# Bigraded differential algebra for vertex algebra complexes

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Praha<EMAIL_ADDRESS>
###### Abstract.

For the bicomplex structure of grading-restricted vertex algebra cohomology
defined in [6], we show that the orthogonality and double grading conditions
applied endow it with the structure of a bigraded differential algebra with
respect to a natural multiplication. The generators and commutation relations
of the bigraded differential algebra form a continual Lie algebra
$\mathcal{G}(V)$ with the root space provided by a grading-restricted vertex
algebra $V$. We prove that the differential algebra generates non-vanishing
cohomological invariants associated to a vertex algebra $V$. Finaly, we
provide examples associated to various choices of the vertex algebra bicomplex
subspaces.

AMS Classification: 53C12, 57R20, 17B69

###### Key words and phrases:

Vertex algebras; cohomological invariants; cohomology classes; bi-differential
algebras; continual Lie algebras

##  1\. Introduction

The cohomology theory for vertex operator algebras is an important and
attractive theme for studies. In [6] the cohomology theory for a grading-
restricted vertex algebra [9] was introduced. The definition of bicomplex
spaces and coboundary operators uses an interpretation of vertex algebras in
terms of rational functions constructed from matrix elements [8] for a
grading-restricted vertex algebra [4]. The notion of composability (see
Section 3.1) of bicomplex space elements with a number of vertex operators is
essentially involved in the formulation. The cohomology of such complexes
defines a cohomology of a grading-restricted vertex algebras in the standard
way. In this paper we develop ideas concerning algebraic structures following
from the cohomology construction [6] for grading-restricted vertex algebras.
We show that the orthogonality and double grading conditions applied to the
bicomplex associated to a grading-restricted vertex algebra brings about the
structure of the bigraded differential algebra with respect the commutator of
bicomplex mappings. We show that the orthogonality being applied to the
bicomplex spaces leads to relations among mappings and actions of coboundary
operators. Using this condition we then find further explicit examples of
continual Lie algebras [12] associated to vertex algebras. We derive also the
simplest cohomological classes for the bicomplex for a grading-restricted
vertex algebra. Such cohomological classes are non-vanishing and independent
of the choice of the bicomplex space elements.

As for possible applications of the material presented in this paper, we would
like to mention computations of higher cohomologies for grading-restricted
vertex algebras [7], search for more complicated cohomological invariants, and
applications to differential geometry. In particular, since vertex algebras is
a useful computational tool, it would be interesting to study possible
relations to cohomology of manifolds. One can show that such cohomological
invariants possess analytical (with respect to the notion of composability) as
well as geometrical meaning. In addition to the natural orthogonality
condition, ona can consider variations of multiplications defined for
bicomplex spaces, and, therefore, more advanced examples of graded
differential algebras. In differential geometry there exist various approaches
to the construction of cohomological classes (cf., in particular, [11]). We
hope to use these techniques to derive counterparts in the cohomology theory
of vertex algebras.

##  2\. The grading-restricted vertex algebra

In this section, we recall [6] properties of grading-restricted vertex
algebras and their grading-restricted generalized modules over the base field
$\mathbb{C}$ of complex numbers. A vertex algebra $(V,Y_{V},\mathbf{1}_{V})$,
cf. [9], consists of a $\mathbb{Z}$-graded complex vector space

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

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

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

with components

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

where

$Y_{V}(v,z)\mathbf{1}=v+O(z).$

A grading-restricted vertex algebra [6], defining is subject to the following
conditions:

  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, i.e., $Y_{V}(u,z)v\in V((z))$ (the space
of formal Laurent series in $z$ with coefficients in $V$).

  3. (3)

Identity property: Let ${\rm Id}_{V}$ 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

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

($c$ 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 $v\in V$,

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

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

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

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,
i.e., $W=\coprod_{\alpha\in\mathbb{C}}W_{(\alpha)}$, 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, i.e., $Y_{W}(u,z)w\in W((z))$.

  3. (3)

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

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

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

A unique symmetric invertible invariant bilinear form $\langle.,.\rangle$ with
normalization

$\langle\mathbf{1}_{V},\mathbf{1}_{V}\rangle=1,$

where [2, 10]

$\langle Y^{\dagger}(a,z)\;b,c\rangle=\left\langle b,Y(a,z)c\right\rangle,$
(2.1)

for

$\displaystyle Y^{\dagger}(a,z)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}}a^{\dagger}(n)z^{-n-1}$ (2.2)
$\displaystyle=$ $\displaystyle
Y\left(e^{zL_{V}(1)}\left(-z^{-2}\right)^{L_{V}(0)}a,z^{-1}\right).$

##  3\. $\overline{W}$-valued rational functions over torsors

Let us recall the notion of multiple torsors to formulate definitions of
chain-cochain double complex structure associated to grading-restricted vertex
algebras. First, we recall here the general definition of ordinary torsors
[1]. Let $G$ be a group, and $S$ a non–empty set. Then $S$ is called a
$G$-torsor, if it is equipped with a simply transitive right action of $G$,
i.e., given $x$, $y\in S$, there exists a unique $g\in G$ such that

$y=x\cdot g,$

where the right action is given by

$x\cdot(gh)=(x\cdot g)\cdot h.$

The choice of any $x\in S$ allows us to identify $S$ with $G$ by sending
$x\cdot g$ to $g$.

Let $V$ be a grading-restricted vertex algebra, and $W$ a grading-restricted
generalized $V$-module. One defines the configuration spaces [6]:

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

for $n\in\mathbb{Z}_{+}$. Let $V$ be a grading-restricted vertex algebra, and
$W$ a grading-restricted generalized $V$-module. By $\overline{W}$ we denote
the algebraic completion of $W$,

$\overline{W}=\prod_{n\in\mathbb{C}}W_{(n)}=(W^{\prime})^{*}.$

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

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

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

$\langle w^{\prime},f(z_{1},\dots,z_{n})\rangle,$ (3.1)

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 one calls $\overline{W}$-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
$\overline{W}_{z_{1},\dots,z_{n}}$. One defines a left action of $S_{n}$ on
$\overline{W}_{z_{1},\dots,z_{n}}$ by

$(\sigma(f))(z_{1},\dots,z_{n})=f(z_{\sigma(1)},\dots,z_{\sigma(n)}),$

for $f\in\overline{W}_{z_{1},\dots,z_{n}}$.

The idea to use torsors [1] is to represent the action of an element $\rho$ of
the group of automorphism of a set of formal variables $(z_{1},\ldots,z_{n})$.
by action of $V$-operators on a set of vertex algebra element $v\in V$. In
particular, one consider a vector $(v_{i},z_{1},\ldots,z_{n})$ with $v_{i}\in
V$. Then the same vector equals

$\left(R^{-1}\left(\rho\right)v_{i},w_{1},\ldots,w_{n}\right),$

i.e., it is identified with

$R^{-1}\left(\rho\right)v_{i}\in V,$

using the transformed formal parameters $(w_{1},\ldots,w_{n})$. Here
$R\left(\rho\right)$ is an operator representing transformation of $z_{i}\to
w_{i}$, as an action on $V$. Therefore if we have an operator which is equal
to a torsor $S$ of the group of automorphism of $(z_{1},\ldots,z_{n})$. Then
this operator equals

$R\left(\rho\right)\;S\;R^{-1}\left(\rho\right).$

Thus, in terms of the coordinates $(v_{i},z_{1},\ldots,z_{n})$, the
differential $Y_{W}(v_{i},w_{i})\;dw_{i}^{\mbox{\rm wt}\ (v_{i})}$ becomes

$Y_{W}(v_{i},z_{i})\;dz_{i}^{\mbox{\rm wt}\
(v_{i})}=R(\rho)\;Y_{W}\left(v_{i},\rho(z_{1},\ldots,z_{n})\right)\;R^{-1}(\rho)\;dw_{i}^{\mbox{\rm
wt}\ (v_{i})}.$

In what follows, we will use corresponding formalism of multiple torsors,
i.e., we consider vectors $(v_{1}\otimes\ldots\otimes
v_{n})(z_{1},\ldots,z_{n})$ for $v_{i}\in V$, $1\leq i\leq n$. Then,
$\overline{W}$-valued function $\Phi$ are multiple element $v_{i}\in V$
torsors with respect to to group of automorphisms of $(z_{1},\ldots,z_{n})$.

For $w\in W$, the $\overline{W}$-valued function [8]
$E^{(n)}_{W}(v_{1}\otimes\cdots\otimes v_{n};w)$ 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 [6]. 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

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

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

$l_{1}=\cdots=l_{i-1}=l_{i+1}=1,$ $l_{i}=m-n-1,$

for some $1\leq i\leq n$, we will use
$\Phi\circ_{i}E^{(l_{i})}_{V;\;\mathbf{1}}$ to denote
$\Phi\circ(E^{(l_{1})}_{V;\;\mathbf{1}}\otimes\cdots\otimes
E^{(l_{n})}_{V;\;\mathbf{1}})$.

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

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

for $\sigma\in S_{n}$ and $v_{1},\dots,v_{n}\in V$. 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 $j=1,\dots,n$. In [6] one finds:

######  Proposition 1.

The subspace of $\mbox{\rm Hom}(V^{\otimes
n},\overline{W}_{z_{1},\dots,z_{n}})$ 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 $S_{n}$.

###  3.1. Maps composable with vertex operators

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 [6, 4] 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 $w^{\prime}\in W^{\prime}$. Introduce

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

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 $i,j=1,\dots,k$, $i\neq j$,.

