# Quasi-lisse vertex (super)algebras

Hao Li  Yau Mathematical Sciences Center<EMAIL_ADDRESS>
###### Abstract.

We study the twisted arc spaces and the vertex superalgebras. We obtain new
finiteness conditions for the convergence of genus-0 and genus-1 $n$-point
correlation functions of quasi-lisse vertex (super)algebras.

###### Contents

  1. 1 Introduction
  2. 1.1 Notations and conventions
  3. 2 Vertex superalgebras
    1. 2.1 Untwisted arc superspaces
    2. 2.2 Twisted arc superspaces
  4. 3 Quasi-lisse vertex superalgebras
    1. 3.1 Definitions
    2. 3.2 Twisted modules
    3. 3.3 Ordinary twisted modules
    4. 3.4 Twisted affine Kac-Moody Lie algebras
    5. 3.5 Twisted arc superalgebras and superconformal vertex algebras
  5. 4 Genus-0 correlation functions
    1. 4.1 Intertwining operators among weak $g$-twisted modules
    2. 4.2 Convergence of genus-0 correlation functions
  6. 5 Genus-$1$ correlation functions
    1. 5.1 Identities for formal $q$-traces
    2. 5.2 The ring $R^{\prime}$
    3. 5.3 Partial differential equations satisfied by genus-$1$ $n$ point correlation functions
    4. 5.4 Genus-$1$ $1$-point functions
    5. 5.5 Example of affine vertex operator algebras
  7. A Coordinate transformation
  8. B Modular forms
  9. C Twisted modular forms

##  1\. Introduction

Vertex algebras and their representations are the rigorous mathematical
formulation of two-dimensional conformal field theory of a certain type. In
the seminal work [Zhu96], Zhu first introduced the concept of the
$C_{2}$-algebra of a vertex operator algebra $V$, which has a Poisson algebra
structure. The finite dimensionality of the $C_{2}$-algebra is used by Zhu to
derive some modular linear differential equations (MLDEs) satisfied by genus-1
correlation functions associated to $V$. This idea is then generalized by
Dong-Li-Mason [DLM00] to the case of twisted representations of $V$, by
Miyamoto [Miy00] and later by Huang [Hua05b] to the case of $q$-traces of
products of intertwining operators. All these works assume the vertex operator
algebras and their modules to be $C_{2}$-cofinite.

In order to give an analogous finiteness condition defined by Beilinson,
Feigin and Mazur [BFM91], [Ara12] Arakawa studied the $C_{2}$-algebra of $V$
from the geometric perspective by thinking of them as the coordinate ring of
associated variety of $V$; $\text{dim}(X_{V})=0$ is equivalent to the
$C_{2}$-cofiniteness and he showed that the $C_{2}$-cofiniteness condition is
equivalent to the condition that the dimension of singular support of $V$ is
trivial. Later, Arakawa and Kawasetsu [AK18] discovered that the argument by
Zhu to derive MLDE still works if one enlarges $X_{V}$ to have finitely many
symplectic leaves. The corresponding vertex algebra is called quasi-lisse
vertex algebra. The crucial step is to use the property of $\text{Specm}{R}$
with finitely many symmplectic leaves [ES10]:

(1) $\displaystyle\text{dim}R/\\{R,R\\}<\infty.$

Arakawa [Ara15] proved that the associated variety of the affine vertex
operator algebra at the admissible level is the closure of some nilpotent
orbit. Thus quasi-lisse vertex algebras can be viewed as the vertex algebraic
generalization of affine VOA at the admissible level.

Meanwhile, Beem-Lemos-Liendo-Peelars-Rastelli-van Rees [BLL+15] constructed a
remarkable map from four-dimension superconformal field theory (4d SCFT) to
two-dimension conformal field theory (2d CFT). It was conjectured by Beem and
Rastelli that the Higgs branch of 4d SCFT should coincide with the associated
variety of corresponding 2d CFT. One can identify the 4d Higgs branch with the
3d Coulomb branch via 3D mirror symmetry. If the mirror theory is quiver gauge
theory, Braveman-Finkelberg-Nakajima defined the Coulomb branch rigorously
([Nak16], [BFN18], [Bra]) which is expected to have finitely many symplectic
leaves. Thus, there is a demand to systematically study the quasi-lisse vertex
operator algebras. In the present paper, we make some modest efforts in this
direction.

Recently McRae [McR21] proved that for some proper $C_{2}$-cofinite vertex
operator algebra, its tensor category of grading-restricted generalized
$V$-modules is rigid. He used the method of Huang-Moore-Seiberg by expanding
$2$-point genus-$1$ correlation functions. Here, the $2$-point function means
the $q$-trace of the product of two intertwining operators. Thus, in order to
prove rigidity the convergence of genus-$1$ $2$-point functions is crucial.
The convergence result for genus-$1$ correlation functions was obtained by
Huang [Hua05b] for $C_{2}$-cofinite algebras. Trying to relax this convergence
condition under the setting of quasi-lisse vertex operator (super)algebra is
one of the main motivations of this paper.

Now let us give the description of the main results in this paper. We extend
the notion of quasi-lisse vertex algebras to the supercase, and show that it
has only finitely many simple ordinary $g$-twisted modules with the finite
order $g$. The singular support of a $V$-module is a subscheme of the arc
space of the associated scheme $\tilde{X}_{V}$ of $V$. For twisted
$V$-modules, we do not have the Li’s s filtration. We try to abelianize some
twisted modules using twisted arc spaces for some familiar examples, i.e.,
twisted affine Lie algebras, N=1 superconformal vertex algebras, N=2
superconformal vertex algebras. We believe that this would play a role in our
future work of studying convergence problems in orbifold conformal field
theory.

Huang [Hua05a] showed that for $C_{1}$-cofinite vertex operator algebras and
its $C_{1}$-cofinite modules, the genus-$0$ $n$-point functions satisfy
certain differential equations with regular singularity. We prove that the
$C_{1}$-cofiniteness condition can be replaced by
$\text{dim}R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}/\\{R_{V},R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}\\}<\infty$
for certain vertex operator algebras.

######  Theorem 1.1.

Suppose $V$ is of CFT type, self-dual, finitely generated by vectors of weight
$1$. Let $W_{i}$ for $i=0,1,2,3$ be weak $\mathbb{Q}$-graded $V$-modules.
Suppose
$\text{dim}R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}/\\{R_{V},R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}\\}<\infty$.
Then for any $w_{i}\in W_{i}$ $(i=0,1,2,3),$ there exist

$a_{k}(z_{1},z_{2}),b_{l}(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm},(z_{1}-z_{2})^{-1}]$

for $k=1,...,m$ and $l=1,...,n$ such that for any discretely weak
$\mathbb{Q}$-graded $V$-modules $W_{4},W_{5}$ and $W_{6}$, any intertwining
operators
$\mathcal{Y}_{1},\mathcal{Y}_{2},\mathcal{Y}_{3},\mathcal{Y}_{4},\mathcal{Y}_{5}$
and $\mathcal{Y}_{6}$ of types

$\binom{W_{0}^{\prime}}{W_{1}W_{4}},\binom{W_{4}}{W_{2}W_{3}},\binom{W_{5}}{W_{1}W_{2}},\binom{W_{0}^{\prime}}{W_{5}W_{3}},\binom{W_{0}^{\prime}}{W_{2}W_{6}}\,\,\,\,\text{and}\,\,\,\,\binom{W_{6}}{W_{1}W_{3}},$

respectively, the series

(2) $\displaystyle\langle
w_{0},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle,$
(3) $\displaystyle\langle
w_{0},\mathcal{Y}_{4}(\mathcal{Y}_{3}(w_{1},z_{1}-z_{2})w_{2},z_{2})w_{3}\rangle,$

and

(4) $\displaystyle\langle
w_{0},\mathcal{Y}_{5}(w_{2},z_{2})\mathcal{Y}_{6}(w_{1},z_{1})w_{3}\rangle,$

are convergent in the regions $|z_{1}|>|z_{2}|>0,$ $|z_{2}|>|z_{1}-z_{2}|>0$
and $|z_{2}|>|z_{1}|>0,$ respectively; if in addition that the first two
conditions in [Hua05a][Theorem 3.5] are satisfied, they satisfy the
convergence and extension properties (see [Hua05a][Definition 3.4] for
definition).

We also make a conjecture about the above convergence result for the
intertwining operators of type $\binom{W_{3}}{W_{1}\;W_{2}}$, where $W_{1}$ is
a finitely strongly generated $V$-module and $W_{2}$, $W_{3}$ are ordinary
twisted $V$-modules.

For genus-$1$ case, we consider the weak $g$-twisted $V$-module
$\tilde{W}_{n}$. Let $h$ be a current satisfying

$L(n)h=\delta_{n,0}h,\quad h_{(n)}h=k\delta_{n,1}\mathbf{1},$

for all $n\geq 0$ and for $k\in\mathbb{C}$ such that $h_{(0)}$ acts semisimply
on $\tilde{W}_{n}$. We assume that
$\text{tr}_{\tilde{W}_{n}}q^{h_{(0)}}_{s}q^{L_{(0)}},$ where $q_{s}=e^{2\pi
is}$, is a well-defined $(q_{s},q)$-series. In the future, we will call such
$h$ the Cartan element and such weak $g$-twisted module to be $h$-stable.
Under the assumption that duality property holds, we also get a similar weaker
finiteness condition

######  Theorem 1.2.

Let $g$ be an automorphism of $V$ of finite order. Let $h$ be the Virasoro
primary vector. Assume $g(V\setminus C_{2}(V))\subset V\setminus C_{2}(V)$.
Let $W_{i}$, $i=1,2,...,n$ be weak $V$-modules, $\tilde{W}_{i}$, $i=1,2,...,n$
be $h$-stable weak $g$-twisted $V$-modules. Assume
$\tilde{W}_{0}=\tilde{W}_{n}$. If
$\text{dim}(R_{W_{1}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{n}}/\\{(R_{V})^{g},R_{W_{1}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{n}})\\}<\infty$
and $\mathcal{Y}_{i},$ $i=1,\cdots,n,$ are intertwining operators of type
$\displaystyle\binom{\tilde{W}_{i-1}}{W_{i}\;\;\tilde{W}_{i}}$.

  * •

For any homogeneous elements $w_{i}\in W_{i}$, $i=1,...,n,$ there exist

$a_{p,i}(z_{1},...,z_{n};q;h)\in
R^{\prime}_{p},\,\,\,b_{p,i}(z_{1},...,z_{n};q;h)\in
R^{\prime}_{2p}\,\,\,c_{p,i}(z_{1},...,z_{n};q;h)\in
R^{\prime}_{l}\otimes(\mathbb{C}[z_{1},...,z_{n}])_{m}\;\;(l+m=p)$

for $p=1,...,m$ and $i=1,...,n$, where $R^{\prime}$ is defined and studied in
Section 6.2, such that genus-$1$ $n$-point function

$\displaystyle
F_{\mathcal{Y}_{1},...,\mathcal{Y}_{n}}(w_{1},...,w_{n};z_{1},...,z_{n};q;h)=\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})v_{1},q_{z_{1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})v_{n},q_{z_{n}})e^{2\pi
ish_{(0)}}q^{L_{(0)}}$

(denoted by $\psi$) satisfies the following system of differential equations:

(5) $\displaystyle\frac{\partial^{m}\psi}{\partial
z_{i}^{m}}+\displaystyle\sum_{p=1}^{m}a_{p,i}(z_{1},...,z_{n};q;h)\frac{\partial^{m-p}\psi}{\partial
z_{i}^{m-p}}=0,$ (6)
$\displaystyle\begin{split}\displaystyle\prod_{k=1}^{m}&\mathcal{O}^{h}_{i}(\sum_{i=1}^{n}\text{wt}w_{i}+2(m-k))\psi\\\
&+\sum_{p=1}^{m}b_{p,i}(z_{1},...,z_{n};q;h)\prod_{k=1}^{m-p}\mathcal{O}^{h}_{i}(\sum_{i=1}^{n}\text{wt}w_{i}+2(m-p-k))\psi=0,\end{split}$
(7) $\displaystyle\begin{split}\displaystyle(q_{s}\frac{\partial}{\partial
q_{s}})^{m}\psi+\sum_{p=1}^{m}c_{p,i}(z_{1},...,z_{n};q;h)(q_{s}\frac{\partial}{\partial
q_{s}})^{m-p}\psi=0,\end{split}$

$i=1,...,n,$ in the regions
$1>|q_{z_{1}}|>\cdots>|q_{z_{n}}|>|q|>0,0<|q_{s}|<1$ where for any
$\alpha\in\mathbb{C}$

$\displaystyle\mathcal{O}_{j}^{h}(\alpha)=(2\pi
i)^{2}q\frac{\partial}{\partial
q}+\tilde{G}_{2}(q)\alpha+\tilde{G}_{2}(q)\displaystyle\sum_{i=1}^{n}z_{i}\frac{\partial}{\partial
z_{i}}-\sum_{i\neq
j}\sum_{k=0}^{N-1}\tilde{\wp}_{1}(z_{i}-z_{j};q)\frac{\partial}{\partial
z_{i}},$

for $j=1,\cdots,n,$ and

$\displaystyle\prod_{j=1}^{m}\mathcal{O}^{h}(\alpha_{j})=\mathcal{O}^{h}(\alpha_{1})\cdots\mathcal{O}^{h}(\alpha_{m}).$

  * •

In the region
$\left\\{(z_{1},...,z_{n},\tau,s)|1>|q_{z_{1}}|>\cdots>|q_{z_{n}}|>|q_{\tau}|>0,0<|q_{s}|<1\right\\}$,
$F_{\mathcal{Y}_{1},...,\mathcal{Y}_{n}}(w_{1},...,w_{n};z_{1},...,z_{n};q_{\tau};h)$
is absolutely convergent and can be analytically extended to a multivalued
analytic function in the region given by $\tau\in\mathbb{H}$, $0<|q_{s}|<1$,
$z_{i}\neq z_{j}+k\tau+l$ for $i,j=1,...,n,$ $i\neq j$, $k,l\in\mathbb{Z}$.

In particular, for a quasi-lisse vertex algebra, thanks to the result obtained
by Etingof and Shedler Theorem 3.1 its finitely strongly generated modules
satisfy all above finiteness conditions when $g$ is identity. As a special
case of Theorem 1.2, we have

######  Corollary 1.3.

If $V$ is a quasi-lisse vertex superalgebra, then the supercharacter of its
simple $g$-twisted module satisfies the twisted modular linear differential
equation.

Let us now describe the main idea of the proof of Theorem 1.1 and 1.2 which
was originated in [Zhu96], [Hua05a], [Hua05b] and [AK18]. In general, one can
define a map $\psi$ from $W_{1}\otimes\cdots\otimes W_{n}$ to $n$-point
correlation functions. Using Jacobi identity would produce the kernel of
$\psi$. The more relations one obtains, the weaker convergence condition one
gets. By finding more relations in the kernel and using the proper filtration,
we obtain the surjective map:

(8) $\displaystyle
R_{W_{1}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{n}}/\\{(R_{V})^{g},R_{W_{1}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{n}}\\}\twoheadrightarrow
T/gr(\ker(\psi)).$

The finite dimensionality of the source of this map would imply the finite
dimensionality of the target. In particular, it would imply that
$T/\ker(\psi)$ is finitely generated over a Noetherian ring. By using
$L(-1)$-derivative property and constructing a proper submodule of
$T/\ker(\psi)$, one can obtain the desired differential equations satisfied by
the correlation functions.

The organization of this paper is as follows. In Section 2, we recall the
definition and basic notations of vertex operator (super)algebras. We
introduce some geometric concepts including arc superspaces and twisted arc
superspaces in Section 3, which will be used in the next section. In Section
4, we introduce the definition of quasi-lisse vertex (super)algebras and show
that it has only finitely many simple ordinary $g$-twisted modules. We also
study the relation between the twisted arc (super)spaces and the twisted
modules of some VOAs, i.e., affine VOAs, superconformal VOAs. We investigate
the convergence of genus-0 $n$-point correlation functions and derive a
necessary condition for it in Section 5. The main part of this paper is in
Section 6, where we obtain a necessary condition for genus-1 correlation
functions to be convergent. We check that the highest weight modules and their
contragredient modules of $L_{k}(\mathfrak{sl}_{2})$ at the admissible level
satisfy the condition for $n$-point correlation functions to be convergent. In
Appendix A, geometrically-modified intertwining operators are introduced and
their geometric meaning is explained. In Appendix B and C, we collect some
facts about modular forms.

## Acknowledgement

I would like to thank Antun Milas for suggesting me to study quasi-lisse
vertex algebras, Yi-Zhi Huang for the motivating questions about the relation
between the quasi-lisse vertex operator algebras and some of his previous
works at the conference organized by Corina Calinescu, and Robert Mcrae for
the motivating questions and useful discussions on the convergence problem. I
would like to thank Drazen Adamovic, Antun Milas and Tomoyuki Arakawa for the
opportunity to present this work in Dubrovnik. I would also like to thank
Bohan Li for reading the earlier version of this paper and giving me some
useful suggestions on the organization of this paper.

###  1.1. Notations and conventions

$\delta(x)=\sum_{n\in\mathbb{Z}}x^{n}$,

$\mathbb{N}:$ all natural number,

$q_{z}=e^{2\pi iz}$,

$\mathbb{Z}_{-}:$ all negative integers,

$\mathbb{Z}_{+}:$ all positive integers,

$\mathbb{H}$: upper half plane,

$\mathbb{C}((x))$: ring of all Laurent series in $x$,

$\mathbb{C}\\{x\\}$: the space of all series of the form
$\sum_{n\in\mathbb{R}}a_{n}x^{n}$, $n,a_{n}\in\mathbb{Q}$.

$\mathbb{G}_{|z_{1}|>\cdots>|z_{n}|>0}:$ the space of all multivalued analytic
functions in $z_{1},,...,z_{n}$ defined on the region
$|z_{1}|>\cdots>|z_{n}|>0$ with preferred branches in the simply-connected
region $|z_{1}|>\cdots>|z_{n}|>0,$ $0\leq\text{arg}z_{i}<2\pi$, $i=1,...,n$.

##  2\. Vertex superalgebras

######  Definition 2.1.

Let $V$ be a superspace, i.e., a $\mathbb{Z}_{2}$-graded vector space,
$V=V_{\overline{0}}\oplus V_{\overline{1}}$, where
$\\{{\overline{0}},\overline{1}\\}=\mathbb{Z}_{2}$. If $a\in V_{p(a)}$, we say
that the element $a$ has parity $p(a)\in\mathbb{Z}_{2}$.

A field is a formal series of the form
$a(z)=\displaystyle\sum_{n\in\mathbb{Z}}a_{(n)}z^{-n-1}$ where $a_{(n)}\in$
End$(V)$ and for each $v\in V$ one has

$a_{(n)}v=0$

for $n\gg 0$.

We say that a field $a(z)$ has parity $p(a)\in\mathbb{Z}_{2}$ if

$a_{(n)}V_{\alpha}\in V_{{\alpha}+p(a)}$

for all $\alpha\in\mathbb{Z}_{2}$, $n\in\mathbb{Z}.$

A vertex superalgebra contains the following data: a vector space of states
$V$, the vacuum vector $\mathbf{1}\in V_{\overline{0}},$ derivation $T$, and
the state-field correspondence map

$a\longmapsto Y(a,z)=\sum_{n\in\mathbb{Z}}a_{(n)}z^{-n-1},$

satisfying the following axioms:

  * •

(translation coinvariance): $[T,Y(a,z)]=\partial Y(a,z)$,

  * •

(vacuum): $Y(\mathbf{1},z)=Id_{V}$, $Y(a,z)\mathbf{1}|_{z=0}=a,$

  * •

(locality): $(z-w)^{N}Y(a,z)Y(b,w)=(-1)^{p(a)p(b)}(z-w)^{N}Y(b,w)Y(a,z)$ for
$N\gg 0$.

Define the $\lambda$-bracket between two fields
$[a_{\lambda}b]:=\displaystyle\sum_{j\in\mathbb{Z}_{+}}\frac{\lambda^{j}}{j!}(a_{(j)}b)$.

A vertex algebra $V$ is called graded if there is an even semisimple operator
$H$ (Hamiltonian) on it such that

$\displaystyle[H,a_{(n)}]=-(n+1)a_{(n)}+(Ha)_{(n)}.$

Let $V_{\Delta}=\\{v\in V|\;Hv=\Delta a\\}$, so that
$V=\sum_{\Delta\in\mathbb{C}}V_{\Delta}$. One defines the character of $V$ by

${\rm ch}[V](q)=\sum_{\Delta\in\mathbb{C}}{\rm dim}(V_{(\Delta)})q^{\Delta}.$

In particular, a vertex superalgebra $V$ is called supercommutative if
$a_{(n)}=0$ for $n\in\mathbb{N}$. It is well-known that the category of
commutative vertex superalgebras is equivalent with the category of unital
commutative associative superalgebras equipped with an even derivation.

