# Conformal blocks on smoothings
via mode transition algebras

Chiara Damiolini  Chiara Damiolini
Department of Mathematics, University of Texas at Austin, Austin, TX 78712
<EMAIL_ADDRESS>,  Angela Gibney  Angela Gibney
Department of Mathematics, University of Pennsylvania, Phil, PA 19104
<EMAIL_ADDRESS>and  Daniel Krashen  Daniel Krashen
Department of Mathematics, University of Pennsylvania, Phil, PA 19104
<EMAIL_ADDRESS>
###### Abstract.

Here we introduce a series of associative algebras attached to a vertex
operator algebra $V$, called mode transition algebras, and show they reflect
both algebraic properties of $V$ and geometric constructions on moduli of
curves. Pointed and coordinatized curves, labeled by modules over $V$, give
rise to sheaves of coinvariants. We show that if the mode transition algebras
admit multiplicative identities satisfying certain natural properties (called
strong identity elements), then these sheaves deform as wanted on families of
curves with nodes. This provides new contexts in which coherent sheaves of
coinvariants form vector bundles. We also show that mode transition algebras
carry information about higher level Zhu algebras and generalized Verma
modules. To illustrate, we explicitly describe the higher level Zhu algebras
of the Heisenberg vertex operator algebra, proving a conjecture of
Addabbo–Barron.

###### Key words and phrases:

Vertex algebras, factorization, sewing, conformal blocks, vector bundles on
moduli of curves, logarithmic conformal field theory

###### 2020 Mathematics Subject Classification:

14H10, 17B69 (primary), 81R10, 81T40,14D21 (secondary)

Mode transition algebras, introduced here, are a series of associative
algebras that give insight into algebraic structures on moduli of stable
pointed curves, and representations of the vertex operator algebras from which
they are derived.

Modules over vertex operator algebras (VOAs for short) give rise to vector
bundles of coinvariants on moduli of smooth, pointed, coordinatized curves
[FBZ04]. To extend these to singular curves, coinvariants must deform as
expected on smoothings of nodes, maintaining the same rank for singular curves
as for smooth ones. By 5.0.3, this holds when coinvariants form coherent
sheaves and the mode transition algebras (defined below) admit multiplicative
identities with certain properties. Consequently, by 5.2.6 one obtains a
potentially rich source of vector bundles, including as in Remark 3.4.6 (b),
the well-known class given by rational and $C_{2}$-cofinite VOAs [TUY89,
BFM91, NT05, DGT22a], and by 7.4.1, a new family on moduli of stable pointed
rational curves from modules over the Heisenberg VOA, which is neither
$C_{2}$-cofinite nor rational. Vector bundles are valuable—their
characteristic classes, degeneracy loci, and section rings have been
instrumental in the understanding of moduli of curves (e.g. [HM82, Mum83,
EH87, ELSV01, Far09, BCHM10]).

Known as essential to the study of the representation theory of VOAs, basic
questions about the structure of higher level Zhu algebras remain open. Via
6.0.1, the mode transition algebras also give a new perspective on these
higher level Zhu algebras. As an application, we prove [AB23a, Conjecture
8.1], thereby giving an explicit description of the higher level Zhu algebras
for the Heisenberg VOA. This is done in Section 7 by analyzing the mode
transition algebras associated to this VOA.

To describe our results more precisely, we set a small amount of notation,
with more details given below. We assume that $V$ is a vertex operator algebra
of CFT type. While they have applications in both VOA theory and algebraic
geometry, we begin by describing the geometric problem, which motivated the
definition of the mode transition algebras. By [DGK22], the sheaves of
coinvariants are coherent when defined by modules over a $C_{2}$-cofinite VOA.
By [DG23], coherence is also known to hold for some sheaves given by
representations of VOAs that are $C_{1}$-cofinite and not $C_{2}$-cofinite. It
is natural then to ask when such coherent sheaves are vector bundles, as they
were shown to be if $V$ is both $C_{2}$-cofinite and rational [TUY89, BFM91,
NT05, DGT22a].

One may check coherent sheaves are locally free by proving they are flat. This
may be achieved using Grothendieck’s valuative criteria of flatness. The
standard procedure first carried out by [TUY89, Theorem 6.2.1], and then
followed in [NT05, Theorem 8.4.5], and [DGT22a, VB Corollary]), is to argue
inductively, using the factorization property ([TUY89, Theorem 6.2.6], [NT05,
Theorem 8.4.3], [DGT22a, Theorem 7.0.1], [DGK22, Theorem 3.2]). However, here
factorization is not available, since we do not assume that Zhu’s algebra
$A(V)$ is finite and semi-simple ([DGK22, Proposition 7.1]).

Instead, as explained in the proof of 5.2.6, the geometric insight here, is
that in place of factorization, one may show that ranks of coinvariants are
constant as nodes are smoothed in families. This relies on the mode transition
algebras admitting multiplicative identities that also act as identity
elements on modules (we call these strong identity elements). The base case
then follows from the assumption of coherence and that sheaves of coinvariants
support a projectively flat connection on moduli of smooth curves [FBZ04,
DGT21]. We refer to this process as smoothing of coinvariants. Let us now
summarize the strategy.

For simplicity, let $\mathscr{C}_{0}$ be a projective curve over $\mathbb{C}$
with a single node $Q$, $n$ smooth points $P_{\bullet}=(P_{1},\ldots,P_{n})$,
and formal coordinates $t_{\bullet}=(t_{1},\ldots,t_{n})$ at $P_{\bullet}$.
Let $W^{1},\ldots,W^{n}$ be an $n$-tuple of $V$-modules, or equivalently of
smooth $\mathscr{U}$-modules, where $\mathscr{U}$ is the universal enveloping
algebra associated to $V$ (defined in detail in Section 2).

We assume each $V$-module $W^{i}$ is generated by a module $W^{i}_{0}$ over
(the zero level) Zhu algebra $\mathsf{A}_{0}(V):=\mathsf{A}$. The vector space
of coinvariants $[W^{\bullet}]_{(\mathscr{C}_{0},P_{\bullet},t_{\bullet})}$ is
the largest quotient of $W^{\bullet}=W^{1}\otimes\cdots\otimes W^{n}$ on which
the Chiral Lie algebra $\mathcal{L}_{\mathscr{C}_{0}\setminus P_{\bullet}}(V)$
acts trivially (described here in Section 4). Coinvariants at
$\mathscr{C}_{0}$ are related to those on the normalization
$\eta:\widetilde{\mathscr{C}}_{0}\rightarrow\mathscr{C}_{0}$ of
$\mathscr{C}_{0}$ at $Q$. Namely, by [DGT22a] the map

$\alpha_{0}\colon
W^{\bullet}_{0}{\overset{}{\rightarrow}}W^{\bullet}_{0}\otimes\mathsf{A},\qquad
u\mapsto u\otimes 1^{\mathsf{A}},$

gives rise to an $\mathcal{L}_{\mathscr{C}_{0}\setminus
P_{\bullet}}(V)$-module map, inducing a map between spaces of coinvariants

$[\alpha_{0}]\colon[W^{\bullet}]_{\left(\mathscr{C}_{0},P_{\bullet},t_{\bullet}\right)}{\overset{}{\rightarrow}}[W^{\bullet}\otimes\Phi(\mathsf{A})]_{\left(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}.$

Here $\Phi(A)$ is a $\mathscr{U}$-bimodule assigned at points $Q_{\pm}$ lying
over $\eta^{-1}(Q)$, and $s_{\pm}$ are formal coordinates at $Q_{\pm}$. By
[DGK22], the map $[\alpha_{0}]$ is an isomorphism if $V$ is $C_{1}$-cofinite.

One may extend the nodal curve $\mathscr{C}_{0}$ to a smoothing family
$(\mathscr{C},P_{\bullet},t_{\bullet})$ over the scheme
$S={\operatorname{\mathrm{Spec}}}(\mathbb{C}[\\![q]\\!])$, with special fiber
$(\mathscr{C}_{0},P_{\bullet},t_{\bullet})$, and smooth generic fiber, while
one may trivially extend $\widetilde{\mathscr{C}}_{0}$ to a family
$(\widetilde{\mathscr{C}},P_{\bullet}\sqcup Q_{\pm},t_{\bullet}\sqcup
s_{\pm})$ over $S$. While the central fibers of these two families of curves
are related by normalization, there is no map between
$\widetilde{\mathscr{C}}$ and $\mathscr{C}$. However, for $V$ rational and
$C_{2}$-cofinite, sheaves of coinvariants on $S$ are naturally isomorphic, an
essential ingredient in the proof that such sheaves are locally free under
these assumptions [DGT22a].

To obtain an analogous isomorphism of coinvariants under less restrictive
conditions, our main idea is to generalize the algebra structure of
$X_{d}\otimes X_{d}^{\vee}\subset\Phi(\mathsf{A})$, which exists for simple
admissible $V$-modules $X=\bigoplus_{d}X_{d}$, and for all $d\in\mathbb{N}$
(see Remark 3.4.6). Namely, we show that $\Phi(\mathsf{A})$ has the structure
of a bi-graded algebra, which we call the mode transition algebra and denote
$\mathfrak{A}=\bigoplus_{d_{1},d_{2}\in\mathbb{Z}}\mathfrak{A}_{d_{1},d_{2}}$.
We show that $\mathfrak{A}$ acts on generalized Verma modules
$\Phi^{\mathsf{L}}(W_{0})=\bigoplus_{d}W_{d}$, such that the subalgebras
$\mathfrak{A}_{d}:=\mathfrak{A}_{d,-d}$, which we refer to as the $d$th mode
transition algebras, act on the degree $d$ components $W_{d}$ (these terms are
defined in Section 3.1 and Section 3.2).

We say that $V$ satisfies smoothing (5.0.1), if for every pair
$(W^{\bullet},\mathscr{C}_{0})$, consisting of $n$ admissible $V$-modules
$W^{\bullet}$, not all trivial, a stable $n$-pointed curve $\mathscr{C}_{0}$
with a node, there exist an element $\mathscr{I}=\sum_{d\geq
0}\mathscr{I}_{d}q^{d}\in\mathfrak{A}[\\![q]\\!]$, such that the map

$\alpha\colon
W^{\bullet}[\\![q]\\!]\longrightarrow(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!],\qquad
u\mapsto\,u\otimes\mathscr{I},$

is an $\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$-module homomorphism
which extends $\alpha_{0}$.

In 5.0.3, we equate smoothing for $V$ with a property of multiplicative
identity elements in the $d$th mode transition algebras $\mathfrak{A}_{d}$,
when they exist. Specifically, if $\mathfrak{A}_{d}$ admit identity elements
$\mathscr{I}_{d}\in\mathfrak{A}_{d}$ for all $d\in\mathbb{N}$, satisfying any
of the equivalent properties of 3.3.1, then we say that
$\mathscr{I}_{d}\in\mathfrak{A}_{d}$ is a strong identity element.

######  Theorem (5.0.3).

Let $V$ be a VOA of CFT-type. The algebras
$\mathfrak{A}_{d}=\mathfrak{A}_{d}(V)$ admit strong identity elements for all
$d\in\mathbb{N}$ if and only if $V$ satisfies smoothing.

We remark that the analogue to $\alpha$ is called the sewing map in [TUY89,
NT05, DGT22a]. As an application of 5.0.3, we obtain geometric consequences
stated as 5.2.1 and 5.2.6. A particular case of which is as follows:

######  Corollary (5.2.6).

If $V$ is $C_{2}$-cofinite and satisfies smoothing, then
$\mathbb{V}(V;W^{\bullet})$ is a vector bundle on
${\overline{\mathcal{M}}_{g,n}}$ for simple $V$-modules $W^{1},\dotsm,W^{n}$.

By Remark 3.4.6, rational VOAs satisfy smoothing, so 5.2.6 specializes to
[DGT22a, VB Corollary]. As is shown in 7.4.1, one can apply the full statement
of 5.2.6 to show that modules over VOAs derived from Heisenberg lie algebras
of dimension one (which are $C_{1}$-cofinite, but neither $C_{2}$-cofinite nor
rational), define vector bundles on moduli of stable pointed rational curves
(see 7.4.1).

6.0.1, described next, gives further tools for investigating other VOAs which
may or may not satisfy smoothing by providing information about the
relationship between mode transition algebras and higher level Zhu algebras
and their representations.

Recall that in [Zhu96] Zhu defines a two step induction functor, which in the
first part takes $\mathsf{A}=\mathsf{A}(V)$-modules to a $V$-modules through a
Verma module construction, and then in the second step takes a quotient. In
LABEL:{def:PhiL} we describe this first step with a different, although
naturally isomorphic functor $\Phi^{\mathsf{L}}$, a crucial ingredient to this
work. Through this functor (naturally isomorphic to) $\Phi^{\mathsf{L}}$, Zhu
shows that there is a bijection between simple $\mathsf{A}$-modules and simple
$V$-modules, so that if $\mathsf{A}$ is finite dimensional, and semi-simple,
$\Phi^{\mathsf{L}}$ describes the category of admissible $V$-modules. However,
if $\mathsf{A}$ is either not finite-dimensional or is not semi-simple, then
there are indecomposable, but non-simple $V$-modules not induced from simple
indecomposable modules over $\mathsf{A}$ via $\Phi^{\mathsf{L}}$. To describe
such modules, [DLM98] defined the higher level Zhu algebras $\mathsf{A}_{d}$
for $d\in\mathbb{N}$ , further studied in [BVWY19a].

The mode transition algebras $\mathfrak{A}_{d}$ are related to the higher
level Zhu algebras $\mathsf{A}_{d}$. For instance,
$\mathfrak{A}_{0}=\mathsf{A}_{0}=\mathsf{A}$ (Remark 3.2.4), and by B.3.1,
there is an exact sequence

(1)
$\textstyle{\mathfrak{A}_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{d}}$$\textstyle{\mathsf{A}_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{d}}$$\textstyle{\mathsf{A}_{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

When $\mathfrak{A}_{d}$ admits an identity element (and not necessarily a
strong identity element), $\mu_{d}$ injective and the sequence splits by Part
(a) of 6.0.1. In particular, if $V$ is $C_{2}$-cofinite, as observed in
[Buh02, GN03, Miy04, He17], the $d$th Zhu algebras $\mathsf{A}_{d}$ are finite
dimensional, hence if $\mathfrak{A}_{d}$ has an identity element, it is finite
dimensional as well.

We note that (1) may be exact when $\mathfrak{A}_{d}$ does not admit an
identity element. For instance, in Section 8 we show exactness of (1) when
$d=1$ for the Virasoro VOA $\mathrm{Vir}_{c}$, for any values of $c$, and use
it to show that $\mathfrak{A}_{1}$ does not admit an identity element. In
particular, by 5.0.3, one finds that $\mathrm{Vir}_{c}$ never satisfies
smoothing. When $c$ is in the discrete series, the maximal simple quotient
$L_{c}$ of $\mathrm{Vir}_{c}$ is rational and $C_{2}$-cofinite, hence by
Remark 3.4.6 b (or by [DGT22a]) it will satisfy smoothing.

6.0.1 allows one to use the mode transition algebras to obtain other valuable
structural information about the higher level Zhu algebras.

######  Theorem (6.0.1).

  1. (a)

If the $d$th mode transition algebra $\mathfrak{A}_{d}$ admits an identity
element, then (1) is split exact, and
$\mathsf{A}_{d}\cong\mathfrak{A}_{d}\times\mathsf{A}_{d-1}$ as rings. In
particular, if $\mathfrak{A}_{d}$ admits an identity element for every
$d\in\mathbb{Z}_{\geq 0}$ then
$\mathsf{A}_{d}\cong\mathfrak{A}_{d}\oplus\mathfrak{A}_{d-1}\oplus\cdots\oplus\mathfrak{A}_{0}.$

  2. (b)

For $\mathfrak{A}=\mathfrak{A}(V)$, if $\mathfrak{A}_{d}$ admits a strong
identity for all $d\in\mathbb{N}$, so that smoothing holds for $V$, then given
any generalized Verma module
$W=\Phi^{\mathsf{L}}(W_{0})=\bigoplus_{d\in\mathbb{N}}W_{d}$ where $L_{0}$
acts on $W_{0}$ as a scalar with eigenvalue $c_{W}\in\mathbb{C}$, there is no
proper submodule $Z\subset W$ with $c_{Z}-c_{W}\in\mathbb{Z}_{>0}$ for every
eigenvalue $c_{Z}$ of $L_{0}$ on $Z$.

We refer to Section 3.1 for a discussion about generalized Verma modules.

We note that by B.3.1 and B.3.3, the exact sequence (1) as well as Part (a) of
6.0.1 hold for generalized higher Zhu algebras and generalized $d$th mode
transition algebras (see B.1.1 and B.2.6). For further discussion see Section
9.3.

We now describe some further consequences of 5.0.3 and 6.0.1.

In Section 7 we describe the $d$th mode transition algebras $\mathfrak{A}_{d}$
for the Heisenberg vertex algebra $M_{a}(1)$ defined by a one dimensional
Heisenberg Lie algebra (denoted $\pi$ in [FBZ04]), and show the
$\mathfrak{A}_{d}$ admit strong identity elements for all $d\in\mathbb{N}$. In
particular, 6.0.1 and 7.2.1 imply that the conjecture of Addabbo and Barron
[AB23a, Conj 8.1] holds, and one can write

(2)
$\mathsf{A}_{d}(\pi)=\mathsf{A}_{d}(M_{a}(1))\cong\prod_{j=0}^{d}\mathrm{Mat}_{p(j)}(\mathbb{C}[x]),$

where $p(j)$ is the number of ways to decompose $j$ into a sum of positive
integers, with $p(0)=1$. The level one Zhu algebra $A_{1}(M_{a}(1))$ was first
constructed in the paper [BVWY19b], and then later announced in [BVWY19a]. In
[AB23b] the authors determine $A_{2}(M_{a}(1))$ using the infrastructure for
finding generators and relations for higher level Zhu algebras they had
developed in [AB23a].

To illustrate their computability, we note that in [DGK24] the $d$-th mode
transition algebras for VOAs derived from Heisenberg Lie algebras of arbitrary
rank are shown to admit strong identities for all $d$, and further results
developed there allow us to compute higher Zhu algebras for these and other
examples.

In Section 9.1.2, we use Part (b) of 6.0.1 to show that the family of triplet
vertex operator algebras $\mathcal{W}(p)$ does not satisfy smoothing. We do
this by giving an explicit pair of modules $W\subset Z$ with
$W=\Phi^{\mathsf{L}}(W_{0})$ and such that $c_{Z}>c_{W}$. The actual pair of
modules used was suggested to us by Thomas Creutzig (with some details filled
in by Simon Wood). Dražen Adamović had also sketched for us the existence of
such an example. The importance of this example is that it establishes that
smoothing is not guaranteed to hold for a $C_{2}$-cofinite VOA if rationality
is not assumed. In particular, while sheaves of coinvariants defined by the
representations of $C_{2}$-cofinite VOAs are coherent, this can be seen as an
indication that they may not necessarily be locally free. Taken together with
the family of Heisenberg vector bundles from 7.4.1, this example illustrates
the subtlety of the problem of determining which sheaves of coinvariants
define vector bundles.

