mathcal{F}$ is $\mathcal{I_{\mathcal{Z}}}$,
in this case one has
$\operatorname{ch}(\mathcal{F}\otimes(\operatorname{det}\mathcal{F})^{-1})=\operatorname{ch}(\mathcal{F})$.
Therefore, in the case of $S^{[n]}$, the above defined Virasoro operators
become the Virasoro operators of Moreira in [6]: in [6], the geometric
realization $\xi_{I_{\mathcal{Z}}}$ is used. If one considers a surface $S$
with only $(p,p)$ cohomology and a moduli space $M$ of Gieseker semistable
sheaves of of rank $r\geq 1$ as the case in [2], then the above defined
Virasoro operators become the Virasoro operators of van Bree [2].

One of the main result of this paper is:

######  Theorem 2.2.

Let $S$ be a non-singular and projective K3 surface. Let $D\in\mathbb{D}^{S}$.
Let $M_{H}(v)$ be a moduli of sheaves on $S$ as above with
$r:=\operatorname{rk}(v)\geq 1$, and let $\mathcal{F}$ be a universal sheaf.
Define
$\overline{\mathcal{F}}:=\mathcal{F}\otimes(\operatorname{det}\mathcal{F})^{-1/r}$
as an element of the rational K-theory of $M_{H}(v)\times S$333
$(\operatorname{det}\mathcal{F})^{-1/r}$ might not exist as a line bundle,
also the geometric realization is well define is this case, since it only
involves Chern characters ($\overline{\mathcal{F}}$ is independent of the
choice of the universal sheaf $\mathcal{F}$ as in [2]). Then for $k\geq-1$, we
have:

$\int_{M_{H}(v)}\xi_{\overline{\mathcal{F}}}\left(\mathcal{L}_{k}D\right)=0.$

##  3\. Markman’s monodromy operator

Let $S$ be a non-singular projective K3 surface and let $M:=M_{H}(v)$ be a
moduli of sheaves on $S$ as the section 2, but in this section, it is possible
to take $\operatorname{rk}(v)=0$. Morphisms $\pi_{M}$ and $\pi_{S}$ are
defined in the same way as the section 2.

Define the morphism $\theta_{\mathcal{F}}:\Lambda\to H^{2}(M,\mathbb{Q})$ as
follows:

$\theta_{\mathcal{F}}(x)=[\pi_{M\ast}(\operatorname{ch}(\mathcal{F})\pi_{S}^{\ast}(\sqrt{\text{td}_{S}}\cdot
x^{\vee}))]_{2},$

where $[\bullet]_{k}$ means take the real degree $k$ component of a cohomology
class.

The morphism $B:H^{\ast}(S,\mathbb{Q})\to H^{\ast}(M,\mathbb{Q})$ is defined
by

(7) $\displaystyle B(x)=\pi_{M\ast}(u_{v}\cdot x^{\vee}),$

where

$u_{v}:=\exp\left(\frac{\theta_{\mathcal{F}}(v)}{(v,v)}\right)\cdot\operatorname{ch}(\mathcal{F})\cdot\sqrt{\text{td}_{S}}\in
H^{\ast}(M\times S,\mathbb{Q}),$

where some pull-backs $\pi_{M}^{*}$, $\pi_{S}^{*}$ are suppressed. One can
check that $B$ is independent of the choice of the (twisted) universal sheaf
$\mathcal{F}$.

Let $S_{1}$ and $S_{2}$ be two non-singular projective K3 surfaces with
polarizations $H_{1},H_{2}$. Let $g:H^{\ast}(S_{1},\mathbb{Z})\to
H^{\ast}(S_{2},\mathbb{Z})$ be an isometry of Mukai lattices, assume $v_{1}$,
$v_{2}$ are two vectors in the Mukai lattice satisfying assumptions of the
section 2 and $v_{2}=g(v_{1})$. Let $M_{i}:=M_{H_{i}}(v_{i})$ for $i=1,2$.
Markman defined the transformation $\gamma(g):H^{\ast}(M_{1},\mathbb{C})\to
H^{\ast}(M_{2},\mathbb{C})$ in [5]:

(8)
$\displaystyle\gamma(g)(x)=\pi_{2\ast}\left(c_{\operatorname{dim}(M)}\big{[}-\pi_{13\ast}\left(\pi_{12}^{\ast}\left((1\otimes
g)u_{v_{1}}\right)^{\vee}\cdot\pi_{23}^{\ast}u_{v_{2}}\right)\big{]}\cdot\pi_{1}^{\ast}x\right),$

where $\pi_{ij}$ ($\pi_{k}$) is the projection of $M_{1}\times S_{2}\times
M_{2}$ to the $(i,j)$-th ($k$-th) factor. The main properties of $\gamma(g)$
are given in the following theorem:

######  Theorem 3.1 (Markman).

Let $S_{1},S_{2}$ and $v_{1},v_{2}$ as above. For any isometry
$g:H^{\ast}(S_{1},\mathbb{C})\to H^{\ast}(S_{2},\mathbb{C})$ such that
$g(v_{1})=v_{2}$, $\gamma(g)$ is the unique operator such that:

  1. $\mathrm{(i)}$

$\gamma(g)$ is a degree-preserving ring isomorphism and is a isometry with
respect to the Poincaré pairing: $\langle x,y\rangle=\int_{M}xy$ for all
$x,y\in H^{\ast}(M,\mathbb{Q})$.

  2. $\mathrm{(ii)}$

$(\gamma(g)\otimes g)(u_{v_{1}})=u_{v_{2}}$.

  3. $\mathrm{(iii)}$

$\gamma(g_{1})\circ\gamma(g_{2})=\gamma(g_{1}g_{2})$ and
$\gamma(g)^{-1}=\gamma(g^{-1})$(if it makes sens), where $g_{1},g_{2}$ are two
isometries.

  4. $\mathrm{(iv)}$

$\gamma(g)\left(c_{k}(T_{M_{1}})\right)=c_{k}(T_{M_{2}})$.

The above properties imply that

(9) $\int_{M_{1}}\sigma=\int_{M_{2}}\gamma(g)(\sigma)\quad\forall\sigma\in
H^{\ast}(M_{1},\mathbb{Q}).$

In fact the property $(\gamma(g)\otimes g)(u_{v_{1}})=u_{v_{2}}$ can be
expressed in a nicer way using the $B$ morphism defined in (7):

######  Lemma 3.2 (Lemma 2.7 of [8]).

Let $M_{i},v_{i}$ with $i=1,2$ and $g$ as above. Let the canonical morphism
$B:H^{\ast}(S,\mathbb{Q})\to H^{\ast}(M_{i},\mathbb{Q})$ be defined as (7) and
write $B_{k}(x)$ for the component in degree $2k$. Let
$f:H^{\ast}(M_{1},\mathbb{Q})\to H^{\ast}(M_{2},\mathbb{Q})$ be a degree
preserving isometric ring isomorphism. Then the following are equivalent:

  1. $\mathrm{(i)}$

$(f\otimes g)(u_{v_{1}})=u_{v_{2}}$,

  2. $\mathrm{(ii)}$

$f(B(\gamma))=B(g\gamma)$ for all $\gamma\in H^{\ast}(S_{1},\mathbb{Q})$.

Therefore, we have the following commutative diagram.

${H^{\ast}(S_{1},\mathbb{Q})}$${H^{\ast}(S_{2},\mathbb{Q})}$${H^{\ast}(M_{1},\mathbb{Q})}$${H^{\ast}(M_{2},\mathbb{Q})}$$\scriptstyle{g}$$\scriptstyle{B}$$\scriptstyle{B}$$\scriptstyle{\gamma(g)}$

Now, let $x_{1}\dots x_{k}\in\Lambda$, consider the following descendent
integrals over $M$:

$\int_{M}P(B_{i}(x_{j}),c_{l}(T_{M})),$

where $P(t_{ij},u_{l})$ is a polynomial with $\mathbb{Q}$-coefficients
depending on the variables $t_{ij}$, with $j=1,\dots,k$, $i\geq 0$ and
$u_{l}$, $l\geq 1$.

Using Markman’s operator, Oberdieck proved a universality property for
integrals over $M$; this roughly means that integral
$\int_{M}P(B_{i}(x_{j}),c_{l}(T_{M}))$ only depends on the polynomial $P$, the
dimension of $M$ and pairings $(v,x_{i})$, $(x_{i},x_{j})$ for all $i,j$ i.e.
the intersection matrix

$\displaystyle\begin{pmatrix}(v,v)&(v,x_{i})_{i=1}^{k}\\\
(x_{i},v)_{i=1}^{k}&(x_{i},x_{j})_{i,j=1}^{k}\end{pmatrix}.$

I will now explain this in detail. In fact, given the data $(M(v),x_{i})$ as
above, the Lemma 2.11 of [8] shows that there exists
$y_{i}\in\Lambda_{\mathbb{C}}$ which have the same above intersection matrix,
and satisfy

$\int_{M}P(B_{i}(x_{j}),c_{l}(T_{M}))=\int_{M}P(B_{i}(y_{j}),c_{l}(T_{M}))$

and $\operatorname{Span}(v,y_{1},\dots,y_{k})$ is a non-degenerate subspace of
$\Lambda_{\mathbb{C}}$ (i.e. the restriction of the inner product of
$\Lambda_{\mathbb{C}}$, induced by the Mukai pairing, onto the subspace is
non-degenerate). The condition $\operatorname{dim}M>2$ is used to obtain the
Lemma 2.11 of [8], therefore if one always assume $\operatorname{dim}M>2$ then
given two pairs $(M(v),x_{i})$ and $(M(v^{\prime}),x_{i}^{\prime})$ with the
same intersection matrix, one may always assume that
$\operatorname{Span}(v,x_{1},\dots,x_{k})$ and
$\operatorname{Span}(v^{\prime},x_{1}^{\prime},\dots,x_{k}^{\prime})$ are non-
degenerate.

Then, one can use the following lemma:

######  Lemma 3.3 (Lemma 2.13 of [8] ).

Let $V$ be a finite-dimensional $\mathbb{C}$-vector space with a
$\mathbb{C}$-linear inner product. Let $v_{1},\dots,v_{k}\in V$ and
$w_{1},\dots,w_{k}\in V$ be lists of vectors such that

  1. $\mathrm{(i)}$

$\operatorname{Span}(v_{1},\dots,v_{k})$ is non-degenerate,

  2. $\mathrm{(ii)}$

$\operatorname{Span}(w_{1},\dots,w_{k})$ is non-degenerate,

  3. $\mathrm{(iii)}$

$\langle v_{i},v_{j}\rangle=\langle w_{i},w_{j}\rangle$ for all $i,j$.

Then there exists an isometry $\phi:V\to V$ such that $\phi(v_{i})=w_{i}$ for
all $i$.

Since I always assume that $\operatorname{Span}(v,x_{1},\dots,x_{k})$ and
$\operatorname{Span}(v^{\prime},x_{1}^{\prime},\dots,x_{k}^{\prime})$ are non-
degenerate, by above lemma, there exists an isometry
$g:H^{\ast}(S,\mathbb{C})\to H^{\ast}(S^{\prime},\mathbb{C})$ taking
$(v,x_{1},\dots,x_{k})$ to
$(v^{\prime}=gv,x_{1}^{\prime}=gx_{1},\dots,x_{k}^{\prime}=gx_{k})$. Therefore
by the properties of the Markman operator and morphism $B$ one has:

$\displaystyle\int_{M(v)}P(B_{i}(x_{j}),c_{l}(T_{M(v)}))$
$\displaystyle=\int_{M(v^{\prime})}\gamma(g)P(B_{i}(x_{j}),c_{l}(T_{M(v^{\prime})}))$
$\displaystyle=\int_{M(v^{\prime})}P(B_{i}(gx_{j}),c_{l}(T_{M(v^{\prime})}))$
$\displaystyle=\int_{M(v^{\prime})}P(B_{i}(x_{j}^{\prime}),c_{l}(T_{M(v^{\prime})})),$

where the first equality is by the Theorem 3.1 and the second equality is by
the Lemma 3.2. This leads to the following theorem:

######  Theorem 3.4 (Oberdieck [8]).

let $P(t_{ij},u_{r})$ be a polynomial depending on the variables $t_{ij}$,
$j=1,...,k,$ $i\geq 0$, and $u_{l}$, $l\geq 1$. Let also
$A=(a_{ij})^{k}_{i,j=0}$ be a $(k+1)\times(k+1)$-matrix. Then there exists
$I(P,A)\in\mathbb{Q}$ ($I(P,A)$ is a rational number only depending on $P,A$)
such that for any $M=M_{H}(v)$ with $dim(M)>2$ and for
$x_{1},...,x_{k}\in\Lambda$ also with $v\in\Lambda$ as above.

