# On holonomy groupoid of vertex operator algebra bundles on foliations

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Zitna 25, Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

For a foliation $\mathcal{F}$ defined on a smooth complex manifold $M$ we
introduce the category of vertex operator algebra $V$ bundles with sections
provided by vectors of elements of the space of algebraically extended
$V$-module $W$-valued differentials. An intrinsic coordinate-independent
formulation for such bundles is given. Finally, we identify the cohomology of
the spaces of sections for a vertex operator algebra $V$ bundle with vertex
operator algebra cohomology of the holonomy groupoid $Hol(M,\mathcal{F})$.

AMS Classification: 53C12, 57R20, 17B69

###### Key words and phrases:

Holonomy groupoids, fiber bundles, vertex operator algebras, cohomology

##  1\. Conflict of Interest

The author states that:

1.) The paper does not contain any potential conflicts of interests.

##  2\. Data availability statement

The author confirms that:

1.) All data generated or analysed during this study are included in this
published article.

2.) Data sharing not applicable to this article as no datasets were generated
or analysed during the current study.

##  3\. Introduction: results of the paper

The theory of foliated manifolds incorporates a few main approaches [7, 8, 6,
11, 17, 18, 33]. The idea of studies of foliations cohomology, cohomology of
related bundles, and connections to to cohomology of foliated manifolds
themselves was proposed in [8]. Let $Vect(M)$ be the Lie algebra of vector
fields on $M$. In [8] is was proven that the Gelfand-Fuks cohomology
$H^{*}(Vect(M))$ [17] is isomorphic to the singular cohomology $H^{*}(E)$ of
the space $E$ of continuous cross sections of a certain fiber bundle
$\mathcal{E}$ over $M$. In [36, 39] they continued to use advanced topological
methods of [8] for cases of more general cosimplicial spaces of maps. In [41]
it was demonstrated that the ordinary theory of vector fields on a complex
manifold $M$ was not always the most effective way to study cohomology of $M$.
One has to $M$ consider more complicated algebraic and geometrical structures
to arrive at non-trivial cohomology theories associated to such structures.
One of possible candidate for such structures is given by vertex operator
algebras with formal parameters considered as local coordinates on complex
manifolds. Vertex operator algebras [5, 12, 30] represent generalizations of
ordinary Lie algebras and constitute an essential part of conformal field
theory [14, 16, 29].

The main motivation for studies of this paper is to develop a vertex operator
algebra approach to cohomology of auxiliary bundle defined on leaf spaces
$M/\mathcal{F}$ and transversal sections of foliations. The ground idea it to
use well-developed and powerful machinery and structural and computation
properties of vertex operator algebras to cohomology of non-commutative
objects attached to $M/\mathcal{F}$ to describe its leaves in terms of
corresponding invariants. By taking into account the standard methods of
defining canonical cosimplicial object [13, 41] as well as the
${\rm\check{C}}$ech-de Rham cohomology construction [11], we construct in this
paper in intrinsic coordinate-independent way canonical fiber bundles
associated to $\mathcal{F}$ on $M$. Our main purpose then is to demonstrate in
Lemma 2 that there exist a computational vertex operator algebra based way to
determine the cohomology of the holonomy groupoid $Hol(M,\mathcal{F})$ for a
foliation $\mathcal{F}$ on $M$. Developing [8], the cohomology of foliations
for a complex smooth manifold $M$ is expressed in terms of cohomology of a
canonical complex for an auxiliary bundle with intrinsic action of the
coboundary operator. The construction of bundles with canonical sections
defined over abstract discs on a smooth complex manifold $M$ is grounded on
the structure of admissible vertex operator algebra $V$ modules $W$.
Corresponding cohomology is considerd in terms of spaces of rational functions
provided by values of non-degenerate bilinear pairings on $W$ with specified
analytic behavior, and satisfying certain symmetry properties.

The content and main results of this article are as follows. In order to give
a local description of leaves of a foliation $\mathcal{F}$ of an
$n$-dimensional smooth manifold $M$ we use the classical approach of
transversal sections as well as algebraic and analytic properties of vertex
operator algebras. We chose two sets of points on $M$ and on a basis
$\mathcal{U}$ of transversal sections with corresponding domains of local
coordinates. Points on $M$ and $\mathcal{U}$ are then endowed with sets of a
$V$ vertex operator algebra elements. By taking algebraic completions
$\overline{W}$ of elements $W$ of the category ${\mathcal{W}}$ of $V$-modules,
we formulate the definition of spaces $\widehat{W}^{q}_{r}$, $q$, $r\geq 0$,
of special vectors ${\bf X}$ (exlicitely defined in (3.1)) of
$\overline{W}$-valued rational forms combined with sets of vertex operators.
For a set of formal complex variables $(z_{1},\ldots,z_{s})$ we introduce the
space $\widehat{W}_{(z_{1},\ldots,z_{s})}$ of algebraic completion of the
graded (with respect to Virasoro algebra $L_{W}(0)$-mode) space of
differential form-valued vectors

${\bf X}(v_{1},z_{1};\ldots;v_{s},z_{s})=\left[X\left(v_{1},z_{1}dz_{{\it
i}(1)};\ldots;v_{s},z_{s}\;dz_{{\it i}(s)}\right)\right],$ (3.1)

where ${\it i}(j)$, $j=1,\ldots,s$, are cycling permutations of $(1,\dots,s)$
starting with $j$, and we denote by $\left[.\right]$ the vector with elements
given by mappings $X$. In cases where it is clear which set of formal
variables is used we skip $(z_{1},\ldots,z_{s})$ from notations and denote
$\widehat{W}_{(z_{1},\ldots,z_{s})}$ as $\widehat{W}$. Assuming that there
exists a non-degenerate bilinear pairing $(.,.)$ on
$\widehat{W}_{(z_{1},\ldots,z_{s})}$, we denote by
$\widehat{W}^{*}_{(z_{1},\ldots,z_{s})}$ the dual to
$\widehat{W}_{(z_{1},\ldots,z_{s})}$ with respect to $(.,.)$. In case when
elements $(z_{1},\ldots,z_{s})$ are associated to certain local coordinates of
$l$ points $(p_{1},\ldots,p_{l})$ on $M$, we denote
$\widehat{W}_{(z_{1},\ldots,z_{s})}$ by $\widehat{W}_{(p_{1},\ldots,p_{l})}$,
and when $(z_{1},\ldots,z_{s})$ are substituted by local coordinates
$(t_{p_{1}},\ldots,t_{p_{s}})$ in vicinities of $(p_{1},\ldots,p_{l})$, we
replace $\widehat{W}_{(z_{1},\ldots,z_{s})}$ by by
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{s}})}$. For fixed
$\theta\in\widehat{W}^{*}_{(z_{1},\ldots,z_{s})}$, and varying elements of
$\widehat{W}_{(z_{1},\ldots,z_{s})}$ we consider a vector of matrix elements
of the form

$\Omega(X(v_{1},z_{1};\ldots;v_{s},z_{s}))=\left(\theta,X(v_{1},z_{1};\ldots;v_{s},z_{s})\right)\in\mathbb{C}((z)),$
(3.2)

where $X(v_{1},z_{1};\ldots;v_{s},z_{s})$ r depends implicitly on $v_{i}\in
V$, $1\leq i\leq s$. We may view the vector ${\bf
X}(v_{1},z_{1};\ldots;v_{s},z_{s})$ of the space $\widehat{W}$ as a section of
a fiber bundle over a collection of non-intersecting punctured discs
$(D^{\times}_{z_{1}},\ldots,D^{\times}_{z_{s}})=(Spec_{z_{j}}\;\mathbb{C}((z_{j}))$,
$1\leq j\leq s$, with an ${\rm
End}\left(\widehat{W}_{(z_{1},\ldots,z_{s})}\right)$-valued fiber
$X(v_{1},z_{1};\ldots;v_{s},z_{s})\in\widehat{W}_{(z_{1},\ldots,z_{s})}$. In
this paper we explain how to construct the vertex operator algebra $V$-bundle
mentioned above in the case when it carries an action of the group ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$ of local coordinates changes in vicinities of
$l$ points on $M$. This means that the action of the group ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}={\rm
Aut}_{1}\;{{\mathcal{O}}}^{(n)}\times\ldots\times{\rm
Aut}_{l}\;{{\mathcal{O}}}^{(n)}$ comes about by exponentiation of the action
of vertex operator algebra $\left({\it Der}_{j}\;{\mathcal{O}}^{(n)}\right)$,
$1\leq j\leq l$, via the action on $\widehat{W}_{(z_{1},\ldots,z_{s})}$. The
representation in term of formal series in $(t_{p_{1}},\ldots,t_{p_{l}})$
allows us to find the precise transformation formula for all elements of
$\widehat{W}_{(p_{1},\ldots,p_{l})}$ under the action of ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$. We then use this formula to give an intrinsic
geometric meaning to sections ${\bf X}(p_{1},\ldots,p_{s})$ of the fiber
bundle in coordinate-free formulation. Namely, we attach to each admissible
vertex operator algebra module $V$-module $W$ (i.e., satisfying certain
properties) a fiber bundle $\mathcal{W}_{M}$ on an arbitrary smooth manifold
$M$. In Section 6 we show that the bundle $\mathcal{W}_{M/\mathcal{F}}$
constructed is canonical, i.e., its sections do not depend on changes
$(t_{p_{1}},\ldots,t_{p_{l}})\mapsto(\widetilde{t}_{p_{1}},\ldots,\widetilde{t}_{p_{l}})$
of coordinates around points $(p_{1},\ldots,p_{l})$ on $M$. To keep elements
of $\widehat{W}^{q}_{r}$ coherent with respect to actions of the coboundary
operators $\Delta^{q}_{r}$ shifting indexes $q$ and $r$, we apply certain
analytic restriction on their characteristics provided by values of non-
degenerate bilinear forms of entries in $\widehat{W}^{q}_{r}$-vectors. The
spaces $\widehat{W}^{q}_{r}$ are defined on cosimplicial domains chosen on
transversal sections of $\mathcal{F}$. Then we formulate definition of the
category of vertex operator algebra bundles defined on $M/\mathcal{F}$. The
spaces $\widehat{W}^{q}_{r}$ associated to the category ${\mathcal{W}}$ of
admissible $V$-modules defined in Section 7. The spaces $C^{q}_{r}$ of vectors
of characteristics $[\Omega{X}]$ of entries of vectors ${\bf X}$ form [28] a
double chain-cochain complex $(C^{q}_{r},\delta^{q}_{r})$ where
$\delta^{q}_{r}{\bf X}=[\Omega(\Delta^{q}_{r}X)]$. The standard definition of
cohomology of this complex is taken as cohomology of $Hol(M,\mathcal{F})$. We
show that elements of the spaces $\widehat{W}$ are invariant torsors with
respect to the group of foliation preserving changes of transversal basis and
local coordinates. Though the construction of
$\mathcal{W}_{M/\mathcal{F}}$-bundle does not depend neither on the choice of
transversal basis nor on the choice of coordinates on $M$, it does depend on
the choice of vertex operator algebra elements as well as on a particular
element of the category $\mathcal{W}$ of admissible $V$-modules. The
construction involves torsors and twists of a vertex operator algebra modules
by the group of automorphisms of local coordinates transformations
(independent for each chosen point on leaves of a foliation $\mathcal{F}$) of
non-intersecting domains of a number of points on $M$.

The plan of the paper is the following. Section 4 contains information on
vertex operator algebras, their modules and properties. In Section 5 we
recall, following [16], the standard definitions of differentials and rational
functions considered on abstract and standard discs. In Section 6 we consider
the general notion of a vertex operator algebra bundle
$\mathcal{W}_{M/\mathcal{F}}$ defined on the leaf space of a foliated smooth
complex manifold $M$. In Section 7 the category of vertex operator algebra
bundles on leafs of $M/\mathcal{F}$ and transversal sections for a foliation
$\mathcal{F}$ on $M$ is considered.