We also expect that smoothing will not hold for the family of symplectic
fermion algebras ${\mathrm{SF}^{+}_{d}}$ which are $C_{2}$-cofinite and not
rational, since ${\mathrm{SF}^{+}_{1}}=\mathcal{W}(2)$. It is natural to ask
whether there is an example of a vertex operator algebra that is
$C_{2}$-cofinite, is not rational, and satisfies smoothing (see Section 9.1).

Finally, we emphasize that our procedure to use smoothing to show that sheaves
of coinvariants are locally free is just one approach to this problem (see
Section 9.2).

### Plan of the paper

In Section 1, we set the terminology used here for vertex operator algebras
and their representations. In Section 2, we provide detailed descriptions of
the universal enveloping algebra $\mathscr{U}$ associated to a vertex operator
algebra $V$. Technical details are given in Appendix A, where an axiomatic
treatment of the constructions of the graded and filtered enveloping algebras
as topological or semi-normed algebras is given. The concepts discussed
involving filtered and graded completions can be found throughout the VOA
literature (for instance in [TUY89, FZ92, FBZ04, Fre07, NT05, MNT10]), but
little is said about how they relate to one another. We discuss these
relations in Section 2. In Section 3 we give an alternative construction of
the generalized Verma module functor $\Phi^{\mathsf{L}}$ (and the right-
analogue $\Phi^{\mathsf{R}}$) from the category of $\mathsf{A}$-modules, to
the category of smooth left (and right) $\mathscr{U}$-modules. We use a
combination of $\Phi^{\mathsf{L}}$ and $\Phi^{\mathsf{R}}$ to define the mode
transition algebras $\mathfrak{A}_{d}\subset\mathfrak{A}$. More general
versions of these constructions are defined in Appendix B, where their
analogous properties are proved. In Section 4, smoothing is formally defined,
and we describe sheaves of coinvariants on families of pointed and
coordinatized curves in general terms, and cite the relevant references. In
Section 5 we prove 5.0.3, 5.2.1, and 5.2.6. In Section 6 we prove 6.0.1 Part
(b), while Part (a) is detailed in Appendix B. In Section 7 we compute the
mode transition algebras $\mathfrak{A}_{d}$ for the Heisenberg algebra for all
$d$. In Section 8 we compute the 1st mode transition algebras for the non-
discrete series Virasoro VOAs. We ask a number of questions in Section 9. In
Section 9.1 and in Section 9.2 questions are discussed about $C_{2}$-cofinite
and non-rational VOAs that may not satisfy smoothing, and whether their
induced sheaves of coinvariants may still define vector bundles. Finally, as
noted, many of the results here are stated and proved for generalizations of
higher level Zhu algebras and of mode transition algebras, and in Section 9.3
we raise the question of finding other examples and applications of such
algebraic structures, beyond those naturally associated to a vertex operator
algebra.

### Acknowledgements

We would like to thank Dražen Adamović, Katrina Barron, Thomas Creutzig,
Haisheng Li, Jianqi Liu, Antun Milas, Rahul Pandharipande, Dennis Sullivan,
and Simon Wood for helpful discussions and for answering our questions. Thomas
Creutzig and Simon Wood gave crucial assistance with the details of Section
9.1.2. We thank Katrina Barron for her valuable feedback at various points.
Most of all, we thank Yi-Zhi Huang who encouraged us from the beginning to
consider questions regarding sewing, when rationality is not assumed. Gibney
was supported by NSF DMS – 2200862, and Krashen was supported by NSF
DMS–1902237.

##  1\. Background on VOAs and their modules

In Section 1.1 we state the conventions we follow for vertex operator algebras
and their representations. Throughout this paper, by an algebra we mean an
associative algebra which is not necessarily commutative and by a ring we mean
an algebra over $\mathbb{Z}$. We refer to [FZ92, Zhu96, BFM91, NT05] for more
details about vertex operator algebras and their modules.

###  1.1. VOAs and their representations

We recall here the definition of a vertex operator algebra of CFT type, which
in the paper will be simply denoted VOA.

######  Definition 1.1.1.

A vertex operator algebra of CFT-type is four-tuple
$(V,{\textbf{1}},\omega,Y(\cdot,z))$:

  1. (a)

$V=\bigoplus_{i\in\mathbb{N}}V_{i}$ is a vector space with $\dim
V_{i}<\infty$, and $\dim V_{0}=1$;

  2. (b)

1 is an element in $V_{0}$, called the vacuum vector;

  3. (c)

$\omega$ is an element in $V_{2}$, called the conformal vector;

  4. (d)

$Y(\cdot,z)\colon V\rightarrow\textrm{End}(V)[\\![z,z^{-1}]\\!]$ is a linear
map $a\mapsto Y(a,z):=\sum_{m\in\mathbb{Z}}a_{(m)}z^{-m-1}$. The series
$Y(a,z)$ is called the vertex operator assigned to $a\in V$,

satisfying the following axioms:

  1. (a)

(vertex operators are fields) for all $a$, $b\in V$, $a_{(m)}b=0$, for $m\gg
0$;

  2. (b)

(vertex operators of the vacuum) $Y({\bm{1}},z)=\mathrm{id}_{V}$, that is

${\bm{1}}_{(-1)}=\mathrm{id}_{V}\qquad\mbox{and}\qquad{\bm{1}}_{(m)}=0,\quad\mbox{for
$m\neq-1$,}$

and for all $a\in V$, $Y(a,z){\bm{1}}\in a+zV[\\![z]\\!]$, that is

$a_{(-1)}{\bm{1}}=a\qquad\mbox{and}\qquad a_{(m)}{\bm{1}}=0,\qquad\mbox{for
$m\geq 0$};$

  3. (c)

(weak commutativity) for all $a$, $b\in V$, there exists an $N\in\mathbb{N}$
such that

$(z_{1}-z_{2})^{N}\,[Y(a,z_{1}),Y(b,z_{2})]=0\quad\mbox{in
}\textrm{End}(V)[\\![z_{1}^{\pm 1},z_{2}^{\pm 1}]\\!];$

  4. (d)

(conformal structure) for
$Y(\omega,z)=\sum_{m\in\mathbb{Z}}\omega_{(m)}z^{-m-1}$,

$\left[\omega_{(p+1)},\omega_{(q+1)}\right]=(p-q)\,\omega_{(p+q+1)}+\frac{c}{12}\,\delta_{p+q,0}\,(p^{3}-p)\,\textrm{id}_{V}.$

Here $c\in\mathbb{C}$ is the central charge of $V$. Moreover:

$\omega_{(1)}|_{V_{m}}=m\cdot\textrm{id}_{V},\quad\text{for all
}m,\qquad\mbox{and}\qquad Y\left(\omega_{(0)}a,z\right)=\frac{d}{dz}Y(a,z).$

######  Definition 1.1.2.

An admissible $V$-module is an $\mathbb{C}$-vector space $W$ together with a
linear map

$Y^{W}(\cdot,z)\colon V\rightarrow\operatorname{End}(W)[\\![z,z^{-1}]\\!],\
a\in V\mapsto Y^{W}(a,z):=\sum_{m\in\mathbb{Z}}a^{W}_{(m)}z^{-m-1},$

which satisfies the following axioms:

  1. (a)

(vertex operators are fields) if $a\in V$ and $u\in W$, then $a^{W}_{(m)}u=0$,
for $m\gg 0$;

  2. (b)

(vertex operators of the vacuum)
$Y^{W}\left({\bm{1}},z\right)=\mathrm{id}_{W}$;

  3. (c)

(weak commutativity) for all $a$, $b\in V$, there exists an $N\in\mathbb{N}$
such that for all $u\in W$

$(z_{1}-z_{2})^{N}\left[Y^{W}(a,z_{1}),Y^{W}(b,z_{2})\right]u=0;$

  4. (d)

(weak associativity) for all $a\in V$ and $u\in W$, there exists an
$N\in\mathbb{N}$, such that for all $b\in V$, one has

$(z_{1}+z_{2})^{N}\left(Y^{W}(Y(a,z_{1})b,z_{2})-Y^{W}(a,z_{1}+z_{2})Y^{W}(b,z_{2})\right)u=0;$

  5. (e)

(conformal structure) for
$Y^{W}(\omega,z)=\sum_{m\in\mathbb{Z}}\omega_{(m)}^{W}z^{-m-1}$, one has

$\left[\omega^{W}_{(p+1)},\omega^{W}_{(q+1)}\right]=(p-q)\,\omega^{W}_{(p+q+1)}+\frac{c}{12}\,\delta_{p+q,0}\,(p^{3}-p)\,\textrm{id}_{W}.$

where $c\in\mathbb{C}$ is the central charge of $V$. Moreover
$Y^{W}\left(L_{-1}a,z\right)=\frac{d}{dz}Y^{W}(a,z)$;

  6. (f)

($\mathbb{N}$-gradability) $W$ admits a grading
$W=\bigoplus\limits_{n\in\mathbb{N}}W_{n}$ with $a^{W}_{(m)}W_{n}\subset
W_{n+\deg(a)-m-1}$.

As one can see in the literature, e.g. by [DL93, FHL93, LL04, Li96], weak
associativity and weak commutativity together are equivalent to the Jacobi
identity: for $\ell$, $m$, $n\in\mathbb{Z}$, and $a$, $b\in V$

$\sum_{i\geq
0}(-1)^{i}{\binom{\ell}{i}}\left(a^{W}_{(m+\ell-i)}b^{W}_{(n+i)}-(-1)^{\ell}b^{W}_{(n+\ell-i)}a^{W}_{(m+i)}\right)=\sum_{i\geq
0}{\binom{m}{i}}(a_{(\ell+i)}(b))^{W}_{(m+n-i)}.$

Moreover, by [DLM97, Lemma 2.2], axiom (e) is redundant.

##  2\. The universal enveloping algebra of a VOA

Here we describe constructions of the universal enveloping algebra associated
to a VOA $V$, as quotients of certain graded, as well as (left and right)
filtered completions of the universal enveloping algebra of the Lie algebra
associated to $V$. Filtered completions are essential to our constructions, as
they are compatible with crucial restriction maps from the Chiral Lie algebra
to certain ancillary Lie algebras, allowing for the definition of the action
of the Chiral Lie algebra on (tensor products of) $V$-modules. The graded
completion, on the other hand, allows both for ease in computation, and
simpler descriptions of induced modules, and bimodules, and in Section 3.2 of
the mode transition algebras. While these concepts are treated in one way or
another throughout the VOA literature, for instance in [FZ92, FBZ04, Fre07,
NT05, MNT10], we provide here and in Appendix A, a uniform description, where
many details are given, clarifications are made, and the different
constructions are compared to one another. We further remark that, although
this section assumes that $V$ is a vertex operator algebra, however all the
arguments and construction here in Section 2 hold assuming only that $V$ is a
graded vertex algebra since the conformal structure does not play a role (see
also Section 9.3).

###  2.1. Graded and filtered completions

We recall the constructions of the universal enveloping algebra [FZ92] and the
current algebra [NT05] associated to a VOA $V$.

###  2.2. Split filtrations

The underlying vector spaces of the objects we will need to consider will
either be graded or filtered (sometimes both), and these filtrations and
gradings will be related to each other. The basic example of this is the space
of Laurent polynomials $\mathbb{C}[t,t^{-1}]$ which we choose to grade by
$\mathbb{C}[t,t^{-1}]_{n}=\mathbb{C}t^{-n-1}$, and the space of Laurent series
$\mathbb{C}(\\!(t)\\!)$, which admits an increasing filtration by setting
$\mathbb{C}(\\!(t)\\!)_{\leq n}:=t^{-n-1}\mathbb{C}[\\![t]\\!]$. We will refer
to this filtration as a left filtration of $\mathbb{C}(\\!(t)\\!)$ (see
A.1.1). In this situation, when we have a graded subspace of a filtered space
$\mathbb{C}[t,t^{-1}]\subset\mathbb{C}(\\!(t)\\!)$ which identifies the degree
$n$ part of the graded subspace with the degree $n$ part of the associated
graded space, we say that this pair gives a split filtration (see A.1.6).
Similarly, we refer to the filtration of $\mathbb{C}(\\!(t^{-1})\\!)$ given by
$\mathbb{C}(\\!(t^{-1})\\!)_{\geq n}=t^{-n-1}\mathbb{C}[\\![t^{-1}]\\!]$ as a
right filtration, and the pair
$\mathbb{C}[t,t^{-1}]\subset\mathbb{C}(\\!(t^{-1})\\!)$ is also a split
filtration.

###  2.3. Lie algebras

Let $V$ be a VOA (or a graded vertex algebra). Since it is a graded vector
space, it admits a trivial left (respectively right) split-filtration, given
by $V_{\leq n}=\bigoplus_{d\leq n}V_{d}$ (respectively $V_{\geq
n}=\bigoplus_{d\geq n}V_{d}$). In view of A.3.3, tensor products of split-
filtered modules are naturally split-filtered, and consequently

$V\otimes_{\mathbb{C}}\mathbb{C}[t,t^{-1}]\subset
V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t)\\!)\quad\text{ and }\quad
V\otimes_{\mathbb{C}}\mathbb{C}[t,t^{-1}]\subset
V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t^{-1})\\!)$

define splittings of their left and right filtrations.

######  Remark 2.3.1.

Concretely, we can define the map $\mathsf{val}\colon
V\otimes\mathbb{C}(\\!(t)\\!)\to\mathbb{Z}$ by

$\mathsf{val}(a\otimes f(t))=\deg(a)-N-1.$

for a homogeneous element $a\in V$ and $f(t)\in
t^{N}\mathbb{C}[\\![t]\\!]\setminus t^{N-1}\mathbb{C}[\\![t]\\!]$. The natural
left filtration on $V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t)\\!)$ is then given
by $(V\otimes\mathbb{C}(\\!(t)\\!))_{\leq n}:=\mathsf{val}^{-1}(-\infty,n]$.

The linear map
$\nabla=L_{-1}\otimes\mathrm{id}+\mathrm{id}\otimes\frac{d}{dt}$ is a linear
endomorphism of each of these spaces of degree $1$ (see A.1.10). We define

${\mathfrak{L}}(V)^{\mathsf{L}}=\dfrac{V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t)\\!)}{\mathrm{Im}(\nabla)},\quad{\mathfrak{L}}(V)^{\mathsf{R}}=\dfrac{V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t^{-1})\\!)}{\mathrm{Im}(\nabla)},\
\mbox{ and }\
{\mathfrak{L}}(V)^{\mathsf{f}}=\dfrac{V\otimes_{\mathbb{C}}\mathbb{C}[t,t^{-1}]}{\mathrm{Im}(\nabla)}.$

These have induced split-filtrations via
${\mathfrak{L}}(V)^{\mathsf{f}}\subset{\mathfrak{L}}(V)^{\mathsf{L}},{\mathfrak{L}}(V)^{\mathsf{R}}$
by A.1.14. These filtered and graded vector spaces admit (filtered and graded)
Lie algebra structures, with Lie brackets defined by:

$\left[a\otimes f(t),b\otimes g(t)\right]:=\sum_{k\geq
0}\frac{1}{k!}\left(a_{(k)}(b)\right)\otimes g(t)\dfrac{d^{k}(f(t))}{dt^{k}},$

for all $a,b\in V$ and $f(t),g(t)\in\mathbb{C}(\\!(t)\\!)$ or
$f(t),g(t)\in\mathbb{C}(\\!(t^{-1})\\!)$.

More concretely in the case of ${\mathfrak{L}}(V)^{\mathsf{f}}$, for $a\in V$
and $i\in\mathbb{Z}$ we denote by $a_{[i]}$ the class of the element $a\otimes
t^{i}$ in
${\mathfrak{L}}(V)^{\mathsf{f}}\subset{\mathfrak{L}}(V)^{\mathsf{L}},{\mathfrak{L}}(V)^{\mathsf{R}}$.
The restriction of the Lie bracket on ${\mathfrak{L}}(V)^{\mathsf{f}}$ then is
given by the following formula

$\left[a_{[i]},b_{[j]}\right]:=\sum_{k\geq
0}\binom{i}{k}\left(a_{(k)}(b)\right)_{[i+j-k]},$

for all $a,b\in V$ and $i,j\in\mathbb{Z}$. Extending the notation introduced
in [DGT22a], we call ${\mathfrak{L}}(V)^{\mathsf{L}}$ and
${\mathfrak{L}}(V)^{\mathsf{R}}$ the left and right ancillary Lie algebras and
${\mathfrak{L}}(V)^{\mathsf{f}}$ the finite ancillary Lie algebra. Note that
${\mathfrak{L}}(V)^{\mathsf{L}}$ is isomorphic to the current Lie algebra
$\mathfrak{g}(V)$ from [NT05].

###  2.4. Enveloping algebras

We now let $\mathit{U}^{\mathsf{L}},\mathit{U}^{\mathsf{R}},\mathit{U}$ be the
universal enveloping algebras of
${\mathfrak{L}}(V)^{\mathsf{L}},{\mathfrak{L}}(V)^{\mathsf{R}},{\mathfrak{L}}(V)^{\mathsf{f}}$
respectively. These enveloping algebras are left filtered, right filtered, and
graded respectively, and we note that
$\mathit{U}\subset\mathit{U}^{\mathsf{L}},\mathit{U}^{\mathsf{R}}$ again give
splittings to the respective filtrations (see A.3.6). In the language of A.9.1
we will say that
$(\mathit{U}^{\mathsf{L}},\mathit{U},\mathit{U}^{\mathsf{R}})$ forms a good
triple of associative algebras.

######  Example 2.4.1.

These induced filtrations can be explicitly described. For instance we have
that $\mathit{U}_{d}$ is linearly spanned by elements $\ell^{1}\cdots\ell^{k}$
such that $\ell^{i}\in{\mathfrak{L}}(V)^{\mathsf{f}}_{d_{i}}$ and
$\sum_{i}d_{i}=d$ (and $k$ possibly zero if $d=0$). Analogously
$(\mathit{U}^{\mathsf{L}})_{\leq d}$ is linearly spanned by elements
$\ell^{1}\cdots\ell^{k}$ such that $\ell^{i}\in{\mathfrak{L}}(V)_{\leq d_{i}}$
and $\sum_{i}d_{i}=d$ (and $k$ possibly zero if $d\geq 0$).

These enveloping algebras have an additional structure of a topology induced
by seminorms, which can be described in terms of systems of neighborhoods of
$0$ (as in A.4.1 and Remark A.4.4). These neighborhoods of the identity are
given by left ideals
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}$ of
$\mathit{U}^{\mathsf{L}}$ and $\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}$
of $\mathit{U}$ and right ideals
$\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}^{\mathsf{R}}$ of
$\mathit{U}^{\mathsf{R}}$ and $\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}$
of $\mathit{U}$ defined by:

$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}=\mathit{U}^{\mathsf{L}}\mathit{U}^{\mathsf{L}}_{\leq-n},\quad\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}=\mathit{U}\mathit{U}_{\leq-n},\quad\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}^{\mathsf{R}}=\mathit{U}^{\mathsf{R}}_{\geq
n}\mathit{U}^{\mathsf{R}},\quad\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}=\mathit{U}_{\geq
n}\mathit{U}.$

This definition coincides with that of a canonical seminorm on a
(split-)filtered algebra, as described in A.6.1, and in particular gives a
good seminorm on the triple (see A.9.3 and Remark A.9.7). Most useful for us
is that the category of good triples of algebras with good seminorms is closed
under quotients and completions (A.9.9 and A.9.10).