$\begin{pmatrix}(v,v)&(v,x_{i})_{i=1}^{k}\\\
(x_{i},v)_{i=1}^{k}&(x_{i},x_{j})_{i,j=1}^{k}\end{pmatrix}=A,$

such that

$\int_{M}P(B_{i}(x_{j}),c_{l}(T_{M}))=I(P,A).$

##  4\. Positive rank Virasoro constraints

###  4.1. $\operatorname{dim}M>2$ case

As in Section 2, let $S$ be a non-singular projective K3 surface, let
$v=(r,D,n)\in\Lambda$ be a Mukai vector with $r=\operatorname{rk}(v)>0$.
Assume $\operatorname{dim}(M_{H}(v))>2$. I would like to use the universality
result the Theorem 3.4 introduced in the previous section to prove the
Virasoro constraints for positive rank on $M_{H}(v)$. Recall that, for all
$\gamma\in H^{\ast}(S,\mathbb{C})$ the symbol
$\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$ is mapped to $H^{\ast}(M)$ by:

$\operatorname{ch}^{\mathrm{H}}_{\bullet}(\gamma)=\pi_{M\ast}\left(\operatorname{ch}\left(\mathcal{F}\otimes\det{(\mathcal{F})}^{-1/\text{rk}(v)}\right)\pi_{S}^{\ast}(\gamma)\right),\forall\gamma\in
H^{\ast}(S,\mathbb{C}).$

We now use following brief calculation to express this in terms of
$B(\bullet)$.

(10)
$\displaystyle\begin{split}&\pi_{M\ast}(\operatorname{ch}(\mathcal{F}\otimes\det{(\mathcal{F})}^{-1/\text{rk}(v)})\pi_{S}^{\ast}(\gamma))\\\
&=\pi_{M\ast}(\operatorname{ch}(\mathcal{F})\exp{\left(-\frac{c_{1}(\mathcal{F})}{\text{rk}(v)}\right)\pi_{S}^{\ast}(\gamma)})\\\
&=\pi_{M\ast}\left(\operatorname{ch}(\mathcal{F})\exp{\left(-\frac{\theta_{\mathcal{F}}(\boldsymbol{\operatorname{p}})+c_{1}(v)}{\text{rk}(v)}\right)}\pi_{S}^{\ast}(\gamma)\right)\\\
&=\exp{\left(\frac{\theta_{\mathcal{F}}(\boldsymbol{\operatorname{p}})}{(\boldsymbol{\operatorname{p}},v)}-\frac{\theta_{\mathcal{F}}(v)}{(v,v)}\right)}\exp{\left(\frac{\theta_{\mathcal{F}}(v)}{(v,v)}\right)}\pi_{M\ast}\left(\operatorname{ch}(\mathcal{F})\pi_{S}^{\ast}\left(\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)}\gamma^{\vee}\sqrt{\text{td}_{S}}^{-1}\right)^{\vee}\sqrt{\text{td}_{S}}\right)\\\
&=\exp{\left(B_{1}\left(\frac{\boldsymbol{\operatorname{p}}}{(\boldsymbol{\operatorname{p}},v)}-\frac{v}{(v,v)}\right)\right)B\left(\left(\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)}\gamma^{\vee}\sqrt{\text{td}_{S}}^{-1}\right)\right)},\end{split}$

where the notation (3) is used, also notice that $\operatorname{dim}(M)>2$
implies $(v,v)>0$. The second equality used the fact that, by Künneth
decomposition,
$c_{1}(\mathcal{F})=\pi_{M}^{\ast}(\theta_{\mathcal{F}}(\boldsymbol{\operatorname{p}}))+\pi_{S}^{\ast}(c_{1}(v))$,
the last line used the fact that $B_{1}(x)=\theta_{\mathcal{F}}(x)$ for $x\in
v^{\perp}$ (see [8]).

Therefore $\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$ for $\gamma\in
H^{p,q}(S)$ is the degree $2i-p+q$ component of

(11)
$\displaystyle\exp{\left(B_{1}\left(\frac{\boldsymbol{\operatorname{p}}}{(\boldsymbol{\operatorname{p}},v)}-\frac{v}{(v,v)}\right)\right)B\left(\left(\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)}\gamma^{\vee}\sqrt{\text{td}_{S}}^{-1}\right)\right)}.$

Consider the evaluation of the integral over $M(v)$ of a polynomial of
$\operatorname{ch}_{i}^{\mathrm{H}}(\gamma)$’s. By Theorem 3.4, one only need
to look at the intersection matrix of the classes appearing in the arguments
of $B$ and $B_{1}$ in (11).

Therefore, for $L\in H^{2}(S,\mathbb{Q})$, one needs to keep track of

$L\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},\quad\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)},\quad
v.$

For $\boldsymbol{\operatorname{p}}$, one needs to keep track of

$\boldsymbol{\operatorname{p}},\quad\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)},\quad
v.$

For $\mathbbm{1}$, one needs to keep track of

$\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)}\sqrt{\text{td}_{S}}^{-1},\quad\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)},\quad
v.$

Recall that for K3 surface, I have
$\sqrt{\operatorname{td}_{S}}=1+\boldsymbol{\operatorname{p}}$. Now, also
consider the descendent of the form
$\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$.
$\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\boldsymbol{\operatorname{p}}})$
is the degree $2k$ component of the following class

$\left[\pi_{M\ast}\left(\operatorname{ch}(\mathcal{F}\otimes\det{(\mathcal{F})}^{-1/\text{rk}(v)})\pi_{S}^{\ast}(\mathbbm{1}+\boldsymbol{\boldsymbol{\operatorname{p}}})\right)\right]_{k},$

which equals
$\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1})+\operatorname{ch}^{\mathrm{H}}_{k}(\boldsymbol{\boldsymbol{\operatorname{p}}})$.
Using the fact
$(\mathbbm{1}+\boldsymbol{\operatorname{p}})\sqrt{\mathrm{td_{S}}}^{-1}=\mathbbm{1}$;
for
$\operatorname{ch}_{k}(\mathbbm{1}+\boldsymbol{\boldsymbol{\operatorname{p}}})$
one needs to track

$\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},\quad\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)},\quad
v.$

One can choose the generators of $\mathbb{D}^{S}$ as

(12)
$\displaystyle\\{\operatorname{ch}_{j}^{\mathrm{H}}(L_{i}),\operatorname{ch}^{\mathrm{H}}_{k}(\boldsymbol{\operatorname{p}}),\operatorname{ch}_{l}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})|j,k,l\in\mathbb{N},\\{L_{i}\\}_{i}\text{
forms a basis of }H^{2}(S,\mathbb{C})\\}.$

Using this set of generators, one only needs to keep track of the intersection
matrix of the following elements to evaluate descendent integrals.

(13) $\displaystyle
L_{i}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},\quad\text{with
}L_{i}\in
H^{2}(S,\mathbb{C});\quad\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)};\quad\boldsymbol{\operatorname{p}};\quad\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)};\quad
v.$

One can calculate Mukai pairings between the above classes, the only
interesting pairings are:

$\displaystyle(v,L_{i}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=0$
$\displaystyle(\boldsymbol{\operatorname{p}}/\text{rk}(v),L_{i}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=0$
$\displaystyle(L_{i}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},L_{j}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=L_{i}L_{j}$
$\displaystyle(\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},L_{i}\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=0$
$\displaystyle(v,\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=\frac{D^{2}}{2r}-n,$
$\displaystyle(\frac{\boldsymbol{\operatorname{p}}}{\text{rk}(v)},\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=-\frac{1}{r}$
$\displaystyle(\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)},\exp{\left(\frac{c_{1}(v)}{\text{rk}(v)}\right)})$
$\displaystyle=0$ $\displaystyle(\boldsymbol{\operatorname{p}},v)$
$\displaystyle=-r$

Consider the following integral:

$\int_{M(v)}P\left(\operatorname{ch}_{i}(L_{j}),\operatorname{ch}_{l}(\boldsymbol{\operatorname{p}}),\operatorname{ch}_{k}(\mathbbm{1}+\boldsymbol{\boldsymbol{\operatorname{p}}})\right),$

where $P$ is a polynomial, $L_{j}\in H^{2}(S,\mathbb{Q})$ and $v=(r,D,n)$.
This integral is defined for general $v\in\Lambda$ that satisfies assumptions
of Section 2, one can now take a specific value by setting
$v=\mathbbm{1}-(n-1)\boldsymbol{\operatorname{p}}$, which leads to the
following

######  Lemma 4.1.

Let $P$ be a polynomial with variables in (12), and $v=(r,D,n)$

(14)
$\displaystyle\begin{split}\int_{M(v)}&P\left(\operatorname{ch}^{\mathrm{H}}_{i}(L_{j}),\operatorname{ch}^{\mathrm{H}}_{l}(\boldsymbol{\mathrm{p}}),\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\mathrm{p}})\right)\\\
&=\int_{S^{[N]}}P\left(\operatorname{ch}^{\mathrm{H}}_{i}(L_{j}),r\operatorname{ch}^{\mathrm{H}}_{l}(\boldsymbol{\mathrm{p}}),\frac{1}{r}\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\mathrm{p}})\right),\end{split}$

where $2N-2=(v,v)=D^{2}-2rn$.

###### Proof.

We now consider the RHS of (4.1). By setting
$v=\mathbbm{1}-(N-1)\boldsymbol{\operatorname{p}}$, I have:

$\displaystyle\begin{split}(\mathbbm{1}-(N-1)\boldsymbol{\operatorname{p}},\mathbbm{1}-(N-1)\boldsymbol{\operatorname{p}})=2N-2=(v,v)=D^{2}-2rn.\end{split}$

The classes in (13) become (up to multiplication of $1/r$ or $r$):

(15) $\displaystyle L_{i},\quad\text{for }L_{i}\in
H^{2}(S);\quad\frac{\mathbbm{1}}{r};\quad
r\boldsymbol{\operatorname{p}};\quad\boldsymbol{\operatorname{p}};\quad\mathbbm{1}-(N-1)\boldsymbol{\operatorname{p}}.$

Notice that $\mathbbm{1}$ is multiplied with $1/r$ and
$\boldsymbol{\operatorname{p}}$ is multiplied with $r$, this is harmless since
$r>0$. One can calculate Mukai pairings among these classes, the only
interesting pairings are the following:

$\displaystyle(L_{i},L_{j})$ $\displaystyle=L_{i}L_{j}$
$\displaystyle(v,\frac{\mathbbm{1}}{r})$
$\displaystyle=\frac{1}{r}(N-1)=\frac{1}{r}(\frac{1}{2}D^{2}-rn)$
$\displaystyle(\boldsymbol{\operatorname{p}},\frac{\mathbbm{1}}{r})$
$\displaystyle=-\frac{1}{r}$ $\displaystyle(\boldsymbol{\operatorname{p}},v)$
$\displaystyle=-r.$

By comparing with the pairings of (13) one can see that (13) and (15) have the
same pairing matrix.