A linear map

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

is said to have the $L(0)$-conjugation property if for $(v_{1},\dots,v_{n})\in
V$, $w^{\prime}\in W^{\prime}$, $(z_{1},\dots,z_{n})\in F_{n}\mathbb{C}$ and
$z\in\mathbb{C}^{\times}$, so that $(zz_{1},\dots,zz_{n})\in F_{n}\mathbb{C}$,

$\displaystyle\langle w^{\prime},z^{L_{W}(0)}(\Phi(v_{1}\otimes\cdots\otimes
v_{n}))(z_{1},\dots,z_{n})\rangle$ (3.2) $\displaystyle=\langle
w^{\prime},(\Phi(z^{L(0)}v_{1}\otimes\cdots\otimes
z^{L(0)}v_{n}))(zz_{1},\dots,zz_{n})\rangle.$

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

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

is said to have the $L(-1)$-derivative property if (i)

$\displaystyle\frac{\partial}{\partial z_{i}}\langle
w^{\prime},(\Phi(v_{1}\otimes\cdots\otimes v_{n}))(z_{1},\dots,z_{n})\rangle$
(3.3) $\displaystyle=\langle w^{\prime},(\Phi(v_{1}\otimes\cdots\otimes
v_{i-1}\otimes L_{V}(-1)v_{i}\otimes v_{i+1}\otimes\cdots\otimes
v_{n}))(z_{1},\dots,z_{n})\rangle,$

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

$\displaystyle\left(\frac{\partial}{\partial
z_{1}}+\cdots+\frac{\partial}{\partial z_{n}}\right)\langle
w^{\prime},(\Phi(v_{1}\otimes\cdots\otimes v_{n}))(z_{1},\dots,z_{n})\rangle$
(3.4) $\displaystyle\quad=\langle
w^{\prime},L_{W}(-1)(\Phi(v_{1}\otimes\cdots\otimes
v_{n}))(z_{1},\dots,z_{n})\rangle,$

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

##  4\. Vertex algebra bicomplexes

Let us recall [6] here the definition of shuffles. 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

$J_{l;s}^{-1}=\\{\sigma\;|\;\sigma\in J_{l;s}\\}.$

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

$\sum_{\sigma\in
J_{l;s}^{-1}}(-1)^{|\sigma|}\sigma\left(\Phi(v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(l)})\right)=0.$ (4.1)

We also find in [6]

######  Proposition 2.

Let ${C}_{m}^{0}(V,W)=W$. Then we have

$C_{m}^{n}(V,W)\subset{C}_{m-1}^{n}(V,W),$

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

In [6] the coboundary operator for the bicomplex spaces ${C}_{m}^{n}(V,W)$ was
introduced:

${\delta}^{n}_{m}:{C}_{m}^{n}(V,W)\to{C}_{m-1}^{n+1}(V,W).$ (4.2)

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

${\delta}^{n}_{m}(\Phi)=E^{(1)}_{W}\circ_{2}\Phi+\sum_{i=1}^{n}(-1)^{i}\Phi\circ_{i}E^{(2)}_{V;\mathbf{1}}+(-1)^{n+1}\sigma_{n+1,1,\dots,n}(E^{(1)}_{W}\circ_{2}\Phi),$
(4.3)

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

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

are absolutely convergent in the regions

$|z_{1}-\zeta|>|z_{2}-\zeta|,$ $|z_{2}-\zeta|>0,$ $|\zeta-
z_{3}|>|z_{1}-\zeta|,$ $|z_{2}-\zeta|>0,$

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

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

for $m\in\mathbb{Z}_{+}$. The coboundary operator

${\delta}^{2}_{\frac{1}{2}}:{C}_{\frac{1}{2}}^{2}(V,W)\to{C}_{0}^{3}(V,W),$
(4.4)

is defined in [6] by

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

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

Using the operators $\delta^{n}_{m}$ and $\delta^{2}_{\frac{1}{2}}$, for
$m\in\mathbb{Z}_{+}$ and $n\in\mathbb{N}$, one introduces [6] 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

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

Then one has [6]

######  Proposition 3.

For $n\in\mathbb{N}$ and $m\in\mathbb{Z}_{+}+1$, the coboundary operators
(4.3) and (4.5) satisfy the cochain complex conditions, i.e.,

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

By Proposition 3, we have complexes for $m\geq 0$,

$0\longrightarrow
C_{m}^{0}(V,W)\stackrel{{\scriptstyle\delta_{m}^{0}}}{{\longrightarrow}}C^{1}_{m-1}(V,W)\stackrel{{\scriptstyle\delta^{1}_{m-1}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\delta_{1}^{m-1}}}{{\longrightarrow}}C_{0}^{m}(V,W)\longrightarrow
0,$ (4.6) $0\longrightarrow
C_{3}^{0}(V,W)\stackrel{{\scriptstyle\delta_{3}^{0}}}{{\longrightarrow}}C_{2}^{1}(V,W)\stackrel{{\scriptstyle\delta_{2}^{1}}}{{\longrightarrow}}C_{\frac{1}{2}}^{2}(V,W)\stackrel{{\scriptstyle\delta_{\frac{1}{2}}^{2}}}{{\longrightarrow}}C_{0}^{3}(V,W)\longrightarrow
0.$ (4.7)

The first and last mappings are trivial embeddings and projections. For
simplicity, we call derivatives the actions of $\delta_{n}^{m}$ and
$\delta_{2}^{\frac{1}{2}}$ on the mappings.

##  5\. Bigraded differential algebras associated to a vertex algebra

A natural task now is to introduce an appropriate multiplication on bicomplex
spaces of (4.6) and (4.7), and derived an analogue of Leibniz formula. Let us
consider two mappings

$\Phi(v_{1}\otimes\ldots\otimes v_{n})(z_{1},\ldots,z_{n})\in C_{m}^{n}(V,W),$
$\Psi(v_{n+1}\otimes\ldots\otimes
v_{n+n^{\prime}})(z_{n+1},\ldots,z_{n+n^{\prime}})\in
C_{m^{\prime}}^{n^{\prime}}(V,W),$

which have $r$ common vertex algebra elements and formal variables, and $t$
common vertex operators that mappings $\Phi$ and $\Psi$ are composable with.
Note that when applying the coboundary operators (4.3) and (4.5) to a map

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

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

For our particular purposes of introduction of a bigraded differential
structure associated to double chain-cochain complexes (4.3) and (4.5), we
define the product of $\Phi$ and $\Psi$ above as

$\displaystyle\Phi\cdot\Psi:V^{\otimes(n+n^{\prime})}\to
C^{n+n^{\prime}-r}_{m+m^{\prime}-t}(V,W),$ (5.1)
$\displaystyle\Phi\cdot\Psi=\left[\Phi,\Psi\right]=\Phi*\Psi-\Psi*\Phi,$ (5.2)

where brackets mean the commutator with respect to the geometrically defined
product $*$ of elements of the spaces $C_{m}^{n}(V,W)$ and
$C_{m^{\prime}}^{n^{\prime}}(V,W)$, introduced in Appendix 8. In the case

$\Psi(v_{n+1}\otimes\ldots\otimes
v_{n+n^{\prime}})(z_{n+1},\ldots,z_{n+n^{\prime}})=\Phi(v_{1}\otimes\ldots\otimes
v_{n})(z_{1},\ldots,z_{n}),$

we obtain from (5.2) that

$\displaystyle\Phi(v_{1}\otimes\ldots\otimes
v_{n})(z_{1},\ldots,z_{n})\cdot\Phi(v_{1}\otimes\ldots\otimes
v_{n})(z_{1},\ldots,z_{n})=0.$ (5.3)

The coboundary operators $\delta^{n}_{m}$ and $\delta^{2}_{\frac{1}{2}}$
possess a variation of Leibniz law with respect to the product (5.2).

In Appendix 8 we prove the following

######  Proposition 4.

For arbitrary $w^{\prime}\in W^{\prime}$ one has

$R\left(\langle
w^{\prime},\delta_{m+m^{\prime}-t}^{n+n^{\prime}-r}\left(\Phi\cdot\Psi\right)\rangle\right)=R\left(\langle
w^{\prime},\delta^{n}_{m}\left(\Phi\right)\cdot\Psi\rangle\right)+(-1)^{n}R\left(\langle
w^{\prime},\Phi\cdot\delta^{n^{\prime}}_{m^{\prime}}\left(\Psi\right)\rangle\right).$
(5.4)

In this section we provide the main results of the paper by deriving relations
for the double graded differential algebras and the cohomological invariants
associated to bicomplexes (4.6) and (4.7) 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 $C_{n}^{m}(V,W)$. In analogy with differential forms, we call
a map $\Phi\in C_{m}^{n}(V,W)$ closed if

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

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

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

For $\Phi\in{C}^{n}_{m}(V,W)$ 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 $\chi\in{C}^{n-1}_{m+1}$,

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

where it is assumed that both parts of the last formula belongs to the same
space ${C}^{n}_{m}(V,W)$.

Under a natural extra condition, the bicomplexes (4.6) and (4.7) allow us to
establish relations among elements of bicomplex spaces. By analogy with the
notion of integrability for differential forms [5], we introduce here the
notion of orthogonality for spaces of a complex. Suppose we consider a complex
given by

$0\longrightarrow
C_{0}\stackrel{{\scriptstyle\delta_{0}}}{{\longrightarrow}}C_{1}\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}C_{2}\stackrel{{\scriptstyle\delta_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\delta_{n-1}}}{{\longrightarrow}}C_{n}\stackrel{{\scriptstyle\delta_{n}}}{{\longrightarrow}}\cdots$
(5.5)

In particular, let us require that for a pair of a bicomplex spaces $C_{i}$
and $C_{j}$, $i,j\geq 0$. there exist subspaces $C^{\prime}_{i}\subset C_{i}$
and such that for $\Phi_{i}\in C^{\prime}_{i}$ and $\Phi_{j}\in
C^{\prime}_{j}$,

$\Phi_{i}\cdot\delta_{j}\Phi_{j}=0,$ (5.6)

namely, $\Phi_{i}$ supposed to be orthogonal to $\delta_{j}\Phi_{j}$ (i.e.,
commutative with respect to the product (5.2)). We call this the orthogonality
condition for mappings of a complex.

It is easy to see that the assumption to belong to the same bicomplex space
for both sides of equations following from orthogonality condition applies the
double grading condition on bicomplex 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
bicomplex (4.6).

We formulate the first main result of this paper:

######  Proposition 5.

The orthogonality condition for the bicomplexes (4.6) and (4.7) brings about
the structure of a double graded differential algebra with respect to the
multiplication (5.2).

###### Proof.

Let us consider the most general case. For non-negative $n_{0}$, $n$, $n_{1}$,
$m_{0}$, $m$, $m_{1}$, let $\chi\in C^{n_{0}}_{m_{0}}(V,W)$, $\Phi\in
C^{n}_{m}(V,W)$, and $\alpha\in C^{n_{1}}_{m_{1}}(V,W)$. For $\Phi$ and
$\alpha$, let $r_{0}$ be the number of common vertex algebra elements (and
formal parameters), and $t_{0}$ be the number of common vertex operators
$\Phi$ and $\alpha$ are composable to. Note that we assume $n$, $n_{1}\geq
r_{0}$, $m$, $m_{1}\geq t_{0}$. Applying the ortogonality condition

$\Phi\cdot\delta^{n_{0}}_{m_{0}}\chi=0,$

implies that there exist $\alpha_{1}\in C^{n_{1}}_{m_{1}}(V,W)$, such that

$\delta^{n_{0}}_{m_{0}}\chi=\Phi\cdot\alpha_{1}.$

From the last equations we obtain

$n_{0}+1=n+n_{1}-r_{0},$ $m_{0}-1=m+m_{1}-t_{0}.$

Note that we have extra conditions following from the last identities:
$n_{0}+1\geq 0$, $m_{0}-1\geq 0$. The conditions above for indexes express the
double grading condition for the bicomplexes (4.6) and (4.7). As a result, we
have a system in integer variables satisfying the grading conditions above.
Consequently applying corresponding derivatives we obtain the full structure
of differential relations

$\displaystyle\Phi\cdot\delta^{n_{0}}_{m_{0}}\chi=0,$ (5.7)
$\displaystyle\delta^{n_{0}}_{m_{0}}\chi=\Phi\cdot\alpha_{1},$ $\displaystyle
0=\delta^{n}_{m}\Phi\cdot\alpha_{1}+(-1)^{n}\Phi\cdot\delta^{n_{1}}_{m_{1}}\alpha_{1},$
$\displaystyle 0=\delta^{n}_{m}\Phi\cdot\delta^{n_{0}}_{m_{0}}\chi,$
$\displaystyle\delta^{n_{0}}_{m_{0}}\chi=\delta^{n}_{m}\Phi\cdot\alpha_{2},$
$\displaystyle 0=\delta^{n}_{m}\Phi\cdot\delta^{n_{i}}_{m_{i}}\alpha_{i},\;\;$
$\displaystyle\delta^{n_{i}}_{m_{i}}\alpha_{i}=\delta^{n}_{m}\Phi\cdot\alpha_{i+1},\;\;i\geq
2,$

where $\alpha_{i}\in C^{n_{i}}_{m_{i}}(V,W)$, and $n_{i}$, $m_{i}$, $i\geq 2$
satisfy relations

$n_{i}=n+n_{i+1}-r_{i+1},$ $m_{i}=m+m_{i+1}-t_{i+1}.$

The sequence of relations (5.7) does not cancel until the conditions on
indeces given above fullfil. ∎

Thus, we see that the orthogonality condition for the bicomplexes (4.6) and
(4.7) together with the action of coboundary operators $\delta^{n}_{m}$ and
$\delta^{2}_{\frac{1}{2}}$, and the multiplication formulas (5.2)–(5.4),
define a differential algebra depending on vertex algebra elements and formal
parameters. In particular, in that way we obtain 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.
[12]) with the continual root space represented by a grading-restricted vertex
algebra $V$.

