# Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras

Bohan Li, Hao Li, Wenbin Yan  Yau Mathematics Sciences Center, Tsinghua
University, Beijing, 100084,China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We calculate the fusion rules among $\mathbb{Z}_{2}$-twisted modules
$L_{\mathfrak{sl}_{2}}(\ell,0)$ at admissible levels. We derive a series MLDEs
for normalized characters of ordinary twisted modules of quasi-lisse vertex
algebras. Examples include affine VOAs of type $A_{1}^{(1)}$ at boundary
admissible level, admissible level $k=-1/2$, $A^{(1)}_{2}$ at boundary
admissible level $k=-3/2$, and $\mathrm{BP}^{k}$-algebra with special value
$k=-9/4$. We also derive characters of some non-vacuum modules for affine VOA
of type $D_{4}$ at non-admissible level $-2$ from spectral flow automorphism.

###### Contents

  1. 1 Introduction
  2. 2 Character formulae at admissible level and their modularity
    1. 2.1 Twisted modular transformation
    2. 2.2 Twisted characters
    3. 2.3 Twisted modules
    4. 2.4 Spectral Flow Automorphsim
    5. 2.5 Li’s delta operator, spectral flow and twisted modules
    6. 2.6 Half-integer spectral flow for $A_{1}^{(1)}$ at boundary admissible level
    7. 2.7 Modular transformation
  3. 3 Twisted Zhu’s Bimodule of highest weight modules
    1. 3.1 Examples
    2. 3.2 Relation between Dong-Li-Mason’s Zhu’s bimodules and twisted Zhu’s bimodules
    3. 3.3 Fusion rules among twisted modules
    4. 3.4 Verlinde formula for $L_{-\frac{4}{3}}(\mathfrak{sl}_{2})$
  4. 4 Twisted Zhu’s bimodules of contragredient modules of highest weight modules
    1. 4.1 Motivation
    2. 4.2 Untwisted Zhu’s bimodules
    3. 4.3 The contragredient modules of the highest weight modules
    4. 4.4 Fusion rules
    5. 4.5 Fusion rules among twisted modules
  5. 5 Twisted modules from spectral flow and their MLDEs
    1. 5.1 $A^{(1)}_{1}$ at boundary admissible levels $k=-2+\frac{2}{u}$
    2. 5.2 $A^{(1)}_{1}$ at admissible level $k=-\frac{1}{2}$
    3. 5.3 $A^{(1)}_{2}$ at boundary admissible level $k=-\frac{3}{2}$
    4. 5.4 Bershadsky-Polyakov Algebra $\mathrm{BP}^{k}$ with $k=-\frac{9}{4}$
  6. 6 $\mathfrak{d}_{4}$ with non-admissible level $k=-2$
  7. A Theta functions
  8. B Proof in Proposition 4.4

##  1\. Introduction

Four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) in
physics have rich mathematical structures. In [BLL+15], the authors propose a
correspondence between the Schur sectors of $4d$ $\mathcal{N}=2$ SCFTs and
$2d$ vertex operator algebras (VOAs). This correspondence has fueled a lot of
work in the past years, including some conjectures about the chiral algebra in
the context of theories of Class $\mathcal{S}$ [Gai12, GMN13, BPRvR15, LP15].
For the genus zero case, the conjecture has been proved in terms of a
functorial construction [Ara18]. Class $\mathcal{S}$ theories have the Coulomb
branch operators with integral scaling dimension, yet there is another class
of $\mathcal{N}=2$ SCFTs called Argyres-Douglas (AD) theories [AD95, Xie13]
which usually have fractional scaling dimensions for the Coulomb branch
operators. These AD theories can be constructed by compactifying $6d$ $(2,0)$
theory on a Riemann surface with irregular singularities. The corresponding
VOAs of a class of AD theories are identified with certain affine Kac-Moody
algebras $L_{k}(\mathfrak{g})$ at admissible level $k$, or affine
$\mathcal{W}_{k}(\mathfrak{g},f)$-algebras [CS16, BN16b, BN16a, SXY17, XYY21,
XY21c]. Dualities of $4d$ theories imply nontrivial isomorphism and collapsing
levels of VOAs [XY21c, XY21a, LXY22], some of which were proved rigorously
[AVEM21, AMP21].

One consequence of this SCFT/VOA correspondence is that the Schur index of the
4d SCFT is equal to the normalized vacuum character of the corresponding VOA,
hence the character formula provides a valuable tool to study the spectrum of
4d SCFTs. The character here means that the trace
$\chi_{\lambda}(\tau,z)=\operatorname{tr}_{L(\ell,\lambda)}e^{2\pi
i\tau(L(0)-\frac{1}{2}zh(0)-\frac{1}{24}c_{\ell})}$ over the
$L(\ell,\lambda)$. In [KW88], the authors derived character formulas for
admissible representations of an affine Kac-Moody Lie algebra
$\hat{\mathfrak{g}}$ at a rational level $\ell$, i.e., $L(\ell,\lambda)$, and
also investigated the modular property of these characters. In particular, the
transformed character $\chi_{\lambda}(-\frac{1}{\tau},\frac{z}{\tau})$ can be
written as a linear combination of characters of admissible representations
with a shifted conformal vector, while $\chi_{\lambda}(-\frac{1}{\tau},z)$ is
a linear combination of the characters of some $\mathbb{Z}_{2}$-twisted
modules. The character formulas were used to derive the Schur index of a large
class of AD theory [XY21b], while the modular properties also have
applications in physics [Raz12].

Another conjecture of the SCFT/VOA correspondence is the identification
between the Higgs branch of vacua of an $\mathcal{N}=2$ SCFT and the
associated variety of the corresponding VOA [BR18, SXY17], which were also
used to propose lisse VOAs from $4d$ SCFTs [XY21a]. The VOA corresponding to a
4d SCFT is often of the quasi-lisse type whose associated variety has finitely
many symplectic leaves. The normalized character of an ordinary representation
of a quasi-lisse VOA was shown to satisfy a modular linear differential
equation (MLDE), and solving MLDE gives explicit expression for the characters
of the affine Lie algebra of the Deligne-Cvitanovic (DC) series [AK18]. The
MLDE for VOAs corresponding to several families of AD theories and
$\mathcal{N}=4$ super Yang-Mills with $\mathfrak{su}(n)$ gauge group were also
discussed in [BR18]. Recently in [ZPW22] the authors constructed flavored
MLDEs for the Schur index of $\mathfrak{a}_{1}$ Class $\mathcal{S}$ theories
based on a compact formula for the index they found earlier [PP22]. Both works
used the Higgs branch structure to probe the singular vector of the
corresponding VOA and then derive the MLDE.

The SCFT/VOA correspondence also goes beyond the Schur index and the vacuum
module. One generalization is to consider the lens space index [RY13] instead
of the normal index. In [FS18], the lens space index was identified with the
characters of the twisted modules. Given an automorphism $g$ of a VOA $V$ of
finite order, the basic properties of $g$-twisted modules for $V$ are
systematically studied by Haisheng Li in [Li96] using the twisted local
systems. In particular, he showed that $\mathbb{Z}_{2}$-twisted modules of the
affine VOA mentioned above can be obtained from its untwisted modules via Li’s
$\Delta$-operators. Later on, in the important work [DLM00], authors showed
that the trace functions of the $g$-twisted modules for the $C_{2}$-cofinite
rational VOA satisfy certain twisted modular linear differential equations
(MLDE) and possess some modular invariance properties. In [Li23], the author
generalized some of the results obtained by Dong, Li, and Mason to the quasi-
lisse vertex operator (super)algebras case, as while as proved that characters
of twisted modules of quasi-lisse vertex algebras satisfy certain twisted
MLDEs. Another generalization is to consider the index in the presence of
defects, which were identified with the twisted modules using spectral flow
[CS16, CGS16, CGS17]. The spectral flow used in these works has a long history
in the conformal field theory (see e.g. [SS87, LMRS02, Rid09, Rid10, CR12,
CR13a, KRW22]).

As reviewed above, there is an increasing interests in understanding twisted
modules and spectral flowed modules of VOAs from the perspective of SCFT/VOA
correspondence, as it provides the knowledge of lens space indices and defects
of the corresponding SCFTs. Especially if one can show that characters of
these modules are solutions of certain MLDEs, closed form expression might
also within the grasp as in the ordinary module case. Since conventional
methods in physics usually give only power series, such closed form
expressions are valuable and may review more interesting properties.
Mathematically, people introduce the twisted modules of $V$ to study the
module category of $G$ invariants $V^{G}$, where $G$ is finite automorphism
group of $V$. In the present work we study the $\mathbb{Z}_{2}$-twisted
modules denoted by $\sigma^{-\frac{1}{2}}(L(\ell,\lambda))$ for affine VOA
$L_{\mathfrak{sl}_{2}}(\ell,0)$ associated with Lie algebra
$\mathfrak{sl}_{2}$ at admissible level and the fusion rules among them by
using the twisted Zhu’s bimodule recently introduced by [Zhu22] and a
conjectural twisted version of Frenkel-Zhu’s bimodule theorem. One of our main
results is the fusion rules among admissible modules and twisted modules.
Firstly, we showed:

######  Theorem 1.1.

Let $\ell=-2+\frac{p}{q}$ be the admissible level where $p$ and $q$ coprime
positive integers with $p\geq 2$. Then $\mathbb{Z}_{2}$-twisted Zhu’s algebra
for $L_{\ell}(\mathfrak{sl}_{2})$ is
$\mathbb{C}[x]/\langle\prod_{r=0}^{p-2}\prod_{s=0}^{q-1}(x+\frac{1}{2}\ell-r+st)\rangle$,
where $0\leq r\leq p-2$, $0\leq s\leq q-1$. In particular, the Dynkin labels
(eigenvalues of $h_{(0)}$ on the highest weight vector) of all
$\mathbb{Z}_{2}$-twisted modules in category $\mathcal{O}$ are
$\\{r-st-\frac{1}{2}\ell|0\leq r\leq p-2,0\leq s\leq q-1\\}$.

######  Theorem 1.2.

All irreducible $\mathbb{Z}_{2}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$
at admissible level in category $\mathcal{O}$ can be obtained by using
$\ell=-\frac{1}{2}$ spectral flow on the untwisted modules in category
$\mathcal{O}$. In, particular, all of those irreducible twisted modules are
ordinary modules at boundary admissible level.

Then we obtain the fusion rules among these $\mathbb{Z}_{2}$-twisted modules.

######  Theorem 1.3.

For admissible weights $j_{i}=n_{i}-(k_{i}-1)t$ $(i=1,2)$ of the vertex affine
algebra $L_{k}(\mathfrak{sl}_{2})$, there are following fusion rules between
one admissible module and one twisted module:

(1) $\displaystyle
L(k,j_{1})\times\sigma^{-\frac{1}{2}}(L(k,j_{2}))=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}\sigma^{-\frac{1}{2}}(L(k,j_{1}+j_{2}-2i))$

if $0\leq k_{2}-1\leq q-k_{1}$, and
$L(k,j_{1})\times\sigma^{-\frac{1}{2}}(L(k,j_{2}))=0$ otherwise.

######  Remark 1.4.

According to the results in [Xu95], one can generalize the symmetries of
fusion rules in [FHL93] to the twisted case:

$\displaystyle N_{jk}^{i}=N_{kj}^{i}$ $\displaystyle N_{jk}^{i}=N_{ji}^{k}.$

Therefore, using above results one can obtain the fusion rules for
$\sigma(L)\times\sigma(L)$.

We also prove the following theorem.

######  Theorem 1.5.

The category of the ordinary $\mathbb{Z}/2\mathbb{Z}$-twisted modules vertex
operator algebra $L_{\ell}(\mathfrak{sl}_{2})$ at the boundary admissible
level $\ell=-2+\frac{2}{q}$ $(\mathrm{gcd}(q,2)=1)$ is semi-simple.

Following the idea in [DLM97] we compute the fusion rules among admissible
modules of $L_{k}(\mathfrak{sl}_{2})$ and their contragredient modules.

######  Theorem 1.6.

For admissible weight $j_{i}=n_{i}-1-(k_{i}-1)t$ $(i=1,2)$ of the vertex
affine algebra $L_{k}(\mathfrak{sl}_{2})$, there are following fusion rules:

(2) $\displaystyle L(k,j_{1})\times
L(k,j_{2})=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}L(k,j_{1}+j_{2}-2i),$
(5) $\displaystyle(L(k,j_{1}))^{*}\times
L(k,j_{2})=L(k,j_{2})\times(L(k,j_{1}))^{*}=\left\\{\begin{array}[]{ll}L(k,-j_{1}+j_{2}),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\ (L(k,j_{1}-j_{2}))^{*},&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$ (6)
$\displaystyle(L(k,j_{1}))^{*}\times(L(k,j_{2}))^{*}=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}(L(k,j_{1}+j_{2}-2i))^{*}.$

where $(L(k,j))^{*}$ denote the contragredient module of the irreducible
highest weight module $L(k,j)$.