There exists a bunch of ways to apply the construction of this paper of a
vertex operator algebra bundle on the space of leaves for a foliation defined
on a smooth manifold. The first obvious aim is to apply this study to
techniques of codimension one foliations discussions reflected in [24, 9, 2,
19]. The problem of finding non-vanishing cohomological invariants for the
space of a foliation leaves, and the problem of distinguishing kinds of
compact and non-compact leaves examples of foliations (such as the Reeb
foliation of the full torus), are among important questions in the theory of
foliations. The category ${\mathfrak{W}}_{M/\mathcal{F}}$ introduced in this
paper for vertex operator algebra bundles $\mathcal{W}_{M/\mathcal{F}}$
defined on leaf spaces of foliations will be used in establishing
corresponding characteristic classes theory. One would be interested in
finding possible relations of the cohomology theory of this paper with the
chiral de-Rham complex on a smooth manifold introduced in [34]. We are also
able to provide applications of vertex operator algebra $V$-bundles
$\mathcal{W}_{M/\mathcal{F}}$ for foliations of complex manifolds [7, 13, 41,
14, 40] in deformation theory [35, 9, 27, 23, 31], and algebraic topology in
general. Constructions introduced in this paper will be useful for purposes of
cosimplitial cohomology [41] of manifolds. Vertex operator algebra bundles on
complex manifolds can be used in construction of various generalizations of
the Bott–Segal theorem [8]. Finally, we would like to mention possible
connections to the Losik’s theory of foliated manifolds. In [32] Losik has
introduced a smooth structure on the leaf space $M/\mathcal{F}$ of a foliation
$\mathcal{F}$ of codimension $p$ on a smooth manifold $M$ that allows to apply
to $M/\mathcal{F}$ the same techniques as to smooth manifolds. Characteristic
classes for foliations as elements of the cohomology of certain bundles over
the leaf space $M/\mathcal{F}$ were defined. We hope to develop this approach
by applying vertex operator algebra techniques and constructions provided in
this paper.

##  4\. Vertex operator algebras and their modules

In this Section we recall definitions and basic properties of vertex operator
algebras and their generalized modules [5, 16, 12, 15, 16, 29, 30]. A vertex
operator algebra $(V,Y_{V},\mathbf{1}_{V},c)$, of Virasoro algebra central
charge $c$, consists of a $\mathbb{Z}$-graded complex vector space
$V=\bigoplus_{s\in\mathbb{Z}}V_{(s)}$, with finite-dimensional grading
subspaces $V_{(s)}$ $\dim V_{(s)}<\infty$ for each $s\in\mathbb{Z}$, equipped
with a linear map $Y_{V}:V\rightarrow{\rm End}(V)[[z,z^{-1}]]$, for a formal
complex parameter $z$ and a distinguished vector $\mathbf{1}_{V}\in V$. The
vertex operator for $v\in V$ is given by
$Y_{V}(v,z)=\sum_{s\in\mathbb{Z}}v(s)z^{-s-1}$, with components
$(Y_{V}(v))_{s}=v(s)\in{\rm End\;}(V)$, with the property
$Y_{V}(v,z)\mathbf{1}_{V}=v+O(z)$. In this paper we apply the following
restrictions on the grading of a vertex operator algebra $V$ or its module
$W$. A vertex operator algebra $V$-module $W$ is a vector space $W$ equipped
with a vertex operator map $Y_{W}:V\otimes W\to W[[z,z^{-1}]]$, and $v\otimes
w\mapsto Y_{W}(v,z)w=\sum_{s\in\mathbb{Z}}(Y_{W})_{s}(v,w)z^{-s-1}$. $W$ is
also subject of actions of and linear operators $L_{W}(0)$ and $L_{W}(-1)$
($0$ and $-1$ Virasoro modes) satisfying the following conditions. One assumes
that $V_{(s)}=0$ for $s\ll 0$. The vector space $W$ is $\mathbb{C}$-graded,
that is, $W=\bigoplus_{\alpha\in\mathbb{C}}W_{(\alpha)}$, such that
$W_{(\alpha)}=0$, when the real part of $\alpha$ is sufficiently negative. The
result of a vertex operator $Y_{V,W}(u,z)(v,w)$, $u$, $v\in V$, $w\in W$,
contains only finitely many negative power terms, that is
$Y_{V,W}(u,z)(v,w)\in(V,W)((z))$, i.e., belongs to the space of formal Laurent
series in $z$ with coefficients in $(V,W)$. Here $(V,W)$ and subscript V,W
mean corresponding expression either for vertex operator algebra $V$ elements
or its module $W$. Let ${\rm Id}_{V,W}$ be the identity operator on $(V,W)$.
Then $Y_{V,W}(\mathbf{1}_{V},z)={\rm Id}_{V,W}$. For $v\in V$,
$Y_{V}(v,z)\mathbf{1}_{V}\in V[[z]]$ and $\lim_{z\to
0}Y_{V}(v,z)\mathbf{1}_{V}=v$. We assume that for $W$ there exist non-
degenerate bilinear pairing $(.,.)$, $W^{\prime}\otimes W\to\mathbb{C}$, where
$W^{\prime}$ denotes the dual $V$-module to $W$. For $s\in\mathbb{Z}_{+}$,
denote by $F_{s}\mathbb{C}$ the configuration space of $s$ ordered points in
$\mathbb{C}$,
$F_{s}\mathbb{C}=\\{(z_{1},\ldots,z_{s})\in\mathbb{C}^{s},z_{i}\neq
z_{j},i\neq j\\}$. For a space $A$ and arbitrary $\theta\in A^{*}$, for
$A^{*}$ dual to $A$, a meromorphic function of several complex variables
$(z_{1},\dots,z_{s})$ defined by a map $f:F_{s}\mathbb{C}\rightarrow A$,
$(z_{1},\ldots,z_{s})\mapsto R(f(z_{1},\ldots,z_{s}))$, is called an
$A$-valued rational function if its characteristic
$\Omega(f(z_{1},\ldots,z_{s}))=(\theta,f(z_{1},\ldots,z_{s}))$, extends to a
rational function denoted $R(f(z_{1},\ldots,z_{s}))$ in $(z_{1},\ldots,z_{r})$
on a larger domain and admits poles at $z_{i}=z_{j}$, $i\neq j$, only. In
particular, in this paper we consider the cases $A=\overline{W}$,
$\widehat{W}$. We assume that for $u$, $u_{1}$, $u_{2}\in V$, the
characteristics
$\Omega\left(Y_{V,W}(v_{1},z_{1})Y_{V,W}(v_{2},z_{2})v\right)$,
$\Omega\left(Y_{V,W}(v_{2},z_{2})Y_{V,W}(v_{1},z_{1})(v,w)\right)$, and
$\Omega\left(Y_{V,W}(Y_{V}(v_{1},z_{1}-z_{2})v_{2},z_{2})(v,w)\right)$,
converge absolutely in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, correspondingly to a common rational function in
$z_{1}$, $z_{2}$. Poles of these characteristics are only allowed at
$z_{1}=0=z_{2}$, and $z_{1}=z_{2}$. The role of a grading operator for $V$ is
played by the zero Virasoro mode with $L_{V}(0)v=rv$ for $v\in V_{(r)}$. Then
for $v\in V$ one has

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

For $w\in W_{(\alpha)}$, there exists $n_{0}\in\mathbb{Z}_{+}$ such that
$(L_{W}(0)-\alpha)^{n_{0}}w=0$. For $v\in V$ the operator $L_{V}(-1)$ is given
by

$L_{V}(-1)v=\mbox{\rm Res}_{z}z^{-2}Y_{V}(v,z)\mathbf{1}_{V}=Y_{(-2)}(v){\bf
1}_{V},$
$\frac{d}{dz}Y_{V}(v,z)=Y_{V,W}(L_{V}(-1)v,z)=[L_{V,W}(-1),Y_{V}(v,z)].$

We denote $\textup{wt}(v)=k$ the weight for $v\in V_{(k)}$. For $v\in V$, the
translation property for vertex operators can be written as

$Y_{W}(v,z)=e^{-z^{\prime}L_{W}(-1)}Y_{W}(v,z+z^{\prime})e^{z^{\prime}L_{W}(-1)},$

$z^{\prime}\in\mathbb{C}$. For $v\in V$, it follows

$\frac{d}{dz}Y(v,z)=Y(L_{V}(-1)v,z).$

For $a\in\mathbb{C}$, the conjugation property with respect to the grading
operator $L_{W}(0)$ is given by

$a^{L_{W}(0)}\;Y_{W}(v,z)\;a^{-L_{W}(0)}=Y_{W}\left(a^{L_{W}(0)}v,az\right).$

A vertex operator algebra $V$ satisfying conditions above is called conformal
of central charge $c\in\mathbb{C}$, if there exists a non-zero conformal
vector $\omega\in V_{2}$ such that the Fourier coefficients $L_{V}(r)$ of the
corresponding vertex operator
$Y(\omega,z)=\sum_{s\in\mathbb{Z}}L_{V}(k)z^{-s-2}$, is determined by Virasoro
modes $L_{V}(r):V\to V$ subject to the commutation relations

$[L_{V}(s),L_{V}(r)]=(s-r)L_{V}(s+r)+\frac{c}{12}(s^{3}-s)\delta_{s,-r}\;{\rm
Id}_{V}.$

In [16], $v\in V$, the following formula was derived

$\left[L_{W}(r),Y_{W}(v,z)\right]=\sum_{r\geq-1}\frac{1}{(r+1)!}\partial^{r+1}_{z}z^{r+1}\;Y_{W}(L_{V}(r)v,z).$

For a vector field
$\beta(z)\partial_{z}=\sum_{r\geq-1}\beta_{r}z^{r+1}\partial_{z}$,
$\beta(z)\partial_{z}\in{\rm Der}{\mathcal{O}}^{(1)}$, which belongs to the
local Lie algebra of the group ${\rm Aut}\;{\mathcal{O}}^{(n)}$, let us
introduce the operator ${\overline{\beta}}=-\sum_{r\geq-1}\beta_{r}L_{W}(r)$.
In [16] they prove the following formula:

$\left[{\overline{\beta}},Y_{W}(v,z)\right]=\sum_{r\geq-1}\frac{1}{(r+1)!}\left(\partial^{r+1}_{z}\beta(z)\right)\;Y_{W}(L_{V}(r)v,z).$

A vertex operator algebra $V$-module $W$ is called quasi-conformal if it
carries an action of ${\rm Der}\;{\mathcal{O}}^{(n)}$ on an $n$-dimensional
smooth manifold $M$ such that commutation formula above holds for any $v\in
V$, and $z=z_{j}$, $1\leq j\leq n$, the element $L_{W}(-1)=-\partial_{z}$ acts
as the translation operator $L_{W}(0)=-z\partial_{z}$, acts semi-simply with
integral eigenvalues, and the Lie subalgebra ${\rm
Der}_{+}\;{\mathcal{O}}^{(n)}$ acts locally nilpotently on $M$. A vector $w\in
W$ of a quasi-conformal vertex operator algebra $V$ is called primary of
conformal dimension $\nu\in\mathbb{Z}_{+}$ if $L_{W}(k)w=0$, $k>0$,
$L_{W}(0)\cdot w=\nu w$. In addition to that, we assume that $V$-module $W$
admits an action of $Der_{j}\;{\mathcal{O}}^{(n)}$. The element
$\left(-\partial_{t_{p}}\right)$ plays a role of the translation operator on
$W_{t_{p}}$ with integral eigenvalues, and the Lie subalgebra
$\left(Der_{+}\right)_{j}\;{\mathcal{O}}^{(n)}$ acts locally nilpotently. The
$\mathbb{C}$-grading operator is provided by the mode $L_{W}(0)$, i.e.,
$L_{W}(0)=\left(-t_{p}\;\partial_{t_{p}}\right)$. Finally, let us assume that
the action of the Lie algebra $Der_{j}\;{\mathcal{O}}^{(n)}$ on
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ can be exponentiated to an action
of the group ${\rm Aut}_{j}\;{\mathcal{O}}^{(n)}$.

Denote by ${\mathfrak{\mathcal{W}}}$ the category of $V$-vertex operator
algebra admissible modules $W$ that satisfy these properties in addition to
all related properties of Section 4. Let $\overline{W}$ denote the algebraic
completion of $W$,
$\overline{W}=\prod_{r\in\mathbb{C}}W_{(r)}=(W^{\prime})^{*}$. We assume that
the space $\overline{W}$ is endowed with a grading
$\overline{W}=\bigcup_{s\in\mathbb{Z},s>s_{0}}\overline{W}_{(s)}$ bounded from
below with respect to the grading operator $L_{W}(0)$.

##  5\. Differentials and rational functions on abstract discs

In this Section we partially follow [16] and describe the setup needed for
formulation of further results. Let $p$ be a point on $M$, and $t_{p}$ be a
local coordinate in a vicinity of $p$. We replace the field of Laurent series
$\mathbb{C}((t_{p}))$ by any complete topological algebra non-canonically
isomorphic to $\mathbb{C}((t_{p}))$.