######  Proposition 6.

For the bicomplex (4.6) the generators

$\left\\{\chi\right.,\Phi,\alpha_{i},\;\delta^{n_{0}}_{m_{0}}\chi,\delta^{n_{1}}_{m_{1}}\Phi,\\\
\left.\delta^{n_{i}}_{m_{i}}\alpha_{i},\;\right\\},$

with $i\geq 0$, and commutation relations (5.7) form a continual Lie algebra
$\mathcal{G}(V)$ with a root space provided by the grading-restricted vertex
algebra $V$.

###### Proof.

With the redefinition (we suppress here the dependence on vertex algebra
elements and formal parameters),

$H_{0}=\delta^{0}_{3}\chi,$ $H^{*}_{0}=\chi,$ $X_{+1}=\Phi,$
$X_{-i}=\alpha_{i},$ $Y_{+1}=\delta^{1}_{2}\Phi,$
$Y_{-i}=\delta^{1}_{t}\alpha_{i},$

we arrive at the commutation relations:

$\displaystyle\left[H_{0},X_{+1}\right]=0,\;\;$
$\displaystyle\left[X_{+1},X_{-1}\right]=H_{0},$
$\displaystyle\left[Y_{+1},X_{-1}\right]=(-1)^{n}\left[Y_{-1},X_{+1}\right],\;\;$
$\displaystyle\left[Y_{+1},X_{-2}\right]=0,$
$\displaystyle\left[Y_{+1},Y_{-i}\right]=0,\;\;$
$\displaystyle\left[Y_{+1},X_{-(i+1)}\right]=Y_{-i}.$

One easily checks Jacobi identities for generators. ∎

From the above proposition we see that, under the orthogonality and grading
conditions, the bicomplexes (4.6) and (4.7) provide the set of relations among
mappings and their derivatives.

######  Proposition 7.

The set of commutation relations generates a sequence of non-vanishing
cohomological classes:

$\left[\left(\delta^{n_{0}}_{m_{0}}\chi\right)\cdot\chi\right],\;\left[\left(\delta^{n}_{m}\Phi\right)\cdot\Phi\right],\;\left[\left(\delta^{n_{i}}_{n_{i}}\alpha_{i}\right)\cdot\alpha_{i}\right],$

for $i=1,\ldots,L$, for some $L\in\mathbb{N}$, with non-vanishing
$\left(\delta^{n_{0}}_{m_{0}}\chi\right)\cdot\chi$,
$\left(\delta^{n}_{m}\Phi\right)\cdot\Phi$, and
$\left(\delta^{n_{i}}_{m_{i}}\alpha_{i}\right)\cdot\alpha_{i}$. These classes
are independent on the choices of $\chi\in C^{n_{0}}_{m_{0}}(V,W)$, $\Phi\in
C^{n}_{m}(V,W)$, and $\alpha_{i}\in C^{n_{i}}_{m_{i}}(V,W)$.

###### Proof.

Let $\phi$ be one of generators $\chi,\Phi,\alpha_{i},\beta$, $1\leq i\leq L$.
Let us show now the non-vanishing property of
$\left(\left(\delta^{1}_{2}\phi\right)\cdot\phi\right)$. Indeed, suppose

$\left(\delta^{n}_{m}\phi\right)\cdot\phi=0.$

Then there exists $\gamma\in C^{n^{\prime}}_{m^{\prime}}(V,W)$, such that

$\delta^{n}_{m}\phi=\gamma\cdot\phi.$

Both sides of the last equality should belong to the same bicomplex space but
one can see that it is not possible since we obtain $m^{\prime}=t-1$, i.e.,
the number of common vertex operators for the last equation is greater than
for one of multipliers. Thus, $\left(\delta^{n}_{m}\phi\right)\cdot\phi$ is
non-vanishing.

Now let us show that $\left[\left(\delta^{n}_{m}\phi\right)\cdot\phi\right]$
is invariant, i.e., it does not depend on the choice of $\Phi\in
C^{n}_{m}(V,W)$. Substitute $\phi$ by $\left(\phi+\eta\right)\in
C^{n}_{m}(V,W)$. We have

$\displaystyle\left(\delta^{n}_{m}\left(\phi+\eta\right)\right)\cdot\left(\phi+\eta\right)$
$\displaystyle=$
$\displaystyle\left(\delta^{n}_{m}\phi\right)\cdot\phi+\left(\left(\delta^{n}_{m}\phi\right)\cdot\eta-\phi\cdot\delta^{n}_{m}\eta\right)$
(5.8) $\displaystyle+$
$\displaystyle\left(\phi\cdot\delta^{n}_{m}\eta+\delta^{n}_{m}\eta\cdot\phi\right)+\left(\delta^{n}_{m}\eta\right)\cdot\eta.$

Since

$\left(\phi\cdot\delta^{n}_{m}\eta+\left(\delta^{n}_{m}\eta\right)\cdot\phi\right)=\phi\delta^{n}_{m}\eta-(\delta^{n}_{m}\eta)\phi+\left(\delta^{n}_{m}\eta\right)\phi-\phi\;\delta^{n}_{m}\eta=0,$

then (5.8) represents the same cohomology class
$\left[\left(\delta^{n}_{m}\phi\right)\cdot\phi\right]$. ∎

##  6\. Examples

In this section we consider particularly interesting examples of algebras
described in Proposition 6. The orthogonality condition for a bicomplex
sequence (4.7), together with the action of coboundary operators
$\delta^{n}_{m}$ and $\delta^{2}_{\frac{1}{2}}$, and the multiplication
formulas (5.2)–(5.4), define a differential bigraded algebra depending on
vertex algebra elements and formal parameters. In particular, for the
bicomplex (4.6), 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. [12]) with the
continual root space represented by a grading-restricted vertex algebra $V$.

###  6.1. Invariants associated with $C^{2}_{\frac{1}{2}}(V,W)$

Due to non-trivial action of the derivative

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

the case when $\Phi\in C^{2}_{\frac{1}{2}}$ is exceptional among the relations
coming from the double grading condition for a vertex algebra bicomplex, and
allow to reconstruct classical invariants. Let us consider this case. Let
$\Phi\in C^{2}_{\frac{1}{2}}(V,W)$, and $\chi\in C^{n}_{m}(V,W)$. Then we
require the orthogonality:

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

Thus there exist $\beta\in C^{n^{\prime}}_{m^{\prime}}(V,W)$ such that

$\delta^{2}_{\frac{1}{2}}\Phi=\chi\cdot\beta.$

We then get

$3=n+n^{\prime}-r,$ $0=m+m^{\prime}-t,$

Let $n=r+\alpha$, $0\leq\alpha\leq 3$, $n^{\prime}=3-\alpha$;
$m^{\prime}=t-m\geq 0$, i.e., $t=m$, thus $m=t=m^{\prime}=0$. Thus, $\chi\in
C^{r+\alpha}_{0}(V,W)$, $\beta\in C^{3-\alpha}_{0}(V,W)$. For
$r+\alpha=3-\alpha=2$ we obtain $\alpha=1$, $r=1$. If we require $\chi=\Phi\in
C^{2}_{k}(V,W)\subset C^{2}_{\frac{1}{2}}(V,W)$, $k>0$, then the equation

$\delta^{2}_{\frac{1}{2}}\Phi=\Phi\cdot\beta,$

corresponds to a generalization of Gadbillon-Vey invariant [5] for
differential forms. In general, we obtain the commutations

$\left[H,X_{+2}\right]=0,$ $\left[H,Y_{-}\right]=X_{+2},$
$\left[X_{-2},X_{+2}\right]=H,$ $\left[Y_{+},H_{0}\right]=X_{+2},$

for generators

$H=\chi,$ $X_{+1}=\Phi,$ $X_{+2}=\delta^{2}_{\frac{1}{2}}\Phi,$ $Y_{-}=\beta.$

It is easy to see that since all mappings have zero operators composable with,
then all further actions of the derivatives vanish. Nevertheless, recall that

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

$k>0$, thus we can consider the most general case when $\chi\in
C^{r+\alpha}_{m_{0}}(V,W)$, $\Phi\in C^{2}_{m_{1}}(V,W)$, $\beta\in
C^{3-\alpha}_{m_{2}}(V,W)$. Then the grading condition requires
$m_{1}-1=m_{0}+m_{2}-t^{\prime}$, where $t^{\prime}$ is the number of common
vertex operators for $\chi\in C^{r+\alpha}_{m_{0}}(V,W)$ and $\beta\in
C^{3-\alpha}_{m_{2}}(V,W)$. Thus on applying derivatives we obtain further
commutation relations of the form (5.7).

###  6.2. Algebra associated with the short bicomplex (4.7)

One can also develop another example associated to the bicomplex (4.7).
Consider $\chi\in C^{0}_{3}(V,W)$, $\Phi\in C^{1}_{2}(V,W)$. The orthogonality
condition gives

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

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

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

which gives $\alpha\in C^{1}_{t}(V,W)$. For the short sequence (4.7) we get a
continual Lie algebra $\mathcal{G}(V)$ with generators

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

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

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

With the redefinition

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

the commutation relations become:

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

i.e., the orthogonality condition brings about a representation of an
affinization [9] of continual counterpart of the Lie algebra $sl_{2}$. We
prove here that that the cohomological classes:

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

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
$\left(\delta^{1}_{t}\alpha\right)\cdot\alpha$, are independent on the choice
of $\Phi\in C^{1}_{2}(V,W)$, $\chi\in C^{0}_{3}(V,W)$, and $\alpha\in
C^{1}_{t}(V,W)$. Vertex algebra elements 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, P. Somberg, and P.
Zusmanovich for related discussions. Research of the author was supported by
the GACR project 18-00496S.