We further use this result to obtain the fusion rules among
$\mathbb{Z}_{2}$-twisted modules and their contragredient modules:

######  Theorem 1.7.

(7)
$\displaystyle(L(\ell,j_{1}))^{*}\times\sigma^{\frac{1}{2}}((L(\ell,j_{2}))^{*})=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}\sigma^{\frac{1}{2}}((L(\ell,j_{1}+j_{2}-2i))^{*}).$
(10)
$\displaystyle(L(\ell,j_{1}))^{*}\times\sigma^{\frac{1}{2}}(L(\ell,j_{2}))=\left\\{\begin{array}[]{ll}\sigma^{\frac{1}{2}}(L(\ell,-j_{1}+j_{2})),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\
\sigma^{\frac{1}{2}}((L(\ell,j_{1}-j_{2}))^{*}),&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$ (13) $\displaystyle
L(\ell,j_{2})\times\sigma^{-\frac{1}{2}}((L(\ell,j_{1}))^{*})=\left\\{\begin{array}[]{ll}\sigma^{-\frac{1}{2}}(L(\ell,-j_{1}+j_{2})),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\
\sigma^{-\frac{1}{2}}((L(\ell,j_{1}-j_{2}))^{*}),&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$

We show that the ordinary twisted modules satisfy the twisted MLDEs. For
$A^{(1)}_{1}$ with boundary admissible level, if we only consider $q$-series,
the normalized characters of those modules form the complete solutions of a
$(u+1)/2$-order $\Gamma^{0}(2)$ MLDE. For $A^{(1)}_{2}$ at boundary admissible
level, we construct some ordinary $e^{2\pi iv_{(0)}}$-twisted modules by the
spectral flow of untwisted modules along different directions in the weight
lattice (see Section 2.5), whose normalized characters satisfy a second-order
$\Gamma^{0}(2)$ MLDE. For $\mathrm{BP}^{k}$-algebra with $k=-9/4$, we show
that characters of twisted modules are solutions of a third-order MLDE under
full $SL(2,\mathbb{Z})$ group. Finally we study characters of
$\mathbb{Z}_{2}$-twisted modules and spectral flowed modules of the affine
vertex algebra $\mathcal{L}_{\mathfrak{d}_{4}}(-2,0)$. Since it is non-
admissible, one can not use Kac-Wakimoto formula to write down the characters
of simple modules directly. Fortunately, we can get some simple modules of
$\mathcal{L}_{\mathfrak{d}_{4}}(-2,0)$ from spectral flowed modules. We also
get one ordinary $\mathbb{Z}_{2}$-twisted module, whose character satisfies a
second-order $\Gamma^{0}(2)$-MLDE. In general, for the simple affine vertex
algebra $L_{k}(\mathfrak{g})$ with $\mathfrak{g}$ being a Lie algebra of DC
series and $k=-h^{\vee}/6-1$, we make the following conjecture on the MLDE of
characters of its $\mathbb{Z}_{2}$-twisted modules.

######  Conjecture 1.8.

Let $V$ be a simple affine vertex algebras associated with the $\mathrm{DC}$
series,

$A_{1}\subset A_{2}\subset G_{2}\subset D_{4}\subset F_{4}\subset E_{6}\subset
E_{7}\subset E_{8}$

at level $-h^{\vee}/6-1$. The normalized $\mathbb{Z}_{2}$-twisted characters
$\chi^{\mathrm{twi}}(q)$ are solutions of a second-order
$\Gamma^{0}(2)$-$\mathrm{MLDE}$ with suitable coefficients $a_{1}$ and $a_{2}$
but without a $D^{(1)}_{q}$ term,

(14)
$\displaystyle\left(D^{(2)}_{q}+a_{1}\Theta_{0,2}+a_{2}\Theta_{1,1}\right)\chi^{\mathrm{twi}}(q)=0$

We show that this conjecture is true when $\mathfrak{g}$ is a classical Lie
algebra.

######  Remark 1.9.

For affine VOA of type $A_{1}$ and $A_{2}$ at boundary admissible level, one
can get ordinary $\mathbb{Z}_{2}$-twisted module
$\sigma^{-\frac{1}{2}\Lambda}(M_{\mathrm{vac}})$ by taking a spectral flow on
the vacuum module $M_{\mathrm{vac}}$ along a special direction
$-\frac{1}{2}\Lambda$ in the weights lattice. We believe that this phenomenon
is universal for all affine VOAs associated with the semisimple Lie algebra
$\mathfrak{g}$ at admissible level $k$. So far, for a general VOA $V$, we do
not have a systematic approach to get the ordinary $\mathbb{Z}_{2}$-twisted
modules from irreducible non-twisted $V$-modules in the category
$\mathcal{O}$.

In this work, the usage of the spectral flow is crucial, as we use it to
obtain twisted modules and new modules. In particular, we hope to prove
Conjecture 1.8 and address some of the questions raised in above Remark 1.9 in
the future work. It is also interesting to investigate the tensor category
structures for some subcategory of the relaxed highest weight category for the
affine VOA at fractional level.

We summarize the main contents in this paper. In Section 2, we recall some
basic notions of admissible representations, their (twisted) characters, the
modularity of their characters under the transformations
$\tau\rightarrow-\frac{1}{\tau}$ , and semi-simplicity of the
$\mathbb{Z}/2\mathbb{Z}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$ at
boundary admissible level. In Section 3, we study the twisted Zhu’s bimodules
of the $\mathbb{Z}_{2}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$ at
admissible level, then use it to classify twisted modules and compute their
fusion rules. In Section 4, we further calculate fusion rules among the
highest weight modules and their contragredient modules for
$L_{k}(\mathfrak{sl}_{2})$ at admissible level. In Section 5, we study the
$e^{2\pi iv_{(0)}}$-twisted modules of affine VOAs at admissible level, then
derive MLDEs their characters satisfy. In Section 6, we discuss on the
generalization of the result in the previous section to
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$.

## Acknowledgement

The authors wish to express their gratitude to Peng Shan for numerous fruitful
discussions. HL would like to thank Antun Milas for some motivating questions
and suggestions back in 2021. The authors would also like to thank Tomoyuki
Arakawa, Thomas Creutzig, Antun Milas for valuable suggestions to improve some
statements in the paper. BL, HL and WY are supported by national key research
and development program of China (NO. 2020YFA0713000) and Dushi program of
Tsinghua. WY is also supported by NNSF of China with Grant NO: 11847301 and
12047502.

##  2\. Character formulae at admissible level and their modularity

Let $\hat{\Delta}$, $\hat{\Delta}_{+}$, $\hat{\Delta}_{+}^{re}$ be all roots,
positive roots, positive real roots of an affine Lie algebra
$\hat{\mathfrak{g}}$. Let $\ell=\text{dim}(\mathfrak{g})$. Given
$\lambda\in\mathfrak{h}^{*}$, the set of $\lambda$-integral roots is
$\hat{\Delta}^{\lambda}:=\\{\alpha\in\hat{\Delta}^{re}|\lambda(\alpha^{\vee})\in\mathbb{Z}\\}$.
$\lambda$ is an admissible weight if the following two properties hold

  * •

$(\lambda+\rho)(\alpha^{\vee})\notin\mathbb{Z}_{\leq 0}$ for all
$\alpha\in\hat{\Delta}_{+}^{re}$,

  * •

$\mathbb{Q}\hat{\Delta}^{\lambda}=\mathbb{Q}\hat{\Delta}.$

Roughly speaking, the admissible weight is integrable with respect to
$\hat{\mathfrak{g}}_{\hat{\Delta}^{\lambda}}$ after a shift by the Weyl vector
$\rho$ of $\hat{\mathfrak{g}}$. If there exists a further isometry $\phi$ of
$\mathfrak{h}^{*}$ such that $\phi(\hat{\Delta}^{\lambda})=\hat{\Delta}$,
$\lambda$ is called a principal admissible weight, i.e., a principal
admissible weight is integrable with respect to $\hat{\mathfrak{g}}$ after a
shift of a Weyl vector.

######  Theorem 2.1.

[KW88] Let $\hat{\mathfrak{g}}$ be a Kac-Moody Lie algebra with a
symmetrizable generalized Cartan matrix, and $\lambda\in\mathfrak{h}^{*}$ be
an admissible weight. Then the character $L(\lambda)$ is given by the
following formula:

(15)
$\displaystyle\mathrm{ch}L(\lambda)=\frac{1}{R}\cdot\displaystyle\sum_{w\in
W_{\lambda}}\varepsilon(w)e^{w(\lambda+\rho)},$

where
$R:=e^{\rho}\prod_{n=1}^{\infty}(1-q^{n})^{\ell}\prod_{\alpha\in\hat{\Delta}_{+}}(1-e^{\alpha}q^{n})(1-e^{-\alpha}q^{n-1})$
is the Kac-Weyl denominator, and
$W^{\lambda}:=\\{r_{\alpha}|\alpha\in\hat{\Delta}^{\lambda}\\}$.

From now on we focus on the case of $A_{1}^{(1)}$ until stated otherwise. It
is known that all admissible weights of $A_{1}^{(1)}$ are principal
admissible. The level $\lambda(c)=k=\frac{t}{u}$ of an admissible weight
satisfies the following condition:

(16) $\displaystyle
k+h^{\vee}\geq\frac{h^{\vee}}{u}\quad\text{and}\quad\text{gcd}(u,h^{\vee})=\gcd(u,r^{\vee})=1,$

with the dual Coxeter number $h^{\vee}=2$ and the lacety number $r^{\vee}=1$.
All admissible weights at level $m=\frac{t}{u}$ are given by [KW88]:

$\displaystyle P^{m}$
$\displaystyle=\\{\lambda_{m,k,n}:=(m-n+k(m+2))\Lambda_{0}+(n-k(m+2))\Lambda_{1}|n,k\in\mathbb{N},n\leq
2u+t-2,k\leq u-1.\\}$
$\displaystyle=\\{t_{-\frac{k\alpha}{2}}(\widetilde{\Lambda^{0}}-(u-1)(m+2)\Lambda_{0}\\},$

where $\widetilde{\Lambda^{0}}=(u(m+2)-2-n)\Lambda_{0}+n\Lambda_{1}.$

One can write down the normalized character $\chi_{\lambda}(\tau,z,t)$ for
$L(\lambda_{m,k,n})$ [KW88]:

(17)
$\displaystyle\begin{split}\chi_{\lambda}(\tau,z,t)&=\frac{A_{\lambda+\rho}(h)}{A_{\rho}(h)}\\\
&=\frac{(\Theta_{a^{+},b}-\Theta_{a^{-},b})(\tau,\frac{z}{u},\frac{t}{u^{2}})}{(\Theta_{1,2}-\Theta_{-1,2})(\tau,z,t))}\\\
&=\frac{(\Theta_{a^{+},b}-\Theta_{a^{-},b})(\tau,\frac{z}{u},\frac{t}{u^{2}})}{-ie^{-4\pi
it}\vartheta_{11}(\tau,z)}\end{split}$

where theta functions $\Theta_{m,n}$ are defined in Appendix and $A_{\lambda}$
is defined as

(18) $\displaystyle A_{\lambda}(h):=\sum_{w\in
W^{\lambda}}\varepsilon(w)\Theta_{w(\lambda)}(h).$

Here $W^{\lambda}:=\\{r_{\alpha}|\alpha\in{\hat{\Delta}^{\lambda}}\\}$, and

$\displaystyle a^{+}:=u((n+1)-k(m+2)),\quad a^{-}:=u(-(n-1)-k(m+2)),\quad
b:=u^{2}(m+2).$

The modular $S$ transformation property of $\chi_{\lambda}$ is

(19)
$\displaystyle\begin{split}\chi_{\lambda_{m,k,n}}(-\frac{1}{\tau},\frac{z}{\tau},t-\frac{|z|^{2}}{2\tau})=\displaystyle\sum_{\begin{subarray}{c}0\leq
k^{\prime}\leq u-1\\\ 0\leq n^{\prime}\leq
u(m+2)-2\end{subarray}}a^{(m)}_{(k,n),(k^{\prime},n^{\prime})}\chi_{\lambda_{m,k^{\prime},n^{\prime}}}(\tau,z,t),\end{split}$

where

(20) $\displaystyle
a_{(k,n),(k^{\prime},n^{\prime})}^{(m)}:=\sqrt{\frac{2}{u^{2}(m+2)}}e^{i\pi(k^{\prime}(n+1)+k(n^{\prime}+1))}e^{-i\pi
kk^{\prime}(m+2)}\times\sin\frac{(n+1)(n^{\prime}+1)\pi}{m+2}.$

###  2.1. Twisted modular transformation

Let
$N:=\overline{\mathfrak{h}}_{\mathbb{R}}\times\overline{\mathfrak{h}}_{\mathbb{R}}\times
i\mathbb{R}$ be the Heisenberg group with multiplication

(21)
$\displaystyle(\alpha,\beta,u)\cdot(\alpha^{\prime},\beta^{\prime},u^{\prime}):=(\alpha+\alpha^{\prime},\beta+\beta^{\prime},u+u^{\prime}+\pi
i(\langle\alpha|\beta^{\prime}\rangle-\langle\alpha^{\prime},\beta\rangle)).$

One defines the action of modular group $SL_{2}(\mathbb{Z})$ and Heisenberg
group $N$ on the space $\mathfrak{h}^{*}$ as follows:

(22) $\displaystyle\begin{split}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot(\tau,z,t);=(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d},t-\frac{c|z|^{2}}{2(c\tau+d)}),\end{split}$

and

(23) $\displaystyle(\alpha,\beta,u)\cdot h:=t_{\beta}h+2\pi i\alpha+(u-\pi
i\langle\alpha,\beta\rangle)\delta,$

where $t_{\beta}$ is the translation operator.