By the last line of (LABEL:calculation4.1), the class
$\boldsymbol{\operatorname{p}}$ in (13) comes from the $B_{1}$ part of
$\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})$, it is
multiplied by $r$ in (15), therefore it comes from the $B_{1}$ part of
$r\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})$;
analogously, $\mathbbm{1}/r$ in (15) comes from
$1/r\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$.
Therefore, the classes in (15) are the arguments of $B$ and $B_{1}$ function
of

$\\{\operatorname{ch}_{j}^{\mathrm{H}}(L_{i}),r\operatorname{ch}^{\mathrm{H}}_{k}(\boldsymbol{\operatorname{p}}),\frac{1}{r}\operatorname{ch}_{l}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})|j,k,l\in\mathbb{N},\\{L_{i}\\}_{i}\text{
forms a basis of }H^{2}(S,\mathbb{C})\\}.$

Therefore by Theorem 3.4, the lemma is true. ∎

In the following paragraphs, I will use the above lemma to prove the Virasoro
constraints on $M(v)$ in the positive rank case. Without of lose of generality
one can always assume $D\in\mathbb{D}^{S}$ is a monomial and $D$ does not
contain $\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})$’s, since one can
always use the relation

(16)
$\displaystyle\operatorname{ch}_{k}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})=\operatorname{ch}_{k}^{\mathrm{H}}(\mathbbm{1})+\operatorname{ch}_{k}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})$

to rewrite $D$. Then by the Lemma 4.1, one has

$\displaystyle\int_{M(v)}R_{k}D=r^{N_{p}}\int_{S^{[N]}}R_{k}D,$

where $N_{p}$ is the number of
$\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})$ minus the
number of
$\mathrm{ch}^{\mathrm{H}}_{j}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$ in
$D$. Next, I claim that

(17) $\displaystyle\int_{M(v)}T_{k}D=r^{N_{p}}\int_{S^{[N]}}T_{k}D.$

For $T_{k}$, the terms which are of interest are the terms containing
$\operatorname{ch}_{i}(\mathbbm{1})$ or
$\operatorname{ch}_{i}(\boldsymbol{\operatorname{p}})$, in $T_{k}$ those terms
are:

$\displaystyle\sum_{a+b=k}\left(a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})+a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\mathbbm{1})\right).$

and

$\displaystyle\sum_{a+b=k}a!b!\operatorname{ch}^{\mathrm{H}}_{a}\operatorname{ch}^{\mathrm{H}}_{b}\left(2\boldsymbol{\operatorname{p}}\right)$

All the contributions from $H^{2}(S,\mathbb{Q})\otimes H^{2}(S,\mathbb{Q})$
cause no problem, since they only contains descendants of the form
$\operatorname{ch}^{\mathrm{H}}_{i}(L)$ with $i\in\mathbb{N},L\in
H^{2}(S,\mathbb{Q})$ and therefore by lemma 4.1, their contributions remain
the same.

For the first term, one can use the relation (16) to rewrite it as:

(18) $\displaystyle\sum_{a+b=k}$
$\displaystyle\bigg{(}a!b!\bigg{[}\operatorname{ch}_{a}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})-\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\bigg{]}\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})$
(19)
$\displaystyle\quad\quad\quad\quad+a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\bigg{[}\operatorname{ch}^{\mathrm{H}}_{b}(\mathbbm{1}+\boldsymbol{\operatorname{p}})-\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})\bigg{]}\bigg{)}.$

Using lemma 4.1 to pass from integral over $M(v)$ to $S^{[N]}$, (18) becomes:

(20) $\displaystyle\sum_{a+b=k}$
$\displaystyle\bigg{(}a!b!\bigg{[}\frac{1}{r}\operatorname{ch}_{a}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})-r\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\bigg{]}r\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})$
(21)
$\displaystyle\quad\quad\quad\quad+a!b!r\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\bigg{[}\frac{1}{r}\operatorname{ch}^{\mathrm{H}}_{b}(\mathbbm{1}+\boldsymbol{\operatorname{p}})-r\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})\bigg{]}\bigg{)}$
(22)
$\displaystyle=\sum_{a+b=k}\bigg{(}2a!b!(1-r^{2})\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})\bigg{)}$
(23)
$\displaystyle\quad+\sum_{a+b=k}\bigg{(}a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})+a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\mathbbm{1})\bigg{)}.$

Next, consider the term
$\sum_{a+b=k}a!b!\operatorname{ch}^{\mathrm{H}}_{a}\operatorname{ch}^{\mathrm{H}}_{b}\left(2\boldsymbol{\operatorname{p}}\right)$,
when passing from integral over $M(v)$ to $S^{[N]}$ using lemma 4.1, this term
becomes:

(24) $\displaystyle\sum_{a+b=k}2a!b!r^{2}$
$\displaystyle\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})$
(25)
$\displaystyle=\sum_{a+b=k}2a!b!\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})+\sum_{a+b=k}2a!b!(r^{2}-1)\operatorname{ch}^{\mathrm{H}}_{a}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{b}(\boldsymbol{\operatorname{p}})$

Therefore (20), (24) and the fact that other künneth component cause no
problems imply (17).

Now, let us look at the $S_{k}$ term. The classes in $H^{0,q}(S)$ are
$\mathbbm{1}\in H^{0,0}(S)$ and $\sigma\in H^{0,2}(S)$. There is only one such
$\sigma$ up to scaling, since $\text{dim }H^{0,2}(S)=1$. Let us first look at
the term

$-\frac{1}{r}(k+1)!R_{-1}[\mathbbm{1}]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})D\right).$

Evidently, when passing from integral over $M(v)$ to $S^{[N]}$, this term
changes to

$-r^{N_{p}}(k+1)!R_{-1}[\mathbbm{1}]\left(\operatorname{ch}^{\mathrm{H}}_{k+1}(\boldsymbol{\operatorname{p}})D\right).$

Now consider the term

(26)
$\displaystyle-\frac{1}{r}(k+1)!R_{-1}[\sigma]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})D\right),$

where $\overline{\sigma}\cdot\sigma=\boldsymbol{\operatorname{p}}$. Notice
that for $\delta\in H^{\ast}(S)$, $\sigma\cdot\delta\neq
0\implies\delta=\overline{\sigma}\text{ or }\mathbbm{1}$ or a linear
combination of those. Because we assumed that
$\operatorname{ch}_{i}(\mathbbm{1})$ does not appear in $D$, the terms that
survive in (26) are terms produced when $R_{-1}[\sigma]$ acts on different
$\operatorname{ch}_{i}^{\mathrm{H}}(\overline{\sigma})$’s or on different
$\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$’s.
In the first case, it will create an additional
$\operatorname{ch}_{i-1}(\boldsymbol{\operatorname{p}})$; in the second case,
it will destroy an
$\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$,
in either case it will produce a factor $r$. Thus when passing from integral
over $M(v)$ to $S^{[N]}$, (26) changes to

$-r^{N_{p}}(k+1)!R_{-1}[\sigma]\operatorname{ch}^{\mathrm{H}}_{k+1}(\overline{\sigma})D.$

Therefore we find:

$\displaystyle\int_{M(v)}S_{k}D=r^{N_{p}}\int_{S^{[N]}}S_{k}D.$

Because the Virasoro constraints on $S^{[N]}$ are proven [6], the Virasoro
constraints on $M(v)$ are also valid. This proves Theorem 2.2 for
$\operatorname{dim}M>2$.

###  4.2. $\operatorname{dim}M=2$ case

As before, let $S$ be a non-singular projective K3 surface, let $v$ be a Mukai
vector with $\operatorname{rk}(v)>0$. In this case, I will recall some facts
following Section 2.4 of [8]. Let $M:=M_{H}(v)$ be a 2-dimensional moduli
space of stable sheaves, hence it is a K3 surface. There is the following well
known isometry of Mukai lattices:

(27) $\displaystyle\widetilde{\Phi}:H^{\ast}(S,\mathbb{Q})\to
H^{\ast}(M,\mathbb{Q}),\quad\gamma\mapsto\pi_{M\ast}(e^{-c_{1}(\mathcal{F})/rk(v)}v(F)\pi_{S}^{\ast}(\gamma)),$

where
$v(\mathcal{F}):=\operatorname{ch}(\mathcal{F})\sqrt{\operatorname{td}_{M\times
S}}.$ The fact that $\widetilde{\Phi}$ is a Hodge isometry implies:

$\displaystyle\widetilde{\Phi}(\boldsymbol{\operatorname{p}})=\mathrm{rk}(v)\mathbbm{1},\quad\widetilde{\Phi}(L)=\phi(L),\quad\widetilde{\Phi}(\mathbbm{1})=\frac{1}{\mathrm{rk}(v)}\boldsymbol{\operatorname{p}},$

where $\phi:H^{2}(S,\mathbb{Q})\to H^{2}(M,\mathbb{Q})$ is a Hodge isometry,
which means $\phi$ is an isometry with respect to the Mukai paring of $S$ and
$M$, since $M$ is also a K3 surface.

One has the following identities by direct calculation:

(28) $\displaystyle\operatorname{ch}_{i}^{\mathrm{H}}(\gamma)$
$\displaystyle=\bigg{[}\frac{1}{\sqrt{\operatorname{td}_{M}}}\widetilde{\Phi}\left(\frac{\gamma}{\sqrt{\operatorname{td}_{S}}}\right)\bigg{]}_{2i-p+q}\text{for
}\gamma\in H^{p,q}(S).$ (29) $\displaystyle\text{ch}_{i}(\gamma)$
$\displaystyle=\bigg{[}\frac{1}{\sqrt{\operatorname{td}_{M}}}\widetilde{\Phi}\left(\frac{\gamma}{\sqrt{\operatorname{td}_{S}}}\right)\bigg{]}_{2\operatorname{deg}(\gamma)+2i-4}\text{for
}\gamma\in H^{\ast}(S,\mathbb{Q}).$

Here the new symbole $\operatorname{ch}_{k}(\gamma)$ has its geometric
realization as:

$\operatorname{ch}_{k}(\gamma)=\pi_{M\ast}\left(\operatorname{ch}_{k}(\mathcal{F}\otimes(\operatorname{det}\mathcal{F})^{-1/r})\cdot\pi_{S}^{\ast}\gamma\right).$

For $\gamma\in H^{p,q}(S)$ one has:

(30)
$\displaystyle\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)=\operatorname{ch}_{i+1+\frac{q-p}{2}}(\gamma).$

Similarly as before:

$\displaystyle\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})=\bigg{[}\frac{1}{\sqrt{\operatorname{td}_{M}}}\widetilde{\Phi}\left(\frac{\mathbbm{1}+\boldsymbol{\operatorname{p}}}{\sqrt{\operatorname{td}_{S}}}\right)\bigg{]}_{2i}=\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1})+\operatorname{ch}_{i}^{\mathrm{H}}(\boldsymbol{\operatorname{p}}).$

Because $\text{dim }M=2$, one can explicitly calculate the cohomology class of
$\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$. The non zero ones among them
are:

(31)
$\displaystyle\begin{split}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})&=\text{rk}(v)\mathbbm{1},\quad\\\
\operatorname{ch}_{2}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})&=-\text{rk}(v)\boldsymbol{\operatorname{p}},\\\
\operatorname{ch}_{2}(\gamma)&=\phi(\gamma),\quad\text{where }\gamma\in
H^{2}(S)\\\
\operatorname{ch}_{2}^{\mathrm{H}}(\mathbbm{1})&=-\text{rk}(v)\mathbbm{1}\\\
\operatorname{ch}_{4}^{\mathrm{H}}(\mathbbm{1})&=\left(\frac{1}{\text{rk}(v)}+\text{rk}(v)\right)\boldsymbol{\operatorname{p}}\\\
\operatorname{ch}^{\mathrm{H}}_{2}(\mathbbm{1}+\boldsymbol{\operatorname{p}})&=\frac{1}{\text{rk}(v)}\boldsymbol{\operatorname{p}}.\end{split}$

In fact, the geometric realization of
$\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})$,
$\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})$ are the same as
$\operatorname{ch}_{i}(\boldsymbol{\operatorname{p}})$,
$\operatorname{ch}_{i}(\mathbbm{1})$. For $\gamma\in H^{2}(S,\mathbb{C})$,
using $\operatorname{ch}_{2}(\gamma)$ will make notation simpler since only
$\operatorname{ch}_{2}(\gamma)\neq 0$ whereas there are potentially several
$i$ such that $\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)\neq 0$.