##  7\. Appendix: Continual Lie algebras

Continual Lie algebras were introduced in [12] and then studied in [13, 14].
Suppose $\mathcal{E}$ is an associative algebra (which we call the base
algebra) over $\mathbb{R}$ or $\mathbb{C}$, and

$K_{0},\;K_{\pm},\;K_{0,0}:\mathcal{E}\times\mathcal{E}\to\mathcal{E},$

are bilinear mappings. The local Lie part of a continual Lie algebra is
defined as

${\widehat{\mathcal{G}}}=\mathcal{G}_{-1}\oplus\mathcal{G}_{0}\oplus\mathcal{G}_{+1},$

where $\mathcal{G}_{i}$, $i=0,\pm 1$, are isomorphic to $\mathcal{E}$ and
parametrized by its elements. The subspaces $\mathcal{G}_{i}$ consist of the
elements

$\left\\{X_{i}(\phi),\phi\in\mathcal{E}\right\\},i=0,\pm 1.$

The generators $X_{i}(\phi)$ are subject to the commutation relations

$\left[X_{0}(\phi),X_{0}(\psi)\right]=X_{0}(K_{0,0}(\phi,\psi)),$
$\left[X_{0}(\phi),X_{\pm 1}(\psi)\right]=X_{\pm 1}(K_{\pm}(\phi,\psi)),$
$\left[X_{+1}(\phi),X_{-1}(\psi)\right]=X_{0}(K_{0}(\phi,\psi)),$

for all $\phi$, $\psi\in\mathcal{E}$. It is also assumed that Jacobi
identities are satisfied. Then the conditions on mappings $K_{0,0}$,
$K_{0,\;\pm}$ follow:

$K_{\pm}(K_{0,0}(\phi,\psi),\chi)=K_{\pm}(\phi,K_{\pm}(\psi,\chi))-K_{\pm}(\psi,K_{\pm}(\phi,\chi)),$
$K_{0,0}(\psi,K_{0}(\phi,\chi))=K_{0}(K_{+}(\psi,\phi),\chi))+K_{0}(\phi,K_{-}(\psi,\chi)),$

for all $\phi$, $\psi$, $\chi\in\mathcal{E}$. An infinite dimensional algebra

$\mathcal{G}(\mathcal{E};K)=\mathcal{G}^{\prime}(\mathcal{E};K)/J,$

is called a continual contragredient Lie algebra, where
$\mathcal{G}^{\prime}(\mathcal{E};K)$ is a Lie algebra freely generated by
${\widehat{\mathcal{G}}}$, and $J$ is the largest homogeneous ideal with
trivial intersection with $\mathcal{G}_{0}$ (consideration of the quotient is
equivalent to imposing the Serre relations in an ordinary Lie algebra case)
[13, 14].

##  8\. Appendix: a multiplication of $C^{n}_{m}(V,W)$-spaces

In this appendix we recall the definition of the simplest variant of
multiplication $*$ of elements of two double complex spaces with the image in
another double complex space coherent with respect to the original
differential (4.2), and satisfying the symmetry (4.1), $L_{V}(0)$-conjugation
(3.2), and $L_{V}(-1)$-derivative (3.3)–(3.4) properties described in Section
1. We prove also an analogue of Leibniz formula (5.4).

###  8.1. Geometrical products of $C^{n}_{m}(V,W)$-spaces

Let us first clarify the geometrical origin of the multiplication. The
structure of $C^{n}_{m}(V,W)$-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 $C^{n}_{m}(V,W)$-spaces we first have to interpret
them geometrically. Basically, a $C^{n}_{m}(V,W)$-space must be associated
with a certain model space, the algebraic $\overline{W}$-language should be
transferred to a geometrical one, two model spaces should be ”connected”
appropriately, and, finally, a product should be defined.

For two spaces $C^{k}_{m}(V,W)$ and $C^{n}_{m^{\prime}}(V,W)$ 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 $W_{z_{1},\ldots,z_{k+n}}$ 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
$F_{k+n}\mathbb{C}$. In general, it may happen that a number $r$ of formal
parameters $(x_{1},\ldots,x_{k})$ of $C^{k}_{m}(V,W)$ coincides with $r$
formal parameters $(y_{1},\ldots,y_{n})$ of $C^{n}_{m^{\prime}}(V,W)$ 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

$(z_{1},\ldots,z_{k+n-r})=(x_{1},\ldots,{x}_{i_{l}},\ldots
x_{k};y_{1},\ldots,\widehat{y}_{i_{l}},\ldots,y_{n}),$ (8.1)

for $1\leq i\leq r$, 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

$\widehat{R}\;\Phi(v_{1}\otimes\ldots\otimes
v_{k};v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}).$

###  8.2. Product of matrix elements

The simplest possible product of elements of two $C^{n}_{m}(V,W)$-spaces is
defined by products of matrix elements of the form (3.1) summed over a
$V_{(l)}$-basis for $l\in\mathbb{Z}$. In geometrical language it corresponds
to sewing of two Riemann spheres associated to $C^{n}_{m}(V,W)$-spaces. We
have the following definition. For $\Phi(v_{1}\otimes\ldots\otimes
v_{k})(x_{1},\ldots,x_{k})\in C^{k}_{m}(V,W)$, and
$\Psi(v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n})\in C_{m^{\prime}}^{n}(V,W)$, the product

$\displaystyle\Phi(v_{1}\otimes\ldots\otimes
v_{k})(x_{1},\ldots,x_{k})*\Psi(v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n})$ (8.2)
$\displaystyle\qquad\qquad\mapsto\widehat{R}\;\Theta\left(v_{1}\otimes\ldots\otimes
v_{k}\otimes v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n}\right)\left(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}\right),$

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

$\displaystyle\langle
w^{\prime},\widehat{R}\;\Theta\left(v_{1}\otimes\ldots\otimes v_{k}\otimes
v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n}\right)\left(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}\right)\rangle$
(8.3) $\displaystyle\quad=\sum_{l\in\mathbb{Z}}\epsilon^{l}\sum_{u_{l}\in
V_{(l)}}\langle w^{\prime},Y^{W}_{WV}\left(\Phi(v_{1}\otimes\ldots\otimes
v_{k})(x_{1},\ldots,x_{k}),\zeta_{1}\right)\;u_{l}\rangle$
$\displaystyle\quad\langle
w^{\prime},Y^{W}_{WV}\left(\Psi(v^{\prime}_{1}\otimes\ldots\otimes{v^{\prime}}_{i_{1}}\otimes\ldots{v^{\prime}}_{j_{r}}\otimes\ldots\otimes
v^{\prime}_{n})\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad(y_{1}\otimes\ldots\otimes{y}_{i_{1}}\otimes\ldots{y}_{j_{r}}\otimes\ldots\otimes
y_{n}),\zeta_{2}\right)\;\overline{u}_{l}\rangle,$

parametrized by $\zeta_{1}$, $\zeta_{2}\in\mathbb{C}$. The sum is taken over
any $V_{l}$-basis $\\{u_{l}\\}$, where $\overline{u}_{l}$ is the dual of
$u_{l}$ with respect to a non-degenerate bilinear form $\langle.\ ,.\rangle$,
(2.1) over $V$. The operation $\widehat{R}$ eliminates $r$ formal parameters
from $\Theta$ from the set $(y_{1},\ldots,y_{n})$ coinciding with $r$ formal
parameter of the set $(x_{1},\ldots,x_{k})$, and excludes all monomials
$(x_{i_{l}}-y_{j_{l}})$, $1\leq l\leq r$, from (8.3). By the standard
reasoning [2, 18], (8.3) does not depend on the choice of a basis of $u\in
V_{l}$, $l\in\mathbb{Z}$. The form of the product defined above is natural in
terms of the theory of charaters for vertex operator algebras [15, 3, 18]. Due
to the symmetry of the geometrical interpretation described above, we could
exclude $r$ formal parameters from the set $(x_{1},\ldots,x_{k})$ in (8.3)
which belong to coinciding pairs resulting to the same definition of the
$*$-product.

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

$\displaystyle\langle
w^{\prime},\sigma(\widehat{R}\;\mathcal{F})(v_{1}\otimes\ldots\otimes
v_{k}\otimes v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}))\rangle$
$\displaystyle\qquad=\langle
w^{\prime},\Theta(\widetilde{v}_{\sigma(1)}\otimes\ldots\otimes\widetilde{v}_{\sigma(k+n-r)})(z_{\sigma(1)},\ldots,z_{\sigma(k+n-r)})\rangle$
$\displaystyle\qquad=\sum\limits_{l\in\mathbb{Z}}\sum_{u_{l}\in
V_{(l)}}\langle
w^{\prime},Y^{W}_{WV}\left(\Phi(\widetilde{v}_{\sigma(1)}\otimes\ldots\otimes\widetilde{v}_{\sigma(k)})(z_{\sigma(1)},\ldots,z_{\sigma(k)}),\zeta_{1}\right)\;u_{l}\rangle$
$\displaystyle\qquad\langle
w^{\prime},Y^{W}_{WV}\left(\Psi(\widetilde{v}_{\sigma(k+1)}\otimes\ldots\otimes\widetilde{v}_{\sigma(k+n-r)})z_{\sigma(k+1)},\ldots,z_{\sigma(k+n-r)}),\zeta_{2}\right)\;\overline{u}_{l}\rangle,$

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

$(\widetilde{v}_{1},\ldots,\widetilde{v}_{k+n-r})=(v_{1},\ldots;v_{k};v^{\prime}_{1},\ldots,\widehat{v}^{\prime}_{j_{1}},\ldots,\widehat{v}^{\prime}_{j_{r}},\ldots,v^{\prime}_{n}).$
(8.5)

Let $t$ be the number of common vertex operators the mappings
$\Phi(v_{1}\otimes\ldots v_{k})(x_{1}$, $\ldots$, $x_{k})\in C^{k}_{m}(V,W)$
and $\Psi(v^{\prime}_{1}\otimes\ldots,\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n})\in C^{n}_{m^{\prime}}(V,W)$, are
composable with. Using the definition of $C^{n}_{m}(V,W)$-space and the
definition of mappings composable with vertex operators, we then have

######  Proposition 8.

For $\Phi(v_{1}\otimes\ldots\otimes v_{k})(x_{1},\ldots,x_{k})\in
C_{m}^{k}(V,W)$ and $\Psi(v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n})\in C_{m^{\prime}}^{n}(V,W)$, the product

$\widehat{R}\;\Theta\left(v_{1}\otimes\ldots\otimes v_{k}\otimes
v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n}\right)\left(x_{1},\ldots,x_{k};y_{1},\ldots,y_{n}\right),$

(8.3) belongs to the space $C^{k+n-r}_{m+m^{\prime}-t}(V,W)$, i.e.,

$*:C^{k}_{m}(V,W)\times C_{m^{\prime}}^{n}(V,W)\to
C_{m+m^{\prime}-t}^{k+n-r}(V,W).$ (8.6)

The proof of this proposition is as follows. Using the geometrical
construction of sewing two Riemann spheres to form another Riemann sphere, we
prove that the product (8.3) belongs to the space
$C_{m+m^{\prime}-t}^{k+n-r}(V,W)$. We show that (8.3) converges to a
$\overline{W}$-valued rational function defined on the configuration space
$F\mathbb{C}_{k+n-r}$, for formal variables with only possible poles at

$(z_{1},\ldots,z_{k+n-r})=(x_{1},\ldots,x_{k};y_{1},\ldots,\widehat{y}_{i_{l}},\ldots,y_{n}),$

satisfies (4.1), $L_{V}(0)$-symmetry (3.2), and $L_{V}(-1)$-derivative
(3.3)–(3.4) conditions, and composable with $m+m^{\prime}-t$ vertex operators.