One can check that the action of Heisenberg group $N$ on $\mathfrak{h}^{*}$ is
an group action, i.e.,

(24)
$\displaystyle((\alpha,\beta,u)(\alpha^{\prime},\beta^{\prime},u^{\prime}))\cdot
h=(\alpha,\beta,u)\cdot((\alpha^{\prime},\beta^{\prime},u^{\prime})\cdot h).$

######  Lemma 2.2.

[Wak01] For $(\alpha,\beta,u)\in N$ and $h=(\tau,z,t)=2\pi
i(-\tau\Lambda_{0}+z+t\delta)\in\mathfrak{h}^{*},$ the following holds:

(25)
$\displaystyle(\alpha,\beta,u)\cdot(\tau,z,t)=(\tau,z+\alpha-\tau\beta,t+\frac{u}{2\pi
i}-\frac{\langle\alpha,\beta\rangle}{2}+\frac{\tau}{2}|\beta|^{2}-\langle\beta,z\rangle).$

###### Proof.

By direct calculation, one has

$\displaystyle\begin{split}&t_{\beta}(\Lambda_{0})=\Lambda_{0}+\beta-\frac{|\beta|^{2}}{2}\delta,\\\
&t_{\beta}(\delta)=\delta,\\\ &t_{\beta}(z)=z-\langle
z,\beta\rangle\delta.\end{split}$

Thus,

$\displaystyle t_{\beta}(h)$ $\displaystyle=t_{\beta}(-2\pi i\Lambda_{0}+2\pi
iz+2\pi i\delta t)$ $\displaystyle=-2\pi
i\tau(\Lambda_{0}+\beta-\frac{|\beta|^{2}}{2}\delta)+2\pi i(z-\langle
z,\beta\rangle\delta)+2\pi it(\delta)$ $\displaystyle=-2\pi
i\Lambda_{0}\tau+2\pi i(z-\tau\beta)+2\pi i\delta(t-\langle
z,\beta\rangle+\tau\frac{|\beta|^{2}}{2}).$

Furthermore,

$\displaystyle(\alpha,\beta,u)\cdot h$ $\displaystyle=-2\pi
i\Lambda_{0}\tau+2\pi i(z+\alpha-\tau\beta)+2\pi i\delta(t+\frac{u}{2\pi
i}-\frac{\langle\alpha,\beta\rangle}{2}-\langle
z,\beta\rangle+\tau\frac{|\beta|^{2}}{2}).$

We are done. ∎

The action of groups $SL_{2}(\mathbb{Z})$ and $N$ on $\mathfrak{h}^{*}$ is
compatible:

######  Lemma 2.3.

For $(\alpha,\beta,u)\in N$ and $h\in\mathfrak{h}^{*}$ and
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{Z})$, the following
holds:

(26) $\displaystyle\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot(\alpha,\beta,u)\cdot
h=(a\alpha+b\beta,c\alpha+d\beta,u)\cdot\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot h.$

The metaplectic group is defined as

$Mp_{2}(\mathbb{Z}):=\left\\{(A,j)\left|\begin{array}[]{c}A\in
SL_{2}(\mathbb{Z}),\\\ j\;\;\text{is a holomorphic function
in}\;\;\tau\in\mathbb{H}\\\ \text{such
that}\;\;j(\tau)^{2}=c\tau+d\end{array}\right.\right\\}.$

Given a holomorphic function $F$ on
$Y=\mathbb{H}\times\mathbb{C}\times\mathbb{C}$, one has the right action of
$Mp_{2}(\mathbb{Z})$ and $N$ on $F$:

$\displaystyle F|_{(A,j)}(\tau,z,t):=\frac{1}{j(\tau)^{\ell}}\cdot
F(A\cdot(\tau,z,t)),$ $\displaystyle
F|_{(\alpha,\beta,u)}(\tau,z,t):=F((\alpha,\beta,u)\cdot(\tau,z,t)).$

Then one defines very important functions for
$\alpha,\beta\in\overline{\mathfrak{h}}^{*}$:

(27) $\displaystyle
F^{\alpha,\beta}(\tau,z,t):=F((\alpha,\beta,0)\cdot(\tau,z,t)),$

namely

$\displaystyle
F^{\alpha,\beta}(\tau,z,t)=F(\tau,z+\alpha-\tau\beta,t-\frac{\langle\alpha,\beta\rangle}{2}-\langle\beta,z\rangle+\frac{\tau}{2}|\beta|^{2}).$

Modular transformation of these functions are given as follows:

######  Lemma 2.4.

[Wak01] Under the action of $(A,j)\in Mp_{2}(\mathbb{Z})$,

  * •

$(F|_{(A,j)})^{\alpha,\beta}=F^{a\alpha+b\beta,c\alpha+d\beta}|_{(A,j)},$

  * •

$F^{\alpha,\beta}|_{(A,j)}=(F|_{(A,j)})^{a^{\prime}\alpha+b^{\prime}\beta,c^{\prime}\alpha+d^{\prime}\beta},$

where $A^{-1}=\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}$.

###  2.2. Twisted characters

Now we consider the normalized character of $L(\lambda)$ at level $k$,
$\chi_{\lambda}(h)=q^{m_{\Lambda}}\operatorname{tr}_{L(\Lambda)}e^{2\pi ih}$
evaluated at $h=2\pi i(-\tau d+z+tc)$, i.e., $\chi_{\lambda}(\tau,z,t)$. Then
given $\alpha,\beta\in\overline{\mathfrak{h}}^{*}$, one has

(28)
$\displaystyle\begin{split}\chi_{\lambda}^{\alpha,\beta}(\tau,z,t)&=q^{m_{\lambda}}\operatorname{tr}_{L(\lambda)}e^{-2\pi
i\tau d+2\pi i(z+\alpha-\tau\beta)+2\pi
ik(t-\frac{\langle\alpha,\beta\rangle}{2}-\langle
z,\beta\rangle+\tau\frac{|\beta|^{2}}{2})}\\\ &=e^{2\pi
ikt}\operatorname{tr}_{L(\lambda)}e^{2\pi
i((z+\alpha)+k(-\frac{\langle\alpha,\beta\rangle}{2}-\langle
z,\beta\rangle))}q^{-\beta+k\frac{|\beta|^{2}}{2}}q^{L_{(0)}-\frac{k}{24}}\\\
&=e^{2\pi ikt}\operatorname{tr}_{L(\lambda)}e^{2\pi
i(z+\alpha-k\frac{\langle\alpha,\beta\rangle}{2}-k\langle
z,\beta\rangle))}q^{L_{(0)}-\beta+k\frac{|\beta|^{2}}{2}-\frac{k}{24}}\end{split}$

The twisted character (28) is very important. When $z=0$, multiplying (28)
with $\eta(\tau)^{k}$ produces preciously the theta function defined on vertex
algebra by Miyamoto [Miy00]. When $\alpha=0$, (28) can be written as

(29)
$\displaystyle\begin{split}\chi_{\lambda}^{0,\beta}(\tau,z,t)&=(\mathbf{y}e^{-2\pi
i\langle
z,\beta\rangle}q^{\frac{|\beta|^{2}}{2}})^{k}\operatorname{tr}_{L(\Lambda)}e^{2\pi
iz}q^{-\beta}q^{L_{(0)}-\frac{k}{24}}\end{split}$

where $\mathbf{y}:=e^{2\pi it}.$

###  2.3. Twisted modules

Let $g$ be the automorphism of $V$ of finite order $T$. We first recall the
definition of $g$-twisted module:

######  Definition 2.5.

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

######  Remark 2.6.

If the grading condition is dropped, one calls it the weak $g$-twisted module.

Now we review some basic facts about twisted modules under the vertex algebra
setting. Let $(V,\omega)$ be a $\mathbb{Z}$-graded conformal vertex algebra.
Let $v\in V_{1}$ be an even vector satisfying the Heisenberg $\lambda$-bracket
relation

(30)
$\displaystyle\begin{split}[v_{\lambda}v]=k\lambda,\quad[\omega_{\lambda}v]=(T+\lambda)v.\end{split}$

Suppose further that $v_{(0)}$ acts semisimply on $V$ such that the
eigenvalues of $v_{(0)}$ belong to $\frac{1}{T}\mathbb{Z}$. Li’s
$\Delta$-operator is defined as

(31)
$\displaystyle\Delta(z):=z^{v_{(0)}}\exp(\sum_{n=1}^{\infty}\frac{v_{(n)}}{-n}(-z)^{-n}).$

When $g(v)=v$, one can obtain an $ge^{2\pi iv_{(0)}}$-twisted module from a
$g$-twisted $V$-module $M$ by using Li’s $\Delta$-operator as the following

######  Proposition 2.7.

[Li96] $(M,Y_{M}(\Delta(z)\cdot,z))$ is a weak $ge^{2\pi iv_{(0)}}$-twisted
module.

When $g=\text{id}$, $e^{2\pi iv_{(0)}}$ is an automorphism of $V$ of order
$T$. Then $(M,Y_{M}(\Delta(z)\cdot,z))$ is a $e^{2\pi iv_{(0)}}$-twisted
module. Let $v^{\prime}\in V_{1}$ satisfy $v_{(0)}v^{\prime}=0$. Define

(32)
$\displaystyle\begin{split}&\hat{L}_{(0)}:=\text{Res}_{z}zY_{M}(\Delta(z)\omega,z),\\\
&\hat{v^{\prime}}_{(0)}:=\text{Res}_{z}Y_{M}(\Delta(z)v^{\prime},z).\end{split}$

By direct calculation, one has

$\displaystyle\hat{L}_{(0)}=L_{(0)}+v_{(0)}+\frac{1}{2}\langle
v,v\rangle,\quad\hat{v^{\prime}}_{(0)}=v^{\prime}_{(0)}+\langle
v,v^{\prime}\rangle.$

The nomalized character of weak $V$-module $M$ is defined as

(33) $\displaystyle\text{ch}(M)(q,z)=\operatorname{tr}_{M}e^{2\pi
izv^{\prime}_{(0)}}q^{L_{(0)}-\frac{c}{24}}.$

Then the normalized character of weak $e^{2\pi iv_{(0)}}$-twisted module
$\hat{M}=(M,Y_{M}(\Delta(z)\cdot,z))$ is

(34) $\displaystyle\text{ch}(\hat{M})(q,z)=\operatorname{tr}_{M}e^{2\pi
iz(v^{\prime}_{(0)}+k\langle
v,v^{\prime}\rangle)}q^{L_{(0)}+v_{(0)}+\frac{1}{2}k\langle
v,v\rangle-\frac{c}{24}}.$

######  Remark 2.8.

In the affine vertex algebra case, let $M=L(\Lambda)$, $v^{\prime}=z$ and
$v=-\beta$. One has

$\mathrm{ch}(\hat{M})(q,z)=\chi_{\Lambda}^{0,-\beta}(\tau,z,0),$

where $q=e^{2\pi i\tau}.$ It is also worth noting that the central charge of
the conformal element $\omega+L_{(-1)}v$ is $c-12\langle v,v\rangle$. Also
$(\omega+L_{(-1)}v)_{(0)}=L_{(0)}+v_{(0)}$. Thus,

(35) $\displaystyle\operatorname{tr}_{M}e^{2\pi
izv^{\prime}_{(0)}}q^{L_{(0)}+v_{(0)}+\frac{1}{2}k\langle
v,v\rangle-\frac{c}{24}}$

can be understood as the trace function on $M$ with respect to the new
conformal vector, $\omega+L_{(-1)}v$. Careful readers may notice the
difference between (34) and (35); this is because the first one changes the
module structure and the second one changes the conformal structure.

###  2.4. Spectral Flow Automorphsim

Let us give a brief introduction to the spectral flow automorphisms (see
[Rid09, CR12, CR13a] for details). Let $\hat{\mathfrak{g}}$ be the (non-
twisted) affine Lie algebra associated with the semisimple complex finite Lie
algebra ($\mathfrak{g}$,$(~{}|~{}))$. Let $\Delta$ be the set of roots of
$\mathfrak{g}$. Let $\alpha\in\Delta$ denote a root of $\mathfrak{g}$ with
root vector $e^{\alpha}$ and coroot $\alpha^{\vee}$, and let $W$ be the Weyl
group of finite Lie algebra $\mathfrak{g}$. Each element $w\in W$ permutes the
roots and induces an automorphism of $\mathfrak{g}$ via
$w(e^{\alpha})=e^{w(\alpha)}$, this action can be generalized to an affine Lie
algebra $\hat{\mathfrak{g}}$ as follows, the corresponding root vector of real
roots are $e^{\alpha}_{n}$, and the roots vector corresponding to imaginary
roots are denoted by $h^{i}_{n}$, one can associate $h^{i}$ with the simple
coroot $\alpha^{\vee}_{i}$ of $\mathfrak{g}$. Let $W\subset
GL(\mathfrak{h}^{*})$ be the Weyl group of $\mathfrak{g}$ generated by the
reflections $s_{\alpha}$ with $\alpha\in\hat{\Delta}^{\text{re}}$, where
$s_{\alpha}(\lambda)=\lambda-\langle\lambda,\alpha^{\vee}\rangle\alpha$ for
$\lambda\in\mathfrak{h}^{*}$. Let the affine Weyl group be
$\widehat{W}=W\ltimes\bar{Q}^{\vee}$. The coroot lattice acts on the roots of
$\hat{\mathfrak{g}}$ by translation in the imaginary direction. Then each
simple coroot $\alpha^{\vee}_{i}$ of $\mathfrak{g}$ defines an independent
transformation $\tau_{i}$ on the root vector of an affine Lie algebra
$\hat{\mathfrak{g}}$ via,