######  Lemma 4.2.

Let $S$ be a non-singular and projective K3 surface, and let $v,M_{H}(v)$ be
defined as above. Suppose $M_{H}(v)$ is 2 dimensional, and suppose
$r:=\operatorname{rk}(v)>0$. Then

$\int_{M}P\left(\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}}),\operatorname{ch}^{\mathrm{H}}_{j}(\gamma_{l}),\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\operatorname{p}})\right)=\int_{S^{[1]}}P\left(r\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}}),\operatorname{ch}^{\mathrm{H}}_{j}(\gamma_{l}),\frac{1}{r}\operatorname{ch}^{\mathrm{H}}_{k}(\mathbbm{1}+\boldsymbol{\operatorname{p}})\right),$

where $P$ is a polynomial and $\gamma_{l}\in H^{2}(S,\mathbb{Q})$.

One can use (30) to rewrite $\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$ as
$\operatorname{ch}_{2}(\gamma)$. The only monomials whose integral on $M$ are
non zero have one of the following three forms (up to multiplication by a
constant):

(32)
$\displaystyle\big{[}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}(\gamma_{1})\operatorname{ch}_{2}(\gamma_{2})$
(33)
$\displaystyle\big{[}\operatorname{ch}^{\mathrm{H}}_{0}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})$
(34)
$\displaystyle\big{[}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}^{\mathrm{H}}_{2}(\mathbbm{1}+\boldsymbol{\operatorname{p}}),$

where $N$ is some non-negative integer and $\gamma_{1},\gamma_{2}\in
H^{2}(S,\mathbb{C})$. Their integral over $M$ are:

(35)
$\displaystyle\int_{M}\big{[}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}(\gamma_{1})\operatorname{ch}_{2}(\gamma_{2})$
$\displaystyle=r^{N}\cdot(\phi(\gamma_{1}),\phi(\gamma_{2}))$ (36)
$\displaystyle=r^{N}\cdot(\gamma_{1},\gamma_{2})$ (37)
$\displaystyle\int_{M}\big{[}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})$
$\displaystyle=-r^{N+1}$ (38)
$\displaystyle\int_{M}\big{[}\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$
$\displaystyle=r^{N-1}.$

One can also evaluate integrals on $S^{[1]}$:

(39)
$\displaystyle\int_{S^{[1]}}\big{[}r\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\operatorname{ch}_{2}(\gamma_{1})\operatorname{ch}_{2}(\gamma_{2})$
$\displaystyle=r^{N}\cdot(\phi(\gamma_{1}),\phi(\gamma_{2}))$ (40)
$\displaystyle=r^{N}\cdot(\gamma_{1},\gamma_{2})$ (41)
$\displaystyle\int_{S^{[1]}}\big{[}r\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\big{[}r\operatorname{ch}_{2}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}$
$\displaystyle=-r^{N+1}$ (42)
$\displaystyle\int_{S^{[1]}}\big{[}r\operatorname{ch}_{0}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\big{]}^{N}\cdot\frac{1}{r}\operatorname{ch}^{\mathrm{H}}_{2}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$
$\displaystyle=r^{N-1}.$

By comparing the two evaluations, lemma 4.2 is proven. Therefore one can use
the same argument as in subsection 4.1 to show the Virasoro constraints are
valid in case of $\text{dim }M=2$. Combined with the result of previous
subsection, Theorem 2.2 is proven.

##  5\. Rank 0 case

The Virasoro operator in the section 4 only works for the strictly positive
rank case, since the definition of $S_{k}$ has the constant $\frac{1}{r}$. To
solve this, one needs the concept of $\delta$-normalized universal sheaf [1]
which will be presented in the following subsection.

###  5.1. $\delta$-normalized universal sheaf and the invariant Virasoro
operator

Let me recall the notion of $\delta$-normalized universal sheaf in [1]. Let
$v\in\Lambda$ $M:=M_{H}(v)$ and $\pi_{M},\pi_{S}$ being defined as above. And
I assume that there exists a universal sheaf on $M\times S$. Let $\alpha\in
K_{0}(X)_{\mathbb{Q}}$ be the topological type of sheaves that $M$
parameterizes, i.e. one has
$\operatorname{ch}(\alpha)\cdot\sqrt{\text{td}_{S}}=v$. Suppose
$\delta\in\mathrm{H}^{\bullet}(S,\mathbb{Z})$ is an algebraic class such that
$\int_{S}\delta\cdot\operatorname{ch}(\alpha)\neq 0$. We say a universal sheaf
$\mathcal{G}$ is $\delta$-normalized if

$\xi_{\mathcal{G}}(\operatorname{ch}_{1}^{\mathrm{H}}(\delta))=0.$

By the remark 2.14 of [1], the $\delta$-normalized universal sheaf always
exists and is unique as an element of the rational K-theory of $M\times X$.
Let $\mathcal{F}$ be any universal sheaf, its unique $\delta$-normalized
universal sheaf is:

(43) $\displaystyle\mathcal{F}_{\delta}=\mathcal{F}\otimes
e^{-\xi_{\mathcal{F}}(\operatorname{ch}^{\mathrm{H}}_{1}(\delta))/\int_{S}\delta\cdot\operatorname{ch}(\alpha)}.$

Also by the same remark, $\xi_{\mathbb{G}_{\delta}}=\xi_{\mathbb{G}}\circ\eta$
where $\eta:\mathbb{D}^{S}\to\mathbb{D}^{S}$ is defined as:

$\eta=\sum_{j\geq
0}\left(-\frac{\operatorname{ch}^{\mathrm{H}}_{1}(\delta)}{\int_{S}\delta\cdot\operatorname{ch}(\alpha)}\right)^{j}R^{j}_{-1}.$

Therefore, the geometric realization with respect to $\delta$-normalized sheaf
is still valid even such a sheaf does not exist in usual sense. By direct
calculation one can check that
$(\mathcal{F}\otimes\mathcal{L})_{\delta}=\mathcal{F}_{\delta}$ for any line
bundle $\mathcal{L}$ on $M$. Since we want that the geometric realisation of
any $D\in\mathbb{D}^{S}$ is independent of choice of the universal sheaf,
$\delta$-normalised universal sheaf will be a good choice to formulate the
Virasoro constraint. For rank zero sheaves,
$\boldsymbol{\operatorname{p}}$-normalised universal sheaf is clearly not
possible, since
$\int_{S}\boldsymbol{\operatorname{p}}\cdot\operatorname{ch}(\alpha)=0$. A
good choice, in this case, would be a $Y$-normalised universal sheaf, for some
$Y\in H^{1,1}(S)$, which will be used in the next subsection.

Now, as in previous sections, let $v=(r,d,n)$ with $r>0$. For later use, let
me introduce the following notations:

(44) $\displaystyle L_{k}$ $\displaystyle:=R_{k}+T_{k}$ (45) $\displaystyle
S_{k,0}$
$\displaystyle:=-\frac{(k+1)!}{r}R_{-1}\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})$
(46) $\displaystyle S_{k,2}$
$\displaystyle:=-\frac{(k+1)!}{r}R_{-1}[\sigma]\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma}),$

Where $\sigma\in H^{0,2}(S)$ with $S$ a K3 surface and
$\sigma\cdot\overline{\sigma}=\boldsymbol{\operatorname{p}}$. As before, for a
$D\in\mathbb{D}^{S}$,

$S_{k,0}(D)=-\frac{(k+1)!}{r}R_{-1}(\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})D),$

similar for $S_{k,2}$. Also, one has $S_{k}=S_{k,0}+S_{k,2}$.

######  Proposition 5.1.

As before, let $\mathcal{L}_{k}=R_{k}+T_{k}+S_{k}$. The commutation relation
$[R_{-1},\mathcal{L}_{k}]=(k+1)\mathcal{L}_{k-1}$ is true.

###### Proof.

The relation $[R_{-1},L_{k}]=(k+1)L_{k-1}$ is already checked in [1]. Let us
calculate $[R_{-1},S_{k}]$, using
$\sum_{t}\delta_{t}^{L}\otimes\delta_{t}^{R}$ as the Künneth decomposition of
$\Delta_{\ast}\mathbbm{1}\in H^{\ast}(S\times S)$ :

$\displaystyle[R_{-1},S_{k}]D$ $\displaystyle=R_{-1}S_{k}D-S_{k}R_{-1}D$
$\displaystyle=-\frac{(k+1)!}{r}\sum_{\\{t|p^{L}_{t}=0\\}}R_{-1}[\delta^{L}_{t}]\bigg{(}\operatorname{ch}_{k}^{\mathrm{H}}(\delta^{R}_{t})D+\operatorname{ch}_{k+1}^{\mathrm{H}}(\delta^{R}_{t})R_{-1}D\bigg{)}$
$\displaystyle\quad\quad-S_{k}R_{-1}D$
$\displaystyle=-\frac{(k+1)!}{r}\sum_{\\{t|p^{L}_{t}=0\\}}R_{-1}[\delta^{L}_{t}]\bigg{(}\operatorname{ch}_{k}^{\mathrm{H}}(\delta^{R}_{t})D\bigg{)}$
$\displaystyle=(k+1)S_{k-1}D,$

where in the second line, I used that $R_{-1}$ commutes with $R_{-1}[\gamma]$.
Therefore I have $[R_{-1},\mathcal{L}_{k}]=(k+1)\mathcal{L}_{k-1}$. ∎

Define the following invariant Virasoro operator as in [1].

(47)
$\displaystyle\mathcal{L}_{\text{inv}}:=\sum_{j\geq-1}\frac{(-1)^{j}}{(j+1)!}\mathcal{L}_{j}R^{j+1}_{-1}$

Using the commutation relation

(48) $\displaystyle[R_{-1},\mathcal{L}_{k}]=(k+1)\mathcal{L}_{k-1}$

one has:

(49) $\displaystyle
R_{-1}\mathcal{L}_{\text{inv}}=\sum_{j\geq-1}\frac{(-1)^{j}}{(j+1)!}\mathcal{L}_{j}R^{j+2}_{-1}+\sum_{j\geq-1}\frac{(-1)^{j}}{(j+1)!}(j+1)\mathcal{L}_{j-1}R^{j+1}_{-1}=0.$

Therefore the geometric realization of
$\mathcal{L}_{\text{inv}}D,D\in\mathbb{D}^{S}$ does not depend on the choice
of universal sheaf, by lemma 2.8 of [1]. More precisely, this means the
following: given two universal sheaf $\mathcal{F}$ and
$\mathcal{F}^{\prime}=\mathcal{F}\otimes\pi_{M}^{\ast}L$ where $L$ is a line
bundle on $M$, then

$\xi_{\mathcal{F}}(\mathcal{L}_{\text{inv}}D)=\xi_{\mathcal{F}^{\prime}}(\mathcal{L}_{\text{inv}}D)\quad\forall
D\in\mathbb{D}^{S}.$

Therefore I could omit the geometric realization and write

$\int_{M}\mathcal{L}_{\text{inv}}D.$

In the positive rank case, I have used
$\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}$ in the geometric realization,
which is not a universal sheaf. One can adapt lemma 2.19 of [1] to show that
the Virasoro constraint is equivalent when using
$\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}$ or
$\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}\otimes\pi_{S}^{\ast}\Delta^{1/r}$,
where $\Delta\in\mathrm{Pic}(S)$ such that $c_{1}(\Delta)=c_{1}(v)$. Also one
can check that the latter sheaf is a
$\boldsymbol{\operatorname{p}}$-normalised universal sheaf.

######  Proposition 5.2.

$\int_{M}\xi_{\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}}\left(\mathcal{L}_{k}D\right)=0\quad\forall
k\geq-1,\forall D\in\mathbb{D}^{S}$
$\iff\int_{M}\xi_{\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}\otimes\pi_{S}^{\ast}\Delta^{1/r}}\left(\mathcal{L}_{k}D\right)=0\quad\forall
k\geq-1,\forall D\in\mathbb{D}^{S},$

where $\Delta\in\operatorname{Pic}(S)$ such that $c_{1}(\Delta)=c_{1}(v)$.
Also
$\mathcal{F}\otimes\det(\mathcal{F})^{-1/r}\otimes\pi_{S}^{\ast}\Delta^{1/r}$
is a $\boldsymbol{p}$-normalised sheaf.