###  5.1. Abstract discs

To introduce abstract discs on $M/\mathcal{F}$ it is possible to consider the
scheme underlying the $\mathbb{C}$-algebra $\mathbb{C}[[t_{p}]]$.
$\mathbb{C}[[t_{p}]]$ is the ring of complex-valued functions on the affine
scheme $D_{t_{p}}={\rm Spec}\;\mathbb{C}[[t_{p}]]$ which we call the standard
disc $D_{t_{p}}$. As a topological space, $D_{p}$ can be described by the
origin corresponding to the maximal ideal $t_{p}\;\mathbb{C}[[t_{p}]]$ and the
generic point. A morphism from $D$ to an affine scheme $Z={\rm
Spec}\;\mathcal{R}$, where $\mathcal{R}$ is a $\mathbb{C}$-algebra, is a
homomorphism of algebras $\mathcal{R}\to\mathbb{C}[[t_{p}]]$. Such a
homomorphism can be constructed by realizing $\mathbb{C}[[t_{p}]]$ as a
completion of $\mathcal{R}$. Geometrically, this is an identification of the
disc $D_{p}$ with the formal neighborhood of a point on $M$. An abstract disc
is an affine scheme ${\rm Spec}\;\mathcal{R}$, where $\mathcal{R}$ is a
$\mathbb{C}$-algebra isomorphic to $\mathbb{C}[[t_{p}]]$. On the abstract
disc, the maximal ideal $t_{p}\mathbb{C}[[t_{p}]]$ has a preferred generator
$t_{p}$. In contrast to that, on an abstract disc there is no preferred
generator in the maximal ideal of ${\mathcal{R}}$, and there is no preferred
coordinate. Denote by ${\mathcal{O}}_{p}$ the completion of the local ring of
$M$. Then $\mathcal{O}_{p}$ is non-canonically isomorphic to
${\mathcal{O}}=\mathbb{C}[[t_{p}]]$. To specify such an isomorphism, or
equivalently, an isomorphism between $D_{p}={\rm Spec}\;{\mathcal{O}}_{p}$,
and $D_{t_{p}}={\rm Spec}\;\mathbb{C}[[t_{p}]]$, we need to choose a formal
coordinate $t_{p}$ at $p\in M$, i.e., a topological generator of the maximal
ideal $\mathfrak{m}_{p}$ of ${\mathcal{O}}_{p}$. In general there is no
preferred formal coordinate at $p\in M$, and $D_{p}$ is an abstract disc.

###  5.2. Rational functions attached to discs

To construct a $\widehat{W}$-valued vertex operator algebra bundle on
$M/\mathcal{F}$ we would like to attach elements of $\widehat{W}$ to the both
standard $D_{t_{p}}={\rm Spec}\;\mathbb{C}[[t_{p}]]$ and abstract discs
$D_{p}$, where $p$ is a point on $M/\mathcal{F}$. For a cohomology theory
purposes we attach also characteristics of $\widehat{W}$-elements represented
in terms of rational functions on discs. Let $\mathcal{K}_{x}$ be the field of
fractions of the ring of integers $\mathbb{Z}$ is the rational field
$\mathbb{Q}$. We denote also by ${\mathcal{K}}[(t_{p_{1}},\ldots,t_{p_{s}})]$
the field of fractions of the polynomial ring over a field ${\mathcal{K}}$ as
the field of rational functions
${\mathcal{K}}((t_{p_{1}},\ldots,t_{p_{s}}))=\left\\{(R_{1}(t_{p_{1}},\ldots,t_{p_{s}}))/(R_{2}(t_{p_{1}},\ldots,t_{p_{s}})):R_{1},R_{2}\in{\mathcal{K}}[(t_{p_{1}},\ldots,t_{p_{s}})]\right\\}$.
For a coordinate $t_{p}$ on $D_{p}$, there exist isomorphisms
$\mathcal{O}_{p}=\mathbb{C}[[t_{p}]]$ and
$\mathcal{K}_{p}=\mathbb{C}((t_{p}))$. We denote by $D_{p}$ and
$D^{\times}_{p}$ at $p$ the disc and punctured disc defined as ${\rm{\rm
Spec}\;}{\mathcal{O}}_{p}^{(n)}$ and ${\rm{\rm Spec}\;}\;{\mathcal{K}}_{p}$)
correspondingly.

###  5.3. Rational power differentials

In this Subsection we recall basic definitions related to differentials [16,
37]. Let $k$ be a rational number. A $k$-differential defined on a manifold
$M$ is a section of the $k$-th tensor power of the canonical line bundle
$\omega$. Choosing a local coordinate $t_{p}$ arround a point $p\in M$ we may
trivialize $\omega^{\otimes k}$ by the non-vanishing section
$(dt_{p})^{\otimes k}$. Any section of $\omega^{\otimes k}$ may then be
written as $f(t_{p})(dt_{p})^{\otimes k}$. For another coordinate
$\widetilde{t}_{p}=\rho(t_{p})$, the same section will be written as
$g(\widetilde{t}_{p})(d\widetilde{t}_{p})^{\otimes k}$, where
$f(t_{p})=g(\rho(t_{p}))(\rho^{\prime}(t_{p}))^{\otimes k}$. Now let us
suppose that we have a section of $\omega^{\otimes k}$ whose representation by
a function does not depend on the choice of local coordinate, i.e.,
$g(\widetilde{t}_{p})=f(\widetilde{t}_{p})$, and
$f(t_{p})=f(\rho(t_{p}))(\rho^{\prime}(t_{p}))^{\otimes k}$ for any change of
variable $\rho(t_{p})$. We call $f(t_{p})(dt_{p})^{\otimes k}$ a canonical
$k$-differential. Let us denote by $\omega_{p}$ the space of differentials on
$D^{\times}_{p}$. Given a linear map $\rho:{\mathcal{K}}_{p}\to{\rm
End}\left(\widehat{W}_{t_{p}}\right)$, such that for any
$x\in\widehat{W}_{t_{p}}$ and large enough $l$, we have
$\rho(\mathfrak{m}_{p})^{l}\cdot x=0$, where $\mathfrak{m}_{p}$ is the maximal
ideal of ${\mathcal{O}}_{p}$ at $p$. Then, according to [16], the vertex
operator
$Y(\rho,t_{p})=\sum\limits_{s\in\mathbb{Z}}\rho(t^{s}_{p})\;t_{p}^{-s-1}\;dt_{p}$,
is a canonical ${\rm End}\left(\widehat{W}_{t_{p}}\right)$-valued differential
on $D^{\times}_{t_{p}}$, i.e., it is independent of the choice of coordinate
$t_{p}$.

##  6\. The vertex operator algebra bundle on $M/\mathcal{F}$

In this Section we provide the construction of $\widehat{W}$-valued vector
bundle $\mathcal{W}_{M/\mathcal{F}}$ on $M/\mathcal{F}$.

###  6.1. Torsors and twists under groups of automorphisms

For an admissible $V$-module $W$ we have the filtration $W_{t_{p_{j}},\leq
m}=\bigoplus\limits_{i\geq{\rm Re}(\kappa)}^{m}W_{t_{p_{j}},i}$, of
$W_{t_{p_{j}}}$ by finite-dimensional ${\rm
Aut}_{p_{j}}\;{\mathcal{O}}^{(n)}$-submodules, $j\geq 1$. Suppose $W$ is an
admissible vertex operator algebra $V$-module as in the definition given in
Section 4. We now explain how to collect elements of the space $\widehat{W}$
into an intrinsic object on a collection of abstract discs on $M/\mathcal{F}$.
We consider a configuration of $l$-points $(p_{1},\ldots,p_{l})$ on
$M/\mathcal{F}$ lying in non-intersecting local discs, and we assume that at
each point of $(p_{1},\ldots,p_{l})$ a coordinate changes independently of
changing of coordinates on other discs. Therefore, the general element of the
group of independent automorphisms of coordinates of $l$ points on
$M/\mathcal{F}$ ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}_{p_{1},\ldots,p_{l}}$ has
the form
$(t_{p_{1}},\ldots,t_{p_{l}})\mapsto(\rho_{1},\ldots,\rho_{l})(t_{p_{1}},\ldots,t_{p_{l}})$.

Let us remind the definition of a torsor [16]. Let $\mathfrak{G}$ be a group,
and $S$ a non-empty set. Then $S$ is called a $\mathfrak{G}$-torsor if it is
equipped with a simply transitive right action of $\mathfrak{G}$. For $s_{1}$,
$s_{2}\in S$, there exists a unique $\mu\in\mathfrak{G}$ such that
$s_{1}\cdot\mu=s_{2}$, where the right action is given by
$s_{1}\cdot(\mu\mu^{\prime})=(s_{1}\cdot\mu)\cdot\mu^{\prime}$. The choice of
any $s_{1}\in S$ allows us to identify $S$ with $\mathfrak{G}$ by sending
$s_{1}\cdot\mu$ to $\mu$. Applying the definition of a group twist [16] to the
group ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$ and its module $\widehat{W}$ we
obtain following the definition. Given a ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-module $\widehat{W}_{(z_{1},\ldots,z_{l})}$ and
a ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor $\mathcal{X}$, one defines the
$\mathcal{X}$-twist of $\widehat{W}_{(z_{1},\ldots,z_{l})}$ as the set

$\mathcal{V}_{\mathcal{X}}=\widehat{W}_{(z_{1},\ldots,z_{l})}\;{{}_{\times}\atop{}^{{\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}}}\mathcal{X}=\widehat{W}_{(z_{1},\ldots,z_{l})}\times\mathcal{X}/\left\\{(w,a\cdot\xi)\sim(aw,\xi)\right\\}.$

for $\xi\in\mathcal{X}$, $a\in{\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$, and
$w\in\widehat{W}_{(z_{1},\ldots,z_{l})}$. Given $\xi\in\mathcal{X}$, we may
identify $\widehat{W}_{(z_{1},\ldots,z_{l})}$ with
$\mathcal{V}_{\mathcal{X}}$, by $w\mapsto(\xi,w)$. This identification depends
on the choice of $\xi$. Since ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$ acts on
$\widehat{W}_{(z_{1},\ldots,z_{l})}$ by linear operators, the vector space
structure induced by the above identification does not depend on the choice of
$\xi$, and $\mathcal{V}_{\mathcal{X}}$ is canonically a vector space. If one
thinks of ${\mathcal{X}}$ as a principal ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-bundle over a set of points, then
$\mathcal{V}_{\mathcal{X}}$ is simply the associated vector bundle
corresponding to $\widehat{W}_{(z_{1},\ldots,z_{l})}$. Any structure on
$\widehat{W}_{(z_{1},\ldots,z_{l})}$ (e.g., a bilinear pairing or
multiplicative structure) that is preserved by ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$ will be inherited by
$\mathcal{V}_{\mathcal{X}}$.

Now we wish to attach to any disc a certain twist
$\mathcal{V}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ of
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$, so that
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ is attached to the standard
discs, and for any set of coordinates $(t_{p_{1}},\ldots,t_{p_{l}})$ on
$(D_{p_{1}},\ldots,D_{p_{l}})$ we have an isomorphism

$i_{(t_{p_{1}},p_{1};\ldots;t_{p_{l}},p_{l})}:\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{1}})}\;\widetilde{\to}\;\mathcal{V}_{(t_{p_{1}},\ldots,t_{p_{l}})}.$
(6.1)

We then associate sections of some bundles on
$(D^{\times}_{t_{p_{1}}},\ldots,D^{\times}_{t_{p_{l}}})$ to elements of
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$. The system of isomorphisms
$i_{(t_{p_{1}},p_{1};\ldots;t_{p_{l}},p_{l})}$ should satisfy certain
compatibility condition. Namely, if $(t_{p_{1}},\ldots,t_{p_{l}})$ and
$(\widetilde{t}_{p_{1}},\ldots,\widetilde{t}_{p_{l}})$ are two sets of
coordinates on $(D_{p_{1}}$, $\ldots$, $D_{p_{l}})$ such that
$(\widetilde{t}_{p_{1}};\ldots;\widetilde{t}_{p_{l}})=(\rho_{1},\ldots,\rho_{l})(t_{p_{1}},\ldots,t_{p_{l}})$,
then we obtain an automorphism
$(i^{-1}_{(\widetilde{t}_{p_{1}},p_{1},\ldots,\widetilde{t}_{p_{l}},p_{l})}\circ
i_{(t_{p_{1}},p_{1};\ldots;t_{p_{1}},p_{1})}$ of
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$. The condition is that the
assignment $(\rho_{1},\ldots,\rho_{l})(z_{1},\ldots,z_{l})\mapsto
i^{-1}_{(\widetilde{t}_{p_{1}},p_{1};\ldots;\widetilde{t}_{p_{l}},p_{l})}\circ
i_{(t_{p_{1}},p_{1};\ldots,t_{p_{l}},p_{l})}$, defines a representation of the
group ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$ of independent changes of
coordinates on $\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$. If this condition
is satisfied, then $\mathcal{V}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ is canonically
identified with the twist of $\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ by
the ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor of formal coordinates at
$(p_{1},\ldots,p_{l})$.