(36)
$\displaystyle\tau_{i}(e_{n}^{\alpha})=e^{\alpha}_{n-\langle\alpha,\alpha^{\vee}_{i}\rangle}\quad(n\in\mathbb{Z}),\qquad\tau_{i}(h^{j}_{n})=h^{j}_{n}\quad(n\neq
0)$

We finally obtain the spectral flow automorphisms $\tau_{i}$ act on generators
of the affine Lie algebra $\hat{\mathfrak{g}}$ and Virasoro element $L_{0}$ as
follows,

(37) $\displaystyle\tau_{i}(e^{\alpha}_{n})$
$\displaystyle=e^{\alpha}_{n-\langle\alpha,\alpha^{\vee}_{i}\rangle},\quad\tau_{i}(h^{j}_{n})=h^{j}_{n}-(\alpha^{\vee}_{i}|\alpha^{\vee}_{j})K$
(38) $\displaystyle\tau_{i}(K)$
$\displaystyle=K,\quad\tau_{i}(L_{0})=L_{0}-h^{i}_{0}+\frac{2}{(\alpha_{i}|\alpha_{i})}K$

For the $\hat{\mathfrak{g}}=A^{(1)}_{1}$, the spectral flow automorphism
$\sigma$ which may be regarded as a square root of the affine Weyl translation
by the simple coroot $\alpha^{\vee}_{i}$ of finite Lie algebra $\mathfrak{g}$,
the powers of $\sigma$ acts as follows,

(39) $\displaystyle\sigma^{\ell}(e_{n})$
$\displaystyle=e_{n-\ell},\quad\sigma^{\ell}(h_{n})=h_{n}-\ell\delta_{n,0}K,\quad\sigma^{\ell}(f_{n})=f_{n+\ell}$
(40) $\displaystyle\sigma^{\ell}(K)$
$\displaystyle=K,\quad\sigma^{\ell}(L_{0})=L_{0}-\frac{1}{2}\ell
h_{0}+\frac{1}{4}\ell^{2}K$

One can use spectral flow automorphsim to modify the action of $A^{(1)}_{1}$
on any module $M$, thereby obtaining new modules $\sigma^{*}(M)$. Explicitly,
the modified algebra action defining these new modules is given by,

(41)
$X\cdot\sigma^{*}|v\rangle=\sigma^{*}(\sigma^{-1}(X)|v\rangle),\quad(X\in\mathfrak{sl}_{2}).$

For example, if $|\lambda,\Delta\rangle$ is a vector of weight $\lambda$ and
conformal dimension $\Delta$, then the vector
$(\sigma^{\ell})^{*}|\lambda,\Delta\rangle\in(\sigma^{\ell})^{*}(M)$, the
weight and conformal dimension of this vector becomes,

(42) $\displaystyle
h_{0}(\sigma^{\ell})^{*}|\lambda,\Delta\rangle=(\lambda+\ell
K)(\sigma^{\ell})^{*}|\lambda,\Delta\rangle$ (43) $\displaystyle
L_{0}(\sigma^{\ell})^{*}|\lambda,\Delta\rangle=(L_{0}+\frac{1}{2}\ell
h_{0}+\frac{1}{4}\ell^{2}K)(\sigma^{\ell})^{*}|\lambda,\Delta\rangle$

(From now on, we denote the new module as $\sigma^{\ell}(M)$). In order to
obtain the character of new module $\sigma^{\ell}(M)$, we need to compute the
character of module $M$. The normalized character of irreducible highest
weight $\hat{\mathfrak{g}}$ module $M$ at level $k$ is defined as,

(44)
$\text{ch}[M](\mathbf{y},\mathbf{z},q)=\text{tr}_{M}\,\mathbf{y}^{k}\mathbf{z}^{h_{0}}q^{L_{0}-c/24}$

where $\mathbf{y}=e^{2\pi it}$, $\mathbf{z}=e^{2\pi iz}$ and $q=e^{2\pi
i\tau}$.

The character of new module $\sigma^{\ell}(M)$ can be written in terms of the
character of module $M$ as follows,

(45)
$\text{ch}[\sigma^{\ell}(M)](\mathbf{y},\mathbf{z},q)=\text{ch}[M]\left(\mathbf{y}\mathbf{z}^{\ell}q^{\ell^{2}/4},\mathbf{z}q^{\ell/2},q\right),$

One can check that the character of module $M$ satisfies the following
relation,

(46)
$\text{ch}[\sigma^{\ell+\ell^{\prime}}(M)](\mathbf{y},\mathbf{z},q)=\text{ch}[\sigma^{\ell}\circ\sigma^{\ell^{\prime}}(M)](\mathbf{y},\mathbf{z},q)=\text{ch}[M]\left(\mathbf{y}\mathbf{z}^{\ell+\ell^{\prime}}q^{(\ell+\ell^{\prime})^{2}/4},\mathbf{z}q^{(\ell+\ell^{\prime})/2},q\right).$

###  2.5. Li’s delta operator, spectral flow and twisted modules

Let $M$ be a highest weight $\hat{\mathfrak{g}}$-module or its contragredient
module. Let $v\in\mathfrak{h}_{\mathbb{R}}^{*}$. We have

$\displaystyle\Delta(z)e_{-1}^{\alpha}\mathbf{1}=z^{\langle
v,\alpha\rangle}e_{-1}\mathbf{1},$
$\displaystyle\Delta(z)h_{-1}^{j}\mathbf{1}=h_{-1}^{j}\mathbf{1}+\langle
v,h^{j}\rangle kz^{-1}\mathbf{1}$

Thus, it induces the spectral flow $\tau$:

$\displaystyle\tau(e_{n}^{\alpha})=e_{n+\langle v,\alpha\rangle}^{\alpha}$
$\displaystyle\tau(h_{n}^{j})=h_{n}^{j}+\langle v,h^{j}\rangle k\delta_{n,0}.$

If $\langle v,\alpha\rangle\in\frac{1}{T}\mathbb{Z}$ for all roots $\alpha$,
according to Proposition 2.7, $(M,Y_{M}(\Delta(z)\cdot,z))$ is a weak $e^{2\pi
iv_{(0)}}$ -twisted ($\mathbb{Z}_{T}$-twisted) module. But the
$\mathbb{Z}_{T}$-twisted modules are not necessarily coming from spectral
flow.

In the future, we shall call $(M,Y_{M}(\Delta(z)\cdot,z))$ the spectral flowed
module or $e^{2\pi iv_{(0)}}$-twisted module.

######  Proposition 2.9.

For $A^{(1)}_{1}$ at admissible level $k=-2+\frac{v}{u}$, let
$\lambda(i,j,v,u)$ be the Dynkin label for the finite Lie algebra
$\mathfrak{sl}_{2}$ of an irreducible highest weight module and
$\Delta(i,j,v,u)$ its conformal weight determined by the Sugawara
construction. They become $\sigma^{-\frac{1}{2}}(\lambda(i,j,v,u))$ and
$\sigma^{-\frac{1}{2}}(\Delta(i,j,v,u))$ after taking $\ell=-\frac{1}{2}$
spectral flow action on $\lambda(i,j,v,u)$ and $\Delta(i,j,v,u)$ as follows,

$\displaystyle\sigma^{-\frac{1}{2}}(\lambda(i,j,v,u))=i-1-(k+2)s-\frac{k}{2},\quad(i=1,\cdots,v-1,j=0,\cdots,u-1)$
$\displaystyle\sigma^{-\frac{1}{2}}(\Delta(i,j,v,u))=\frac{1}{16}\left(4+k-4i+4(2+k)j+\frac{4(-1+(i-(2+k)j)^{2})}{2+k}\right),\quad(i=1,\cdots,v-1,j=0,\cdots,u-1)$

###### Proof.

The statement follows from (42), (43) and theorem (3.5.3) in [AM95]. ∎

One can compute the $\mathbb{Z}_{2}$-twisted Zhu’s algebra
$A_{\sigma}(L_{k}(\mathfrak{sl}_{2}))$ at admissible levels $k=-2+\frac{v}{u}$
(The proof will be given in Proposition 3.4), and find that all irreducible
$\mathbb{Z}_{2}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$ from
$\ell=-\frac{1}{2}$ spectral flow on the untwisted modules in category
$\mathcal{O}$.

###  2.6. Half-integer spectral flow for $A_{1}^{(1)}$ at boundary admissible
level

For $A_{1}^{(1)}$, $h^{\vee}=2$, the boundary admissible levels are
$k=-2+\frac{2}{u}$, where $u=2n+1$ is a positive odd integer. All admissible
weights are,

(47)
$\Lambda_{k,j}:=t_{-\frac{j}{2}}\cdot(k\Lambda_{0})=\left(k+\frac{2j}{u}\right)\Lambda_{0}-\frac{2j}{u}\Lambda_{1},\quad
j=0,1,\cdots,u-1,$

where $\Lambda_{0}$ and $\Lambda_{1}$ are fundamental weights of affine Lie
algebra $A^{(1)}_{1}$. All characters of irreducible modules can be written in
terms of Jacobi theta function $\theta_{1}(\mathbf{z};q)$ as follows,

(48)
$\text{ch}[L(\Lambda_{k,j})](\mathbf{y},\mathbf{z},q)=\mathbf{y}^{k}\mathbf{z}^{-\frac{2j}{u}}q^{\frac{j^{2}}{2u}}\frac{\theta_{1}(\mathbf{z}^{2}q^{-j};q^{u})}{\theta_{1}(\mathbf{z}^{2};q)},\quad(j=0,1,\dots,u-1)$

We should emphasize that the vaccum character $(j=0)$ of these VOAs denoted by
$L_{\frac{-4n}{2n+1}}(\mathfrak{sl}_{2})$ coincide with superconformal indices
of the $4d$ supersymmetric gauge theories which called $(A_{1},D_{2n+1})$
theories. Especially, when $u=3$, level $k=-4/3$, the
$L_{-\frac{4}{3}}(\mathfrak{sl}_{2})$ coincides with the $\mathfrak{a}_{1}$ DC
series of simple Lie algebras.

One can generalize the integer flow parameter $\ell$ to half integer, (45)
stays the same. For the $A_{1}^{(1)}$ at boundary admissible levels
$k=-2+\frac{2}{u}$, using (45), all characters of these twisted modules take
the following form,

(49)
$\text{ch}[\sigma^{-\frac{1}{2}}({L({\Lambda_{k,j}}}))](\mathbf{y},\mathbf{z},q)=(\mathbf{y}\mathbf{z}^{-\frac{1}{2}}q^{\frac{1}{16}})^{k}(\mathbf{z}q^{-\frac{1}{4}})^{-\frac{2j}{u}}q^{\frac{j^{2}}{2u}}\frac{\theta_{1}(\mathbf{z}^{2}q^{-j-\frac{1}{2}};q^{u})}{\theta_{1}(\mathbf{z}^{2}q^{-\frac{1}{2}};q)},\quad(j=0,1,\dots,u-1)$

Now, we use $\lambda(j,u)$ instead of $\lambda(1,j,2,u)$ (Similar to
$\Delta(j,u)$), and get,

$\displaystyle\sigma^{-\frac{1}{2}}(\lambda(j,u))=\frac{u-2j-1}{u},\quad\sigma^{-\frac{1}{2}}(\Delta(j,u))=\frac{1+4j(1+j-u)-u}{8u},\quad(j=0,1,\dots
u-1)$

In particular, we observe that the $\sigma^{-\frac{1}{2}}(\lambda(j,u))$ and
$\sigma^{-\frac{1}{2}}(\Delta(j,u))$ satisfy the following relations
respectively,

(50)
$\sigma^{-\frac{1}{2}}(\lambda(j,u))=-\sigma^{-\frac{1}{2}}(\lambda(u-j-1,u)),\quad\sigma^{-\frac{1}{2}}(\Delta(j,u))=\sigma^{-\frac{1}{2}}(\Delta(u-1-j,u))$

Fix a boundary admissible level $k$, all
$\sigma^{-\frac{1}{2}}({L({\Lambda_{k,j}}}))$ are ordinary modules. Then the
normalized characters satisfy a modular linear differential equation (Section
5). We shall use the fact that $A_{\hat{\sigma}}(L_{k}(\mathfrak{sl}_{2}))$ is
semi-simple to prove semisimplicity of $\mathbb{Z}_{2}$-twisted modules of
$L_{k}(\mathfrak{sl}_{2})$.

######  Lemma 2.10.

Let $\sigma^{-\frac{1}{2}}(L(\Delta_{1},\lambda_{1}))$ and
$\sigma^{-\frac{1}{2}}(L(\Delta_{2},\lambda_{2}))$ be irreducible
$\mathbb{Z}_{2}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$. Suppose that
there is a nontrivial extension of

$\displaystyle
0\longrightarrow\sigma^{-\frac{1}{2}}(L(\Delta_{1},\lambda_{1}))\overset{\iota}{\longrightarrow}M\overset{\pi}{\longrightarrow}\sigma^{-\frac{1}{2}}(L(\Delta_{2},\lambda_{2}))\longrightarrow
0$

Then $L_{0}$ acts locally finitely on $M$.