###### Proof.

One need to show that in K3 case, the algebra isomorphism
$\operatorname{F}:\mathbb{D}^{S}\to\mathbb{D}^{S}$ in lemma 2.19 of [1]
satisfies
$\mathcal{L}_{k}\circ\operatorname{F}=\operatorname{F}\circ\mathcal{L}_{k}$
for all $k\geq-1$. More precisely: recall that $\operatorname{F}$ is defined
as
$\operatorname{F}\left(\operatorname{ch}_{i}^{\mathrm{H}}(\gamma)\right)=\operatorname{ch}_{i}^{\mathrm{H}}\left(e^{c_{1}(L)}\gamma\right)$
with $L\in\operatorname{Pic}(S)$, and It is already shown in [1] that
$L_{k}\circ F=F\circ L_{k}$ for $k\geq-1$. I need to show
$\operatorname{F}\circ S_{k}=S_{k}\circ\operatorname{F}$ in addition. This is
true since $\operatorname{F}$ commutes with $R_{-1}$, $R_{-1}[\gamma]$ and

(50)
$\displaystyle\begin{split}\operatorname{F}S_{k}D&=-\frac{(k+1)!}{r}\operatorname{F}\bigg{(}R_{-1}[\sigma]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})D\right)+R_{-1}\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})D\right)\bigg{)}\\\
&=-\frac{(k+1)!}{r}\bigg{(}R_{-1}[\sigma]\left(\operatorname{F}\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})D\right)+R_{-1}\left(\operatorname{F}\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})D\right)\bigg{)}\\\
&=-\frac{(k+1)!}{r}\bigg{(}R_{-1}[\sigma]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})\operatorname{F}D\right)+R_{-1}\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\boldsymbol{\operatorname{p}})\operatorname{F}D\right)\bigg{)}\\\
&=S_{k}\operatorname{F}D,\end{split}$

where $\sigma$ is the only class in $H^{0,2}(S)$ and
$e^{c_{1}(H)}\cdot\overline{\sigma}=\overline{\sigma}$ is used. ∎

Consider Mukai vector $v=(n,d,0)$ with $n,d\neq 0$. I want to find a Virasoro
operators $\mathcal{L}^{Y}_{k}=R_{k}+T_{k}+S^{Y}_{k}$ which satisfy the
Virasoro constraints when using geometric realization with a $Y$-normalised
universal sheaf $\mathcal{F}_{Y}$, where $Y\in H^{1,1}(S)$. This means I want
to find Virasoro operators $\mathcal{L}^{Y}_{k}$ which satisfy:

(51)
$\displaystyle\int_{M(n,d,0)}\xi_{\mathcal{F}_{Y}}(\mathcal{L}^{Y}_{k}D)=0,\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1.$

Let me define the $S^{Y}_{k}$ operator as follows:

(52)
$\displaystyle\begin{split}S^{Y}_{k,0}&:=-\frac{(k+1)!}{\int_{S}Y\cdot\operatorname{ch}(\alpha)}R_{-1}\operatorname{ch}_{k+1}^{\mathrm{H}}(Y)\\\
S^{Y}_{k,2}&:=-\frac{(k+1)!}{n}R_{-1}[\sigma]\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})\\\
S^{Y}_{k}&:=S^{Y}_{k,0}+S^{Y}_{k,2},\end{split}$

where $\sigma\in H^{0,2}(X)$,
$\operatorname{ch}(\alpha)\cdot\sqrt{\text{td}_{S}}=v$ and
$\overline{\sigma}\sigma=\boldsymbol{\operatorname{p}}$. It turns out that
this is the correct definition of $S^{Y}_{k}$ in order for the Virasoro
operators to satisfy the desired property (51). I will prove this statement in
the following paragraphs.

######  Proposition 5.3.

Let $v=(n,d,r)$ with $n,d\neq 0$ be a Mukai vector, and let $Y\in H^{1,1}(S)$,
then:

$\int_{M(v)}\xi_{\mathcal{F}_{Y}}(\mathcal{L}^{Y}_{k}D)=0\text{ for any
$k\geq-1$, $D\in\mathbb{D}^{S}$}$
$\Longleftrightarrow\int_{M(v)}\mathcal{L}_{\mathrm{inv}}D=0.$

To prove this proposition, I need the following

######  Claim 5.4.

$\mathcal{L}^{Y}_{k}(JD)=J\mathcal{L}^{Y}_{k}D$ where
$J=\operatorname{ch}_{1}^{\mathrm{H}}(Y)\in\mathbb{D}^{S}$.

###### proof of claim.

(53) $\displaystyle R_{k}(JD)$
$\displaystyle=(k+1)!\operatorname{ch}_{k+1}^{\mathrm{H}}(Y)D+JR_{k}D$ (54)
$\displaystyle T_{k}(JD)$ $\displaystyle=JT_{k}D$ (55) $\displaystyle
S^{Y}_{k,0}(JD)$
$\displaystyle=-\frac{(k+1)!}{\int_{S}Y\cdot\operatorname{ch}(\alpha)}R_{-1}\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(Y)JD\right)$
(56)
$\displaystyle=JS^{Y}_{k,0}(D)-\frac{(k+1)!}{\int_{S}Y\cdot\operatorname{ch}(\alpha)}\operatorname{ch}_{k+1}^{\mathrm{H}}(Y)\operatorname{ch}_{0}^{\mathrm{H}}(Y)D$
(57)
$\displaystyle=JS^{Y}_{k,0}(D)-(k+1)!\operatorname{ch}_{k+1}^{\mathrm{H}}(Y)D$
(58) $\displaystyle S^{Y}_{k,2}(JD)$
$\displaystyle=-\frac{(k+1)!}{n}R_{-1}[\sigma]\left(\operatorname{ch}_{k+1}^{\mathrm{H}}(\overline{\sigma})JD\right)$
(59) $\displaystyle=JS^{Y}_{k,2}(D)\text{ since $R_{-1}[\gamma]J=0$}.$

Summing all terms gives the claim. ∎

###### Proof of proposition.

This proof of the Proposition 5.3 adapts the proof of the Proposition 2.16 of
[1]. To be more complete, I write the entire proof. First note:

(60)
$\displaystyle\begin{split}\sum_{j\geqslant-1}&\frac{(-1)^{j}}{(j+1)!}S^{Y}_{j,0}R_{-1}^{j+1}\\\
&=-\frac{1}{\int_{S}Y\cdot\operatorname{ch}(\alpha)}\sum_{j\geqslant-1}(-1)^{j}\left(\operatorname{ch}_{j}^{\mathrm{H}}(Y)R_{-1}^{j+1}+\operatorname{ch}_{j+1}^{\mathrm{H}}(Y)R_{-1}^{j+2}\right)=0,\end{split}$

where the convention $\mathrm{ch}_{i<0}^{H}(Y)=0$ is used as before.
Analogously, one also has

(61) $\sum_{j\geqslant-1}\frac{(-1)^{j}}{(j+1)!}S_{j,0}R_{-1}^{j+1}=0.$

Therefore

(62)
$\displaystyle\begin{split}\mathcal{L}_{\mathrm{inv}}&=\sum_{j\geqslant-1}\frac{(-1)^{j}}{(j+1)!}(L_{j}+S_{j,0}+S_{j,2})R_{-1}^{j+1}\\\
&=\sum_{j\geqslant-1}\frac{(-1)^{j}}{(j+1)!}(L_{j}+S^{Y}_{j,0}+S^{Y}_{j,2})R_{-1}^{j+1}\\\
&=\sum_{j\geqslant-1}\frac{(-1)^{j}}{(j+1)!}\mathcal{L}^{Y}_{j}R_{-1}^{j+1},\end{split}$

where the second equality uses (60) and (61) and $S^{Y}_{k,2}=S_{k,2}$.
Therefore $\Longrightarrow$ is obvious.

Let us prove $\Longleftarrow$ next. The idea is to prove by induction, suppose
the first part of the proposition holds for every $k^{\prime}>k$ (it certainly
holds for $k^{\prime}>\operatorname{dim}(M)$ for degree reasons). Let
$J=\operatorname{ch}_{1}^{\mathrm{H}}(Y)\in\mathbb{D}^{S}$ as before, then the
fact that $\mathcal{F}_{Y}$ is $Y$-normalised implies:
$\xi_{\mathcal{F}_{Y}}(J)=0$. Applying $\mathcal{L}_{\mathrm{inv}}$ in (62) to
$J^{k+1}D$ gives

(63) $\displaystyle
0=\int_{M(v)}\xi_{\mathcal{F}_{Y}}\left(\mathcal{L}_{\mathrm{inv}}\left(J^{k+1}D\right)\right)=\sum_{j\geqslant-1}\frac{(-1)^{j}}{(j+1)!}\int_{M(v)}\xi_{\mathcal{F}_{Y}}\left(\mathcal{L}_{j}^{Y}R_{-1}^{j+1}\left(J^{k+1}D\right)\right)$

Since $R_{-1}$ is a derivation and
$R_{-1}(J)=\operatorname{ch}_{0}^{\mathrm{H}}(Y)=\int_{S}Y\cdot\operatorname{ch}(\alpha)$,
I have

(64) $\displaystyle R_{-1}^{j+1}\left(J^{k+1}D\right)$
$\displaystyle=\sum_{s=0}^{j+1}\binom{j+1}{s}R_{-1}^{s}\left(J^{k+1}\right)R_{-1}^{j+1-s}(D)$
(65)
$\displaystyle=\sum_{s=0}^{\min(k+1,j+1)}\binom{j+1}{s}\frac{(k+1)!}{(k+1-s)!}z^{s}J^{k+1-s}R_{-1}^{j+1-s}(D),$

where I denote $z=\int_{S}\delta\cdot\operatorname{ch}(\alpha)$. Using the
fact that $\mathcal{L}_{k}^{Y}$ commutes with $J$, i.e. the Claim 5.4 and
$\xi_{\mathcal{F}_{Y}}(J)=0$, one has that the only non vanishing terms in
(63) come from the terms with $k=s-1$ in (65). Therefore:

(66) $\displaystyle 0=\sum_{j\geqslant
k}\frac{(-1)^{j}}{(j-k)!}z^{k+1}\int_{M(v)}\xi_{\mathcal{F}_{Y}}\left(\mathcal{L}^{Y}_{j}R_{-1}^{j-k}(D)\right).$

By induction hypothesis, all terms with $j>k$ vanishes, therefore one has:

(67) $\displaystyle
0=\int_{M(v)}\xi_{\mathcal{F}_{Y}}\left(\mathcal{L}_{k}^{Y}D\right).$

This finishes the proof. ∎

In section 3, I have proven:

$\int_{M(n,d,0)}\xi_{\mathcal{F}\otimes(\operatorname{det}\mathcal{F})^{-1/n}}(\mathcal{L}_{k}D)=0,\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1,$

which, by the Proposition 5.2, implies

$\int_{M(n,d,0)}\xi_{\mathcal{F}_{\boldsymbol{\operatorname{p}}}}(\mathcal{L}_{k}D)=0,\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1.$

This implies, by the definition (47) of $\mathcal{L}_{\text{inv}}$, that:

$\int_{M(n,d,0)}\xi_{\mathcal{F}_{\boldsymbol{\operatorname{p}}}}(\mathcal{L}_{\text{inv}}D)=\int_{M(n,d,0)}\mathcal{L}_{\text{inv}}D=0,\quad\forall
D\in\mathbb{D}^{S}.$

Above, by the Proposition 5.3, implies:

$\int_{M(n,d,0)}\xi_{\mathcal{F}_{Y}}(\mathcal{L}^{Y}_{k}D)=0,\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1.$

###  5.2. Using Markman operator to deduce Virasoro for rank zero case.

Now write the sheaf $\mathcal{F}_{Y}$ in terms of the language of [8]:

(68) $\displaystyle\mathcal{F}_{Y}$ $\displaystyle=\mathcal{F}\otimes
e^{-\xi_{\mathcal{F}}(\operatorname{ch}^{\mathrm{H}}_{1}(Y))/\int_{S}Y\cdot\operatorname{ch}(\alpha)}$
(69)
$\displaystyle=\mathcal{F}\otimes\exp{\left(\frac{\theta_{\mathcal{F}}(Y)}{(Y,v)}\right)}$
(70)
$\displaystyle=\mathcal{F}\otimes\exp{\left(\frac{\theta_{\mathcal{F}}(v)}{(v,v)}\right)}\otimes\exp{\left(\frac{\theta_{\mathcal{F}}(Y)}{(Y,v)}-\frac{\theta_{\mathcal{F}}(v)}{(v,v)}\right)},$

where those are identities in $K$-group of coherent sheaves on $M\times S$.
Therefore, by similar calculations as (LABEL:calculation4.1), one has:

(71)
$\displaystyle\xi_{\mathcal{F}_{Y}}(\operatorname{ch}^{\mathrm{H}}_{\bullet}(\gamma))=\exp{\left(B_{1}\left(\frac{Y}{(Y,v)}-\frac{v}{(v,v)}\right)\right)}B\left(\gamma^{\vee}\sqrt{\text{td}_{S}}^{-1}\right).$

Therefore in order to use the Markmann operator one only needs to keep track
of intersection matrix of
$(v,\boldsymbol{\operatorname{p}},\mathbbm{1},L_{i})$, where $L_{i}\in
H^{2}(S)$ and $Y=L_{i}$ for one of the $i$’s.