In order to prove convergence of a product of elements of two spaces
$C_{m}^{k}(V,W)$ and $C_{m^{\prime}}^{n}(V,W)$ we use a geometrical
interpretation [8, 17]. Recall that a $C_{m}^{k}(V,W)$ -space is defined by
means of matrix elements [2] of the form (3.1), and satisfying
$L(0)$-conjugation, $L(-1)$-derivsative conditions, (4.1), and composable with
$m$ vertex operators. For a vertex algebra $V$, and it module $W$, satisying
certain extra conditions [16], one associate elements of a space
$C_{m}^{k}(V,W)$ with the data on the Riemann sphere. In particular, formal
parameters of $C_{m}^{k}(V,W)$-elements and vertex operators they are
composable to, are identified with local coordinates of marked points on a
sphere. For a pair of spaces $C_{m}^{k}(V,W)$ and $C_{m^{\prime}}^{n}(V,W)$,
we consider data on two Riemann spheres. Two extra points are chosen for
centers of annuli used in order to sew spheres [17, 8] to obtain another
sphere. The resulting product (8.3) represents a sum of products of matrix
elements originated from two original Riemann spheres. Two complex parameters
$\zeta_{1}$, $\zeta_{2}$ of (8.3) are identified with coordinates on annuli.
After identification of annuli $r$ coinciding coordinates may occur. This
takes into account case of coinciding formal parameters.

The sewing parameter condition is [17] $\zeta_{1}\zeta_{2}=\epsilon$. In two
sphere $\epsilon$-sewing formulation, the complex parameters $\zeta_{a}$,
$a=1$, $2$ are coordinates inside identified annuluses, and $|\zeta_{a}|\leq
r_{a}$. The product (8.3) converges for various cases of $V$ and $W$. In
particular, it is converges in case of a vertex algebra decomposable into
Heisenberg vertex operator algebras [16]. Some further converging examples of
$V$ and it modules $W$ will be considered elsewhere. The matrix elements in
(8.3) are absolutely convergent in powers of $\epsilon$ with some radii of
convergence $R_{a}\leq r_{a}$, with $|\zeta_{a}|\leq R_{a}$. By expanding the
product (8.3) as power series in $\epsilon$ for $|\zeta_{a}|\leq R_{a}$, where
$|\epsilon|\leq r$ for $r<r_{1}r_{2}$. By applying Cauchy’s inequality to
coefficients for $x$\- and $y$-depending parts of the product we find that
(8.3) is absolute convergent as a formal series in $\epsilon$ is defined for
$|\zeta_{a}|\leq r_{a}$, and $|\epsilon|\leq r$ for $r<r_{1}r_{2}$, with extra
poles only at $z_{i}$, $1\leq i\leq k+n-r$.

When (8.3) is convergent, using geometrical procedure [8, 17] of sewing of two
Riemann spheres, we prove that the limiting function is analytically
extendable to a $\overline{W}$-valued function defined on the configuration
space

$F\mathbb{C}_{k+n-r}=\left\\{(z_{1},...,z_{k+n-r}):z_{i}\neq z_{j},i\neq
j\right\\}.$

Element $\Phi\in C^{k}_{m}(V,W)$ and $\Psi\in C^{n}_{m^{\prime}}(V,W)$ are
defined on the configuration spaces $F\mathbb{C}_{k}$ and $F\mathbb{C}_{n}$
correspondingly. Due to the construction of the product (8.3), for every
$u_{l}\in V_{(l)}$, each summand in (8.3) defines a rational function on the
configuration space $F\mathbb{C}_{k+n-r}$. Recall the sewing relation
$\zeta_{1}\zeta_{2}=\epsilon$ for two Riemann spheres.

By construction, it is assumed that annular regions are not intersecting.
After the identification of annular regions the analytic extension of the
limiting function of (8.3) is given by the the sewing geometrical procedure of
[8, 17]. The extension is from two original Riemann spheres to the sphere
formed by sewing. The construction of (8.3) provides that it gives the
$\overline{W}$-valued rational function on the configuration space
$F\mathbb{C}_{k+n-r}$. Similar, the construction of the product (8.3) provides
that the limiting function is a $\overline{W}$-valued rational function with
the only possible poles at $z_{i}=z_{j}$, $1\leq i<j\leq k+n-r$.

We define the action of

$\partial_{p}=\partial_{z_{p}}={\partial}/{\partial_{z_{p}}},$

for $1\leq p\leq k+n-r$, the differentiation of

$\Theta(v_{1}\otimes\ldots\otimes v_{k}\otimes
v^{\prime}_{1}\otimes\ldots\otimes\widehat{v}^{\prime}_{i_{l}}\otimes\ldots\otimes
v^{\prime}_{n})(x_{1},\ldots,x_{k};y_{1},\ldots;\widehat{y}_{i_{l}},\ldots,y_{n}),$

for $1\leq l\leq r$, with respect to the $p$-th entry of
$(x_{1},\ldots,x_{k};y_{1},\ldots,\widehat{y}_{i_{l}},\ldots,y_{n})$ as
follows

$\displaystyle\langle w^{\prime},\partial_{p}\Theta(v_{1}\otimes\ldots\otimes
v_{k}\otimes
v^{\prime}_{1}\otimes\ldots\otimes\widehat{v}^{\prime}_{i_{l}}\otimes\ldots\otimes
v^{\prime}_{n})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(x_{1},\ldots,x_{k};y_{1},\ldots,\widehat{y}_{i_{l}},\ldots,y_{n})\rangle$
$\displaystyle=\sum_{q\in\mathbb{Z}}\epsilon^{q}\sum_{u_{q}\in V_{q}}\langle
w^{\prime},\partial^{\delta_{p,i}}_{x_{i}}Y^{W}_{WV}\left(\Phi(v_{1}\otimes\ldots\otimes
v_{k})(x_{1},\ldots,x_{k}),\zeta_{1}\right)\;u_{q}\rangle$
$\displaystyle\qquad\qquad\langle
w^{\prime},\partial^{\delta_{p,j}}_{y_{j}}Y^{W}_{WV}\left(\Phi(v^{\prime}_{1}\otimes\ldots\otimes\widehat{v}^{\prime}_{i_{l}}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,\widehat{y}_{i_{l}},\ldots,y_{n}),\zeta_{2}\right)\overline{u}_{q}\rangle.$

By direct substitution we prove that the product (8.3) satisfies the
$L_{V}(-1)$-derivative (3.3)–(3.4) and $L_{V}(0)$-conjugation (3.2)
properties. Using the definition of the action of an element $\sigma\in
S_{k+n-r}$ on the product (8.3), we prove (4.1) for (8.3).

Next, we show that (8.3) is composable with $m+m^{\prime}-t$ vertex operators.
Recall that $\Phi(v_{1}\otimes\ldots\otimes v_{k})(x_{1},\ldots,x_{k})$ is
composable with $m$ vertex operators, and
$\Psi(v^{\prime}_{1}\otimes\ldots\otimes v^{\prime}_{n})(y_{1},\ldots,y_{n})$
is composable with $m^{\prime}$ vertex operators. Let us consider the first
condition of composability for the product (8.3) with a number of vertex
operators. We redefine the notations for the set

$\displaystyle(v^{\prime\prime}_{1},\ldots,v^{\prime\prime}_{k};v^{\prime\prime}_{k+1},\ldots,v^{\prime\prime}_{k+m};v^{\prime\prime}_{k+m+1},\dots,v^{\prime\prime}_{k+n-r+m+m^{\prime}-t};v^{\prime\prime}_{n-r+1},\ldots,v^{\prime\prime}_{n-r+m^{\prime}-t})$
$\displaystyle\qquad\qquad=(v_{1},\ldots,v_{k};v_{k+1},\ldots,v_{k+m};v^{\prime}_{1},\dots,v^{\prime}_{n};v^{\prime}_{n-r+1},\ldots,v^{\prime}_{n-r+m^{\prime}-t}),$
$\displaystyle(z_{1},\ldots,z_{k};z_{k+1},\dots,z_{k+n-r})=(x_{1},\ldots,x_{k};y_{1},\ldots,\widehat{y}_{i_{l}}\ldots,y_{n}),$

of vertex algebra $V$ elements. Introduce
$l^{\prime\prime}_{1},\dots,l^{\prime\prime}_{k+n-r}\in\mathbb{Z}_{+}$, such
that
$l^{\prime\prime}_{1}+\ldots+l^{\prime\prime}_{k+n-r}=k+n-r+m+m^{\prime}-t$.
Define

$\displaystyle\Psi^{\prime\prime}_{i}$ $\displaystyle=$ $\displaystyle
E^{(l^{\prime\prime}_{i^{\prime\prime}})}_{V}(v^{\prime\prime}_{k^{\prime\prime}_{1}}\otimes\ldots\otimes
v^{\prime\prime}_{k^{\prime\prime}_{i^{\prime\prime}}};\mathbf{1}_{V})(z_{k^{\prime\prime}_{1}}-\zeta^{\prime\prime}_{i^{\prime\prime}},\ldots,z_{k^{\prime\prime}_{i^{\prime\prime}}}-\zeta^{\prime\prime}_{i^{\prime\prime}}),$
(8.8)

where

$\displaystyle{k^{\prime\prime}_{1}}={l^{\prime\prime}_{1}+\ldots+l^{\prime\prime}_{i^{\prime\prime}-1}+1},\quad\ldots,\quad{k^{\prime\prime}_{i^{\prime\prime}}}={l^{\prime\prime}_{1}+\ldots+l^{\prime\prime}_{i^{\prime\prime}-1}+l^{\prime\prime}_{i^{\prime\prime}}},$
(8.9)

for $i^{\prime\prime}=1,\dots,k+n-r$, and we take

$(\zeta^{\prime\prime}_{1},\ldots,\zeta^{\prime\prime}_{k+n-r})=(\zeta_{1},\ldots,\zeta_{k};\zeta^{\prime}_{1},\ldots,\zeta^{\prime}_{n}).$

