# The simplest cohomological invariants for vertex algebras

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences

Zitna 25, 11567
Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

For the double complex structure of grading-restricted vertex algebra
cohomology defined in [5], we introduce a multiplication of elements of double
complex spaces. We show that the orthogonality and bi-grading conditions
applied on double complex spaces, provide in relation among mappings and
actions of co-boundary operators. Thus, we endow the double complex spaces
with structure of bi-graded differential algebra. We then introduce the
simples cohomology classes for a grading-restricted vertex algebra, and show
their independence on the choice of mappings from double complex spaces. We
prove that its cohomology class does not depend on mappings representing of
the double complex spaces. Finally, we show that the orthogonality relations
together with the bi-grading condition bring about generators and commutation
relations for a continual Lie algebra.

AMS Classification: 53C12, 57R20, 17B69

###### Key words and phrases:

Vertex algebras, cohomological invariants, cohomology classes

##  1\. Introduction: $\overline{W}$-valued rational functions

In [5] the cohomology theory for a grading-restricted vertex algebra [8] (see
Appendix 5) was introduced. The definition of double complex spaces and co-
boundary operators, uses an interpretation of vertex algebras in terms of
rational functions constructed from matrix elements [7] for a grading-
restricted vertex algebra. The notion of composability (see Section 1.2) of
double complex space elements with a number of vertex operators, is
essentially involved in the formulation. Then the cohomology of such complexes
defines in the standard way a cohomology of a grading-restricted vertex
algebras. It is an important problem to study possible cohomological classes
for vertex algebras. In this paper we do the first steps to discover simplest
cohomological invariants associated to the setup described above. For that
purpose we first endow the double complex spaces with natural product, derive
a counterpart of Leibniz formula for the action of co-boundary operators. Then
we introduce the notion of a cohomological class for a vertex algebra. The
orthogonality condition of double complex space is then defined. We show that
the orthogonality being applied to the double complex spaces leads to
relations among mappings and actions of co-boundary operators. The simplest
non-vanishing cohomology classes for a grading-restricted vertex algebra is
then derived. We show that such classes are independent of the choice of
elements of the double complex spaces. Finally, we discuss occurring relations
of a vertex algebra double complex relations with a continual Lie algebra [9].
For further applications of material introduced in this paper, we would
mention the natural question of searching for more general cohomological
invariants for a grading-restricted vertex algebra. Concerning possible
applications, one can use the cohomological classes we derive to compute
higher cohomologies of grading-restricted vertex algebras.

Let $V$ be a grading-restricted vertex algebra, and $W$ a a grading-restricted
generalized $V$-module (see Appendix 5). One defines the configuration space
[5]:

$F_{n}\mathbb{C}=\\{(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\;|\;z_{i}\neq
z_{j},i\neq j\\},$

for $n\in\mathbb{Z}_{+}$.

######  Definition 1.

A $\overline{W}$-valued rational function $\mathcal{F}$ in
$(z_{1},\dots,z_{n})$ with the only possible poles at $z_{i}=z_{j}$, $i\neq
j$, is a map

$\displaystyle\mathcal{F}:F_{n}\mathbb{C}$ $\displaystyle\to$
$\displaystyle\overline{W},$ $\displaystyle(z_{1},\dots,z_{n})$
$\displaystyle\mapsto$ $\displaystyle\mathcal{F}(z_{1},\dots,z_{n}),$

such that for any $w^{\prime}\in W^{\prime}$,

$\langle w^{\prime},\mathcal{F}(z_{1},\dots,z_{n})\rangle,$ (1.1)

is a rational function in $(z_{1},\dots,z_{n})$ with the only possible poles
at $z_{i}=z_{j}$, $i\neq j$. Such map is called in what fallows
$\overline{W}$-valued rational function in $(z_{1},\dots,z_{n})$ with possible
other poles. Denote the space of all $\overline{W}$-valued rational functions
in $(z_{1},\dots,z_{n})$ by $\overline{W}_{z_{1},\dots,z_{n}}$.

Namely, if a meromorphic function $f(z_{1},\dots,z_{n})$ on a region in
$\mathbb{C}^{n}$ can be analytically extended to a rational function in
$(z_{1},\dots,z_{n})$, then the notation $R(f(z_{1},\dots,z_{n}))$, is used to
denote such rational function. Note that the set of a grading-restricted
vertex algebra elements $(v_{1},\ldots,v_{n})$ associated with corresponding
$(z_{1},\ldots,z_{n})$ play the role of non-commutative parameters for a
function $\mathcal{F}$ in (1.1). Let us introduce the definition of a
$\mathcal{W}_{z_{1},\ldots,z_{n}}$-space:

######  Definition 2.

We define the space $\mathcal{W}_{z_{1},\dots,z_{n}}$ of
$\overline{W}_{z_{1},\dots,z_{n}}$-valued rational forms $\Phi$ with each
vertex algebra element entry $v_{i}$, $1\leq i\leq n$ of a quasi-conformal
grading-restricted vertex algebra $V$ tensored with power $\mbox{\rm wt}\
(v_{i})$-differential of corresponding formal parameter $z_{i}$, i.e.,

$\displaystyle\Phi\left(v_{1},z_{1};\ldots;v_{n},z_{n}\right)$ (1.2)
$\displaystyle\qquad=\mathcal{F}\left(dz_{1}^{{\rm\mbox{\rm wt}\
}(v_{1})}\otimes v_{1},z_{1};\ldots;dz_{n}^{{\rm\mbox{\rm wt}\
}(v_{n})}\otimes v_{n},z_{n}\right)\in\mathcal{W}_{z_{1},\dots,z_{n}}.$

where $\mathcal{F}\in\overline{W}_{z_{1},\dots,z_{n}}$.

######  Definition 3.

One defines an action of $S_{n}$ on the space $\mbox{\rm Hom}(V^{\otimes
n},\mathcal{W}_{z_{1},\dots,z_{n}})$ of linear maps from $V^{\otimes n}$ to
$W_{z_{1},\dots,z_{n}}$ by

$\sigma(\Phi)(v_{1}\otimes\cdots\otimes
v_{n})(z_{1},\dots,z_{n}),=\Phi(v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(n)})(z_{\sigma(1)},\dots,z_{\sigma(n)}),$ (1.3)

for $\sigma\in S_{n}$ and $v_{1},\dots,v_{n}\in V$,
$\Phi\in\mathcal{W}_{z_{1},\dots,z_{n}}$. We will use the notation
$\sigma_{i_{1},\dots,i_{n}}\in S_{n}$, to denote the the permutation given by
$\sigma_{i_{1},\dots,i_{n}}(j)=i_{j}$, for $j=1,\dots,n$.

######  Definition 4.

For $n\in\mathbb{Z}_{+}$, a linear map

$\mathcal{F}(v_{1},z_{1};\ldots;v_{n},z_{n})=V^{\otimes
n}\to\mathcal{W}_{z_{1},\dots,z_{n}},$

is said to have the $L_{V}(-1)$-derivative property if

$(i)\qquad\partial_{z_{i}}\mathcal{F}(v_{1},z_{1};\ldots;v_{n},z_{n})=\mathcal{F}(v_{1},z_{1};\ldots;L_{V}(-1)v_{i},z_{i};\ldots;v_{n},z_{n}),$
(1.4)

for $i=1,\dots,n$, $(v_{1},\dots,v_{n})\in V$, $w^{\prime}\in W$, and

$\displaystyle(ii)\qquad\sum\limits_{i=1}^{n}\partial_{z_{i}}\mathcal{F}(v_{1},z_{1};\ldots;v_{n},z_{n})=L_{W}(-1).\mathcal{F}(v_{1},z_{1};\ldots;v_{n},z_{n}),$
(1.5)

with some action $"."$ of $L_{W}(-1)$ on
$\mathcal{F}(v_{1},z_{1};\ldots;v_{n},z_{n})$.

######  Definition 5.

A linear map

$\mathcal{F}:V^{\otimes n}\to\mathcal{W}_{z_{1},\dots,z_{n}}$

has the $L_{W}{(0)}$-conjugation property if for $(v_{1},\dots,v_{n})\in V$,
$(z_{1},\dots,z_{n})\in F_{n}\mathbb{C}$, and $z\in\mathbb{C}^{\times}$, such
that $(zz_{1},\dots,zz_{n})\in F_{n}\mathbb{C}$,

$\displaystyle
z^{L_{W}(0)}\mathcal{F}\left(v_{1},z_{1};\ldots;v_{n},z_{n}\right)=\mathcal{F}\left(z^{L_{V}{(0)}}v_{1},zz_{1};\ldots;z^{L_{V}{(0)}}v_{n},zz_{n}\right).$
(1.6)

###  1.1. E-elements

For $w\in W$, the $\overline{W}$-valued function
$E^{(n)}_{W}(v_{1}\otimes\cdots\otimes v_{n};w)$ is given by

$E^{(n)}_{W}(v_{1}\otimes\cdots\otimes
v_{n};w)(z_{1},\dots,z_{n})=E(Y_{W}(v_{1},z_{1})\cdots Y_{W}(v_{n},z_{n})w),$

where an element $E(.)\in\overline{W}$ is given by

$\langle w^{\prime},E(.)\rangle=R(\langle w^{\prime},.\rangle),$

and $R(.)$ denotes the rationalization in the sense of [5]. Namely, if a
meromorphic function $f(z_{1},\dots,z_{n})$ on a region in $\mathbb{C}^{n}$
can be analytically extended to a rational function in $(z_{1},\dots,z_{n})$,
then the notation $R(f(z_{1},\dots,z_{n}))$ is used to denote such rational
function. One defines

$E^{W;(n)}_{WV}(w;v_{1}\otimes\cdots\otimes
v_{n})=E^{(n)}_{W}(v_{1}\otimes\cdots\otimes v_{n};w),$

where $E^{W;(n)}_{WV}(w;v_{1}\otimes\cdots\otimes v_{n})$ is an element of
$\overline{W}_{z_{1},\dots,z_{n}}$. One defines

$\Phi\circ\left(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\cdots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}\right):V^{\otimes
m+n}\to\overline{W}_{z_{1},\dots,z_{m+n}},$

by

$\displaystyle(\Phi\circ(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\cdots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}}))(v_{1}\otimes\cdots\otimes v_{m+n-1})$
$\displaystyle=E(\Phi(E^{(l_{1})}_{V;\mathbf{1}}(v_{1}\otimes\cdots\otimes
v_{l_{1}})\otimes\cdots
E^{(l_{n})}_{V;\mathbf{1}}(v_{l_{1}+\cdots+l_{n-1}+1}\otimes\cdots\otimes
v_{l_{1}+\cdots+l_{n-1}+l_{n}}))),$

and

$E^{(m)}_{W}\circ_{m+1}\Phi:V^{\otimes
m+n}\to\overline{W}_{z_{1},\dots,z_{m+n-1}},$

is given by

$\displaystyle(E^{(m)}_{W}\circ_{m+1}\Phi)(v_{1}\otimes\cdots\otimes v_{m+n})$
$\displaystyle=E(E^{(m)}_{W}(v_{1}\otimes\cdots\otimes
v_{m};\Phi(v_{m+1}\otimes\cdots\otimes v_{m+n}))).$

Finally,

$E^{W;(m)}_{WV}\circ_{0}\Phi:V^{\otimes
m+n}\to\overline{W}_{z_{1},\dots,z_{m+n-1}},$

is defined by

$\displaystyle(E^{W;(m)}_{WV}\circ_{0}\Phi)(v_{1}\otimes\cdots\otimes
v_{m+n})=E(E^{W;(m)}_{WV}(\Phi(v_{1}\otimes\cdots\otimes
v_{n});v_{n+1}\otimes\cdots\otimes v_{n+m})).$

In the case that $l_{1}=\cdots=l_{i-1}=l_{i+1}=1$ and $l_{i}=m-n-1$, for some
$1\leq i\leq n$, we will use $\Phi\circ_{i}E^{(l_{i})}_{V;\;\mathbf{1}}$ to
denote $\Phi\circ(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\cdots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}})$.