These neighborhoods can also be described alternately as follows:

######  Lemma 2.4.2.

One has an identification of left ideals
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}=\mathit{U}\mathit{U}_{\leq-n}=\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$.
Similarly, one has
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}=\mathit{U}^{\mathsf{L}}{\mathfrak{L}}(V)^{\mathsf{L}}_{\leq-n}$,
$\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}^{\mathsf{R}}={\mathfrak{L}}(V)^{\mathsf{R}}_{\geq
n}\mathit{U}^{\mathsf{R}}$ and
$\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}={\mathfrak{L}}(V)^{\mathsf{f}}_{\geq
n}\mathit{U}$.

We note that in 2.4.2, we identify ${\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$
with its image in $\mathit{U}$.

###### Proof.

We will prove only the first equality, the others following by similar
methods. Clearly $\mathit{U}\mathit{U}_{\leq-n}$ contains
$\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$, so we need only prove the
reverse inclusion, which amounts to showing
$\mathit{U}_{-n}\subset\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$. For
this, suppose we have an element $\alpha=a^{1}_{n_{1}}a^{2}_{n_{2}}\cdots
a^{r}_{n_{r}}\in\mathit{U}_{-n}$. By inductively using [He17, Lemma A.2.1],
for $M>>0$, we may write $\alpha=a^{1}_{n_{1}}a^{2}_{n_{2}}\cdots
a^{r}_{n_{r}}=\beta+\gamma$, with
$\beta\in{\mathfrak{L}}(V)^{\mathsf{f}}_{-n}$ and
$\gamma\in\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-M}$. In particular,
choosing $M\geq n$, we find
$\gamma\in\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$, showing
$\alpha\in\mathit{U}{\mathfrak{L}}(V)^{\mathsf{f}}_{\leq-n}$ as desired. ∎

We can restrict these seminorms to various filtered and graded parts of these
algebras as in A.4.6. For example, we write
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}_{\leq p}$ to
denote
$\operatorname{N}\\!_{\mathsf{L}}^{n}(\mathit{U}^{\mathsf{L}})\cap\mathit{U}^{\mathsf{L}}_{\leq
p}$. Concretely we obtain systems of neighborhoods as follows (see A.3.4 for
more details):

$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}_{\leq
p}=(\mathit{U}^{\mathsf{L}}\mathit{U}^{\mathsf{L}}_{\leq-n})_{\leq
p}=\sum_{j\leq-n}\mathit{U}^{\mathsf{L}}_{\leq
p-j}\mathit{U}^{\mathsf{L}}_{\leq
j},\qquad\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}^{\mathsf{R}}_{\geq
p}=(\mathit{U}^{\mathsf{R}}_{\geq n}\mathit{U}^{\mathsf{R}})_{\geq
p}=\sum_{i\geq n}\mathit{U}^{\mathsf{R}}_{\geq i}\mathit{U}^{\mathsf{R}}_{\geq
p-i}$
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}_{p}=(\mathit{U}\mathit{U}_{\leq-n})_{p}=\sum_{j\leq-n}\mathit{U}_{p-j}\mathit{U}_{j},\
\mbox{ and
}\qquad\operatorname{N}\\!_{\mathsf{R}}^{n}\mathit{U}_{p}=(\mathit{U}_{\geq
n}\mathit{U}^{\mathsf{R}})_{p}=\sum_{i\geq n}\mathit{U}_{i}\mathit{U}_{p-i}.$

We note that in particular, we have
$\operatorname{N}\\!_{\mathsf{R}}^{n+p}\mathit{U}_{p}=\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}_{p}$.
Through the restriction of the seminorm to these subspaces, we then define a
filtered completions of $\mathit{U}^{\mathsf{L}}$ and
$\mathit{U}^{\mathsf{R}}$, both containing a graded completion of $\mathit{U}$
(see A.5.6). Specifically, we set

$\widehat{\mathit{U}}^{\mathsf{L}}_{d}:=\varprojlim_{n}\dfrac{\mathit{U}^{\mathsf{L}}_{\leq
d}}{\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{L}}_{\leq
d}},\qquad\widehat{\mathit{U}}^{\mathsf{R}}_{d}:=\varprojlim_{n}\dfrac{\mathit{U}^{\mathsf{R}}_{\geq
d}}{\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}^{\mathsf{R}}_{\geq
d}},\qquad\widehat{\mathit{U}}_{d}:=\varprojlim_{n}\dfrac{\mathit{U}_{d}}{\operatorname{N}\\!_{\mathsf{L}}^{n}\mathit{U}_{d}}=\varprojlim_{n}\dfrac{\mathit{U}_{d}}{\operatorname{N}\\!_{\mathsf{R}}^{n+d}\mathit{U}_{d}},$

And set

$\widehat{\mathit{U}}^{\mathsf{L}}:=\bigcup_{d}\widehat{\mathit{U}}^{\mathsf{L}}_{d},\qquad\widehat{\mathit{U}}^{\mathsf{R}}:=\bigcup_{d}\widehat{\mathit{U}}^{\mathsf{R}}_{d},\qquad\widehat{\mathit{U}}:=\bigoplus_{d}\widehat{\mathit{U}}_{d}.$

As previously mentioned, it follows from A.9.10 that this will result in a
good triple
$(\widehat{\mathit{U}}^{\mathsf{L}},\widehat{\mathit{U}},\widehat{\mathit{U}}^{\mathsf{R}})$
of associative algebras with good seminorms.

Finally, one may construct a graded ideal $J$ of $\widehat{\mathit{U}}$
generated by the Jacobi relations, namely $J$ is generated by for $\ell$, $m$,
$n\in\mathbb{Z}$, and $a$, $b\in V$, by

$\sum_{i\geq
0}(-1)^{i}{\binom{\ell}{i}}\left(a_{[m+\ell-i]}b_{[n+i]}-(-1)^{\ell}b_{[n+\ell-i]}a_{[m+i]}\right)=\sum_{i\geq
0}{\binom{m}{i}}(a_{(\ell+i)}(b))_{[m+n-i]}.$

If we let $J^{\mathsf{R}}$ and $J^{\mathsf{L}}$ be the ideals of
$\widehat{\mathit{U}}^{\mathsf{R}}$ and $\widehat{\mathit{U}}^{\mathsf{L}}$
generated by $J$, and we let
$\overline{J},\overline{J}^{\mathsf{R}},\overline{J}^{\mathsf{L}}$ be the
respective closures (see A.5.10), then we find that
$(\overline{J}^{\mathsf{L}},\overline{J},\overline{J}^{\mathsf{R}})$ form a
good triple (of nonunital algebras) by A.9.5 and A.9.6. Finally, by A.9.9, we
find that the resulting quotient algebras

$\mathscr{U}^{\mathsf{L}}=\widehat{\mathit{U}}^{\mathsf{L}}/\overline{J}^{\mathsf{L}},\quad\mathscr{U}=\widehat{\mathit{U}}/\overline{J},\quad\mathscr{U}^{\mathsf{R}}=\widehat{\mathit{U}}^{\mathsf{R}}/\overline{J}^{\mathsf{R}},$

form a good triple of associative algebras with good seminorms (actually
norms).

######  Definition 2.4.3.

We call $\mathscr{U}^{\mathsf{L}},\mathscr{U}^{\mathsf{R}},\mathscr{U}$ the
left, right and finite universal enveloping algebras of $V$, respectively.

We note that a $V$-module corresponds, in this language, to a
$\mathscr{U}$-module $W$ (or a $\mathscr{U}^{\mathsf{L}}$-module) such that
the action of this normed (and hence topological) algebra is continuous. That
is, such that the multiplication map

$\mathscr{U}\times W\to W\quad\text{ or equivalently,
}\quad\mathscr{U}^{\mathsf{L}}\times W\to W$

is continuous, where $W$ is given the discrete topology, and $\mathscr{U}$ and
$\mathscr{U}^{\mathsf{L}}$ are topologized according to their norms.

####  2.4.4. Relation to the literature

We note that $\mathscr{U}$ coincides with the universal enveloping algebra of
$V$ introduced in [FZ92], while we can identify $\mathscr{U}^{\mathsf{L}}$
with the current algebra introduced in [NT05] or with the universal enveloping
algebra $\widetilde{U}(V)$ introduced in [FBZ04], with a minor modification
(see [Fre07, footnote on p.74]).

###  2.5. Subalgebras and subquotient algebras

We describe here some algebras built from the algebras
$\mathscr{U},\mathscr{U}^{\mathsf{L}},\mathscr{U}^{\mathsf{R}}$ which will
play a special role. By definition,
$\mathscr{U}^{\mathsf{L}}_{\leq-n}\triangleleft\mathscr{U}^{\mathsf{L}}_{\leq
0}$ and $\mathscr{U}^{\mathsf{R}}_{\geq
n}\triangleleft\mathscr{U}^{\mathsf{R}}_{\geq 0}$ are two-sided ideals when
$n>0$ with

$\mathscr{U}^{\mathsf{L}}_{\leq
0}/\mathscr{U}^{\mathsf{L}}_{\leq-1}\cong\mathscr{U}_{0}\cong\mathscr{U}^{\mathsf{R}}_{\geq
0}/\mathscr{U}^{\mathsf{R}}_{\geq 1},$

by the fact that these algebras are part of a good triple. We now look more
closely at $\mathscr{U}_{0}$, which forms a subring of $\mathscr{U}$. As our
triple of algebras is good, the seminorms on our algebras are almost canonical
(A.6.8), and in particular by A.6.8(c), it follows that
$\operatorname{N}\\!_{\mathsf{L}}^{n}\mathscr{U}_{0}=\operatorname{N}\\!_{\mathsf{R}}^{n}\mathscr{U}_{0}$
for every $n$, so that there is no ambiguity in denoting these neighborhoods
by $\operatorname{N}\\!^{r}\mathscr{U}_{0}$. We also see in the same way that
$\operatorname{N}\\!^{r}\mathscr{U}_{0}$ is a two-sided ideal of
$\mathscr{U}_{0}$.

######  Definition 2.5.1.

The $d$th higher level Zhu algebra of $V$ is the quotient

$\mathsf{A}_{d}(V)=\mathscr{U}_{0}/\operatorname{N}\\!^{d+1}\mathscr{U}_{0}.$

For an element $\alpha\in\mathscr{U}_{0}$, we will write $[\alpha]_{d}$ for
its image in $\mathsf{A}_{d}(V)$. When $V$ is understood, we will denote
$\mathsf{A}_{d}(V)$ simply by $\mathsf{A}_{d}$.

####  2.5.2. Relation to the literature

In [Zhu96] the author defines an associative algebra, now referred to as the
(zeroth) Zhu algebra as the quotient of $V$ by an appropriate subspace $O(V)$.
In [FZ92, NT05] it is shown that this algebra is isomorphic to an appropriate
quotient of the degree zero piece of the universal enveloping algebra of $V$
(or of the current algebra of $V$). As mentioned in the introduction, higher
level Zhu algebras $\mathsf{A}_{d}$ have been introduced in [DLM98] as
quotients of $V$ by subspaces $O_{d}(V)$, and proved to be realized as
quotients of the degree zero piece of the universal enveloping algebra of $V$
in [He17]. We notice further that the map that realizes the isomorphism
between $V/O_{d}(V)$ and $\mathsf{A}_{d}$ is explicitly realized by
identifying $[a]\in V/O_{d}(V)$ with the class of the element $a_{\deg(a)-1}$
in $\mathit{U}_{0}/\operatorname{N}\\!^{d+1}\mathit{U}_{0}$ for every
homogeneous element $a\in V$.

##  3\. Induced modules and the mode transition algebra

The constructions and results discussed here are true in greater generality,
as detailed in Section B.1 and in Section B.2. For instance, as in Section 2,
the constructions mentioned in Section 3.1 and in Section 3.2 hold for a
graded vertex algebra. The conformal structure is however used in Section 3.4.

###  3.1. Induced modules

As is the convention, throughout we denote $\mathsf{A}_{0}$ by $\mathsf{A}$.

######  Definition 3.1.1.

For a left module $W_{0}$ over $\mathsf{A}$, we define the left generalized
Verma module $\Phi^{\mathsf{L}}(W_{0})$ as

(3)
$\Phi^{\mathsf{L}}(W_{0})=\bigoplus_{p=0}^{\infty}\Phi^{\mathsf{L}}(W_{0})_{p}=\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}\right)\otimes_{\mathscr{U}_{0}}W_{0}\cong\left(\mathscr{U}^{\mathsf{L}}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}^{\mathsf{L}}\right)\otimes_{\mathscr{U}_{0}}W_{0}.$

For $Z_{0}$ a right module over $\mathsf{A}$, we define the right generalized
Verma module $\Phi^{\mathsf{L}}(W_{0})$ as

(4)
$\Phi^{\mathsf{R}}(Z_{0})=\bigoplus_{p=0}^{\infty}\Phi^{\mathsf{L}}(Z_{0})_{-p}=Z_{0}\otimes_{\mathscr{U}_{0}}\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}\right)\cong
Z_{0}\otimes_{\mathscr{U}_{0}}\left(\mathscr{U}^{\mathsf{R}}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}^{\mathsf{R}}\right).$

We note that this is well defined. In fact, the claimed isomorphisms of (3)
and (4) follow from A.8.1, while the grading is explained in Remark B.1.6.

Moreover, from A.8.1 we have that
$\mathscr{U}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}\cong\mathscr{U}^{\mathsf{L}}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}^{\mathsf{L}}$,
so that this quotient can be regarded both as a
$(\mathscr{U},\mathscr{U}_{0})$ bimodule and as a
$(\mathscr{U}^{\mathsf{L}},\mathscr{U}_{0})$ bimodule. In particular, this
shows that the ancillary algebra acts on the left on
$\Phi^{\mathsf{L}}(W_{0})$.

The functors $\Phi^{\mathsf{L}}_{d}$ have a universal property, described in
3.1.2, and proved more generally in B.1.4. Namely, given a continuous left
$\mathscr{U}$-module $W$, we define an $\mathsf{A}_{d}$-module $\Omega_{d}(W)$
by

$\Omega_{d}(W)={\mathrm{rAnn}}_{W}(\operatorname{N}\\!_{\mathsf{L}}^{d+1}U)=\\{w\in
W\mid(\operatorname{N}\\!_{\mathsf{L}}^{d+1}U)w=0\\},$

where by 2.4.2, one has that the right annihilator
${\mathrm{rAnn}}_{W}(\operatorname{N}\\!_{\mathsf{L}}^{d+1}U)$ coincides with
${\mathrm{rAnn}}_{W}({\mathfrak{L}}(V)^{\mathsf{L}}_{\leq-d-1})$. It therefore
follows that the definition of $\Omega_{d}(W)$ agrees with

$\Omega_{d}(W)=\\{w\in W\mid{\mathfrak{L}}(V)^{\mathsf{L}}_{-d-1}w=0\\},$

as originally given in [DLM98].

######  Proposition 3.1.2.

Let $M$ be a $\mathscr{U}$-module and $W_{0}$ an $\mathsf{A}_{d}$-module. Then
there is a natural isomorphism of bifunctors:

$\mathrm{Hom}_{\mathsf{A}_{d}}(W_{0},\Omega_{d}(M))=\mathrm{Hom}_{\mathscr{U}}(\Phi^{\mathsf{L}}_{d}(W_{0}),M).$

######  Remark 3.1.3.

If $W_{0}$ if finite dimensional over $\mathbb{C}$, and if $V$ is
$C_{1}$-cofinite, then there are a finite number of elements $x^{1}$, $x^{2}$,
$\ldots$, $x^{r}\in V$ such that $\Phi^{\mathsf{L}}(W_{0})$ is spanned by
elements of the form

$x^{1}_{(-m_{1})}\cdot x^{2}_{(-m_{2})}\cdots x^{r}_{(-m_{r})}\otimes u,$

for some $u\in W_{0}$ and positive integers $m_{1}\geq m_{2}\geq\cdots\geq
m_{r}\geq 1$.

####  3.1.4. Relation to the literature

By the universal property in 3.1.2, the functor $\Phi^{\mathsf{L}}$ is
naturally isomorphic the functor denoted $\overline{M}_{0}$ in [Zhu96, page
258], [DLM98, (4.4)], and [BVWY19a, page 3301]. Moreover, one can relate
$\Phi^{\mathsf{L}}$ to $Q_{n}(d)$ from [MNT10], which in the language used
here, can be written
$Q_{n}(d)={\mathscr{U}_{d}}/{\operatorname{N}\\!_{\mathsf{L}}^{n}\mathscr{U}_{d}}$.
In particular for $n=1$ this gives the degree $d$ part of a generalized Verma
module

$\Phi^{\mathsf{L}}(W_{0})_{d}={\mathscr{U}_{d}}/{\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}_{d}}\otimes_{\mathscr{U}_{0}}W_{0}=Q_{1}(d)\otimes_{\mathscr{U}_{0}}W_{0}.$

In [MNT10, Eq 2.6.1] a series of sub-algebras (called Quasi-finite algebras)
of the universal universal enveloping algebra is defined for all
$d\in\mathbb{N}$, and by [MNT10, Thm 3.3.4] if $V$ is $C_{2}$-cofinite, then
for $d\gg 0$ there is an equivalence of categories of $A_{d}$-modules and
(admissible) $V$-modules.

###  3.2. Mode transition algebras and their action on modules

In this section we introduce the mode transition algebra $\mathfrak{A}(V)$
associated with a vertex operator algebra $V$. A general treatment of these
algebras is developed in Appendix B, while we will list here the principal
consequences of the general theory. We begin by introducing the space
underlying $\mathfrak{A}(V)$, often denoted $\mathfrak{A}$ when $V$ is
understood.

######  Definition 3.2.1.

Let $V$ be a VOA and $\mathsf{A}=\mathsf{A}_{0}$ be the Zhu algebra associated
to $V$. We define $\mathfrak{A}=\mathfrak{A}(V)$ to be the vector space

$\displaystyle\mathfrak{A}$
$\displaystyle=\Phi^{\mathsf{R}}(\Phi^{\mathsf{L}}(\mathsf{A}))=\Phi^{\mathsf{L}}(\Phi^{\mathsf{R}}(\mathsf{A}))=\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}\right)\otimes_{\mathscr{U}_{0}}\mathsf{A}\otimes_{\mathscr{U}_{0}}\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}\right).$

Moreover, using the notation
$\mathfrak{A}_{d_{1},d_{2}}=\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}\right)_{d_{1}}\otimes_{\mathscr{U}_{0}}\mathsf{A}\otimes_{\mathscr{U}_{0}}\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}\right)_{d_{2}}$
we write

$\mathfrak{A}=\bigoplus_{d_{1}\in\mathbb{Z}_{\geq 0}}\
\bigoplus_{d_{2}\in\mathbb{Z}_{\leq 0}}\mathfrak{A}_{d_{1},d_{2}}.$

The isomorphism described in the following Lemma is crucial to the definition
of an algebra structure on $\mathfrak{A}$. We refer to B.2.1 for its proof.