We say that a vertex superalgebra ${V}$ is generated by a subset
$\mathcal{U}\subset{V}$ (or the fields correspond to elements in
$\mathcal{U}$) if any element of ${V}$ can be written as a finite linear
combination of terms of the form

$\displaystyle
b^{1}_{(i_{1})}{b^{2}}_{(i_{2})}\cdots{b^{n}}_{(i_{n})}\mathbf{1}$

for $b^{k}\in\mathcal{U}$, $i_{k}\in\mathbb{Z}$, and $n\in\mathbb{N}$. If
every element of $V$ can be written with $i_{k}\in\mathbb{Z}_{-}$, we write
${V}=\langle\mathcal{U}\rangle_{S}$ and say $V$ is strongly generated by
$\mathcal{U}$ (or $V$ is strongly generated by the fields correspond to
$\mathcal{U}$).

######  Example 2.2.

[Kac98][Zhe17] Let $\mathfrak{g}$ be a finite-dimensional Lie superalgebra
with a nondegenerate even supersymmetric invariant bilinear form
$\langle\cdot,\cdot\rangle$. One can associate the affine Lie superalgebra
$\widehat{\mathfrak{g}}$ to the pair
$(\mathfrak{g},\langle\cdot,\cdot\rangle)$.

Its universal vacuum representation of level $k\in\mathbb{C}$,
$V^{k}(\mathfrak{g})$, is a vertex superalgebra. In particular, when
$\mathfrak{g}$ is a simple Lie algebra or Lie superalgebra of type
$A(m,n),C(n),B(m,n),D(m,n),D(2,1,a),F(4),G(3)$ [KW01] $V^{k}(\mathfrak{g})$
has a unique maximal ideal $I_{V^{k}(\mathfrak{g})},$ and
$L_{k}(\mathfrak{g})=V^{k}(\mathfrak{g})/I_{V^{k}(\mathfrak{g})}$ is also a
vertex superalgebra.

######  Example 2.3.

[Kac98] To any $n$ dimensional superspace $A$ with a non-degenerate anti-
supersymmetric bilinear form $\langle\cdot,\cdot\rangle$, one can associate a
Lie superalgebra $C_{A}$. If we fix a basis of $A$:

$\left\\{\phi^{1},\cdots,\phi^{n}\right\\},$

the free fermionic vertex algebra $\mathcal{F}$ associated to $A$ is a vertex
superalgebra strongly generated by $\phi^{i}_{(-\frac{1}{2})}\mathbf{1}$
$(i=1,\cdots,n)$, where
$Y(\phi^{i}_{(-\frac{1}{2})}\mathbf{1},z)=\displaystyle\sum_{n\in\frac{1}{2}+\mathbb{Z}}\phi_{(n)}^{i}z^{-n-\frac{1}{2}}.$

######  Definition 2.4.

A vertex superalgebra $V$ is called a $\frac{1}{2}\mathbb{Z}$-graded (resp.
$\mathbb{Z}$-graded) vertex operator superalgebra (VOSA) if it is
$\frac{1}{2}\mathbb{Z}$-graded (resp. $\mathbb{Z}$-graded),

$V=\oplus_{n\in\frac{1}{2}\mathbb{Z}}V_{(n)},\quad(\text{resp.\;\;}V=\oplus_{n\in\mathbb{Z}}V_{(n)})$

with a conformal vector $\omega$ such that the set of operators
$\left\\{L_{(n)},id_{V}\right\\}_{n\in\mathbb{Z}}$ with
$L_{(n)}=\omega_{(n+1)}$ defines a representation of Virasoro algebra on $V$;
that is

(9)
$\displaystyle[L_{(n)},L_{(m)}]=(m-n)L_{(m+n)}+\frac{m^{3}-m}{12}\delta_{m+n,0}c_{V}$

for $m,n\in\mathbb{Z}$. In terms of $\lambda$-bracket, (10) translates as

(10) $\displaystyle[L_{\lambda}L]=(T+2\lambda)L+\frac{\lambda^{3}}{12}c.$

We call $c_{V}$ the central charge of $V$. We require that $L_{(0)}$ is
diagonalizable and it defines the $\frac{1}{2}\mathbb{Z}$ grading (resp.
$\mathbb{Z}$) - its eigenvalues are called (conformal) weights. A field
$Y(v,z)$ of $V$ is called primary of conformal weight $\Delta$ if
$[L_{\lambda}v]=(T+\Delta\lambda)v$; write $Y(v,z)=\sum
v_{(n)}z^{-n-\Delta(v)}$. The vertex operator algebra is said to be of CFT
type if $V_{(n)}=0$ for $n<0$ and $V_{(0)}=\mathbb{C}\mathbf{1}$.

######  Example 2.5.

[Kac98] The universal vertex algebras associated with conformal superalgebras,
Virasoro, Neveu-Schwarz, $N=2$, and $N=4$ are $\frac{1}{2}\mathbb{Z}$-graded
vertex operator superalgebras denoted by
$V_{Vir}(c,0),V_{c}^{N=1},V_{c}^{N=2},$ and $V_{c}^{N=4}$.

######  Definition 2.6.

A commutative vertex superalgebra $V$ is called a vertex Poisson superalgebra
if it is equipped with a linear operation,

$V\rightarrow{\rm Hom}(V,z^{-1}V[z^{-1}]),\quad a\rightarrow
Y_{-}(a,z)=\sum_{n\in\mathbb{N}}a_{(n)}z^{-n+1},$

such that

  * •

$(Ta)_{(n)}=-na_{(n-1)}$,

  * •

$a_{(n)}b=\displaystyle\sum_{j\in\mathbb{N}}(-1)^{n+j+1}\frac{(-1)^{p(a)p(b)}}{j!}T^{j}(b_{(n+j)}a),$

  * •

$[a_{(m)},b_{(n)}]=\displaystyle\sum_{j\in\mathbb{N}}\binom{m}{j}(a_{(j)}b)_{(m+n-j)},$

  * •

$a_{(n)}(b\cdot c)=(a_{(n)}b)\cdot c+(-1)^{p(a)p(b)}b\cdot(a_{(n)}c),$

for $a,b,c\in V$ and $n,m\in\mathbb{N}$.

A vertex Lie superalgebra structure on $V$ is given by $(V,Y_{-},T)$. So one
also says that a vertex Poisson superalgebra is a commutative vertex
superalgebra equipped with a vertex Lie superalgebra structure. In fact, one
can obtain a vertex Poisson superalgebra from any vertex superalgebra through
standard filtration or Li’s filtration. Following [Li05], one can define a
decreasing sequence of subspaces $\left\\{F_{n}(V)\right\\}$ of the
superalgebra $V$, where for $n\in\mathbb{Z}$, $F_{n}(V)$ is linearly spanned
by the vectors

$u_{(-1-k_{1})}^{(1)}\cdots u_{(-1-k_{r})}^{(r)}\bf{1}$

for $r\in\mathbb{Z_{+}}$, $u^{(1)},\cdots,u^{(r)}\in V,$
$k_{1},\cdots,k_{r}\in\mathbb{N}$ with $k_{1}+\cdots+k_{r}\geq n.$ Then Li’s
filtration of $V$ is given by

$V=F_{0}(V)\supset F_{1}(V)\supset\cdots$

satisfying

$\displaystyle u_{(n)}v\in F_{r+s-n-1}(V)\quad{\rm for}\quad u\in
F_{r}(V),v\in F_{s}(V),r,s\in\mathbb{N},n\in\mathbb{Z},$ $\displaystyle
u_{(n)}v\in F_{r+s-n}(V)\quad{\rm for}\quad u\in F_{r}(V),v\in
F_{r}(V),r,s,n\in\mathbb{N}.$

The corresponding associated graded algebra
$gr^{F}(V)=\displaystyle\coprod_{n\in\mathbb{N}}F_{n}(V)/F_{n+1}(V)$ is a
vertex Poisson superalgebra. Its vertex Lie superalgebra structure is given
by:

$T(u+F_{r+1}(V))=Tu+F_{r+2}(V),$
$Y_{-}(u+F_{r+1}(V),z)(v+F_{s+1}(V))=\sum_{n\in\mathbb{N}}(u_{(n)}v+F_{r+s-n+1}(V))z^{-n-1},$

for $u\in F_{r}(z),v\in F_{s}(z)$ with $r,s\in\mathbb{N}$. For the standard
increasing filtration $\left\\{G_{n}(V)\right\\}$, we also have the associated
graded vertex superalgebra $gr^{G}(V)$. In [Ara12, Proposition 2.6.1], it was
shown that

$gr^{F}(V)\cong gr^{G}(V)$

as vertex Poisson superalgebras. Thus we sometimes drop the upper index $F$ or
$G$ for brevity.

According to [Li05], we know that

$F_{n}(V)=\left\\{u_{(-1-i)}v|u\in V,i{\in\mathbb{Z}_{+}},v\in
F_{n-i}(V)\right\\}.$

In particular, $F_{0}(V)/F_{1}(V)=V/C_{2}(V)=R_{V}$ is a Poisson superalgebra
according to [Zhu96]. Its Poisson structure is given by

$\overline{u}\cdot\overline{v}=\overline{u_{(-1)}v},$
$\left\\{\overline{u},\overline{v}\right\\}=\overline{u_{(0)}v},$

for $u,v\in V$ where $\overline{u}=u+C_{2}(V)$. It was shown in [Li05,
Corallary 4.3] that $gr^{F}(V)$ is generated by $R_{V}$ as a differential
algebra.

Next, let us compute the $C_{2}$-algebras for some simple examples.

######  Example 2.7.

Following notations in Example 2.3, let $\mathcal{F}$ be a free fermionic
vertex superalgebra associated with an $n$-dimensional superspace $A$.
Clearly, the $C_{2}$-algebra of $\mathcal{F}$ is

$R_{\mathcal{F}}=\mathbb{C}[\overline{\phi^{1}_{(-\frac{1}{2})}\mathbf{1}},\cdots,\overline{\phi^{n}_{(-\frac{1}{2})}\mathbf{1}}],$

where $\overline{\phi^{i}_{(-\frac{1}{2})}\mathbf{1}}$ is even (resp. odd) if
$\phi^{i}$ is even (resp. odd) in $A$.

######  Example 2.8.

According to [vEH21, 16.16], for a simple affine vertex algebras
$L_{k}(\mathfrak{g})$, $k\in\mathbb{N}$, where $\mathfrak{g}$ is a simple Lie
algebra, one has:

$R_{L_{k}(\mathfrak{g})}=\mathbb{C}[u^{1}_{(-1)}\mathbf{1},u^{2}_{(-1)}\mathbf{1},\cdots,u^{n}_{(-1)}\mathbf{1}]/\langle
U(\mathfrak{g})\circ((e_{\theta})_{(-1)})^{k+1}\mathbf{1}\rangle,$

where $\left\\{u^{1},u^{2}\cdots,u^{n}\right\\}$ is a basis of $\mathfrak{g}$,
$\theta$ is the highest root of $\mathfrak{g}$, and $\circ$ represents the
adjoint action. In particular, when $\mathfrak{g}=\mathfrak{sl}_{2}$, one has

$R_{L_{k}(\mathfrak{g})}\cong\mathbb{C}[e,f,h]/\langle f^{i}\circ
e^{k+1}|i=0,\cdots,2k+2\rangle,$

where $e,f,h$ correspond to
$e_{(-1)}\mathbf{1},f_{(-1)}\mathbf{1},h_{(-1)}\mathbf{1}$.

######  Example 2.9.

For any simple Virasoro algebras $L_{Vir}(c_{(p,p^{\prime})},0)$, where
$c_{(p,p^{\prime})}=1-\frac{6(p-p^{\prime})^{2}}{pp^{\prime}}$ where
$p>p^{\prime}\geq 2$ and $p,p^{\prime}$ are coprime, according to [BFM91,
vEH21] its $C_{2}$-algebra is isomorphic to $\mathbb{C}[x]/\langle
x^{\frac{(p-1)(p^{\prime}-1)}{2}}\rangle,$ where $x$ corresponds to
$\omega=L_{(-2)}\mathbf{1}$.

######  Example 2.10.

The $C_{2}$-algebra of $V_{c}^{N=1},V_{c}^{N=2},$ and $V_{c}^{N=4}$ are
$\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}],\mathbb{C}[L_{(-2)},J_{(-1)},G^{+}_{(-\frac{3}{2})},G^{-}_{(-\frac{3}{2})}]$
and

$\mathbb{C}[L_{(-2)},J^{0}_{(-1)},J^{+}_{(-1)},J^{-}_{(-1)},G^{+}_{(-\frac{3}{2})},G^{-}_{(-\frac{3}{2})},\overline{G}^{+}_{(-\frac{3}{2})},\overline{G}^{-}_{(-\frac{3}{2})}],$

respectively. Here, $L_{(-2)}$ is an even variable of weight $2$;
$J_{(-1)},J^{0}_{(-1)},J^{+}_{(-1)},J^{-}_{(-1)}$ are even variables of weight
$1$; and
$G_{(-\frac{3}{2})},G^{+}_{(-\frac{3}{2})},G^{-}_{(-\frac{3}{2})},\overline{G}^{+}_{(-\frac{3}{2})},\overline{G}^{-}_{(-\frac{3}{2})}$
are odd variables of weight $\frac{3}{2}$.

######  Example 2.11.

According to [Ada05], the maximal submodule of
$V^{-\frac{3}{4}}(\mathfrak{sl}_{2})$ is generated by the singular vector

$v_{\text{sing}}=(e_{(-1)}L_{(-2)}+\frac{1}{3}e_{(-3)}-\frac{1}{2}h_{-1}e_{-2}+\frac{1}{2}h_{(-2)}e_{(-1)})\mathbf{1},$

where
$L_{(-2)}\mathbf{1}=\frac{3}{4}(e_{(-1)}f_{(-1)}+f_{(-1)}e_{(-1)}+\frac{1}{2}h_{(-1)}^{2})$
is the conformal vector in $V_{-\frac{3}{4}}(\mathfrak{sl}_{2})$. The
$C_{2}$-algebra of $L_{-\frac{4}{3}}(\mathfrak{sl}_{2})$ is

$R_{L_{-\frac{4}{3}}(\mathfrak{sl}_{2})}=\mathbb{C}[x,y,z]/\langle
x(xy-z^{2}),y(xy-z^{2}),z(xy-z^{2})\rangle.$

###  2.1. Untwisted arc superspaces

A general introduction to the theory of supermanifolds and superschemes can be
found in [Man13]. We are going to follow the setting in [KV11].

######  Definition 2.12.

A superspace is a locally ringed space $(X,\mathcal{O}_{X})$, where $X$ is a
topological space and $\mathcal{O}_{X}$ is a sheaf of supercommutative rings
on it such that the stalk $\mathcal{O}_{x}=\mathcal{O}_{X,x}$ at any point
$x\in X$ is a local ring. A morphism between two superspaces is a graded
morphism (with respect to $\mathbb{Z}_{2}$-grading) of locally ringed spaces.
A superscheme is a superspace $(X,\mathcal{O}_{X})$ such that
$(X,\mathcal{O}_{X,\overline{0}})$ is an ordinary scheme, while
$\mathcal{O}_{M,\overline{1}}$ is a quasi-coherent sheaf of
$\mathcal{O}_{X,\overline{0}}$-module. A superscheme is quasicompact if the
underlying topological space is quasicompact. We denote the category of
superspaces and superschemes by $\mathbf{Ssp}$ and $\mathbf{Ssch}$.

Given a superscheme $X$ we denote its underlying topological space by
$\underline{X}$. The even part of $X$ is defined to be
$X_{rd}=(\underline{X},\mathcal{O}_{X}/(\mathcal{O}_{X,\overline{1}}+(\mathcal{O}_{X,\overline{1}})^{2}))$
and the reduced scheme is
$X_{red}=(\underline{X},\mathcal{O}_{X,\overline{0}}/\sqrt{\mathcal{O}_{X,\overline{0}}})$,
where $\sqrt{\mathcal{O}_{X,\overline{0}}}$ is the sheaf of all nilpotent
elements in structure sheaf $\sqrt{\mathcal{O}_{X,\overline{0}}}$. Note
$X_{rd}$ and $X_{red}$ are both ordinary schemes.

######  Example 2.13.

Let $R^{n|m}=\mathbb{C}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]$ be a
polynomial superalgebra where $x_{1},\ldots,x_{n}$ are even variables and
$y_{1},\ldots,y_{m}$ are odd variables, then
$\mathbb{A}^{n|m}=(\text{Spec}(R^{n|m}),R^{n|m})$ is a quasicompact affine
superscheme of dimension $n|m$. It is clear that every superscheme of
dimension $n|m$ is locally isomorphic to an affine superscheme. In particular,
every $N=m$ supercurve is locally isomorphic to $\mathbb{A}^{1|m}$. Readers
can refer to [Hel07] for the general definition of supercurves.

One defines a super-ind-scheme as a filtering inductive limit of quasicompact
superschemes and their closed immersions. And the category $\mathbf{Ssch}$ is
a full subcategory of super-ind-schemes $\mathbf{Isch}$. Let
$\epsilon:R^{1|m}\rightarrow\mathbb{C}$ be the augmentation morphism which
maps all variables to $0.$ Let
$(D^{1|m})^{(n)}=\text{Spec}(R^{1|m}/(ker(\epsilon))^{n+1}).$ We denote by
$\widehat{R^{1|m}}$ the $ker(\epsilon)$-completion of the polynomial ring. The
truncation morphism $R^{1|m}/(ker(\epsilon))^{n+2}\rightarrow
R^{1|m}/(ker(\epsilon))^{n+1}$ induces a closed embedding
$(D^{1|m})^{(n)}\rightarrow(D^{1|m})^{(n+1)}$. Then the ind-super-scheme
$D^{1|m}=\text{Spec}(\widehat{R^{1|m}})=\displaystyle\varinjlim_{n\rightarrow\infty}(D^{1|m})^{(n)}$
is called a $N=m$ superdisc.

Fix a separated superscheme of finite type over $k$. Let $\mathbf{Ssch}/k$ be
the category of superschemes over a field $k$, and $\mathbf{Set}$ be a
category of sets. Let $D=D^{1|0}$. Define a family of contravariant functors
$F_{m}:\mathbf{Ssch}/k\rightarrow\mathbf{Set}$ by

$F_{m}(Y)=Hom_{k}(Y\times_{k}(D)^{(m)},X).$

The proof of the following Proposition under non-super setting is in [ISH07],
and it can be proved similarly in super case.

######  Proposition 2.14.

The functors $F_{m}$ is representable by superschemes $X_{m}.$

Note $X_{1}$ is the total tangent space of $X.$ We call the superschemes
$X_{m}$ the $m$-jet superscheme. The embeddings $D^{(m)}\hookrightarrow
D^{(m+1)}$ induces the projections $X_{m+1}\rightarrow X_{m}$, and one defines
the arc superspace of $X$ as

$X_{\infty}=\displaystyle\varprojlim_{m\rightarrow\infty}X_{m}.$

The embedding $D^{(0)}\hookrightarrow D^{(m)}$ induces the projection
$\pi_{\infty}:X_{m}\rightarrow X$. We call the coordinate rings of $X_{m}$,
$X_{\infty}$ and the fiber of $\pi_{\infty}$ over $\mathbf{p}\in X$ the
$m$-jet superalgebra, arc superalgebra and focussed arc superalgebra at
$\mathbf{p}$ of $X$, respectively.

######  Remark 2.15.

If we replace $D$ with $D^{1|n}$ in above definition of arc superspace, one
can similarly define $N=n$ arc superspace. The situation would become more
complicated, since we need to introduce extra $n$ differentiations besides
$T$.

For an affine superscheme, one can explicitly describe its jet superscheme and
arc superspace. Here we closely follow [Ara12] and [BMS13].