In the next Subsection we will show that given the space $\widehat{W}$ one can
attach to it a vector bundle $\mathcal{W}_{M/\mathcal{F}}$ on the space of
leaves $M/\mathcal{F}$ for a foliation $\mathcal{F}$ defined on any smooth
complex manifold $M$. I.e., the elements of $\widehat{W}$ give rise to a
collection of coordinate-independent sections $X(p_{1},\ldots,p_{l})$ of the
bundle $\mathcal{W}^{*}_{M/\mathcal{F}}$ in the neighborhoods of a collection
of points $(p_{1},\ldots,p_{l})\in M/\mathcal{F}$. The construction is based
on the principal bundle for the group ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$,
which naturally exists on an arbitrary smooth curve and on any collection
$(D_{p_{1}},\ldots,D_{p_{l}})$ of non-intersecting discs. We denote by ${\it
Aut}_{(p_{1},\ldots,p_{l})}$ the set of all coordinates
$(t_{p_{1}},\ldots,t_{p_{l}})$ on discs $(D_{p_{1}},\ldots,D_{p_{l}})$,
centered at points $(p_{1},\ldots,p_{l})$. It comes equipped with a natural
right action of the group of automorphisms ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$. If $t_{p_{i}}\in{\it Aut}_{p_{i}}$, $1\leq
i\leq l$, and $\rho(z_{i})\in{\rm Aut}_{i}\;{\mathcal{O}}^{(n)}$, then
$\rho_{i}(t_{p_{i}})\in{\it Aut}_{p_{i}}$. Furthermore, as it was shown in
[16] that $(\rho_{i}*\mu_{i})(t_{p_{i}})=\mu_{i}(\rho_{i}(t_{p_{i}}))$, for
$1\leq i\leq l$, it defines a right simply transitive action of ${\rm
Aut}_{i}\;{\mathcal{O}}^{(n)}$ on ${\it Aut}_{p_{i}}$. Thus we see that the
group ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$ acts naturally on ${\it
Aut}_{(p_{1},\ldots,p_{l})}$, and is a ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor. Thus, we can define the following
twist. We can introduce the ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-twist of
$\overline{W}_{(p_{1},\ldots,p_{l})}$
$\mathcal{V}_{(p_{1},\ldots,p_{l})}=\overline{W}_{(p_{1},\ldots,p_{l})}\;{{}_{\times}\atop{}^{{\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}}}\;{\it Aut}_{(p_{1},\ldots,p_{l})}$. The
original definition was given in [3, 42]. For each set of formal coordinates
$(t_{p_{1}},\ldots,t_{p_{l}})$ at points $(p_{1},\ldots,p_{l})$,
$(w_{1},\ldots,w_{l})\in\widehat{W}_{(p_{1},\ldots,p_{l})}$, any element of
the twist $\mathcal{V}_{(p_{1},\ldots,p_{s})}$ may be written uniquely as a
pair $\left((w_{1},\ldots,w_{l})\right.$,
$\left.(t_{p_{1}},\ldots,t_{p_{l}})\right)$.

###  6.2. Definition of $\mathcal{W}_{M/\mathcal{F}}$-bundle of
$\widehat{W}$-elements

Now let us formulate the definition of fiber bundle associated through vectors
of elements ${\bf X}\in\widehat{W}$ defined on any set of standard discs
$U=(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})$ around points $(p_{1},\ldots,p_{l})$
on $M/\mathcal{F}$ with local coordinates $(t_{p_{1}},\ldots,t_{p_{l}})$. We
construct an analog of a principal ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-bundle
for $M/\mathcal{F}$. The fiber space is provided by vectors ${\bf X}$ of
elements $X(t_{p_{1}},\ldots,t_{p_{l}})$, given by a fiber bundle
$\mathcal{W}_{M/\mathcal{F}}|_{(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})}$ defined
by trivializations $i_{(t_{p_{1}},\ldots,t_{p_{l}})}:{\bf
X}(p_{1},\ldots,p_{l})=\left[X(p_{1},\ldots,p_{l})\right]\to(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})$,
with a continuous ${\bf X}(p_{1},\ldots,p_{l})$-preserving right action ${\bf
X}(p_{1},\ldots,p_{l})\times{\rm Aut}_{l}\;{\mathcal{O}}^{(n)}\to{\bf
X}(p_{1},\ldots,p_{l})$. Namely, for two sections $\zeta$, $\zeta.a$ of
$\mathcal{W}_{M/\mathcal{F}}|_{(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})}$, the map
$a\mapsto\zeta.a$ is a homeomorphism for all $a\in{\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$. Then, according to the definition of a torsor,
the fiber of such bundle at points $(p_{1},\ldots,p_{l})$ is the ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor ${\it Aut}_{(p_{1},\ldots,p_{l})}$.

Denote by ${\it Aut}_{l}$ the set of $l$-tuples of local coordinates ${\it
Aut}_{(p_{1},\ldots,p_{l})}$ all over leaves of $\mathcal{F}$. Given a finite-
dimensional ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-module
$\widehat{W}_{i,(p_{1},\ldots,p_{l})}$, let
$\mathcal{W}_{M/\mathcal{F}}|_{(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})}=\widehat{W}_{i,(t_{p_{1}},\ldots,t_{p_{l}})}{\times\atop\;{\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}}\;{\it Aut}_{l}$, be the fiber bundle associated
to $\widehat{W}_{i,(t_{p_{1}},\ldots,t_{p_{l}})}$ and ${\it Aut}_{l}$. Then,
$\mathcal{W}_{M/\mathcal{F}}|_{(D_{t_{p_{1}}},\ldots,D_{t_{p_{1}}})}$ is a
finite-rank bundle over
$M/\mathcal{F}|_{(D_{t_{p_{1}}},\ldots,D_{t_{p_{1}}})}$ whose fiber at a
collection of points $(p_{1},\ldots,p_{l})\in M/\mathcal{F}$ is given by the
vector $[X(p_{1},\ldots,p_{l})]$. In a vicinity of every point of
$(p_{1},\ldots,p_{l})$ on $M/\mathcal{F}$ we can choose discs
$(D_{p_{1}},\ldots,D_{p_{l}})$ such that the bundle
$\mathcal{W}_{M/\mathcal{F}}$ over $(D_{p_{1}},\ldots,D_{p_{l}})$ is
$(D_{p_{1}},\ldots,D_{p_{l}})\times{\bf X}(p_{1},\ldots,p_{l})$, where ${\bf
X}(p_{1},\ldots,p_{l})$ is a section of $\mathcal{W}_{M/\mathcal{F}}$. The
fiber bundle $\mathcal{W}_{M/\mathcal{F}}$ with fiber
$\left[X(p_{1},\ldots,p_{l})\right]$ is a map
$\mathcal{W}_{M/\mathcal{F}}:\widehat{W}\rightarrow M/\mathcal{F}$ where
$M/\mathcal{F}$ is $\mathcal{W}_{M/\mathcal{F}}$-bundle base space. For every
set of points $(p_{1},\ldots,p_{l})\in M/\mathcal{F}$ with local discs
$(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})$ $i_{(t_{p_{1}},\ldots,t_{p_{l}})}^{-1}$
is homeomorphic to $(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})\times\widehat{W}$.
Namely, we have for
$\left[X(p_{1},\ldots,p_{l})\right]:i_{(t_{p_{1}},\ldots,t_{p_{l}})}^{-1}\rightarrow(D_{t_{p_{1}}},\ldots,D_{t_{p_{1}}})\times\widehat{W}_{t_{p_{1}},\ldots,t_{p_{l}}}$,
that $\mathcal{P}$ $\circ$ $\left[X(p_{1},\ldots,p_{l})\right]$ =
$i_{(t_{p_{1}},\ldots,t_{p_{l}})}$ $|_{i^{-1}_{(t_{p_{1}},\ldots,t_{p_{l}})}}$
$(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})$, where ${\mathcal{P}}$ is the
projection map on $(D_{t_{p_{1}}},\ldots,D_{t_{p_{l}}})$. For an ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-module
$\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ which has a filtration by finite-
dimensional submodules $\widehat{W}_{s,(t_{p_{1}},\ldots,t_{p_{l}})}$, $s\geq
0$, we consider the directed inductive limit $\mathcal{W}_{M/\mathcal{F}}$ of
a system of finite rank bundles $\mathcal{W}_{s,M/\mathcal{F}}$ on
$M/\mathcal{F}$ defined by embeddings
$\mathcal{W}_{s,M/\mathcal{F}}\rightarrow\mathcal{W}_{s^{\prime},M/\mathcal{F}}$,
for $s\leq s^{\prime}$, i.e., $\mathcal{W}_{M/\mathcal{F}}$ it as a fiber
bundle of infinite rank over $M/\mathcal{F}$.

###  6.3. Explicit construction of canonical intrinsic setup for
$\mathcal{W}_{M}$

Let $W$ be a quasi-conformal vertex operator algebra $V$-module $W$ defined in
Section 4. In order to be able to introduce a section $X(p_{1},\ldots,p_{l})$
of the vertex operator algebra bundle $\mathcal{W}_{M/\mathcal{F}}$ defined on
abstract discs $(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})$ in the
coordinate independent description, we associate $X(p_{1},\ldots,p_{l})$ to
coordinate independent vector ${\bf X}(v_{1},z_{1};\ldots;v_{l},z_{l})$.

Now let us give the following definition. For each set of points
$(p_{1},\ldots,p_{l})$ and elements
$(w_{1},\ldots,w_{l})\in\overline{W}_{z_{1},\ldots,z_{l}}^{\otimes l}$, we
define an intrinsic $\widehat{W}$-valued meromorphic section
$X(p_{1},\ldots,p_{l})$ on the punctured discs
$(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})$ by an operation
$(w_{1},\ldots,w_{l})$, $(p_{1},\ldots,p_{l})\mapsto X(p_{1},\ldots,p_{l})$,
assigning to a vector $X(p_{1},\ldots,p_{l})$ of
$\mathcal{W}_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$ an element of
${\bm{\mathcal{K}}}_{(p_{1},\ldots,p_{l})}$ (i.e., rational
$\widehat{W}$-valued functions on
$(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})$), defined by the
$\mathcal{W}^{*}_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$-fiber ${\bf
X}_{i_{(t_{p_{1}},\ldots,t_{p_{l}})}}\in\widehat{W}_{(p_{1},\ldots,p_{l})}$.
Consider the operator ${\rm
R}(\rho_{1},\ldots,\rho_{s})=\left[\widehat{\partial}_{J}\rho_{i(I)}\right]=\left[\widehat{\partial}_{J}\rho_{i_{1}(I)},\widehat{\partial}_{J}\rho_{i_{2}(I)},\cdots,\\\
\widehat{\partial}_{J}\rho_{i_{s}(I)}\right]^{T}$. The index operator $J$
takes the value of index $z_{j}$ of arguments in the vector (6.2), while the
index operator $I$ takes values of index of differentials $dz_{i}$ in each
entry of the vector ${\bf X}$ (3.1). Thus, the index operator
$i(I)=(i_{I},\ldots,i_{s}(I))$ is given by consequent cycling permutations of
$I$. We define the operator $\widehat{\partial}_{J}\rho_{a}=\exp(-\sum_{{\bf
r}_{n},\;\sum\limits_{i=1}^{n}r_{i}\geq 1}r_{J}\;\beta^{(a)}_{{\bf
r}_{s}}\;\zeta^{r_{1}}_{1}\;\ldots\;\zeta^{r_{J}}_{J}\ldots\zeta^{r_{s}}_{s}\;\partial_{z_{J}})$,
which contains index operators $J$ as index of a dummy variable $\zeta_{J}$
turning into $z_{j}$, $j=1,\ldots,s$. In the last formula
$\widehat{\partial}_{J}$ acts on each argument of maps $X$ in the vector ${\bf
X}$. In [16] it was shown that the mappings
$(\rho_{1},\ldots,\rho_{l})(z_{1},\ldots,z_{j})\mapsto
R\left(\rho_{1},\ldots,\rho_{l}\right)$, for $1\leq j\leq l$, define a
representation of ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$ on
$\widehat{W}_{(z_{1},\ldots,z_{l})}$ by ${\rm
R}\left(\rho\circ\widetilde{\rho}\right)={\rm R}\left(\rho\right)\;{\rm
R}\left(\widetilde{\rho}\right)$, for $\rho$, $\widetilde{\rho}\in{\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$. Then we see that for generic elements ${\bf
X}\left(v_{1},z_{1};\ldots;v_{s},z_{s}\right)\in\widehat{W}_{U}$, for an
admissible vertex operator algebra $V$-module $W$, ${\bf
X}(v_{1},z_{1};\ldots;v_{s},z_{s})$ are independent on changes
$(z_{1},\ldots,z_{s+s^{\prime}})\mapsto(\widetilde{z}_{1},\ldots,\widetilde{z}_{s+s^{\prime}})=\left((\rho_{1},\ldots,\rho_{s+s^{\prime}})(z_{1},\ldots,z_{s+s^{\prime}})\right)$,
for $1\leq i\leq s+s^{\prime}$, of local coordinates of $(z_{1},\ldots,z_{s})$
and $(\widetilde{z}_{1},\ldots,\widetilde{z}_{s^{\prime}})$, at points
$(p_{1},\ldots,p_{s})$ and
$(\widetilde{p}_{1},\ldots,\widetilde{p}_{s^{\prime}})$.