###  1.2. Maps composable with vertex operators

Since $\overline{W}$-valued rational functions above are valued in
$\overline{W}$, for $z\in\mathbb{C}^{\times}$, $u,v\in V$, $w\in W$,
$Y_{V}(u,z)v\in\overline{V}$, and $Y_{W}(u,z)v\in\overline{W}$, one might not
be able to compose in general a linear map from a tensor power of $V$ to
$\overline{W}_{z_{1},\dots,z_{n}}$ with vertex operators. Thus in [5] they
consider linear maps from tensor powers of $V$ to
$\overline{W}_{z_{1},\dots,z_{n}}$ such that these maps can be composed with
vertex operators in the sense mentioned above.

######  Definition 6.

For a $V$-module $W=\coprod_{n\in\mathbb{C}}W_{(n)}$ and $m\in\mathbb{C}$, let
$P_{m}:\overline{W}\to W_{(m)}$ be the projection from $\overline{W}$ to
$W_{(m)}$. Let $\Phi:V^{\otimes n}\to\overline{W}_{z_{1},\dots,z_{n}}$ be a
linear map. For $m\in\mathbb{N}$, $\Phi$ is said [5] to be composable with $m$
vertex operators if the following conditions are satisfied:

  1. (1)

Let $l_{1},\dots,l_{n}\in\mathbb{Z}_{+}$ such that $l_{1}+\cdots+l_{n}=m+n$,
$v_{1},\dots,v_{m+n}\in V$ and $w^{\prime}\in W^{\prime}$. Set

$\displaystyle\Psi_{i}$ $\displaystyle=$ $\displaystyle
E^{(l_{i})}_{V}(v_{k_{1}}\otimes\cdots\otimes
v_{k_{i}};\mathbf{1}_{V})(z_{k_{1}},\dots,z_{k_{i}}),$

where ${k_{1}}={l_{1}+\cdots+l_{i-1}+1}$, …,
$v_{k_{i}}={l_{1}+\cdots+l_{i-1}+l_{i}}$, for $i=1,\dots,n$. Then there exist
positive integers $N^{n}_{m}(v_{i},v_{j})$ depending only on $v_{i}$ and
$v_{j}$ for $i,j=1,\dots,k$, $i\neq j$ such that the series

$\sum_{r_{1},\dots,r_{n}\in\mathbb{Z}}\langle
w^{\prime},(\Phi(P_{r_{1}}\Psi_{1}\otimes\dots\otimes
P_{r_{n}}\Psi_{n}))(\zeta_{1},\dots,\zeta_{n})\rangle,$

is absolutely convergent when
$|z_{l_{1}+\cdots+l_{i-1}+p}-\zeta_{i}|+|z_{l_{1}+\cdots+l_{j-1}+q}-\zeta_{i}|<|\zeta_{i}-\zeta_{j}|$,
for $i,j=1,\dots,k$, $i\neq j$ and for $p=1,\dots,l_{i}$ and
$q=1,\dots,l_{j}$. The sum must be analytically extended to a rational
function in $(z_{1},\dots,z_{m+n})$, independent of
$(\zeta_{1},\dots,\zeta_{n})$, with the only possible poles at $z_{i}=z_{j}$,
of order less than or equal to $N^{n}_{m}(v_{i},v_{j})$, for $i,j=1,\dots,k$,
$i\neq j$.

  2. (2)

For $v_{1},\dots,v_{m+n}\in V$, there exist positive integers
$N^{n}_{m}(v_{i},v_{j})$, depending only on $v_{i}$ and $v_{j}$, for
$i,j=1,\dots,k$, $i\neq j$, such that for $w^{\prime}\in W^{\prime}$, and
${\bf v}_{n,m}=(v_{1+m}\otimes\cdots\otimes v_{n+m})$, ${\bf
z}_{n,m}=(z_{1+m},\dots,z_{n+m})$, such that

$\sum_{q\in\mathbb{C}}\langle
w^{\prime},(E^{(m)}_{W}(v_{1}\otimes\cdots\otimes v_{m};P_{q}((\Phi({\bf
v}_{n,m}))({\bf z}_{n,m})))\rangle,$

is absolutely convergent when $z_{i}\neq z_{j}$, $i\neq j$ $|z_{i}|>|z_{k}|>0$
for $i=1,\dots,m$, and $k=m+1,\dots,m+n$, and the sum can be analytically
extended to a rational function in $(z_{1},\dots,z_{m+n})$ with the only
possible poles at $z_{i}=z_{j}$, of orders less than or equal to
$N^{n}_{m}(v_{i},v_{j})$, for $i,j=1,\dots,k$, $i\neq j$,.

In [5] one finds:

######  Proposition 1.

The subspace of $\mbox{\rm Hom}(V^{\otimes
n},\mathcal{W}_{z_{1},\dots,z_{n}})$ consisting of linear maps having the
$L(-1)$-derivative property, having the $L(0)$-conjugation property or being
composable with $m$ vertex operators is invariant under the action of $S_{n}$.

##  2\. Chain complexes and cohomologies

Let us recall the definition of shuffles [5].

######  Definition 7.

For $l\in\mathbb{N}$ and $1\leq s\leq l-1$, let $J_{l;s}$ be the set of
elements of $S_{l}$ which preserve the order of the first $s$ numbers and the
order of the last $l-s$ numbers, i.e.,

$J_{l,s}=\\{\sigma\in
S_{l}\;|\;\sigma(1)<\cdots<\sigma(s),\;\sigma(s+1)<\cdots<\sigma(l)\\}.$

The elements of $J_{l;s}$ are called shuffles. Let
$J_{l;s}^{-1}=\\{\sigma\;|\;\sigma\in J_{l;s}\\}$.

Now we introduce the notion of a $C^{n}_{m}(V,\mathcal{W})$-space:

######  Definition 8.

Let $V$ be a vertex operator algebra and $W$ a $V$-module. For
$n\in\mathbb{Z}_{+}$, let ${C}_{0}^{n}(V,\mathcal{W})$ be the vector space of
all linear maps from $V^{\otimes n}$ to $\mathcal{W}_{z_{1},\dots,z_{n}}$
satisfying the $L(-1)$-derivative property and the $L(0)$-conjugation
property. For $m$, $n\in\mathbb{Z}_{+}$, let ${C}_{m}^{n}(V,\mathcal{W})$ be
the vector spaces of all linear maps from $V^{\otimes n}$ to
$\mathcal{W}_{z_{1},\dots,z_{n}}$ composable with $m$ vertex operators, and
satisfying the $L(-1)$-derivative property, the $L(0)$-conjugation property,
and such that

$\sum_{\sigma\in
J_{l;s}^{-1}}(-1)^{|\sigma|}\sigma\left(\Phi(v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(l)})\right)=0.$ (2.1)

Using a generalization of the construciton of the vertex algebra bundle and
coordinate-free formulation of vertex operators in [1] for the case of
$\mathcal{W}$-valued forms, we obtain following

######  Lemma 1.

that an element (1.2) of $C^{n}_{m}(V,\mathcal{W})$ is invariant with respect
the group ${\rm Aut}_{z_{1},\ldots,z_{n}}{\mathcal{O}}^{(n)}$ of
$n$-dimensional independent changes of formal parameters

$(z_{1},\ldots,z_{n})\mapsto(\rho_{1}(z_{1},\ldots,z_{n}),\ldots,\rho_{n}(z_{1},\ldots,z_{n})).$

$\square$

We also find in [5]

######  Proposition 2.

Let ${C}_{m}^{0}(V,\mathcal{W})=\mathcal{W}$. Then we have
$C_{m}^{n}(V,\mathcal{W})\subset{C}_{m-1}^{n}(V,\mathcal{W})$, for
$m\in\mathbb{Z}_{+}$.

In [5] the co-boundary operator for the double complex spaces
${C}_{m}^{n}(V,\mathcal{W})$ was introduced:

${\delta}^{n}_{m}:{C}_{m}^{n}(V,\mathcal{W})\to{C}_{m-1}^{n+1}(V,\mathcal{W}).$
(2.2)

For $\Phi\in{C}_{m}^{n}(V,\mathcal{W})$, it is given by

${\delta}^{n}_{m}(\Phi)=E^{(1)}_{W}\circ_{2}\Phi+\sum_{i=1}^{n}(-1)^{i}\Phi\circ_{i}E^{(2)}_{V;\mathbf{1}}+(-1)^{n+1}\sigma_{n+1,1,\dots,n}(E^{(1)}_{W}\circ_{2}\Phi),$
(2.3)

where $\circ_{i}$ is defined in Subsection 1. Explicitly, for
$v_{1},\dots,v_{n+1}\in V$, $w^{\prime}\in W^{\prime}$ and
$(z_{1},\dots,z_{n+1})\in F_{n+1}\mathbb{C}$,

$\displaystyle\langle
w^{\prime},(({\delta}^{n}_{m}(\Phi))(v_{1}\otimes\cdots\otimes
v_{n+1}))(z_{1},\dots,z_{n+1})\rangle$ $\displaystyle=R(\langle
w^{\prime},Y_{W}(v_{1},z_{1})(\Phi(v_{2}\otimes\cdots\otimes
v_{n+1}))(z_{2},\dots,z_{n+1})\rangle)$
$\displaystyle\quad+\sum_{i=1}^{n}(-1)^{i}R(\langle
w^{\prime},(\Phi(v_{1}\otimes\cdots\otimes v_{i-1}\otimes
Y_{V}(v_{i},z_{i}-z_{i+1})v_{i+1}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\otimes\cdots\otimes
v_{n+1}))(z_{1},\dots,z_{i-1},z_{i+1},\dots,z_{n+1})\rangle)$
$\displaystyle\quad+(-1)^{n+1}R(\langle
w^{\prime},Y_{W}(v_{n+1},z_{n+1})(\Phi(v_{1}\otimes\cdots\otimes
v_{n}))(z_{1},\dots,z_{n})\rangle).$

In the case $n=2$, there is a subspace of ${C}_{0}^{2}(V,\mathcal{W})$
containing ${C}_{m}^{2}(V,\mathcal{W})$ for all $m\in\mathbb{Z}_{+}$ such that
${\delta}^{2}_{m}$ is still defined on this subspace. Let
${C}_{\frac{1}{2}}^{2}(V,\mathcal{W})$ be the subspace of
${C}_{0}^{2}(V,\mathcal{W})$ consisting of elements $\Phi$ such that for
$v_{1},v_{2},v_{3}\in V$, $w^{\prime}\in W^{\prime}$,