######  Lemma 3.2.2.

There is an isomorphism:

$\displaystyle\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}\right)\otimes_{\mathscr{U}}\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}\right)$
$\displaystyle\to\mathsf{A}$
$\displaystyle\overline{\alpha}\otimes\overline{\beta}$
$\displaystyle\mapsto\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta$

where, for $\alpha,\beta\in\mathscr{U}$ homogeneous, we define
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta$
as :

$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta=\begin{cases}0&\text{if
}\deg(\alpha)+\deg(\beta)\neq 0\\\ [\alpha\beta]_{0}&\text{if
}\deg(\alpha)+\deg(\beta)=0\end{cases}$

and we extend the definition to general products by linearity.

######  Example 3.2.3.

We explicitly describe the element
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta\in\mathsf{A}$
when $\alpha$ and $\beta$ are homogeneous elements of opposite degrees. Three
cases can occur:

  * •

$\deg(\alpha)<0$. It follows that $\deg(\beta)>0$ and so
$\alpha\beta\in\operatorname{N}\\!^{1}\mathscr{U}_{0}$, which gives
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta=0$.

  * •

$\deg(\alpha)=0=\deg(\beta)$. We have that
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta=[\alpha]_{0}\cdot[\beta]_{0}$
since the map $\mathscr{U}_{0}\to\mathsf{A}_{0}$ is a ring homomorphism.

  * •

$\deg(\alpha)>0$. We first rewrite $\alpha\beta=\beta\alpha+[\alpha,\beta]$.
Since $\beta\alpha\in\operatorname{N}\\!^{1}\mathscr{U}_{0}$, we have that
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta$
coincide with $[\,[\alpha,\beta]\,]_{0}$.

Note that if $\alpha,\beta\in{\mathfrak{L}}(V)^{\mathsf{f}}$, then
$[\alpha,\beta]\in{\mathfrak{L}}(V)^{\mathsf{f}}_{0}$, so the above
description tells us that
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta$
is computed via the standard map
${\mathfrak{L}}(V)^{\mathsf{f}}_{0}\to\mathsf{A}$ described in [Li94].

######  Remark 3.2.4.

We note that one has that $\mathfrak{A}_{0,0}=\mathsf{A}$. Indeed, by the
definitions

$\mathfrak{A}_{0,0}=\left({\mathscr{U}}/{\operatorname{N}\\!_{\mathsf{L}}^{1}\mathscr{U}}\right)_{0}\otimes_{{\mathscr{U}}_{0}}\left({\mathscr{U}_{0}}/{\operatorname{N}\\!^{1}\mathscr{U}_{0}}\right)\otimes_{\mathscr{U}_{0}}\left({\mathscr{U}}/{\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}}\right)_{0}\cong\mathsf{A}\otimes_{\mathsf{A}}\mathsf{A}\otimes_{\mathsf{A}}\mathsf{A}\cong\mathsf{A}.$

We will next simultaneously describe the algebra structure on $\mathfrak{A}$,
and the action of this algebra $\mathfrak{A}$ on generalized Verma modules.

######  Definition 3.2.5.

Let $W_{0}$ be an $\mathsf{A}$-module. Following B.2.4 we define the map
$\mathfrak{A}\times\Phi^{\mathsf{L}}(W_{0})\to\Phi^{\mathsf{L}}(W_{0})$ as
follows. For $\mathfrak{a}=u\otimes a\otimes u^{\prime}\in\mathfrak{A}$ and
$\beta\otimes w\in\Phi^{\mathsf{L}}(W_{0})$ we set

$\mathfrak{a}\star(\beta\otimes w):=u\otimes
a(u^{\prime}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.7}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.7}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\beta)w.$

By 3.2.5 this map defines an algebra structure on $\mathfrak{A}$ and
$\Phi^{\mathsf{L}}(W_{0})$ becomes a left $\mathfrak{A}$-module. Moreover the
subspace $\mathfrak{A}_{d}:=\mathfrak{A}_{d,-d}$ is closed under
multiplication, hence it defines a subalgebra of $\mathfrak{A}$. The following
is a special case of B.2.6.

######  Definition 3.2.6.

We call $\mathfrak{A}(V)=\mathfrak{A}=(\mathfrak{A},+,\star)$ the mode
transition algebra of $V$, and $\mathfrak{A}_{d}=(\mathfrak{A}_{d},+,\star)$
the $d$-th mode transition algebra of $V$.

######  Remark 3.2.7.

More generally, the $d$-th mode transition algebra $\mathfrak{A}_{d}$ acts
naturally on the degree $d$ part of any weak $\mathbb{N}$-graded module in a
way which is compatible with the action of the enveloping algebra (see B.2.9).

######  Remark 3.2.8.

We observe that the underlying vector space and the algebra structure of
$\mathfrak{A}(V)$ does not depend on the existence of a conformal structure on
$V$. Therefore $\mathfrak{A}(V)$ can be defined for every graded vertex
algebra $V$.

We refer to Remark 3.4.6 for an explicit description of the algebra structure
of $\mathfrak{A}(V)$ when $V$ is a $C_{2}$-cofinite and rational vertex
operator algebra. The following assertion is straightforward

######  Remark 3.2.9.

Let $W_{0}$ be an $\mathsf{A}_{0}$-module. Then the action of
$\mathfrak{A}_{d}$ on $\Phi^{\mathsf{L}}(W_{0})_{d}$ factors through the
action of $\mathsf{A}_{d}$ described in 3.2.5 via the map $\mu_{d}$.

####  3.2.10. Relation to the literature

In [DJ08] a series of unital associative algebras $\mathsf{A}_{e,d}$, defined
as quotients of $V$, with $\mathsf{A}_{d,d}\cong\mathsf{A}_{d}$. By 3.2.5, the
$\mathfrak{A}_{d}$ act on the degree $d$ part of an induced module
$W=\Phi^{\mathsf{L}}(W_{0})$, as is true for the $\mathsf{A}_{e,d}$, although
they differ. In [Hua20], a related series of associative algebras $A^{d}(V)$
is defined. These contain higher level Zhu algebras as sub-algebras, and act
on (the sum of) components of a module up to degree $d$. In [Hua21], relations
are established between bimodules for these associative algebras and
(logarithmic) intertwining operators, and using these, in [Hua23], modular
invariance of (logarithmic) intertwining operators is proved.

###  3.3. Strong unital action of $\mathfrak{A}_{d}$ on modules

The $\mathfrak{A}_{0}=\mathsf{A}$ has an identity element given by the image
of ${\bm{1}}_{[-1]}$, denoted $1$. On the other hand, as we show in Section 8,
for $d\in\mathbb{Z}_{>0}$, the $\mathfrak{A}_{d}$ may not admit multiplicative
identity elements. However, if there are unities in $\mathfrak{A}_{d}$ for all
$d\in\mathbb{N}$, we have the following results about them.

######  Definition/Lemma 3.3.1.

Let $M$ be an $\mathsf{A}$-module, and assume that for every $d\in\mathbb{N}$
the ring $\mathfrak{A}_{d}$ is unital, with unity
$\mathscr{I}_{d}\in\mathfrak{A}_{d}$. We say that $\mathscr{I}_{d}$ is a
strong identity element for every $d$ if one of the following equivalent
conditions is verified:

  1. (1)

For every $d$,$n$,$m\in\mathbb{N}$, for all $u\in{\mathfrak{L}}(V)_{d}$, and
$\mathfrak{a}\in\mathfrak{A}_{n,-m}$ one has

$(u\cdot\mathscr{I}_{n})\star\mathfrak{a}=u\cdot\mathfrak{a}\qquad\text{ and
}\qquad\mathfrak{a}\star(\mathscr{I}_{m}\cdot u)=\mathfrak{a}\cdot u.$

  2. (2)

For every $n,m\in\mathbb{N}$ and for every
$\mathfrak{a}\in\mathfrak{A}_{n,-m}$ one has
$\mathscr{I}_{n}\star\mathfrak{a}=\mathfrak{a}=\mathfrak{a}\star\mathscr{I}_{m}$.

  3. (3)

For every $d\in\mathbb{N}$ the homomorphism
$\mathfrak{A}_{d}\to\operatorname{End}(\Phi^{\mathsf{L}}(\mathsf{A})_{d})$ is
unital;

  4. (4)

For every $d\in\mathbb{N}$, the homomorphism
$\mathfrak{A}_{d}\to\operatorname{End}(\Phi^{\mathsf{L}}(\mathsf{A})_{d})$ is
unital and injective.

  5. (5)

For every $d\in\mathbb{N}$ and $M$ an $\mathsf{A}$-module, the homomorphism
$\mathfrak{A}_{d}\to\operatorname{End}(\Phi^{\mathsf{L}}(M)_{d})$ is unital.

###### Proof.

We prove these conditions are equivalent. Since (4) implies (3) and (5)
implies (3), it will be enough to show the following implications:

(1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) $\Rightarrow$ (5) and (3)
$\Rightarrow$ (4)

(1) $\Rightarrow$ (2): this follows by taking $d=0$ and $u=1$.

(2) $\Rightarrow$ (1): This follows from B.2.5.

(2) $\Rightarrow$ (3): This follows from the identification of
$\Phi^{\mathsf{L}}(\mathsf{A})_{d}$ with $\mathfrak{A}_{d,0}$.

(3) $\Rightarrow$ (2): By linearity, we can assume that
$\mathfrak{a}\in\mathsf{A}_{n,-m}$ is represented by an element of the form
$u\otimes a\otimes v$ with $u\in\mathscr{U}^{\mathsf{L}}_{n}$,
$v\in\mathscr{U}^{\mathsf{R}}_{-m}$ and $a\in\mathsf{A}$. Then

$\mathscr{I}_{n}\star(u\otimes a\otimes
v)=\mathscr{I}_{n}\star(u\otimes\mathfrak{a}\otimes
1)\cdot(v)=(\mathscr{I}_{n}\star(u\otimes a))\otimes v=(u\otimes a\otimes
1)\cdot v=u\otimes a\otimes v$

where (3) ensures that the third equality holds.

(3) $\Rightarrow$ (5): By linearity we can assume that an element of
$\Phi^{\mathsf{L}}(M)_{d}$ is given by $u\otimes m$ for
$u\in\mathscr{U}^{\mathsf{L}}_{d}$ and $m\in M$. Hence we obtain
$\mathscr{I}_{d}\star(u\otimes m)=(\mathscr{I}_{d}\star u)\otimes m=u\otimes
m$.

(3) $\Rightarrow$ (4): By 3.2.5 the action of an element
$\mathfrak{a}\in\mathfrak{A}_{d}$ via $\star$ on
$\mathfrak{A}_{d}\subseteq\mathfrak{A}=\Phi^{\mathsf{L}}(\Phi^{\mathsf{R}}(\mathsf{A}))=\Phi^{\mathsf{L}}(\mathsf{A})\otimes_{\mathscr{U}_{0}}\left(\mathscr{U}/\operatorname{N}\\!_{\mathsf{R}}^{1}\mathscr{U}\right)$
is determined by the action of $\mathfrak{a}$ on
$\Phi^{\mathsf{L}}(\mathsf{A})$. In particular, as the former is injective
when we have an identity element, the latter must be injective in this case as
well. ∎

###  3.4. Relation to the functor $\Phi$

To define the mode transition algebra $\mathfrak{A}$, we used the map
$\Phi^{\mathsf{L}}\Phi^{\mathsf{R}}=\Phi^{\mathsf{L}}\Phi^{\mathsf{R}}$, which
we can interpret as a functor from the category of $\mathsf{A}$-bimodules to
the category of $\mathscr{U}$-bimodules. Its properties in a more general
framework are described in B.2.5.

We now show that this functor agrees with the functor denoted $\Phi$ in
[DGK22, Definition 2.2] which assigns to an $\mathsf{A}$-bimodule $M$ the
$(\mathscr{U}^{\mathsf{L}})^{\otimes 2}$-module

$\Phi(M):=(\mathscr{U}^{\mathsf{L}}\otimes\mathscr{U}^{\mathsf{L}})\underset{\mathscr{U}^{\mathsf{L}}_{\leq
0}\otimes\mathscr{U}^{\mathsf{L}}_{\leq
0}}{{{\widehat{\otimes}}^{\textsf{f}}}}M,$

where $\mathscr{U}^{\mathsf{L}}_{\leq 0}\otimes\mathscr{U}^{\mathsf{L}}_{\leq
0}$ acts on $M$ as follows:

$(u\otimes v)(m)=\begin{cases}u\cdot m\cdot\theta(v)&\text{ if
}u,v\in\mathscr{U}_{0}\\\ 0&\text{ otherwise}.\end{cases}$

Here $\theta$ is the natural involution of $\mathscr{U}_{0}$, from e.g.
[DGK22, Eq.(7)]), and which we describe briefly in Section 3.4.1. In 3.4.5 we
describe the relation between functors $\Phi^{\mathsf{L}}$ and
$\Phi^{\mathsf{R}}$ to $\Phi$.

####  3.4.1. The involutions, left and right actions

As we explain here, there is an anti-Lie algebra isomorphism $\theta$ used to
transport the universal enveloping algebra, considered as an object that acts
on modules on the left (denoted here by $\mathscr{U}^{\mathsf{L}}$), to an
analogous completion $\mathscr{U}^{\mathsf{R}}$ that acts on modules on the
right.

The map $\theta\colon V\otimes_{\mathbb{C}}\mathbb{C}(\\!({t})\\!)\to
V\otimes_{\mathbb{C}}\mathbb{C}(\\!({t^{-1}})\\!)$ is defined, for $a\in V$
homogeneous, by

(5) $a\otimes\sum_{i\geq N}c_{i}t^{i}\mapsto(-1)^{\deg(a)}\sum_{j\geq
0}^{\deg(a)}\left(\dfrac{1}{j!}\left(L_{1}^{j}(a)\right)\otimes\sum_{i\geq
N}c_{i}t^{2\deg(a)-i-j-2}\right),$

and extended linearly.

The map $\theta$ is related to the involution
$\gamma=(-1)^{L_{0}}e^{L_{1}}\colon V\rightarrow V$ defined, for $a\in V$
homogeneous by

(6) $a\mapsto(-1)^{\deg a}\sum_{i\geq 0}\frac{1}{i!}L_{1}^{i}(a),$

and extended by linearity. To state the relation succinctly, for every
homogeneous $a\in V$ we set

(7) $J_{n}(a):=a_{[\deg(a)-1+n]}.$

This notation, used in [NT05], has the property that $\deg(J_{n}(v))=-n$, so
that the degree of such an element is easily read.

######  Lemma 3.4.2.

For $a\in V$, homogeneous, $\theta(J_{n}(a))=J_{-n}(\gamma(a))$.

###### Proof.

This follows by combining (5) and (6) and using linearity. ∎

######  Lemma 3.4.3.

The map $\theta$ induces a Lie algebra anti-isomorphism
${\mathfrak{L}}(V)^{\mathsf{L}}\to{\mathfrak{L}}(V)^{\mathsf{R}}$, which
restricts to a Lie algebra involution on ${\mathfrak{L}}(V)^{\mathsf{f}}$,
such that $\theta({\mathfrak{L}}(V)^{\mathsf{L}}_{\leq
d})={\mathfrak{L}}(V)^{\mathsf{R}}_{\geq-d}$ and
$\theta({\mathfrak{L}}(V)^{\mathsf{f}}_{d})={\mathfrak{L}}(V)^{\mathsf{f}}_{-d}$.

###### Proof.

One can check that this restricts to an endomorphism of
$V\otimes\mathbb{C}[t,t^{-1}]$, which by [NT05, Proposition 4.1.1], defines a
Lie algebra involution of ${\mathfrak{L}}(V)^{\mathsf{f}}$, that is,
$\theta([\ell_{1},\ell_{2}])=-[\theta(\ell_{1}),\theta(\ell_{2})]$, and
$\theta^{2}=\mathrm{id}$. Moreover, it is easy to verify that
$\theta({\mathfrak{L}}(V)^{\mathsf{f}}_{d})\subset{\mathfrak{L}}(V)^{\mathsf{f}}_{-d}$.
As the Lie algebras
${\mathfrak{L}}(V)^{\mathsf{L}},{\mathfrak{L}}(V)^{\mathsf{R}}$ carry
exhaustive and separated split filtrations by the graded subalgebra
${\mathfrak{L}}(V)^{\mathsf{f}}$, they are naturally equipped with norms via
Remark A.4.5 – that is, by declaring that elements of large positive degree
are large in ${\mathfrak{L}}(V)^{\mathsf{L}}$ and small in
${\mathfrak{L}}(V)^{\mathsf{R}}$. With this definition, it follows that
$\theta$ is continuous, and as noted in Remark A.4.8, that the multiplication
on the Lie algebras is continuous. Finally, the fact that
${\mathfrak{L}}(V)^{\mathsf{f}}$ induces a splitting of the filtrations, it
follows that ${\mathfrak{L}}(V)^{\mathsf{f}}$ is simultaneously dense in
${\mathfrak{L}}(V)^{\mathsf{L}}$ and ${\mathfrak{L}}(V)^{\mathsf{R}}$.
Consequently, we see by continuity that $\theta$ induces an anti-homomorphism
from ${\mathfrak{L}}(V)^{\mathsf{L}}$ to ${\mathfrak{L}}(V)^{\mathsf{R}}$,
which is an anti-isomorphism as $\theta^{2}=\mathrm{id}$. ∎

If $R=(R,+,\cdot)$ is a ring, denote by $R^{\mathrm{op}}$ its opposite ring,
that is $R^{\mathrm{op}}=(R,+,*)$ where $a*b:=b\cdot a$. Similarly, if
$(L,[\,,\,])$ is a Lie algebra, we denote by $L^{\mathrm{op}}$ its opposite
Lie algebra, where $[a,b]_{L^{\mathrm{op}}}:=[b,a]_{L}$.

######  Lemma 3.4.4.

For $\mathit{U}({\mathfrak{L}}(V)^{\mathsf{R}})$ the universal enveloping
algebra of ${\mathfrak{L}}(V)^{\mathsf{R}}$,

$\theta\colon\mathit{U}({\mathfrak{L}}(V)^{\mathsf{L}})\to\mathit{U}({\mathfrak{L}}(V)^{\mathsf{R}})^{\mathrm{op}}$

is an isomorphism of rings.