Indeed, consider the vector ${\bf
X}(v_{1},\widehat{z}_{1};\ldots;v_{s},\widehat{z}_{s})$ =
$\left[X\left(v_{1},\widehat{z}_{1}\;d\widehat{z}_{{\it
i}(1)};\ldots;v_{s},\widehat{z}_{s}\;d\widehat{z}_{{\it i}(s)}\right)\right]$.
Note that
$d\widehat{z}_{j}=\sum\limits_{i=1}^{n}dz_{i}\;{\partial_{z_{i}}\rho_{j}}$,
$\partial_{z_{i}}\rho_{j}=\frac{\partial\rho_{j}}{\partial z_{i}}$. By the
definition of the action of ${\rm Aut}_{s}\;{\mathcal{O}}^{(n)}$, when
rewriting $d\widehat{z}_{i}$, we have

$\displaystyle{\bf
X}(v_{1},\widehat{z}_{1}\;d\widehat{z}_{1};\ldots;g_{s},\widehat{z}_{s}\;d\widehat{z}_{s})$
$\displaystyle=$ $\displaystyle{\rm
R}(\rho_{1},\ldots,\rho_{s})\;\left[X\left(v_{1},z_{1}\;d\widehat{z}_{i(1)};\ldots;v_{s},z_{s}\;d\widehat{z}_{i(s)}\right)\right]$
$\displaystyle=$ $\displaystyle{\rm
R}(\rho_{1},\ldots,\rho_{s})\;\left[X\left(v_{i},z_{i}\;\sum\limits_{j=1}^{s}\partial_{j}\rho_{i(s)}\;dz_{j}\right)\right].$

By linearity of the mapping $X$, we obtain from the last equation

${\bf
X}(\widehat{v}_{1},\widehat{z}_{1};\ldots;\widehat{v}_{s},\widehat{z}_{s})={\bf
X}(v_{1},\widehat{z}_{1}\;d\widehat{z}_{1};\ldots;v_{2},\widehat{z}_{s}\;d\widehat{z}_{s})=\left[X\left(v_{1},z_{1}\;dz_{i(1)};\ldots;v_{s},z_{s}\;dz_{i(s)}\right)\right],$
(6.2)

Due to properties of a vertex operator algebra $V$ admissible module $W$, the
action of operators $R\left(\rho_{1},\ldots,\rho_{s}\right)$ on
$(v_{1},\ldots,v_{s})\in V^{\otimes s}$ results in a sum of finitely many
terms. We then conclude that the vector ${\bf X}$ is invariant, i.e.,

$\displaystyle{\bf
X}(\widehat{v}_{1},\widehat{z}_{1};\ldots;\widehat{v}_{s},\widehat{z}_{s})$
$\displaystyle=$ $\displaystyle{\bf
X}\left(v_{1},\widehat{z}_{1}\;d\widehat{z}_{1};\ldots;v_{s},\widehat{z}_{s}\;d\widehat{z}_{s}\right)$
$\displaystyle=$ $\displaystyle{\bf
X}\left(v_{1},z_{1}\;dz_{1};\ldots;v_{s},z_{s}\;dz_{s}\right)={\bf
X}(v_{1},z_{1};\ldots;v_{s},z_{s}).$

The insertions of $k$ vertex operators
$\left(v_{k},t_{p_{k}}\;dt_{p_{k}}\right)$, $k\geq 0$, which are present in
the definition of sections of a vertex operator algebra bundle
$\mathcal{W}_{M/\mathcal{F}}$, keep elements ${\bf X}$ invariant with respect
to coordinate changes. Thus, the construction of spaces $\widehat{W}$ are
invariant under the action of the group ${\rm Aut}_{s}\;{\mathcal{O}}^{(n)}$.

We now formulate the following Lemma used later for the main result Lemma 2 of
this paper for the category of vertex operator algebra bundles
$\mathcal{W}_{M/\mathcal{F}}$ on $M/\mathcal{F}$.

######  Lemma 1.

A $\widehat{W}$-valued, independent of the choice of coordinates
$(t_{i,p_{j}})$, $1\leq i\leq n$, $1\leq j\leq l$, on a set of non-
intersecting discs $(D^{\times}_{i,p_{j}})$, section $X(p_{1},\ldots,p_{n})$
of the bundle $\mathcal{W}^{*}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{j}})}$ on
the $\overline{W}_{(p_{1},\ldots,p_{l})}$-valued fibers
$X_{i_{(t_{p_{1}},\ldots,t_{p_{l}})}}$ defined by (6.1) on
$(D^{\times}_{i,p_{j}})$ dual to
$\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{j}})}$ is given by the
formula

$X(p_{1},\ldots,p_{l})=\left[X_{i_{(t_{p_{1}},\ldots,t_{p_{l}})}}(w_{1},\ldots,w_{l})\right]=\left[X(v_{1},z_{1};\ldots;v_{l},z_{l})\right]={\bf
X}(v_{1},z_{1};\ldots;v_{l},z_{l}),$ (6.3)

$[X(v_{1},z_{1};\ldots;v_{l},z_{l})]\in\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$,
where $(t_{p_{1}},\ldots,t_{p_{l}})$ are coordinates on the discs
$(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})$, and
$(w_{1},\ldots,w_{l})\in\overline{W}_{(z_{1},\ldots,z_{l})}$.

###### Proof.

Now let us proceed with the explicit construction of ${\bf
X}_{i_{(t_{p_{1}},\ldots,t_{p_{l}})}}$. By choosing coordinates
$(t_{p_{1}},\ldots,t_{p_{l}})$ on a collection of discs
$(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})$, we obtain a trivialization
$i_{(t_{p_{1}},\ldots,t_{p_{l}})}:{\bf
X}\left(\widehat{W}[[(t_{p_{1}},\ldots,t_{p_{l}})]]\right)\;{\widetilde{}\atop\rightarrow}\;\Gamma\left(\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}\right)$,
of the bundle $\mathcal{W}_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$
which we call the $(t_{p_{1}},\ldots,t_{p_{l}})$-trivialization.

We also obtain trivializations of the fiber
$\widehat{W}_{(p_{1},\ldots,p_{l})}\;{\widetilde{}\atop\rightarrow}\;\gamma\left(\mathcal{W}_{M}|_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}\right)$,
and its dual
$\widehat{W}^{*}_{(p_{1},\ldots,p_{l})}\;{\widetilde{}\atop\rightarrow}\;\gamma\left(\mathcal{W}^{*}_{M/\mathcal{F}}|_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}\right)$.
Let us denote by $\left(w_{1},t_{p_{1}};\ldots;w_{1},t_{p_{l}}\right)$ the
image of $(w_{1},\ldots,w_{l})\in\widehat{W}_{z_{1},\ldots,z_{l}}$ in
$\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$
and by $\left(t_{p_{1}};\ldots;t_{p_{l}}\right)$ of
$\mathcal{W}^{*}_{M/\mathcal{F}}|_{(D^{\times}_{t_{p_{1}}},\ldots,D^{\times}_{t_{p_{l}}})}$
under $(t_{p_{1}},\ldots,t_{p_{l}})$-trivialization. In order to define the
required section ${\bf X}(p_{1},\ldots,p_{l})$ with respect to these
trivializations we need to attach an element of
$(\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$ to each
$\left(v_{1},t_{p_{1}};\ldots;v_{l},t_{p_{l}}\right)\in\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{t_{p_{1}}},\ldots,D^{\times}_{t_{p_{l}}})}$,
and a section $i_{(t_{p_{1}},\ldots,t_{p_{l}})}(x_{1},\ldots,x_{l})$ of
$\mathcal{W}|_{(D^{\times}_{t_{p_{1}}},\ldots,D^{\times}_{t_{p_{l}}})}$ for
$(x_{1},\ldots,x_{l})\in\widehat{W}_{(t_{p_{1}},\ldots,t_{p_{l}})}$. It is
sufficient to assign a function to the sets
$(v_{1},z_{1};\ldots;v_{l},z_{l})$,
$(w_{1},\ldots,w_{l})\in\widehat{W}_{(z_{1},\ldots,z_{l})}$ in the
$(t_{p_{1}},\ldots,t_{p_{l}})$-trivialization. Thus, we identify a
$\widehat{W}$-valued section $\widetilde{\bf X}(p_{1},\ldots,p_{l})$ of
$\mathcal{W}^{*}_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$, with the
section ${\bf X}(v_{1},z_{1};\ldots;v_{l},z_{l})$ of
$\mathcal{W}_{(D^{\times}_{p_{1}},\ldots,D^{\times}_{p_{l}})}$ by means of
formula (6.3).

Let
$(\widetilde{t}_{p_{1}},\ldots,\widetilde{t}_{p_{l}})=(\rho_{1},\ldots,\rho_{l})(t_{p_{1}},\ldots,t_{p_{l}})$
be another set of coordinates. Then, using the above arguments, we construct
analogously a section $\widetilde{\bf X}(p_{1},\ldots,p_{l})$ by the formula

$\widetilde{\bf
X}(p_{1},\ldots,p_{l})=\left[\widetilde{X}_{i_{(\widetilde{t}_{p_{1}},\ldots,\widetilde{t}_{p_{l}})}}(w_{1},\ldots,w_{l})\right]=\left[X(\widetilde{v}_{1},\widetilde{z}_{1};\ldots;\widetilde{v}_{l},\widetilde{z}_{l})\right]={\bf
X}(\widetilde{v}_{1},\widetilde{z}_{1};\ldots;\widetilde{v}_{l},\widetilde{z}_{l}).$

Since $\left(i^{-1}_{(t_{p_{1}},\ldots,t_{p_{l}})}\circ
i_{(t_{p_{1}},\ldots,t_{p_{l}})}\right)$ is an automorphism of
$\widehat{W}_{(p_{1},\ldots,p_{l})}$, we represent a change of variables
$\widetilde{t}_{p_{j}}=\rho_{j}(z_{j})$, $1\leq j\leq l$, in terms of
composition of trivializations

$\rho_{j}(z_{j})\mapsto i^{-1}_{\widetilde{t}_{j,p}}\circ i_{t_{j,p}},$ (6.4)

and, therefore, relate ${\bf
X}_{i_{(\widetilde{t}_{p_{1}},\ldots,\widetilde{t}_{p_{l}})}}(\widetilde{w}_{1},\ldots,\widetilde{w}_{l})$
with ${\bf X}({i_{(t_{p_{1}},\ldots,t_{p_{l}})}}(w_{1},\ldots,w_{l})$. Since
(6.4) defines a representation on $\widehat{W}$ of the group ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$ of changes of coordinates, then
$\widehat{W}_{(p_{1},\ldots,p_{l})}$ is canonically identified with the twist
of $\overline{W}$ by the ${\rm Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor of formal
coordinates at $(p_{1},\ldots,p_{l})$. Using definition of a torsor one sees
that elements of the space $\widehat{W}^{q}_{r}|U$ can be treated as ${\rm
Aut}_{l}\;{\mathcal{O}}^{(n)}$-torsor of the product of groups of a coordinate
transformation, namely, that
$(v_{1},z_{1};\ldots;v_{l},z_{l})=\left(R(\rho_{l})^{-1}.(v_{1},\widetilde{t}_{p_{1}};\ldots;v_{l},\widetilde{t}_{p_{l}}\right)$,
Thus, we relate the l.h.s and r.h.s. of (6.3). Since the element ${\bf
X}(v_{1},z_{1};\ldots;v_{l},z_{l})$ is invariant with respect to changes of
coordinates, Lemma follows. ∎

##  7\. Category of vertex algebra bundles on leaves of $M/\mathcal{F}$ and
transversal sections

In this Section we construct canonical twisted vertex algebra $V$-module
bundle $\mathcal{W}_{M/\mathcal{F}}$ on leaves and transversal sections of a
codimension $p$ foliation $\mathcal{F}$ defined on a smooth $n$-dimensional
manifold $M$.