###### Proof.

Since the Zhu’s algebra
$A_{\hat{\sigma}}(L_{k}(\mathfrak{sl}_{2}))\cong\mathbb{C}[h]/\langle
I\rangle$ is a commutative algebra and semisimple. According to characters
$\text{ch}[\sigma^{-\frac{1}{2}}({L(\Lambda_{k,j}){}})]$ of all twisted
modules. We have $\sigma^{-\frac{1}{2}}(L(\Delta_{1},\lambda_{1}))$ and
$\sigma^{-\frac{1}{2}}(L(\Delta_{2},\lambda_{2}))$ are $L_{0}$-diagonalizable
and the $L_{0}$-finite dimensional. The proof is similar to [Fas22] (Lemma
7.3). Finally, we have that, $m\in M$ belongs to some $L_{0}$-stable finite
dimensional vector subspace of

$\displaystyle\bigoplus_{i=1}^{r}\ker(L_{0}-\nu_{i}Id)\oplus\bigoplus_{j=1}^{s}\ker(L_{0}-\mu_{j}Id).$

where $\nu_{1},\cdots,\nu_{r}$ and $\mu_{1},\cdots,\mu_{s}$ are eigenvalues of
some eigenvectors
$v_{1},\cdots,v_{r}\in\sigma^{-\frac{1}{2}}(L(\Delta_{1},\lambda_{1}))$ and
$w_{1},\cdots,w_{s}\in\sigma^{-\frac{1}{2}}(L(\Delta_{2},\lambda_{2}))$
respectively, then the assertion follows. ∎

Then we can prove the following theorem which is Theorem 1.5 in the
introduction.

######  Theorem 2.11.

The category of $\mathbb{Z}/2\mathbb{Z}$-twisted modules of the vertex
operator algebra $L_{k}(\mathfrak{sl}_{2})$ at boundary level $k$ is semi-
simple.

###### Proof.

For any distinct conformal dimension
$\sigma^{-\frac{1}{2}}(\Delta_{m}),\sigma^{-\frac{1}{2}}(\Delta_{n})\in\sigma^{-\frac{1}{2}}(\Delta(j,u))$
which correspond to two distinct simple modules. We have
$\sigma^{-\frac{1}{2}}(\Delta_{m})\neq\sigma^{-\frac{1}{2}}(\Delta_{n})\,(\mathrm{mod}\,\mathbb{Z})$.
It is known [Fas22] (Lemma 7.4) that if there exists a nontrivial extension of
two distinct simple $\mathbb{Z}_{2}$-twisted
$L_{k}(\mathfrak{sl}_{2})$-modules

$\displaystyle
0\longrightarrow\sigma^{-\frac{1}{2}}(L(\Delta_{m},\lambda_{m}))\overset{\iota}{\longrightarrow}M\overset{\pi}{\longrightarrow}\sigma^{-\frac{1}{2}}(L(\Delta_{n},\lambda_{n}))\longrightarrow
0$

then $\sigma^{-\frac{1}{2}}(\Delta_{m})$ and
$\sigma^{-\frac{1}{2}}(\Delta_{n})$ coincide modulo $\mathbb{Z}$. Then, we
have
$\mathrm{Ext}^{1}[\sigma^{-\frac{1}{2}}(L(\Delta_{m},\lambda_{m})),\sigma^{-\frac{1}{2}}(L(\Delta_{n},\lambda_{n}))]=0$
for $\sigma^{-\frac{1}{2}}(\Delta_{m})\neq\sigma^{-\frac{1}{2}}(\Delta_{n})$.
For the simple modules which have the same conformal dimension
$\sigma^{-\frac{1}{2}}(\Delta_{m})=\sigma^{-\frac{1}{2}}(\Delta_{n})$. One can
consider the top space of these modules, the exact sequence

$\displaystyle
0\longrightarrow\sigma^{-\frac{1}{2}}(L(\Delta_{m},\lambda_{m}))_{\mathrm{top}}\overset{\iota}{\longrightarrow}M_{\mathrm{top}}\overset{\pi}{\longrightarrow}\sigma^{-\frac{1}{2}}(L(\Delta_{n},\lambda_{n}))_{\mathrm{top}}\longrightarrow
0$

has non-trivial extension of $A_{\hat{\sigma}}(L_{k}(\mathfrak{sl}_{2}))$
modules, This contradicts the fact that Zhu’s algebra
$A_{\hat{\sigma}}(L_{k}(\mathfrak{sl}_{2})$ is semisimple. So, there is no
nontrivial extension between two distinct simple modules and nontrivial self-
extension. This completes the proof. ∎

###  2.7. Modular transformation

Applying (2.4) to (28), one has the following $S$-transformation for twisted
characters:

(51)
$\displaystyle\begin{split}\chi_{\Lambda}^{\alpha,\beta}(\tau,z,t)|_{S}=(\chi_{\Lambda}(\tau,z,t)|_{S})^{\beta,-\alpha},\end{split}$

where $S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$.

######  Example 2.12.

(Boundary admissible level) We follow the same notations in Section 2.6. Let
$\text{Spec}(T)$ be the eigenvalues of $T$. First, note that
$\text{Spec}((\epsilon h_{1})_{(0)})\in 2\epsilon\mathbb{Z}$,
$(\epsilon\in\mathbb{Q})$. The twisted character (29) evaluated at $h=2\pi
i(-\tau d+\alpha_{1}z+ct)$ is

(52)
$\displaystyle\begin{split}\chi_{\Lambda}^{0,-\frac{\ell}{2}\alpha_{1}}(\tau,z,t)&=(\mathbf{y}e^{-2\pi
i\langle\alpha_{1}z,-\frac{\ell}{2}\alpha_{1}\rangle}q^{\frac{|-\frac{\ell}{2}\alpha_{1}|^{2}}{2}})^{m}\operatorname{tr}_{L(\Lambda)}e^{2\pi
i\alpha_{1}z}q^{\frac{\ell}{2}\alpha_{1}}q^{L_{(0)}-\frac{m}{24}}\\\
&=(\mathbf{y}\mathbf{z}^{\ell}q^{\frac{\ell^{2}}{4}})^{m}\operatorname{tr}_{L(\Lambda)}(\mathbf{z}q^{\frac{\ell}{2}})^{h_{0}}q^{L_{(0)}-\frac{m}{24}},\end{split}$

where we identity $(h_{1})_{(0)}$ with $\alpha_{1}$. (52) is the same as (45).

We split the discussion into two cases:

  * •

$\text{Spec}((\epsilon h_{1})_{0})\in\mathbb{Z}$, i.e.,
$\epsilon=\frac{\ell}{2},$ $\ell\in\mathbb{Z}$. Let $\sigma$ be spectral flow
automorphism on $M$ (see [CR13b] for more details). In this case, (52) is the
character of $\sigma^{\ell}(M)$.

  * •

$\text{Spec}((\epsilon h_{1})_{0})\in\frac{1}{T}\mathbb{Z},$
$T\in\mathbb{Z}_{>0}$, i.e., $\epsilon=\frac{\ell}{2}$,
$\ell\in\frac{1}{T}\mathbb{Z},$ $T\in\mathbb{Z}_{>0}$. (52) now becomes the
character of the $e^{2\pi i(\epsilon h_{1})_{(0)}}$-twisted module $\hat{M}$
introduced in Section 2.3, where $v=\epsilon h_{1}$. The $S$-transformation of
$\chi_{\lambda_{m,k}}^{0,-\frac{\ell}{2}\alpha_{1}}(\tau,z,t)$ is

(53)
$\displaystyle\begin{split}\chi_{\lambda_{m,k}}^{0,-\frac{\ell}{2}\alpha_{1}}(\tau,z,t)|_{S}&=(\chi_{\lambda_{m,k}}(\tau,z,t)|_{S})^{-\frac{\ell}{2}\alpha_{1},0}\\\
&=\displaystyle\sum_{\lambda_{m,k^{\prime}}\in
P_{m}}a_{\lambda_{m,k},\lambda_{m,k^{\prime}}}(\chi_{\lambda_{m,k^{\prime}}}^{-\frac{\ell}{2}\alpha_{1},0}(\tau,z,t))\end{split}$

where

$a_{\lambda_{m,k},\lambda_{m,k^{\prime}}}=(-1)^{k+k^{\prime}}e^{-\frac{2\pi
ikk^{\prime}}{u}}\frac{1}{\sqrt{u}}\sin\frac{u\pi}{2},$

and

(54)
$\displaystyle\begin{split}\chi_{\lambda_{m,k^{\prime}}}^{-\frac{\ell}{2}\alpha_{1},0}(\tau,z,t)=\chi_{\lambda_{m,k^{\prime}}}(\tau,z-\frac{\ell}{2},t).\end{split}$

As one can see, the $S$ transformation of the character of $e^{2\pi i(\sigma
h_{1})_{(0)}}$-twisted module $L(\lambda_{m,k})$ can be written as linear
combination of the characters of untwisted modules with the same $S$-matrix as
in untwisted case.

##  3\. Twisted Zhu’s Bimodule of highest weight modules

The fusion rules among admissible representations in category $\mathcal{O}$
were studied by many authors ([DLM97, Rid09, Rid10, CR12, CR13a, Gab01], and
etc.). They found that the Verlinde formula is no longer true in the case of
$L_{k}(\mathfrak{g})$, where $k$ is the admissible level and $\mathfrak{g}$ is
a simple Lie algebra, since the negative coefficient would appear. The method
used by Dong-Li-Mason is to apply Frenkel-Zhu’s bimodule theorem [FZ96], while
the other methods involve the use of characters and some machinery from
physics [AY92, Gab01, LMRS02]. In this section, we shall compute the twisted
Zhu’s bimodules recently introduced in [Zhu22]; in particular, the twisted
Zhu’s algebra is a special case, then we use the twisted Zhu’s algebra and
bimodules to classify the twisted modules and also calculate the fusion rules
among the twisted modules.

Let $M^{1},M^{2},M^{3}$ be $g_{1},g_{2},g_{3}$-twisted modules, respectively.
We consider the fusion rules $N^{3}_{1\;2}$, where $1,2,3$ represent modules
$M^{1},M^{2},M^{3}$. Recall the following definition [Zhu22] of
$A_{g_{2}}$-bimodule $A_{g_{1}g_{2},g_{2}}(M^{1})$. Denote the remainder of
$r\in\mathbb{N}$ divided by $T$ by $[r]$.

For homogeneous element $u\in V$ and $w_{1}\in M^{1}$, one defines

$u\circ_{g_{1}g_{2},g_{2}}w_{1}=\text{Res}_{z}{\frac{(1+z)^{\operatorname{wt}u-1+\delta(j_{2})+\frac{j_{2}}{T}}}{z^{1+\delta(j_{1},j_{2})-\frac{j_{1}}{T}}}}$

where $u\in V^{(j_{1},j_{2})}$ and

(55) $\delta(j_{1},j_{2})=\begin{cases}1,&j_{2}=0\\\ 1,&j_{2}\neq
0,\;j_{1}+j_{2}\geq T\\\ 0,&j_{2}\neq 0,\;j_{1}+j_{2}<T.\end{cases}$

Let $O^{\prime}_{g_{1}g_{2},g_{2}}(M^{1})$ be the subspace of $M^{1}$ spanned
by all $u\circ_{g_{1}g_{2},g_{2}}w_{1}$. One defines
$A_{g_{1}g_{2},g_{2}}(M^{1})=M^{1}/O^{\prime}_{g_{1}g_{2},g_{2}}(M^{1})$.

Now we recall [Zhu22] the $A_{g_{1}g_{2}}(V)$-$A_{g_{2}}(V)$-bimodule on
$A_{g_{1}g_{2},g_{2}}(M)$. For homogeneous $u\in V$ and $w\in M$, the left and
right bimodule actions are defined as

(56)
${u*_{g_{1}g_{2},g_{2}}w}=\begin{cases}{\text{Res}_{z}{Y_{M}(u,z)w\frac{(1+z)^{\operatorname{wt}u-1+\delta(j_{2})+\frac{j_{2}}{T}}}{z^{1-\frac{j_{1}}{T}}}}}&{j_{1}+j_{2}\equiv
0\;(\text{mod}\;\;T)}\\\ {0}&\text{otherwise},\end{cases}$

and

(57)
$w*_{g_{2},g_{1}g_{2}}u=\begin{cases}\text{Res}_{z}{Y_{M}(u,z)w\frac{(1+z)^{\operatorname{wt}u-1}}{z^{1-\frac{j_{1}}{T}}}}&{j_{2}=0}\\\
{0},&\text{otherwise}.\end{cases}$

In particular, when $g_{1}=id$, $g_{2}=g$, the $A_{g,g}(V)$ with the
multiplication given by (57) is the same as the $g$-twisted Zhu’s algebra,
$A_{g}(V)$ defined in [DLM00].

######  Conjecture 3.1.

[Zhu22] One has:

$\dim\hom_{A_{g_{1}g_{2}}(V)}(A_{g_{1}g_{2},g_{2}}(M^{1})\otimes
M^{2}(0),M^{3}(0))=N^{3}_{1\;2}.$

######  Remark 3.2.

When $g_{1}=g_{2}=id$, this Conjecture was proved by Frenkel and Zhu in
[FZ96].