Let $f_{1},f_{2},\ldots,f_{k}$ be $\mathbb{Z}_{2}$-homogeneous elements in the
polynomial superalgebra $R^{n|l}$. We will describe the jet superalgebra of
the quotient superalgebra:

$\displaystyle
R=\frac{\mathbb{C}[x_{1},x_{2},\ldots,x_{n},y_{1},\ldots,y_{l}]}{\langle
f_{1},f_{2},\ldots,f_{k}\rangle}.$

Firstly, let us introduce new even variables $x_{j,(-\Delta_{j}-i)}$ and odd
variables $y_{j^{\prime},(-\Delta_{j^{\prime}}-i)}$ for $i=-1,0,\ldots,m$,
where $\Delta_{j}$ and $\Delta_{j^{\prime}}$ are weights of
$x_{j,(-\Delta_{j})}$ and $y_{j^{\prime},(-\Delta_{j^{\prime}})}$. In most
cases, we will assume that the weight of each even variable belongs to
$\mathbb{Z}$, and the weight of the odd variable belongs to
$\frac{1}{2}+\mathbb{Z}$. Define an even derivation $T$ on

$\mathbb{C}[x_{j,(-\Delta_{j}-i)},y_{j^{\prime},(-\Delta_{j^{\prime}}-i))}\;|\;-1\leq
i\leq m,\;1\leq j\leq n,\;1\leq j^{\prime}\leq l]$

by

$T(u_{j,(-\Delta_{j}-i)})=\begin{cases}(-\Delta_{j}-i)u_{j,(-\Delta_{j}-i-1)}&\text{for
$i\leq m-1$}\\\ 0&\text{for $i=m$},\\\ \end{cases}$

where $u=x$ or $y$. Here we identify $x_{j}$ and $y_{j^{\prime}}$ with
$x_{j,(-\Delta_{j}+1)}$ and $y_{j^{\prime},(-\Delta_{j^{\prime}}+1)},$
respectively. Set

$\displaystyle
J_{m}(R)=\frac{\mathbb{C}[x_{j,(-\Delta_{j}-i)},y_{j^{\prime},(-\Delta_{j^{\prime}}-i)}\;|\;-1\leq
i\leq m,\;1\leq j\leq n,\;1\leq j^{\prime}\leq l]}{\langle
T^{j}f_{i}|i=1,\ldots k,\;j\in\mathbb{N}\rangle}.$

Then the $m$-jet superscheme of $X=\text{Spec}(R)$ is ${\rm Spec}(R_{m})$. The
arc superalgebra of $R$ is

$\displaystyle J_{\infty}(R)$
$\displaystyle=\displaystyle\lim_{\underset{m}{\leftarrow}}R_{m}$
$\displaystyle=\frac{\mathbb{C}[x_{j,(-\Delta_{j}-i)},y_{j^{\prime},(-\Delta_{j^{\prime}}-i)}\;|\;-1\leq
i,\;1\leq j\leq n,\;1\leq j^{\prime}\leq l]}{\langle T^{j}f_{i}|i=1,\ldots
k,\;j\in\mathbb{N}\rangle}.$

The arc superalgebra is a differential commutative superalgebra. The focused
arc superalgebra at $0$ is

$J^{0}_{\infty}(R)=J_{\infty}(R)|_{x_{j,(-\Delta_{j}+1)}=y_{j^{\prime},(-\Delta_{j^{\prime}}+1)}=0}.$

We denote the ideal

$\langle T^{j}f_{i};i=1,\ldots,k,j\in\mathbb{N}\rangle$

by $\langle f_{1},\ldots,f_{k}\rangle_{\partial}.$ Later, we sometimes write
$x_{(j)}$ as $x(j).$ The arc superspace of $X$ is

$X_{\infty}={\rm Spec}(J_{\infty}(R)).$

We define the weight of each variable $u_{(-\Delta-j)}$ to be $\Delta+j$,
where $u=x$ or $y$ if $\Delta+j\geq 0$ and $0$ otherwise. Then
$J_{\infty}(R)=\displaystyle\coprod_{m\in\frac{1}{2}\mathbb{Z}}(J_{\infty}(R))_{(m)}$,
where $(J_{\infty}(R))_{(m)}$ is the set of all elements in arc superalgebra
with weight $m$. We define Hilbert series of $J_{\infty}(R)$ as:

$HS_{q}(J_{\infty}(R))=\sum_{m\in\frac{1}{2}\mathbb{Z}}{\rm
dim}(J_{\infty}(R)_{(m)})q^{m}.$

When the quotient ideal of $R$ is generated by homogeneous polynomials, the
arc superspace of $X$ and focussed arc superspace of $X$ over $0$ are
isomorphic.

######  Proposition 2.16.

Let $I$ be a homogeneous ideal of $R$. We have the following differential
algebra isomorphism:

$J_{\infty}(R)\cong J^{0}_{\infty}(R).$

###### Proof.

We assume $R$ has no odd variable, and the general case can be proved
similarly. Suppose $I$ is generated by homogeneous polynomials
$P_{1},\cdots,P_{k}$ of weight $d_{1},\cdots,d_{k}.$ By direct calculation, we
get the following formula:

(11)
$\displaystyle\begin{split}&T^{j}(P_{i}(x_{1},\cdots,x_{n}))|_{x_{1,(-1)}=\cdots=x_{n,(-1)}=0}\\\
&=\begin{cases}0&\text{if $j<d_{i}$ }\\\
\frac{j!}{(j-d_{i}+1)!}T^{j-d_{i}}P_{i}(T(x_{1}),\cdots,T(x_{n}))&\text{if
$j\geq d_{i}$}.\\\ \end{cases}\end{split}$

Therefore, we can define a differential algebra homomorphism:

$\phi:J_{\infty}(\mathbb{C}[x_{1},x_{2},\cdots,x_{n}]/I)\rightarrow
J^{0}_{\infty}(\mathbb{C}[x_{1},x_{2},\cdots,x_{n}]/I)$

by sending $x_{i,(0)}$ to $x_{i,(-1)}.$ Clearly, this map is a bijection. The
result follows.

∎

Following [Ara12], $J_{\infty}(R)$ has a unique Poisson vertex superalgebra
structure such that

$u_{(n)}v=\begin{cases}\left\\{u,v\right\\},&\mbox{if }n=0\\\ 0,&\mbox{if
}n>0\end{cases}$

for $u,v\in R\in J_{\infty}(R)$.

Furthermore, one can extend the embedding $R_{V}\hookrightarrow gr^{F}(V)$ to
a surjective differential superalgebra homomorphism
$J_{\infty}(R_{V})\twoheadrightarrow gr^{F}(V)$. It is obvious that the map is
a differential superalgebra homomorphism. It is surjective since $gr^{F}(V)$
is generated by $R_{V}$ as a differential algebra. Moreover, it was shown in
[Ara12] that this map is actually a Poisson vertex superalgebra epimorphism.
From now on, we call this map $\psi.$ The supervariety $\text{Specm}(R_{V})$
and the superscheme $X_{V}=\text{Spec}(R_{V})$ are called associated variety
and associated scheme of vertex superalgebra $V$. These important objects are
first studied and defined in [Ara12]. Above argument implies that the arc
superspace of the associated scheme of $V$ contains a closed subscheme
$\text{Spec}({gr^{F}(V)})$, which is called the singular support of the $V$
and denoted by $SS(V).$

The map $\psi$ is not necessarily injective, and it is an open problem to
characterize rational vertex algebras for which $\psi$ is injective. Following
[AvEH], we give the following definition.

######  Definition 2.17.

We call a vertex (super)algebra classically free if the map $\psi$ is an
isomorphism.

The reason why we only have surjectivity for $\psi$ is that the Poisson
algebra structure on arc algebra $J_{\infty}(R_{V})$ loses some information
about OPE. To be more precise, we consider the affine vertex algebra
$V^{k}(\mathfrak{g})$ and its simple quotient $L_{k}(\mathfrak{g})$. Let
$\left\\{u^{i}\right\\}_{i=1}^{n}$ be the basis of $\mathfrak{g}$. The arc
algebra only records the following relation

$[u^{i}_{(n)},u^{j}_{(m)}]=[u^{i},u^{j}]_{(n+m)}.$

Meanwhile, in affine vertex algebra, one has

$[u^{i}_{(n)},u^{j}_{(m)}]=[u^{i},u^{j}]_{(n+m)}+kn\langle
u^{i},u^{j}\rangle.$

Suppose $R_{L_{k}(\mathfrak{g})}=\mathbb{C}[\mathfrak{g}]/I_{k}$. The
classically freeness of $L_{k}(\mathfrak{g})$ means that all its quotient
relations are generated by successive action by $T$, $u^{i}_{(0)}$ and
$u^{i}_{(n)}$ $n\leq-1$ on $I_{k}$. The only possible way to break the
classically freeness is that there might exist some extra quotient relations
coming from $u_{(m)}T^{n}a$, where $0<m<n$, $a\in I_{k}$. Therefore we obtain

######  Proposition 2.18.

Given a simple affine vertex operator algebra $L_{k}(\mathfrak{g})$, it is
classically free if and only if for any $0<m<n$ and $a\in I_{k}$ there exists
$b\in I_{k}$ such that $u_{(m)}T^{n}a=T^{n-m}b$.

We summarize the results about classically freeness of affine vertex algebras:

######  Theorem 2.1.

  * •

[vEH21, Theorem 15.21] $L_{k}(\mathfrak{sl}_{2})$ are classically free at all
positive integer level.

  * •

[F+08, Thereom 1] $L_{1}(\mathfrak{g})$ is classically free for all finite
dimensional simple Lie algebra $\mathfrak{g}.$

  * •

[LS21, Corollary 6.2] $L_{n}(\mathfrak{sp}_{2r})$ are classically free for all
positive integer level $n$.

  * •

[CLS22, Corollary 3.1] For $-\frac{m}{2}+r+n+1>0$, the simple affine vertex
superalgebra $L_{n}(\mathfrak{osp}_{m|2r})$ is classically free.

######  Remark 2.19.

It is also expected by physicists that all the vertex algebras coming from the
4D-2D correspondence are all classically free.

###  2.2. Twisted arc superspaces

Let $X$ be a separated superscheme of finite type over $\mathbb{C}$. Let
$g:X\rightarrow X$ be a linear automorphism of order $m$. In this section, we
introduce the concept $g$-twisted arc superspace of $X$.

According to the definition of arc superspaces, we have

$\text{Hom}_{\mathbf{Ssch}}({\rm
Spec}(\mathbb{C}),X_{\infty})=\text{Hom}_{\mathbf{Ssch}}(D,X).$

For $m$-th root of unity $\xi_{m}$ we have an automorphism $\varphi_{\lambda}$
of $\mathbb{C}\llbracket t\rrbracket$ determined by $t\mapsto\xi_{m}t$. Since

$\text{Hom}_{\mathbf{Ssch}}(D,X)\cong\text{Hom}_{\mathbf{Rings}}(\Gamma(X,\mathcal{O}_{X}),\mathbb{C}\llbracket
x\rrbracket),$

$\varphi_{\lambda}$ induces a natural automorphism $\phi_{\lambda}$ of
superarcs in $X$ by composing

$\gamma\in\text{Hom}{\mathbf{Rings}}(\Gamma(X,\mathcal{O}_{X}),\mathbb{C}\llbracket
x\rrbracket)$

with $\varphi_{\lambda}$.

######  Definition 2.20.

The $g$-twisted arc superspaces of $X$, $X_{\infty}^{g},$ is the fixed sub-
superscheme of $X_{\infty}$ under the automorphism
$g^{-1}\circ\phi_{\lambda}.$

######  Example 2.21 (Affine case).

The non-super case was also discussed in [Szc18]. Let $R^{n|l}$ be the
quotient polynomial ring whose quotient is generated by homogeneous
polynomials as we defined in the previous section. Suppose $g$ is a linear
automorphism of $R$ of order $m$. After a linear change of coordinates we may
diagonalize $g$ such that its action is given by
$g(x_{j})=\xi^{\alpha_{j}}_{m}x_{j}$ $(1\leq j\leq n)$ and
$g(y_{j^{\prime}})=\xi^{\beta_{j^{\prime}}}_{m}y_{j^{\prime}}$ $(1\leq
j^{\prime}\leq l)$. For the arc superalgebra $J_{\infty}(R)$, we have the
following identification $(n\geq-1)$:

$\displaystyle x_{j,(-\Delta_{j}-n)}\longrightarrow x_{j}\otimes
t^{-\Delta_{j}-n},\quad
y_{j^{\prime},(-\Delta_{j^{\prime}}-n)}\longleftrightarrow
y_{j^{\prime}}\theta\otimes t^{-\Delta_{j^{\prime}}-n-\frac{1}{2}},$

where $\Delta_{j^{\prime}}\in\frac{1}{2}+\mathbb{Z}.$ Then

$\displaystyle(J_{\infty}(\mathbb{C}[x_{j},y_{j^{\prime}}|1\leq j\leq n,1\leq
j^{\prime}\leq l])^{g^{-1}\circ\phi_{\lambda}}$
$\displaystyle=\mathbb{C}[x_{j}\otimes
t^{\alpha_{j}+km},y_{j^{\prime}}\theta\otimes t^{\beta_{j^{\prime}}+km}|$
$\displaystyle\quad k\in\mathbb{Z},1\leq j\leq n,1\leq j^{\prime}\leq
l,\alpha_{j}+km\leq-\Delta_{j}+1,\beta_{j^{\prime}}+km\leq-\Delta_{j^{\prime}}+\frac{1}{2}]$

For simplicity, we assume $\Delta_{j}=1$, $j=1,...,n$ and
$\Delta_{j^{\prime}}=\frac{1}{2}$, $j^{\prime}=1,...,l.$ Then
$(J_{\infty}(\mathbb{C}[x_{j},y_{j^{\prime}}|1\leq j\leq n,1\leq
j^{\prime}\leq l])^{g^{-1}\circ\phi_{\lambda}}$ equals

$\displaystyle\mathbb{C}[x_{j}\otimes
t^{\alpha_{j}-km},y_{j^{\prime}}\theta\otimes
t^{\beta_{j^{\prime}}-km}|k\in\mathbb{Z}_{+}].$

We change the coordinate of superdisk from $t$ to $\frac{1}{t^{m}}$ and let
$\theta=t^{\frac{1}{2}}$. We introduce even derivation $T_{g}$ on

$\displaystyle\mathbb{C}[x_{j,(\frac{\alpha_{j}}{m}-k)},y_{j^{\prime},(\frac{\beta_{j^{\prime}}}{m}-k+\frac{1}{2})}|k\in\mathbb{Z}_{+}]$

by

$T_{g}(x_{j,(\frac{\alpha_{j}}{m}-k)})=-(\frac{\alpha_{j}}{m}-k-1)x_{j,(\frac{\alpha_{j}}{m}-k-1)}\quad$

and

$T_{g}(y_{j^{\prime},(\frac{\beta_{j^{\prime}}}{m}-k+\frac{1}{2})})=-(\frac{\beta_{j^{\prime}}}{m}-k-\frac{1}{2})y_{j^{\prime},(\frac{\beta_{j^{\prime}}}{m}-k-\frac{1}{2})}\quad.$

By identifying $x_{j}$, $y_{j^{\prime}}$, $j=1,...,n$, $j^{\prime}=1,...,l$
with $x_{j,(\frac{\alpha_{j}}{m}-1)}$,
$y_{j^{\prime},(\frac{\beta_{j^{\prime}}}{m}-\frac{1}{2})}$, respectively, we
get the coordinate ring of $({\rm Spec}R)^{g}_{\infty}$:

$J^{g}_{\infty}(R)=\frac{\mathbb{C}[x_{j,(\frac{\alpha_{j}}{m}-k)},y_{j^{\prime},(\frac{\beta_{j^{\prime}}}{m}+\frac{1}{2}-k)}\;|\;k\in\mathbb{Z}_{+},\;1\leq
j\leq n,\;1\leq j^{\prime}\leq l]}{\langle T_{g}^{j}f_{i}|i=1,\ldots
k,\;j\in\mathbb{N}\rangle}.$

######  Example 2.22.

Recall Example 2.11, we consider the automorphism $g$ on
$R_{L_{-\frac{4}{3}}(\mathfrak{sl}_{2})}$ by sending $x$ to $-x$, $y$ to $-y$
and $z$ to $z$. Then

$\displaystyle
J_{\infty}^{g}(R_{L_{-\frac{4}{3}}}(\mathfrak{sl}_{2}))=\mathbb{C}[x_{(-\frac{1}{2})},y_{(-\frac{1}{2})},z_{(0)}]/$
$\displaystyle\langle
T^{j}_{g}(x_{(-\frac{1}{2})}(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2})),T^{j}_{g}(y_{(-\frac{1}{2})}(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2})),T^{j}_{g}(z(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2}))|j\in\mathbb{N}\rangle,$

where

$\displaystyle
T_{g}(x_{(-\frac{1}{2})}(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2}))=3x_{(-\frac{3}{2})}x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}+\frac{3}{2}x_{(-\frac{1}{2})}x_{(-\frac{1}{2})}y_{(-\frac{3}{2})}-2z_{(0)}z_{(-1)}x_{(-\frac{1}{2})}-\frac{3}{2}z_{(0)}^{2}x_{(-\frac{3}{2})},$
$\displaystyle
T_{g}(y_{(-\frac{1}{2})}(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2}))=3y_{(-\frac{3}{2})}y_{(-\frac{1}{2})}x_{(-\frac{1}{2})}-\frac{3}{2}y_{(-\frac{1}{2})}y_{(-\frac{1}{2})}x_{(-\frac{3}{2})}-2z_{(0)}z_{(-1)}y_{(-\frac{1}{2})}-\frac{3}{2}z_{(0)}^{2}y_{(-\frac{3}{2})},$
$\displaystyle
T_{g}(z_{(0)}(x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}-z_{(0)}^{2}))=z_{(-1)}x_{(-\frac{1}{2})}y_{(-\frac{1}{2})}+\frac{3}{2}z_{(0)}x_{(-\frac{3}{2})}y_{(-\frac{1}{2})}+\frac{3}{2}z_{(0)}x_{(-\frac{1}{2})}y_{(-\frac{3}{2})}-3z_{(-1)}z_{(0)}^{2}.$

##  3\. Quasi-lisse vertex superalgebras

In this section, we first define the quasi-lisse vertex superalgebras and then
discuss their $g$-twisted modules.

###  3.1. Definitions

We recall the concept of module of a Poisson algebra.

######  Definition 3.1.

Let $A$ be a Poisson algebra, then $M$ is an $A$-module if $M$ is an
$A$-module in the usual associative algebra sense and equipped with a bilinear
map

$\left\\{\cdot,\cdot\right\\}:A\times M\rightarrow M$

such that for all $x,y\in A$ and $v\in M,$

  * •

$\left\\{x,y\right\\}m=x(yv)-y(xv),$

  * •

$\left\\{x,ym\right\\}=\left\\{x,y\right\\}m+y\left\\{x,m\right\\}$,

  * •

$\left\\{xy,m\right\\}=x\left\\{y,m\right\\}+y\left\\{x,m\right\\}.$

Suppose we have two $A$-modules $M_{1}$ and $M_{2}$, then the tensor product
$M_{1}\otimes M_{2}$ is again a Poisson algebra module over $A$ with the
bracket action given by

$\left\\{a,m_{1}\otimes m_{2}\right\\}=\left\\{a,m_{1}\right\\}\otimes
m_{2}+m_{1}\otimes\left\\{a,m_{2}\right\\},$

for any $a\in A$ and $m_{i}\in M_{i}$, $i=1,2.$

Let $X$ be an affine Poisson variety, i.e., the global sections
$\mathcal{O}(X)$ is a Poisson algebra. Given any element $f\in{X}$, one can
define a Hamilton vector field
$\mathcal{E}_{f}:=\text{ad}f=\left\\{f,\cdot\right\\}$ on $X$. So we have a
natural map

$\displaystyle\alpha:\mathcal{O}(X)\rightarrow\text{Vect}(X),$

given by $\alpha(f)=\mathcal{E}_{f}$, where $\text{Vect}(X)$ are vector fields
on $X$.

######  Definition 3.2.

A maximally locally closed connected subvariety $Z\subset X$ is called a
symplectic leave if $\alpha_{x}:\mathcal{O}(X)\rightarrow T_{x}(X)$ is
surjective for all $x\in Z$.

We recall some results from [ES10].

######  Theorem 3.1.

Let $X$ be an affine Poisson variety. Let $G$ be a finite group of Poisson
automorphisms of $X$. If $X$ has finitely many symplectic leaves, then

  * •

$N/\left\\{\mathcal{O}_{X},N\right\\}$ is finite dimensional for any coherent
sheaf of Poisson modules $N$ over $\mathcal{O}_{X}.$

  * •

$\mathcal{O}_{X}/\left\\{\mathcal{O}_{X/G},\mathcal{O}_{X}\right\\}$ is finite
dimensional. In particular, the $G$-invariants,

$(\mathcal{O}_{X}(X))^{G}/\left\\{(\mathcal{O}_{X}(X))^{G},(\mathcal{O}_{X}(X))^{G}\right\\}$

is finite-dimensional.