$\displaystyle\sum_{r\in\mathbb{C}}\big{(}\langle
w^{\prime},E^{(1)}_{W}(v_{1};P_{r}((\Phi(v_{2}\otimes
v_{3}))(z_{2}-\zeta,z_{3}-\zeta)))(z_{1},\zeta)\rangle$
$\displaystyle\quad+\langle w^{\prime},(\Phi(v_{1}\otimes
P_{r}((E^{(2)}_{V}(v_{2}\otimes
v_{3};\mathbf{1}))(z_{2}-\zeta,z_{3}-\zeta))))(z_{1},\zeta)\rangle\big{)},$

and

$\displaystyle\sum_{r\in\mathbb{C}}\big{(}\langle
w^{\prime},(\Phi(P_{r}((E^{(2)}_{V}(v_{1}\otimes
v_{2};\mathbf{1}))(z_{1}-\zeta,z_{2}-\zeta))\otimes
v_{3}))(\zeta,z_{3})\rangle$ $\displaystyle\quad+\langle
w^{\prime},E^{W;(1)}_{WV}(P_{r}((\Phi(v_{1}\otimes
v_{2}))(z_{1}-\zeta,z_{2}-\zeta));v_{3}))(\zeta,z_{3})\rangle\big{)}$

are absolutely convergent in the regions
$|z_{1}-\zeta|>|z_{2}-\zeta|,|z_{2}-\zeta|>0$ and $|\zeta-
z_{3}|>|z_{1}-\zeta|,|z_{2}-\zeta|>0$, respectively, and can be analytically
extended to rational functions in $z_{1}$ and $z_{2}$ with the only possible
poles at $z_{1},z_{2}=0$ and $z_{1}=z_{2}$. It is clear that
${C}_{m}^{2}(V,\mathcal{W})\subset{C}_{\frac{1}{2}}^{2}(V,\mathcal{W})$ for
$m\in\mathbb{Z}_{+}$. The co-boundary operator

${\delta}^{2}_{\frac{1}{2}}:{C}_{\frac{1}{2}}^{2}(V,\mathcal{W})\to{C}_{0}^{3}(V,\mathcal{W}),$
(2.4)

is defined in [5] by

$\displaystyle{\delta}^{2}_{\frac{1}{2}}(\Phi)=E^{(1)}_{W}\circ_{2}\Phi+\sum\limits_{i=1}^{2}(-1)^{i}E^{(2)}_{V,\mathbf{1}_{V}}\circ_{i}\Phi+E^{W;(1)}_{WV}\circ_{2}\Phi,$
(2.5) $\displaystyle\langle
w^{\prime},(({\delta}^{2}_{\frac{1}{2}}(\Phi))(v_{1}\otimes v_{2}\otimes
v_{3}))(z_{1},z_{2},z_{3})\rangle$ $\displaystyle=R(\langle
w^{\prime},(E^{(1)}_{W}(v_{1};\Phi(v_{2}\otimes
v_{3}))(z_{1},z_{2},z_{3})\rangle$ $\displaystyle\quad\quad+\langle
w^{\prime},(\Phi(v_{1}\otimes E^{(2)}_{V}(v_{2}\otimes
v_{3};\mathbf{1})))(z_{1},z_{2},z_{3})\rangle)$ $\displaystyle\quad-R(\langle
w^{\prime},(\Phi(E^{(2)}_{V}(v_{1}\otimes v_{2};\mathbf{1}))\otimes
v_{3}))(z_{1},z_{2},z_{3})\rangle$ $\displaystyle\quad\quad+\langle
w^{\prime},(E^{W;(1)}_{WV}(\Phi(v_{1}\otimes
v_{2});v_{3}))(z_{1},z_{2},z_{3})\rangle)$

for $w^{\prime}\in W^{\prime}$, $\Phi\in{C}_{\frac{1}{2}}^{2}(V,\mathcal{W})$,
$v_{1},v_{2},v_{3}\in V$ and $(z_{1},z_{2},z_{3})\in F_{3}\mathbb{C}$.

Consider the short sequence of the double complex spaces

$0\longrightarrow
C_{3}^{0}(V,\mathcal{W})\stackrel{{\scriptstyle\delta_{3}^{0}}}{{\longrightarrow}}C_{2}^{1}(V,\mathcal{W})\stackrel{{\scriptstyle\delta_{2}^{1}}}{{\longrightarrow}}C_{\frac{1}{2}}^{2}(V,\mathcal{W})\stackrel{{\scriptstyle\delta_{\frac{1}{2}}^{2}}}{{\longrightarrow}}C_{0}^{3}(V,\mathcal{W})\longrightarrow
0,$ (2.6)

of (2.2). The first and last arrows are trivial embeddings and projections.

In [5] we find:

######  Proposition 3.

For $n\in\mathbb{N}$ and $m\in\mathbb{Z}_{+}+1$, the co-boundary operators
(2.3) and (2.5) satisfy the chain complex conditions, i.e.,

${\delta}^{n+1}_{m-1}\circ{\delta}^{n}_{m}=0,$
${\delta}^{2}_{\frac{1}{2}}\circ{\delta}^{1}_{2}=0.$

Since

${\delta}_{2}^{1}({C}_{2}^{1}(V,\mathcal{W}))\subset{C}_{1}^{2}(V,\mathcal{W})\subset{C}_{\frac{1}{2}}^{2}(V,\mathcal{W}),$

the second formula follows from the first one, and

${\delta}^{2}_{\frac{1}{2}}\circ{\delta}^{1}_{2}={\delta}^{2}_{1}\circ{\delta}^{1}_{2}=0.$

Using the double complexes (2.2) and (2.4), for $m\in\mathbb{Z}_{+}$ and
$n\in\mathbb{N}$, one introduces in [5] the $n$-th cohomology $H^{n}_{m}(V,W)$
of a grading-restricted vertex algebra $V$ with coefficient in $W$, and
composable with $m$ vertex operators to be

$H_{m}^{n}(V,\mathcal{W})=\ker\delta^{n}_{m}/\mbox{\rm
im}\;\delta^{n-1}_{m+1},$
$H^{2}_{\frac{1}{2}}(V,\mathcal{W})=\ker\delta^{2}_{\frac{1}{2}}/\mbox{\rm
im}\;\delta_{2}^{1}.$

##  3\. The $\epsilon$-product of $C^{n}_{m}(V,\mathcal{W})$-spaces

In this section we introduce definition of the $\epsilon$-product of double
complex spaces $C^{n}_{m}(V,\mathcal{W})$ with the image in another double
complex space coherent with respect to the original differential (2.2), and
satisfying the symmetry (2.1), $L_{V}(0)$-conjugation (1.6), and
$L_{V}(-1)$-derivative (1.4) properties and derive an analogue of Leibniz
formula.

###  3.1. Motivation and geometrical interpretation

The structure of $C^{n}_{m}(V,\mathcal{W})$-spaces is quite complicated and it
is difficult to introduce algebraically a product of its elements. In order to
define an appropriate product of two $C^{n}_{m}(V,\mathcal{W})$-spaces we
first have to interpret them geometrically. Basically, a
$C^{n}_{m}(V,\mathcal{W})$-space must be associated with a certain model
space, the algebraic $\mathcal{W}$-language should be transferred to a
geometrical one, two model spaces should be ”connected” appropriately, and,
finally, a product should be defined.

For two $\mathcal{W}_{x_{1},\ldots,x_{k}}$\- and
$\mathcal{W}_{y_{1},\ldots,y_{n}}$-spaces we first associate formal complex
parameters in the sets $(x_{1},\ldots,x_{k})$ and $(y_{1},\ldots,y_{n})$ to
parameters of two auxiliary spaces. Then we describe a geometric procedure to
form a resulting model space by combining two original model spaces. Formal
parameters of $\mathcal{W}_{z_{1},\ldots,z_{k+n}}$ should be then identified
with parameters of the resulting space.

Note that according to our assumption, $(x_{1},\ldots,x_{k})\in
F_{k}\mathbb{C}$, and $(y_{1},\ldots,y_{n})\in F_{n}\mathbb{C}$. As it follows
from the definition of the configuration space $F_{n}\mathbb{C}$ in Subsection
1, in the case of coincidence of two formal parameters they are excluded from
$F_{n}\mathbb{C}$. In general, it may happen that some number $r$ of formal
parameters of $\mathcal{W}_{x_{1},\ldots,x_{k}}$ coincide with some $r$ formal
parameters of $\mathcal{W}_{y_{1},\ldots,y_{n}}$ on the whole $\mathbb{C}$ (or
on a domain of definition). Then, we exclude one formal parameter from each
coinciding pair. We require that the set of formal parameters

$(z_{1},\ldots,z_{k+n-r})=(\ldots,{x}_{i_{1}},\ldots,{x}_{i_{r}},\ldots;\ldots,\widehat{y}_{j_{1}},\ldots,\widehat{y}_{j_{r}},\ldots),$
(3.1)

where $\;\widehat{.}\;$ denotes the exclusion of corresponding formal
parameter for $x_{i_{l}}=y_{j_{l}}$, $1\leq l\leq r$, for the resulting model
space would belong to $F_{k+n-r}\mathbb{C}$. We denote this operation of
formal parameters exclusion by
$\widehat{R}\;\mathcal{F}(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n};\epsilon)$.

Now we formulate the definition of the $\epsilon$-product of two
$C^{n}_{m}(V,\mathcal{W})$-spaces:

######  Definition 9.

For $\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\in C^{k}_{m}(V,\mathcal{W})$,
and $\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$ the product

$\displaystyle\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\cdot_{\epsilon}\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})$
(3.2)
$\displaystyle\qquad\qquad\mapsto\widehat{R}\;\mathcal{F}\left(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon\right),$

is a $\mathcal{W}_{z_{1},\ldots,z_{k+n-r}}$-valued rational form

$\displaystyle\langle
w^{\prime},\widehat{R}\;\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon)\rangle$
(3.3) $\displaystyle\quad=\sum_{u\in V}\langle
w^{\prime},Y^{W}_{WV}\left(\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k}),\zeta_{1}\right)\;u\rangle$
$\displaystyle\quad\langle
w^{\prime},Y^{W}_{WV}\left(\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;{v^{\prime}}_{i_{1}},\widehat{y}_{i_{1}};\ldots;\ldots;{v^{\prime}}_{j_{r}},\widehat{y}_{j_{r}};\ldots;v^{\prime}_{n},y_{n}),\zeta_{2}\right)\;\overline{u}\rangle,$

via (1.1), parametrized by $\zeta_{1}$, $\zeta_{2}\in\mathbb{C}$, and we
exclude all monomials $(x_{i_{l}}-y_{j_{l}})$, $1\leq l\leq r$, from (3.5).
The sum is taken over any $V_{l}$-basis $\\{u\\}$, where $\overline{u}$ is the
dual of $u$ with respect to a non-degenerate bilinear form $\langle.\
,.\rangle_{\lambda}$, (5.8) over $V$, (see Appendix 5).