###### Proof.

We have established in 3.4.3 that
$\theta\colon{\mathfrak{L}}(V)^{\mathsf{L}}\to{\mathfrak{L}}(V)^{\mathsf{R}}$
is an anti-isomorphism of Lie algebras, so that
$\theta\colon{\mathfrak{L}}(V)^{\mathsf{L}}\to({\mathfrak{L}}(V)^{\mathsf{R}})^{\mathrm{op}}$
is a Lie-algebra isomorphism. Moreover, as
$\mathit{U}(({\mathfrak{L}}(V)^{\mathsf{R}})^{\mathrm{op}})=\mathit{U}({\mathfrak{L}}(V)^{\mathsf{R}})^{\mathrm{op}}$,
it follows that $\theta$ induces an isomorphism between
$\mathit{U}({\mathfrak{L}}(V)^{\mathsf{L}})$ and
$\mathit{U}({\mathfrak{L}}(V)^{\mathsf{R}})^{\mathrm{op}}$, as wanted. Here we
note that $\theta(\alpha\cdot\beta)=\theta(\beta)\cdot\theta(\alpha)$ for
every $\alpha,\beta\in{\mathfrak{L}}(V)^{\mathsf{L}}$ (and where $\cdot$ is
the usual product in the $\mathit{U}({\mathfrak{L}}(V)^{\mathsf{L}})$ and
$\mathit{U}({\mathfrak{L}}(V)^{\mathsf{R}})$). ∎

In particular $\theta$ is an isomorphism between
$\mathit{U}({\mathfrak{L}}(V)^{\mathsf{f}})$ and
$\mathit{U}({\mathfrak{L}}(V)^{\mathsf{f}})^{\mathrm{op}}$.

######  Lemma 3.4.5.

Let $B$ be an associative ring, $W^{1}$ an $(\mathsf{A},B)$-bimodule and
$W^{2}$ a $(B,\mathsf{A})$-bimodule. Then we have a natural identification

$\Phi^{\mathsf{L}}(W^{1})\otimes_{B}\Phi^{\mathsf{R}}(W^{2})\cong\Phi(W^{1}\otimes_{B}W^{2})$

of $(\mathscr{U}^{\mathsf{L}},\mathscr{U}^{\mathsf{R}})$-bimodules. In
particular we have $\mathfrak{A}=\Phi(\mathsf{A})$.

###### Proof.

We first note that there is a natural equivalence of categories between left
$(\mathscr{U}^{\mathsf{L}})^{\otimes 2}$-modules and
$(\mathscr{U}^{\mathsf{L}},(\mathscr{U}^{\mathsf{L}})^{\text{op}})$-modules.
Moreover, as described in 3.4.4, the involution $\theta$ provides an
identification
$\mathscr{U}^{\mathsf{R}}\cong(\mathscr{U}^{\mathsf{L}})^{\text{op}}$. It
follows that the map
$\Phi^{\mathsf{L}}(W^{1})\otimes_{B}\Phi^{\mathsf{R}}(W^{2})\to\Phi(W^{1}\otimes_{B}W^{2})$
induced by

$(u\otimes w_{1})\otimes(w_{2}\otimes
v)\mapsto(u\otimes\theta(v))\otimes(w_{1}\otimes w_{2})$

for all $u\in\mathscr{U}^{\mathsf{L}}$, $v\in\mathscr{U}^{\mathsf{R}}$ and
$w_{i}\in W_{i}$ is indeed an isomorphism.∎

######  Remark 3.4.6.

Let $V$ be a rational VOA. Then the mode transition algebras
$\mathfrak{A}_{d}$ have strong identity elements. To see this, we note that
the rationality of $V$ implies that $\mathsf{A}$ is finite and semi-simple,
and thus has a bimodule decomposition

$\mathsf{A}\cong\prod_{k=1}^{m}I^{k}=\prod_{k=1}^{m}W_{0}^{k}\otimes(W_{0}^{k})^{\vee},$

where $\\{1,\dots,m\\}$ is the set indexing the isomorphism classes of simple
$V$-modules. Equivalently, $W_{0}^{k}$ runs over all isomorphism classes of
simple left $\mathsf{A}$-modules (so that $W_{0}^{\vee}$ is the corresponding
dual right $\mathsf{A}$-module). By 3.4.5 we therefore have:

$\displaystyle\mathfrak{A}=\Phi(\mathsf{A})=\Phi\left(\prod_{k=1}^{m}W_{0}^{k}\otimes(W_{0}^{k})^{\vee}\right)=\prod_{k=1}^{m}\Phi^{\mathsf{L}}(W_{0}^{k})\otimes_{\mathbb{C}}\Phi^{\mathsf{R}}(W_{0}^{k})^{\vee}.$

Note that since $V$ is rational, the functor $\Phi^{\mathsf{L}}$ takes simple
$\mathsf{A}$-modules to simple $V$-modules. This follows from the fact that
$\Phi^{\mathsf{L}}$ preserves indecomposable modules [DGK22, Lemma 1.2], and
that for a rational VOA, indecomposable modules are simple. Hence for $W_{0}$
a simple $\mathsf{A}$-module, $W:=\Phi^{\mathsf{L}}(W_{0})$ must be simple as
well. Similarly, we have that also $\Phi^{\mathsf{R}}(W_{0}^{\vee})$ must be a
simple $V$-module. We give some details about this: the involution $\theta$
identifies left and right $\mathsf{A}$-modules allowing us to consider
$(W^{k}_{0})^{\vee}$ as a left $\mathsf{A}$-module which we denote as
$\mbox{}^{\theta}(W^{k}_{0})^{\vee}$. From this perspective, and the above
discussions relating $\Phi^{\mathsf{L}}$, $\Phi^{\mathsf{R}}$ and $\theta$, we
thus have a natural identification

$\Phi^{\mathsf{L}}(\mbox{}^{\theta}(W^{k}_{0})^{\vee})=\Phi^{\mathsf{R}}((W^{k}_{0})^{\vee})\qquad\text{and}\qquad\Phi^{\mathsf{L}}(\mbox{}^{\theta}(W^{k}_{0})^{\vee})_{d}=\Phi^{\mathsf{R}}((W^{k}_{0})^{\vee})_{-d},$

for every $d\in\mathbb{N}$. Since $W^{k}_{0}$ was simple, also
$\mbox{}^{\theta}(W^{k}_{0})^{\vee}$ is simple, and thus we conclude that
$\Phi^{\mathsf{R}}((W^{k}_{0})^{\vee})$ is simple as well.

We now show that $\Phi^{\mathsf{R}}((W_{0}^{k})^{\vee})$ is isomorphic to the
contragredient module $(W^{k})^{\prime}$ of $W^{k}$. The contragredient module
$(W^{k})^{\prime}$ has zero part
$(W^{k})^{\prime}_{0}=\mbox{}^{\theta}(W^{k}_{0})^{\vee}$ inducing a natural
map
$\Phi^{\mathsf{R}}((W^{k}_{0})^{\vee})=\Phi^{\mathsf{L}}(\mbox{}^{\theta}(W^{k}_{0})^{\vee})\to(W^{k})^{\prime}$.
As the contragredient of a simple $V$-module is simple, it follows that this
map is an isomorphism. In particular one has
$\Phi^{\mathsf{R}}((W^{k}_{0})^{\vee})_{-d}=(W^{k}_{d})^{\vee}$ and thus

(8) $\mathfrak{A}=\bigoplus_{d\in\mathbb{Z}_{\geq
0}}\prod_{k=1}^{m}W^{k}_{d}\otimes_{\mathbb{C}}(W^{k}_{d})^{\vee}.$

Using this decomposition, the $\star$-product is induced, by linearity, from

$(a_{W^{k}}\otimes\varphi_{W^{k}_{d}})\star(b_{W^{j}_{e}}\otimes\psi_{W^{j}})=\begin{cases}\varphi_{W^{k}_{d}}(b_{W^{j}_{e}})(a_{W^{k}}\otimes\psi_{W^{j}})&\text{if
}k=j\text{ and }e=d\\\ 0&\text{otherwise},\end{cases}$

where $\varphi_{W^{k}_{d}}\colon W^{k}_{d}\to\mathbb{C}$ and $b_{W_{e}^{j}}\in
W_{e}^{j}$. Note that, by rationality, the spaces $W^{k}_{d}$ are necessarily
finite dimensional [Zhu96, Definition 1.2.4]. It then follows that, for all
$d\in\mathbb{Z}_{\geq 0}$, we have an isomorphism of rings

$\mathfrak{A}_{d}\cong\prod_{i=1}^{m}\text{End}_{\mathbb{C}}(W^{k}_{d})$

and thus ${\textbf{1}}_{d}:=\prod_{i=1}^{m}\text{Id}_{W^{k}_{d}}$ is its
strong identity element. We remark that [NT05, Proposition 7.2.1] states,
without a detailed proof and using a different notation, that (8) holds, under
the assumption that $V$ is not only rational, but also $C_{2}$-cofinite. From
our argument it is apparent how $C_{2}$-cofinitess is not needed.

##  4\. Smoothing, limits, and coinvariants

In Section 4.1 we describe the sheaf of coinvariants on schemes $S$
parametrizing families of pointed and coordinatized curves in general terms,
while in Section 4.2, we explain what we mean by sheaves defined over a scheme
$S=\operatorname{\mathrm{Spec}}R$, where $R$ is a ring complete with respect
to some ideal $I$. In Section 4.3, we describe the setup for considering
coinvariants on smoothings of nodal curves, establishing some results needed
for our geometric applications. In particular, in Section 4.4, for the proof
of 5.1.2, we explicitly describe the sheaf $\mathcal{L}_{\mathscr{C}\setminus
P_{\bullet}}(V)$ of Chiral Lie algebras.

Throughout this section, $V$ is a VOA with no additional finiteness
assumptions.

###  4.1. Coinvariants

Let $S$ be a scheme and let $\mathscr{W}$ be a quasi-coherent sheaf of
$\mathscr{O}_{S}$-modules. Let $\mathscr{L}$ be a quasi-coherent sheaf of Lie
algebras on $S$ acting on $\mathscr{W}$. We define the sheaf of coinvariants
$[\mathscr{W}]_{\mathscr{L}}$ on S as the cokernel

${\mathscr{L}}\otimes_{\mathscr{O}_{S}}{\mathscr{W}}\to{\mathscr{W}}\to[{\mathscr{W}}]_{{\mathscr{L}}}\to
0.$

For future use, it will be helpful to note that the formation of the sheaves
of coinvariants commutes with base change.

######  Lemma 4.1.1.

Let $\mathscr{L}$ be a quasi-coherent sheaf of Lie algebras on a scheme $S$
acting on a quasi-coherent sheaf $\mathscr{W}$. For any morphism
$S^{\prime}\to S$, we have
$\left([{\mathscr{W}}]_{{\mathscr{L}}}\right)_{S^{\prime}}\cong[{\mathscr{W}}_{S^{\prime}}]_{{\mathscr{L}}_{S^{\prime}}}$.

###### Proof.

This follows from right exactness of pullback of quasi-coherent sheaves
(equivalently right exactness of tensor). ∎

######  Remark 4.1.2.

Let $\pi\colon\mathscr{C}\to S$ be a projective curve, with $n$ distinct
smooth sections $P_{\bullet}\colon S\to\mathscr{C}$ and formal coordinates
$t_{\bullet}$ at $P_{\bullet}$. Assume further that
$\mathscr{C}\setminus\sqcup P_{\bullet}(S)\to S$ is affine. This assumption is
possible by Propagation of Vacua [Cod19, Thm 3.6] (see also [DGT22a, Theorem
4.3.1]). Denote by $W^{\bullet}=W^{1}\otimes\dots\otimes W^{n}$ the tensor
product of an $n$-tuple of $V$-modules and let
${\mathscr{W}}:=W^{\bullet}\otimes\mathcal{O}_{S}$. The sheaf of Chiral Lie
algebras $\mathscr{L}:=\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$,
originally defined in this context for families of stable curves with
singularities in [DGT21, DGT22a], is explicitly described with more details
here in Section 4.4. The sheaf of coinvariants $[\mathscr{W}]_{\mathscr{L}}$
defined above will also be denoted
$[W^{\bullet}]_{(\mathscr{C},P_{\bullet},t_{\bullet})}$. While quasi-coherent
[DGT21], for $V$ $C_{2}$-cofinite, or if generated in degree $1$, this sheaf
is coherent [DGK22, DG23].

###  4.2. Completions

As in Section 4.1, we consider coinvariants over
$S=\operatorname{\mathrm{Spec}}(R)$, where $R$ is a ring that is complete with
respect to some ideal $I$. For $k\in\mathbb{Z}_{\geq 0}$, setting
$S_{k}=\operatorname{\mathrm{Spec}}(R_{k})=\operatorname{\mathrm{Spec}}(R/I^{k+1})$,
pullbacks ${\mathscr{L}}_{k}$ and ${\mathscr{W}}_{k}$ of $\mathscr{L}$ and
$\mathscr{W}$ to $S_{k}$ respectively, we work with coinvariants
$[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}$ for any $k\in\mathbb{Z}_{\geq 0}$.
Due to quasicoherence, each of these can be thought of as a module over
$R_{k}$, with maps
$[{\mathscr{W}}_{k+1}]_{{\mathscr{L}}_{k+1}}\to[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}$.

######  Definition 4.2.1.

In the above situation, we define the formal coinvariants, denoted
$\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}$ to be the $I$-adically complete
$R$-module

$\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}=\varprojlim[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}.$

######  Proposition 4.2.2.

We have an identification

$\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}=\operatorname{coker}\left[\varprojlim\pi_{*}{\mathscr{L}}_{k}\otimes_{\mathscr{O}_{S_{k}}}{\mathscr{W}}_{k}\longrightarrow\varprojlim{\mathscr{W}}_{k}\right]$

######  Proposition 4.2.3.

Suppose $[{\mathscr{W}}]_{{\mathscr{L}}}$ is finitely generated over an
$I$-adically complete Noetherian ring $R$. Then the natural map
$[{\mathscr{W}}]_{{\mathscr{L}}}\to\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}$
is an isomorphism.

###### Proof.

Consider the exact sequence of $R$-modules (omitting the $\pi_{*}$ from the
notation, and identifying the quasicoherent sheaves with the corresponding
$R$-modules):

${\mathscr{L}}\otimes_{R}{\mathscr{W}}\longrightarrow{\mathscr{W}}\longrightarrow[{\mathscr{W}}]_{{\mathscr{L}}}\longrightarrow
0.$

Tensoring with $R/I^{k}$ (or geometrically base-changing along $S_{k}\to S$)
is a right exact operation, hence it yields an exact sequence

${\mathscr{L}}_{k}\otimes_{R_{n}}{\mathscr{W}}_{k}\longrightarrow{\mathscr{W}}_{k}\longrightarrow\left([{\mathscr{W}}]_{{\mathscr{L}}}\right)_{R_{k}}\longrightarrow
0,$

which shows that we can identify
$[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}=\left([{\mathscr{W}}]_{{\mathscr{L}}}\right)_{R_{k}}$.
In particular, the composition

$[{\mathscr{W}}]_{{\mathscr{L}}}\longrightarrow\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}\longrightarrow[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}$

coincides with the surjection
$[{\mathscr{W}}]_{{\mathscr{L}}}\to[{\mathscr{W}}]_{{\mathscr{L}}}\otimes_{R}R/I^{k}$.
Since $[{\mathscr{W}}]_{{\mathscr{L}}}$ is finitely generated over a complete
Noetherian ring, it is $I$-adically complete by [Sta23, Tag 00MA(3)].
Therefore we can identify
$[{\mathscr{W}}]_{{\mathscr{L}}}=\varprojlim[{\mathscr{W}}]_{{\mathscr{L}}}\otimes_{R}R/I^{k}=\varprojlim[{\mathscr{W}}_{k}]_{{\mathscr{L}}_{k}}=\widehat{[{\mathscr{W}}]_{{\mathscr{L}}}}$,
giving the desired isomorphism. ∎

###  4.3. Smoothing setup

In order to introduce the smoothing property for $V$, we will recall the
notion of a smoothing of a nodal curve, and set a small amount of notation
used throughout. Let $R=\mathbb{C}[\\![q]\\!]$ and write
$S=\operatorname{\mathrm{Spec}}(R)$. Let $\mathscr{C}_{0}$ be a projective
curve over $\mathbb{C}$ with at least one node $Q$, smooth and distinct points
$P_{\bullet}=(P_{1},\ldots,P_{n})$ such that $\mathscr{C}_{0}\setminus
P_{\bullet}$ is affine, and formal coordinates
$t_{\bullet}=(t_{1},\ldots,t_{n})$ at $P_{\bullet}$. Let
$\eta\colon\widetilde{\mathscr{C}}_{0}\rightarrow\mathscr{C}_{0}$ be the
partial normalization of $\mathscr{C}_{0}$ at $Q$, which is naturally pointed
by $Q_{\pm}:=\eta^{-1}(Q)$. We also suppose we have chosen formal coordinates
at $Q_{\pm}$ and we call them $s_{\pm}$.

The choice of our formal coordinates $s_{\pm}$ determine a smoothing family
$(\mathscr{C},P_{\bullet},t_{\bullet})$ over $S$, with the central fiber given
by $(\mathscr{C}_{0},P_{\bullet},t_{\bullet})$. Let
$(\widetilde{\mathscr{C}},P_{\bullet}\sqcup Q_{\pm},t_{\bullet}\sqcup
s_{\pm})$ denote the trivial extension $\widetilde{\mathscr{C}}_{0}\times S$
with its corresponding markings. We will now discuss the relationship between
coinvariants for $(\mathscr{C},P_{\bullet},t_{\bullet})$ and
$(\widetilde{\mathscr{C}},P_{\bullet}\sqcup Q_{\pm},t_{\bullet}\sqcup
s_{\pm})$.

Let $W^{1},\dots,W^{n}$ be an $n$-tuple of $V$-modules, or equivalently,
smooth $\mathscr{U}$-modules for $\mathscr{U}$ the universal enveloping
algebra of $V$ (defined in Section 2), and $W^{\bullet}$ their tensor product.
As is described above in Remark 4.1.2, we may also consider the sheaf of
coinvariants $[W^{\bullet}]_{(\mathscr{C},P_{\bullet},t_{\bullet})}$.

As mentioned in the introduction, there is a map $\alpha_{0}\colon
W^{\bullet}\to W^{\bullet}\otimes\Phi(\mathsf{A})$ which induces a map between
coinvariants

$[\alpha_{0}]\colon[W^{\bullet}]_{\left(\mathscr{C}_{0},P_{\bullet},t_{\bullet}\right)}{\overset{\cong}{\longrightarrow}}[W^{\bullet}\otimes\Phi(\mathsf{A})]_{\left(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}.$

Moreover, if $V$ is $C_{1}$-cofinite, then we will show in 4.4.4 that
$[\alpha_{0}]$ is an isomorphism. We recall that
$\Phi(\mathsf{A})=\mathfrak{A}$, so we will generally use the notation
$\mathfrak{A}$ below.

The following result, which is a consequence of 4.2.3, allows us to describe
coinvariants over $\widetilde{\mathscr{C}}$ whenever they are finite
dimensional. The assumptions of the following result are satisfied when $V$ is
$C_{2}$-cofinite, for all $V$-modules $W^{\bullet}$, and also more generally
(by [DGK22]).