######  Example 3.2.

  * •

[CG97] the algebra $\mathcal{O}(M)$ of regular functions (with pointwise
multiplication) on a symplectic manifold $M$ has a canonical Poisson algebra
structure. By the definition of symplectic leaves, the connected component of
$M$ is a single symplectic leave. Examples include $\mathbb{C}^{2n}$,
cotangent bundle of any smooth manifold, the coadjoint orbit
$\mathbb{O}\in\mathfrak{g}^{*}$, etc.

  * •

[ES18] Given a normal variety $X$, if it admits a symplectic resolution
$\rho:\tilde{X}\rightarrow X$, i,e,, $\tilde{X}$ admits a global nondegenerate
closed two-form, $X$ has finitely many symplectic leaves. Examples include
Nakajima quiver variety (Nilcone $\mathcal{N}=\coprod\mathbb{O}_{i}$ and
Slodowy slices in a complex semisimple Lie algebra, Kleinian singularities,
$\mathbb{C}^{2}/S_{n}$, etc.).

  * •

(Conjecture) The associated variety of the vertex algebra coming from $4D$
theory has finitely many symplectic leaves, including genus zero Moore-
Tachikawa symplectic variety.

The following definition in non-super case was given in [Ara12] and [AK18].

######  Definition 3.3.

A finitely strongly generated vertex (super)algebra $V$ is called lisse if
${\rm dim}{\rm Spec}(R_{V})=0$. It is called quasi-lisse if the poisson
variety $(\tilde{X}_{V})_{red}$ has finitely many symplectic leaves.

######  Example 3.4.

[GK07] $L_{k}({\mathfrak{g}})$ is lisse if and only if $\mathfrak{g}$ is a
simple Lie algebra or a Lie superalgebra of type $B(0,n)$, where $n,k$ are
positive integers.

######  Theorem 3.5.

[AM21] For a vertex (super)algebra $V$ its singular support is isomorphic with
the arc (super)space of $X_{V}$ as topological space if $V$ is quasi-lisse.

For a weak $V$-module $M$ (see Definition 3.7 and 3.8), set

(12) $\displaystyle
C_{1}(M)=\text{span}_{\mathbb{C}}\\{\omega_{(0)}m,a_{(-1)}m;a\in\oplus_{\Delta>0}V_{\Delta},m\in
M\\}$ (13) $\displaystyle C_{2}(M)=\text{span}_{\mathbb{C}}\\{a_{(-2)}m;a\in
V,m\in M\\}.$

We say that $M$ is $C_{1}$-cofinite and $C_{2}$-cofinite if
$\text{dim}M/C_{1}(M)<\infty$ and $\text{dim}M/C_{2}(M)<\infty.$
$R_{M}:=M/C_{2}(M)$ is a $R_{V}$-module. One calls $M$ strongly generated over
$V$ if $R_{M}$ is finitely generated over $R_{V}$.

###  3.2. Twisted modules

Following [DZ06], one can define an automorphism $g$ of a vertex operator
superalgebra $V$ and an ordinary $g$-twisted V-module.

######  Definition 3.6.

An automorphism $g$ of $V$ is a linear automorphism of $V$ preserving $\omega$
such that the actions of $g$ and $Y(v,z)$ on $V$ are compatible in the sense
that

$gY(v,z)g^{-1}=Y(gv,z)$

for $v\in V.$

We denote the automorphism group of $V$ by ${\rm Aut}V$. Let $\sigma$ be the
canonical automorphism of $V$, which is defined as
$\sigma|_{V_{\overline{i}}}=(-1)^{i}id_{V}$. Let the order of $g$ and
$g\sigma$ be $T$ and $T^{\prime}$, respectively. We have the following
eigenspace decompositions:

$\displaystyle V=\oplus_{r\in\mathbb{Z}/T^{\prime}\mathbb{Z}}V^{r*}$
$\displaystyle V=\oplus_{r\in\mathbb{Z}/T\mathbb{Z}}V^{r},$

where $V^{r*}=\left\\{v\in V|g\sigma v=e^{-2\pi ir/T^{\prime}}\right\\}$ and
$V^{r}=\left\\{v\in V|gv=e^{-2\pi ir/T}v\right\\}.$

######  Definition 3.7.

An (ordinary) $g$-twisted $V$-module is a $\mathbb{C}$-linear space $M$
equipped with a linear map

$\displaystyle V\rightarrow{\rm End}(V)[[z^{\frac{1}{T}},z^{-\frac{1}{T}}]],$
$\displaystyle v\mapsto
Y_{M}(v,z)=\displaystyle\sum_{n\in\mathbb{Q}}v_{(n)}z^{-n-1}$

satisfying

  * •

For $v,w\in M$, $v_{(m)}w=0$ if $m$ is large enough.

  * •

$Y_{M}(\mathbf{1},z)=id_{V}.$

  * •

For $v\in V^{r}=\left\\{v\in V|gv=e^{\frac{2\pi ir}{T}}v\right\\}$, and $0\leq
r\leq T-1$

$Y_{M}(v,z)=\displaystyle\sum_{n\in\frac{r}{T}+\mathbb{Z}}v_{(n)}z^{-n-1}.$

  * •

(Jacobi identity) For $u\in V^{r}$

$\displaystyle
z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})Y_{M}(v,z_{1})Y_{M}(w,z_{2})-(-1)^{|v||w|}z_{0}^{-1}\delta(\frac{z_{0}-z_{1}}{-z_{0}})Y_{M}(v,z_{2})Y_{M}(u,z_{1})$
$\displaystyle=z_{2}^{-1}(\frac{z_{1}-z_{0}}{z_{2}})^{-\frac{r}{T}}\delta(\frac{z_{1}-z_{0}}{z_{2}})Y_{M}(Y(u,z_{0})v,z_{2}).$

  * •

(grading)

    * –

$M=\oplus_{\lambda\in\mathbb{C}}M_{\lambda},$ where $M_{\lambda}=\left\\{w\in
M|L_{(0)}w=\lambda w\right\\}$,

    * –

$M_{\lambda}$ is finite dimensional,

    * –

for a fixed $\lambda$, $M_{\frac{n}{T^{\prime}}+\lambda}=0$ for all small
enough integers $n$.

######  Remark 3.8.

If the grading condition is dropped, one calls it the weak $g$-twisted module.
If the module carries a $\frac{1}{T^{\prime}}$-grading $M=\oplus_{n\geq
0}M_{(h+n)}=:\oplus_{0\leq n\in\frac{1}{T^{\prime}}}M{(n)}$ such that

$v_{(m)}M(n)\in M(n+\text{wt}v-m-1),$

for homogeneous $v\in V$, one calls $V$ an admissible $g$-twisted $V$-module.

Let $u,v\in V$ such that $g(u)=e^{\frac{2\pi ir_{u}}{T}}u$ and
$g(v)=e^{\frac{2\pi ir_{v}}{T}}v$. From the definition one can derive the
twisted Borcherds commutator [DZ06] and iterated formula [Li96]:

(14)
$\displaystyle[u_{(m)},v_{(n)}]=\sum_{i=0}^{\infty}\binom{m}{i}(u_{(i)}v)_{(m+n-i)},\,\,\,\text{for}\,\,\,m\in\frac{r_{u}}{T}+\mathbb{Z},\,\,n\in\frac{r_{v}}{T}+\mathbb{Z}.$
(15)
$\displaystyle\begin{split}(u_{(n)}v)_{(m)}&=\sum_{j=0}^{\infty}\sum_{w=0}^{\infty}(-1)^{j}\binom{\frac{r_{u}}{T}}{j}\binom{n+j}{w}\\\
&((-1)^{w}u_{(\frac{r_{u}}{T}+n-w)}v_{(-\frac{r_{u}}{T}+m+w)}-(-1)^{n+j-w+|u||v|}v_{(-\frac{r_{u}}{T}+m+n+j-w)}u_{(\frac{r_{u}}{T}-j+w)})\end{split}$

for $n\in\mathbb{Z}$ and $m\in\frac{r_{u}+r_{v}}{T}+\mathbb{Z}$.

Set $Y_{M}(\omega,z)=\displaystyle\sum_{n\in\mathbb{Z}}L_{(n)}z^{-n-2},$ then
$Y_{M}(L_{(-1)}v,z)=\frac{d}{dz}Y_{M}(v,z)$ for $v\in V$ and $L_{(n)}$ satisfy
the Virasoro algebra relations with central charge $c$ (cf. [DLM97]).

###  3.3. Ordinary twisted modules

Given a vertex superalgebra
$V=\displaystyle\coprod_{n\in\frac{1}{2}\mathbb{Z}}V_{(n)}$ where
$V_{\overline{0}}=\displaystyle\coprod_{n\in\mathbb{Z}}V_{(n)}$ and
$V_{\overline{1}}=\displaystyle\coprod_{n\in\mathbb{Z}+\frac{1}{2}}V_{(n)},$
there are two binary operations defined as following: for homogeneous $a\in V$
and $b\in V$,

(16) $\displaystyle
a\ast_{g}b=\begin{cases}\displaystyle\sum_{i\in\mathbb{N}}\binom{wt(a)}{i}a_{(i-1)}b,&\mbox{if
}a,b\in V^{0*}\\\ 0,&\mbox{if }a\;\text{or}\;b\in V^{r*},r\neq 0\end{cases},$

and

(17) $\displaystyle
a\circ_{g}b=\begin{cases}\displaystyle\sum_{i\in\mathbb{N}}\binom{wt(a)}{i}a_{(i-2)}b,&\mbox{if
}a\in V^{0*}\\\
\displaystyle\sum_{i\in\mathbb{N}}\binom{wt(a)-1+\frac{r}{T^{\prime}}}{i}a_{(i-1)}b,&\mbox{if
}a\in V^{r*},r\neq 0\end{cases}.$

Let $O_{g}(V)$ be the linear span of elements of the form $a\circ_{g}b$ in
$V.$ Then twisted Zhu’s algebra $A_{g}(V)$ is defined as the quotient space
$V/O_{g}(V)$ with the multiplication from $\ast_{g}$. According to [DSK06],
there is a filtration $\left\\{\overline{F}_{k}(A_{g}(V))\right\\}$ on
$A_{g}(V)$ where
$\overline{F}_{k}(A_{g}(V)):=(\displaystyle\bigoplus_{i\in\frac{1}{2}\mathbb{Z},i\leq
k}V_{(i)}+O_{g}(V))/O_{g}(V)$. Its associated graded algebra,

$gr^{\overline{F}}(A_{g}(V))=\displaystyle\bigoplus_{i=0}^{\infty}\overline{F}_{k}(A_{g}(V))/\overline{F}_{k-1}(A_{g}(V)),$

is a commutative algebra. Define
$\overline{F}_{k}(V)=\displaystyle\oplus_{i\in\frac{1}{2},i\leq k}V_{(i)}$.

######  Lemma 3.9.

We have a surjective Poisson algebra homomorphism:

(18) $\displaystyle\phi:(R_{V})^{\sigma g}\mapsto
gr^{\overline{F}}(A_{g}(V)),$

where $\phi(a+(C_{2}(V))_{(p)})=a+O_{g}(V)+\overline{F}_{p-1}(V)$ for
$a\in(V)^{\sigma g}$.

###### Proof.

First, we show that the map $\phi$ is a well-defined map. It is enough to
prove

$(C_{2}(V))_{(p)}\subset O_{g}(V)+\overline{F}_{p-1}(V).$

The space $(C_{2}(V))_{(p)}$ is spanned by the weight $p$ elements of the form
$a_{(-2)}b$ with homogeneous elements $a$ and $b$. For such $a$ and $b$, we
have $wt(a_{(i-2)}b)\leq p-1$ if $i\in\mathbb{Z}_{+}.$ Hence

$a_{(-2)}b\equiv\begin{cases}\displaystyle a\circ b\,\,\,({\rm
mod}\,\,\,\,\overline{F}_{p-1}(V))&\mbox{if }a\in V^{0*}\\\ 0\,\,\,({\rm
mod}\,\,\,\,\overline{F}_{p-1}(V)+O_{g}(V))&\mbox{if }a\in V^{r*}\;\;r\neq
0\end{cases}.$

Therefore, $(C_{2}(V))_{(p)}\subset O_{g}(V)+\overline{F}_{p-1}(V),$ and the
map $\phi$ is well-defined.

Next, let us show that $\phi$ is a Poisson algebra homomorphism. Given two
elements $a\in(V^{\sigma g})_{(p)}$ and $b\in(V^{\sigma g})_{(q)},$ for $u\in
a+(C_{2}(V))_{(p)}$ and $v\in b+(C_{2}(V))_{(q)}$, we have

$u_{(-1)}v\in a_{(-1)}b+(C_{2}(V))_{(p+q)}$

and

$u_{(0)}v\in a_{(0)}b+(C_{2}(V))_{(p+q-1)}.$

Moreover, according to [Zhu96] and (16), (17), for $u\in
a+O_{g}(V)+\overline{F}_{p-1}(V)$ and $v\in b+O_{g}(V)+\overline{F}_{q-1}(V)$,
we have

$u\ast v\in a_{(-1)}b+O_{g}(V)+\overline{F}_{p+q-1}(V)$

and

$u\ast v-v\ast u\in a_{(0)}b+O_{g}(V)+\overline{F}_{p+q-2}(V).$

So $\phi$ is a Poisson algebra homomorphism. And the surjectivity follows from
the fact that $V^{r*}\subset O_{g}(V)$ when $r\neq 0$. ∎

######  Theorem 3.10.

Let $V$ be a quasi-lisse vertex operator superalgebra. Then it has finitely
many simple ordinary $g$-twisted modules.

###### Proof.

By previous Lemma,

${\rm Specm}(gr^{\overline{F}}(A_{g}(V)))$

is a Poisson subvariety of ${\rm Specm}(((R_{V})^{\sigma
g})=(X_{V})_{red}^{\langle\sigma g\rangle}$, and hence has finitely many
symplectic leaves. According to [ES10], $A_{g}(V)$ has finitely many simple
finite-dimensional representations. The result follows from [DLM98, Theorem
7.2]. ∎

######  Example 3.1.

[LLY] The $e^{\frac{1}{2}\pi ih_{(0)}}$-twisted Zhu’s algebra of of
$L_{-\frac{4}{3}}(\mathfrak{sl}_{2})$ is $C[x]/\langle
x(x-\frac{2}{3})(x+\frac{2}{3})\rangle$. There are three simple twisted
modules with Dynkin labels (the eigenvalue of $h_{(0)}$ of the highest weight
vector), i.e., $0,\pm\frac{2}{3}$.

###  3.4. Twisted affine Kac-Moody Lie algebras

Let $L(\tilde{X})$ ($L(\tilde{X}^{2})$ or $L(\tilde{X}^{3})$) be (twisted)
affine Kac-Moody algebra of type $\tilde{X}$ ($\tilde{X}^{2}$ or
$\tilde{X}^{3}$), and $V^{k}(X)$ and $L_{k}(X)$ be the vacuum vertex algebra
of type $X$ and its simple quotient. We recall some facts in [Li96]:

######  Theorem 3.11.

An irreducible $L(\tilde{X}^{2})$ or $L(\tilde{X}^{3})$-module is a
$\mu$-twisted module for $L_{k}(X)$, $k\in\mathbb{Z}_{+}$ if and only if it is
an integrable representation of $L(\tilde{X}^{2})$ or $L(\tilde{X}^{3})$,
where $\mu$ is the twisted graph automorphism of $L(\tilde{X})$.

######  Example 3.12.

[Li96] (Twisted affine vertex algebra of type $A_{n}^{(2)}$) Let
$q\in\mathbb{Z}_{+}$.

If $n=2q-1$, the Dynkin diagram of $\mathfrak{sl}_{n+1}$ is

$1$$q-1$$2q-1$$q+1$

The Dynkin diagram permutations $\sigma(i)=2q-i$ induce an automorphism
$\sigma$ on $\mathfrak{sl}_{n+1}.$

If $n=2q$, the Dynkin diagram of $\mathfrak{sl}_{n+1}$ is

$1$$q$$2q$$q+1$

The Dynkin diagram permutations $\sigma(i)=2q+1-i$ induce an automorphism
$\sigma$ on $\mathfrak{sl}_{n+1}.$

The automorphism $\sigma$ induces a twisted graph automorphism $\mu$ on
$L_{k}(A_{n})$, $k\in\mathbb{Z}_{+}$ by

$\mu(t^{s}\otimes x)=(-1)^{s}t^{s}\otimes\sigma(x)\quad\text{for}\quad
x\in\mathfrak{sl}_{n+1}.$

Then irreducible $L(\tilde{A_{n}}^{(2)})$-module $L(\lambda)$, where $\lambda$
is the highest weight, is a $\mu$-twisted $L_{k}(A)$-module, where the module
action when $n=2q+1$ is given by:

(19) $\displaystyle
Y_{M}((E_{i,i+1}+E_{{2q-i,2q-i+1}})_{(-1)}\mathbf{1},z)=\displaystyle\sum_{n\in\mathbb{Z}}(E_{i,i+1}+E_{{2q-i,2q-i+1}})_{(2n)}z^{-2n-1},$
(20) $\displaystyle
Y_{M}((E_{i,i+1}-E_{{2q-i,2q-i+1}})_{(-1)}\mathbf{1},z)=\displaystyle\sum_{n\in\mathbb{Z}}(E_{i,i+1}-E_{{2q-i,2q-i+1}})_{(n)}z^{-n-1}.$

And module action for the case when $n=2q$ can be given similarly.

We cannot use Li’s filtration to abelianize the twisted module, since the
twisted commutator formula (14) and iterated formula (15) are not compatible
with the filtration. Nevertheless, we can still abelianize the twisted module
from a different perspective. We consider the simple vertex operator algebra
$L_{k}(X)$. Its $C_{2}$-algebra is $\mathbb{C}[\mathfrak{g}]/\langle
U(\mathfrak{g})e_{\theta}^{k+1}\rangle.$ On one hand, let $\mu$ be the action
on arcs in $\text{Spec}(R_{L_{k}(X)})$ given by
$\mu(x(t^{1/m})):=\sigma^{-1}x(\varepsilon_{m}t^{1/m})$, where $m$ is the
order of $\sigma$ and $\varepsilon_{m}=e^{\frac{2\pi i}{m}}.$ Then we have the
twisted arc algebra of $R_{L_{k}(X)}$, $J_{\infty}^{\sigma}(R_{L_{k}(X)})$. On
the other hand, we can think of $L_{k}(X)$ as a module over $L(\tilde{X})$. We
also have $(L(\tilde{X}))^{\mu}\cong L(\tilde{X}^{(2)})$ or
$(L(\tilde{D_{4}}))^{\mu}\cong L(\tilde{D_{4}}^{(3)})$ depending on the order
of Dynkin diagram automorphism. Then we get a $(L(\tilde{X}))^{\mu}$-module in
$L_{k}(X)$ generated by its highest weight vector $v_{k}$,
$U((L(\tilde{X}))^{\mu})v_{k}$. We can show that

######  Proposition 3.13.

For the irreducible module over $(L(\tilde{X})^{(2)})$ or
$(L(\tilde{X})^{(3)})$ with highest weight $k\Lambda_{0}$
$(k\in\mathbb{Z}_{+})$, $L(k\Lambda_{0})$, we have the following isomorphisms
or epimorphisms:

  * 1.

$L(k\Lambda_{0})\cong U((L(\tilde{X}))^{\mu})v_{k}$ when $X=A_{2n}$, $n\geq
1$;

  * 2.

when $X=A_{2n-1},D_{n},E_{6}\,\,\,(n\geq 3),$ the natural map
$U((L(\tilde{X}))^{\mu})v_{2k}\twoheadrightarrow L(k\Lambda_{0})$ has a
nontrivial kernel;

  * 3.

when $X=D_{4}$, and the natural map
$U((L(\tilde{X}))^{\mu^{\prime}})v_{3k}\twoheadrightarrow L(k\Lambda_{0})$ has
a nontrivial kernel.

###### Proof.

Let $V(k\Lambda_{0})$ be the Verma module of $L(\tilde{X}^{(2)})$ (or
$(L(\tilde{X}^{(3)}))$). And let $m^{k}_{1}$ and $m^{k}_{2}$ be the singular
vectors of $V^{k}(X)$ and $V(k\Lambda_{0})$, respectively. Firstly note that
$V(k\Lambda_{0})\cong V^{k}(X)\cap U(L(\tilde{X})^{\mu})v_{k}$ in the case 1,
and $V(k\Lambda_{0})\cong V^{k}(X)\cap U(L(\tilde{X})^{\mu})v_{2k}$,
$V(k\Lambda_{0})\cong V^{k}(X)\cap U(L(\tilde{X})^{\mu})v_{3k}$ in case 2 and
3, respectively. Denote the maximal ideals of $V(k\Lambda_{0})$ and $V^{k}(X)$
by $I_{L(k\Lambda_{0})}$ and $I_{L_{k}(X)}$.