Then we consider

$\displaystyle\mathcal{I}^{k+n-r}_{m+m^{\prime}-t}(\widehat{R}\;\Theta)=\sum_{r^{\prime\prime}_{1},\dots,r^{\prime\prime}_{k+n-r}\in\mathbb{Z}}\langle
w^{\prime},\widehat{R}\;\Theta(P_{r^{\prime\prime}_{1}}\Psi^{\prime\prime}_{1}\otimes\ldots\otimes
P_{r^{\prime\prime}_{k+n-r}}\Psi^{\prime\prime}_{k+n-r})$ (8.10)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(\zeta^{\prime\prime}_{1},\ldots,\zeta^{\prime\prime}_{k+n-r})\rangle,$

and prove it is absolutely convergent with some conditions. The condition

$\displaystyle|z_{l^{\prime\prime}_{1}+\ldots+l^{\prime\prime}_{i-1}+p^{\prime\prime}}-\zeta^{\prime\prime}_{i}|+|z_{l^{\prime\prime}_{1}+\ldots+l^{\prime\prime}_{j-1}+q^{\prime\prime}}-\zeta^{\prime\prime}_{i}|<|\zeta^{\prime\prime}_{i}-\zeta^{\prime\prime}_{j}|,$
(8.11)

of absolute convergence for (8.10) for $i^{\prime\prime}$,
$j^{\prime\prime}=1,\dots,k+n-r$, $i\neq j$ and for
$p^{\prime\prime}=1,\dots,l^{\prime\prime}_{i}$ and
$q^{\prime\prime}=1,\dots,l^{\prime\prime}_{j}$, follows from corresponding
conditions for $\Phi$ and $\Psi$. We obtain

$\displaystyle\left|\mathcal{I}^{k+n-r}_{m+m^{\prime}-t}(\widehat{R}\;\Theta)\right|\leq\left|\mathcal{I}^{k}_{m}(\Phi)\right|\;\left|\mathcal{I}^{n}_{m^{\prime}}(\Psi)\right|.$

Thus, we infer that (8.10) is absolutely convergent. Recall that the maximal
orders of possible poles of (8.10) are $N^{k}_{m}(v_{i},v_{j})$,
$N^{n}_{m^{\prime}}(v^{\prime}_{i^{\prime}},v^{\prime}_{j^{\prime}})$ at
$x_{i}=x_{j}$, $y_{i^{\prime}}=y_{j^{\prime}}$. From the last expression we
deduce that there exist positive integers
$N^{k+n-r}_{m+m^{\prime}-t}(v^{\prime\prime}_{i^{\prime\prime}},v^{\prime\prime}_{j^{\prime\prime}})$
for $i$, $j=1,\dots,k$, $i\neq j$, $i^{\prime}$, $j^{\prime}=1,\dots,n$,
$i^{\prime}\neq j$, depending only on $v^{\prime\prime}_{i^{\prime\prime}}$
and $v^{\prime\prime}_{j^{\prime\prime}}$ for $i^{\prime\prime}$,
$j^{\prime\prime}=1,\dots,k+n$, $i^{\prime\prime}\neq j^{\prime\prime}$ such
that the series (8.10) can be analytically extended to a rational function in
$(x_{1},\dots,x_{k};y_{1},\ldots,y_{n})$, independent of
$(\zeta^{\prime\prime}_{1},\dots,\zeta^{\prime\prime}_{k+n-r})$, with extra
possible poles at and $x_{i}=y_{j}$, of order less than or equal to
$N^{k+n-r}_{m+m^{\prime}-t}(v^{\prime\prime}_{i^{\prime\prime}},v^{\prime\prime}_{j^{\prime\prime}})$,
for $i^{\prime\prime}$, $j^{\prime\prime}=1,\dots,n$, $i^{\prime\prime}\neq
j^{\prime\prime}$.

Let us proceed with the second condition of composability. For the product
(8.3) we obtain
$(v^{\prime\prime}_{1},\dots,v^{\prime\prime}_{k+n-r+m+m^{\prime}-t})\in V$,
and $(z_{1},\ldots$ , $z_{k+n-r+m+m^{\prime}-t})\in\mathbb{C}$, we find
positive integers $N^{k+n-r}_{m+m^{\prime}-t}(v^{\prime}_{i},v^{\prime}_{j})$,
depending only on $v^{\prime}_{i}$ and $v^{\prime\prime}_{j}$, for
$i^{\prime\prime}$, $j^{\prime\prime}=1,\dots,k+n-r$, $i^{\prime\prime}\neq
j^{\prime\prime}$, such that for arbitrary $w^{\prime}\in W^{\prime}$. Under
conditions

$\displaystyle z_{i^{\prime\prime}}\neq z_{j^{\prime\prime}},\quad
i^{\prime\prime}\neq j^{\prime\prime},\quad$
$\displaystyle|z_{i^{\prime\prime}}|>|z_{k^{\prime\prime\prime}}|>0,$ (8.12)

for $i^{\prime\prime}=1,\dots,m+m^{\prime}-t$, and
$k^{\prime\prime\prime}=m+m^{\prime}-t+1,\dots,m+m^{\prime}-t+k+n-r$, let us
introduce

$\displaystyle\mathcal{J}^{k+n-r}_{m+m^{\prime}-t}(\widehat{R}\;\Theta)=\sum_{q\in\mathbb{C}}\langle
w^{\prime},E^{(m+m^{\prime}-t)}_{W}\Big{(}v^{\prime\prime}_{1}\otimes\ldots\otimes
v^{\prime\prime}_{m+m^{\prime}-t};$ (8.13) $\displaystyle\qquad
P_{q}(\widehat{R}\;\Theta(v^{\prime\prime}_{m+m^{\prime}-t+1}\otimes\ldots\otimes
v^{\prime\prime}_{m+m^{\prime}-t+k+n-r})$
$\displaystyle\qquad\qquad(z_{m+m^{\prime}-t+1},\ldots,z_{m+m^{\prime}-t+k+n-r}))\Big{)}(z_{1},\ldots,z_{m+m^{\prime}-t})\rangle.$

We then obtain

$\displaystyle|\mathcal{J}^{k+n}_{m+m^{\prime}}(\widehat{R}\;\Theta)|\leq\left|\mathcal{J}^{k}_{m}(\Phi)\right|\;\left|\mathcal{J}^{n}_{m^{\prime}}(\Psi)\right|,$

where we have used the invariance of (8.3) with respect to $\sigma\in
S_{m+m^{\prime}-t+k+n-r}$. $\mathcal{J}^{k}_{m}(\Phi)$ and
$\mathcal{J}^{n}_{m^{\prime}}(\Psi)$ in the last expression are absolute
convergent. Thus, we infer that
$\mathcal{J}^{k+n-r}_{m+m^{\prime}-t}(\widehat{R}\;\Theta)$ is absolutely
convergent, and the sum (8.10) is analytically extendable to a rational
function in $(z_{1},\dots,z_{k+n-r+m+m^{\prime}-t})$ with the only possible
poles at $x_{i}=x_{j}$, $y_{i^{\prime}}=y_{j^{\prime}}$, and at
$x_{i}=y_{j^{\prime}}$, i.e., the only possible poles at
$z_{i^{\prime\prime}}=z_{j^{\prime\prime}}$, of orders less than or equal to
$N^{k+n-r}_{m+m^{\prime}-t}(v^{\prime\prime}_{i^{\prime\prime}},v^{\prime\prime}_{j^{\prime\prime}})$,
for $i^{\prime\prime}$, $j^{\prime\prime}=1,\dots,k^{\prime\prime\prime}$,
$i^{\prime\prime}\neq j^{\prime\prime}$. This finishes the proof of the
proposition.

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

The product admits the action ot the differential operator
$\delta^{k+n-r}_{m+m^{\prime}-t}$ defined in (4.3) and (4.5) where $r$ is the
number of common formal parameters, and $t$ the number of commpon composable
vertex operators for $\Phi\in C^{k}_{m}(V,W)$ and $\Psi\in
C^{n}_{m^{\prime}}(V,W)$. The co-boundary operators (4.3) and (4.5) possesse a
variation of Leibniz law with respect to the $*$-product. By direct
computation we check

######  Proposition 9.

For $\Phi(v_{1}\otimes\ldots\otimes v_{k})(x_{1}\ldots x_{k})\in
C_{m}^{k}(V,W)$ and $\Psi(v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n})\in C_{m^{\prime}}^{n}(V,W)$, the action of
$\delta_{m+m^{\prime}-t}^{k+n-r}$ on their product (8.3) is given by

$\displaystyle\delta_{m+m^{\prime}-t}^{k+n-r}(\Phi(v_{1}\otimes\ldots\otimes
v_{k})(x_{1},\ldots,x_{k})*\Psi(v^{\prime}_{1}\otimes\ldots\otimes
v^{\prime}_{n})(y_{1},\ldots,y_{n}))$ (8.14)
$\displaystyle\qquad=\left(\delta^{k}_{m}\Phi(\widetilde{v}_{1}\otimes\ldots\otimes\widetilde{v}_{k})(z_{1},\ldots,z_{k})\right)*\Psi(\widetilde{v}_{k+1}\otimes\ldots\otimes\widetilde{v}_{k+n-r})(z_{k+1},\ldots,z_{k+n-r})$
$\displaystyle\;+(-1)^{k}\Phi(\widetilde{v}_{1}\otimes\ldots\otimes\widetilde{v}_{k})(z_{1},\ldots,z_{k})$
$\displaystyle\qquad\qquad*\left(\delta^{n-r}_{m^{\prime}-t}\Psi(\widetilde{v}_{1}\otimes\ldots\otimes\widetilde{v}_{k+n-r})(z_{k+1},\ldots,z_{k+n-r})\right).$

$\square$

Finally, we have the following

######  Corollary 1.

The multiplication (8.3) extends the chain-cochain complex structure of
Proposition 3 to all products $C^{k}_{m}(V,W)\times C^{n}_{m^{\prime}}(V,W)$,
$k$, $n\geq 0$, $m$, $m^{\prime}\geq 0$. ∎

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

######  Corollary 2.

The product of elements of the spaces $C^{2}_{ex}(V,W)$ and $C^{n}_{m}(V,W)$
is given by (8.3),