Let $e,f,h$ be the basis of $\mathfrak{sl}_{2}$. Define
$\hat{\sigma}=e^{\frac{\pi ih_{(0)}}{2}}$. One can obtain irreducible
$\hat{\sigma}$-twisted ($\mathbb{Z}_{2}$-twisted) modules of
$L_{\mathfrak{sl}_{2}}(\ell,0)$ in category $\mathcal{O}$ by using Li’s Delta
operator:

$\Delta_{-\frac{1}{2}}(z)=z^{\frac{1}{4}h_{(0)}}\exp\left(\sum_{n=1}^{\infty}\frac{\frac{1}{4}h_{(n)}}{-n}(-z)^{-n}\right),$

i.e., $\sigma^{-\frac{1}{2}}(L_{\mathfrak{sl}_{2}}(\ell,j))$, where we define

$(\sigma^{-\frac{1}{2}}(M),Y^{-\frac{1}{2}}_{M}(\cdot,z)):=(M,Y_{M}(\Delta_{-\frac{1}{2}}(z)\cdot,z)).$

Now we compute the twisted Zhu’s bimodule in the case where $M^{1}$ is
untwisted and $M^{2},M^{3}$ are $\hat{\sigma}$-twisted modules. We have

(58)
$u\circ_{\hat{\sigma},\hat{\sigma}}w=\text{Res}_{z}{\frac{(1+z)^{\operatorname{wt}u-1+\delta(j_{2})+\frac{j_{2}}{2}}}{z^{1+\delta(j_{2})}}}Y_{M}(u,z)w,$

where $u\in V^{(0,j_{2})}$. If $u\in V_{1}^{(0,1)}$, one has

(59) $\displaystyle\begin{split}&u\circ_{\hat{\sigma},\hat{\sigma}}w\\\
&=\text{Res}_{z}{\frac{(1+z)^{\frac{1}{2}}}{z^{m+1}}}Y_{M}(u_{(-1)}\mathbf{1},z)w\\\
&=u(-m-1)w+\frac{1}{2}u(-m)w-\frac{1}{8}u(-m+1)w+\frac{1}{16}u(-m+2)w+\cdots\in
O_{\hat{\sigma},\hat{\sigma}}(M),\end{split}$

where $m\geq 0$.

In our case, we have the following bimodule action [DLM97]

######  Proposition 3.3.

The $A_{\hat{\sigma}}(V)$-bimodule
$A_{\hat{\sigma},\hat{\sigma}}(\sigma^{-\frac{1}{2}}(M))$ is isomorphic to
$\mathbb{C}[x,y]$ with bimodule action as follows:

(60) $\displaystyle x*f(x,y)=(x+j-2y\frac{\partial}{\partial y})f(x,y),\quad
f(x,y)*x=xf(x,y)$

for any $f(x,y)\in\mathbb{C}[x,y]$, where $h_{(0)}v=jv$.

###### Proof.

By Definition, we have

$\displaystyle h_{(-1)}*(h_{(-1)}^{m}f_{(0)}^{n}v)$
$\displaystyle=(h_{(-1)}+h_{(0)})h_{(-1)}^{m}f_{(0)}^{n}v$
$\displaystyle=(h_{(-1)}+j-2n)h_{(-1)}^{m}f_{(0)}^{n}v$

and

$\displaystyle(h_{(-1)}^{m}f_{(0)}^{n})*h_{(-1)}=h_{(-1)}(h_{(-1)}^{m}f_{(0)}^{n})=h_{(-1)}^{m+1}f_{(0)}^{n}v.$

The Proposition follows immediately if we set
$x=h_{(-1)}+O(M(\ell,j)),y=f(0)+O(M(\ell,j))$. ∎

###  3.1. Examples

Let $g$ be an automorphism of the universal affine vertex algebra
$V^{k}(\mathfrak{g})$ and denote by $\mathfrak{g}^{0}$ the fixed point
subalgebra of $\mathfrak{g}$ under the action of $g$. The $g$-twisted Zhu’s
algebra of $V^{k}(\mathfrak{g})$ is the universal enveloping algebra of
$\mathfrak{g}^{0}$, $U(\mathfrak{g}^{0})$, via the map [Yan17]

$\displaystyle F:A_{g}(V^{k}(\mathfrak{g}))\mapsto U(\mathfrak{g}^{0})$
$\displaystyle F([x^{1}_{(-n_{1}-1)}x^{2}_{(-n_{2}-1)}\cdots
x^{m}_{(-n_{m}-1)}\mathbf{1}])=(-1)^{n_{1}+n_{2}+\cdots+n_{m}}x^{m}x^{m-1}\cdots
x^{1},$

where $x^{1},x^{2},...,x^{m}\in\mathfrak{g}^{0}$. Moreover, $g$-twisted Zhu’s
algebra of a simple affine vertex algebra $L_{k}(\mathfrak{g})$ is
$U(\mathfrak{g}^{0})/\langle[U(\mathfrak{g})v_{{\text{sing}}}]\rangle$, where
$[v_{{\text{sing}}}]$ means the equivalence class of singular vector
$v_{sing}$ generating the maximal submodule of $V^{k}(\mathfrak{g})$ in
$A_{g}(L_{k}(\mathfrak{g})).$

Now we compute the $\hat{\sigma}$-twisted Zhu’s algebra
$A_{\hat{\sigma}}(L_{-\frac{4}{3}}(\mathfrak{sl}_{2}))$. We denote by
$L_{\mathfrak{sl}_{2}}(\ell,0)$ or $L(\ell,0)$ the vertex algebra
$L_{\ell}(\mathfrak{sl}_{2})$ at level $\ell$, and denote the admissible
irreducible $L_{\ell}(\mathfrak{sl}_{2})$-module by
$L_{\mathfrak{sl}_{2}}(\ell,j)$ or $L(\ell,j)$, where $j$ is the Dynkin label
of finite part of the admissible weight of $\widehat{\mathfrak{sl}_{2}}$. Note
that $A_{\hat{\sigma}}(L_{-\frac{4}{3}}(\mathfrak{sl}_{2}))$ coincides with
$A_{\hat{\sigma},\hat{\sigma}}(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0))$. By
above general argument, we have

$\displaystyle A_{\hat{\sigma}}(L_{-\frac{4}{3}}(\mathfrak{sl}_{2}))\cong
U(h)/\langle[U(\mathfrak{g})v_{{\text{sing}}}]\rangle.$

The weight zero singular vector is [CR13b]

(61) $\displaystyle
9h_{(-1)}^{3}+18h_{(-2)}h_{(-1)}-16h_{(-3)}-36f_{(-1)}h_{(-1)}e_{(-1)}-24e_{(-2)}f_{(-1)}+96f_{(-2)}e_{(-1)}\mathbf{1}.$

Using (59), the equivalence class of (61) in
$A_{\hat{\sigma}}(L_{-\frac{4}{3}}(\mathfrak{sl}_{2}))$ can be written in
terms of $h$. To that end, for $[f_{(-1)}h_{(-1)}e_{(-1)}]$ we have:

$\displaystyle[f_{(-1)}h_{(-1)}e_{(-1)}]$
$\displaystyle=[(-\frac{1}{2}f_{(0)}+\frac{1}{8}f_{(1)}-\frac{1}{16}f_{(2)}+\cdots)h_{(-1)}e_{(-1)}]$
$\displaystyle=[\frac{1}{2}h_{(-1)}^{2}-\frac{1}{2}h_{(-1)}+\frac{1}{6}-\frac{5}{12}h_{(-1)}+\frac{1}{6}]$
$\displaystyle=[\frac{1}{2}h_{(-1)}^{2}-\frac{11}{12}h_{(-1)}+\frac{1}{3}];$

for $[e_{(-2)}f_{(-1)}]$ we have:

$\displaystyle[e_{(-2)}f_{(-1)}]$
$\displaystyle=[(-\frac{1}{2}e_{(-1)}+\frac{1}{8}e_{(0)}-\frac{1}{16}e_{(1)}+\cdots)f_{(-1)}]$
$\displaystyle=[\frac{3}{8}h_{(-1)}+\frac{1}{6}];$

for $[f_{(-2)}e_{(-1)}]$ we have:

$\displaystyle[f_{(-2)}e_{(-1)}]=[-\frac{3}{8}h_{(-1)}+\frac{1}{6}].$

Combining everything together, we get (61) equals

(62)
$\displaystyle[9h_{(-1)}^{3}+18h_{(-2)}h_{(-1)}-16h_{(-3)}+18h_{(-1)}^{2}+12h_{(-1)}].$

The image of $(\ref{sing21})$ under $F$ is

$\displaystyle h(h+\frac{2}{3})(h-\frac{2}{3}).$

Thus,
$A_{\hat{\sigma}}(L_{-\frac{4}{3}}(\mathfrak{sl}_{2}))\cong\mathbb{C}[h]/\langle
h(h+\frac{2}{3})(h-\frac{2}{3})\rangle$.

Consider the $\hat{\sigma}$-invariant subspace of $M^{1}$ and $M^{2}$. One has
usual Zhu’s bimodule
$A((M^{i})^{\hat{\sigma}})=(M^{i})^{\hat{\sigma}}/O((M^{i})^{\hat{\sigma}})$,
where $i=2,3$. From above definition, we have

$\displaystyle
A_{\hat{\sigma},\hat{\sigma}}(M^{2})=M^{2}/O\left((M^{2})^{\hat{\sigma}}\right),$
$\displaystyle
A_{\hat{\sigma},\hat{\sigma}}(M^{3})=M^{3}/O\left((M^{3})^{\hat{\sigma}}\right).$

###  3.2. Relation between Dong-Li-Mason’s Zhu’s bimodules and twisted Zhu’s
bimodules

Let $\omega$ be the original Sugawara Virasoro vector of $L(\ell,0)$. Set
$\omega_{z}=\omega+\frac{1}{2}zh(-2)\mathbf{1}\in L(\ell,0),$ where $z$ is a
complex number. Then $\omega_{z}$ is a Virasoro vector of $L(\ell,0)$ with the
central charge $c_{\ell,z}=c_{\ell}-6\ell z^{2}$. Let $z$ be a positive
rational number less than $1$. Note that the vertex operator algebra
$(L(\ell,0),Y,\mathbf{1},\omega_{z})$ is $\mathbb{Q}$-graded instead of
$\mathbb{Z}$-graded. In [DLM97] authors extend the definition of Zhu’s
$A(V)$-theory of one-to-one correspondence between the set of equivalence
classes of irreducible admissible $V$-modules and the set of equivalence
classes of irreducible $A(V)$-modules and Zhu-Frenkel’s $A(M)$-theory for
fusion rules to any $\mathbb{Q}$-graded vertex operator algebra $V$. Denote
the Zhu’s algebra and bimodule of $V$ by
$A^{\text{dlm}}(V)=V/V\circ_{\text{dlm}}V$ and
$A^{\text{dlm}}(M)=M/V\circ_{\text{dlm}}M.$

Assume

$\exp\left(\sum_{n\geq
1}\frac{\frac{1}{4}h_{(n)}}{-n}(-z)^{-n}\right)=\sum_{n\geq 0}u_{(n)}z^{-n}.$

For brevity, we suppress the subscript of $\Delta_{-\frac{1}{2}}$ and denote
it by $\Delta$. Let $\epsilon(e)=\epsilon(f)=0$ and $\epsilon(h)=1$. We have

$\displaystyle\Delta(1)(a\circ_{\text{dlm}}m)$
$\displaystyle=\Delta(1)\text{Res}_{z=0}{\left(Y(a,z)\frac{(1+z)^{[\operatorname{wt}(a)_{\text{}}]}}{z^{1+\epsilon(a)}}m\right)}$
$\displaystyle=\text{Res}_{z=0}{\left(\Delta(1)Y(a,z)\frac{(1+z)^{[\operatorname{wt}(a)_{\text{}}]}}{z^{1+\epsilon(a)}}m\right)}$
$\displaystyle=\text{Res}_{z=0}{Y(\Delta(z+1)a,z)\frac{(1+z)^{[\operatorname{wt}(a)_{\text{}}]}}{z^{1+\epsilon(a)}}\Delta(1)m}$
$\displaystyle=\sum_{n\geq
0}\text{Res}_{z=0}{Y(u_{(n)}a,z)\frac{(1+z)^{[\operatorname{wt}(a)_{\text{}}]+\lambda-n}}{z^{1+\epsilon(a)}}\Delta(1)m}$
$\displaystyle=\sum_{n\geq
0}\text{Res}_{z=0}{Y(u_{(n)}a,z)\frac{(1+z)^{[\operatorname{wt}(u_{(n)}a)]+\lambda}}{z^{1+\epsilon(a)}}\Delta(1)m}$
$\displaystyle=(\Delta(1)a)\circ_{\hat{\sigma},\hat{\sigma}}(\Delta(1)m).$

where $\frac{h_{(0)}}{4}a=\lambda a.$ Using similar arguments, we get

$\displaystyle\Delta(1)(a*_{\text{dlm}}m)$
$\displaystyle=(\Delta(1)a)*_{\hat{\sigma},\hat{\sigma}}(\Delta(1)m),$
$\displaystyle\Delta(1)(m*_{\text{dlm}}a)$
$\displaystyle=(\Delta(1)m)*_{\hat{\sigma},\hat{\sigma}}(\Delta(1)a),$

where $*_{\text{dlm}}$ is the bimodule action defined by [DLM97]. Thus we have
the following result.