The case 1 follows from the fact that
$m^{k}_{1}=m^{k}_{2}=(e_{\theta})_{(-1)}^{k+1}v_{k},$ where $\theta$ is the
longest root of the corresponding Lie algebra. Next, we prove the case when
$X=D_{4}$, and all other cases can be proved similarly. Let
$\alpha_{1},\alpha_{2},\alpha_{3}$ and $\alpha_{4}$ be positive simple roots
of the Lie algebra of type $D_{4}$. The Dynkin diagram automorphism is given
by

$\displaystyle\sigma(\alpha_{1})=\alpha_{3},\quad\sigma(\alpha_{3})=\alpha_{4},\quad\sigma(\alpha_{4})=\alpha_{1}.$

In this case,

$\displaystyle m^{3k}_{1}=(e_{\theta})_{(-1)}^{3k+1}v_{3k}^{\prime}$
$\displaystyle m^{k}_{2}=(e_{\theta^{0}}+\omega\sigma
e_{\theta^{0}}+\omega^{2}\sigma^{2}e_{\theta^{0}})_{(-1)}^{3k+1}v_{k}^{\prime\prime},$

where $v_{k}^{\prime\prime}$ is the highest weight vector in
$V(k\Lambda_{0})$, $\theta$ is the highest root,
$\theta^{0}=\alpha_{1}+\alpha_{2}+\alpha_{3}$, and $\omega=e^{2\pi i/3}.$ By
calculation, we have

$\displaystyle(e_{-\alpha_{2}-\alpha_{4}}+\omega
e_{-\alpha_{1}-\alpha_{2}}+\omega^{2}e_{-\alpha_{2}-\alpha_{3}})_{(0)}^{3k+1}(e_{\theta})_{(-1)}^{3k+1}v_{3k}^{\prime}$
$\displaystyle=(e_{\theta^{0}}+\omega\sigma
e_{\theta^{0}}+\omega^{2}\sigma^{2}e_{\theta^{0}})_{(-1)}^{3k+1}v_{3k}^{\prime}+\text{extra
terms},$

where the extra terms involves with $(e_{\alpha_{1}})_{(-1)}$ or
$(e_{\alpha_{2}})_{(-1)}$ or $(e_{\alpha_{3}})_{(-1)}$ which are not fixed by
$\mu$. Therefore,

$L(k\Lambda_{0})\cong U((L(\tilde{X})^{\mu^{\prime}}))v_{3k}^{\prime}/\langle
U((L(\tilde{X})^{\mu^{\prime}}))(e_{\theta^{0}}+\omega\sigma
e_{\theta^{0}}+\omega^{2}\sigma^{2}e_{\theta^{0}})_{(-1)}^{3k+1}v_{3k}^{\prime}\rangle.$

The result follows.

∎

######  Conjecture 3.14.

The vertex operator algebras $L_{k}(X)$ for $k\in\mathbb{Z}_{\geq 1}$ are
classically free.

If the conjecture is true, we have

$J^{\sigma}_{\infty}(\mathbb{C}[\mathfrak{g}]/\langle
U(\mathfrak{g})e_{\theta}^{k+1})\cong gr^{F}(U((L(\tilde{X}))^{\mu})v_{k})$

where $\mathfrak{g}$ is of type $X$, and
$gr^{F}(U((L(\tilde{X}))^{\mu})v_{k})$ is equivalence classes of all elements
in $U((L(\tilde{X}))^{\mu})v_{k}$ in $gr^{F}(L_{k}(X)).$ From above
discussion, we can view $J^{\sigma}_{\infty}(\mathbb{C}[\mathfrak{g}]/\langle
U(\mathfrak{g})e_{\theta}^{k+1}\rangle)$ as the strict chiral quantization of
$U((L(\tilde{X}))^{\mu})v_{k}$ .

###  3.5. Twisted arc superalgebras and superconformal vertex algebras

Recall Example (2.5), $V_{c}^{N=1}$ is the Neveu-Schwarz vertex algebra. The
Ramond algebra with central charge $c$ is

$\displaystyle\displaystyle\mathcal{R}^{c}=\oplus_{m\in\mathbb{Z}}\mathbb{C}L_{(m)}\oplus\oplus_{n\in\mathbb{Z}}\mathbb{C}F_{(n)}\oplus\mathbb{C}c.$

for $m,n\in\mathbb{Z}$. Recall the following fact:

######  Proposition 3.15.

[Li96] Let $c$ be a complex number. Then any module of $\mathcal{R}^{c}$ such
that $u_{(n)}=0$ when $u\in\mathcal{R}^{c}$ and $n$ is large enough, is a weak
$\sigma$-twisted $V_{c}^{N=1}$-module, where $\sigma$ is the canonical
automorphism of vertex superalgebra.

Therefore, we have a $\sigma$-twisted $V_{c}^{N=1}$-module
$\overline{N}(c,0)={N}(c,0)/\langle F_{(-1)}\mathbf{1}\rangle$, where
${N}(c,0)$ is the Verma module of Ramond algebra. The $C_{2}$-algebra of
$V_{c}^{N=1}$ is $\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}]$, and its arc
algebra is $J_{\infty}(\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}])$. Then we
define an action on superarcs in
$\text{Spec}(\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}])$ by
$\sigma(G\otimes\theta t^{n})=-G\otimes\theta(-t)^{n}$, where we identify
$G_{(n+\frac{1}{2})}$ with $G\otimes\theta t^{n}$. The $\sigma$-twisted arc
algebra of $\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}]$ is
$J^{\sigma}_{\infty}(\mathbb{C}[L_{(-2)},G_{(-\frac{3}{2})}])=\mathbb{C}[L_{(-n)},F_{(-n)}|n\geq
2]$, which we can view it as a _strict chiral quantization_ of
$\overline{N}(c,0)$.

Next, we consider $V_{c}^{N=2}$ (see Example 2.5). Let $\sigma$ be canonical
automorphsim of $V_{c}^{N=2}$, then

$J^{\sigma}_{\infty}(\mathbb{C}[L_{(-2)},J_{(-1)},G^{\pm}_{(-\frac{3}{2})}])$

is the strict chiral quantization of module
$\overline{N^{\prime}}(c,0)=N^{\prime}(c,0)/\langle G^{\pm}_{(-1)}\rangle$,
where $N^{\prime}(c,0)$ is the Verma module of the $N=2$ Ramond algebra. More
generally, we have an automorphisms on $V_{c}^{N=2}$ given as following:

$\displaystyle\sigma_{\varepsilon}:\,\,G_{(n+\frac{1}{2})}^{\pm}\mapsto\varepsilon^{\pm}G_{(n+\frac{1}{2})}^{\pm},\,\,J_{(n)}\mapsto
J_{(n)},\,\,L_{(n)}\mapsto J_{(n)},\,\,c\mapsto c,$

where $\varepsilon=e^{2\pi it/s},$ and $0\leq\frac{t}{s}<1$ with $(t,s)=1$.
This corresponds to an action on superarcs of $\text{Spec}(R_{V_{c}^{N=2}})$
given by
$\sigma_{\varepsilon}(G_{(n+\frac{1}{2})}^{\pm})=\varepsilon^{\pm}G^{\pm}\otimes\theta(\varepsilon^{-1}t)^{n},$
where we identify $G_{(n+\frac{1}{2})}^{\pm}$ with
$G^{\pm}\otimes\theta(\varepsilon^{-1}t)^{n}$. Then the twisted arc algebra
$J^{\sigma_{\varepsilon}}(R_{V_{c}^{N=2}})$ is the strict chiral quantization
of the image of $V_{c}^{N=2}$ under spectral flow map $D(t/s)$:

$\displaystyle D(t/s):\quad$ $\displaystyle L_{(n)}\mapsto
L_{(n)}+t/sJ_{(n)}+\frac{(t/s)^{2}}{6}\delta_{n,0}c,\quad J_{(n)}\mapsto
J_{(n)}+\frac{t/s}{3}\delta_{n,0}c,$ $\displaystyle c\mapsto c,\quad
G^{\pm}_{(n+\frac{1}{2})}\mapsto G^{\pm}_{(n+\frac{1}{2}\pm t/s)}.$

Note that $J^{\sigma_{-1}}(R_{V_{c}^{N=2}})\cong J^{\sigma}(R_{V_{c}^{N=2}}).$
The Virasoro-preserving automorphisms of $V_{c}^{N=2}$ is generated by
$\sigma_{\epsilon}$ and mirror map $\kappa$ [Bar13]:

$\displaystyle\kappa:\,\,G^{\pm}_{(n+\frac{1}{2})}\mapsto
G^{\mp}_{(n+\frac{1}{2})},\,\,J_{(n)}\mapsto-J_{(n)},\,\,L_{(n)}\mapsto
L_{(n)},\,\,c\mapsto c.$

Then the twisted arc algebra $J^{\kappa}_{\infty}(R_{V_{c}^{N=2}})$ is the
strict chiral quantization of the Verma module of mirror twisted $N2$ algebra
quotient out $\langle G^{1}_{(-\frac{1}{2})},G^{2}_{(-1)}\rangle$.

##  4\. Genus-0 correlation functions

###  4.1. Intertwining operators among weak $g$-twisted modules

Let $(W_{1},Y_{1})$ be a weak module of vertex operator superalgebra $V$, and
$(W_{j},Y_{j}),$ $j=2,3$ be $g$-twisted $V$-modules, where the order of $g$ is
$T$. Assume that $W_{i}$, $i=1,2,3$ are $\mathbb{Z}_{2}$-graded such that
$v_{(n)}w\in W_{i}^{|v|+|w|}$ for $v\in V^{|v|}$ and $w\in W_{i}^{|w|}$.

######  Definition 4.1.

An intertwining operator of type $\binom{W_{3}}{W_{1}\;W_{2}}$ is a linear map

$\displaystyle\mathcal{Y}(\cdot,x)\cdot:W_{1}\otimes W_{2}\rightarrow
W_{3}\\{x\\}$

such that for $w^{1}\in W_{1}$, $w^{2}\in W_{2}$,

(21) $\displaystyle w^{1}_{(n)}w^{2}=0\quad\text{for $n$ sufficiently large},$

where $\mathcal{Y}(w^{1},x)=\sum_{n\in\mathbb{Q}}w^{1}_{(n)}x^{-n-1};$ the
Jacobi identity

(22)
$\displaystyle\begin{split}\hfil\displaystyle\begin{split}&x_{0}^{-1}\delta(\frac{x_{1}-x_{2}}{x_{0}})Y_{3}(u,x_{1})\mathcal{Y}(w^{1},x_{2})\\\
&\quad\quad-(-1)^{|u||w^{1}|}x_{0}^{-1}\delta(\frac{-x_{2}+x_{1}}{x_{0}})\mathcal{Y}(w^{1},x_{2})Y_{2}(u,x_{1})\\\
&=x_{2}^{-1}\delta(\frac{x_{1}-x_{0}}{x_{2}})(\frac{x_{1}-x_{0}}{x_{2}})^{-\frac{k}{T}}\mathcal{Y}(Y_{1}(u,x_{0})w^{1},x_{2})\end{split}\end{split}$

holds for $u\in V^{k}$;

(23)
$\displaystyle\frac{d}{dz}\mathcal{Y}(w^{1},x)=\mathcal{Y}(L_{(-1)}w^{1},x),$

for $w^{1}\in W_{1}$.

Taking $\text{Res}_{x_{0}}$ of (22), one obtains the commutator formula:

(24)
$\displaystyle\begin{split}&Y_{3}(u,x_{1})\mathcal{Y}(w^{1},x_{2})-(-1)^{|u||w^{1}|}\mathcal{Y}(w^{1},x_{2})Y_{2}(u,x_{1})\\\
&\quad\quad=\text{Res}_{x_{0}}x_{1}^{-1}\delta(\frac{x_{2}+x_{0}}{x_{1}})(\frac{x_{2}+x_{0}}{x_{1}})^{\frac{k}{T}}\mathcal{Y}(Y_{1}(u,x_{0})w^{1},x_{2}).\end{split}$

Taking $\text{Res}_{x_{1}}$ of (22), one obtains the associator formula:

(25)
$\displaystyle\begin{split}&(x_{2}+x_{0})^{\frac{k}{T}}\mathcal{Y}(Y_{1}(u,x_{0})w_{1},x_{2})-Y_{3}^{\circ}(u,x_{0}+x_{2})\mathcal{Y}(w_{1},x_{2})\\\
&\quad\quad=-\text{Res}_{x_{1}}(-1)^{|u||w^{1}|}x_{0}^{-1}\delta(\frac{-x_{2}+x_{1}}{x_{0}})\mathcal{Y}(w^{1},x_{2})Y_{2}^{\circ}(u,x_{1}),\end{split}$

where $Y^{\circ}_{i}(u,x)=z^{\frac{k}{T}}Y_{i}(u,z)\in\text{End}M\llbracket
x,x^{-1}\rrbracket.$

By using the similar argument in [Hua10, Theorem 2.4], one has:

######  Theorem 4.1.

The twisted Jacobi identity (22) is equivalent to the following property: for
any $u\in V$, $w^{1}\in W_{1}$, $w^{2}\in W_{2}$ and
$w^{\prime}\in(W_{3})^{\prime}$, there exists a multivalued analytic function
of the form

$\displaystyle
f(z_{1},z_{2})=\sum_{r,s=N_{1}}^{N_{2}}a_{rs}z_{1}^{r/k}z_{2}^{s/k}(z_{1}-z_{2})^{-N}$

for $N_{1},N_{2}\in\mathbb{Z}$ and $N\in\mathbb{Z}_{+}$, such that the series

(26) $\displaystyle\langle
w^{\prime},Y_{3}(u,z_{1})\mathcal{Y}(w^{1},z_{2})w\rangle,$ (27)
$\displaystyle(-1)^{|u||w^{1}|}\langle
w^{\prime},\mathcal{Y}^{(}w^{1},z_{2})Y_{2}(u,z_{1})w\rangle,$ (28)
$\displaystyle\langle
w^{\prime},\mathcal{Y}(Y_{1}(u,z_{1}-z_{2})w^{1},z_{2})w\rangle$

are absolutely convergent in the regions $|z_{1}|>|z_{2}|>0$,
$|z_{2}|>|z_{1}|>0$, $|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to the branch

$\displaystyle\sum_{r,s=N_{1}}^{N_{2}}a_{rs}e^{(r/s)\log(z_{1})}e^{(s/k)\log(z_{2})}(z_{1}-z_{2})^{-N}.$

of $f(z_{1},z_{2})$, where $\log z_{i}=\log|z|+i\text{arg}z_{i}$, $i=1,2$ and
$0\leq\text{arg}z_{i}<2\pi.$

This property is called duality property.

######  Conjecture 4.2.

Let $V$ be quasi-lisse, $W_{1}$ and $W_{2}$ be finitely strongly generated
$V$-modules and $W_{3}$ be ordinary $g$-twisted modules, the intertwining
operators $\mathcal{Y}_{1}$ and $\mathcal{Y}_{2}$ of type
$\binom{W_{4}}{W_{1}\;W_{5}}$ and $\binom{W_{5}}{W_{2}\;W_{3}}$ satisfy the
duality property, where $W_{3}$, $W_{4}$, $W_{5}$ are weak $g$-twisted
$V$-modules.

###  4.2. Convergence of genus-0 correlation functions

In this subsection, we prove Conjecture 4.2 in the case where $g$ is identity.
Let $V$ be the quasi-lisse vertex operator superalgebra. Let $W_{i}$,
$i=0,1,2,3$ be weak $\mathbb{Q}$-graded $V$-modules, $R=\mathbb{C}[z_{1}^{\pm
1},z_{2}^{\pm},(z_{1}-z_{2})^{-1}]$, and $T=R\otimes W_{0}\otimes W_{1}\otimes
W_{2}\otimes W_{3}$, which has a natural $R$-module structure. Let
$W^{\prime}=\coprod_{n\in\mathbb{Q}}W(n)^{*}$ be the contragredient module of
$W$, whose vertex operator map $Y^{\prime}$ is defined by

(29) $\displaystyle\langle Y^{\prime}(u,x)w^{\prime},w\rangle=\langle
w^{\prime},Y_{W}(e^{xL_{(1)}}(-x^{-2})^{L_{(0)}}u,x^{-1})w\rangle,$

for any $u\in V$, $w^{\prime}\in W^{\prime}$ and $w\in W$. In particular, for
$u\in V$ and $n\in\mathbb{Q}$, we have the operators $u_{(n)}$ on $W$. Let
$u_{(n)}^{*}$ be the adjoint of $u_{(n)}$ acting on $W^{\prime}$. Note
$\text{wt}u_{(n)}=\text{wt}u-n-1$ and $\text{wt}u_{(n)}^{*}=-\text{wt}u+n+1$.

Let $\mathcal{Y}_{1}$ and $\mathcal{Y}_{2}$ be intertwining operators of types
$\binom{W_{0}}{W_{1}\,\,W_{5}}$ and $\binom{W_{5}}{W_{2}\,\,W_{3}}$,
respectively. By (24):

(30)
$\displaystyle\begin{split}\mathcal{Y}_{2}(w_{2},x_{2})Y_{W_{3}}(u,y_{1})w_{3}&=Y_{W_{5}}(u,y_{1})\mathcal{Y}_{2}(w_{2},x_{2})w_{5}\\\
&+\text{Res}_{y_{2}}y_{1}^{-1}\delta(\frac{x_{2}+y_{2}}{y_{1}})\mathcal{Y}_{2}(Y_{W_{2}}(u,y_{2})w_{2},x_{2})w_{5}.\end{split}$

By taking $\text{Res}_{y_{1}}$ on both sides of (30), we get

$\displaystyle\begin{split}\mathcal{Y}_{2}(w_{2},x_{2})u_{(0)}w_{3}=u_{(0)}\mathcal{Y}_{2}(w_{2},x_{2})w_{5}+\mathcal{Y}_{2}(u_{(0)}w_{2},x_{2})w_{5}.\end{split}$

Applying the commutator formula again, we get:

(31) $\displaystyle\begin{split}&\langle
w_{4}^{\prime},\mathcal{Y}_{1}(w_{1},x_{1})\mathcal{Y}_{2}(w_{2},x_{2})u_{(0)}w_{3}\rangle\\\
&=\langle
u_{(0)}^{*}w_{4}^{\prime},\mathcal{Y}_{1}(w_{1},x_{1})\mathcal{Y}_{2}(w_{2},x_{2})w_{3}\rangle-\langle
w_{4}^{\prime},\mathcal{Y}_{1}(u_{(0)}w_{1},x_{1})\mathcal{Y}_{2},x_{2})w_{3}\rangle\\\
&-\langle
w_{4}^{\prime},\mathcal{Y}_{1}(w_{1},x_{1})\mathcal{Y}_{2}(u_{(0)}w_{2},x_{2})w_{3}\rangle.\end{split}$

For any $u\in\coprod_{n>0}V_{(n)}$ and $w_{i}\in W_{i}$, $i=0,1,2,3,$ let $J$
be the submodule of $T$ generated by elements of the form:

$\displaystyle\mathcal{A}($ $\displaystyle u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle=\displaystyle\sum_{k\geq
0}\binom{-1}{k}(-z_{1})^{k}u_{(-1-k)}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}-w_{0}\otimes u_{(-1)}w_{1}\otimes w_{2}\otimes w_{3}$
$\displaystyle-\displaystyle\sum_{k\geq
0}\binom{-1}{k}(-(z_{1}-z_{2}))^{-1-k}\otimes w_{0}\otimes w_{1}\otimes
u_{(k)}w_{2}\otimes w_{3}$ $\displaystyle-\displaystyle\sum_{k\geq
0}\binom{-1}{k}(-z_{1})^{-1-k}\otimes w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{(k)}w_{3},$ $\displaystyle\mathcal{B}($ $\displaystyle
u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle=\displaystyle\displaystyle\sum_{k\geq
0}\binom{-1}{k}(-z_{2})^{k}u^{*}_{(-1-k)}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle-\displaystyle\displaystyle\sum_{k\geq
0}\binom{-1}{k}(z_{1}-z_{2})^{-1-k}\otimes w_{0}\otimes u_{(k)}w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle-w_{0}\otimes w_{1}\otimes
u_{(-1)}w_{2}\otimes w_{3}$
$\displaystyle-\displaystyle\displaystyle\sum_{k\geq
0}\binom{-1}{k}(-z_{2})^{-1-k}\otimes w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{(k)}w_{3},$ $\displaystyle\mathcal{C}($ $\displaystyle
u,w_{0},w_{1},w_{2},w_{3})$ $\displaystyle=u_{(-1)}^{*}w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3}-\displaystyle\displaystyle\sum_{k\geq
0}\binom{-1}{k}z_{1}^{-1-k}\otimes w_{0}\otimes u_{(k)}w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle-\displaystyle\sum_{k\geq
0}\binom{-1}{k}z_{2}^{-1-k}\otimes w_{0}\otimes w_{1}\otimes
u_{(k)}w_{2}\otimes w_{3}$ $\displaystyle-w_{0}\otimes w_{1}\otimes
w_{2}\otimes u_{(-1)}w_{3}$ $\displaystyle\mathcal{D}($ $\displaystyle
u,w_{0},w_{1},w_{2},w_{3})$ $\displaystyle=u_{(-1)}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle-\displaystyle\sum_{k\geq
0}\binom{-1}{k}z_{1}^{1+k}w_{0}\otimes
e^{z_{1}^{-1}L_{(1)}}(-z_{1}^{2})^{L_{(0)}}u_{(k)}(-z_{1}^{-2})^{L_{(0)}}e^{-z_{1}^{-1}L_{(1)}}w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle-\displaystyle\sum_{k\geq
0}\binom{-1}{k}z_{2}^{1+k}w_{0}\otimes w_{1}\otimes
e^{z_{2}^{-1}L_{(1)}}(-z_{2}^{2})^{L(0)}u_{(k)}(-z_{1}^{-2})^{L_{(0)}}e^{-z_{2}^{-1}L_{(1)}}w_{2}\otimes
w_{3}$ $\displaystyle+w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{(-1)}^{*}w_{3},$

$\displaystyle\mathcal{E}($ $\displaystyle u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle=u_{(0)}^{*}w_{0}\otimes w_{1}\otimes w_{2}\otimes
w_{3}-w_{0}\otimes w_{1}\otimes w_{2}\otimes u_{(0)}w_{3}$ $\displaystyle-
w_{0}\otimes u_{(0)}w_{1}\otimes w_{2}\otimes w_{3}-w_{0}\otimes w_{1}\otimes
u_{(0)}w_{2}\otimes w_{3}.$

The weights on $W_{i}$ for $i=0,1,2,3$ induce a weight on $W_{0}\otimes
W_{1}\otimes W_{2}\otimes W_{3}$. Let $T_{(r)}$ be the homogeneous subspace of
weight $r$ for $r\in\mathbb{N}$. Then we have a filtration $I$ on $T$ by
letting $I^{n}(T)=\displaystyle\coprod_{i\leq n}I_{(i)}$. Now we have an
associated graded space with respect to the filtration $I$, $gr^{I}(T)$.
Moreover, the filtration $I$ on $T$ also induces a filtration on $J$ by
$I^{n}(J)=I^{n}(T)\cap J$.