###  7.1. Holonomy and transversal basis for a foliation

Let us first recall [11] definitions of transversal basis and holonomy
embeddings for a foliation $\mathcal{F}$. Transversal sections $U_{i}$ of a
foliation $\mathcal{F}$ passing through points $p_{i}$, $i\geq 0$, are
neighborhoods of the leaves through $p_{i}$ in the leaf space $M/\mathcal{F}$.
Suppose we are given a path $\alpha$ between two points $p_{1}$ and $p_{2}$
which belong to the same leaf of $\mathcal{F}$. For two transversal sections
$U_{1}$ and $U_{2}$ passing through $p_{1}$ and $p_{2}$ one defines a
transport $\alpha$ along the leaves from a neighborhood of $p_{1}\in U_{1}$ to
a neighborhood of $p_{2}\in U_{2}$. Then it is assumed that there exists a
germ of a diffeomorphism ${\rm
hol}(\alpha):(U_{1},p_{1})\longrightarrow(U_{2},p_{2})$ called the holonomy of
$\alpha$. When the transport $\alpha$ is defined in all of $U_{1}$ and embeds
into $U_{2}$ then $h:U_{1}\hookrightarrow U_{2}$ is denoted by ${\rm
hol}(\alpha):U_{1}\hookrightarrow U_{2}$ and it is called a holonomy
embedding. A composition of paths induces a composition of corresponding
holonomy embeddings. Two homotopic paths always define the same holonomy. The
holonomy groupoid [10, 26, 42] is the groupoid $Hol(M,\mathcal{F})$ over $M$
where arrows $p_{1}\longrightarrow p_{2}$ are such germs $hol(\alpha)$. A
transversal basis $\mathcal{U}$ for $\mathcal{F}$ is a set of transversal
sections $U_{i}\subset M$ such that for a section $U_{i}$ passing through a
point $p_{i}$, and for any transversal section $U_{j}$ passing through
$p_{j}\in M$, one can find a holonomy embedding $h:U_{i}\hookrightarrow U_{j}$
with $U_{i}\in\mathcal{U}$ and $p_{j}\in h(U_{i})$.

###  7.2. Spaces of sections of $V$-bundles

Let $M$ be endowed with a coordinate chart
$\mathcal{V}=\left\\{V_{r},r\in\mathbb{Z}\right\\}$. Consider a (possibly
infinite) arbitrary set $p_{l}$, $l\geq 0$, of $l$ distinct points and
corresponding domains $V_{l}\subset M$. Let
$\mathcal{U}=\left\\{U_{k}\right\\}$, $k\geq 0$, be a transversal basis of
$\mathcal{F}$. We chose a (possibly infinite) set $(p_{l+1},\ldots,p_{l+m})$
of arbitrary distinct $m\geq 0$ points on a set of sections
$U_{i^{\prime},b}$, $1\leq i^{\prime}\leq m$, $1\leq b\leq k$ of
$\mathcal{U}$. Let us associate to each point of $(p_{1},\ldots,p_{l})$ and
$(p_{l+1},\ldots,p_{l+m})$ vertex operator algebra elements
$(v_{i,1},\ldots,v_{i,n})$, $1\leq i\leq l$, and $(v_{j,1},\ldots,v_{j,p})$,
$l+1\leq j\leq l+m$ correspondingly. For the set of all choices of $ln+mp$
vertex operator algebra $V$ elements and $\widetilde{n}=ln+mp$, it is
convenient to renotate the vertex operator algebra elements as

$(\widetilde{v}_{1},\ldots,\widetilde{v}_{\widetilde{n}})=(v_{1,1},\ldots,v_{l,1},\ldots,v_{1,n},\ldots,v_{l,n},v_{l+1,},\ldots,v_{l+1,p},\ldots,v_{l+m,1},\ldots,v_{l+m,p}).$

Endow each of points among $(p_{1},\ldots,p_{l})$ and
$(p_{l+1},\ldots,p_{l+m})$ with sets $(z_{i,1},\ldots,z_{i,n})$, $1\leq i\leq
l$ and $(z_{j,1},\ldots,z_{j,p})$, $l+1\leq j\leq l+m$, of local coordinates
on domains $V_{l}$ and $V_{l+m}$. Denote also

$(\widetilde{z}_{1},\ldots,\widetilde{z}_{\widetilde{n}})=(z_{1,1},\ldots,z_{l,1},\ldots,z_{1,n},\ldots,z_{l,n},z_{l+1,},\ldots,z_{l+1,p},\ldots,z_{l+m,1},\ldots,z_{l+m,p}).$

Now, taking into account the content of Section 6, and, in particular, Lemma
1, we are on a position to introduce the spaces of sections of vertex operator
algebra $V$ bundle $\mathcal{W}_{M/\mathcal{F}}$ over leaves and transversal
sections of a codimension $p$ foliation $\mathcal{F}$ defined on $M$. Note
that the space of $\mathcal{F}$-leaves is not in general a manifold.
Nevertheless, one can always consider local coordinates in appropriate domains
on leaves of $M/\mathcal{F}$ induced by local coordinates on a chart defined
on $M$. In this Section we provide the specific form of canonical sections
${\bf X}$ of a vertex operator algebra bundle $\mathcal{W}_{M/\mathcal{F}}$ as
elements of the spaces $\overline{W}$ considered on specific domains on
$M/\mathcal{F}$ and $\mathcal{U}$. In order to work with objects having
coordinate invariant formulation, we consider elements of $\overline{W}$ with
local coordinates $z$ multiplied by powers of corresponding differentials
$dz$. For all choices of $l$ points and all choices of vertex operator algebra
elements for $ln\geq 0$ complex variables
$(\widetilde{z}_{1},\ldots,\widetilde{z}_{ln})$ defined in domains $V_{k}$,
$1\leq k\leq ln$ of the coordinate chart $\mathcal{V}$ on $M$, let us consider
the vector of the form (3.1) with variables
$(\widetilde{v}_{1},\widetilde{z}_{1};\ldots;\widetilde{v}_{ln},\widetilde{z}_{ln})$,
containing $\overline{W}$-rational functions $X$. In [16], in the case $n=1$,
they proved for primary $u$ that the vertex operator
$Y_{W}(u,z)\;dz^{\textup{wt}(u)}$ is an invariant object with respect to
changes of the local coordinate. In previous Section we proved that the
vectors ${\bf X}$ introduced above as well as vertex operators
$\mathcal{Y}(u,z_{i})=Y_{W}(u,z_{i})\;dz_{i}^{\textup{wt}(u)}$, $i\geq 0$, for
primary $u\in V$, are invariant with respect to changes of coordinates, i.e.,
to the group of coordinate transformations ${\rm
Aut}_{s}\;{\mathcal{O}}^{(n)}$ on $M/\mathcal{F}$
$\left(w_{1},\ldots,w_{s}\right)\mapsto(z_{1},\ldots,z_{s})$, and
corresponding differentials.

In [20] the classical approach to cohomology of vector fields of manifolds was
initiated. In [13, 41] we find an alternative way to describe cohomology of
Lie algebra of vector fields on a manifold in the cosimplicial setup. Taking
into account the standard methods of defining canonical (i.e., independent of
the choice of covering $\mathcal{U}$ and coordinates) cosimplicial object [13,
41] as well as the ${\rm\check{C}}$ech-de Rham cohomology construction [11],
we formulate here the vertex operator algebra approach to cohomology of a
foliation. Let $I_{q}=\left\\{z_{i_{s},j_{s}}\right\\}$, $1\leq s\leq q$, be a
subset (with no repetitions) of the set $(z_{i,1},\ldots,z_{i,n})$, $1\leq
i\leq l$ of local variables corresponding to of $l$ points
$(p_{1},\ldots,p_{l})$ taken on the same leaf $\mathfrak{f}$ of
$M/\mathcal{F}$. Similarly, let
$J_{r}=\left\\{z_{i_{s^{\prime}},j_{s^{\prime}}}\right\\}$, $1\leq
s^{\prime}\leq r$, be a subset (with no repetitions) of the set
$(z_{i,1},\ldots,z_{i,p})$, $l+1\leq i\leq l+m$, of local variables
corresponding to of $m$ points $(p_{l+1},\ldots,p_{l+m})$ on sections of a
transversal basis $\mathcal{U}$ of $\mathcal{F}$.

Assume that all points $(p_{1},\ldots,p_{l+m})$ belong to the same leaf
$\mathfrak{f}$ of $M/\mathcal{F}$. Consider a subspace
$\widehat{W}^{q}(V,W,U,\mathcal{F})$ of vectors ${\bf
X}(v_{i_{1},j_{1}},z_{i_{1},j_{1}};\ldots;v_{i_{q},j_{q}},z_{i_{q},j_{q}})$
associated to $I_{q}$ for all sets of vertex operator algebra elements
$(v_{i_{1},j_{1}},\ldots,v_{i_{q},j_{q}})$ with local coordinates defined on a
domain $U$ of $M$. A vertex operator algebra $V$ bundle
$\mathcal{W}_{M/\mathcal{F}}$ consists of the union of the spaces
$\widehat{W}^{q}_{r}=\widehat{W}^{q}_{r}(V,W,\mathcal{F})$, $q$, $r\geq 0$.
The spaces $\widehat{W}^{q}_{r}$ consist of of sections of
$\mathcal{W}_{M/\mathcal{F}}$ defined as the space
$\widehat{W}^{q}(V,W,\mathcal{F})$ of vectors ${\bf X}$ on each leaf
$\mathfrak{f}$ of $\mathcal{F}$, and any subset $J_{r}$ of $r$ vertex
operators for all sets of vertex operator algebra elements
$v_{i^{\prime}_{a},j^{\prime}_{a}}$
${\mathcal{Y}}_{W}(v_{i^{\prime}_{a},j^{\prime}_{a}},z_{i^{\prime}_{a},j^{\prime}_{a}})$,
$1\leq a\leq r$, with local coordinates $z_{i^{\prime}_{a},j^{\prime}_{a}}$
defined in $r$ subdomains $U_{i^{\prime}_{a},b}\subset U_{b}$, $1\leq b\leq k$
of $k$ transversal sections $U_{b}$ of a transversal basis $\mathcal{U}$. Here
the domain $U$ is defined as
$U=\bigcap_{{U_{i^{\prime}_{1},1}\stackrel{{\scriptstyle
h_{i^{\prime}_{1},1}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle
h_{i^{\prime}_{r},k}}}{{\longrightarrow}}U_{i^{\prime}_{r},k},1\leq a\leq
r,\;1\leq b\leq k,}\;}U_{i^{\prime}_{a},b}$, where the intersection ranges
over $r$ subdomains $U_{i^{\prime}_{a},b}\subset U_{b}\subset\mathcal{U}$ of
$r$ local coordinates for any choice of $l$ points $(p_{1},\ldots,p_{l})$ on
the same leaf $\mathfrak{f}$ of $M/\mathcal{F}$ related by the holonomy
embeddings $h_{i^{\prime}_{a},b}$, $1\leq a\leq r$, $1\leq b\leq k$. In the
case $k=0$ the sequence of holonomy embeddings is empty. It is easy to see
that the definition of $\widehat{W}^{q}_{r}$ does not depend on the choice of
$\mathcal{U}$.