######  Corollary 4.3.1.

Assume that the sheaf
$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$ is coherent over $S$. Then one has
identifications

$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{\left(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}\\\
\cong[W^{\bullet}\otimes\mathfrak{A}]_{\left(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}[\\![q]\\!]\\\
\cong[W^{\bullet}\otimes\mathfrak{A}]_{\left(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup
s_{\pm}\right)}\otimes_{\mathbb{C}}\mathbb{C}[\\![q]\\!].$

###### Proof.

The second isomorphism holds because the coherence assumption implies that
$[W^{\bullet}\otimes\mathfrak{A}]_{(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$ is a finite dimensional vector space. To
prove the first isomorphism, we consider, for $R=\mathbb{C}[\\![q]\\!]$, the
$R$-module and $R$-Lie algebra

$\mathscr{W}=(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!],\ \mbox{ and }\
\mathscr{L}=\mathcal{L}_{\widetilde{\mathscr{C}}\setminus\\{P_{\bullet}\sqcup
Q_{\pm}\\}}(V).$

Since $[\mathscr{W}]_{\mathscr{L}}$ is a finite dimensional $R$-module, for
$R_{k}=\mathbb{C}[\\![q]\\!]/q^{k+1}$, and $S_{k}=\mathrm{Spec}(R_{k})$, and
one can show by 4.2.3, that

(9)
$[\mathscr{W}]_{\mathscr{L}}=\varprojlim\left(\left[\mathscr{W}\otimes_{R}R_{k}\right]_{\mathscr{L}\otimes_{R}R_{k}}\right).$

Note further that
$\mathscr{W}\otimes_{R}R_{k}=\left(W^{\bullet}\otimes\mathfrak{A}\right)\otimes_{\mathbb{C}}R_{k}$,
and similarly,

$\mathscr{L}\otimes_{R}R_{k}=\mathcal{L}_{\widetilde{\mathscr{C}_{0}}\setminus\\{P_{\bullet}\sqcup
Q_{\pm}\\}}(V)\otimes_{\mathbb{C}}R_{k}.$

Using this, together with 4.2.2, we deduce that (9) is isomorphic to

$\varprojlim\left([W^{\bullet}\otimes\mathfrak{A}]_{(\widetilde{\mathscr{C}_{0}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}\otimes_{\mathbb{C}}R_{k}\right)$

which is indeed
$[W^{\bullet}\otimes\mathfrak{A}]_{(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}[\\![q]\\!]$, as was asserted. ∎

######  Remark 4.3.2.

Corollary 4.3.1 implies that, up to some assumptions of coherence, the sheaf
of coinvariants associated with $W^{\bullet}\otimes\mathfrak{A}$ over
$\widetilde{\mathscr{C}}_{0}$ deforms trivially to the sheaf of coinvariants
over the trivial deformation $\widetilde{\mathscr{C}}$ of
$\widetilde{\mathscr{C}}_{0}$. Consequently, the target of the induced map
$[\alpha]$, which extends the map $[\alpha_{0}]$ is therefore identified with
the sheaf of coinvariants associated with $\widetilde{\mathscr{C}}$ (and not
only with a completion thereof).

We conclude this section with some criteria to show coherence of sheaves
coinvariants over $S$. Throughout we will use the notation
$R_{k}=\mathbb{C}[\\![q]\\!]/q^{k+1}$ and
$S_{k}=\operatorname{\mathrm{Spec}}(R_{k})$ for every $k\in\mathbb{N}$.

######  Lemma 4.3.3.

For $M$ any module over $R_{k}$, let $m_{1},\ldots,m_{r}\in M$ be elements
whose images generate $M\otimes_{R_{k}}R_{0}$. Then the elements
$m_{1},\ldots,m_{r}$ also generate $M$.

###### Proof.

We induct on $k$, the case $k=0$ being automatic. For the induction step,
suppose $m\in M$ and consider the $R_{k-1}$ module
$\overline{M}=M\otimes_{R_{k}}R_{k-1}$. By the induction hypothesis, the
elements $m_{1},\ldots,m_{r}$ generate $\overline{M}$. Therefore we can find
$a_{1},\ldots a_{r}\in R_{k}$ so that

$m^{\prime}=m-\sum a_{i}m_{i}\in M,$

maps to $0$ in $\overline{M}$.

Now consider the submodule $M^{\prime}=q^{k}M\subset M$. As $q^{k}M$ is
exactly the kernel of the map $M\to\overline{M}=M\otimes_{R_{k}}R_{k-1}$ we
find that $m^{\prime}\in M^{\prime}$, and therefore we can write
$m^{\prime}=q^{k}x$ for some $x\in M$. Writing $x=\sum b_{i}m_{i}\pmod{q}$, we
find $x-\sum b_{i}m_{i}=qy$ for some $y\in M$. But now we have

$m=\left(\sum a_{i}m_{i}\right)+m^{\prime}=\left(\sum
a_{i}m_{i}\right)+q^{k}\left(\left(\sum
b_{i}m_{i}\right)+qy\right)=\sum(a_{i}+q^{k}b_{i})m_{i},$

as desired. ∎

######  Proposition 4.3.4.

If $[W^{\bullet}]_{(\mathscr{C}_{0},P_{\bullet},t_{\bullet})}$ is a finite
dimensional vector space, then both

$[W^{\bullet}[\\![q]\\!]]_{(\mathscr{C},P_{\bullet},t_{\bullet})}\quad\text{
and
}\quad[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{\left(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}$

are coherent.

###### Proof.

For every $k\in\mathbb{N}$ and for every scheme $X$ over $S$, denote the
pullback of $S$ to $S_{k}$ by $X_{k}$. Define

$\displaystyle
M_{k}:=[W^{\bullet}_{R_{k}}]_{\left(\mathscr{C}_{k},P_{\bullet},t_{\bullet}\right)}\quad\mbox{
and
}\quad\widetilde{M}_{k}:=[(W^{\bullet}\otimes\mathfrak{A})_{R_{k}}]_{\left(\widetilde{\mathscr{C}}_{k},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}.$

Let us first show that $M_{k}$ and $\widetilde{M}_{k}$ are coherent. As we are
considering modules over the Noetherian ring $R_{k}$, we only need to show
that they are finitely generated. But by 4.3.3, for this it suffices to show
that $\widetilde{M}_{0}$ and $M_{0}$ are finitely generated. This holds
because by assumption $M_{0}$ is finitely generated and $\alpha_{0}\colon
M_{0}\to\widetilde{M}_{0}$ is an isomorphism by [DGT22a].

For simplicity, denote

$\displaystyle
M=[W^{\bullet}[\\![q]\\!]]_{\left(\mathscr{C},P_{\bullet},t_{\bullet}\right)}\quad\mbox{
and
}\quad\widetilde{M}=[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{\left(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)}.$

By 4.1.1, it follows that $M_{k}=M\otimes_{R}R_{k}$ and
$\widetilde{M}_{k}=\widetilde{M}\otimes_{R}R_{k}$. Consequently the natural
maps $\widetilde{M}_{k}\to\widetilde{M}_{k-1}$ and $M_{k}\to M_{k-1}$ are
surjective. It follows therefore from [Sta23, Lemma 087W] that $M$ and
$\widetilde{M}$ will be finitely generated over $R$ whenever $M_{k}$ and
$\widetilde{M}_{k}$ are finitely generated over $R_{k}$ for every $k$. This is
what we have just shown and so $M$ and $\widetilde{M}$ are coherent. ∎

###  4.4. The sheaf of Chiral Lie algebras

The sheaf of Chiral Lie algebras $\mathcal{L}_{\mathscr{C}\setminus
P_{\bullet}}(V)$ can be identified with a quotient of the space of sections of
the sheaf
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$
on the affine open set $\mathscr{C}\setminus P_{\bullet}\subset\mathscr{C}$.
Here, for later use in the proof of 5.1.2, in order to describe the action of
$\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$, we explicitly describe
the sheaf
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$,
where $\mathcal{V}_{\mathscr{C}}$ is the contracted product
$\left(V\otimes_{\mathbb{C}}\mathcal{O}_{C}\right)\times_{\mathcal{A}ut\mathcal{O}}\mathcal{A}ut_{\mathscr{C}}$
(see Remark 4.4.2).

For this, suppose we are given a relative curve $\mathscr{C}$, projective over
$S=\operatorname{\mathrm{Spec}}\mathbb{C}[\\![q]\\!]$, with closed fiber
$\mathscr{C}_{0}$ (cut out by the ideal generated by $q$), and an
$(n+1)$-tuple of distinct closed points $P_{0},\ldots,P_{n}\in\mathscr{C}_{0}$
with affine complement $\mathscr{C}_{0}\setminus
P_{\bullet}=\mathscr{C}_{0}\setminus\bigcup_{i}P_{i}$. Let
$B=\mathcal{O}_{\mathscr{C}}(\mathscr{C}_{0}\setminus P_{\bullet})$ denote
those rational functions on $\mathscr{C}$ which are regular at every scheme-
theoretic point of $\mathscr{C}_{0}\setminus P_{\bullet}$ and let
$\widehat{B}$ denote its $q$-adic completion. By [Pri00, Theorem 3.4],
coherent sheaves on $\mathscr{C}$ may be described by specifying coherent
sheaves $M_{U}$ on $U=\operatorname{\mathrm{Spec}}\widehat{B}$, coherent
sheaves $M_{i}$ on
$D_{i}:=\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},P_{i}}$
for each $i$, together with “gluing data on the overlaps.”

The overlaps in this case are described as the formal completions
$D^{\times}_{i}$ of the fiber products
$\operatorname{\mathrm{Spec}}\widehat{B}\times_{\mathscr{C}}\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},P_{i}}$,
and the gluing data is a choice of an isomorphism
$(M_{i})_{D_{i}^{\times}}\cong(M_{U})_{D_{i}^{\times}}$. More concretely, the
$D^{\times}_{i}$ can be described as follows. In a given complete local ring
$\widehat{\mathcal{O}}_{\mathscr{C},P_{i}}$, the ideal generated by $q$ which
describes the closed fiber will factor into a product of of primes
$\wp_{i,j}$. For each of these we can consider the localization and completion
at the prime. We find that $D^{\times}_{i}$ is the disjoint union of the
formal spectra of the rings
$\left((\widehat{\mathcal{O}}_{\mathscr{C},P_{i}})_{\wp_{i,j}}\right)^{\widehat{\
\ }\wp_{i,j}}$. In particular, a coherent sheaf over $D^{\times}_{i}$ is the
data of a finitely generated module over the Noetherian ring
$\left((\widehat{\mathcal{O}}_{\mathscr{C},P_{i}})_{\wp_{i,j}}\right)^{\widehat{\
\ }\wp_{i,j}}$.

In our case, we consider a semistable family of curves $\mathscr{C}/S$, such
that $\mathscr{C}$ is a regular scheme and the closed fiber is reduced. We
focus our attention on an isolated node $Q$, and choose points $P_{\bullet}$
with $Q=P_{0}$ and with $\mathscr{C}_{0}\setminus P_{\bullet}$ smooth. We then
find that in $\widehat{\mathcal{O}}_{\mathscr{C},Q}$, the complete (regular)
local ring at $Q$, we may factor $q=s_{+}s_{-}$. Consequently, we may write
$\widehat{\mathcal{O}}_{\mathscr{C},Q}\cong\mathbb{C}[\\![s_{+},s_{-}]\\!]$.
That is, we have

$\widehat{\mathcal{O}}_{\mathscr{C},Q}\cong\mathbb{C}[\\![s_{+},s_{-},q]\\!]/(s_{+}s_{-}-q)\cong\mathbb{C}[\\![s_{+},s_{-}]\\!].$

In this case, if we let $\wp_{+}$ be the prime generated by $s_{-}$ and
$\wp_{-}$ be the prime generated by $s_{+}$ (in
$\widehat{\mathcal{O}}_{\mathscr{C},Q}$), then we find

$\left((\widehat{\mathcal{O}}_{\mathscr{C},P_{i}})_{\wp_{\pm}}\right)^{\widehat{\
\ }\wp_{\pm}}=\mathbb{C}(\\!(s_{\pm})\\!)[\\![q]\\!].$

As $\mathcal{V}_{\mathscr{C}}$ (and similarly
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$)
is a limit of coherent sheaves $(\mathcal{V}_{\mathscr{C}})_{\leq k}$, we may
use the above procedure to describe it.

We choose $U$ so that the torsor $\mathcal{A}ut_{\mathscr{C}/S}$ is trivial
over $\operatorname{\mathrm{Spec}}\widehat{B}$ via the choice of a function
$s\in\widehat{B}$ such that $ds$ is a free generator of
$\omega_{\mathscr{C}/S}(\operatorname{\mathrm{Spec}}\widehat{B})$ as an
$\widehat{B}$-module. In other words, $s$ is a coordinate on $U$. In
particular, sections of
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$
on $\operatorname{\mathrm{Spec}}\widehat{B}$ can be described as the
$\widehat{B}$ module:

(10)
$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\operatorname{\mathrm{Spec}}\widehat{B})=\bigoplus_{k\in\mathbb{N}}V_{k}\otimes_{\mathbb{C}}\widehat{B}\
(d/ds)^{k-1}.$

######  Remark 4.4.1.

It is important to note that these expressions are not intrinsic to
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$
as a sheaf on $\mathscr{C}$, but rather depend on a choice of parameter $s$.
Different choices give different identifications which correspond to
inhomogeneous isomorphisms between the direct sums, but which do preserve the
filtrations
$(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S})_{\leq
k}$.

Similarly, on $D_{Q}={\mathrm{Spec}}(\widehat{\mathcal{O}}_{\mathscr{C},Q})$,
either $s_{+}$ or $s_{-}$ can be used to define a trivialization of the torsor
$\mathcal{A}ut_{\mathscr{C}}$, this time corresponding to the two possible
choices of generators $ds_{+}/s_{+}$ or $ds_{-}/s_{-}$ of
$\omega_{\mathscr{C}/S}$. These choices allow us to give the following
expressions for the sections of our sheaf on $D_{Q}$ as a
$\widehat{\mathcal{O}}_{\mathscr{C},Q}$-module:

(11)
$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(D_{Q})=\bigoplus_{k\in\mathbb{N}}V_{k}\otimes_{\mathbb{C}}\mathbb{C}[\\![s_{+},s_{-}]\\!]s_{\pm}^{k-1}(d/ds_{\pm})^{k-1}.$

In particular, we may express a section $\sigma$ on $D_{Q}$ with respect to
either the trivialization given by $s_{+}$ or by $s_{-}$. Since
$\gamma(s_{+})=s_{-}$, the trivializations of $\mathcal{A}ut_{\mathscr{C}}$
associated to the coordinates $s_{+}$ and $s_{-}$ (regarded as sections of the
torsor) are related by the order $2$ element
$(-1)^{L_{0}}e^{L_{1}}\in\mathcal{A}ut\mathcal{O}$, which acts on $V$ via the
involution $\gamma$ described in Eq. 6. Hence, we can write sections of the
contracted product
$(V\otimes_{\mathbb{C}}\mathcal{O}_{\mathscr{C}})\times_{\mathcal{A}ut\mathcal{O}}\mathcal{A}ut_{\mathscr{C}}$
over $D_{Q}$ as

$\left(v\otimes f,s_{+}\right)=\left(v\otimes f,\gamma
s_{-}\right)\sim\left(\gamma(v)\otimes f,s_{-}\right),$

for $f\in{\mathcal{O}}_{\mathscr{C}}$. Choosing $v\in V_{\ell}$, the element
of (11) which in the $s_{+}$ trivialization is represented by

$\sum_{i,j\geq 0}v\otimes
x_{i,j}s_{+}^{i}s_{-}^{j}s_{+}^{\ell-1}(d/ds_{+})^{\ell-1},$

is represented with respect to the $s_{-}$ trivialization as

$\sum_{i,j\geq 0}\sum_{m=0}^{\ell}\frac{1}{m!}L_{1}^{m}v\otimes
x_{i,j}s_{+}^{i}s_{-}^{j}s_{-}^{\ell-m-1}(d/ds_{-})^{\ell-m-1}.$

More generally, one should consider a sum of such terms for various values of
$\ell$.

Finally we consider the sheaf
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$
on $D_{\pm}^{\times}=\mathrm{Spec}(\mathbb{C}(\\!(s_{\pm})\\!)[\\![q]\\!])$.
In $D_{\pm}^{\times}$, as in $D_{Q}$, we may use the functions $s_{\pm}$ to
trivialize our torsor. Consequently we have:

(12)
$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(D_{\pm})=\bigoplus_{k\in\mathbb{N}}V_{k}\otimes_{\mathbb{C}}\mathbb{C}(\\!(s_{\pm})\\!)[\\![q]\\!]\
s_{\pm}^{k-1}(d/ds_{\pm})^{k-1}.$

Without loss of generality the trivializing coordinate $s$ on $U$ maps to our
previously chosen trivializing coordinate $s_{+}$ in $D_{+}^{\times}$. That
is, the map $i_{+}\colon D_{+}^{\times}\hookrightarrow U$ corresponds to maps
of rings

(13) $\widehat{B}\rightarrow\mathbb{C}(\\!(s_{+})\\!)[\\![q]\\!],\ s\mapsto
s_{+}.$

Although it is unnecessary here, to map $s$ to both $s_{+}$ and $s_{-}$
simultaneously, one could work étale locally.

For notational convenience, it is useful to consider the action of
$\mathcal{A}ut\mathcal{O}$ as on ${\mathfrak{L}}(V)^{\mathsf{f}}_{0}$, the
degree $0$ part of the ancillary algebra and to recall the notation (7). For
$\rho\in\mathcal{A}ut\mathcal{O}$ and a homogeneous element $a\in V$, we have
$\rho J_{0}(a)=J_{0}(\rho a)$. Further, when we use a coordinate $s$ to
trivialize our torsor $\mathcal{A}ut_{\mathscr{C}}$, we will identify the
expression $a_{[\deg(a)-1+k]}$ with the element
$J_{k}(a)\in{\mathfrak{L}}(V)^{\mathsf{f}}_{-k}$. Finally, we simplify
notation further by omitting the factors of the form $d/ds$ from our
presentations.

Given
$\sigma_{U}\in\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\operatorname{\mathrm{Spec}}\widehat{B})$
we write $(\sigma_{U})_{\pm}$ for its restriction to $D_{\pm}^{\times}$. Using
the notation above, following the explicit expressions of (10) and (12), we
find that if

$\sigma_{U}=\sum_{\ell=0}^{k}v_{\ell}\otimes f_{\ell},$

then writing $f_{+}$ for the expansion (restriction) of the regular function
$f$ to $\mathbb{C}(\\!(s_{+})\\!)[\\![q]\\!]$, we have (as the coordinates are
compatible)

$(\sigma_{U})_{+}=\sum_{\ell=0}^{k}v_{\ell}\otimes(f_{\ell})_{+}=\sum
v_{\ell}\otimes(g_{\ell})_{+}s_{+}^{\ell-1}.$

On the other hand, if
$\sigma_{Q}\in\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(D_{Q})$,
is written as $\sum_{\ell=0}^{k}\sum_{i,j\geq 0}v_{\ell}\otimes
x_{i,j}^{\ell}s_{+}^{i+\ell-1}s_{-}^{j}$. If the section $\sigma_{Q}$, so
represented, is to be compatible and glue together with the section
$\sigma_{U}$ above, we find that

(14) $\sum_{\ell=0}^{k}\sum_{i,j\geq
0}J_{0}(v_{\ell})x_{i,j}^{\ell}s_{+}^{i}s_{-}^{j}=\sum_{i,j,\ell}J_{0}(v_{\ell})x_{i,j}^{\ell}s_{+}^{i-j}q^{j}=\sum_{i,j,\ell}J_{i-j}(v_{\ell})x_{i,j}^{\ell}q^{j}$

must represent the expression for $\sigma_{U}$ restricted to $D_{+}^{\times}$.