######  Lemma 4.3.

Suppose $V$ is of CFT type, self-dual, finitely generated by weight $1$
vectors. If

(32) $\displaystyle\text{dim}R_{W_{0}}\otimes R_{W_{1}}\otimes
R_{W_{2}}\otimes R_{W_{3}}/\left\\{R_{V},R_{W_{0}}\otimes R_{W_{1}}\otimes
R_{W_{2}}\otimes R_{W_{3}}\right\\}<0,$

where the tensor product is taken over $R_{V}$, then $R$-module $T/J$ is
finitely generated.

###### Proof.

The equivalence classes of

$\displaystyle\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3}),$
$\displaystyle\mathcal{B}(u,w_{0},w_{1},w_{2},w_{3}),$
$\displaystyle\mathcal{C}(u,w_{0},w_{1},w_{2},w_{3}),$
$\displaystyle\mathcal{D}(u,w_{0},w_{1},w_{2},w_{3})$

in $gr^{I}(T)$ are

$\displaystyle-w_{0}\otimes u_{(-1)}w_{1}\otimes w_{2}\otimes w_{3},$
$\displaystyle-w_{0}\otimes w_{1}\otimes u_{(-1)}w_{2}\otimes w_{3},$
$\displaystyle-w_{0}\otimes w_{1}\otimes w_{2}\otimes u_{(-1)}w_{3},$
$\displaystyle u_{(-1)}w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}.$

Thus by replacing $u$ with $L_{(-1)}u$, we get that $w_{0}\otimes
u_{(-2)}w_{1}\otimes w_{2}\otimes w_{3}$, $w_{0}\otimes w_{1}\otimes
u_{(-2)}w_{2}\otimes w_{3}$, $w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{(-2)}w_{3}$ and $u_{(-2)}w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$ all
belong to $gr^{I}(J)$.

Now we define a map

$\displaystyle\phi:R\otimes R_{W_{0}}\otimes R_{W_{1}}\otimes R_{W_{2}}\otimes
R_{W_{3}}/\left\\{R_{V},R_{W_{0}}\otimes R_{W_{1}}\otimes R_{W_{2}}\otimes
R_{W_{3}}\right\\}\rightarrow T/gr^{I}(J)$

by
$\phi(\overline{a_{0}}\otimes\overline{a_{1}}\otimes\overline{a_{2}}\otimes\overline{a_{3}})=\overline{a_{0}\otimes
a_{1}\otimes a_{2}\otimes a_{3}}$, where $a_{i}\in W_{i}$, $i=0,1,2,3$. In
order to show that this map is well-defined, we have to prove that

(33) $\displaystyle\begin{split}&C_{2}(W_{0})\otimes W_{1}\otimes W_{2}\otimes
W_{3}+W_{0}\otimes C_{2}(W_{1})\otimes W_{2}\otimes W_{3}\\\ &+W_{0}\otimes
W_{1}\otimes C_{2}(W_{2})\otimes W_{3}+W_{0}\otimes W_{1}\otimes W_{2}\otimes
C_{2}(W_{3})\end{split}$

and

(34) $\displaystyle\begin{split}&\\{u_{(0)}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}+w_{0}\otimes u_{(0)}w_{1}\otimes w_{2}\otimes
w_{3}+w_{0}\otimes w_{1}\otimes u_{(0)}w_{2}\otimes w_{3}\\\ &+w_{0}\otimes
w_{1}\otimes w_{2}\otimes u_{(0)}w_{3}|u\in V_{(1)},w_{i}\in
W_{i},i=0,1,2,3\\}\end{split}$

belong to $gr^{I}(J)$. We have already obtained (33). It suffices to show
(34). Assume that the weight of $u$ is $1.$ Recall that we have

$\displaystyle\langle
u^{*}_{(m)}w_{1},w_{2}\rangle=-\displaystyle\sum_{i=0}^{\infty}\frac{1}{i!}\langle
w_{1},(L_{(1)}^{i}u)_{(-m-i)}w_{2}\rangle,$

where $w_{1}\in W^{\prime}$ and $w_{2}\in W$. By [Li94], if $V$ is of CFT
type, $V$ is self-dual is equivalent with that all weight $1$ vectors in $V$
are primary. We have $u_{(0)}^{*}=-u_{(0)}$. And
$\mathcal{E}(u,w_{0},w_{1},w_{2},w_{3})\in J$ implies (34).

It is obvious that the map $\phi$ is surjective. Thus the finite
dimensionality of

$R_{W_{0}}\otimes R_{W_{1}}\otimes R_{W_{2}}\otimes
R_{W_{3}}/\left\\{R_{V},R_{W_{0}}\otimes R_{W_{1}}\otimes R_{W_{2}}\otimes
R_{W_{3}}\right\\}$

implies that $T/gr^{I}(J)$ is finitely generated as $R$-module. ∎

For convenience, we shall denote
$\text{wt}w_{0}+\text{wt}w_{1}+\text{wt}w_{2}+\text{wt}w_{3}=\sigma$. For
$n\in\mathbb{Z}_{+}$, let $F_{n}^{z_{1}=z_{2}}(R)$ be the vector space spanned
by elements of the form $f(z_{1},z_{2})(z_{1}-z_{2})^{-n}$ for
$f(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}].$ Let
$F_{r}^{(z_{1}=z_{2})}$ with $r\in\mathbb{Q}$ be the subspace of $T$ spanned
by elements of the form $f(z_{1},z_{2})(z_{1}-z_{2})^{-n}w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3}$, where
$f(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}],$ and $n+\sigma\leq r$.
Let $F_{r}^{(z_{1}=z_{2})}(J)=F_{r}^{(z_{1}=z_{2})}(T)\cap J$ for
$r\in\mathbb{Q}$. Suppose that $T/J$ is finitely generated over $R$, then we
have:

######  Proposition 4.4.

There exists a positive integer $M$ such that for any $r\in\mathbb{N}$,
$F_{r}^{(z_{1}=z_{2})}(T)\subset F_{r}^{(z_{1}=z_{2})}(J)+I_{M}(T).$

###### Proof.

Since $T/J$ is finitely generated, there exists a positive integer $M>0$ such
that when $m>M$ we have $T\setminus I_{m}(T)\subset J.$ When $r\leq M,$ it is
clear that $F_{r}^{(z_{1}=z_{2})}(T)\subset I_{M}(T).$ For $r=s,$ $s>M,$ let

$(z_{1}-z_{2})^{-i}g(z_{1},z_{2})w_{0}\otimes w_{1}\otimes w_{2}\otimes
w_{3}\in F_{r}^{(z_{1}=z_{2})}(T),$

where $w_{i},$ $i=0,1,2,3$ are homogeneous elements in $W_{i},$ $i=0,1,2,3$,
$g(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$. If $r-i\leq M$, it is
clear that $F_{r}^{(z_{1}=z_{2})}(T)\subset I_{M}(T).$ Now we assume that
$r-i>M$. Then,

$w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}\in T\setminus I_{M}(T)\subset
I_{r-i}(J)$

and therefore,

$(z_{1}-z_{2})^{-i}g(z_{1},z_{2})w_{0}\otimes w_{1}\otimes w_{2}\otimes
w_{3}\in F_{r}^{(z_{1}=z_{2})}(J).$

∎

One considers the $\mathbb{C}$-module

$T^{(z_{1}-z_{2})}=\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]\otimes W_{0}\otimes
W_{1}\otimes W_{2}\otimes W_{3}.$

And let $T_{(r)}^{(z_{1}-z_{2})}$ for $r\in\mathbb{Q}$ be the space of
elements of $T^{(z_{1}-z_{2})}$ of weight $r.$ Let $w_{i}\in W_{i}$ for
$i=0,1,2,3$. Then by Proposition 4.4,

$w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}\otimes
w_{4}=\mathcal{W}_{1}+\mathcal{W}_{2}$

where $\mathcal{W}_{1}\in F_{r}^{(z_{1}=z_{2})}(J)$ and $\mathcal{W}_{2}\in
I_{M}(T).$ By using the same arguments in [Hua05a, Lemma 2.2], we have

######  Lemma 4.5.

For any $s\in[0,1),$ there exist $S\in\mathbb{R}$ such that
$s+S\in\mathbb{Z}_{+}$ and for any $w_{i}\in W_{i},i=0,1,2,3,$ satisfying
$\sigma\in s+\mathbb{Z},$ $(z_{1}-z_{2})^{\sigma+S}\mathcal{W}_{2}\in
T^{(z_{1}=z_{2})}$, where

$T^{(z_{1}=z_{2})}=\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]\otimes W_{0}\otimes
W_{1}\otimes W_{2}\otimes W_{3}.$

######  Theorem 4.6.

Let $V$ be the same as in Lemma 4.3 and $W_{i}$ for $i=0,1,2,3$ satisfy the
finiteness condition (32). Then for any $w_{i}\in W_{i}$ $(i=0,1,2,3),$ there
exist

$a_{k}(z_{1},z_{2}),b_{l}(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm},(z_{1}-z_{2})^{-1}]$

for $k=1,...,m$ and $l=1,...,n$ such that for any discretely
$\mathbb{Q}$-graded $V$-modules $W_{4},W_{5}$ and $W_{6}$, any intertwining
operators
$\mathcal{Y}_{1},\mathcal{Y}_{2},\mathcal{Y}_{3},\mathcal{Y}_{4},\mathcal{Y}_{5}$
and $\mathcal{Y}_{6}$ of types

$\binom{W_{0}^{\prime}}{W_{1}W_{4}},\binom{W_{4}}{W_{2}W_{3}},\binom{W_{5}}{W_{1}W_{2}},\binom{W_{0}^{\prime}}{W_{5}W_{3}},\binom{W_{0}^{\prime}}{W_{2}W_{6}}\,\,\,\,\text{and}\,\,\,\,\binom{W_{6}}{W_{1}W_{3}},$

respectively, the series

(35) $\displaystyle\langle
w_{0},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle,$
(36) $\displaystyle\langle
w_{0},\mathcal{Y}_{4}(\mathcal{Y}_{3}(w_{1},z_{1}-z_{2})w_{2},z_{2})w_{3}\rangle,$

and

(37) $\displaystyle\langle
w_{0},\mathcal{Y}_{5}(w_{2},z_{2})\mathcal{Y}_{6}(w_{1},z_{1})w_{3}\rangle,$

satisfy the expansions of the system of differential equations with regular
singular points

(38) $\displaystyle\frac{\partial^{m}\phi}{\partial
z_{1}^{m}}+a_{1}(z_{1},z_{2})\frac{\partial^{m-1}\phi}{\partial
z_{1}^{m-1}}+\cdots+a_{m}(z_{1},z_{2})\phi=0,$ (39)
$\displaystyle\frac{\partial^{n}\phi}{\partial
z_{2}^{n}}+b_{1}(z_{1},z_{2})\frac{\partial^{n-1}\phi}{\partial
z_{2}^{n-1}}+\cdots+b_{m}(z_{1},z_{2})\phi=0$

in the regions $|z_{1}|>|z_{2}|>0,$ $|z_{2}|>|z_{1}-z_{2}|>0$ and
$|z_{2}|>|z_{1}|>0,$ respectively.

###### Proof.

Let $\Delta=\text{wt}w_{0}-\text{wt}w_{1}-\text{wt}w_{2}-\text{wt}w_{3}$. We
consider the map

$\displaystyle\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}:T\rightarrow
z_{1}^{\Delta}\mathbb{C}(\\{z_{2}/z_{1}\\})[z_{1}^{\pm},z_{2}^{\pm}],$
$\displaystyle\psi_{\mathcal{Y}_{3},\mathcal{Y}_{4}}:T\rightarrow
z_{2}^{\Delta}\mathbb{C}(\\{(z_{1}-z_{2})/z_{1}\\})[z_{2}^{\pm},(z_{1}-z_{2})^{\pm}],$
$\displaystyle\varepsilon_{\mathcal{Y}_{5},\mathcal{Y}_{6}}:T\rightarrow
z_{2}^{\Delta}\mathbb{C}(\\{z_{1}/z_{2}\\})[z_{1}^{\pm},z_{2}^{\pm}],$

defined by

$\displaystyle\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}(f(z_{1},z_{2})w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3})$
$\displaystyle\quad=\iota_{|z_{1}|>|z_{2}|>0}(f(z_{1},z_{2}))\rangle
w_{0},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle,$
$\displaystyle\psi_{\mathcal{Y}_{3},\mathcal{Y}_{4}}(f(z_{1},z_{2})w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3})$
$\displaystyle\quad=\iota_{|z_{2}|>|z_{1}-z_{2}|>0}(f(z_{1},z_{2}))\rangle
w_{0},\mathcal{Y}_{4}(\mathcal{Y}_{3}(w_{1},z_{1}-z_{2})w_{2},z_{2})w_{3}\rangle,$
$\displaystyle\xi_{\mathcal{Y}_{5},\mathcal{Y}_{6}}(f(z_{1},z_{2})w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3})$
$\displaystyle\quad=\iota_{|z_{2}|>|z_{1}|>0}(f(z_{1},z_{2}))\rangle
w_{0},\mathcal{Y}_{5}(w_{2},z_{2})\mathcal{Y}_{6}(w_{1},z_{1})w_{3}\rangle,$

respectively, where

$\displaystyle\iota_{|z_{1}|>|z_{2}|>0}:R\rightarrow\mathbb{C}[[z_{2}/z_{1}]][z_{1}^{\pm},z_{2}^{\pm}],$
$\displaystyle\iota_{|z_{2}|>|z_{1}-z_{2}|>0}:R\rightarrow\mathbb{C}[[(z_{1}-z_{2})/z_{2}]][z_{2}^{\pm},(z_{1}-z_{2})^{\pm}]$
$\displaystyle\iota_{|z_{2}|>|z_{1}|>0}:R\rightarrow\mathbb{C}[[z_{1}/z_{2}]][z_{1}^{\pm},z_{2}^{\pm}],$

are the maps expanding elements of $R$ as series in the regions
$|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}-z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
respectively.

By (31), we know that $\mathcal{E}(u,w_{0},w_{1},w_{2},w_{3})$ are in the
intersection of the kernels of
$\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}},\psi_{\mathcal{Y}_{3},\mathcal{Y}_{4}},\xi_{\mathcal{Y}_{5},\mathcal{Y}_{6}}$;
it was also noted in [Hua05a] that all
$\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3}),\mathcal{B}(u,w_{0},w_{1},w_{2},w_{3}),\mathcal{C}(u,w_{0},w_{1},w_{2},w_{3}),\mathcal{D}(u,w_{0},w_{1},w_{2},w_{3})$
are in the intersection of the kernels. Thus we have induced maps:

$\displaystyle\overline{\phi}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}:T/J\rightarrow
z_{1}^{\Delta}\mathbb{C}(\\{z_{2}/z_{1}\\})[z_{1}^{\pm},z_{2}^{\pm}],$
$\displaystyle\overline{\psi}_{\mathcal{Y}_{3},\mathcal{Y}_{4}}:T/J\rightarrow
z_{2}^{\Delta}\mathbb{C}(\\{(z_{1}-z_{2})/z_{1}\\})[z_{2}^{\pm},(z_{1}-z_{2})^{\pm}],$
$\displaystyle\overline{\varepsilon}_{\mathcal{Y}_{5},\mathcal{Y}_{6}}:T/J\rightarrow
z_{2}^{\Delta}\mathbb{C}(\\{z_{1}/z_{2}\\})[z_{1}^{\pm},z_{2}^{\pm}].$

Since the $T/J$ is finitely generated over $R$, by using $L(-1)$-derivative
property for the intertwining operator it follows from the same argument in
[Hua05a, Theorem 1.4] that (35), (36), (37) satisfy the equations (38), (39).

Lastly, by using (4.4), (4.5) and the arguments in [Hua05a, Theorem 2.3], any
possible singular point of equations (38), (39) is regular.

∎

######  Remark 4.7.

Using the similar argument in [Hua05a][Theorem 3.5], if $V$ further satisfies
the conditions 1 and 2 in Huang’s Theorem 3.5, (35), (36) and (37) satisfy the
convergence and extension properties (see [Hua05a][Definition 3.4] for
definition).

######  Remark 4.8.

Let $W_{i}$, $i=0,1,2,3$ be finitely strongly generated $V$-modules. Then

$\displaystyle\text{dim}R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}/\\{R_{V},R_{W_{0}}\otimes_{R_{V}}\cdots\otimes_{R_{V}}R_{W_{3}}\\}<\infty.$

This follows from Theorem 3.1.

######  Remark 4.9.

The finitely strongly generated condition for $M$ is weaker than Huang’s
$C_{1}$-cofinite condition. Indeed, by [Ara12] $M$ is strongly finitely
generated over $V$ of $CFT$ type is equivalent to the Li’s $C_{1}$-cofinite
condition. Since the Li’s $C_{1}(M)$ equals
$\text{span}_{\mathbb{C}}\\{\omega_{(0)}m,a_{(-1)}m;a\in\oplus_{\Delta>0}V_{\Delta},m\in
M\\}$ containing Huang’s $C_{1}(M)$ which equals
$\text{span}_{\mathbb{C}}\\{a_{(-1)}m;a\in\oplus_{\Delta>0}V_{\Delta},m\in
M\\}$, Li’s $C_{1}(M)$-cofinite condition is weaker.

##  5\. Genus-$1$ correlation functions

Let $V$ be a vertex superalgebra, $W_{1}$ be its weak $V$-module and $W_{2}$
and $W_{3}$ be its weak $g$-twisted modules, where $g$ is of finite order $T$.
Let $u\in V^{r}$ and $w\in W_{1}$ be $\mathbb{Z}_{2}$-homogeneous elements. We
first study some basic properties of geometrically-modified twisted
intertwining operators which are introduced in Appendix A.