$*:C^{2}_{ex}(V,W)\times C^{n}_{m}(V,W)\to C^{n+2-r}_{m}(V,W),$ (8.15)

and, in particular,

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

$\square$

## References

  * [1] Frenkel, E.; Ben-Zvi, D. Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88\. American Mathematical Society, Providence, RI, 2001. xii+348 pp.
  * [2] I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989; Memoirs Amer. Math. Soc. 104, 1993.
  * [3] Ph. Francesco, P. Mathieu, and D. Senechal. Conformal Field Theory. Graduate Texts in Contemporary Physics. 1997.
  * [4] F. Qi. Representation theory and cohomology theory of meromorphic open string vertex algebras, Ph.D. dissertation, (2018).
  * [5] E. Ghys L’invariant de Godbillon-Vey. Seminaire Bourbaki, 41–eme annee, n 706, S. M. F. Asterisque 177–178 (1989)
  * [6] Y.-Zh. Huang A cohomology theory of grading-restricted vertex algebras. Comm. Math. Phys. 327 (2014), no. 1, 279–307.
  * [7] Y.-Z. Huang, The first and second cohomologies of grading-restricted vertex algebras, Comm. Math. Phys. 327 (2014), no. 1, 261–278.
  * [8] Y.-Z. Huang, Two-Dimensional Conformal Geometry and Vertex Operator Algebras, Progress in Mathematics, Vol. 148, Birkhäuser, Boston, 1997.
  * [9] V. Kac: Vertex Operator Algebras for Beginners, University Lecture Series 10, AMS, Providence 1998. Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–307.
  * [10] Li, H.: Symmetric invariant bilinear forms on vertex operator algebras, J. Pure. Appl. Alg. 96 (1994) 279–297.
  * [11] M.V.Losik, On some generalization of a manifold and its characteristic classes (Russian), Funcional. Anal. i Prilozhen. 24(1990), no 1, 29-37 ; English translation in Functional Anal. Appl. 24 (1990), 26–32.
  * [12] M. V. Saveliev, Integro-differential nonlinear equations and continual Lie algebras. Comm. Math. Phys. 121 (1989), no. 2, 283–290.
  * [13] M. V. Saveliev, A. M. Vershik. Continuum analogues of contragredient Lie algebras. Commun. Math. Phys. 126, 367, 1989;
  * [14] M. V. Saveliev, A. M. Vershik. New examples of continuum graded Lie algebras. Phys. Lett. A, 143, 121, 1990.
  * [15] A. Tsuchiya, K. Ueno, and Y. Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure. Math. 19 (1989), 459–566.
  * [16] M. P. Tuite, A. Zuevsky, A generalized vertex operator algebra for Heisenberg intertwiners. J. Pure Appl. Algebra 216 (2012), no. 6, 1442–1453.
  * [17] A. Yamada, Precise variational formulas for abelian differentials. Kodai Math.J. 3 (1980), 114–143.
  * [18] Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–307.
  * [19] A. Zuevsky. Prescribed rational functions cohomology of foliations on smooth manifolds. To appear.


# PRASEMap: A Probabilistic Reasoning and Semantic Embedding based Knowledge
Graph Alignment System

Zhiyuan Qi  Tencent Jarvis LabShenzhenChina<EMAIL_ADDRESS>,  Ziheng Zhang
Tencent Jarvis LabShenzhenChina<EMAIL_ADDRESS>,  Jiaoyan Chen
University of OxfordOxfordUK<EMAIL_ADDRESS>,  Xi Chen  Tencent
Platform and Content GroupShenzhenChina<EMAIL_ADDRESS>and  Yefeng
Zheng  Tencent Jarvis LabShenzhenChina<EMAIL_ADDRESS>
(2021)

###### Abstract.

Knowledge Graph (KG) alignment aims at finding equivalent entities and
relations (i.e., mappings) between two KGs. The existing approaches utilize
either reasoning-based or semantic embedding-based techniques, but few studies
explore their combination. In this demonstration, we present PRASEMap, an
unsupervised KG alignment system that iteratively computes the Mappings with
both Probabilistic Reasoning (PR) And Semantic Embedding (SE) techniques.
PRASEMap can support various embedding-based KG alignment approaches as the SE
module, and it also enables easy human computer interaction that additionally
provides an option for users to feed the mapping annotations back to the
system for better results. The demonstration showcases these features via a
stand-alone Web application with user friendly interfaces. The demo is
available at https://prasemap.qizhy.com.

knowledge graph, knowledge graph alignment, probability reasoning, embedding

††copyright: acmcopyright††journalyear: 2021††copyright:
acmcopyright††conference: Proceedings of the 30th ACM International Conference
on Information and Knowledge Management; November 1–5, 2021; Virtual Event,
QLD, Australia††booktitle: Proceedings of the 30th ACM International
Conference on Information and Knowledge Management (CIKM ’21), November 1–5,
2021, Virtual Event, QLD, Australia††price: 15.00††doi:
10.1145/3459637.3481972††isbn: 978-1-4503-8446-9/21/11††submissionid:
3069††ccs: Computing methodologies Knowledge representation and reasoning

##  1\. Introduction

Knowledge Graph (KG) alignment attempts to discover equivalent entities,
relations, and attributes (i.e., mappings) between two KGs for constructing
larger and more comprehensive KGs. The previous approaches can be roughly
categorized as reasoning-based and embedding-based ones (Sun et al., 2020;
Zhang et al., 2020). The traditional reasoning-based approaches, such as PARIS
(Suchanek et al., 2012) and LogMap (Jiménez-Ruiz and Grau, 2011), often
exploit logical reasoning together with lexical and graph matching techniques.
However, they are usually weak at utilizing the KG structural information. In
contrast, the embedding-based models, such as GCNAlign (Wang et al., 2018) and
MultiKE (Zhang et al., 2019), can well explore the graph structures by
utilizing KG embedding techniques but they lack holistic analysis or logical
consistency check for the discovered mappings (Sun et al., 2020).

In light of the complementarity between the reasoning-based and the embedding-
based approaches, we propose to demonstrate PRASEMap, a stand-alone Web
application of the KG alignment system PRASE, which is recently proposed in
(Qi et al., 2021), based on Probabilistic Reasoning (PR) and Semantic
Embedding (SE). As suggested by the name, PRASEMap consists of a PR module, an
SE module, and an iterative algorithm for the interaction between two modules
(Qi et al., 2021). Its PR module is written in C++ and wrapped in Python,
while its SE module can be constructed in Python based on various embedding-
based KG alignment algorithms. It is worth mentioning that PRASEMap provides
flexible APIs to allow users to easily develop and evaluate their own SE
module implementations. PRASEMap features several innovative components:

  * •

A fully functional knowledge graph alignment system computing/predicting
entity mappings across the pre-defined or user-uploaded KG pair.

  * •

A registration and login interface helping users initialize, monitor, and
resume their assigned task.

  * •

An interface visualizing the representative subgraphs of the resultant aligned
KG pair and highlighting the mapping relationships.

Note that the whole system can be performed in two modes, i.e., an automatic
mode without any seed mappings or human intervention and a semi-automatic mode
accepting the human annotated mappings for augmentation with an annotation
interface specifically designed to identify entity mappings. To summarize, the
Web application PRASEMap allows users to test the widely-used benchmarks or
upload their own KGs, monitor the execution, annotate uncertain mappings,
submit their feedback, and download the resultant mappings. The demonstration
of PRASEMap can be experienced online at https://prasemap.qizhy.com, and its
source code is made available at https://github.com/qizhyuan/PRASEMap.

##  2\. System Design

We consider the knowledge graph alignment that aims at discovering the entity
mappings, i.e., equivalent entities referring to the same real-world object,
between two KGs $\mathcal{KG}_{1}$ and $\mathcal{KG}_{2}$. This section first
presents the architecture and pipeline of PRASEMap and then briefly introduces
the two core modules — Probabilistic Reasoning (PR) and Semantic Embedding
(SE).

Figure 1. The Architecture of PRASEMap.

###  2.1. Architecture and Pipeline

As shown in Figure 1, PRASEMap consists of the data preprocessing module
(denoted as PRE module), the PR module, and the SE module. These modules work
in an iterative pipeline whose steps are described as follow.

[Step 1. Initialization] The PRE module parses the two KG files that are
either pre-stored in the system or uploaded by the user, and constructs two KG
objects that contain standardized relation and attribute triples. Note that
for cross-lingual KGs, the PRE module can call the translation API to
translate the text into the same language for subsequent PR and SE modules.

[Step 2. Running PR Module] The PR module receives the KG objects, and
iteratively discovers cross-KG literal mappings, entity mappings, and relation
mappings. After several self-iterations, it outputs the entity mappings (cf.
Section 2.2).

[Step 3. Running SE Module] The SE module uses the entity mappings from the PR
module as seeds for training, and calculates entity embeddings and more entity
mappings as the output (cf. Section 2.3).

[Step 4. Rerunning PR Module] The PR module is performed again with the entity
embeddings and mappings from the SE module. Different from Step 2, entity
embeddings are utilized to enable more accurate probabilistic reasoning, and
the received entity mappings are utilized to disclose more potential mappings.

Given a task, Step 1 and Step 2 are executed once, while Step 3 and Step 4 can
be repeated for several times (which can be configured by the user) to fully
explore the benefits of the two modules. This is the automatic mode of
PRASEMap relying on no seed mapping or human interaction. In the semi-
automatic mode, the PR module selects and presents some uncertain mappings to
the user and receives the manual annotations, based on which it further
discovers more mappings and deletes some wrong mappings.

###  2.2. Probabilistic Reasoning Module

Figure 2. Mapping Visualization.

A famous reasoning-based alignment system termed PARIS (Suchanek et al., 2012)
is adopted and upgraded as the PR module. PARIS was originally implemented in
Java.111http://webdam.inria.fr/paris/ To be compatible with many embedding-
based KG alignment algorithms implemented in Python, we re-implement PARIS
using C++ and create its Python bindings using the PyBind11 library, which
makes PRASEMap much more effective than our original PRASE that is implemented
in Python and evaluated in (Qi et al., 2021).

Briefly, the original PARIS first computes the functionality of the KG
relations and attributes. Given a head entity, the functionality can be used
to determine whether two tail entities or literals across two KGs are
identical. Next, PARIS discovers initial literal mappings and entity mappings
by lexical matching, and further iteratively updates the probabilities of
these mappings as well as the subsumption relationships between relations and
attributes by reasoning with the functionality. PARIS finally outputs entity
mappings along with other mappings after several iterations when the mappings
converge. Please refer to (Suchanek et al., 2012) for more details.

In Step 2, the PR module uses the original PARIS while in Step 4, the PR
module uses the upgraded PARIS that has several new features. First, it can
exploit the semantic embeddings output by the SE module to adjust the
estimated probability of entity mappings during its reasoning process, so as
to improve the recall and the precision of the resultant mappings.
Specifically, the new formulation of computing the equivalent probability of
two entities $e_{1}\in{\mathcal{KG}_{1}}$ and $e_{2}\in{\mathcal{KG}_{2}}$ is
given by $\text{Pr}_{\text{new}}(e_{1}\equiv
e_{2}):=\alpha\text{Pr}_{\text{old}}(e_{1}\equiv
e_{2})+(1-\alpha)\text{sim}(\bm{e_{1}},\bm{e}_{2})$ with $\alpha\in(0,1)$,
where the first term corresponds to the original probability estimation
formulation of PARIS; the second term measures the (cosine) similarity of
entity embeddings used to adjust $\text{Pr}_{\text{old}}$. We use the Eigen
library to manipulate embeddings in the process of reasoning. Second, the
upgraded PARIS directly adopts those high-quality entity mappings calculated
by the SE module into its initial entity mapping set for further exploring
more potential mappings during its self-iterations. Please refer to (Qi et
al., 2021) for detailed explanations of the PR module.

Considering the difficulty in aligning two complex KGs, manual labeling often
plays an important role in the real-world applications (Zhang et al., 2020).
To further leverage manual labeling, the PR module enables optional human
interaction through which those uncertain mappings are sent to users for
feedback. Specifically, the PR module randomly selects a number of the
mappings with low scores for labeling. The users can mark a score of $0$
(i.e., false) or $1$ (i.e., true) for each entity mapping. Note that PRASEMap
regards the labels from user feedback as ground truth and will not update
these labeled mappings’ scores during the subsequent iterations.