######  Remark 1.

Due to the symmetry of the geometrical interpretation describe above, we could
exclude from the set $(x_{1},\ldots,x_{k})$ in (3.5) $r$ formal parameters
which belong to coinciding pairs resulting to the same definition of the
$\epsilon$-product.

By the standard reasoning [2, 12], (3.5) does not depend on the choice of a
basis of $u\in V_{l}$, $l\in\mathbb{Z}$. In the case when multiplied forms
$\mathcal{F}$ do not contain $V$-elements, i.e., for $\Phi$,
$\Psi\in\mathcal{W}$, (3.5) defines the product $\Phi\cdot_{\epsilon}\Psi$
associated to a rational function:

$\displaystyle{\mathcal{R}(\epsilon)}=\sum_{l\in\mathbb{Z}}\epsilon^{l}\sum_{u\in
V_{l}}\langle
w^{\prime},Y^{W}_{WV}\left(\Phi,\zeta_{1}\right)\;u\rangle\langle
w^{\prime},Y^{W}_{WV}\left(\Psi,\zeta_{2}\right)\;\overline{u}\rangle,$ (3.4)

which defines $\mathcal{F}(\epsilon)\in\mathcal{W}$ via
$\mathcal{R}(\epsilon)=\langle w^{\prime},\mathcal{F}(\epsilon)\rangle$.

###  3.2. Convergence and properties of of the $\epsilon$-product

In order to prove convergence of a product of elements of two spaces
$\mathcal{W}_{x_{1},\ldots,x_{k}}$ and $\mathcal{W}_{y_{1},\ldots,y_{n}}$ of
rational $\mathcal{W}$-valued forms, we have to use a geometrical
interpretation [7, 11]. Recall that a $\mathcal{W}_{z_{1},\ldots,z_{n}}$-space
is defined by means of matrix elements of the form (1.1). For a vertex algebra
$V$, this corresponds [2] to a matrix element of a number of $V$-vertex
operators with formal parameters identified with local coordinates on a
Riemann sphere. Geometrically, each space $\mathcal{W}_{z_{1},\ldots,z_{n}}$
can be also associated to a Riemann sphere with a few marked points, and local
coordinates vanishing at these points [7]. An extra point can be associated to
a center of an annulus used in order to sew the sphere with another sphere.
The product (3.5) has then a geometric interpretation. The resulting model
space would also be associated to a Riemann sphere formed as a result of
sewing procedure. In Appendix 6 we describe explicitly the geometrical
procedure of sewing of two spheres [11].

Let us identify (as in [7, 11, 12, 10, 3, 1]) two sets $(x_{1},\ldots,x_{k})$
and $(y_{1},\ldots,y_{n})$ of complex formal parameters, with local
coordinates of two sets of points on the first and the second Riemann spheres
correspondingly. Identify complex parameters $\zeta_{1}$, $\zeta_{2}$ of (3.5)
with coordinates (6.1) of the annuluses (6.3). After identification of
annuluses $\mathcal{A}_{a}$ and $\mathcal{A}_{\overline{a}}$, $r$ coinciding
coordinates may occur. This takes into account case of coinciding formal
parameters. In this way, we construct the map (3.2).

As we see in (3.5), the product is defined by a sum of products of matrix
elements [2] associated to each of two spheres. Such sum is supposed to
describe a $\mathcal{W}$-valued rational differential form defined on a sphere
formed as a result of geometrical sewing [11] of two initial spheres. Since
two initial spaces $\mathcal{W}_{x_{1},\ldots,x_{k}}$ and
$\mathcal{W}_{y_{1},\ldots,y_{n}}$ are defined through rational-valued forms
expressed by matrix elements of the form (1.1). We then arrive at the
resulting product defines a $\mathcal{W}_{z_{1},\ldots,z_{k+n-r}}$-valued
rational form by means of an absolute convergent matrix element on the
resulting sphere. The complex sewing parameter, parameterizing the module
space of sewin spheres, parametrizes also the product of $\mathcal{W}$-spaces.

Next, we formulate

######  Definition 10.

We define the action of an element $\sigma\in S_{k+n-r}$ on the product of
$\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\in\mathcal{W}_{x_{1},\ldots,x_{k}}$,
and
$\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in\mathcal{W}_{y_{1},\ldots,y_{n}}$,
as

$\displaystyle\langle
w^{\prime},\sigma(\widehat{R}\;\mathcal{F})(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon)\rangle$
(3.5) $\displaystyle\qquad=\langle
w^{\prime},\mathcal{F}(\widetilde{v}_{\sigma(1)},z_{\sigma(1)};\ldots;\widetilde{v}_{\sigma(k+n-r)},z_{\sigma(k+n-r)};\epsilon)\rangle$
$\displaystyle\qquad=\sum_{u\in V}\langle
w^{\prime},Y^{W}_{WV}\left(\mathcal{F}(\widetilde{v}_{\sigma(1)},z_{\sigma(1)};\ldots;\widetilde{v}_{\sigma(k)},z_{\sigma(k)}),\zeta_{1}\right)\;u\rangle$
$\displaystyle\qquad\langle
w^{\prime},Y^{W}_{WV}\left(\mathcal{F}(\widetilde{v}_{\sigma(k+1)},z_{\sigma(k+1)};\ldots;\widetilde{v}_{\sigma(k+n-r)},z_{\sigma(k+n-r)}),\zeta_{2}\right)\;\overline{u}\rangle,$

where by $(\widetilde{v}_{\sigma(1)},\ldots,\widetilde{v}_{\sigma(k+n-r)})$ we
denote a permutation of

$(\widetilde{v}_{1},\ldots,\widetilde{v}_{k+n-r})=(v_{1},\ldots;v_{k};\ldots,\widehat{v}^{\prime}_{j_{1}},\ldots,\widehat{v}^{\prime}_{j_{r}},\ldots).$
(3.6)

Let $t$ be the number of common vertex operators the mappings
$\mathcal{F}(v_{1},x_{1}$; $\ldots$; $v_{k},x_{k})\in
C^{k}_{m}(V,\mathcal{W})$ and
$\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in
C^{n}_{m^{\prime}}(V,\mathcal{W})$, are composable with. The rational form
corresponding to the $\epsilon$-product
$\widehat{R}\mathcal{F}\left(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon\right)$
converges in $\epsilon$, and satisfies (2.1), $L_{V}(0)$-conjugation (1.6) and
$L_{V}(-1)$-derivative (1.4) properties. Using Definition 8 of
$C^{n}_{m}(V,\mathcal{W})$-space and Definition 6 of mappsings composable with
vertex operators, we then have

######  Proposition 4.

For $\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\in C_{m}^{k}(V,\mathcal{W})$
and $\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$, the product
$\widehat{R}\mathcal{F}\left(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon\right)$
(3.5) belongs to the space $C^{k+n-r}_{m+m^{\prime}-t}(V,\mathcal{W})$, i.e.,

$\cdot_{\epsilon}:C^{k}_{m}(V,\mathcal{W})\times
C_{m^{\prime}}^{n}(V,\mathcal{W})\to
C_{m+m^{\prime}-t}^{k+n-r}(V,\mathcal{W}).$ (3.7)

$\square$

######  Remark 2.

Note that due to (5.3), in Definition (3.5) it is assumed that
$\mathcal{F}(v_{1},x_{1}$; $\ldots$ ; $v_{k},x_{k})$ and
$\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})$ are composable
with the $V$-module $W$ vertex operators $Y_{W}(u,-\zeta_{1})$ and
$Y_{W}(\overline{u},-\zeta_{2})$ correspondingly. The product (3.5) is
actually defined by a sum of products of matrix elements of ordinary
$V$-module $W$ vertex operators acting on $\mathcal{W}$-elements. The elements
$u\in V$ and $\overline{u}\in V^{\prime}$ are connected by (5.9), and
$\zeta_{1}$, $\zeta_{2}$ are related by (6.4). The form of the product defined
above is natural in terms of the theory of chacaters for vertex operator
algebras [10, 3, 12].

######  Remark 3.

For purposes of construction of cohomological invariant, we do not exclude in
this paper the case of $r$ pais of common formal parameters $x_{i}=y_{j}$,
$1\leq i\leq k$, $1\leq j\leq n$, for
$\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\in C_{m}^{k}(V,\mathcal{W})$ and
$\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$ in Proposition 1. Such formal parameter
pairs are excluded from the right hand side of the map (3.7).

We then have two corollaries:

######  Corollary 1.

For the spaces $\mathcal{W}_{x_{1},\ldots,x_{k}}$ and
$\mathcal{W}_{y_{1},\ldots,y_{n}}$ with the product (3.5)
$\mathcal{F}\in\mathcal{W}_{z_{1},\ldots,z_{k+n-r}}$, the subspace of
$\mbox{\rm Hom}(V^{\otimes n},{\mathcal{W}}_{z_{1},\ldots,z_{k+n-r}}$
consisting of linear maps having the $L_{W}(-1)$-derivative property, having
the $L_{V}(0)$-conjugation property or being composable with $m$ vertex
operators is invariant under the action of $S_{k+n-r}$.

######  Corollary 2.

For a fixed set $(v_{1},\ldots v_{k};v_{k+1},\ldots,v_{k+n})\in V$ of vertex
algebra elements, and fixed $k+n$, and $m+m^{\prime}$, the $\epsilon$-product
$\mathcal{F}(v_{1},z_{1};\ldots;v_{k},z_{k};v_{k+1},z_{k+1};\ldots$ ;
$v_{k+n-r},y_{k+n-r};\epsilon)$,

$\cdot_{\epsilon}:C^{k}_{m}(V,\mathcal{W})\times
C^{n}_{m^{\prime}}(V,\mathcal{W})\rightarrow
C^{k+n-r}_{m+m^{\prime}-t}(V,\mathcal{W}),$

of the spaces $C^{k}_{m}(V,\mathcal{W})$ and
$C^{n}_{m^{\prime}}(V,\mathcal{W})$, for all choices of $k$, $n$, $m$,
$m^{\prime}\geq 0$, is the same element of
$C^{k+n-r}_{m+m^{\prime}-t}(V,\mathcal{W})$ for all possible $k\geq 0$.
$\square$

By Lemma 1, elements of the space $C^{k+n-r}_{m+m^{\prime}-t}$ resulting from
the $\epsilon$-product are invariant with respect to changes of formal
parameters of the group ${\rm
Aut}_{z_{1},\ldots,z_{k+n-r}}{\mathcal{O}}^{(k+n-r)}$.

We then have

######  Definition 11.

For fixed sets $(v_{1},\ldots,v_{k})$,
$(v^{\prime}_{1},\ldots,v^{\prime}_{n})\in V$,
$(x_{1},\ldots,x_{k})\in\mathbb{C}$, $(y_{1},\ldots,y_{n}$) $\in\mathbb{C}$,
we call the set of all
$\mathcal{W}_{x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}}$-valued rational forms
$\widehat{R}\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k}$ ;
$v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon)$ defined by (3.5)
with the parameter $\epsilon$ exhausting all possible values, the complete
product of the spaces $\mathcal{W}_{x_{1},\ldots,x_{k}}$ and
$\mathcal{W}_{y_{1},\ldots,y_{n}}$.