######  Proposition 5.1.

Let $\mathcal{Y}$ be an intertwining operator of type
$\binom{W_{3}}{W_{1}\;W_{2}}$. Then

(40)
$\displaystyle\mathcal{U}(1)\mathcal{Y}(w,x)(\mathcal{U}(1))^{-1}=\mathcal{Y}(\mathcal{U}(e^{2\pi
ix})w,e^{2\pi ix}-1)$

and

(41)
$\displaystyle[Y(\mathcal{U}(x_{1})u,x_{1}),\mathcal{Y}(\mathcal{U}(x_{2})w,x_{2}]=2\pi
i\text{Res}_{y}\delta(\frac{x_{1}}{e^{2\pi iy}x_{2}})(\frac{x_{1}}{e^{2\pi
iy}x_{2}})^{r/T}\mathcal{Y}(\mathcal{U}(x_{2})Y(u,y)w,x_{2}).$

###### Proof.

The first identity follows immediately from the same argument in Example A.1.
We prove the second identity. Recall the twisted commutator formula:

$\displaystyle[Y(u,x_{1}),\mathcal{Y}(w,x_{2})]$
$\displaystyle=\text{Res}_{x_{0}}x_{2}^{-1}\delta(\frac{x_{1}-x_{0}}{x_{2}})(\frac{x_{1}-x_{0}}{x_{2}})^{-r/T}\mathcal{Y}(Y(u,x_{0})w,x_{2}).$

Thus

(42)
$\displaystyle\begin{split}&[Y(x_{1}^{L_{(0)}}u,x_{1}),\mathcal{Y}(x_{2}^{L_{(0)}}w,x_{2})]\\\
&=\text{Res}_{x_{0}}x_{2}^{-1}\delta(\frac{x_{1}-x_{0}}{x_{2}})(\frac{x_{1}-x_{0}}{x_{2}})^{-r/T}\mathcal{Y}(Y(x_{1}^{L_{(0)}}u,x_{0})x_{2}^{L_{(0)}}w,x_{2})\\\
&=\text{Res}_{x_{0}}x_{2}^{-1}\delta(\frac{x_{1}-x_{0}}{x_{2}})(\frac{x_{1}-x_{0}}{x_{2}})^{-r/T}\mathcal{Y}(Y((\frac{x_{1}}{x_{2}})^{L_{(0)}}u,\frac{x_{0}}{x_{2}})w,x_{2}),\end{split}$

where we used the $L_{(0)}$-conjugation formula:

(43) $\displaystyle
q^{-L_{(0)}}Y(q^{L_{(0)}}u,x)q^{L_{(0)}}=Y(u,\frac{x}{q}).$

Now we change the variable $x_{0}$ to $y$ as follows:

$\displaystyle x_{0}=\displaystyle\sum_{k\in\mathbb{N}}\frac{x_{2}(2\pi
iy)^{k}}{k!}=(e^{2\pi iy}-1)x_{2}.$

Then RHS of (42) becomes

(44) $\displaystyle\begin{split}&\text{Res}_{y}\frac{\partial x_{0}}{\partial
y}x_{2}^{-1}\delta(\frac{x_{1}-(e^{2\pi
iy}-1)x_{2}}{x_{2}})(\frac{x_{1}-(e^{2\pi
iy}-1)x_{2}}{x_{2}})^{-r/T}\mathcal{Y}(x_{2}^{L_{(0)}}Y((x_{1}/x_{2})^{L_{(0)}}u,e^{2\pi
iy}-1)w,x_{2})\\\
&\stackrel{{\scriptstyle\textnormal{(a)}}}{{\mathstrut{=}}}2\pi
i\text{Res}_{y}\delta(\frac{x_{1}}{e^{2\pi iy}x_{2}})(\frac{x_{1}}{e^{2\pi
iy}x_{2}})^{-r/T}\mathcal{Y}(x_{2}^{L_{(0)}}Y((x_{1}/x_{2})^{L_{(0)}}u,e^{2\pi
iy}-1)w,x_{2})\\\
&\stackrel{{\scriptstyle\textnormal{(b)}}}{{\mathstrut{=}}}2\pi
i\text{Res}_{y}\delta(\frac{x_{1}}{e^{2\pi iy}x_{2}})(\frac{x_{1}}{e^{2\pi
iy}x_{2}})^{-r/T}\mathcal{Y}(x_{2}^{L_{(0)}}Y(e^{2\pi iyL_{(0)}}u,e^{2\pi
iy}-1)w,x_{2})\\\
&\stackrel{{\scriptstyle\textnormal{(c)}}}{{\mathstrut{=}}}2\pi
i\text{Res}_{y}\delta(\frac{x_{1}}{e^{2\pi iy}x_{2}})(\frac{x_{1}}{e^{2\pi
iy}x_{2}})^{-r/T}\mathcal{Y}(x_{2}^{L_{(0)}}\mathcal{U}(1)Y((\mathcal{U}(1))^{-1}u,y)(\mathcal{U}(1))^{-1}w,x_{2}),\end{split}$

where (44) and (44) follow from following two identities: for
$\alpha\in\mathbb{C}$

$\displaystyle
x_{0}^{-1}\delta(\frac{x_{1}-x_{2}}{x_{0}})(\frac{x_{1}-x_{2}}{x_{0}})^{\alpha}=x_{1}^{-1}\delta(\frac{x_{0}+x_{2}}{x_{1}})(\frac{x_{0}+x_{2}}{x_{1}})^{-\alpha}$
$\displaystyle
x^{-1}\delta(\frac{y}{x})(\frac{y}{x})^{\alpha}=y^{-1}\delta(\frac{y}{x})(\frac{y}{x})^{\alpha}=y^{-1}\delta(\frac{x}{y})(\frac{x}{y})^{-\alpha},$

and (44) follows from (40). (42) and (44) together give

(45)
$\displaystyle\begin{split}&[Y(x_{1}^{L_{(0)}}u,x_{1}),\mathcal{Y}(x_{2}^{L_{(0)}}w,x_{2})]\\\
&=2\pi i\text{Res}_{y}\delta(\frac{x_{1}}{e^{2\pi
iy}x_{2}})(\frac{x_{1}}{e^{2\pi
iy}x_{2}})^{-r/T}\mathcal{Y}(\mathcal{U}(x_{2})Y((\mathcal{U}(1))^{-1}u,y)(\mathcal{U}(1))^{-1}w,x_{2}).\end{split}$

(41) can be obtained from (45) by substituting $\mathcal{U}(1)u$ for $u$ and
$\mathcal{U}(1)w$ for $w$ and using the fact that

$\displaystyle\mathcal{U}(xy)=x^{L_{(0)}}\mathcal{U}(y).$

∎

######  Proposition 5.2.

For any $u\in V^{r}$,

(46) $\displaystyle 2\pi
ix\frac{d}{dx}\mathcal{Y}(\mathcal{U}(x)u,x)=\mathcal{Y}(\mathcal{U}(x)L_{(-1)}u,x).$

###### Proof.

Using the $L_{(-1)}$-derivative property for $\mathcal{Y}$, we obtain

(47) $\displaystyle\begin{split}&2\pi
ix\frac{x}{dx}\mathcal{Y}(\mathcal{U}(x)u,x)\\\ &=2\pi
ix\frac{d}{dx}\mathcal{Y}((2\pi ix)^{L_{(0)}}e^{-L_{+}(A)}u,x)\\\
&=\mathcal{Y}((2\pi iL_{(0)}+2\pi ixL_{(-1)})(2\pi
ix)^{L_{(0)}}e^{-L_{+}(A)}u,x)\\\ &=\mathcal{Y}(x^{L_{(0)}}(2\pi iL_{(0)}+2\pi
ixL_{(-1)})\mathcal{U}(1)u,x).\end{split}$

Let $\omega$ be the conformal vector in $V$. Then

(48)
$\displaystyle\begin{split}\mathcal{U}(1)L_{(-1)}&=\text{Res}_{x}\mathcal{U}(1)Y(\omega,x)\\\
&\stackrel{{\scriptstyle\textnormal{(d)}}}{{\mathstrut{=}}}\text{Res}_{x}Y(\mathcal{U}(e^{2\pi
ix})\omega,e^{2\pi x}-1)\mathcal{U}(1)\\\ &=\text{Res}_{x}Y(e^{2\pi
ixL_{(0)}})\mathcal{U}(1)\omega,e^{2\pi ix}-1)\mathcal{U}(1)\\\
&\stackrel{{\scriptstyle\textnormal{(e)}}}{{\mathstrut{=}}}\text{Res}_{x}(2\pi
i)^{2}Y(e^{2\pi ixL_{(0)}}(\omega-\frac{c}{24}\mathbf{1}),e^{2\pi
ix}-1)\mathcal{U}(1)\\\ &=\text{Res}_{x}(2\pi i)^{2}Y(e^{2\pi
ixL_{(0)}}\omega,e^{2\pi ix}-1)\mathcal{U}(1)\\\ &=\text{Res}_{x}(2\pi
i)^{2}e^{4\pi ix}Y(\omega,e^{2\pi ix}-1)\mathcal{U}(1)\\\
&\stackrel{{\scriptstyle\textnormal{(f)}}}{{\mathstrut{=}}}\text{Res}_{y}(2\pi
i+2\pi iy)Y(\omega,y)\mathcal{U}(1)\\\ &=(2\pi iL_{(-1)}+2\pi
iL_{(0)})\mathcal{U}(1).\end{split}$

where (48) follows from (40), (48) follows from the fact that

$\mathcal{U}(1)\omega=(2\pi i)^{2}(\omega-\frac{c}{24}\mathbf{1}),$

and we change the variable from $x$ to $y=e^{2\pi ix}-1$ in (48). Using (48),
RHS of (47) equals

$\displaystyle Y_{M}(\mathcal{U}(x)L_{(-1)}u,x),$

proving the proposition. ∎

Let $W_{1},W_{2},W_{5}$ be finitely strongly generated $V$-modules, and
$W_{0},W_{3},W_{4}$ be ordinary $g$-twisted modules. Let $W_{0}^{\prime}$ be
contragredient module of $W_{0}$ which is a $g^{-1}$-twisted $V$-module. Let
$\mathcal{Y}_{1},\mathcal{Y}_{2},\mathcal{Y}_{3},\mathcal{Y}_{4},\mathcal{Y}_{5}$
be intertwining operators of types
$\binom{W_{0}}{W_{1}\;W_{4}},\binom{W_{4}}{W_{2}\;W_{3}},\binom{W_{5}}{W_{1}\;W_{2}},\binom{W_{0}}{W_{5}\;W_{3}}$,
respectively. If the Conjecture 4.2 is true, for any $w_{0}^{\prime}\in
W_{0}^{\prime}$, $w_{i}\in W_{i}$, $i=1,2,3$, when
$|z_{1}|>|z_{2}|>|z_{1}-z_{2}|>0$, we have the associativity

(49) $\displaystyle\langle
w_{0}^{\prime},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(\mathcal{Y}_{3}(w_{1},z_{1}-z_{2})w_{2},z_{2})w_{3}\rangle.$

Moreover, (49) holds if the $g$ is identity and $W_{0},W_{1},W_{2},W_{3}$
satisfy the finiteness condition (32).

Assume the associative property is true. We also have

######  Proposition 5.3.

When $|q_{(z_{1}-z_{2})}|>1>|q_{(z_{1}-z_{2})}-1|>0,$

$\displaystyle\langle
w_{0}^{\prime},\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})w_{1},q_{z_{2}})\mathcal{Y}_{2}(\mathcal{U}(q_{z_{2}})w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(\mathcal{U}(q_{z_{2}})\mathcal{Y}_{3}(w_{1},z_{1}-z_{2}))w_{2},q_{z_{2}})w_{3}\rangle.$

###### Proof.

By (48), (43), and (40),

$\displaystyle\langle
w_{0}^{\prime},\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})w_{1},q_{z_{2}})\mathcal{Y}_{2}(\mathcal{U}(q_{z_{2}})w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(\mathcal{Y}_{3}(\mathcal{U}(q_{z_{1}})w_{1},q_{z_{1}}-q_{z_{2}})\mathcal{U}(q_{z_{2}})w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(\mathcal{Y}_{3}(\mathcal{U}(q_{z_{1}})w_{1},q_{z_{1}}-q_{z_{2}})q_{z_{2}}^{L_{(0)}}\mathcal{U}(1)w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(q_{z_{2}}^{L_{(0)}}\mathcal{Y}_{3}(\mathcal{U}(q_{z_{1}-z_{2}})w_{1},q_{z_{1}-z_{2}}-1)\mathcal{U}(1)w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(q_{z_{2}}^{L_{(0)}}\mathcal{U}(1)\mathcal{Y}_{3}(\mathcal{U}(w_{1},z_{1}-z_{2})w_{2},q_{z_{2}})w_{3}\rangle$
$\displaystyle=\langle
w_{0}^{\prime},\mathcal{Y}_{4}(\mathcal{U}(q_{z_{2}})\mathcal{Y}_{3}(\mathcal{U}(w_{1},z_{1}-z_{2})w_{2},q_{z_{2}})w_{3}\rangle.$

The result is obtained by noting that
$|q_{z_{1}}|>|q_{z_{2}}|>|q_{z_{1}}-q_{z_{2}}|>0$ or equivalently
$|q_{(z_{1}-z_{2})}|>1>|q_{(z_{1}-z_{2})}-1|>0$. ∎

###  5.1. Identities for formal $q$-traces

In this subsection, we derive some important identities of genus-$1$
correlation functions. These identities involve the twisted modular forms
which are introduced in Appendix $B$ and $C$.

Let $W_{i}$, $i=1,2,...,n$ be weak $V$-modules, $\tilde{W}_{i}$, $i=1,2,...,n$
be weak $g$-twisted $V$-modules. Let $h$ be a Cartan element of $V$. Assume
$\tilde{W}_{0}=\tilde{W}_{n}$ is $h$-stable. We would use this assumption for
the rest of Section $6$.

######  Proposition 5.4.

Let $g$ be of order $T$ and $h_{(0)}(u)=\lambda^{-1}u$ for
$\mathbb{Z}_{2}$-homogeneous element $u\in V^{r}$. For any
$\mathbb{Z}_{2}$-homogeneous elements, $u_{i}\in W_{i}$, $i=1,...,n,$ we
denote $e^{2\pi i\frac{r}{T}}$ by $\mu$ and ${|u|(|u_{1}|+\cdots+|u_{n}|)}$ by
$t$. Let $\mathcal{Y}_{i}$, $i,2,...,n$ be intertwining operator of type
$\binom{\tilde{W}_{i-1}}{W_{i}\;\tilde{W}_{i}}$, respectively. Let
$\theta=e^{2\pi is\lambda^{-1}}$, $\phi=\mu$, and $\phi(h)=e^{2\pi
ish_{(0)}}$. We have

(50)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}Y(\mathcal{U}(x)u,x)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad=\displaystyle\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{x_{i}}{x},q)\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\\\
&\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&+\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

Moreover when $u\in V^{0}$, we also have

(51)
$\displaystyle\begin{split}\sum_{i=1}^{n}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(0)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}=0.\end{split}$

###### Proof.

The LHS of (50) equals

(52)
$\displaystyle\begin{split}&\displaystyle\sum_{i=1}^{n}\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\cdot\\\
&\quad\quad\quad\quad\quad\cdot[Y(\mathcal{U}(x)u,x),\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{i},x_{i})]\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdot\\\
&\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+(-1)^{|u|(|u_{1}|+\cdots+|u_{n}|)}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})Y(\mathcal{U}(x)u,x)\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\stackrel{{\scriptstyle\textnormal{(a)}}}{{\mathstrut{=}}}\text{tr}_{\tilde{W}_{n}}\displaystyle\sum_{i=1}^{n}2\pi
i\text{Res}_{y}\delta(\frac{x}{e^{2\pi iy}x_{i}})(\frac{x}{e^{2\pi
iy}x_{i}})^{-r/T}\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})Y_{i}(\mathcal{U}(x_{i})Y(u,y)u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+(-1)^{|u|(|u_{1}|+\cdots+|u_{n}|)}\text{tr}_{\tilde{W}_{n}}\theta^{-1}Y(\mathcal{U}(\frac{x}{q})u,\frac{x}{q})\mathcal{Y}_{1}(\mathcal{U}(x_{1}w_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})w_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\stackrel{{\scriptstyle\textnormal{(b)}}}{{\mathstrut{=}}}\text{tr}_{\tilde{W}_{n}}\displaystyle\text{Res}_{y}\displaystyle\sum_{i=1}^{n}\sum_{m\geq
0}\frac{(2\pi i)^{m+1}y^{m}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\delta(\frac{x}{x_{i}})(\frac{x}{x_{i}})^{-r/T}))\\\
&\quad\quad\quad\quad\quad\cdot\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots
Y_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})Y_{i}(\mathcal{U}(x_{i})Y(u,y)u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+(-1)^{|u|(|u_{1}|+\cdots+|u_{n}|)}\theta^{-1}q^{-x\frac{\partial}{\partial
x}}\text{tr}_{\tilde{W}_{n}}Y(\mathcal{U}(\mathcal{U}(x),x)\mathcal{Y}_{1}(\mathcal{U}(x_{1}w_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})w_{n},x_{n})\phi(h)q^{L_{(0)}},\\\
\end{split}$

where we use (41) for (52), $L_{(0)}$-conjugation property and trace property
($\text{tr}AB=\text{tr}BA$) for (52). From (52), we obtain

(53)
$\displaystyle\begin{split}&(1-(-1)^{t}\theta^{-1}q^{-x\frac{\partial}{\partial
x}})\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})Y(\mathcal{U}(x)u,x)\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\text{tr}_{\tilde{W}_{n}}\displaystyle\text{Res}_{y}\displaystyle\sum_{i=1}^{n}\sum_{m\geq
0}\frac{(2\pi i)^{m+1}y^{m}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\delta(\frac{x}{x_{i}})(\frac{x}{x_{i}})^{-r/T}))\\\
&\quad\quad\quad\quad\quad\cdot\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots
Y_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})Y_{i}(\mathcal{U}(x_{i})Y(u,y)u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}\sum_{l\in\mathbb{Z}_{+}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{l-r/T}}{x_{i}^{l-r/T}}))\cdot\\\
&\quad\quad\quad\quad\quad\cdot\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}\sum_{l\in\mathbb{N}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{-l-r/T}}{x_{i}^{-l-r/T}}))\cdot\\\
&\quad\quad\quad\quad\quad\cdot\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}.\\\
\end{split}$

Since $(1-q^{-x\frac{\partial}{\partial x}})$ acting on any term independent
of $x$ is $0$, we obtain:

(54)
$\displaystyle\begin{split}&(1-(-1)^{t}\theta^{-1}q^{-x\frac{\partial}{\partial
x}})(\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})Y(\mathcal{U}(x)u,x)\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad\quad\quad-\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}})\\\
&\quad\quad\quad\quad=\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}(\sum_{l\in\mathbb{Z}_{+}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{l-r/T}}{x_{i}^{l-r/T}}))\\\
&\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{l\in\mathbb{N}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{-l-r/T}}{x_{i}^{-l-r/T}})))\\\
&\quad\quad\quad\quad\quad\cdot\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots
Y_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}.\\\
\end{split}$

Note when $r=0$ and $m=0$, RHS contains a term

(55) $\displaystyle\begin{split}2\pi
i\sum_{i=1}^{n}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(0)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

Since LHS of (54) has no constant term as a series in $x$, (55) must be $0$,
and we get (51).

In the rest of the proof, we assume that $\mu\neq 1$, i.e., $r\neq 0$. We next
compute the inverse of $(1-(-1)^{t}\theta^{-1}q^{-x\frac{\partial x}{x}})$ on
the ring $\mathbb{C}((q^{\frac{1}{T}}))$ (We work on
$\mathbb{C}((q^{\frac{1}{T}}))$ because the weights of $g$-twisted module are
always lower truncated).