######  Proposition 3.4.

The map $V\rightarrow V,a\mapsto\Delta(1)a,$ induces an algebra isomorphism

$A^{\text{dlm}}(V)\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}A_{\hat{\sigma}}(V).$

The map $M^{1}\rightarrow M^{1},m\mapsto\Delta(1)m$ induces an
$A_{\hat{\sigma},\hat{\sigma}}(V)(\cong A^{\text{dlm}}(V))$-bimodule
isomorphism

$A^{\text{dlm}}(M^{1})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}A_{\hat{\sigma},\hat{\sigma}}(M^{1})$

######  Example 3.5.

Note $\Delta(1)h(-1)v=(h-\frac{2}{3})v$, and $\Delta(1)f(0)v=f(0)v$. If
$M^{1}=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})$, we have

$A_{\hat{\sigma},\hat{\sigma}}(M^{1})\cong\mathbb{C}[x,y]/\langle
y,(x-\frac{2}{3})x\rangle.$

If $M^{1}=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})$, we have

$A_{\hat{\sigma},\hat{\sigma}}(M^{1})\cong\mathbb{C}[x,y]/\langle
y,(x-\frac{2}{3})\rangle.$

Here, we identify $f(0)$ and $h(-1)$ with $y$ and $x$.

###  3.3. Fusion rules among twisted modules

According to [DLM97],

$A^{\text{dlm}}(L(\ell,0))\cong\mathbb{C}[x]/\langle\prod_{r=0}^{p-2}\prod_{s=0}^{q-1}(x-r+st)\rangle.$

Since $\Delta(1)h(-1)v=(h(-1)+\frac{1}{2}\ell)v$, by Proposition 3.4, we have

(63) $\displaystyle
A_{\hat{\sigma}}(L(\ell,0))\cong\mathbb{C}[x]/\langle\prod_{r=0}^{p-2}\prod_{s=0}^{q-1}(x+\frac{1}{2}\ell-r+st)\rangle.$

Similarly, since $\Delta(1)f(0)v=f(0)v$,
$A_{\hat{\sigma},\hat{\sigma}}(L_{\mathfrak{sl}_{2}}(\ell,j))$ is isomorphic
to the quotient space of $\mathbb{C}[x,y]$ modulo the subspace

$\mathbb{C}[x,y]y^{n}+\mathbb{C}[x]g_{j,0}(x,y)+\mathbb{C}[x]g_{j,1}(x,y)+\cdots+\mathbb{C}[x]g_{j,n-1}(x,y)$

where $g_{j,i}=y^{i}\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(x+\frac{1}{2}\ell-
r-i+st).$ The left and right actions of $A_{\hat{\sigma}}(L(\ell,0))$ is given
by (60).

######  Theorem 3.6.

For admissible weights $j_{i}=n_{i}-(k_{i}-1)t$ $(i=1,2)$, the fusion rules
are given as follows:

(64) $\displaystyle
L(\ell,j_{1})\times\sigma^{-\frac{1}{2}}(L(\ell,j_{2}))=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}\sigma^{-\frac{1}{2}}(L(\ell,j_{1}+j_{2}-2i))$

if $0\leq k_{2}-1\leq q-k_{1}$, and
$L(\ell,j_{1})\times\sigma^{-\frac{1}{2}}(L(\ell,j_{2}))=0$ otherwise.

###### Proof.

For any admissible weight $j$, let $\mathbb{C}v_{j}^{\prime}$ be the one-
dimensional module for Lie algebra $\mathbb{C}h$ such that
$hv_{j}^{\prime}=(j-\frac{1}{2}\ell)v_{j}^{\prime}.$ We then calculate the
$A(L(\ell,0))$-module
$A_{\hat{\sigma},\hat{\sigma}}(L(\ell,j_{1}))\otimes_{A_{\hat{\sigma}}(L(\ell,0))}\mathbb{C}v_{j_{2}}^{\prime}.$
Using the result above this Theorem, we have

$A_{\hat{\sigma},\hat{\sigma}}(L(\ell,j_{1}))\otimes_{A_{\hat{\sigma}}(L(\ell,0))}\mathbb{C}v_{j_{2}}^{\prime}\cong\mathbb{C}[x,y]/J$

where $J$ is the subspace of $\mathbb{C}[x,y]$ spanned by

$\\{x-j_{2}+\frac{1}{2}\ell,\mathbb{C}[x,y]y^{n_{1}},g_{j_{1},i}(j_{2},1)\mathbb{C}[x]y^{i},i=0,1,...,n_{1}-1\\}.$

Then the result follows from the similar argument of [DLM97, Theorem 4.7]. ∎

######  Example 3.7.

let $j_{i}=-k_{i}\frac{2}{3}$ $(i=1,2)$ where $k_{i}=0,1,2$, we have

(65) $\displaystyle
L_{\mathfrak{sl}_{2}}(-\frac{4}{3},j_{1})\times\sigma^{-\frac{1}{2}}(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},j_{2}))=\sigma^{-\frac{1}{2}}(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},j_{1}+j_{2}))$

for $0\leq k_{2}\leq 2-k_{1}.$

###  3.4. Verlinde formula for $L_{-\frac{4}{3}}(\mathfrak{sl}_{2})$

Let
$\\{\lambda_{1}:=-\frac{4}{3}\Lambda_{0},\lambda_{2}:=-\frac{2}{3}\Lambda_{0}-\frac{2}{3}\Lambda_{1},\lambda_{3}:=-\frac{4}{3}\Lambda_{1}\\}$.
The $S$-transformation is given by

$\chi_{\lambda_{i}}(-\frac{1}{\tau})=\sum_{j=1}^{3}a_{ij}\sigma^{-\frac{1}{2}}(\chi_{\lambda_{j}}(\tau))$
$(a_{ij})=\frac{\sqrt{3}}{3}\left({\begin{array}[]{ccc}-1&1&-1\\\
1&-e^{-\frac{2}{3}\pi i}&e^{\frac{2}{3}\pi i}\\\ -1&e^{\frac{2}{3}\pi
i}&-e^{-\frac{2}{3}\pi i}\\\ \end{array}}\right).$

The fusion rules are

$N_{0i}^{j}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{array}\right),\;\;N_{1i}^{j}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
0&0&0\\\ \end{array}\right),\;\;N_{2i}^{j}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 0&0&0\\\ \end{array}\right),$

while the Verlinde formula and $S$-matrix would give us the following fusion
rules:

$N_{0i}^{j}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{array}\right),\;\;N_{1i}^{j}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
-1&0&0\\\ \end{array}\right),\;\;N_{2i}^{j}=\left(\begin{array}[]{ccc}0&0&1\\\
-1&0&0\\\ 0&-1&0\\\ \end{array}\right).$

The cause of the negative coefficients in the above fusion rules is explained
in [CR13b]. In order to obtain the positive coefficients, one need include
more objects in our category, i.e., atyptical and typical admissible modules
and take the convergent regions of the characters of these modules into
consideration.

##  4\. Twisted Zhu’s bimodules of contragredient modules of highest weight
modules

###  4.1. Motivation

Given an admissible irreducible highest weight
$L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0)$-module
$L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})$, one can obtain the
ordinary twisted modules and contragredient modules by taking integer and
half-integer spectral flow respectively (Figure 1). In general, let
$\\{M_{i}\\}$ be the collection of all highest weight modules for
$L_{\mathfrak{sl}_{2}}(\ell,0)$ at the admissible level, one can obtain all
the $\mathbb{Z}_{2}$-twisted modules of $L_{\mathfrak{sl}_{2}}(\ell,0)$ at the
admissible level either from $\\{\sigma^{-\frac{1}{2}}(M_{i})\\}$ or
$\\{\sigma^{\frac{1}{2}}(M_{i}^{*})\\}$.

$|\frac{-4}{3},\frac{-1}{3}\rangle$$|0,0\rangle$$|\frac{2}{3},\frac{-1}{12}\rangle$$|\frac{4}{3},\frac{-1}{3}\rangle$$\sigma$$\sigma^{-\frac{3}{2}}$$\sigma^{\frac{1}{2}}$$\sigma$$\mathrm{contragredient}$
Figure 1. The relation among admissible irreducible highest weight
$L(-4/3,0)$-modules, contragredient modules and ordinary
$\mathbb{Z}_{2}$-twisted modules, where each state labelled by
$|\lambda,\Delta\rangle$, its $\mathfrak{sl}_{2}$-weight $\lambda$ and
conformal dimension $\Delta$.

In this Section, we will calculate the twisted Zhu’s bimodules of the
contragredient modules of highest weight modules and the fusion rules among
them. The idea is to use the similar isomorphism in Proposition 3.4. To that
end, we calculate the untwisted Zhu’s bimodule first and fusion rules among
untwisted modules.

###  4.2. Untwisted Zhu’s bimodules

Let $M$ be a $V$-module. Let $O(M)$ be the linear span of elements of type

$\text{Res}_{z}{\left(Y(a,z)\frac{(z+1)^{\operatorname{wt}a}}{z^{2}}v\right)}.$

In particular, for $V=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0)$. The $O(M)$ is
spanned by

$\displaystyle\text{Res}_{z}{\left(Y(e,z)\frac{(z+1)}{z^{m+1}}\right)}v$
$\displaystyle=(e(-m-1)+e(-m))v,$
$\displaystyle\text{Res}_{z}{\left(Y(f,z)\frac{(z+1)}{z^{m+1}}\right)}v$
$\displaystyle=(f(-m-1)+f(-m))v,$
$\displaystyle\text{Res}_{z}{\left(Y(h,z)\frac{(z+1)}{z^{m+1}}\right)}v$
$\displaystyle=(h(-m-1)+h(-m))v,$

for any positive integer $m$ and for $v\in M$.

By [FZ96], one has the following isomorphism $F$

$\displaystyle F:L(j)\otimes U(\mathfrak{g})$ $\displaystyle\rightarrow
A(V_{\mathfrak{g}}(\ell,j)),$ $\displaystyle F:v\otimes a_{1}\cdots a_{n}$
$\displaystyle\mapsto[a_{n}(-1)\cdots a_{1}(-1)v].$

whose inverse is given by

$\displaystyle F^{-1}:A(V_{\mathfrak{g}}(\ell,j))$ $\displaystyle\rightarrow
L(j)\otimes U(\mathfrak{g}),$ $\displaystyle F^{-1}:[a_{1}(-1-i_{1})\cdots
a_{n}(-1-i_{n})v]$ $\displaystyle\rightarrow(-1)^{i_{1}+\cdots+i_{n}}v\otimes
a_{n}\cdots a_{1},$

where the tensor products are over $U(\mathfrak{g})$.

Consider the example of $L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0)$. It has three
irreducible modules in category $\mathcal{O}$, i.e.,
$L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0),L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})$,
$L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})$. By [MFF86] the maximal
submodule of $V_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})$ is generated
by singular vectors $v_{1}$ and $v_{2}$,

(66) $\displaystyle
v_{1}=\frac{2}{9}e_{(-2)}-\frac{1}{3}e_{(-1)}h_{(-1)}+e_{(-1)}e_{(-1)}f_{(0)}$
(67) $\displaystyle
v_{2}=-\frac{10}{9}f_{(-1)}-\frac{5}{3}h_{(-1)}f_{(0)}+e_{(-1)}f_{(0)}^{2},$

and the maximal submodule of
$V_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})$ is generated by singular
vectors $e_{(-1)}$ and

$\displaystyle\frac{280}{81}f_{(-2)}+\frac{70}{27}h_{(-2)}f_{(0)}-\frac{10}{9}e_{(-2)}f_{(0)}^{2}$
(68)
$\displaystyle+\frac{140}{27}h_{(-1)}f_{(-1)}+\frac{35}{9}h_{(-1)}^{2}f_{(0)}-\frac{5}{3}h_{(-1)}e_{(-1)}f_{(0)}^{2}$
$\displaystyle-\frac{70}{9}e_{(-1)}f_{(-1)}f_{(0)}-\frac{10}{3}e_{(-1)}h_{(-1)}f_{(0)}^{2}+e_{(-1)}^{2}f_{(0)}^{3}$

We next use the singular vectors to compute
$A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})),A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})).$
For $A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3}))$, the preimage of
equivalence classes of $v_{1}$ and $v_{2}$ are

(69) $\displaystyle-\frac{2}{9}v\otimes e-\frac{1}{3}v\otimes he+fv\otimes
e^{2},$ (70) $\displaystyle-\frac{10}{9}v\otimes f-\frac{5}{3}fv\otimes
h+f^{2}v\otimes e.$

Thus

$A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3}))=(L(-\frac{2}{3})\otimes
U(\mathfrak{sl}_{2}))/I_{1},$

where $I_{1}$ is generated by (69) and (70), and the tensor products are over
$U(\mathfrak{g})$. For $A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3}))$,
the preimage of equivalence classes of (68) is

$\displaystyle-\frac{280}{81}v\otimes f-\frac{70}{27}fv\otimes
h+\frac{10}{9}f^{2}v\otimes e$ $\displaystyle+\frac{140}{27}v\otimes
hf+\frac{35}{9}fv\otimes h^{2}$ $\displaystyle-\frac{5}{3}f^{2}v\otimes
eh-\frac{70}{9}fv\otimes fe-\frac{10}{3}f^{2}v\otimes he+f^{3}v\otimes e^{2}.$

Denote by $L(-\frac{2}{3})^{*}$ and $L(-\frac{4}{3})^{*}$ the dual of highest
weight $\mathfrak{sl}_{2}$-modules with weights $-\frac{2}{3}$ and
$-\frac{4}{3}$. We now consider the
$A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0))$-module $L(-\frac{2}{3})^{*}\otimes
A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3}))$. Let $v^{\prime}$ be the
lowest weight vector of $L(-\frac{2}{3})^{*}$. Then $I_{1}\otimes
v^{\prime}\cong\langle-\frac{10}{9}v\otimes ev^{\prime}+fv\otimes
e^{2}v^{\prime},-\frac{10}{9}fv\otimes v^{\prime}+f^{2}v\otimes
ev^{\prime}\rangle.$ It is isomorphic to

$A(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3}))\otimes
L(-\frac{2}{3})^{*}\cong\mathbb{C}(v\otimes v^{\prime}).$

Thus we have

$L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})\times\left(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})\right)^{*}=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0).$

Similarly, we also have

(71) $\displaystyle
L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})\times\left(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})\right)^{*}=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},0),$
(72) $\displaystyle
L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})\times\left(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})\right)^{*}=\left(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})\right)^{*},$
(73) $\displaystyle
L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{4}{3})\times\left(L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3})\right)^{*}=L_{\mathfrak{sl}_{2}}(-\frac{4}{3},-\frac{2}{3}),$

###  4.3. The contragredient modules of the highest weight modules

As one can see from above example, directly computing Zhu’s bimodules depends
on the explicit form of singular vectors. In practice, it is extremely tedious
to convert the singular vectors given in [MFF86] into their normal forms. It
was noted in [DLM97] that the fusion rules among the admissible modules remain
the same after a shift of the conformal vector. By a proper shift of the
conformal vector, there are nice and compact projection formulas for singular
vectors ([Fuc89], [DLM97]) which can help us to compute fusion rules avoiding
finding explicit form of singular vectors.