###  3.3. Coboundary operator acting on the product space

In Proposition 4 we proved that the product (3.5) of elements
$\mathcal{F}_{1}\mathbb{C}_{m}^{k}(V,\mathcal{W})$ and $\mathcal{F}_{2}\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$ belongs to
$C^{k+n-r}_{m+m^{\prime}-t}(V,\mathcal{W})$. Thus, the product admits the
action ot the differential operator $\delta^{k+n-r}_{m+m^{\prime}-t}$ defined
in (2.2) where $r$ is the number of common formal parameters, and $t$ the
number of commpon composable vertex operators for $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$. The co-boundary operator (2.2) possesses a variation of
Leibniz law with respect to the product (3.5). We then have

######  Proposition 5.

For $\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\in C_{m}^{k}(V,\mathcal{W})$
and $\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$, the action of
$\delta_{m+m^{\prime}-t}^{k+n-r}$ on their product (3.5) is given by

$\displaystyle\delta_{m+m^{\prime}-t}^{k+n-r}\left(\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\cdot_{\epsilon}\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})\right)$
$\displaystyle\qquad=\left(\delta^{k}_{m}\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\right)\cdot_{\epsilon}\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n},z_{k+n-r})$
$\displaystyle\;+(-1)^{k}\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\cdot_{\epsilon}\left(\delta^{n-r}_{m^{\prime}-t}\mathcal{F}(\widetilde{v}_{1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r})\right),$

where we use the notation as in (3.1) and (3.6).

Appendix 7 contains the proof of this Proposition.

######  Remark 4.

Checking (2.2) we see that an extra arbitrary vertex algebra element
$v_{n+1}\in V$, as well as corresponding extra arbitrary formal parameter
$z_{n+1}$ appear as a result of the action of $\delta^{n}_{m}$ on
$\mathcal{F}\in C^{n}_{m}(V,\mathcal{W})$ mapping it to
$C^{n+1}_{m-1}(V,\mathcal{W})$. In application to the $\epsilon$-product (3.5)
these extra arbitrary elements are involved in the definition of the action of
$\delta_{m+m^{\prime}-t}^{k+n-r}$ on
$\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k})\cdot_{\epsilon}\mathcal{F}(v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n})$.

Note that both sides of (5) belong to the space
$C_{m+m^{\prime}-t+1}^{n+n^{\prime}-r+1}(V,W)$. The co-boundary operators
$\delta^{n}_{m}$ and $\delta^{n^{\prime}}_{m^{\prime}}$ in (5) do not include
the number of common vertex algebra elements (and formal parameters), neither
the number of common vertex operators corresponding mappings composable with.
The dependence on common vertex algebra elements, parameters, and composable
vertex operators is taken into account in mappings multiplying the action of
co-boundary operators on $\Phi$.

Finally, we have the following

######  Corollary 3.

The multiplication (3.5) extends the chain-cochain complex structure of
Proposition 3 to all products $C^{k}_{m}(V,\mathcal{W})\times
C^{n}_{m^{\prime}}(V,\mathcal{W})$, $k$, $n\geq 0$, $m$, $m^{\prime}\geq 0$. ∎

######  Corollary 4.

The product (3.5) and the product operator (2.2) endow the space
$C^{k}_{m}(V,\mathcal{W})$ $\times$ $C^{n}_{m}(V,\mathcal{W})$, $k$, $n\geq
0$, $m$, $m^{\prime}\geq 0$, with the structure of a bi-graded differential
algebra
$\mathcal{G}(V,\mathcal{W},\cdot_{\epsilon},\delta^{k+n-r}_{m+m^{\prime}-t})$.
∎

For elements of the spaces $C^{2}_{ex}(V,\mathcal{W})$ we have the following

######  Corollary 5.

The product of elements of the spaces $C^{2}_{ex}(V,\mathcal{W})$ and
$C^{n}_{m}(V,\mathcal{W})$ is given by (3.5),

$\cdot_{\epsilon}:C^{2}_{ex}(V,\mathcal{W})\times C^{n}_{m}(V,\mathcal{W})\to
C^{n+2-r}_{m}(V,\mathcal{W}),$ (3.9)

and, in particular,

$\cdot_{\epsilon}:C^{2}_{ex}(V,\mathcal{W})\times C^{2}_{ex}(V,\mathcal{W})\to
C^{4-r}_{0}(V,\mathcal{W}).$

$\square$

###  3.4. The commutator

Let us consider the mappings $\Phi(v_{1},z_{1}$ ; $\ldots$; $v_{n},z_{k})$
$\in$ $C_{m}^{k}(V,\mathcal{W})$, and
$\Psi(v_{k+1},z_{k+1};\ldots;v_{k+n},z_{k+n})\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$, with have $r$ common vertex algebra
elements (and, correspondingly, $r$ formal variables), and $t$ common vertex
operators mappings $\Phi$ and $\Psi$ are composable with. Note that when
applying the co-boundary operators (2.3) and (2.5) to a map
$\Phi(v_{1},z_{1};\ldots;v_{n},z_{n})\in C^{n}_{m}(V,\mathcal{W})$,

$\delta^{n}_{m}:\Phi(v_{1},z_{1};\ldots;v_{n},z_{n})\to\Phi(v^{\prime}_{1},z^{\prime}_{1};\ldots;v^{\prime}_{n+1},z^{\prime}_{n+1})\in
C^{n+1}_{m-1}(V,\mathcal{W}),$

one does not necessary assume that we keep the same set of vertex algebra
elements/formal parameters and vertex operators composable with for
$\delta^{n}_{m}\Phi$, though it might happen that some of them could be common
with $\Phi$.

Let us define an extra product (related to the $\epsilon$-product) the product
of $\Phi$ and $\Psi$,

$\displaystyle\Phi\cdot\Psi:V^{\otimes(k+n-r)}\to\mathcal{W}_{z_{1},\ldots,z_{k+n-r}},\;$
(3.10)
$\displaystyle\Phi\cdot\Psi=\left[\Phi,_{\cdot\epsilon}\Psi\right]=\Phi\cdot_{\epsilon}\Psi-\Psi\cdot_{\epsilon}\Phi,$
(3.11)

where brackets denote ordinary commutator in
$\mathcal{W}_{z_{1},\ldots,z_{k+n-r}}$. Due to the properties of the maps
$\Phi\in C_{m}^{k}(V,\mathcal{W})$ and $\Psi\in
C_{m^{\prime}}^{n}(V,\mathcal{W})$, the map $(\Phi\cdot_{\epsilon}\Psi)$
belongs to the space $C_{m+m^{\prime}-t}^{k+n-r}(V,\mathcal{W})$. For $k=n$
and

$\Psi(v_{n+1},z_{n+1};\ldots;v_{2n},z_{2n})=\Phi(v_{1},z_{1};\ldots;v_{n},z_{n}),$

we obtain from (3.11) and (3.5) that

$\displaystyle\Phi(v_{1},z_{1};\ldots;v_{n},z_{n})\cdot\Phi(v_{1},z_{1};\ldots;v_{n},z_{n})=0.$
(3.12)

##  4\. The invariants

In this section we provide the main result of the paper by deriving the
simplest cohomological invariants associated to the short double complex (2.4)
for a grading-restricted vertex algebra.

Let us give first some further definitions. In this section we skip the
dependence on vertex algebra elements and formal parameters in notations for
elements of $C_{n}^{m}(V,\mathcal{W})$.

######  Definition 12.

In analogy with differential forms, we call a map $\Phi\in
C_{m}^{n}(V,\mathcal{W})$ closed if

$\delta^{n}_{m}\Phi=0.$

For $m\geq 1$, we call it exact if there exists $\Psi\in
C_{m-1}^{n+1}(V,\mathcal{W})$ such that

$\Psi=\delta^{n}_{m}\Phi.$

######  Definition 13.

For $\Phi\in{C}^{n}_{m}(V,\mathcal{W})$ we call the cohomology class of
mappings $\left[\Phi\right]$ the set of all closed forms that differs from
$\Phi$ by an exact mapping, i.e., for $\chi\in{C}^{n-1}_{m+1}$,

$\left[\Phi\right]=\Phi+\delta^{n-1}_{m+1}\chi,$

(we assume that both parts of the last formula belongs to the same space
${C}^{n}_{m}(V,\mathcal{W})$).

Under a natural extra condition, the short double complex (2.6) allows us to
establish relations among elements of double complex spaces. In particular, we
require that for a pair of double complex spaces
$C^{n_{1}}_{k_{1}}(V,\mathcal{W})$ and $C^{n_{2}}_{k_{2}}(V,\mathcal{W})$
there exist subspaces ${C^{\prime}}^{n_{1}}_{k_{1}}(V,\mathcal{W})\subset
C^{n_{1}}_{k_{1}}(V,\mathcal{W})$ and
${C^{\prime}}^{n_{2}}_{k_{2}}(V,\mathcal{W})\subset
C^{n_{2}}_{k_{2}}(V,\mathcal{W})$ such that for
$\Phi_{1}\in{C^{\prime}}^{n_{1}}_{k_{1}}(V,\mathcal{W})$ and $\Phi_{2}\in
C^{\prime n_{2}}_{k_{2}}(V,\mathcal{W})$,

$\Phi_{1}\cdot\delta^{n_{2}}_{k_{2}}\Phi_{2}=0,$ (4.1)

namely, $\Phi_{1}$ supposed to be orthogonal to
$\delta^{n_{2}}_{k_{2}}\Phi_{2}$ (i.e., commutative with respect to the
product (3.11)). We call this the orthogonality condition for mappings and
actions of co-boundary operators for a double complex. It is easy to see that
the assumption to belong to the same double complex space for both sides of
the equations following from the orthogonality condition applies the bi-
grading condition on double complex spaces. Note that in the case of
differential forms considered on a smooth manifold, the Frobenius theorem for
a distribution provides the orthogonality condition. In this Section we derive
algebraic relations occurring from the orthogonality condition on the short
double complex (2.6). We formulate

######  Proposition 6.

The orthogonality condition for the short double complex sequence (2.6)
determines the cohomological classes:

$\left[\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\right],\;\left[\left(\delta^{0}_{3}\chi\right)\cdot\chi\right],\;\left[\left(\delta^{1}_{t}\alpha\right)\cdot\alpha\right],$
(4.2)

for $0\leq t\leq 2$, with non-vanishing
$\left(\delta^{1}_{2}\Phi\right)\cdot\Phi$,
$\left(\delta^{0}_{3}\chi\right)\cdot\chi$, and
$\left(\delta^{1}_{t}\alpha\right)\cdot\alpha$. These classes are independent
on the choice of $\Phi\in C^{1}_{2}(V,\mathcal{W})$, $\chi\in
C^{0}_{3}(V,\mathcal{W})$, and $\alpha\in C^{1}_{t}(V,\mathcal{W})$.

######  Remark 5.

A cohomology class with vanishing
$\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha$ is given by
$\left[\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha\right]$.