Consider another Mukai vector $v^{\prime}=(0,d,n)$,
$\text{Span}_{\mathbb{C}}(v,\boldsymbol{\operatorname{p}},\mathbbm{1},L_{i})$
and
$\text{Span}_{\mathbb{C}}(v^{\prime},\mathbbm{1},\boldsymbol{\operatorname{p}},L_{i})$
with $L_{i}\in H^{2}(X)$ are non-degenerate with respect to the Mukai pairing.
Then $(v,\boldsymbol{\operatorname{p}},\mathbbm{1},L_{i})$ and
$(v^{\prime},\mathbbm{1},\boldsymbol{\operatorname{p}},L_{i})$ have the same
intersection matrix, therefore by Lemma 3.3, there is an isometry sending
$v\mapsto v^{\prime}$, $\mathbbm{1}\mapsto\boldsymbol{\operatorname{p}}$,
$\boldsymbol{\operatorname{p}}\mapsto\mathbbm{1}$ and $L_{i}\mapsto L_{i}$.
Thus let us define operator
$\boldsymbol{\mathrm{M}}:\mathbb{D}^{S}\to\mathbb{D}^{S}$ which is an algebra
isomorphism as

(72)
$\displaystyle\boldsymbol{\mathrm{M}}(\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1}))=\operatorname{ch}_{i}^{\mathrm{H}}(-\mathbbm{1})$
(73)
$\displaystyle\boldsymbol{\mathrm{M}}(\operatorname{ch}_{i}^{\mathrm{H}}(\boldsymbol{\operatorname{p}}))=\operatorname{ch}_{i}^{\mathrm{H}}(\mathbbm{1}+\boldsymbol{\operatorname{p}})$
(74)
$\displaystyle\boldsymbol{\mathrm{M}}(\operatorname{ch}_{i}^{\mathrm{H}}(L_{i}))=\operatorname{ch}_{i}^{\mathrm{H}}(L_{i})$

The operator $\boldsymbol{\mathrm{M}}$ is defined to make the intersection
matrix of the arguments of function $B(\bullet)$ in (71) invariant after the
isometry.

In resume, the following constraint on $M(v^{\prime})$ is proven by the
Theorem 2.2 and section 5.1:

(75)
$\displaystyle\int_{M({v^{\prime}})}\xi_{\mathcal{F}_{Y}}\left(\boldsymbol{\mathrm{M}}\circ\mathcal{L}_{k}^{Y}D\right)=0,\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1.$

We also need the following lemma to transform (75) in to a nicer form.

######  Lemma 5.5.

$\boldsymbol{\mathrm{M}}\circ\boldsymbol{\mathrm{M}}=\operatorname{id}$,
$\boldsymbol{\mathrm{M}}\circ R_{k}=R_{k}\circ\boldsymbol{\mathrm{M}}$ and
$\boldsymbol{\mathrm{M}}\circ T_{k}=T_{k}\circ\boldsymbol{\mathrm{M}}$.

###### Proof of lemma.

$\boldsymbol{\mathrm{M}}\circ\boldsymbol{\mathrm{M}}=\operatorname{id}$ and
$\boldsymbol{\mathrm{M}}\circ R_{k}=R_{k}\circ\boldsymbol{\mathrm{M}}$ are
obvious.

$\displaystyle\boldsymbol{\mathrm{M}}T_{k}$
$\displaystyle=\boldsymbol{\mathrm{M}}\left(\sum_{i+j=k}(-1)^{2-p^{L}}i!j!\operatorname{ch}_{i}^{\mathrm{H}}\operatorname{ch}_{j}^{\mathrm{H}}(\text{td}_{S})\right)$
$\displaystyle=\boldsymbol{\mathrm{M}}\left(\sum_{\begin{subarray}{c}i+j=k\\\
\gamma_{i}^{L},\gamma_{i}^{R}\in
H^{2}(S)\end{subarray}}(-1)^{2-p^{L}}i!j!\operatorname{ch}^{\mathrm{H}}_{i}(\gamma_{i}^{L})\operatorname{ch}^{\mathrm{H}}_{j}(\gamma_{i}^{R})+\sum_{i+j=k}i!j!\left(\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})+\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})\right)\right)$
$\displaystyle\quad\quad+\boldsymbol{\mathrm{M}}\left(2\sum_{i+j=k}i!j!\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})\right)$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=k\\\
\gamma_{i}^{L},\gamma_{i}^{R}\in
H^{2}(S)\end{subarray}}(-1)^{2-p^{L}}i!j!\operatorname{ch}^{\mathrm{H}}_{i}(\gamma_{i}^{L})\operatorname{ch}^{\mathrm{H}}_{j}(\gamma_{i}^{R})$
$\displaystyle\quad\quad+\sum_{i+j=k}i!j!\left[-\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})-\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{j}(\mathbbm{1})-2\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\mathbbm{1})\right]$
$\displaystyle\quad\quad+2\sum_{i+j=k}i!j!\left[\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\mathbbm{1})+\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})+\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{j}(\mathbbm{1})+\operatorname{ch}^{\mathrm{H}}_{i}(\boldsymbol{\operatorname{p}})\operatorname{ch}^{\mathrm{H}}_{j}(\boldsymbol{\operatorname{p}})\right]$
$\displaystyle=T_{k}.$

∎

Therefore the following constraint, in the rank zero case, is true:

######  Proposition 5.6.

Let $v=(0,d,n)$ with $d,n>0$ be a Mukai vector, let $Y\in H^{1,1}(S)$ and let
$S^{Y}_{k}:=S^{Y}_{k,0}+S^{Y}_{k,2}$ as defined in (52), then the following
constraints are true:

$\int_{M(v)}\xi_{\mathcal{F}_{Y}}\bigg{(}\left(R_{k}+T_{k}+\boldsymbol{\mathrm{M}}\circ
S^{Y}_{k}\circ\boldsymbol{\mathrm{M}}\right)D\bigg{)}=0\quad\forall
D\in\mathbb{D}^{S},\forall k\geq-1.$

The lemma 5.5 means that the operator $\boldsymbol{\mathrm{M}}$ keeps the
operator $L_{k}=R_{k}+T_{k}$ invariant, which justify to call above constraint
the Virasoro constraint.

######  Remark 5.7.

Assume $F\in M_{(0,d,n)}$ with $n>0$. On the set level, the map $F\mapsto
F^{D}:=\curly Ext(F,\omega_{S})$ is a bijection, notice that $\curly
Ext(F,\omega_{S})\cong\mathcal{R}\mathcal{H}{\it{om}}(F,\mathcal{O}_{S})[1]=F^{\vee}[1]$.
The type of $F^{\vee}$ is $\operatorname{ch}(F^{\vee}[1])=(0,d,-n)$, therefore
one may expect that the proposition 5 also holds for spaces $M_{(0,d,-n)}$
with $d,n>0$.

##  6\. Negative Virasoro operators

In this section, let $S$ be a K3 surface as before. In [1], the Virasoro
operators are related with lattice vertex algebra operators. Inspired by this
correspondence, I will construct the operators $L_{k<-1}$ which along with
previously defined $L_{k\geq-1}$ form a Virasoro algebra with the central
charge $e(S)=24$, the topological Euler characteristic of $S$.

Let $k>1$ define the following derivations:

(76) $\forall\gamma\in H^{\ast}(S,\mathbb{C}),\quad
R_{-k}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)=\begin{cases}\frac{(i-k)!}{(i-1)!}\operatorname{ch}^{\mathrm{H}}_{i-k}(\gamma),&\text{if}\
i\geq k\\\ 0,&\text{otherwise}\end{cases}$

Define the operator $\operatorname{d}_{i}[v]$ acting on $D\in\mathbb{D}^{S}$
as a derivation as follows:

(77) $\displaystyle\forall i>0,j\geq
0,\quad\operatorname{d}_{i}[v]\operatorname{ch}^{\mathrm{H}}_{j}(\gamma)=\frac{\delta_{ij}}{(i-1)!}\int_{S}\gamma\cdot
v\quad\forall\gamma\in H^{\ast}(S,\mathbb{C}).$

for $v$ a homogeneous element of $H^{\ast}(S,\mathbb{C})$. Let
$\sum_{i}\delta_{i}^{L}\otimes\delta_{i}^{R}$ be the Künneth decomposition of
the diagonal class $\Delta\in H^{\ast}(S\times S,\mathbb{C})$ and
$\delta_{i}^{L},\delta_{j}^{R}$ have Hodge type $(p^{L}_{i},q^{L}_{i})$ and
$(p^{R}_{i},q^{R}_{i})$, respectively. Next, choose such a Künneth
decomposition $\sum_{w}\delta_{w}^{L}\otimes\delta_{w}^{R}$, define the
operator $T_{-k}$ as444For surfaces with only even cohomology,
$(-1)^{\operatorname{dim}(S)-p^{L}}=(-1)^{p^{L}p^{R}}$. The latter format is
used in [6],:

(78) $\displaystyle
T_{-k}=-\frac{1}{4}\sum_{w}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{w}^{L}p_{w}^{R}}\operatorname{d}_{j}[\delta^{L}_{w}]\operatorname{d}_{i}[\delta^{R}_{w}]+\left(\frac{1}{4}\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
i>0\\\
j>0\end{subarray}}\operatorname{d}_{j}[\boldsymbol{\operatorname{p}}]\operatorname{d}_{i}[\boldsymbol{\operatorname{p}}].$

By the same reason why $T_{k},k\geq-1$ are independent of the representations
of the Künneth decomposition, which is since $T_{k}$ was defined in a multi-
linear way, one can see that $T_{-k},k>0$ are also independent of the Künneth
decomposition.

To clarify calculations, I will choose a particular representation of Künneth
decomposition. Let $\\{\alpha_{i}\\}_{i}$ be a basis for
$H^{\ast}(S,\mathbb{C})$, define $\\{\alpha_{i}^{\vee}\\}_{i}$ to be the
unique vectors satisfying $\int_{S}\alpha_{j}\alpha_{i}^{\vee}=\delta_{ij}$.
Then $\sum_{i}\alpha_{i}\otimes\alpha_{i}^{\vee}$ is a Künneth decomposition.
One can choose the basis $B$ of $H^{\ast}(S,\mathbb{C})$ in such a way that:
$\mathbbm{1}\in B$ and $\mathbbm{1}^{\vee}=\mathbf{p}\in B$; also
$\sigma,\sigma^{\vee}=\overline{\sigma}\in B$ where $\sigma\in H^{2,0}(S)$;
$\gamma^{\vee}=\gamma\in B$ for all $\gamma=H^{1,1}(S)$. That means:

(79) $\displaystyle B=\\{\gamma,\gamma^{\vee}|\gamma\in
H^{0}(S,\mathbb{C})\cup H^{2,0}(S,\mathbb{C})\\}\cup B^{1,1},$

where $B^{1,1}=\\{\gamma_{1},\dots,\gamma_{s}\\}$ is a basis of
$H^{1,1}(S,\mathbb{C})$ such that $\int_{S}\gamma_{i}\gamma_{j}=\delta_{ij}$.
This is possible since this symmetric paring is non-degenerate on
$H^{1,1}(S,\mathbb{C})$ and can be diagonalized. The operator $T_{-k}$ now
becomes:

(80) $\displaystyle T_{-k}=-\frac{1}{4}\sum_{v\in
B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{v}}\operatorname{d}_{j}[v]\operatorname{d}_{i}[v^{\vee}]+\left(\frac{1}{4}\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
i>0\\\
j>0\end{subarray}}\operatorname{d}_{j}[\boldsymbol{\operatorname{p}}]\operatorname{d}_{i}[\boldsymbol{\operatorname{p}}],$

where $p_{v}$ is defined by $v\in H^{p_{v},q_{v}}(S,\mathbb{C})$. Also define
the operator $L_{-k}$ as:

(81) $\displaystyle L_{-k}=R_{-k}+T_{-k}.$

In this subsection, I will prove the following proposition:

######  Proposition 6.1.

Let $S$ be a K3 surface and let the operators
$\\{L_{k}:\mathbb{D}^{S}\to\mathbb{D}^{S}\\}_{k\in\mathbb{Z}}$ be defined as
(6) when $k\geq-1$ and be defined as (81) when $k<-1$ then the following
commutation relation is true:

(82) $\displaystyle[L_{l},L_{-k}]=(-l-k)L_{l-k}+\frac{k^{3}-k}{12}\cdot
e(S)\cdot\delta_{k,l},$

where $e(S)$ is the topological Euler characteristic of $S$.