The spaces $\widehat{W}^{q}_{r}$ are related by the shift operators
$\Delta^{q}_{r}:\widehat{W}^{q}_{r}\to\widehat{W}^{q+1}_{r-1}$, increasing the
upper index and decreasing the lower index in elements of
$\widehat{W}^{q}_{r}$. For $I_{q+1}=(i_{k},j_{k})$, $1\leq k\leq q+1$, and
$J_{r}=(i^{\prime}_{k^{\prime}},j^{\prime}_{k^{\prime}})$, $1\leq
k^{\prime}\leq r$, and ${\bf X}\in\widehat{W}_{r}^{q}$ let us define the
operator in the standard way [11, 28]

$\displaystyle\Delta^{q}_{r}{\bf
X}(v_{i_{1},j_{1}},z_{i_{1},j_{1}};\ldots;v_{i_{q},j_{q}},z_{i_{q},j_{q}})$
(7.1)
$\displaystyle\qquad=\mathcal{Y}_{W}(v_{i_{1},j_{1}},z_{i_{1},j_{1}}){\bf
X}\left(v_{i_{2},j_{2}},z_{i_{2},j_{2}};\ldots;v_{i_{q},j_{q}},z_{i_{q},j_{q}}\right)$
$\displaystyle\quad+\sum_{s=1}^{q}(-1)^{s}{\bf
X}\left(v_{i_{1},j_{1}},z_{i_{1},j_{1}};\ldots;\mathcal{Y}_{W}(v_{i_{s-1},j_{s-1}},z_{i_{s-1},j_{s-1}}-\zeta_{s})\right.$
$\displaystyle\quad\qquad\left.\mathcal{Y}_{W}(v_{i_{s+1},j_{s+1}},z_{i_{s+1},j_{s+1}}-\zeta_{s})\mathbf{1}_{V};\ldots;v_{i_{q},j_{q}},z_{i_{q},j_{q}}\right)$
$\displaystyle\qquad+(-1)^{q+1}\mathcal{Y}_{W}(v_{i_{q+1},j_{q+1}},z_{i_{q+1},j_{q+1}}){\bf
X}\left(v_{i_{1},j_{1}},z_{i_{1},j_{1}};\ldots;v_{i_{q},j_{q}},z_{i_{q},j_{q}}\right).$

The shift operator $\Delta^{q}_{r}$ is chosen in such a way that its
characteristics of would have nice analytic and cohomological properties. Note
that, after application of $\Delta^{q}_{r}$ on an element ${\bf X}$ containing
local coordinates and corresponding vertex operator algebra elements
associated to all $n$ dimensions, of $M$, the result is of such action is then
related to submanifold with less number of local coordinates describing points
$(p_{1},\ldots,p_{l})$. For $q=2$, there exists a subspace
$\widehat{W}_{\vartheta}^{2}$ of $\widehat{W}_{0}^{2}$ containing
$\widehat{W}_{r}^{2}$ for all $r\geq 1$ such that $\Delta^{2}_{r}$ is defined
on this subspace. For $J_{3}=(i_{k},j_{k})$, $1\leq k\leq 3$, the operator
$\Delta^{2}_{\vartheta}$ is defined for ${\bf
X}\in\widehat{W}_{\vartheta}^{2}$ by a particular case of (7.1). With the
shift operator $\Delta^{q}_{r}$ we obtain the sequences:
$\widehat{W}_{r}^{0}\stackrel{{\scriptstyle\Delta^{0}_{r}}}{{\longrightarrow}}\widehat{W}_{r-1}^{1}\stackrel{{\scriptstyle\Delta^{1}_{r-1}}}{{\longrightarrow}}(\widehat{W}_{r-2}^{2},\delta_{r,3}\;\widehat{W}^{2}_{\vartheta})\stackrel{{\scriptstyle(\Delta^{r-2}_{2},\;\delta_{r,3}\;\Delta^{2}_{\vartheta})}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\Delta^{r-1}_{1}}}{{\longrightarrow}}\widehat{W}_{0}^{r}$.

The construction of the vertex operator algebra $V$-bundle
$\mathcal{W}_{M/\mathcal{F}}$ provides a description of the holonomy groupoid
$Hom(M,\mathcal{F})$ introduced in Subsection 7.1 in terms of holonomy
embeddings. We consider the spaces of vectors ${\bf
X}(h_{i_{1},1},\ldots,h_{i_{r},k})={\bf X}|_{U}$ taken on all leaves
$\mathfrak{f}$ of $M/\mathcal{F}$. In terms of holonomy embedding, the shift
operator (c.f. [11]) in this case is given by the standard differentials. The
vertical differential
$\widehat{W}^{q}_{r}\longrightarrow\widehat{W}^{q}_{r+1}$ is $(-1)^{q}d^{-1}$
where $d$ is the usual De Rham differential. The horizontal differential
$\Delta:\widehat{W}^{q}_{r}\longrightarrow\widehat{W}^{q+1}_{r}$ is
$\Delta=\sum(-1)^{i}\Delta_{i}$, where

$\displaystyle\Delta_{i}{\bf X}(h_{1},\ldots,h_{q+1})$ $\displaystyle=$
$\displaystyle\delta_{i,0}\;h_{1}{}^{*}\;{\bf
X}(h_{2},\ldots,h_{q+1})+\delta_{i,q+1}{\bf X}(h_{1},\ldots,h_{q})$ (7.2)
$\displaystyle+$ $\displaystyle(1-\delta_{i,0})(1-\delta_{i,q+1}){\bf
X}(h_{1},\ldots,h_{i+1}h_{i},\ldots,h_{k+1}).$

The category $\mathfrak{W}_{M/\mathcal{F}}$ of vertex operator algebra
admissible $V$-bundles $\mathcal{W}_{M/\mathcal{F}}$ for a foliation
$\mathcal{F}$ consists of objects $\mathcal{W}_{M/\mathcal{F}}$ with morphisms
provided by intertwining operators [12].

###  7.3. Characteristics of bundle $\mathcal{W}_{M/\mathcal{F}}$-sections

In the definition of spaces $\widehat{W}$ sequences of holonomy embeddings
$h_{i}$, $i\geq 0$ were involved. For germs $hol(\alpha)$ of the groupoid
$Hol(M,\mathcal{F})$ over $M$ we define the spaces $\widetilde{X}$ of vectors
${\bf X}$ defined in previous subsections. The holonomy groupoid is the
groupoid $Hol(M,\mathcal{F})$ over $M$ where arrows $p_{1}\longrightarrow
p_{2}$ are such germs $hol(\alpha)$. In this Section we prove the main result
of this paper for the category of $V$-bundles for foliations defined on a
complex manifold. Recall that the cohomology of $Hol(V,\mathcal{F})$
determines the cohomology of a foliation $\mathcal{F}$ [11]. The main
advantage of Lemma 2 provided at the end of this Section, is that by using
vertex operator algebra properties we are able to compute explicitly the
cohomology of the holonomy groupoid $Hol(M,\mathcal{F})$ in terms of special
functions. For meromorphic functions of several complex variables defined on
sets of open domains of $M$ with local coordinates $z_{i,j}$ which are
extendable rational functions $f(z_{i,j})$ on larger domains on $M$ we denote
such extensions by $R(f(z_{i,j}))$. For a set of
$\overline{W}_{(z_{i,j})}$-defining elements $(v_{i,j})$ we consider the
converging rational functions $f(v_{i,j},z_{i,j})\in\overline{W}_{(z_{i,j})}$
of $z_{i,j}\in F_{ln}\mathbb{C}$.

By involving the definition of $\widehat{W}^{q}_{r}$ it is possible to
introduce a vertex operator algebra $V$ cohomology of the leaves space
$M/\mathcal{F}$ of a foliation $\mathcal{F}$. Let us consider the spaces
$C^{q}_{r}=C^{q}_{r}(V,W,\mathcal{F})$ containing vectors of rational
functions provided by vectors of characteristics $[\Omega(X)]$ of ${\bf
X}$-entries. For any ${\bf X}\in\widehat{W}^{q}_{r}$, the map $\Delta^{q}_{r}$
induces the map $\delta^{q}_{r}$ by $[\Omega(X)]$. The coboundary operator
$\delta^{q}_{r}$ exhibits the chain-cochain property if characteristics of
entries $X$ of ${\bf X}$ satisfy the following conditions.

For sets $p_{j}$, $1\leq j\leq l$ of $l$ points on the same leaf of
$M/\mathcal{F}$, we consider a map
$X(v_{i,1},z_{i,1};\ldots;v_{i,n},z_{i,n}):V^{\otimes
ln}\in\overline{W}[[z_{1},\ldots,z_{r}]]$, $1\leq i\leq l$, combined with a
set of $mp$ vertex operators at points $p_{l+k}$, $1\leq k\leq m$, such that
its characteristic $\Omega(X)$ satisfy the following properties. We imply
certain conditions on the characteristics
$\Omega(v_{1},z_{1};\ldots;v_{s},z_{s})=\Omega(X(v_{1},z_{1};\ldots;v_{s},z_{s}))$
for elements $X(v_{1},z_{1};\ldots;v_{s},z_{s})$ by to the relations mentioned
below to be coherent with definitions for $\overline{W}$ given in [28]. We
require that for $i=1,\ldots,s$,

$\partial_{z_{i}}\Omega(v_{1},z_{1};\ldots;v_{s},z_{s})=\Omega(v_{1},z_{1};\ldots;v_{i-1},z_{i-1};L_{V}(-1)v_{i},z_{i};v_{i+1},z_{i+1};\ldots;v_{s},z_{s}),$
$\left(\partial_{z_{1}}+\cdots+\partial_{z_{s}}\right)\Omega(v_{1},z_{1};\ldots;v_{s},z_{s})=L_{W}(-1)\Omega(v_{1},z_{1};\ldots;v_{s},z_{s}).$

Since $L_{W}(-1)$ is a weight-one operator on $W$, $e^{zL_{W}(-1)}$ is a
linear operator on $\overline{W}$ for any $z\in\mathbb{C}$. For a linear map
$X$ with $(v_{1},\ldots,v_{s})\in V^{\otimes s}$, $(z_{1},\ldots,z_{s})\in
F_{s}\mathbb{C}$, $z\in\mathbb{C}$ such that $(z_{1}+z,\dots,z_{s}+z)\in
F_{s}\mathbb{C}$, the characteristics

$e^{zL_{W}(-1)}\Omega(v_{1},z_{1};\ldots;v_{s},z_{s})=\Omega(v_{1},z_{1}+z;\ldots;v_{s},z_{s}+z),$

and for $(v_{1},\dots,v_{s})\in V^{\otimes s}$, $(z_{1},\ldots,z_{s})\in
F_{s}\mathbb{C}$, $z\in\mathbb{C}$ and $1\leq i\leq s$ such that
$(z_{1},\ldots,z_{i-1},z_{i}+z,z_{i+1},\dots,z_{s})\in F_{s}\mathbb{C}$, the
power series expansion of the characteristic of the element
$\Omega(v_{1},z_{1};\ldots;v_{i-1},z_{i-1};v_{i},z_{i}+z;v_{i+1},z_{i+1};\ldots;v_{s},z_{s})$,
in $z$ are equal to the power series expansion of the characteristic
$\Omega(v_{1},z_{1};\ldots;v_{i-1},z_{i-1};e^{zL_{V}(-1)}v_{i},z_{i};$
$v_{i+1},z_{i+1};\ldots;v_{s},z_{s})$, in $z$. In particular, the power series
in $z$ is absolutely convergent on the open disc $|z|<\min_{i\neq
j}\\{|z_{i}-z_{j}|\\}$. In addition to that, for $(v_{1},\dots,v_{s})\in
V^{\otimes s}$, $(z_{1},\ldots,z_{s})\in F_{s}\mathbb{C}$ and
$z\in\mathbb{C}^{\times}$ so that $(zz_{1},\ldots,zz_{s})\in F_{s}\mathbb{C}$,
a linear map $X:V^{\otimes s}\to\overline{W}$ the characteristics

$z^{L_{W}(0)}\Omega(v_{1},z_{1};\ldots;v_{s},z_{s})=\Omega(z^{L_{V}(0)}v_{1},zz_{1};\ldots;z^{L_{V}(0)}v_{s},zz_{s}).$

should coincide.

Recall now the definition of shuffles. For $l>0$ 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 $s$ and the last $l-s$ numbers, i. e., $J_{l,s}=\\{\sigma\in
S_{l}\;|\;\sigma(1)<\ldots<\sigma(s),\;\sigma(s+1)<\ldots<\sigma(l)\\}$. The
elements of $J_{l;s}$ are then called shuffles. We will use the notation
$J_{l;s}^{-1}=\\{\sigma\;|\;\sigma\in J_{l;s}\\}$ for them. Finally, define
the left action of the permutation group $S_{r}$ on $\overline{W}$ by
$\sigma(f)(z_{1},\ldots,z_{r})=f(z_{\sigma(1)},\ldots,z_{\sigma(r)})$, for
$f\in\overline{W}$. We require that

$\sum_{\sigma\in
J_{l;s}^{-1}}(-1)^{|\sigma|}\sigma(\Omega(v_{\sigma({1,1})},z_{\sigma({1,1})};\ldots;v_{\sigma({l,1})},\ldots,v_{\sigma({1,n})},\ldots,v_{\sigma(l,n)}))=0.$

Denote by $P_{s}:\overline{W}\to\overline{W}_{(s)}$, the projection of
$\overline{W}$ on $\overline{W}_{(s)}$. Denote by
$(l_{i},\ldots,l_{\widetilde{n}})$ a partition of ${\widetilde{n}}$ of
$\widetilde{n}=\sum_{i\geq 1}l_{i}$, $k_{i}=l_{1}+\cdots+l_{i-1}$, and
$\zeta_{i}\in\mathbb{C}$. Consider the local coordinates
$(\widetilde{z}_{ln+1},\ldots,\widetilde{z}_{\widetilde{n}})$ of points
$(p_{l+1},\ldots,p_{l+m})$ bounded in the domains
$|\widetilde{z}_{k_{i}+k^{\prime}}-\zeta_{i}|+|\widetilde{z}_{k_{j}+k^{\prime\prime}}-\zeta_{j}|<|\zeta_{i}-\zeta_{j}|$,
for $i$, $j=1,\dots,n$, $i\neq j$, and for $k^{\prime}=1,\dots$, $l_{i}$,
$k^{\prime\prime}=1$, $\dots$, $l_{j}$. For $\widetilde{k}_{i}=k_{i}+l_{i}$,
define
$f_{i}=\Omega\left(\mathcal{Y}_{W}(\widetilde{v}_{1},\widetilde{z}_{1}-\zeta_{i})\ldots\mathcal{Y}_{W}(\widetilde{v}_{\widetilde{k}_{i}},\widetilde{z}_{\widetilde{k}_{i}}-\zeta_{i})\right)$,
for $i=1,\dots,ln$.