###### Proof.

Let us consider two maps $\chi\in C^{0}_{3}(V,\mathcal{W})$, $\Phi\in
C^{1}_{2}(V,\mathcal{W})$. We require them to be orthogonal, i.e.,

$\Phi\cdot\delta^{0}_{3}\chi=0.$ (4.3)

Thus, there exists $\alpha\in C^{n}_{m}(V,\mathcal{W})$, such that

$\delta^{0}_{3}\chi=\Phi\cdot\alpha,$ (4.4)

and $1=1+n-r$, $2=2+m-t$, i.e., $n=r$, which leads to $r=1$; $m=t$, $0\leq
t\leq 2$, i.e., $\alpha\in C^{1}_{t}(V,\mathcal{W})$. All other orthogonality
conditions for the short sequence (2.6) does not allow relations of the form
(4.4).

Consider now (4.3). We obtain, using (5)

$\delta^{2-r^{\prime}}_{4-t^{\prime}}(\Phi\cdot\delta^{0}_{3}\chi)=\left(\delta^{1}_{2}\Phi\right)\cdot\delta^{0}_{3}\chi+\Phi\cdot\delta^{1}_{2}\delta^{0}_{3}\chi=\left(\delta^{1}_{2}\Phi\right)\cdot\delta^{0}_{3}\chi=\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha.$

Thus

$0=\delta^{3-r^{\prime}}_{3-t^{\prime}}\delta^{2-r^{\prime}}_{4-t^{\prime}}(\Phi\cdot\delta^{0}_{3}\chi)=\delta^{3-r^{\prime}}_{3-t^{\prime}}\left(\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha.\right),$

and $\left(\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha\right)$ is
closed. At the same time, from (4.3) it follows that

$0=\delta^{1}_{2}\Phi\cdot\delta^{0}_{3}\chi-\Phi\cdot\delta^{1}_{2}\delta^{0}_{3}\chi=\left(\Phi\cdot\delta^{0}_{3}\chi\right).$

Thus

$\delta^{1}_{2}\Phi\cdot\delta^{0}_{3}\chi=\delta^{1}_{2}\Phi\cdot\Phi\cdot\alpha=0.$

Consider (4.4). Acting by $\delta^{1}_{2}$ and substituting back we obtain

$0=\delta^{1}_{2}\delta^{0}_{3}\chi=\delta^{1}_{2}(\Phi\cdot\alpha)=\delta^{1}_{2}(\Phi)\cdot\alpha-\Phi\cdot\delta^{1}_{t}\alpha.$

thus

$\delta^{1}_{2}(\Phi)\cdot\alpha=\Phi\cdot\delta^{1}_{t}\alpha.$

The last equality trivializes on applying $\delta^{3}_{t+1}$ to both sides.

Let us show now the non-vanishing property of
$\left(\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\right)$. Indeed, suppose
$\left(\delta^{1}_{2}\Phi\right)\cdot\Phi=0$. Then there exists $\gamma\in
C^{n}_{m}(V,\mathcal{W})$, such that $\delta^{1}_{2}\Phi=\gamma\cdot\Phi$.
Both sides of the last equality should belong to the same double complex space
but one can see that it is not possible. Thus,
$\left(\delta^{1}_{2}\Phi\right)\cdot\Phi$ is non-vanishing. One proves in the
same way that $\left(\delta^{0}_{3}\chi\right)\cdot\chi$ and
$\left(\delta^{1}_{t}\alpha\right)\cdot\alpha$ do not vanish too. Now let us
show that $\left[\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\right]$ is
invariant, i.e., it does not depend on the choice of $\Phi\in
C^{1}_{2}(V,\mathcal{W})$. Substitute $\Phi$ by $\left(\Phi+\eta\right)\in
C^{1}_{2}(V,\mathcal{W})$. We have

$\displaystyle\left(\delta^{1}_{2}\left(\Phi+\eta\right)\right)\cdot\left(\Phi+\eta\right)$
$\displaystyle=$
$\displaystyle\left(\delta^{1}_{2}\Phi\right)\cdot\Phi+\left(\left(\delta^{1}_{2}\Phi\right)\cdot\eta-\Phi\cdot\delta^{1}_{2}\eta\right)$
(4.5) $\displaystyle+$
$\displaystyle\left(\Phi\cdot\delta^{1}_{2}\eta+\delta^{1}_{2}\eta\cdot\Phi\right)+\left(\delta^{1}_{2}\eta\right)\cdot\eta.$

Since

$\left(\Phi\cdot\delta^{1}_{2}\eta+\left(\delta^{1}_{2}\eta\right)\cdot\Phi\right)=\Phi\delta^{1}_{2}\eta-(\delta^{1}_{2}\eta)\Phi+\left(\delta^{1}_{2}\eta\right)\Phi-\Phi\;\delta^{1}_{2}\eta=0,$

then (4.5) represents the same cohomology class
$\left[\left(\delta^{1}_{2}\Phi\right)\cdot\Phi\cdot\alpha\right]$. The same
folds for $\left[\left(\delta^{0}_{3}\chi\right)\cdot\chi\right]$, and
$\left[\left(\delta^{1}_{t}\alpha\right)\cdot\alpha\right]$. ∎

######  Remark 6.

Due to Proposition 1, all chahomological classes are invariant with respect to
correponding group ${\rm Aut}_{z_{1},\ldots,z_{n}}{\mathcal{O}}^{(n)}$ changes
of formal parameters.

The orthogonality condition for a double complex sequence (2.6), together with
the action of co-boundary operators (2.2) and (2.4), and the multiplication
formulas (3.11)–(5), define a differential bi-graded algebra depending on
vertex algebra elements and formal parameters. In particular, for the short
sequence (2.6), we obtain in this way the generators and commutation relations
for a continual Lie algebra $\mathcal{G}(V)$ (a generalization of ordinary Lie
algebras with continual space of roots, c.f. [9]) with the continual root
space represented by a grading-restricted vertex algebra $V$.

######  Lemma 2.

For the short sequence (2.6) we get a continual Lie algebra $\mathcal{G}(V)$
with generators

$\left\\{\Phi(v_{1}),\;\chi,\;\alpha(v_{2}),\;\delta^{1}_{2}\Phi(v_{1}),\;\delta^{0}_{3}\chi,\;\delta^{1}_{t}\alpha(v_{2}),0\leq
t\leq 2\right\\},$ (4.6)

and commutation relations for a continual Lie algebra $\mathcal{G}(V)$

$\displaystyle\;\;\Phi\cdot\delta^{1}_{t}\alpha$ $\displaystyle=$
$\displaystyle\alpha\cdot\delta^{1}_{2}\Phi\neq 0,$
$\displaystyle\delta^{0}_{3}\chi$ $\displaystyle=$
$\displaystyle\Phi\cdot\alpha,$ (4.7)

with all other relations being trivial. The sum of cohomological classes (4.2)
provides an invariant of $\mathcal{G}(V)$.

###### Proof.

Recall that $\Phi(v_{1})(z_{1})\in C^{1}_{2}(V,\mathcal{W})$, $\chi\in
C^{0}_{3}(V,\mathcal{W})$, $\alpha\in C^{1}_{t}(V,\mathcal{W})$, $0\leq t\leq
2$. One easily checks the commutation relations coming from the orthogonality
and bi-grading conditions. Further applications of (2.2), (2.4), and (4.1) to
(2.6) lead to trivial results. $\Phi\cdot\delta^{1}_{t}\alpha\neq 0$ is proven
by contradiction. It is easy to check Jacobi identities for (4.6) and (2).
With a redefinition

$\displaystyle H$ $\displaystyle=$ $\displaystyle\delta^{0}_{3}\chi,$
$\displaystyle H^{*}$ $\displaystyle=$ $\displaystyle\chi,$ $\displaystyle
X_{+}(v_{1})$ $\displaystyle=$ $\displaystyle\Phi(v_{1}),$ $\displaystyle
X_{-}(v_{2})$ $\displaystyle=$ $\displaystyle\alpha(v_{2}),$ $\displaystyle
Y_{+}(v_{1})$ $\displaystyle=$ $\displaystyle\delta^{1}_{2}\Phi(v_{1}),$
$\displaystyle Y_{-}(v_{2})$ $\displaystyle=$
$\displaystyle\delta^{1}_{t}\alpha(v_{2}),$ (4.8)

the commutation relations (2) become:

$\displaystyle\left[X_{+}(v_{1}),X_{-}(v_{2})\right]$ $\displaystyle=$
$\displaystyle H,$ $\displaystyle\left[X_{+}(v_{1}),Y_{-}(v_{1})\right]$
$\displaystyle=$ $\displaystyle\left[X_{-}(v_{2}),Y_{+}(v_{1})\right],$

i.e., the orthogonality condition brings about a representation of an
affinization [8] of continual counterpart of the Lie algebra $sl_{2}$. Vertex
algebra elements in (4) play the role of roots belonging to continual non-
commutative root space given by a vertex algebra $V$. ∎

## Acknowledgments

The author would like to thank Y.-Z. Huang, H. V. Lê, and P. Somberg for
related discussions. Research of the author was supported by the GACR project
18-00496S and RVO: 67985840.

##  5\. Appendix: Grading-restricted vertex algebras and their modules

In this section, following [5] we recall basic properties of grading-
restricted vertex algebras and their grading-restricted generalized modules,
useful for our purposes in later sections. We work over the base field
$\mathbb{C}$ of complex numbers. A vertex algebra $(V,Y_{V},\mathbf{1})$, cf.
[8], consists of a $\mathbb{Z}$-graded complex vector space

$V=\bigoplus_{n\in\mathbb{Z}}\,V_{(n)},\quad\dim V_{(n)}<\infty\,\,\mbox{for
each}\,\,n\in\mathbb{Z},$

and linear map

$Y_{V}:V\rightarrow{\rm End\;}(V)[[z,z^{-1}]],$

for a formal parameter $z$ and a distinguished vector $\mathbf{1_{V}}\in V$.
The evaluation of $Y_{V}$ on $v\in V$ is the vertex operator

$Y_{V}(v)\equiv Y_{V}(v,z)=\sum_{n\in\mathbb{Z}}v(n)z^{-n-1},$

with components

$(Y_{V}(v))_{n}=v(n)\in{\rm End\;}(V),$

where $Y_{V}(v,z)\mathbf{1}=v+O(z)$. Now we describe further restrictions [5],
defining a grading-restricted vertex algebra:

  1. (1)

Grading-restriction condition: $V_{(n)}$ is finite dimensional for all
$n\in\mathbb{Z}$, and $V_{(n)}=0$ for $n\ll 0$.

  2. (2)

Lower-truncation condition: For $u,v\in V$, $Y_{V}(u,z)v$ contains only
finitely many negative power terms, that is, $Y_{V}(u,z)v\in V((z))$ (the
space of formal Laurent series in $z$ with coefficients in $V$).