###### Proof.

I also write $D\in\mathbb{D}^{S}$ as a monomial containing only descendants of
the form $\operatorname{ch}^{\mathrm{H}}_{i}(v\in B)$. In order to clarify the
action of $T_{-k}$, for any $D\in\mathbb{D}^{S}$, I can rewrite $D$ in the
following form (since $S$ only has even cohomology, all descendants commute
with each other):

(83) $\displaystyle D$ $\displaystyle=\prod_{v\in
B}\prod_{\begin{subarray}{c}k>i>0\end{subarray}}\left(\operatorname{ch}^{\mathrm{H}}_{i}(v)\right)^{n_{i,v}}\cdot
D_{0}\cdot D_{\geq k}$ (84) $\displaystyle=D_{<k}\cdot D_{0}\cdot D_{\geq k},$

where $D_{0},D_{<k},D_{\geq k}$ contains, respectively, only descendent of the
form $\operatorname{ch}^{\mathrm{H}}_{0}(v\in
B)$,$\operatorname{ch}^{\mathrm{H}}_{0<\bullet<k}(v\in B)$ and
$\operatorname{ch}^{\mathrm{H}}_{\geq k}(v\in B)$. The reason why I write $D$
in this form is that $T_{-k}$ will only act non trivially on $D_{<k}$ part,
and $R_{-k}$ will only act non trivially on $D_{\geq k}$ part.

Let $k>1$, $l\geq-1$ I calculate some commutators now:

(85) $\displaystyle\begin{split}[R_{l},T_{-k}](D_{<k}D_{0}D_{\geq
k})&=(R_{l}T_{-k}D_{<k})(D_{0}D_{\geq k})+(T_{-k}D_{<k})(R_{l}D_{0}D_{\geq
k})\\\ &\quad\quad-(T_{-k}R_{l}D_{<k})(D_{0}D_{\geq
k})-(T_{-k}D_{<k})(R_{l}D_{0}D_{\geq k})\\\ &=(R_{l}T_{-k}D_{<k})(D_{0}D_{\geq
k})-(T_{-k}R_{l}D_{<k})(D_{0}D_{\geq k})\\\ \end{split}$

To calculate $R_{l}T_{-k}D_{<k}-T_{-k}R_{l}D_{<k}$, let me calculate the
following first:

(86)
$\displaystyle(R_{l}d_{j}[v]d_{i}[v^{\vee}]-\operatorname{d}_{j}[v]\operatorname{d}_{i}[v^{\vee}]R_{l})D_{<k},$

where $i,j$ are fixed integers satisfying $i+j=k,i>0,j>0$. If $R_{l}$ acts on
$\operatorname{ch}^{\mathrm{H}}_{i-l}(v)$ in $D_{<k}$ then its contribution to
$-\operatorname{d}_{j}[v]\operatorname{d}_{i}[v^{\vee}]R_{l}D_{<k}$ is:

(87)
$\displaystyle-\frac{i!}{(i-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{i-l,v}n_{j,v^{\vee}}(n_{i,v}+1)\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i-l}(v)\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})};$

when $R_{l}$ acts on $\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})$, it
gives:

(88)
$\displaystyle-\frac{j!}{(j-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{j-l,v^{\vee}}(n_{j,v^{\vee}}+1)n_{i,v}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)};$

when $R_{l}$ acts on $\operatorname{ch}^{\mathrm{H}}_{i}(v)$ or on
$\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})$ it gives respectively:

(89)
$\displaystyle-\frac{(l+i)!}{(i-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{i,v}n_{j,v^{\vee}}(n_{i,v}-1)\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{i+l}(v)}{\operatorname{ch}^{\mathrm{H}}_{i}(v)\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)}\quad\text{
and }$ (90)
$\displaystyle-\frac{(l+j))!}{(j-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{j,v^{\vee}}(n_{j,v^{\vee}}-1)n_{i,v}\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{j+l}(v^{\vee})}{\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)}$

when $R_{l}$ acts on other descendants, say
$\operatorname{ch}^{\mathrm{H}}_{p}(w)$, it gives:

(91)
$\displaystyle-\frac{(l+p))!}{(p-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{p,w}n_{j,v^{\vee}}n_{i,v}\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{p+l}(w)}{\operatorname{ch}^{\mathrm{H}}_{p}(w)\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)}.$

Now, let us look at the first term of (86):

(92) $\displaystyle
R_{l}d_{j}[v]d_{i}[v^{\vee}]D_{<k}=\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{j,v^{\vee}}n_{i,v}R_{l}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)},$

It is easy to see that when $R_{l}$ does not act on
$\operatorname{ch}^{\mathrm{H}}_{i-l}(v)$ and not on
$\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})$, but on
$\operatorname{ch}^{\mathrm{H}}_{i}(v),\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee}),\operatorname{ch}^{\mathrm{H}}_{p}(w)$
respectively, it gives $-\eqref{samereslt1}$, $-\eqref{samereslt3}$ and
$-\eqref{samereslt2}$ as above. When $R_{l}$ acts on
$\operatorname{ch}^{\mathrm{H}}_{i-l}(v)$ or on
$\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})$ it gives:

(93)
$\displaystyle\frac{i!}{(i-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{i-l,v}n_{j,v^{\vee}}n_{i,v}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i-l}(v)\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})}\quad\text{
and }$ (94)
$\displaystyle\frac{j!}{(j-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{j-l,v^{\vee}}n_{j,v^{\vee}}n_{i,v}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)}.$

Summing (87), (88), (93) and (94) gives:

(95) $\displaystyle\eqref{onecom}$
$\displaystyle=-\frac{i!}{(i-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{i-l,v}n_{j,v^{\vee}}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i-l}(v)\operatorname{ch}^{\mathrm{H}}_{j}(v^{\vee})}$
(96)
$\displaystyle\quad\quad-\frac{j!}{(j-l-1)!}\frac{1}{(i-1)!}\frac{1}{(j-1)!}n_{j-l,v^{\vee}}n_{i,v}\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{j-l}(v^{\vee})\operatorname{ch}^{\mathrm{H}}_{i}(v)}$
(97)
$\displaystyle=-i\operatorname{d}_{i-l}[v^{\vee}]\operatorname{d}_{j}[v]D_{<k}-j\operatorname{d}_{i}[v^{\vee}]\operatorname{d}_{j-l}[v]D_{<k}$

Summing terms of the form (86) with according coefficients over
$i+j=k,i>0,j>0$ and over $B$ one gets:

(98) $\displaystyle[R_{l},T_{-k}]=\begin{cases}(-l-k)T_{-(k-l)},&\text{if}\
l-k<0\\\ 0,&\text{otherwise}.\end{cases}$

Next, I calculate the following commutator:

(99) $\displaystyle\begin{split}[T_{l},R_{-k}](D_{<k}D_{0}D_{\geq
k})&=T_{l}D_{<k}D_{0}(R_{-k}D_{\geq k})-(R_{-k}T_{l})D_{<k}D_{0}D_{\geq k}\\\
&\quad\quad\quad-T_{l}D_{<k}D_{0}(R_{-k}D_{\geq k})\\\
&=-(R_{-k}T_{l})D_{<k}D_{0}D_{\geq k}\\\ \end{split}.$

If $l<k$, $R_{-k}T_{l}$ vanishes. Suppose now $l\geq k$, then:

(100) $\displaystyle R_{-k}T_{l}$
$\displaystyle=R_{-k}\sum_{i+j=l}(-1)^{\operatorname{dim}(S)-p^{L}}i!j!\operatorname{ch}^{\operatorname{H}}_{i}\operatorname{ch}^{\mathrm{H}}_{j}(\operatorname{td}_{S})$
(101)
$\displaystyle=\sum_{i+j=l}(-1)^{\operatorname{dim}(S)-p^{L}}\left(i!j!\frac{(i-k)!}{(i-1)!}\operatorname{ch}^{\operatorname{H}}_{i-k}\operatorname{ch}^{\mathrm{H}}_{j}(\operatorname{td}_{S})+i!j!\frac{(j-k)!}{(j-1)!}\operatorname{ch}^{\operatorname{H}}_{i}\operatorname{ch}^{\mathrm{H}}_{j-k}(\operatorname{td}_{S})\right)$
(102) $\displaystyle=(l+k)T_{l-k},$

where the convention $\operatorname{ch}^{\mathrm{H}}_{<0}(\bullet)=0$ is used.
Therefore:

(103) $\displaystyle[T_{l},R_{-k}]=\begin{cases}(-l-k)T_{l-k},&\text{if}\
l-k\geq 0\\\ 0,&\text{otherwise}.\end{cases}$

Next, I calculate $[T_{-k},T_{l}]D$. Generally, for two derivations operators
$d_{1},d_{2}$ one has:
$[d_{1}d_{2},T_{l}]D=(d_{1}d_{2}T_{l})D+(d_{1}T_{l})(d_{2}D)+(d_{2}T_{l})(d_{1}D)$.
Because the operator $T_{-k}$ is the sum of compositions of two derivatives,
one can use this equality to calculate $[T_{-k},T_{l}]D$. As mentioned before,
derivations $d_{i}[\bullet]$ with $0<i<k$ only act non trivially on the
component $D_{<k}$, therefore we can perform the following calculations:

(104) $\displaystyle\begin{split}&-\frac{1}{4}\sum_{v\in
B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{v}}(\operatorname{d}_{j}[v]T_{l})(\operatorname{d}_{i}[v^{\vee}]D_{<k})\\\
&=-\frac{1}{4}\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ 1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{2}{(j-1)!}(-1)^{p_{v}}j!(l-j)!\operatorname{ch}^{\mathrm{H}}_{l-j}(v)\right)\left((-1)^{p_{v}}\frac{1}{(i-1)!}\cdot
n_{i,v}\cdot\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i}(v)}\right)\\\
&\quad\quad-\frac{1}{4}\sum_{\begin{subarray}{c}i+j=k\\\ 1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\left(\int_{S}\operatorname{td}_{S}\right)\cdot\frac{2}{(j-1)!}j!(l-j)!\operatorname{ch}^{\mathrm{H}}_{l-j}(\boldsymbol{\operatorname{p}})\right)\left(\frac{1}{(i-1)!}\cdot
n_{i,\mathbbm{1}}\cdot\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})}\right)\\\
&=-\frac{1}{2}\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ 1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}(i-k)\cdot
n_{i,v}\cdot\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{i-(k-l)}(v)}{\operatorname{ch}^{\mathrm{H}}_{i}(v)}\right)\\\
&\quad\quad-\frac{1}{2}\left(\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}jn_{i,\mathbbm{1}}\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{l-j}(\boldsymbol{\operatorname{p}})}{\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})}\right)\end{split}.$

Similarly:

(105) $\displaystyle\begin{split}&-\frac{1}{4}\sum_{v\in
B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{v}}(\operatorname{d}_{j}[v^{\vee}]T_{l})(\operatorname{d}_{i}[v]D_{<k})\\\
&=-\frac{1}{2}\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ 1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}(i-k)\cdot
n_{i,v^{\vee}}\cdot\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{i-(k-l)}(v^{\vee})}{\operatorname{ch}^{\mathrm{H}}_{i}(v^{\vee})}\right)\\\
&\quad\quad-\frac{1}{2}\left(\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}jn_{i,\mathbbm{1}}\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{l-j}(\boldsymbol{\operatorname{p}})}{\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})}\right)\\\
\end{split}.$

There is also the following terms:

(106)
$\displaystyle\begin{split}&\frac{1}{4}\left(\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
i>0\\\
j>0\end{subarray}}(\operatorname{d}_{j}[\boldsymbol{\operatorname{p}}]T_{l})(\operatorname{d}_{i}[\boldsymbol{\operatorname{p}}]D_{<k})\\\
&=\frac{1}{4}\left(\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
i>0\\\
j>0\end{subarray}}\left(\frac{2}{(j-1)!}j!(l-j)!\operatorname{ch}^{\mathrm{H}}_{l-j}(\boldsymbol{\operatorname{p}})\right)\left(\frac{1}{(i-1)!}\cdot
n_{i,\mathbbm{1}}\cdot\frac{D_{<k}}{\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})}\right)\\\
&=\frac{1}{2}\left(\int_{S}\operatorname{td}_{S}\right)\sum_{\begin{subarray}{c}i+j=k\\\
1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}jn_{i,\mathbbm{1}}\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{l-j}(\boldsymbol{\operatorname{p}})}{\operatorname{ch}^{\mathrm{H}}_{i}(\mathbbm{1})}\right)\end{split}$
(107) $\displaystyle\begin{split}&\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(\operatorname{d}_{j}[\boldsymbol{\operatorname{p}}]\operatorname{d}_{i}[\boldsymbol{\operatorname{p}}]T_{k})(D_{<k})\\\
&=0\end{split}$

Summing (104), (105) and (106) one gets:

(108)
$\displaystyle\begin{split}\eqref{TT21}+&\eqref{TT22}+2\times\eqref{TT23}\\\
&=-\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ 1\leq
j\leq\operatorname{min}(k-1,l)\end{subarray}}\left(\frac{(i-(k-l))!}{(i-1)!}(i-k)\cdot
n_{i,v^{\vee}}\cdot\frac{D_{<k}\operatorname{ch}^{\mathrm{H}}_{i-(k-l)}(v^{\vee})}{\operatorname{ch}^{\mathrm{H}}_{i}(v^{\vee})}\right)\\\
&=-R_{l-k}[\bullet-k]D_{<k}\end{split},$

where, for a function $f(\bullet)$ defined on integers, $R_{j}[f(\bullet)]$ is
the derivation acting as
$R_{j}[f(\bullet)]\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)=f(i)R_{j}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$.

There are also the following terms in $[T_{-k},T_{l}]$ which are non-zero only
if $k=l$:

(109) $\displaystyle\begin{split}&-\frac{1}{4}\sum_{v\in
B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{v}}(\operatorname{d}_{j}[v]\operatorname{d}_{i}[v^{\vee}]T_{k})D_{<k}\\\
&=-\frac{1}{4}\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}(-1)^{p_{v}}\left(2(-1)^{p_{v}}ij\right)D_{<k}\\\
&=-\frac{1}{2}\sum_{v\in B}\sum_{\begin{subarray}{c}i+j=k\\\ i>0\\\
j>0\end{subarray}}ijD_{<k}\\\ &=-\frac{1}{2}\sum_{v\in
B}\sum_{\begin{subarray}{c}k>i>0\end{subarray}}(ik-i^{2})D_{<k}\\\
&=-\sum_{v\in
B}\frac{1}{2}\left(k\cdot\frac{k(k-1)}{2}-\frac{(k-1)k(2k-1)}{6}\right)D_{<k}\\\
&=-\frac{k^{3}-k}{12}\cdot e(S)D_{<k},\end{split}$

where $e(S)$ is the topological Euler characteristic and, for K3,
$e(S)=|B|=\operatorname{dim}H^{\ast}(S,\mathbb{C})=24$.

Next, I calculate the following commutator:

(110) $\displaystyle\begin{split}&[R_{l},R_{-k}](D_{<k}D_{0}D_{\geq k})\\\
&=(R_{l}R_{-k}-R_{-k}R_{l})D_{<k}D_{0}D_{\geq k}\\\
&=(R_{l}D_{<k}D_{0})(R_{-k}D_{\geq k})+D_{<k}D_{0}(R_{l}R_{-k}D_{\geq
k})-(R_{-k}R_{l}D_{<k}D_{0})D_{\geq k}\\\
&\quad-(R_{l}D_{<k}D_{0})(R_{-k}D_{\geq k})-D_{<k}D_{0}(R_{-k}R_{l}D_{\geq
k})\\\ &=D_{<k}D_{0}(R_{l}R_{-k}D_{\geq k})-D_{<k}D_{0}(R_{-k}R_{l}D_{\geq
k})-(R_{-k}R_{l}D_{<k}D_{0})D_{\geq k}\\\ &=D_{<k}D_{0}((-k-l)R_{l-k}D_{\geq
k})-(R_{l-k}[(\bullet+l)]D_{<k}D_{0})D_{\geq k}.\end{split}$

Combining (108) with (LABEL:com2), one has (notice that
$R_{k}\operatorname{ch}^{\mathrm{H}}_{0}(\bullet)=0,\forall k\in\mathbb{Z}$):

(111)
$\displaystyle\begin{split}-\eqref{109}+\eqref{com2}&=D_{<k}D_{0}((-k-l)R_{l-k}D_{\geq
k})-(R_{l-k}[(\bullet+l)]D_{<k}D_{0})D_{\geq k}\\\
&\quad\quad+(R_{l-k}[\bullet-k]D_{<k}D_{0})D_{\geq k}\\\
&=D_{<k}D_{0}((-k-l)R_{l-k}D_{\geq k})+((-k-l)R_{l-k}D_{<k}D_{0})D_{\geq k}\\\
&=(-k-l)R_{l-k}(D_{<k}D_{0}D_{\geq k})\end{split}$

Summing the above commutators one has:

(112) $\displaystyle[L_{l},L_{-k}]=(-l-k)L_{l-k}+\frac{k^{3}-k}{12}\cdot
e(S)\cdot\delta_{kl}$

Let me calculate $[L_{-l},L_{-k}]$ next.

(113) $\displaystyle[R_{-l},R_{-k}]\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$
$\displaystyle=(R_{-l}R_{-k}-R_{-k}R_{-l})\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$
(114)
$\displaystyle=\frac{(i-k-l)!}{(i-k-1)!}\frac{(i-k)!}{(i-1)!}\operatorname{ch}^{\mathrm{H}}_{i-k-l}(\gamma)$
(115)
$\displaystyle\quad\quad\quad\quad-\frac{(i-k-l)!}{(i-l-1)!}\frac{(i-l)!}{(i-1)!}\operatorname{ch}^{\mathrm{H}}_{i-k-l}(\gamma)$
(116)
$\displaystyle=(l-k)R_{-k-l}\operatorname{ch}^{\mathrm{H}}_{i}(\gamma),\quad\text{for
$i\geq l+k$}.$ (117)
$\displaystyle[R_{-l},R_{-k}]\operatorname{ch}^{\mathrm{H}}_{i}(\gamma)$
$\displaystyle=0,\quad\text{for $i<l+k$}.$

Therefore $[R_{-l},R_{-k}]=R_{-k-l}$. Next, I calculate $[R_{-l},T_{-k}]D$.
Firstly, consider the following term:

(118)
$\displaystyle(R_{-l}\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{k-j}[v]-\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{k-j}[v]R_{-l})D.$

Similar as the calculation for (98), the contribution to the above term
vanishes unless $R_{-l}$ acts on one of the following symbols:

$\operatorname{ch}^{\mathrm{H}}_{j}(v),\operatorname{ch}^{\mathrm{H}}_{k-j}(v^{\vee}),\operatorname{ch}^{\mathrm{H}}_{j+l}(v),\operatorname{ch}^{\mathrm{H}}_{k-j+l}(v^{\vee}).$

By an easy calculation, one finds that when $R_{-l}$ acts on
$\operatorname{ch}^{\mathrm{H}}_{j}(v),\operatorname{ch}^{\mathrm{H}}_{k-j}(v^{\vee})$,
the contribution to (118) also vanishes. Let me calculate the case when
$R_{-l}$ acts on $\operatorname{ch}^{\mathrm{H}}_{j+l}(v)$:

$\displaystyle R_{-l}$
$\displaystyle\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{k-j}[v]D$
$\displaystyle=\frac{1}{(j-1)!}\frac{1}{(k-j-1)!}n_{j,v}n_{k-j,v^{\vee}}n_{v,j+l}\frac{j!}{(j+l-1)!}\frac{D}{\operatorname{ch}^{\mathrm{H}}_{j+l}(v)\operatorname{ch}^{\mathrm{H}}_{k-j}(v^{\vee})}$
$\displaystyle\operatorname{d}_{j}[v^{\vee}]$
$\displaystyle\operatorname{d}_{k-j}[v]R_{-l}D$
$\displaystyle=\frac{1}{(j-1)!}\frac{1}{(k-j-1)!}(n_{j,v}+1)n_{k-j,v^{\vee}}n_{v,j+l}\frac{j!}{(j+l-1)!}\frac{D}{\operatorname{ch}^{\mathrm{H}}_{j+l}(v)\operatorname{ch}^{\mathrm{H}}_{k-j}(v^{\vee})}$

Therefore:

$\displaystyle\eqref{124}=-j\operatorname{d}[v^{\vee},j+l]\operatorname{d}[v,k-j]D.$

The contribution of $\operatorname{ch}^{\mathrm{H}}_{k-j+l}(v^{\vee})$ can be
calculated similarly. Summing over $i+j=k,i>0,j>0$ those two type of
contributions, one has:

(119) $\displaystyle\sum_{\begin{subarray}{c}i+j=k+l\\\ j\geq l+1\\\ i\leq
k-1\end{subarray}}-(j-l)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]-\sum_{\begin{subarray}{c}i+j=k+l\\\
i\geq l+1\\\ j\leq
k-1\end{subarray}}(i-l)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v].$

For the commutator $[T_{-l},R_{-k}]D$, we have the following term:

(120) $\displaystyle\sum_{\begin{subarray}{c}i+j=k+l\\\ j\geq k+1\\\ i\leq
l-1\end{subarray}}(j-k)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]+\sum_{\begin{subarray}{c}i+j=k+l\\\
i\geq k+1\\\ j\leq
l-1\end{subarray}}(i-k)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]$

Summing the first term of (119) and the second term of (120) and summing the
second term of (119) and the first term of (120):

(121) $\displaystyle\eqref{125}+\eqref{126}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=k+l\\\ j\geq 1\\\ i\geq
1\end{subarray}}-(j-l)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]-\sum_{\begin{subarray}{c}i+j=k+l\\\
j\geq 1\\\ i\geq
1\end{subarray}}(i-l)\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]$
(122) $\displaystyle=(l-k)\sum_{\begin{subarray}{c}i+j=k+l\\\ j\geq 1\\\ i\geq
1\end{subarray}}\operatorname{d}_{j}[v^{\vee}]\operatorname{d}_{i}[v]$

Summing over $v\in B$ with corresponding coefficients, one has:

(123) $\displaystyle[R_{-l},T_{-k}]+[T_{-l},R_{-k}]=(l-k)T_{-l-k}.$

For the commutator $[T_{-l},T_{-k}]$, one can easily see that this vanishes.
Therefore the following commutation relation is proven:

(124) $\displaystyle[L_{-l},L_{-k}]=(l-k)L_{-l-k},\text{ for $l>1,k>1$}.$

For $m,n\geq-1$, it has been shown (lemma 2.11 [2]) that

(125) $\displaystyle[L_{m},L_{n}]=(-m+n)L_{m+n},\text{ for $m,n\geq-1$}.$

Therefore the Proposition 6.1 is proven. ∎

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