To express $\sigma_{U}$ restricted to $D_{-}^{\times}$, following (5), we will
make use of the anti-isomorphism
$\theta\colon{\mathfrak{L}}(V)^{\mathsf{L}}\to{\mathfrak{L}}(V)^{\mathsf{R}}$
described in (5) and related to $\gamma$ via 3.4.2. We then conclude that
$\sigma_{U}$ restricted to $D_{-}^{\times}$ is given by the expression

$\displaystyle\sum_{\ell=0}^{k}\sum_{i,j\geq
0}\gamma(J_{0}(v_{\ell}))x_{i,j}^{\ell}s_{+}^{i}s_{-}^{j}$
$\displaystyle=\sum_{i,j,\ell}J_{0}(\gamma(v_{k}))x_{i,j}^{\ell}s_{-}^{j-i}q^{i}$
$\displaystyle=\sum_{i,j,\ell}J_{j-i}(\gamma(v_{k}))x_{i,j}^{\ell}q^{i}=\sum_{i,j,\ell}\theta(J_{i-j}(v_{\ell}))x_{i,j}^{\ell}q^{i}.$

######  Remark 4.4.2.

Through the above description, we have that the sheaf
$\mathcal{V}_{\mathscr{C}}$ discussed at length in [DGT22a] agrees, even on
the boundary of $\overline{\mathcal{M}}_{g,n}$, with the sheaf
$\mathscr{V}_{\mathscr{C}}$ described in [DGT21].

We conclude with two lemmas which will be useful in our applications in the
next section.

######  Lemma 4.4.3.

Let $\mathscr{C}$ be a family of curves over $S$, possibly with nodal
singularities. Consider a collection of sections $P_{1},\ldots,P_{n}$ such
that $\mathscr{C}\setminus P_{\bullet}=U\subset\mathscr{C}$ is affine, and let
$Q_{1},\ldots Q_{k}\subset\mathscr{C}$ be a finite collection of distinct
closed points in $U$ (possibly including nodes). Let
$D_{Q_{i}}=\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q_{i}}$
be the complete local ring at $Q_{i}$ with maximal ideal
$\widehat{\mathfrak{m}}_{\mathscr{C},Q_{i}}$. Then for any $\ell\geq 0$ and
any invertible sheaf of $\mathcal{O}_{\mathscr{C}}$-modules $\mathcal{L}$, the
natural map

$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)(U)\to\bigoplus_{i}\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)(\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q_{i}}/\left(\widehat{\mathfrak{m}}_{\mathscr{C},Q_{i}}\right)^{\ell})$

is surjective.

###### Proof.

As
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}=\bigcup\limits_{k}\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)_{\leq
k}$, it suffices to show that the map

$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)_{\leq
k}(U)\to\bigoplus_{i}\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)_{\leq
k}(\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q_{i}}/\left(\widehat{\mathfrak{m}}_{\mathscr{C},Q_{i}}\right)^{\ell})$

is surjective for all $k$. Since the sheaf
$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\mathcal{L}\right)_{\leq
k}$ is free of finite rank over $\mathcal{O}_{\mathscr{C}}$, then this holds
true. Indeed, for any coherent sheaf of modules $M$ on $\mathscr{C}$, the
natural map
$M(U)\to\bigoplus_{i}M(\operatorname{\mathrm{Spec}}\mathcal{O}_{\mathscr{C}}(U)/\mathfrak{m}_{\mathscr{C},Q_{i}}(U)^{\ell})=\bigoplus_{i}M(U)\otimes_{\mathcal{O}_{\mathscr{C}}(U)}\mathcal{O}_{\mathscr{C}}(U)/\mathfrak{m}_{\mathscr{C},Q_{i}}(U)^{\ell}$
is seen to be surjective, using the fact that
$\mathcal{O}_{\mathscr{C}}(U)\to\bigoplus_{i}\mathcal{O}_{\mathscr{C}}(U)/\mathfrak{m}_{\mathscr{C},Q_{i}}(U)^{\ell}$
is surjective by the Chinese Remainder Theorem, and that tensoring with $M$ is
right exact. ∎

######  Lemma 4.4.4.

As in Section 4.3, let $\mathscr{C}_{0}$ be a projective curve over
$\mathbb{C}$ with at least one node $Q$, smooth and distinct points
$P_{\bullet}=(P_{1},\ldots,P_{n})$ such that $\mathscr{C}_{0}\setminus
P_{\bullet}$ is affine, and formal coordinates
$t_{\bullet}=(t_{1},\ldots,t_{n})$ at $P_{\bullet}$. Let
$\eta\colon\widetilde{\mathscr{C}}_{0}\rightarrow\mathscr{C}_{0}$ be the
partial normalization of $\mathscr{C}_{0}$ at $Q$, pointed by
$Q_{\pm}:=\eta^{-1}(Q)$, and choose formal coordinates $s_{\pm}$ at $Q_{\pm}$.
Let $W^{1},\dots,W^{n}$ be an $n$-tuple of $V$-modules. Then the map
$\alpha_{0}\colon W^{\bullet}\to W^{\bullet}\otimes\mathfrak{A}$ defined by
$\alpha_{0}(w)=w\otimes 1$ induces a map between the vector spaces of
coinvariants:

$[\alpha_{0}]\colon[W^{\bullet}]_{\left(\mathscr{C}_{0},P_{\bullet},t_{\bullet}\right)}{\overset{\cong}{\longrightarrow}}[W^{\bullet}\otimes\mathfrak{A}]_{\left(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm}\right)},$

which is an isomorphism in case $V$ is $C_{1}$-cofinite.

###### Proof.

Suppose $\mathscr{C}_{0}$ has $m$ nodes in total (including $Q$) and let
$\widetilde{\mathscr{C}_{0}}^{\prime}$ be the (full) normalization of
$\mathscr{C}_{0}$. Following [DGK22, Remark 3.4] we find we have maps

$\textstyle{W^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{0}^{\prime}}$$\scriptstyle{\alpha_{0}}$$\textstyle{W^{\bullet}\otimes\mathfrak{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{0}^{\prime\prime}}$$\textstyle{W^{\bullet}\otimes\mathfrak{A}^{\otimes
m}}$

inducing corresponding maps
$[\alpha_{0}],[\alpha_{0}^{\prime}],[\alpha_{0}^{\prime\prime}]$ on the
respective coinvariants such that $[\alpha_{0}^{\prime}]$ an isomorphism. It
follows that $[\alpha_{0}]$ is injective and therefore remains only to show
that it is also surjective.

For surjectivity, we follow the spirit of the proof of [DGT22a, Prop. 6.2.1].
We may represent an element of $\mathfrak{A}$ as given by an expression
$a^{1}_{[n_{1}]}\cdots a^{k}_{[n_{k}]}\otimes 1\otimes b^{1}_{[m_{1}]}\cdots
b^{r}_{[m_{r}]}$. For simplicity of notation, let us write
$a=a^{1}_{[n_{1}]}\cdots a^{k}_{[n_{k}]}$, $a^{\prime}=a^{2}_{[n_{2}]}\cdots
a^{k}_{[n_{k}]}$ and $b=b^{1}_{[m_{1}]}\cdots b^{r}_{[m_{r}]}$. We will show
that all elements of the form $[w\otimes(a\otimes 1\otimes b)]$ are in the
image of $\alpha_{0}$ by induction on $k-m$, the base case $k-m=0$ being true
by construction (note $b$ has nonpositive degree by definition). For the
induction step, let us suppose that $k>0$ (the case $m<0$ being similar), and
let $d_{+}^{\prime}$ be the degree of $a^{\prime}$ and $d_{-}$ the degree of
$b$. Without loss of generality, we may assume
$\deg(a^{1}_{[n_{1}]})\geq\cdots\geq\deg(a^{k}_{[n_{k}]})\geq 0$. By 4.4.3,
setting $\mathcal{L}=\omega_{\mathscr{C}_{0}}(n_{1}Q_{+}+NQ_{-})$ for
$N>d_{-}-\deg(a^{1})$, we may find a section $\sigma=a^{1}\otimes f$,

$\sigma\in\left(\mathcal{V}_{\widetilde{\mathscr{C}}_{0}}\otimes_{\mathcal{O}_{\mathscr{C}_{0}}}\omega_{\mathscr{C}_{0}}(n_{1}Q_{+}+(d_{-}-1)Q_{-})\right)(\widetilde{\mathscr{C}}_{0}\setminus
P_{\bullet})\subset\left(\mathcal{V}_{\widetilde{\mathscr{C}}_{0}}\otimes_{\mathcal{O}_{\mathscr{C}_{0}}}\omega_{\mathscr{C}_{0}}\right)(\widetilde{\mathscr{C}}_{0}\setminus
P_{\bullet}\sqcup Q_{\pm})$

such that the image $\sigma^{\mathsf{L}}_{Q_{+}}$ of $\sigma$ in
$\left(\mathcal{V}_{\widetilde{\mathscr{C}}_{0}}\otimes_{\mathcal{O}_{\mathscr{C}_{0}}}\omega_{\mathscr{C}_{0}}\right)(\widehat{\mathcal{O}}_{\widetilde{\mathscr{C}}_{0},Q_{+}})\cong{\mathfrak{L}}(V)^{\mathsf{L}}$
has the form $a^{1}_{[n_{1}]}+\widetilde{a}$ where
$\deg(\widetilde{a})<-d_{+}^{\prime}$. By construction,
$\sigma^{\mathsf{L}}_{Q_{-}}$ has degree $<d_{-}$ and consequently
$\sigma^{\mathsf{R}}_{Q_{+}}$ has degree $>-d_{-}$. So we find
$\sigma^{\mathsf{L}}_{Q_{+}}(a^{\prime}\otimes 1\otimes b)=a\otimes 1\otimes
b$ and $(a\otimes 1\otimes b)\sigma^{\mathsf{R}}_{Q_{-}}=0$. This tells us

$\sigma\cdot\big{(}w\otimes(a^{\prime}\otimes 1\otimes b)\big{)}=(\sigma
w)\otimes(a^{\prime}\otimes 1\otimes b)+w\otimes(a\otimes 1\otimes b)$

yielding $[w\otimes(a\otimes 1\otimes b)]=-[(\sigma
w)\otimes(a^{\prime}\otimes 1\otimes b)]$, completing the induction step. ∎

##  5\. Smoothing via strong identity elements

Here we prove 5.0.3, which relates the smoothing property of $V$, described
here in 5.0.1, to the existence of strong identity elements in $\mathfrak{A}$.
5.0.3 relies crucially on 5.1.2. These results are proved in Section 5.1.
Geometric consequences regarding coinvariants are given in Section 5.2.

Throughout this section we will use the notation introduced in Section 4.3,
considering two families of marked, parametrized curves
$(\mathscr{C},P_{\bullet},t_{\bullet})$ and
$(\widetilde{\mathscr{C}},P_{\bullet}\sqcup Q_{\pm},t_{\bullet}\sqcup
s_{\pm})$ over the base scheme
$S=\operatorname{\mathrm{Spec}}(\mathbb{C}[\\![q]\\!])$. As usual,
$\mathfrak{A}_{0}=\mathsf{A}$.

######  Definition 5.0.1.

Given a family $(\mathscr{C},P_{\bullet},t_{\bullet})$, and collection of
$V$-modules $W^{1},\ldots,W^{n}$, an element $\mathscr{I}=\sum_{d\geq
0}\mathscr{I}_{d}q^{d}\in\mathfrak{A}[\\![q]\\!]$ defines a smoothing map for
$W^{\bullet}$ over $(\mathscr{C},P_{\bullet},t_{\bullet})$, if
$\mathscr{I}_{0}=1\in\mathfrak{A}_{0}$, and the map $W^{\bullet}\to
W^{\bullet}\otimes\mathfrak{A}[\\![q]\\!]$, $w\mapsto w\otimes\mathscr{I}$
extends by linearity and $q$-adic continuity to an
$\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$-module homomorphism
$\alpha\colon
W^{\bullet}[\\![q]\\!]\longrightarrow(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]$.
We say that $\mathscr{I}=\sum_{d\geq
0}\mathscr{I}_{d}q^{d}\in\mathfrak{A}[\\![q]\\!]$ defines a smoothing map for
$V$, if it defines a smoothing map for all $V$-modules $W^{\bullet}$, over all
families $(\mathscr{C},P_{\bullet},t_{\bullet})$.

######  Definition 5.0.2.

Smoothing holds for $W^{\bullet}$ over the family
$(\mathscr{C},P_{\bullet},t_{\bullet})$, if there is an element
$\mathscr{I}=\sum_{d\geq 0}\mathscr{I}_{d}q^{d}\in\mathfrak{A}[\\![q]\\!]$
giving a smoothing map for $W^{\bullet}$ over
$(\mathscr{C},P_{\bullet},t_{\bullet})$.
$V$ satisfies smoothing if smoothing holds for all $W^{\bullet}$, over all
families $(\mathscr{C},P_{\bullet},t_{\bullet})$.

######  Theorem 5.0.3.

Let $V$ be a VOA. Then the algebras $\mathfrak{A}_{d}$ admit strong identity
elements for all $d\in\mathbb{N}$ if and only if $V$ satisfies smoothing.

###  5.1. Proof of 5.0.3

Following the idea of 3.3.1(2), we make the following definition:

######  Definition 5.1.1.

We say that a sequence $(\mathscr{I}_{d})_{d\in\mathbb{N}}$, with
$\mathscr{I}_{d}\in\mathfrak{A}$ satisfies the strong identity element
equations if for every homogeneous $a\in V$, and $n\in\mathbb{Z}$ such that
$n\leq d$, we have

(15) $J_{n}(a)\mathscr{I}_{d}=\mathscr{I}_{d-n}J_{n}(a).$

In 5.1.1 there is no assumption on the (bi-)degrees of the elements
$\mathscr{I}_{d}\in\mathfrak{A}$. However, if
$\mathscr{I}_{d}\in\mathfrak{A}_{d}$ is an identity element for each $d$, then
by 3.3.1 they satisfy the strong identity element equations if and only if
they are strong identity elements.

######  Proposition 5.1.2.

Let $V$ be a VOA and let $\mathscr{I}_{d}\in\mathfrak{A}$ for
$d\in\mathbb{N}$. Then $\mathscr{I}=\sum\mathscr{I}_{d}q^{d}$ defines a
smoothing map for $W^{\bullet}$ over $(\mathscr{C},P_{\bullet},t_{\bullet})$
if and only if the sequence $(\mathscr{I}_{d})$ satisfies the strong identity
element equations (15).

###### Proof.

The map $\alpha\colon
W^{\bullet}[\\![q]\\!]\to(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]$ is a map
of $\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$-modules if and only if,
for every $\sigma\in\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$ and
$u\in W^{\bullet}$, one has $\alpha(\sigma(u))=\sigma(\alpha(u))$. Here, the
left hand side equals $(\sigma\cdot u)\otimes\mathscr{I}$. To describe the
right hand side, as is explained in the beginning of Section 4.4, we recall
that elements of the Lie algebra $\mathcal{L}_{\mathscr{C}\setminus
P_{\bullet}}(V)$ are represented by sections of the sheaf
$\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}$
over the affine open set $\mathscr{C}\setminus P_{\bullet}$. Consequently we
can understand the right hand side in terms of the maps

$\displaystyle\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\mathscr{C}\setminus
P_{\bullet})$
$\displaystyle\to\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(D_{\pm}^{\times})\cong
V\otimes_{\mathbb{C}}\mathbb{C}(\\!(t)\\!)[\\![q]\\!]$ $\displaystyle\sigma$
$\displaystyle\mapsto\sigma_{\pm}^{\mathsf{L}}.$

We let
$\sigma_{-}^{\mathsf{R}}=\theta(\sigma_{-}^{\mathsf{L}})\in\mathbb{C}(\\!(t^{-1})\\!)[\\![q]\\!]$.
We then have

$\sigma(u\otimes\mathscr{I})=\sigma(u)\otimes\mathscr{I}+u\otimes(\sigma_{+}^{\mathsf{L}}\otimes
1+1\otimes\sigma_{{}_{-}}^{\mathsf{R}})(\mathscr{I}).$

It follows that $\alpha$ is a map of $\mathcal{L}_{\mathscr{C}\setminus
P_{\bullet}}(V)$-modules if and only if

(16) $\sigma\cdot\mathscr{I}=\left(\sigma_{+}^{\mathsf{L}}\otimes
1+1\otimes\sigma_{-}^{\mathsf{R}}\right)\cdot\mathscr{I}=0.$

We now reframe this in the language developed towards the end of Section 4.4.
For a section
$\sigma\in\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)_{\leq
k}(\mathscr{C}\setminus P_{\bullet})$, writing $s_{+}s_{-}=q$ on
$\widehat{\mathcal{O}}_{\mathscr{C},Q}$, we may write (in terms of the local
trivializations of Section 4.4)

$\sigma|_{D_{Q}}=\sum_{\ell=0}^{k}\sum_{i,j\geq 0}J_{0}(v_{\ell})\
x_{i,j}^{\ell}\ s_{+}^{i}s_{i}^{j}$

and for this section $\sigma$ we have

$\sigma_{+}^{\mathsf{L}}=\sum_{\ell=0}^{k}\sum_{i,j\geq
0}J_{i-j}(v_{\ell})x^{\ell}_{i,j}q^{j}\text{ \quad and
\quad}\sigma_{-}^{\mathsf{R}}=\sum_{\ell=0}^{k}\sum_{i,j\geq
0}J_{i-j}(v_{\ell})x^{\ell}_{i,j}q^{i}.$

Putting this together with (16), we find that smoothing holds if and only if
for all $\sigma$ as above (and for all $k$), we have

$\sum_{\ell=0}^{k}\sum_{i,j,d\geq
0}x_{ij}\left(J_{i-j}(v_{\ell})\cdot\mathscr{I}_{d}\
q^{d+j}-\mathscr{I}_{d}\cdot J_{i-j}(v_{\ell})\ q^{d+i}\right)=0.$

This in turn holds if and only if each coefficient of $q^{m}$ is zero,
translating to the statement

(17) $\sum_{\ell=0}^{k}\ \sum_{0\leq i,j\leq
m}x_{ij}\left(J_{i-j}(v_{\ell})\cdot\mathscr{I}_{m-j}-\mathscr{I}_{m-i}\cdot
J_{i-j}(v_{\ell})\right)=0,$

for every $m\geq 0$.

We note that the systems of equations

$J_{n}(v_{\ell})\mathscr{I}_{d}=\mathscr{I}_{d-n}J_{n}(v_{\ell}),\text{ with
}n\leq d,\text{ and }d\in\mathbb{N},v_{\ell}\in V_{\ell},\ell\in\mathbb{N},$

and

$J_{i-j}(v_{\ell})\cdot\mathscr{I}_{m-j}-\mathscr{I}_{m-i}\cdot
J_{i-j}(v_{\ell})=0,\text{ with }0\leq i,j\leq m,\text{ and
}m\in\mathbb{N},v_{\ell}\in V_{\ell},\ell\in\mathbb{N},$

are equivalent after a change of variables. We then have showed that, if
$(\mathscr{I}_{d})_{d\in\mathbb{N}}$ satisfies the strong identity element
equations, it follows that $\mathscr{I}$ defines a smoothing map. It remains
to show the converse, namely that if (17) holds for every $\sigma$, then the
strong identity element equations hold.