(56) $\displaystyle(1-(-1)^{t}\theta^{-1}q^{-x\frac{\partial}{\partial
x}})^{-1}x^{l-r/T}=\begin{cases}\frac{-(-1)^{t}\theta
q^{l-r/T}x^{l-r/T}}{1-(-1)^{t}\theta q^{l-r/T}}&l>0\\\
\frac{x^{l-r/T}}{1-(-1)^{t}\theta^{-1}q^{-(l-r/T)}}&l\leq 0.\end{cases}$

Then

(57)
$\displaystyle\begin{split}&(1-(-1)^{t}\theta^{-1}q^{-x\frac{\partial}{\partial
x}})^{-1}(\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}(\sum_{l\in\mathbb{Z}_{+}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{l-r/T}}{x_{i}^{l-r/T}}))\\\
&\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{l\in\mathbb{N}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{-l-r/T}}{x_{i}^{-l-r/T}}))))\\\
&\quad\quad\quad\quad=\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}(\sum_{l\in\mathbb{Z}_{+}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial x_{i}})^{m}(\frac{-(-1)^{t}\theta
q^{l-r/T}x^{l-r/T}}{(1-(-1)^{t}\theta q^{l-r/T})x_{i}^{l-r/T}}))\\\
&\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{l\in\mathbb{N}}\frac{(2\pi
i)^{m+1}}{m!}((x_{i}\frac{\partial}{\partial
x_{i}})^{m}(\frac{x^{-l-r/T}}{(1-(-1)^{t}\theta^{-1}q^{(l+r/T)})x_{i}^{-l-r/T}}))))\\\
&\quad\quad\quad\quad=\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}(\sum_{l\in\mathbb{Z}_{+}}-\frac{(-1)^{m+t}(2\pi
i)^{m+1}(l+r/T)^{m}}{m!}(\frac{\theta
q^{l-r/T}(\frac{x_{i}}{x})^{-l+r/T}}{(1-(-1)^{t}\theta q^{l-r/T})})\\\
&\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{l\in\mathbb{N}}\frac{(2\pi
i)^{m+1}(l+r/T)^{m}}{m!}(\frac{(\frac{x_{i}}{x})^{l+\frac{r}{T}}}{1-(-1)^{t}\theta^{-1}q^{(l+r/T)}}))\\\
&\quad\quad\quad\quad=P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{x_{i}}{x},q).\end{split}$

We obtain (50) from (57) and (54).

∎

Assume the Conjecture 4.2 is true or the finiteness condition (32) is
satisfied if $g$ is trivial. Let $W_{i}$, $\tilde{W}_{i}$, $i=1,2,...,n$
satisfy the condition in Conjecture 4.2 or the finiteness condition, for any
$v_{i}\in W_{i}$ and $\tilde{w}_{n}\in\tilde{W}_{n}$,
$\tilde{w}_{0}^{\prime}\in\tilde{W}_{0}^{\prime}$,

$\displaystyle\langle
w_{0}^{\prime},\mathcal{Y}_{1}(v_{1},z_{1})\cdots\mathcal{Y}_{n}(v_{n},z_{n})\tilde{w}_{n}\rangle$

is absolutely convergent when $|z_{1}|>\cdots>|z_{n}|>0$.

######  Theorem 5.5.

Let $W_{i}$, $\tilde{W}_{i}$, $i=1,2,...,n$ be as above, for any $0\leq j\leq
n$, we have

(58)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(x_{j-1}),u_{j-1},x_{j-1})\cdot\mathcal{Y}_{j}(\mathcal{U}(x_{j})Y(u,y)u_{j},x_{j})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(x_{j+1})u_{j+1},x_{j+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{1}{e^{2\pi
iy}},q)\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j-1}(\mathcal{U}(x_{j-1})u_{j-1},x_{j-1})\mathcal{Y}_{j}(\mathcal{U}(x_{j})u_{(m)}u_{j},x_{j})\mathcal{Y}_{j+1}(\mathcal{U}(x_{j+1})u_{j+1},x_{j+1})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\sum_{i\neq
j}\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{x_{i}}{x_{j}e^{2\pi
iy}},q)\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

###### Proof.

We prove by induction on $j$. By (40), $L_{(0)}$-conjugation property, we have

(59)
$\displaystyle\begin{split}\mathcal{U}(x_{1})Y(u,y)&=x_{1}^{L_{(0)}}\mathcal{U}(1)Y(u,y)(\mathcal{U}(1))^{-1}\mathcal{U}(1)\\\
&=x_{1}^{L_{(0)}}Y(\mathcal{U}(e^{2\pi iy})u,e^{2\pi iy}-1)\mathcal{U}(1)\\\
&=Y(x_{1}^{L_{(0)}}\mathcal{U}(e^{2\pi iy}u,x_{1}(e^{2\pi
iy}-1))x_{1}^{L_{(0)}}\mathcal{U}(1)\\\ &=Y(\mathcal{U}(x_{1}e^{2\pi
iy})u,x_{1}(e^{2\pi iy}-1))\mathcal{U}(x_{1}).\end{split}$

Then when $j=1$, the LHS of (58) equals:

(60)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})Y(u,y)u_{1},x_{1})\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad=\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(Y(\mathcal{U}(x_{1}e^{2\pi
iy})u,x_{1}(e^{2\pi iy}-1))\mathcal{U}(x_{1})u_{1},x_{1}))\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}.\end{split}$

Let $y=z$ and $x_{1}=z_{1}$ be complex numbers satisfying
$|z_{1}q_{z}|>|z_{1}|>|z_{1}(q_{z}-1)|>0$, where we choose the values of
$z_{1}^{n}$ for $n\in\mathbb{C}$ be $e^{n\text{log}z_{1}}$,
$0\leq\text{arg}z_{1}<2\pi$. By Proposition (49), we have

(61)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(Y(\mathcal{U}(z_{1}q_{z})u,z_{1}(q_{z}-1))\mathcal{U}(z_{1})u_{1},z_{1}))\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\text{tr}_{\tilde{W}_{n}}Y(\mathcal{U}(z_{1}q_{z})u,z_{1}q_{z})\mathcal{Y}_{1}(\mathcal{U}(z_{1})u_{1},z_{1})\cdots\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

as formal series in $x_{2},...,x_{n},q$ and $q_{s}$. Since $z_{1}$ can be any
nonzero complex number, we replace $z_{1}$ with a formal variable $x_{1}$, and
using (50), we obtain:

(62)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}Y(\mathcal{U}(x_{1}q_{z})u,x)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\text{tr}_{\tilde{W}_{n}}Y((\mathcal{U}(x))(\mathcal{U}(x))^{-1}\mathcal{U}(x_{1}q_{z})u,x)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{i=1}^{n}\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{x_{i}}{x},q)\\\
&\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})((\mathcal{U}(x))^{-1}\mathcal{U}(x_{1}q_{z})u)_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&+\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)(\mathcal{U}(x))^{-1}\mathcal{U}(x_{1}q_{z})u)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

as a formal power series in $x,x_{1},...,x_{n},q$ and $q_{s}$.

For any complex number $z$ satisfying $|q_{z}|>1>|q_{z}-1|>0$, substituting
$x_{1}q_{z}$ for $x$ in (62), we obtain

(63)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}Y(\mathcal{U}(x_{1}q_{z})u,x_{1}q_{z})\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{1}{q_{z}},q)\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{(m)}u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{2}(\mathcal{U}(x_{2})u_{2},x_{2})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n}u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\sum_{i=2}^{n}\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(\frac{x_{i}}{x_{1}q_{z}},q)\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdot\\\
&\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{i-1}(\mathcal{U}(x_{i-1})u_{i-1},x_{i-1})\mathcal{Y}_{i}(\mathcal{U}(x_{i})u_{(m)}u_{i},x_{i})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(x_{i+1})u_{i+1},x_{i+1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(x_{1})u_{1},x_{1})\cdots\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}\end{split}$

as formal power series in $x_{1},...,x_{n},q,q_{s}$ whose coefficients are
convergent in the region $|q_{z}|>1>|q_{z}-1|>0$. Combining (61) with (63),
(58) holds when $j=1$. We now prove the theorem in the general case. Assume
that for $j=k<n$, (58) holds. We need to show that (58) holds for $k+1$. Fix
$z_{i}^{0}\in\mathbb{C}$, $i=1,...,n,$ satisfying
$|z_{1}^{0}|>\cdots>|z_{n}^{0}|>0$, let $\gamma_{k}$ be the path in

$\displaystyle M^{n}=\\{(z_{1},...,z_{n})\in\mathbb{C}^{n}|z_{i}\neq
0,z_{i}\neq z_{l},i\neq l\\}$

from $(z_{1}^{0},...,z_{k}^{0},z_{k+1}^{0},...,z_{n}^{0})$ to
$(z_{1}^{0},...,z_{k+1}^{0},z_{k}^{0},...,z_{n}^{0})$ given by

$\displaystyle\gamma_{k}(t)=(z_{1}^{0},...,e^{(1-t)\text{log}z_{k}^{0}+t\text{log}z_{k+1}^{0}},e^{t\text{log}z_{k}^{0}+(1-t)\text{log}z_{k+1}^{0}},...,z_{n}^{0}).$

Given any branch of a multivalued analytic function of $z_{1},...,z_{n}$ in
the simply connected region $|z_{1}|>\cdots>|z_{n}|$,
$0\leq\text{arg}z_{i}<2\pi$, $i=1,...,n,$ we have a unique analytic extension
of this branch to the region

(64)
$\displaystyle|z_{1}|>\cdots>|z_{k-1}|>|z_{k+1}|>|z_{k}|>|z_{k+2}|>\cdots>|z_{n}|>0,$
(65) $\displaystyle 0\leq\text{arg}z_{i}<2\pi,\quad i=1,...,n,$

along the path $\gamma_{k}$. By duality property of intertwining operators,
the branch of the analytic extension along $\gamma_{k}$ of

$\langle\tilde{u}_{0}^{\prime},\mathcal{Y}_{1}(u_{1},z_{1})\cdots\mathcal{Y}_{k}(u_{k},z_{k})\mathcal{Y}_{k+1}(u_{k+1},z_{k+1})\cdots\mathcal{Y}_{n}(u_{n},z_{n})\tilde{u}_{n}\rangle$

to the region (64) equals

$\langle\tilde{u}_{0}^{\prime},\mathcal{Y}_{1}(u_{1},z_{1})\cdots\mathcal{Y}_{k+1}(u_{k+1},z_{k+1})\mathcal{Y}_{k}(u_{k},z_{k})\cdots\mathcal{Y}_{n}(u_{n},z_{n})\tilde{u}_{n}\rangle.$

Now we consider

(66)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(u_{1},x_{1})\cdots\mathcal{Y}_{k}(\mathcal{U}(x_{k})u_{k},x_{k})\mathcal{Y}_{k+1}(\mathcal{U}(x_{k+1})Y(u,y)u_{k+1},x_{k+1})\cdots\\\
&\quad\quad\quad\quad\quad\quad\mathcal{Y}_{n}(\mathcal{U}(x_{n})u_{n},x_{n})\phi(h)q^{L_{(0)}}.\end{split}$

We substitute $z_{1},...,z_{k},z_{k+1},...,z_{n}$ for
$x_{1},...,x_{k+1},x_{k},...,x_{n}$ in the region (64). (66) becomes a series
in formal variable $q,q_{s}$ and $y$ with single valued analytic coefficient
in the region (64). These coefficients can be analytically extended uniquely
back to the region $|z_{1}|>\cdots>|z_{n}|$, $0\leq\text{arg}z_{i}<2\pi$,
$i=1,...,n,$ along $\gamma^{-1}$ and by using induction assumption we prove
the case when $j=k+1$. The result follows.

∎

Let $\mathbb{G}_{|z_{1}|>\cdots>|z_{n}|>0}$ be the space of all multivalued
analytic functions in $z_{1},...,z_{n}$ defined on the region
$|z_{1}|>\cdots>|z_{n}|>0$ with preferred branches in the simply-connected
region $|z_{1}|>\cdots>|z_{n}|>0$, $0\leq\text{arg}z_{i}<2\pi,$ $i=1,...,n$.
Let $W_{i}$ $\tilde{W}_{i}$, $h$ be as in Proposition 5.5, for any $v_{i}\in
W_{i}$, we see that

$\displaystyle\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})v_{1},q_{z_{1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})v_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}$

is absolutely convergent when $|q_{z_{1}}|>\cdots>|q_{z_{n}}|>0$.

######  Theorem 5.6.

For any integer $j$ satisfying $1\leq j\leq n$ and any $l\in\mathbb{Z}_{+}$,
in $\mathbb{G}_{|q_{z_{1}}|>\cdots>|q_{z_{n}}|>0}$, we have when $\phi=1$ and
$(-1)^{t}\theta\neq 1$,

(67)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(-l)}u_{j},q_{z_{j}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{k\geq
l}((-1)^{l-1}\binom{k-1}{k-l}\tilde{G}_{k}\genfrac{[}{]}{0.0pt}{0}{(-1)^{t}\theta}{\phi}(q)-\delta_{k,l}2\pi
i\frac{1}{1-\theta^{-1}})\\\
&\quad\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(k-l)}u_{j},q_{z_{j}})\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\displaystyle\sum_{i\neq
j}\sum_{m\in\mathbb{N}}((-1)^{m+1}\binom{m+l-1}{l-1}\tilde{P}_{m+l}\genfrac{[}{]}{0.0pt}{0}{(-1)^{t}\theta}{\phi}(z_{i}-z_{j},q)-\delta_{0,m}2\pi
i\frac{1}{1-\theta^{-1}})\\\
&\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdot\cdots\mathcal{Y}_{i-1}(\mathcal{U}(q_{z_{i-1}})u_{i-1},q_{z_{i-1}})\mathcal{Y}_{i}(\mathcal{U}(q_{z_{i}})u_{(m)}u_{i},q_{z_{i}})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(q_{z_{i+1}})u_{i+1},q_{z_{i+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{l,1}\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots
Y_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}},\end{split}$

otherwise, the LHS of (67) remains the same and one omits the terms
$\delta_{k,l}2\pi i\frac{1}{1-\theta^{-1}}$ and $\delta_{0,m}2\pi
i\frac{1}{1-\theta^{-1}}$ on RHS.

###### Proof.

As a $q$-series, the coefficients of
$P_{m}^{\theta,\phi}(\frac{q_{z_{i}}}{q_{z_{j}}q_{z}},q)$ for $m>0$ are
convergent in the region $|\frac{q_{z_{i}}}{q_{z_{j}}q_{z}}|<1$. Let
$z,z_{1},...,z_{n}\in\mathbb{C}$ satisfying $|q_{z_{1}}|>\cdots>|q_{z_{n}}|>0$
and $|\frac{q_{z_{i}}}{q_{z_{j}}q_{z}}|<1$ for $1\leq i,j\leq n$. We
substitute $z,q_{z_{1}},...,q_{z_{n}}$ for $y,x_{1},...,x_{n}$ in (58). Then
both sides of (58) become $q$-series with analytic functions in
$z,z_{1},....,z_{n}$, additionally by using (129), we get

(68)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}}),u_{j-1},q_{z_{j-1}})\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})Y(u,y)u_{j},q_{z_{j}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(-z,q)\\\
&\quad\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(m)}u_{j},q_{z_{j}})\mathcal{Y}_{j+1}(q_{z_{j+1}})u_{(n-l)}u_{j+1},q_{z_{j+1}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\displaystyle\sum_{i\neq
j}\sum_{m\in\mathbb{N}}P_{m+1}^{(-1)^{t}\theta,\phi}(z_{i}-z_{j}-z,q)\\\
&\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdot\cdots\mathcal{Y}_{i-1}(\mathcal{U}(q_{z_{i-1}})u_{i-1},q_{z_{i-1}})\mathcal{Y}_{i}(\mathcal{U}(q_{z_{i}})u_{(m)}u_{i},q_{z_{i}})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(q_{z_{i+1}})u_{i+1},q_{z_{i+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{(-1)^{t}\theta^{-1},1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\end{split}$

By (122), the Taylor expansion of
$P_{m+1}\genfrac{[}{]}{0.0pt}{0}{\theta}{\phi}(z_{i}-z_{j}-z;q)$ as a function
in $z$ at $0$ is

(69)
$\displaystyle\displaystyle\sum_{n=0}^{\infty}\binom{m+n}{m}P_{m+n+1}\genfrac{[}{]}{0.0pt}{0}{\theta}{\phi}(z_{i}-z_{j},q)z^{n}.$

By (127), (69), (132) and taking the coefficients of $z^{l-1}$ on both sides
of (69) we obtain (67). ∎

######  Corollary 5.7.

If $u$ is an even element fixed by $g$, $\tilde{W}_{n}$ is an ordinary module,
we have

(70)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(-l)}u_{j},q_{z_{j}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\displaystyle\sum_{\begin{subarray}{c}2k\geq l\\\ k\geq
2\end{subarray}}(-1)^{l-1}\binom{2k-1}{2k-l}\tilde{G}_{2k}(q)\\\
&\quad\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(2k-l)}u_{j},q_{z_{j}})\mathcal{Y}_{j+1}(\mathcal{U}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\displaystyle\sum_{i\neq
j}\sum_{m\in\mathbb{N}}(-1)^{m+1}\binom{m+l-1}{l-1}\tilde{\wp}_{m+l}(z_{i}-z_{j},q)\\\
&\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdot\cdots
Y_{i-1}(\mathcal{U}(q_{z_{i-1}})u_{i-1},q_{z_{i-1}})\mathcal{Y}_{i}(\mathcal{U}(q_{z_{i}})u_{(m)}u_{i},q_{z_{i}})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(q_{z_{i+1}})u_{i+1},q_{z_{i+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{l,1}\tilde{G}_{2}(q)\sum_{i=1}^{n}\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdot\\\
&\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{i-1}(\mathcal{U}(q_{z_{i-1}})u_{i-1},q_{z_{i-1}})\mathcal{Y}_{i}(\mathcal{U}(q_{z_{i}})(u_{(0)}+u_{(1)}z_{i})u_{i},q_{z_{i}})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(q_{z_{i+1}})u_{i+1},q_{z_{i+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{l,1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}},\end{split}$

###### Proof.

In this case, we have $t=0$ and $\theta=\phi=1$. Now by (126) and (67), one
has

(71)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\cdot\mathcal{Y}_{j}(\mathcal{U}(x_{j})u_{(-l)}u_{j},x_{j})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad=\pi
i\delta_{l,1}\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(0)}u_{j},q_{z_{j}})\mathcal{Y}_{j+1}(\mathcal{U}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}+\displaystyle\sum_{2k\geq
l}(-1)^{l-1}\binom{2k-1}{2k-l}\tilde{G}_{2k}(q)\\\
&\quad\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(2k-l)}u_{j},q_{z_{j}})\mathcal{Y}_{j+1}(\mathcal{U}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad+\displaystyle\sum_{i\neq
j}\sum_{m\in\mathbb{N}}(-1)^{m+1}\binom{m+l-1}{l-1}\tilde{P}_{m+l}\genfrac{[}{]}{0.0pt}{0}{1}{1}(z_{i}-z_{j},q)\\\
&\quad\quad\quad\quad\quad\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdot\cdots\mathcal{Y}_{i-1}(\mathcal{U}(q_{z_{i-1}})u_{i-1},q_{z_{i-1}})\mathcal{Y}_{i}(\mathcal{U}(q_{z_{i}})u_{(m)}u_{i},q_{z_{i}})\cdot\\\
&\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{i+1}(\mathcal{U}(q_{z_{i+1}})u_{i+1},q_{z_{i+1}})\cdots\mathcal{Y}_{n}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad\quad\quad+\delta_{l,1}\text{tr}_{\tilde{W}_{n}}o(\mathcal{U}(1)u)\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{n}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}.\end{split}$

The result follows from (C.3) and (67). ∎

######  Remark 5.8.

Identity (70) recovers Theorem 2.4 in [Hua05b].

######  Corollary 5.9.

For any integer $j$ satisfying $1\leq j,s\leq n$ and any $l\in\mathbb{Z}_{+}$,
in $\mathbb{G}_{|q_{z_{1}}|>\cdots>|q_{z_{n}}|>0}$, we have when $\phi=1$ and
$(-1)^{t}\theta\neq 1$,

(72)
$\displaystyle\begin{split}&\text{tr}_{\tilde{W}_{n}}\mathcal{Y}_{1}(\mathcal{U}(q_{z_{1}})u_{1},q_{z_{1}})\cdots\mathcal{Y}_{j-1}(\mathcal{U}(q_{z_{j-1}})u_{j-1},q_{z_{j-1}})\cdot\mathcal{Y}_{j}(\mathcal{U}(q_{z_{j}})u_{(-1)}u_{j},q_{z_{j}})\cdot\\\
&\quad\quad\quad\quad\quad\quad\cdot\mathcal{Y}_{j+1}(\mathcal{U}(q_{z_{j+1}})u_{j+1},q_{z_{j+1}})\cdots\mathcal{Y}_{n}(\mathcal{U}(q_{z_{n}})u_{n},q_{z_{n}})\phi(h)q^{L_{(0)}}\\\
&\quad\quad\quad-\displaystyle\sum_{k\in\mathbb{Z}_{+}}(\tilde{G}_{k}\genfrac{[}{]}{0.0pt}{0}{(-1)^{t}\theta}{\phi}(q)-\delta_{k,1}2\pi
i\frac{1}{1-\theta^{-1}})\\\