  3. (3)

Identity property: Let $\mathbf{1}_{V}$ be the identity operator on $V$. Then

$Y_{V}(\mathbf{1}_{V},z)={\rm Id}_{V}.$

  4. (4)

Creation property: For $u\in V$, $Y_{V}(u,z)\mathbf{1}_{V}\in V[[z]]$ and

$\lim_{z\to 0}Y_{V}(u,z)\mathbf{1}_{V}=u.$

  5. (5)

Duality: For $u_{1},u_{2},v\in V$, $v^{\prime}\in
V^{\prime}=\coprod_{n\in\mathbb{Z}}V_{(n)}^{*}$ ($V_{(n)}^{*}$ denotes the
dual vector space to $V_{(n)}$ and $\langle\,.,.\rangle$ the evaluation
pairing $V^{\prime}\otimes V\to\mathbb{C}$), the series $\langle
v^{\prime},Y_{V}(u_{2},z_{2})Y_{V}(u_{1},z_{1})v\rangle$, and $\langle
v^{\prime},Y_{V}(Y_{V}(u_{1},z_{1}-z_{2})u_{2},z_{2})v\rangle$, are absolutely
convergent in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common rational function in
$z_{1}$ and $z_{2}$ with the only possible poles at $z_{1}=0=z_{2}$ and
$z_{1}=z_{2}$.

One assumes the existence of Virasoro vector $\omega\in V$: its vertex
operator $Y(\omega,z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2}$ is determined by
Virasoro operators $L(n):V\to V$ fulfilling (notice that with abuse of
notation we denote $L_{V}(n)=L(n)$)

$[L(m),L(n)]=(m-n)L(m+n)+\frac{c}{12}(m^{3}-m)\delta_{m+b,0}{\rm Id_{V}},$

($c$ is called the central charge of $V$). The grading operator is given by
$L(0)u=nu,\quad u\in V_{(n)}$, ($n$ is called the weight of $u$ and denoted by
$\mbox{\rm wt}\ (u)$).

  6. (6)

$L_{V}(0)$-bracket formula: Let $L_{V}(0):V\to V$ be defined by $L_{V}(0)v=nv$
for $v\in V_{(n)}$. Then

$[L_{V}(0),Y_{V}(v,z)]=Y_{V}(L_{V}(0)v,z)+z\frac{d}{dz}Y_{V}(v,z),$

for $v\in V$.

  7. (7)

$L_{V}(-1)$-derivative property: Let $L_{V}(-1):V\to V$ be the operator given
by

$L_{V}(-1)v=\mbox{\rm
Res}_{z}z^{-2}Y_{V}(v,z)\mathbf{1}=Y_{(-2)}(v)\mathbf{1},$

for $v\in V$. Then for $v\in V$,

$\frac{d}{dz}Y_{V}(u,z)=Y_{V}(L_{V}(-1)u,z)=[L_{V}(-1),Y_{V}(u,z)].$

Correspondingly, a grading-restricted generalized $V$-module is a vector space
$W$ equipped with a vertex operator map

$Y_{W}:V\otimes W\to W[[z,z^{-1}]],$ $\displaystyle u\otimes w$
$\displaystyle\mapsto$ $\displaystyle Y_{W}(u,w)\equiv
Y_{W}(u,z)w=\sum_{n\in\mathbb{Z}}(Y_{W})_{n}(u,w)z^{-n-1},$

and linear operators $L_{W}(0)$ and $L_{W}(-1)$ on $W$ satisfying conditions
similar as in the definition for a grading-restricted vertex algebra. In
particular,

  1. (1)

Grading-restriction condition: The vector space $W$ is $\mathbb{C}$-graded,
that is, $W=\coprod_{\alpha\in\mathbb{C}}W_{(\alpha)}$, such that
$W_{(\alpha)}=0$ when the real part of $\alpha$ is sufficiently negative.

  2. (2)

Lower-truncation condition: For $u\in V$ and $w\in W$, $Y_{W}(u,z)w$ contains
only finitely many negative power terms, that is, $Y_{W}(u,z)w\in W((z))$.

  3. (3)

Identity property: Let ${\rm Id}_{W}$ be the identity operator on $W$,
$Y_{W}(\mathbf{1},z)={\rm Id}_{W}$.

  4. (4)

Duality: For $u_{1}$, $u_{2}\in V$, $w\in W$, $w^{\prime}\in
W^{\prime}=\coprod_{n\in\mathbb{Z}}W_{(n)}^{*}$ ($W^{\prime}$ is the dual
$V$-module to $W$), the series

$\displaystyle\langle
w^{\prime},Y_{W}(u_{1},z_{1})Y_{W}(u_{2},z_{2})w\rangle,$
$\displaystyle\langle
w^{\prime},Y_{W}(u_{2},z_{2})Y_{W}(u_{1},z_{1})w\rangle,$
$\displaystyle\langle
w^{\prime},Y_{W}(Y_{V}(u_{1},z_{1}-z_{2})u_{2},z_{2})w\rangle,$ (5.1)

are absolutely convergent in the regions $|z_{1}|>|z_{2}|>0$,
$|z_{2}|>|z_{1}|>0$, $|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common
rational function in $z_{1}$ and $z_{2}$ with the only possible poles at
$z_{1}=0=z_{2}$ and $z_{1}=z_{2}$.

The locality

$Y_{W}(v_{1},z_{1})Y_{W}(v_{2},z_{2})\sim
Y_{W}(v_{2},z_{2})Y_{W}(v_{1},z_{1}),$

and associativity

$Y_{W}(v_{1},z_{1})Y_{W}(v_{2},z_{2})\sim
Y_{W}(Y_{V}v_{1},z_{1}-z_{2})v_{2},z_{2}),$

properties for the vertex operators in a $V$-module $W$ follow from the Jacobi
identity [8].

  5. (5)

$L_{W}(0)$-bracket formula: For $v\in V$,

$[L_{W}(0),Y_{W}(v,z)]=Y_{W}(L(0)v,z)+z\frac{d}{dz}Y_{W}(v,z).$

  6. (6)

$L_{W}(0)$-grading property: For $w\in W_{(\alpha)}$, there exists
$N\in\mathbb{Z}_{+}$ such that $(L_{W}(0)-\alpha)^{N}w=0$.

  7. (7)

$L_{W}(-1)$-derivative property: For $v\in V$,

$\frac{d}{dz}Y_{W}(u,z)=Y_{W}(L_{V}(-1)u,z)=[L_{W}(-1),Y_{W}(u,z)].$

For $v\in V$, and $w\in W$, the intertwining operator

$\displaystyle Y_{WV}^{W}:V\to W,$ (5.2) $\displaystyle v\mapsto
Y_{WV}^{W}(w,z)v,$

is defined by

$\displaystyle Y_{WV}^{W}(w,z)v=e^{zL_{W}(-1)}Y_{W}(v,-z)w.$ (5.3)

###  5.1. Non-degenerate invariant bilinear form on $V$

The subalgebra

$\\{L_{V}(-1),L_{V}(0),L_{V}(1)\\}\cong SL(2,\mathbb{C}),$

associated with Möbius transformations on $z$ naturally acts on $V$, (cf.,
e.g. [8]). In particular,

$\gamma_{\lambda}=\left(\begin{array}[]{cc}0&\lambda\\\ -\lambda&0\\\
\end{array}\right):z\mapsto w=-\frac{\lambda^{2}}{z},$ (5.4)

is generated by

$T_{\lambda}=\exp\left(\lambda
L_{V}{(-1)}\right)\;\exp\left({\lambda}^{-1}L_{V}(1)\right)\;\exp\left(\lambda
L_{V}(-1)\right),$

where

$T_{\lambda}Y(u,z)T_{\lambda}^{-1}=Y\left(\exp\left(-\frac{z}{\lambda^{2}}L_{V}(1)\right)\left(-\frac{z}{\lambda}\right)^{-2L_{V}(0)}u,-\frac{\lambda^{2}}{z}\right).$
(5.5)

In our considerations (cf. Appendix 6) of Riemann sphere sewing, we use in
particular, the Möbius map

$z\mapsto z^{\prime}=\epsilon/z,$

associated with the sewing condition (6.4) with

$\lambda=-\xi\epsilon^{\frac{1}{2}},$ (5.6)

with $\xi\in\\{\pm\sqrt{-1}\\}$. The adjoint vertex operator [8, 2] is defined
by

$Y^{\dagger}(u,z)=\sum_{n\in\mathbb{Z}}u^{\dagger}(n)z^{-n-1}=T_{\lambda}Y(u,z)T_{\lambda}^{-1}.$
(5.7)

A bilinear form $\langle.,.\rangle_{\lambda}$ on $V$ is invariant if for all
$a,b,u\in V$, if

$\langle Y(u,z)a,b\rangle_{\lambda}=\langle
a,Y^{\dagger}(u,z)b\rangle_{\lambda},$ (5.8)

i.e.

$\langle u(n)a,b\rangle_{\lambda}=\langle a,u^{\dagger}(n)b\rangle_{\lambda}.$

Thus it follows that

$\langle L_{V}(0)a,b\rangle_{\lambda}=\langle a,L_{V}(0)b\rangle_{\lambda},$
(5.9)

so that

$\langle a,b\rangle_{\lambda}=0,$ (5.10)

if $wt(a)\not=wt(b)$ for homogeneous $a,b$. One also finds

$\langle a,b\rangle_{\lambda}=\langle b,a\rangle_{\lambda}.$

The form $\langle.,.\rangle_{\lambda}$ is unique up to normalization if
$L_{V}(1)V_{1}=V_{0}$. Given any $V$ basis $\\{u^{\alpha}\\}$ we define the
dual $V$ basis $\\{\overline{u}^{\beta}\\}$ where

$\langle
u^{\alpha},\overline{u}^{\beta}\rangle_{\lambda}=\delta^{\alpha\beta}.$

##  6\. Appendix: A sphere formed from sewing of two spheres

The matrix element for a number of vertex operators of a vertex algebra is
usually associated [2, 3, 10] with a vertex algebra character on a sphere. We
extrapolate this notion to the case of $\mathcal{W}_{z_{1},\ldots,z_{n}}$
spaces. In Section 3 we explained that a space
$\mathcal{W}_{z_{1},\ldots,z_{n}}$ can be associated with a Riemann sphere
with marked points, while the product of two such spaces is then associated
with a sewing of such two spheres with a number of marked points and extra
points with local coordinates identified with formal parameters of
$\mathcal{W}_{x_{1},\ldots,x_{k}}$ and $\mathcal{W}_{y_{1},\ldots,y_{n}}$. In
order to supply an appropriate geometric construction for the product, we use
the $\epsilon$-sewing procedure (described in this Appendix) for two initial
spheres to obtain a matrix element associated with (3.2).

######  Remark 7.

In addition to the $\epsilon$-sewing procedure of two initial spheres, one can
alternatively use the self-sewing procedure [11] for the sphere to get, at
first, the torus, and then by sending parameters to appropriate limit by
shrinking genus to zero. As a result, one obtains again the sphere but with a
different parameterization. In the case of spheres, such a procedure
consideration of the product of $\mathcal{W}$-spaces so we focus in this paper
on the $\epsilon$-formalizm only.