We do this by the following strategy: we will show that for every $0\leq
i_{0},j_{0}\leq m$, $m\in\mathbb{N}$ and $v_{\ell_{0}}^{\prime}\in
V_{\ell_{0}}$, we may find a section
$\sigma\in\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\mathscr{C}\setminus
P_{\bullet})$ so that the expansion of $\sigma$ at $Q$ has the form

(18) $\sigma|_{D_{Q}}=\sum_{\ell=0}^{k}\sum_{i,j\geq 0}J_{0}(v_{\ell})\
x_{i,j}^{\ell}\
s_{+}^{i}s_{i}^{j}=J_{0}(v^{\prime}_{\ell_{0}})s_{+}^{i_{0}}s_{-}^{j_{0}}+\sum_{\ell=0}^{k}\sum_{i,j\geq
m}J_{0}(v_{\ell})\ x_{i,j}^{\ell}\ s_{+}^{i}s_{i}^{j}.$

That is, we argue that the coefficients $x_{i,j}$ in of the terms in (18) of
degree less than $m$ are only nonzero in the case $i=i_{0},j=j_{0}$, and in
this case $x_{i_{0},j_{0}}=1$. For such a section $\sigma$, (17) simply
becomes
$J_{i_{0}-j_{0}}(v_{\ell})\cdot\mathscr{I}_{m-j_{0}}-\mathscr{I}_{m-i_{0}}\cdot
J_{i_{0}-j_{0}}(v_{\ell})=0$, which, as has been noted, is equivalent to the
strong identity element equations once we run this argument for all
$i_{0},j_{0}$ and $m$. For this final step, we note that by 4.4.3 we have a
surjective map

$\textstyle{\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\mathscr{C}\setminus
P_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)_{\leq
k}(\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q}/\left(\widehat{\mathfrak{m}}_{\mathscr{C},Q}\right)^{2m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)_{\leq
k}(\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q})\otimes_{\widehat{\mathcal{O}}_{\mathscr{C},Q}}\widehat{\mathcal{O}}_{\mathscr{C},Q}/(\widehat{\mathfrak{m}}_{\mathscr{C},Q})^{2m}}$

for every $m\geq 0$. Hence there exists
$\sigma\in\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\mathscr{C}\setminus
P_{\bullet})$ whose image in
$\left(\mathcal{V}_{\mathscr{C}}\otimes_{\mathcal{O}_{\mathscr{C}}}\omega_{\mathscr{C}/S}\right)(\operatorname{\mathrm{Spec}}\widehat{\mathcal{O}}_{\mathscr{C},Q})$
is congruent modulo
$(\widehat{\mathfrak{m}}_{\mathscr{C},Q})^{2m}=(s_{+},s_{-})^{2m}$ to
$J_{0}(v^{\prime}_{\ell_{0}})s_{+}^{i_{0}}s_{-}^{j_{0}}$. It follows (18)
holds for this $\sigma$ as desired and the proof is complete. ∎

We note that already 5.1.2 shows that the smoothing property never depends on
modules or specific families of curves:

######  Corollary 5.1.3.

Smoothing holds for $W^{\bullet}$ over a family
$(\mathscr{C},P_{\bullet},t_{\bullet})$ if and only if $V$ satisfies
smoothing.

###### Proof.

If smoothing holds for $W^{\bullet}$ over a family
$(\mathscr{C},P_{\bullet},t_{\bullet})$, then by 5.1.2, the sequence
$(\mathscr{I}_{d})_{d\in\mathbb{Z}}$ satisfies the strong identity element
equations. But then, invoking again 5.1.2, we deduce that this sequence
defines a smoothing map for any choice of modules and family of curves. ∎

In what follows, for an element
$b\in\mathfrak{A}=\bigoplus_{i,j}\mathfrak{A}_{i,j}$, we write
$b_{i,j}\in\mathfrak{A}_{i,j}$ for the corresponding homogeneous component of
$b$.

######  Lemma 5.1.4.

For $V$ a VOA, if the sequence $(\mathscr{I}_{d})_{d\in\mathbb{N}}$ with
$\mathscr{I}_{d}\in\mathfrak{A}$ satisfies the strong identity element
equations (15), then so does the sequence
$(\mathscr{I}_{d}^{\prime})_{d\in\mathbb{N}}$ where
$\mathscr{I}_{d}^{\prime}:=(\mathscr{I}_{d})_{d,-d}$.

###### Proof.

Suppose we have a sequence $(\mathscr{I}_{d})$ satisfying the strong identity
element equations. When we equate terms of like degree in the expression

$J_{n}(a)\cdot\mathscr{I}_{d}=\mathscr{I}_{d-n}\cdot J_{n}(a),$

we obtain

$J_{n}(a)\cdot(\mathscr{I}_{d})_{i+n,j}=(\mathscr{I}_{d})_{i,j-n}\cdot
J_{n}(a)$

for every $i,j$. In particular, for
$\mathscr{I}_{d}^{\prime}=(\mathscr{I}_{d})_{d,-d}$, we find that the strong
identity element equations (15) hold for the sequence
$(\mathscr{I}_{d}^{\prime})_{d\in\mathbb{N}}$, as was claimed. ∎

In what follows we will use the following equalities, which are a direct
consequence of B.2.5. Let $\mathfrak{a},\mathfrak{b}\in\mathfrak{A}$. Then for
every $u,w\in\mathscr{U}$ we have

(19)
$u\cdot(\mathfrak{a}\star\mathfrak{b})=(u\cdot\mathfrak{a})\star\mathfrak{b}\qquad\text{
and }\qquad(\mathfrak{a}\star\mathfrak{b})\cdot
w=\mathfrak{a}\star(\mathfrak{b}\cdot w).$

######  Lemma 5.1.5.

Suppose we have a collection of elements $\mathscr{I}_{d}\in\mathfrak{A}_{d}$
for each $d\geq 0$, with $\mathscr{I}_{0}=1\in\mathfrak{A}_{0}$. Then,
$\mathscr{I}_{d}$ is a strong identity element in
$\mathfrak{A}_{d}\subset\mathfrak{A}$, for all $d\in\mathbb{N}$, if and only
if the sequence $(\mathscr{I}_{d})_{d\in\mathbb{N}}$ satisfies the strong
identity element equations (15).

###### Proof.

3.3.1(2) with $\mathfrak{a}=J_{n}(v)$ implies that strong identity elements
satisfy the strong identity element equations (15), so we are left to prove
the converse statement. To show that $\mathscr{I}_{d}$ is a strong identity
element for each $d$, it suffices to show that $\mathscr{I}_{d}$ acts as the
identity element on $\mathfrak{A}_{d,e}$ for every $e\in\mathbb{Z}$. That is,
for every $\mathfrak{a}\in\mathfrak{A}_{0,e}$, and $n_{1}\leq\cdots\leq
n_{r}<0$ with $\sum n_{i}=-d$, we need to show

$\mathscr{I}_{d}\star\left(J_{n_{1}}(v_{1})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)=J_{n_{1}}(v_{1})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}.$

We argue by induction on $r$. The base case $r=0$ holds since by assumptions
$\mathscr{I}_{0}=1\in\mathfrak{A}_{0}=\mathsf{A}$, hence
$\mathscr{I}_{0}\star\mathfrak{a}=1\cdot\mathfrak{a}=\mathfrak{a}$. For the
inductive step, we write:

$\displaystyle\mathscr{I}_{d}\star\left(J_{n_{1}}(v_{1})\cdot
J_{n_{2}}(v_{2})\cdots J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)$
$\displaystyle=\mathscr{I}_{d}\star\left(\left(J_{n_{1}}(v_{1})\right)\left(J_{n_{2}}(v_{2})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)\right)$
$\displaystyle\scriptsize{\eqref{eq:lemStarUV1}}\quad$
$\displaystyle=\left(\mathscr{I}_{d}\cdot
J_{n_{1}}(v_{1})\right)\star\left(J_{n_{2}}(v_{2})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)$
$\displaystyle\scriptsize{\eqref{eq:StrongUnitEquation}}\quad$
$\displaystyle=\left(J_{n_{1}}(v_{1})\cdot\mathscr{I}_{d+n_{1}}\right)\star\left(J_{n_{2}}(v_{2})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)$
$\displaystyle\scriptsize{\eqref{eq:lemStarUV1}}\quad$
$\displaystyle=J_{n_{1}}(v_{1})\cdot\left(\mathscr{I}_{d+n_{1}}\star\left(J_{n_{2}}(v_{2})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a}\right)\right)$ (by induction)
$\displaystyle=J_{n_{1}}(v_{1})J_{n_{2}}(v_{2})\cdots
J_{n_{r}}(v_{r})\cdot\mathfrak{a},$

where the last identity holds by induction. ∎

We may now complete the proof of 5.0.3.

###### Proof of 5.0.3.

Suppose the algebras $\mathfrak{A}_{d}$ admit strong identity elements.
Writing $\mathscr{I}_{d}$ for these unities, we can apply 5.1.5 to deduce that
the sequence $(\mathscr{I}_{d})_{d\in\mathbb{N}}$ satisfies the strong
identity element equations and therefore, by 5.1.2, the element
$\mathscr{I}=\sum\mathscr{I}_{d}q^{d}$ defines a smoothing map for any family
of marked curves and choice of modules $W^{\bullet}$. Hence $V$ satisfies
smoothing.

Conversely, if $V$ satisfies smoothing, there exists
$\mathscr{I}=\sum\mathscr{I}_{d}q^{d}$ which defines a smoothing map for any
family of marked curves and choice of modules $W^{\bullet}$, then by 5.1.2,
the sequence $(\mathscr{I}_{d})_{d\in\mathbb{N}}$ satisfies the strong
identity element equations. Using 5.1.4 we may find a new sequence
$(\mathscr{I}_{d}^{\prime})_{d\in\mathbb{N}}$ with
$\mathscr{I}_{d}^{\prime}\in\mathfrak{A}_{d}$ which also satisfy the strong
identity element equations. It follows from 5.1.5 that the elements
$\mathscr{I}_{d}^{\prime}$ are strong identity elements. ∎

###  5.2. Geometric results

We describe in this section some statements about coinvariants, most of which
are implications of 5.0.3.

######  Corollary 5.2.1.

For any VOA $V$, let $W^{\bullet}$ be $V$-modules such that the sheaf
$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$ is coherent over $S$. Assume that
$\mathfrak{A}_{d}$ admits a strong identity element $\mathscr{I}_{d}$ for
every $d\in\mathbb{N}$. Set $\mathscr{I}=\sum_{d\geq 0}\mathscr{I}_{d}q^{d}$,
and let $\alpha\colon
W^{\bullet}[\\![q]\\!]\to(W^{\bullet}\otimes\mathfrak{A}){[\\![q]\\!]}$ be the
map induced by $w\mapsto w\otimes\mathscr{I}$ (see 5.0.1). Then the diagram

${{[W^{\bullet}[\\![q]\\!]]_{(\mathscr{C},P_{\bullet},t_{\bullet})}}{}{}}$${{[W^{\bullet}\otimes\mathfrak{A}]_{(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup
s_{\pm})}[\\![q]\\!]}{}}$${{{[W^{\bullet}]}_{(\mathscr{C}_{0},P_{\bullet},t_{\bullet})}}}$${{[W^{\bullet}\otimes\mathfrak{A}]}_{(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup
s_{\pm})}}$$\scriptstyle{[\alpha]}$$\scriptstyle{[\alpha_{0}]}$

commutes, where $\alpha_{0}\colon W^{\bullet}\to W\otimes\mathfrak{A}$ is
given by $w\mapsto w\otimes\mathscr{I}_{0}$.

###### Proof.

The vertical maps are given by imposing the condition $q=0$, and are
surjective. After the identification of $\mathfrak{A}$ with $\Phi(\mathsf{A})$
provided in 3.4.5, we see that the map $[\alpha_{0}]$ is well defined as in
[DGK22, Proposition 3.3].

By the proof of 5.0.3 we deduce that the map $\alpha$ is a map of
$\mathcal{L}_{\mathscr{C}\setminus P_{\bullet}}(V)$-modules and since
$\mathcal{L}_{\mathscr{C}\setminus
P_{\bullet}}(V)\subset\mathcal{L}_{\widetilde{\mathscr{C}}\setminus(P_{\bullet}\sqcup
Q_{\pm})}(V)$, this induces a map of coinvariants

$[W^{\bullet}[\\![q]\\!]]_{(\mathscr{C},P_{\bullet},t_{\bullet})}\longrightarrow[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$

whose reduction modulo $q$ is indeed $[\alpha_{0}]$. Finally, we use Corollary
4.3.1 to identify

$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup
s_{\pm})}\cong[W^{\bullet}\otimes\mathfrak{A}]_{(\widetilde{\mathscr{C}}_{0},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}[\\![q]\\!]$

which concludes the proof. ∎

To state the following consequence, we recall that sheaves of coinvariants
$\mathbb{V}(V;W^{\bullet})$ are attached to coordinatized curves
$(C,P_{\bullet},t_{\bullet})$ such that $C\setminus P_{\bullet}$ is affine. By
Propagation of vacua [Cod19, DGT22a], we may drop the latter condition, so
that $\mathbb{V}(V;W^{\bullet})$ can be considered a sheaf on
${\widetriangle{\mathcal{M}}_{g,n}}$, the stack of stable coordinatized
curves. Depending on $V$ and on $W^{\bullet}$, this further descends to a
sheaf over ${\overline{\mathcal{M}}_{g,n}}$. To formulate our next result it
is convenient to introduce the following notation.

######  Definition 5.2.2.

Let $V$ be a VOA. We say that $V$ has coherent coinvariants if for every
family of stable and pointed coordinatized curves
$(C,P_{\bullet},t_{\bullet})$, and modules $W^{\bullet}$, the sheaf of
coinvariants $[W^{\bullet}]_{(C,P_{\bullet},t_{\bullet})}$ is coherent.

######  Definition 5.2.3.

Let $V$ be a VOA. We say that $V$ has finite gluing if for every stable and
pointed coordinatized curve $(\mathscr{C},P_{\bullet},t_{\bullet})$ with a
node $Q$, and modules $W^{\bullet}$, the space of coinvariants
$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$ is coherent over
$\operatorname{\mathrm{Spec}}(\mathbb{C}[\\![t]\\!])$.

######  Remark 5.2.4.

We make two observations:

  1. (i)

We note that if $V$ is $C_{2}$-cofinite, then by [DGK22, Corollary 4.2], $V$
has coherent coinvariants and finite gluing. As we shall see, this is also
true for VOAs like the Heisenberg which are generated in degree 1, but are not
$C_{2}$-cofinite.

  2. (ii)

By Corollary 4.3.1, if $V$ has finite gluing it follows that
$[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}$ is actually free over
$\mathbb{C}[\\![q]\\!]$.

We begin with an auxiliary result.

######  Lemma 5.2.5.

Let $V$ be a $C_{1}$-cofinite VOA that satisfies smoothing and such that
${[W^{\bullet}[\\![q]\\!]]_{(\mathscr{C},P_{\bullet},t_{\bullet})}}$ and
${[(W^{\bullet}\otimes\mathfrak{A})[\\![q]\\!]]_{(\widetilde{\mathscr{C}},P_{\bullet}\sqcup
Q_{\pm},t_{\bullet}\sqcup s_{\pm})}}$ are coherent over $S$. Then the map
$[\alpha]$ defined in 5.2.1 is an isomorphism.

###### Proof.

Since $V$ is $C_{1}$-cofinite, 4.4.4 ensures that $[\alpha_{0}]$ is an
isomorphism. Since the source and target of $[\alpha]$ is finitely generated
and the target is locally free (see Remark 5.2.4 (ii)), Nakayama’s lemma
ensures that $[\alpha]$ is an isomorphism as well. ∎

To state the next results, we shall refer to the moduli stacks
${\widetriangle{\mathcal{M}}_{g,n}}$, parametrizing families of stable pointed
curves of genus $g$ with coordinates, and $\overline{\mathcal{J}}_{g,n}$, of
stable pointed curves of genus $g$ with first order tangent data, and
projection maps

${\widetriangle{\mathcal{M}}_{g,n}}\rightarrow\overline{\mathcal{J}}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n},$

discussed in detail in [DGT22a, §2]. Recall the notation from Remark 4.1.2.

######  Corollary 5.2.6.

Let $W^{1},\dots,W^{n}$ be simple modules over a $C_{1}$-cofinite vertex
operator algebra $V$, such that coinvariants are coherent for curves of genus
$g$, and such that $\mathfrak{A}_{d}(V)$ admit strong identity elements for
all $d\in\mathbb{Z}_{\geq 0}$. Then sheaves of coinvariants are locally free,
giving rise to a vector bundle
$\mathbb{V}_{g}(V;W^{\bullet})^{\overline{\mathcal{J}}_{g,n}}$ on
$\overline{\mathcal{J}}_{g,n}$. If the conformal dimensions of
$W^{1},\dots,W^{n}$ are rational, these sheaves define vector bundles
$\mathbb{V}_{g}(V;W^{\bullet})$ on ${\overline{\mathcal{M}}_{g,n}}$.

###### Proof.

Since a sheaf of $\mathcal{O}_{S}$-modules is locally free if and only if is
coherent and flat, in order to show a coherent sheaf
$[W^{\bullet}]_{\mathcal{L}}$ is locally free, it suffices to show that it is
flat. For this, we can use the valuative criteria of [Gro67, Thm 11.8.1, §3]
to reduce to the case that our base scheme is
$S=\operatorname{\mathrm{Spec}}(\mathbb{C}[\\![q]\\!])$. By [Har77, Ex.
II.5.8], since $S$ is Noetherian and reduced, and since formation of
coinvariants commutes with base change, by 4.1.1, it suffices to check that
vector spaces of coinvariants have the same dimension over all pointed and
coordinatized curves.

Our strategy for checking this condition holds is to argue by induction on the
number of nodes, reducing to the base case where the curve has no nodes.

To take the inductive step, following the notation of 5.2.1, let
$\mathscr{C}_{0}\rightarrow{\mathrm{Spec}}(k)$ be a nodal curve with $k+1$
nodes, and let $\mathscr{C}\rightarrow{\mathrm{Spec}}(\mathbb{C}[\\![q]\\!])$
be a smoothing family with $\mathscr{C}_{0}$ the special fiber. By 4.3.4 and
by 5.2.5, we deduce that $[\alpha]$ is an isomorphism, so that the dimension
of the space of coinvariants associated with $\mathscr{C}_{0}$ agrees with the
dimension of the space of coinvariants for the partial normalization
$\widetilde{\mathscr{C}}_{0}$, a curve with $k$ nodes. Therefore, by
induction, the vector space
$[W^{\bullet}]_{({\mathscr{C}}_{0},P_{\bullet},t_{\bullet})}$ has the same
dimension as the vector space of coinvariants associated with a smooth curve.

We are then left to show that spaces of coinvariants associated with smooth
curves of the same genus have the same dimensions. This holds since