###  2.3. Semantic Embedding Module

The default SE module in PRASEMap is constructed based on GCNAlign (Wang et
al., 2018), implemented in TensorFlow. GCNAlign is a typical embedding-based
approach using Graph Convolutional Networks (GCNs) to encode the neighbouring
information of entities. It first constructs the adjacency matrices of two KGs
based on the functionality of relations, and then embeds entities into a low-
dimensional space using GCNs, where equivalent entities are close to each
other.

The SE module receives the entity mappings from the PR module and selects the
entity mappings with high scores as its training seeds. After training, the
resultant entity embeddings are used to discover more potential entity
mappings by measuring the distance between them. The newly discovered mappings
and the learned embeddings are fed back to the PR module for iteration (Step
4).

Note that PRASEMap is compatible to different embedding-based KG alignment
approaches if they can output entity embeddings and entity mappings. In
addition to GCNAlign, the current PRASEMap system has already implemented
MTransE (Chen et al., 2017) and MultiKE (Zhang et al., 2019), both of which
often achieve state-of-the-art performance in many benchmarks. More SE modules
will be added in the follow-up development. Users can also easily adopt other
embedding-based implementations from external libraries, such as OpenEA (Sun
et al., 2020), or develop their own embedding-based KG alignment approaches.

##  3\. Demonstration

This section provides a demonstration towards PRASEMap and a brief evaluation
of the human annotation during the iterations.

###  3.1. Web Application

For better human-computer interaction, we construct a stand-alone Web
application with multiple key features and visualization interfaces.
Specifically, the back end is developed based on SpringBoot, Mybatis, and
Redis. The front end is developed using Vue.js and antv/g6 — a graph
visualization engine. The system itself stores some widely-used KG alignment
benchmarks from popular general-purpose KGs, such as DBpedia and Wikidata, as
well as an industry benchmark from Tencent’s medical KGs (Zhang et al., 2020).
The Web application also allows users to upload their own KG files for KG
alignment and visualization. The resultant mappings can be downloaded. We next
introduce three important interfaces and their usage.

[Task Running and Mapping Visualization] The user registers at the Web
application with his/her email, and then receives a unique task ID for his/her
own KG alignment task on the Web application. After login at the Web
application, the user submits a KG alignment task, and the server enqueues the
task on a work queue to maintain the workload. When PRASEMap starts to run for
a task, some logging information is presented for the user to monitor the
running process including the running stage, the time cost, the training loss,
and the number of discovered mappings. To illustrate the alignment result, the
Web application can visualize each mapping, where the corresponding one-hop
neighbourhoods of the two associated entities of the mapping are presented.
With the neighbourhoods, the user can better understand why entities are
matched, and meanwhile the Web application can present more mappings between
their neighbouring entities, which can further justify why the two entities
are matched. Figure 4 shows the visualization of an example mapping between
“Florin_Tene” of DBpedia-EN (blue-circled entities) and “Florin_Tene” of
DBpedia-DE (pink-circled entities), where the blue/pink lines represent the
relations within each single KG, and the orange lines represent all the entity
mappings. Furthermore, the user can click any individual entity with the mouse
to check the corresponding attributes that are also helpful to explain a
mapping.

[Manual Labeling] The Web application enables the user to return their
feedback to PRASEMap for improvement. PRASEMap selects a number of entity
mappings with low confidence and presents them to the user in a table, named
as User Feedback Form in Figure 3. For each mapping, the user can judge and
annotate whether the two entities are equivalent or not. To assist manual
labeling, another visualization interface is developed to provide the user
more information of the involved entities. Specifically, when the user clicks
the visualization icon, the neighbourhood around each entity (e.g., the
connected relations, entities, and attributes) is shown (cf. Figure 2). Manual
labeling is fully optional for the user, and the user can label as many
mappings as they want. Once the user finishes the labeling and submits their
feedback, PRASEMap runs again with these user-delivered mappings and updates
the alignment results.

[KG and Neighbourhood Visualization] The application can present the overall
statistical information of a KG, including the numbers of the entities,
relations, and attributes, and the numbers of the relation and attribute
triples. Meanwhile, as aforementioned, the system can visualize the
neighbourhood of a specific entity in a KG. Figure 2 visualizes the entity
“Mariah_Carey” of DBpedia-EN, where its attributes, its one-hop neighbouring
entities and connected relations are shown. The attributes of a neighbouring
entity can be displayed when the user clicks it.

Figure 3. The Interface for Manual Mapping Labeling. Figure 4. Neighbourhood
Visualization.

###  3.2. Evaluation of Human Annotation

The performance of the automatic mode of PRASEMap has been verified in the
performance evaluation of PRASE (Qi et al., 2021). This work complements the
evaluation on its semi-automatic mode, i.e., the effectiveness of human
annotation feedback.

We present the results of PRASEMap on Tencent’s industry benchmark MED-BBK-9K
(Zhang et al., 2020) using MultiKE as the SE module, with 10, 50 and 100 human
annotations fed back to iteration. Note that we use undiscovered ground truth
mappings as human feedback for convenience in evaluation. The results are
shown in Table 1, where P, R, and F1 denote precision, recall, and F1-score,
respectively. It can be observed that, compared with PARIS, both the precision
and the recall of PRASEMap significantly increase, leading to a higher
F1-score. Besides, the performance of PRASEMap benefits from the human
labeling as the P/R/F1 increases as the number of annotations increases. This
finding also sheds light on the importance and effectiveness of human
interaction in KG alignment. Please refer to (Qi et al., 2021) for more
comprehensive evaluation about PRASEMap.

Table 1. Experimental results on MED-BBK-9K (Zhang et al., 2020). Model | P (%) | R (%) | F1 (%)
---|---|---|---
PARIS | 80.588 | 37.110 | 50.818
PRASEMap | 83.833 | 61.973 | 71.265
PRASEMap w/ 10 Annotations | 84.186 | 62.115 | 71.486
PRASEMap w/ 50 Annotations | 84.402 | 62.606 | 71.889
PRASEMap w/ 100 Annotations | 84.760 | 63.130 | 72.363

##  4\. Conclusion and Discussion

In this demonstration, we present a KG alignment system termed PRASEMap that
leverages both probabilistic reasoning and semantic embedding for the
automatic mapping discovery. The system is implemented with a Web application
with friendly input, output, and visualization interfaces as well as different
settings for fully configurable PRASEMap. It also supports human-computer
interaction that utilizes human feedback for better results.

The Tencent team is actively developing new features, such as involvement of
ontology (Xiang et al., 2021), and meanwhile maintaining PRASEMap for higher
reliability, better performance, and industry deployment. For future work, we
will utilize PRASEMap as a primary component in the construction of large-
scale medical KGs. Moreover, the manual labeling interface in PRASEMap will be
integrated into the annotation platform to assist the human annotators and
improve their productivity in various tasks beyond mapping annotation, such as
KG correction and entity canonicalization (Chen et al., 2020). We also plan to
explore more learning-based strategies for the selection of undiscovered
mappings to reduce the annotation workload.

###### Acknowledgements.

This work was funded by National Key R&D Program of China (2018YFC2000702). We
also acknowledged the kind assistance from Shengzhang Lai and Chun Zhang in
developing and maintaining the Web application.

## References

  * (1)
  * Chen et al. (2020) Jiaoyan Chen, Xi Chen, Ian Horrocks, Erik B. Myklebust, and Ernesto Jimenez-Ruiz. 2020.  Correcting Knowledge Base Assertions. In _Proceedings of The Web Conference 2020_ (Taipei, Taiwan) _(WWW ’20)_. Association for Computing Machinery, New York, NY, USA, 1537–1547.  https://doi.org/10.1145/3366423.3380226
  * Chen et al. (2017) Muhao Chen, Yingtao Tian, Mohan Yang, and Carlo Zaniolo. 2017\.  Multilingual Knowledge Graph Embeddings for Cross-lingual Knowledge Alignment. In _Proceedings of the 26th International Joint Conference on Artificial Intelligence_. Melbourne, Australia, 1511–1517.
  * Jiménez-Ruiz and Grau (2011) Ernesto Jiménez-Ruiz and Bernardo Cuenca Grau. 2011.  LogMap: Logic-based and Scalable Ontology Matching. In _Proceedings of the 10th International Semantic Web Conference_. Bonn, Germany, 273–288.
  * Qi et al. (2021) Zhiyuan Qi, Ziheng Zhang, Jiaoyan Chen, Xi Chen, Yuejia Xiang, Ningyu Zhang, and Yefeng Zheng. 2021.  Unsupervised Knowledge Graph Alignment by Probabilistic Reasoning and Semantic Embedding. In _Proceedings of the 30th International Joint Conference on Artificial Intelligence_. Montreal, Canada.
  * Suchanek et al. (2012) Fabian M. Suchanek, Serge Abiteboul, and Pierre Senellart. 2012\.  PARIS: Probabilistic Alignment of Relations, Instances, and Schema. In _Proceedings of the 38th International Conference on Very Large Databases_. Istanbul, Turkey, 157–168.
  * Sun et al. (2020) Zequn Sun, Qingheng Zhang, Wei Hu, Chengming Wang, Muhao Chen, Farahnaz Akrami, and Chengkai Li. 2020.  A Benchmarking Study of Embedding-based Entity Alignment for Knowledge Graphs. In _Proceedings of the 46th International Conference on Very Large Databases_. Tokyo, Japen, 2326–2340.
  * Wang et al. (2018) Zhichun Wang, Qingsong Lv, Xiaohan Lan, and Yu Zhang. 2018\.  Cross-lingual Knowledge Graph Alignment via Graph Convolutional Networks. In _Proceedings of the Conference on Empirical Methods in Natural Language Processing_. Brussels, Belgium, 349–357.
  * Xiang et al. (2021) Yuejia Xiang, Ziheng Zhang, Jiaoyan Chen, Xi Chen, Zhenxi Lin, and Yefeng Zheng. 2021\.  OntoEA: Ontology-guided Entity Alignment via Joint Knowledge Graph Embedding. In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_. Association for Computational Linguistics, Online, 1117–1128.  https://doi.org/10.18653/v1/2021.findings-acl.96
  * Zhang et al. (2019) Qingheng Zhang, Zequn Sun, Wei Hu, Muhao Chen, Lingbing Guo, and Yuzhong Qu. 2019\.  Multi-view Knowledge Graph Embedding for Entity Alignment. In _Proceedings of the 28th International Joint Conference on Artificial Intelligence_. Macao, China, 5429–5435.
  * Zhang et al. (2020) Ziheng Zhang, Jiaoyan Chen, Xi Chen, Hualuo Liu, Yuejia Xiang, Bo Liu, and Yefeng Zheng. 2020.  An Industry Evaluation of Embedding-based Entity Alignment. In _Proceedings of the 28th International Conference on Computational Linguistics_. Barcelona, Spain, 179–189.