In our particular case of $\mathcal{W}$-values rational functions obtained
from matrix elements (1.1) two initial auxiliary spaces we take Riemann
spheres $\Sigma^{(0)}_{a}$, $a=1$, $2$, and the resulting space is formed by
the sphere $\Sigma^{(0)}$ obtained by the procedure of sewing
$\Sigma^{(0)}_{a}$. The formal parameters $(x_{1},\ldots,x_{k})$ and
$(y_{1},\ldots,y_{n})$ are identified with local coordinates of $k$ and $n$
points on two initial spheres $\Sigma^{(0)}_{a}$, $a=1$, $2$ correspondingly.
In the $\epsilon$ sewing procedure, some $r$ points among
$(p_{1},\ldots,p_{k})$ may coincide with points among
$(p^{\prime}_{1},\ldots,p^{\prime}_{n})$ when we identify the annuluses (6.3).
This corresponds to the singular case of coincidence of $r$ formal parameters.

Consider the sphere formed by sewing together two initial spheres in the
sewing scheme referred to as the $\epsilon$-formalism in [11]. Let
$\Sigma_{a}^{(0)}$, $a=1$, $2$ be to initial spheres. Introduce a complex
sewing parameter $\epsilon$ where

$|\epsilon|\leq r_{1}r_{2},$

Consider $k$ distinct points on $p_{i}\in\Sigma_{1}^{(0)}$, $i=1,\ldots,k$,
with local coordinates $(x_{1},\ldots,x_{k})\in F_{k}\mathbb{C}$, and distinct
points $p_{j}\in\Sigma_{2}^{(0)}$, $j=1,\ldots,n$, with local coordinates
$(y_{1},\ldots,y_{n})\in F_{n}\mathbb{C}$, with

$\left|x_{i}\right|\geq|\epsilon|/r_{2},$
$\left|y_{i}\right|\geq|\epsilon|/r_{1}.$

Choose a local coordinate $z_{a}\in\mathbb{C}$ on $\Sigma^{(0)}_{a}$ in the
neighborhood of points $p_{a}\in\Sigma^{(0)}_{a}$, $a=1$, $2$. Consider the
closed disks

$\left|\zeta_{a}\right|\leq r_{a},$

and excise the disk

$\\{\zeta_{a},\;\left|\zeta_{a}\right|\leq|\epsilon|/r_{\overline{a}}\\}\subset\Sigma^{(0)}_{a},$
(6.1)

to form a punctured sphere

$\widehat{\Sigma}^{(0)}_{a}=\Sigma^{(0)}_{a}\backslash\\{\zeta_{a},\left|\zeta_{a}\right|\leq|\epsilon|/r_{\overline{a}}\\}.$

We use the convention

$\overline{1}=2,\quad\overline{2}=1.$ (6.2)

Define the annulus

$\mathcal{A}_{a}=\left\\{\zeta_{a},|\epsilon|/r_{\overline{a}}\leq\left|\zeta_{a}\right|\leq
r_{a}\right\\}\subset\widehat{\Sigma}^{(0)}_{a},$ (6.3)

and identify $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ as a single region
$\mathcal{A}=\mathcal{A}_{1}\simeq\mathcal{A}_{2}$ via the sewing relation

$\zeta_{1}\zeta_{2}=\epsilon.$ (6.4)

In this way we obtain a genus zero compact Riemann surface

$\Sigma^{(0)}=\left\\{\widehat{\Sigma}^{(0)}_{1}\backslash\mathcal{A}_{1}\right\\}\cup\left\\{\widehat{\Sigma}^{(0)}_{2}\backslash\mathcal{A}_{2}\right\\}\cup\mathcal{A}.$

This sphere form a suitable geometrical model for the construction of a
product of $\mathcal{W}$-valued rational forms in Section 3.

##  7\. Appendix: proof of Proposition 5

###### Proof.

For a vertex operator $Y_{V,W}(v,z)$ let us introduce a notation
$\omega_{V,W}=Y_{V,W}(v,z)\;dz^{{\rm wt}v}$. Let us use notations (3.1) and
(3.6). According to (2.2), the action of $\delta_{m+m^{\prime}-t}^{k+n-r}$ on
$\widehat{R}\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{k},y_{n};\epsilon)$
is given by

$\displaystyle\langle
w^{\prime},\delta_{m+m^{\prime}-t}^{k+n-r}\widehat{R}\;\mathcal{F}(v_{1},x_{1};\ldots;v_{k},x_{k};v^{\prime}_{1},y_{1};\ldots;v^{\prime}_{n},y_{n};\epsilon)\rangle$
$\displaystyle\quad=\langle
w^{\prime},\sum_{i=1}^{k}(-1)^{i}\;\widehat{R}\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{i-1},z_{i-1};\;\omega_{V}(\widetilde{v}_{i},z_{i}-z_{i+1})\widetilde{v}_{i+1},z_{i+1};\;\widetilde{v}_{i+2},z_{i+2};$
$\displaystyle\qquad\qquad\qquad\ldots;\widetilde{v}_{k},z_{k};\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n},z_{k+n};\epsilon)\rangle$
$\displaystyle\qquad+\sum_{i=1}^{n-r}(-1)^{i}\;\langle
w^{\prime},\mathcal{F}\left(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k};\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+i-1},z_{k+i-1};\right.$
$\displaystyle\qquad\qquad\qquad\omega_{V}\left(\widetilde{v}_{k+i},z_{k+i}-z_{k+i+1})\;\widetilde{v}_{k+i+1},z_{k+i+1};\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad\widetilde{v}_{k+i+2},z_{k+i+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r};\epsilon\right)\rangle$
$\displaystyle\qquad+\langle
w^{\prime},\omega_{W}\left(\widetilde{v}_{1},z_{1}\right)\;\mathcal{F}(\widetilde{v}_{2},z_{2};\ldots;\widetilde{v}_{k},z_{k};\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r};\epsilon)\rangle$
$\displaystyle\qquad+\langle
w,(-1)^{k+n+1-r}\omega_{W}(\widetilde{v}_{k+n-r+1},z_{k+n-r+1})\;$
$\displaystyle\qquad\qquad\qquad\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k};\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r};\epsilon)\rangle$
$\displaystyle\quad=\sum\limits_{u\in V}\langle
w^{\prime},\sum_{i=1}^{k}(-1)^{i}\;Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{i-1},z_{i-1};\;\omega_{V}(\widetilde{v}_{i},z_{i}-z_{i+1})\widetilde{v}_{i+1},z_{i+1};\;$
$\displaystyle\qquad\qquad\qquad\widetilde{v}_{i+2},z_{i+2};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad+\sum\limits_{u\in V}\sum_{i=1}^{n-r}(-1)^{i}\;\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}\left(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle\right.$
$\displaystyle\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+i-1},z_{k+i-1};$
$\displaystyle\qquad\qquad\qquad\qquad\omega_{V}(\widetilde{v}_{i},z_{k+i}-z_{k+i+1})\;\widetilde{v}_{k+i+1},z_{k+i+1};\widetilde{v}_{k+i+2},z_{k+i+2};$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad+\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\omega_{W}\left(\widetilde{v}_{1},z_{1}\right)\;\mathcal{F}(\widetilde{v}_{2},z_{2};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad+\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}((-1)^{k+1}\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}\right)\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad-\sum\limits_{u\in V}\langle w^{\prime},(-1)^{k+1}\langle
w^{\prime},Y^{W}_{VW}(\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}\right)\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad+\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\omega_{W}(\widetilde{v}_{k+n-r+1},z_{k+n-r+1})\;$
$\displaystyle\qquad\qquad\qquad\qquad\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad-\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\omega_{W}(\widetilde{v}_{k+n-r+1},z_{k+n-r+1})$
$\displaystyle\qquad\qquad\qquad\qquad\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\rangle$
$\displaystyle\quad=\sum\limits_{u\in V}\langle
w^{\prime},\;Y^{W}_{VW}(\delta^{k}_{m}\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\qquad+(-1)^{k}\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\delta^{n-r}_{m^{\prime}-t}\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle\quad=\langle
w^{\prime},\delta^{k}_{m}\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\cdot_{\epsilon}\langle
w^{\prime},\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r})\rangle$
$\displaystyle\qquad+(-1)^{k}\langle
w^{\prime},\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\cdot_{\epsilon}\delta^{n-r}_{m^{\prime}-t}\mathcal{F}(\widetilde{v}_{k+1},z_{k+1};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r})\rangle,$

since,

$\displaystyle\sum\limits_{u\in V}\langle
w^{\prime},(-1)^{k+1}Y^{W}_{VW}(\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}\right)\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},(-1)^{k+1}e^{\zeta_{1}L_{W}{(-1)}}Y_{W}(u,-\zeta_{1})\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}\right)\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},(-1)^{k+1}e^{\zeta_{1}L_{W}{(-1)}}\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}\right)Y_{W}(u,-\zeta_{1})\;\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},(-1)^{k+1}\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}\right)\;e^{\zeta_{1}L_{W}{(-1)}}Y_{W}(u,-\zeta_{1})\;\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{v\in V}\sum\limits_{u\in V}\langle
v^{\prime},(-1)^{k+1}\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}\right)w\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},e^{\zeta_{1}L_{W}{(-1)}}Y_{W}(u,-\zeta_{1})\;\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},e^{\zeta_{1}L_{W}{(-1)}}Y_{W}(u,-\zeta_{1})\;\;\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k})\rangle$
$\displaystyle\qquad\qquad\sum\limits_{v\in V}\langle
v^{\prime},(-1)^{k+1}\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}\right)w\rangle$
$\displaystyle\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;$
$\displaystyle\qquad\qquad\qquad\qquad\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\;\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},(-1)^{k+1}\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}\right)\;$
$\displaystyle\qquad\qquad
Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\;\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},(-1)^{k+1}\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}\right)$
$\displaystyle\qquad\qquad\;e^{\zeta_{2}L_{W}{(-1)}}Y_{W}(\overline{u},-\zeta_{2})\;\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r})\rangle$
$\displaystyle=\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\;\rangle$
$\displaystyle\qquad\langle
w^{\prime},(-1)^{k+1}\;\;e^{\zeta_{2}L_{W}{(-1)}}\;Y_{W}(\overline{u},-\zeta_{2})\;\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}-\zeta_{2}\right)$
$\displaystyle\qquad\qquad\;\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r})\rangle$
$\displaystyle\qquad\qquad=\sum\limits_{u\in V}\langle
w^{\prime},Y^{W}_{VW}(\mathcal{F}(\widetilde{v}_{1},z_{1};\ldots;\widetilde{v}_{k},z_{k}),\zeta_{1})u\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},Y^{W}_{VW}(\omega_{W}(\widetilde{v}_{k+1},z_{k+1})\;\mathcal{F}(\widetilde{v}_{k+2},z_{k+2};\ldots;\widetilde{v}_{k+n-r},z_{k+n-r}),\zeta_{2})\overline{u}\rangle,$

due to locality (4) of vertex opertors, and arbitrarness of
$\widetilde{v}_{k+1}\in V$ and $z_{k+1}$, we can always put

$\omega_{W}\left(\widetilde{v}_{k+1},z_{k+1}+\zeta_{1}-\zeta_{2}\right)=\omega_{W}(\widetilde{v}_{k+2},z_{k+2}),$

for $\widetilde{v}_{k+1}=\widetilde{v}_{k+2}$,
$z_{k+2}=z_{k+1}+\zeta_{2}-\zeta_{1}$. ∎

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