Assume that there exist positive integers
$\beta(\widetilde{v}_{l^{\prime},i},\widetilde{v}_{l",j})$ depending only on
$\widetilde{v}_{l^{\prime},i}$ and $\widetilde{v}_{l^{\prime\prime},j}$ for
$i$, $j=1,\dots,\widetilde{n}$, $i\neq j$, $1\leq
l^{\prime},l^{\prime\prime}\leq\widetilde{n}$, such that the characteristic

$\Omega\left(\sum\limits_{r_{1},\ldots,r_{\widetilde{n}}\in\mathbb{Z}}X\left(P_{r_{1}}f_{1},\zeta_{1};\ldots;P_{r_{\widetilde{n}}}f_{\widetilde{n}},\zeta_{\widetilde{n}}\right)\right),$
(7.3)

is absolutely convergent in the domains defined above to an analytic extension
in $(\widetilde{z}_{1},\ldots,\widetilde{z}_{\widetilde{n}})$ independently of
complex parameters $(\zeta_{1},\ldots,\zeta_{\widetilde{n}})$, with poles of
order less than or equal to
$\beta(\widetilde{v}_{l^{\prime},i},\widetilde{v}_{l^{\prime\prime},j})$
allowed only on the diagonal of
$(\widetilde{z}_{1},\ldots,\widetilde{z}_{\widetilde{n}})$. We assume that for
$(\widetilde{v}_{1},\ldots,\widetilde{v}_{\widetilde{n}})$, the characteristic

$\Omega\left(\sum_{q\in\mathbb{C}}\mathcal{Y}_{W}(v_{l+1,1},z_{l+1,1})\ldots\mathcal{Y}_{W}(v_{l+m,1},z_{l+m,p})P_{q}X\left(v_{1,1},z_{1,1};\ldots;v_{l,1},z_{l,n}\right)\right),$
(7.4)

(incorporating local coordinates on $M$ and transversal sections) is
absolutely convergent on the domains
$|\widetilde{z}_{i}|>|\widetilde{z}_{s}|>0$, for $i=1,\dots,m$, and
$s=m+1,\dots,m+l$, when $\widetilde{z}_{i}\neq\widetilde{z}_{j}$, $i\neq j$
and the sum is analytically extendable to a rational function in
$(\widetilde{z}_{1},\ldots,z_{\widetilde{n}})$ with poles of orders less than
or equal to
$\beta(\widetilde{v}_{l^{\prime},i},\widetilde{v}_{l^{\prime\prime},j})$
allowed $\widetilde{z}_{i}=\widetilde{z}_{j}$.

For $q=2$, for $\widetilde{v}_{1}$, $\widetilde{v}_{2}$, $\widetilde{v}_{3}\in
V$, the characteristics
$\sum_{s\in\mathbb{C}}\Omega(\mathcal{Y}_{W}(\widetilde{v}_{1},\widetilde{z}_{1})\mathcal{Y}_{W}(P_{s}(X(\widetilde{v}_{2},\widetilde{z}_{2}-\zeta;\widetilde{v}_{3},\widetilde{z}_{3}-\zeta)),\zeta))$
$+\Omega(\widetilde{v}_{1},\widetilde{z}_{1};P_{s}(\mathcal{Y}_{V}(\widetilde{v}_{2},\widetilde{z}_{2}-\zeta)\mathcal{Y}_{V}(\widetilde{v}_{3};\widetilde{z}_{3}-\zeta)\mathbf{1}_{V}),\zeta)$,
and $\sum_{s\in\mathbb{C}}$ $\Omega($
$P_{s}(Y_{V}(\widetilde{v}_{1},\widetilde{z}_{1}-\zeta)\mathcal{Y}_{V}(\widetilde{v}_{2};\widetilde{z}_{2}-\zeta)\mathbf{1}_{V}),\zeta)\mathcal{Y}_{V}(\widetilde{v}_{3},\widetilde{z}_{3})$
$+\mathcal{Y}_{W}(\widetilde{v}_{3},\widetilde{z}_{3})\mathcal{Y}_{W}(P_{s}(X(\widetilde{v}_{1},\widetilde{z}_{1}-\zeta;\widetilde{v}_{2},\widetilde{z}_{2}-\zeta)),\zeta)$,
defined on the domains $|\widetilde{z}_{1}-\zeta|>|\widetilde{z}_{2}-\zeta|$,
$|\widetilde{z}_{2}-\zeta|>0$, and
$|\zeta-\widetilde{z}_{3}|>|\widetilde{z}_{1}-\zeta|,|\widetilde{z}_{2}-\zeta|>0$,
respectively, are absolutely convergent and analytically extendable to
rational functions in $\widetilde{z}_{1}$ and $\widetilde{z}_{2}$ with poles
allowed only at $\widetilde{z}_{1}$, $\widetilde{z}_{2}=0$, and
$\widetilde{z}_{1}=\widetilde{z}_{2}$.

###  7.4. The bundle dual to $\mathcal{W}_{M/\mathcal{F}}$ on $M/\mathcal{F}$

The conditions on characteristics described in the previous Subsection allow
to define a fiber bundle on the transversal sections of $M/\mathcal{F}$ in the
dual formulation. This gives us an idea how to use the notion of a dual vertex
operator algebra bundle associated to transversal sections of a foliation. For
many purposes it is useful when the conditions on grading subspaces of
$\overline{W}_{(p_{1},\ldots,p_{l})}$ are relaxed in the definition of
admissible $V$-module $W$. The advantage of the dual (defined with respect to
an appropriate form) fiber bundle $\mathcal{W}^{\dagger}_{\mathcal{U}}$ is
that we do not need to assume that the $\mathbb{C}$-grading on
$\overline{W}_{(p_{1},\ldots,p_{l})}$ is bounded from below or that the graded
components are finite-dimensional. The condition (7.3) for the grading
together with conditions on orders of poles, and then the canonical pairing
(7.4) give rise to a pairing
$\gamma\left(\mathcal{W}^{\dagger}_{\mathcal{U}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}\right)\times\gamma\left(\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}\right)\rightarrow\mathbb{C}^{l}$,
for corresponding space of fibers. For each fiber $\mu$ of
$\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}$
we obtain a linear operator on
$\mathcal{W}_{\mathcal{U}}^{\dagger}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}$
given by this pairing. Thus, we obtain a well-defined linear map

$\mathcal{W}_{\mathcal{U}}^{\dagger}{}_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}:\gamma\left(\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}\right)\rightarrow{\rm
End}\left(\gamma\left(\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}\right)\right),$
(7.5)

i.e., the fibers expressed in terms of vertex operators defined on transversal
sections. For formal coordinates $(t_{i,p_{j}})$, $1\leq i\leq n-p$, $1\leq
j\leq l$, on $(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})$, a fiber
$X(v_{i,j},z_{i,j})$ of
$\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}$

with elements of $\overline{W}_{(p_{1},\ldots,p_{l})}$ with respect to the
$(t_{i,p_{j}})$-trivialization, the map (7.5) is given by $(v_{i,j},z_{i,j})$.
Starting from an admissible vertex operator algebra $V$-module $W$, and
applying Lemma1 we construct explicitly a fiber bundle
$\mathcal{W}_{M/\mathcal{F}}$ over $M/\mathcal{F}$, with canonical sections
$X(p_{1},\ldots,p_{n})$ of
$\mathcal{W}_{M/\mathcal{F}}|_{(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})}$
and fibers with values in ${\rm
End}\left(\overline{W}_{(p_{1},\ldots,p_{n})}\right)$ for any set of non-
intersecting discs $(D^{\times}_{i,p_{1}},\ldots,D^{\times}_{i,p_{l}})$,
$1\leq i\leq n-p$ on $M/\mathcal{F}$.

###  7.5. A relation for $\mathcal{W}_{M}$ and $Hol(M,\mathcal{F})$
cohomologies

In this Subsection we provide Lemma 2 relating the vertex operator algebra $V$
cohomology of the holonomy groupoid $Hol$( $M$, $\mathcal{F})$ and $V$-bundle
$\mathcal{W}_{M/\mathcal{F}}$. Note that the spaces $\widehat{W}^{q}_{r}$
containing non-commutative elements ${\bf X}$, as well as their cohomology are
described here in terms of their characteristics. In [28] it was proven that
the operator $\delta^{q}_{r}X=\Delta^{q}_{r}\Omega(X)$ possesses the chain-
cochain property. For the holonomy groupoid $Hol(M,\mathcal{F})$ we obtain the
linear maps $\delta^{n}_{r}:C_{r}^{n}\to C_{r-1}^{n+1}$, for each pair $l$,
$r\geq 0$, and $\delta^{2}_{\vartheta}:C_{\vartheta}^{2}\to C_{0}^{3}$. Since
$C_{\infty}^{q}\subset C_{r}^{q}$ for any $r\geq 0$, and $C_{r_{2}}^{q}\subset
C_{r_{1}}^{q}$, for $r_{1}$, $r_{2}\in\geq 0$ with $r_{1}\leq r_{2}$,
$\delta_{r}^{q}.C_{\infty}^{q}$ is independent of $r$. Let
$\delta_{\infty}^{q}=\delta_{r}^{q}.C_{\infty}^{q}:C_{\infty}^{q}\to
C_{\infty}^{l+1}$. Thus, we obtain a double complex
$(C^{q}_{r},\delta^{q}_{r})$, $(C^{2}_{\vartheta}),\delta^{2}_{\vartheta})$
$q$, $r\geq 0$, in particular, with $r=\infty$, $0\longrightarrow
C_{r}^{0}\stackrel{{\scriptstyle\delta^{0}_{r}}}{{\longrightarrow}}C_{r-1}^{1}\stackrel{{\scriptstyle\delta^{1}_{r-1}}}{{\longrightarrow}}(C_{r-2}^{2},\delta_{r,3}\;C^{2}_{\vartheta})\stackrel{{\scriptstyle(\delta^{2}_{r-2},\;\delta_{r,3}\;\delta^{2}_{\vartheta})}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\delta^{r-1}_{1}}}{{\longrightarrow}}C_{0}^{r}\longrightarrow
0$, with $\delta_{r-1}^{q+1}\circ\delta^{q}_{r}=0$,
$\delta^{2}_{\vartheta}\circ\delta^{1}_{2}=0$. Using the above chain complex
one is able to introduce a cohomology of $M/\mathcal{F}$. For $q$, $r\geq 0$
we define the $q$, $r$-th vertex operator algebra cohomology
$H^{q}_{r}=H^{q}_{r}(V,W,\mathcal{F})$ of $M/\mathcal{F}$ with coefficients in
$\overline{W}$ and depending on $mp$ vertex operators introduced on
$\mathcal{U}$ to be $H_{r}^{q}={\rm
Ker}\left\\{\begin{array}[]{c}\delta^{q}_{r}\\\
\delta_{q,1}\;\delta^{2}_{\vartheta}\end{array}/\mbox{\rm
Im}\;\delta^{q-1}_{r+1}\right.$, including $q$, $r=\infty$. Note that in
general $q\leq l$, $m\leq p$. Thus the cohomology $H^{q}_{r}$ describes
subdomains of lower dimensions on leaves of $M/\mathcal{F}$. Taking into
account the content of this and previous Sections we obtain the following

######  Lemma 2.

The vertex operator algebra cohomology of the holonomy groupoid
$Hol(M,\mathcal{F})$ is equivalent to the cohomology of section spaces for
$V$-twisted vertex algebra bundles $\mathcal{W}_{M/\mathcal{F}}$.

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