Since

(74) $\displaystyle[f(0),e(0)^{\gamma}]=-(\gamma
e(0)^{\gamma-1}+\gamma(\gamma-1)e(0)^{\gamma-1}),$ (75)
$\displaystyle[h(0),e(0)^{\gamma}]=2\gamma e(0)^{\gamma},$

then by using the similar argument as in [MFF86] we have

######  Proposition 4.1.

Let $j=n-1-(k-1)$ where $n$ and $k$ are positive integers satisfying $1\leq
n\leq p-1$, $1\leq k\leq q$ and let $v$ be a highest weight vector of the
Verma module $(M(\ell,j))^{*}$. Set

$\displaystyle
E_{1}(n,k)=e(0)^{n+(k-1)t}f(-1)^{n+(k-2)t}e(0)^{n+(k-3)t}f(-1)^{n+(k-4)t}\cdots
f(-1)^{n-(k-2)t}e(0)^{n-(k-1)t},$ $\displaystyle
E_{2}(n,k)=f(-1)^{p-n+(q-k)t}e(0)^{p-n+(q-k-1)t}f(-1)^{p-n+(q-k-2)t}e(0)^{p-n+(q-k-3)t}$
$\displaystyle\quad\cdots e(0)^{p-n-(q-k+1)t}f(-1)^{p-n-(q-k)t}.$

Then $v_{-j,1}=E_{1}(n,k)v,v_{-j,2}=E_{2}(n,k)v$ are singular vectors of
$(M(\ell,j))^{*}$.

Basically, we just interchange $e$ and $f$ based on the corresponding result
in the case of highest weight modules.

Next, we consider the projection formula. Let $P_{1}$ be the projection
$\hat{\mathfrak{g}}$ onto $\mathfrak{g}$ such that $P_{1}(a\otimes t^{n})=a$
for any $a\in\mathfrak{g}$ and $P_{1}(c)=0$. Let $H_{\alpha}=fe-\alpha
h-\alpha(\alpha+1)$.

######  Proposition 4.2.

[MFF86] The following projection formulas hold:

$\displaystyle
P_{1}(E_{1}(n,k))=\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}H_{-r-st}\right)e^{n}$
$\displaystyle
P_{1}(E_{2}(n,k))=\left(\prod_{r=0}^{p-n-1}\prod_{s=1}^{q-k}H_{r+st}\right)f^{p-n}.$

Define subalgebra

$T_{-}=\mathbb{C}e+t^{-1}\mathbb{C}[x^{-1}]\otimes\mathfrak{g}.$

Let $B_{0}=\mathbb{C}(t^{-1}+1)\otimes
e+(x^{-2}+x^{-1})\mathbb{C}[x^{-1}]\otimes\mathfrak{g}.$ Since $B_{0}$ is an
ideal of $N_{-}$, $U(N_{-})B_{0}=B_{0}U(N_{-})$ is an ideal of $U(N_{-})$. Set
$L_{0}=N_{-}/B_{0}$. Define

$T_{+}=e(0)+B_{0},T_{-}=f(-1)+B_{0},T_{0}=h(-1)+B_{0}.$

They obey the following $\mathfrak{sl}_{2}$-relationships

$[T_{0},T_{+}]=-2T_{+},\;\;[T_{0},T_{-}]=2T_{-},\;\;[T_{+},T_{-}]=T_{0}.$

Define $G_{\alpha}=T_{-}T_{+}-\alpha T_{0}+\alpha(\alpha+1)$. They satisfy the
following relationships

$\displaystyle
G_{\alpha}G_{\beta}=G_{\beta}G_{\alpha},\;\;\;T_{+}^{m}G_{\alpha}=G_{\alpha-m}T_{+}^{m},\;\;\;T_{-}^{m}G_{\alpha}=G_{\alpha+m}T_{-}^{m},$
$\displaystyle T_{-}^{m}T_{+}^{m}=G_{0}G_{1}\cdots
G_{m-1},\;\;\;T_{+}^{m}T_{-}^{m}=G_{-1}G_{-2}\cdots G_{-m},$

for any complex numbers $\alpha,\beta$ and for any positive integer $m$.

Let $P$ be the natural quotient map from $U(N_{-})$ onto $U(L_{0})$. Using the
similar method as suggested in [MFF86] we obtain

######  Proposition 4.3.

The following formulas hold:

$\displaystyle
P(E_{1}(n,k))=\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}G_{-r-st}\right)T_{+}^{n}$
$\displaystyle
P(E_{2}(n,k))=\left(\prod_{r=0}^{p-n-1}\prod_{s=1}^{q-k}G_{r+st}\right)T_{-}^{p-n}.$

###  4.4. Fusion rules

For the contragredient modules of the highest weight modules, we choose the
new conformal vector $\omega_{z}=\omega-\frac{1}{2}zh(-2)\bf{1}$, where
$0<z<1$.

Let $M$ be any weak $V(\ell,\mathbb{C})$-module. Since

$\operatorname{wt}h(-1)=1,\;\;\operatorname{wt}e(-1)=1+z,\;\;\operatorname{wt}f(-1)=1-z,$

we have

$\displaystyle\text{Res}_{z}{\frac{(1+z)^{[\operatorname{wt}f]}}{z^{m}}Y(f,z)u}$
$\displaystyle=f(-m)u$
$\displaystyle\text{Res}_{z}{\frac{(1+z)^{[\operatorname{wt}e]}}{z^{m}}Y(e,z)u}$
$\displaystyle=(e(-m)+e(1-m))u,$
$\displaystyle\text{Res}_{z}{\frac{(1+z)^{\operatorname{wt}h}}{z^{m+1}}Y(h,z)u}$
$\displaystyle=(h(-m-1)+h(-m))u$

for any positive integer $m$ and for $u\in M$.

######  Proposition 4.4.

Let $j=n-1-(k-1)t$ be an admissible weight. Then the $A(L(\ell,0))$-bimodule
$A((L(\ell,j))^{*})$ is isomorphic to the quotient space of $\mathbb{C}[x,z]$
modulo the subspace

$\mathbb{C}[x,z]z^{n}+\mathbb{C}[x]f^{\prime}_{j,0}(x,z)+\mathbb{C}[x]f^{\prime}_{j,1}(x,z)+\cdots+\mathbb{C}[x]f^{\prime}_{j,n-1}(x,z)$

where
$f^{\prime}_{j,i}(x,z)=z^{i}\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(x+r+i-st).$
The left and right actions of $A(L(\ell,0))$ on $A(L(\ell,j)^{*})$ are given
by (60).

###### Proof.

First, $(M(\ell,j))^{*}\cong U(N_{-})$ as a vector space.

We have

$O((M(\ell,j))^{*})\cong f(-1)U(N_{-})+B_{0}U(N_{-}).$

Recall $v_{-j,1},v_{-j,2}$ are two singular vectors of $(M(\ell,j))^{*}$. Then
we have

$\displaystyle
A((L(\ell,j))^{*})\cong(M(\ell,j))^{*}/(O((M(\ell,j))^{*}+U(N_{-})v_{-j,1}+U(N_{-})v_{-j,2})$
$\displaystyle\cong
U(N_{-})/(B_{0}U(N_{-})+f(-1)U(N_{-})+U(N_{-})E_{1}(n,k)+U(N_{-})E_{2}(n,k))$

as $A((L(\ell,0))^{*})$-bimodules. Note that $U(N_{-})/B_{0}U(N_{-})\cong
U(L_{0})$. Thus

$A((L(\ell,j))^{*})\cong
U(L_{0})/(U(L_{0})P(E_{1}(n,k))+U(L_{0})P(E_{2}(n,k))+T_{-}U(L_{0})).$

For any nonnegative integers $a,b,d,$ using above relationships, we have (See
Appendix B for the detail)

(76)
$\displaystyle\begin{split}&\;\;\;T_{-}^{a}T_{0}^{b}T_{+}^{d}P(E_{1}(n,k))\\\
&=T_{-}^{a}\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}(r+st+d)(T_{0}+r+st+d-1)\right)T_{0}^{b}T_{+}^{n+d}\;\;\;\mod
T_{-}U(L_{0}).\end{split}$

Noticing that $r+st+d\neq 0$ for any $1\leq r\leq n$, $1\leq s\leq k-1$,
$d\in\mathbb{Z}_{+}$ we obtain

$\displaystyle\begin{split}&\;\;\;U(L_{0})P(E_{1}(n,k))+T_{-}U(L_{0})\\\
&=T_{-}U(L_{0})+\sum_{d=0}^{\infty}\mathbb{C}[T_{0}]\left(\prod_{r=0}^{n-1}\prod_{s=1}^{k-1}(T_{0}+r+st+d)\right)T_{+}^{n+d}.\end{split}$

Similarly, let $a,b,d$ be any nonnegative integers. If $d<p-n$, we have (see
Appendix B for details)

(77)
$\displaystyle\begin{split}&\;\;\;T_{-}^{a}T_{0}^{b}T_{+}^{d}P(E_{2}(n,k))\\\
&=T_{-}^{a+p-n-d}(T_{0}+2(p-n-d))^{b}\prod_{r=0}^{p-n-1}\prod_{s=1}^{q-k}\prod_{i=1}^{d}G_{r+st-p+n}G_{-i-p+n+d}\;\;\;\mod
T_{-}U(L_{0}).\end{split}$

If $d=m+p-n$ for some $m\in\mathbb{Z}_{+}$, we have

$\displaystyle\;\;\;T_{-}^{a}T_{0}^{b}T_{+}^{d}P(E_{2}(n,k))$
$\displaystyle=T_{-}^{a}\prod_{r=1}^{p-n}\prod_{s=0}^{q-k}(st-m-r)(-T_{0}+st-
m-r+1)T_{0}^{b}T_{+}^{m}\;\;\;\mod T_{-}U(L_{0}).$

Since $-st+m+r-1\neq 0$ for any $1\leq r\leq p-n$, $0\leq s\leq q-k$, we
obtain

$\displaystyle\;\;\;U(L_{0})P(E_{2}(n,k))+T_{+}U(L_{0})$
$\displaystyle=T_{-}U(L_{0})+\sum_{m=0}^{\infty}\mathbb{C}[T_{0}]\left(\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(T_{0}-st+m+r)\right)T_{+}^{m}.$

Thus

$\displaystyle\;\;\;U(L_{0})P(E_{1}(n,k))+U(L_{0})P(E_{2}(n,k))+T_{-}U(L_{0})$
$\displaystyle\subset
T_{+}U(L_{0})+U(L_{0})T_{+}^{n}+\sum_{i=0}^{n-1}\mathbb{C}[T_{0}]\left(\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(T_{0}-st+m+r)\right)T_{+}^{m}.$

On the other hand, since $-r-st-d\neq s^{\prime}t^{\prime}-m-r^{\prime}$ for
any $0\leq r\leq n-1,1\leq s\leq k-1$, $0\leq r\leq p-n-1,0\leq s\leq q-k$,
$d,m\in\mathbb{Z}_{+}$, $\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(x-st+m+r)$ and
$\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(x-st+m+r)$ are relatively prime. Then we
obtain

$\mathbb{C}[T_{0}]T_{+}^{n+i}\subset
U(L_{0})P(E_{1}(n,k))+U(L_{0})P(E_{2}(n,k))+T_{-}U(L_{0})$

for any $i\in\mathbb{Z}_{+}$. This shows that

$\displaystyle U(L_{0})P(E_{1}(n,k))+U(L_{0})P(E_{2}(n,k))+T_{-}U(L_{0})$
$\displaystyle\supset
T_{-}U(L_{0})+U(L_{0})T_{+}^{n}+\sum_{i=0}^{n-1}\mathbb{C}[T_{0}]\left(\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}(T_{0}-st+m+r)\right)T_{+}^{i}.$

Set $x=T_{0}$, $y=T_{-}$. Then the Proposition follows from the similar
argument in Proposition 4.3, Lemma 4.5 ([DLM97]). ∎

We have the fusion rules:

######  Theorem 4.5.

For admissible weight $j_{i}=n_{i}-1-(k_{i}-1)t$ $(i=1,2)$, the fusion rules
are given as follows:

(78) $\displaystyle L(\ell,j_{1})\times
L(\ell,j_{2})=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}L(\ell,j_{1}+j_{2}-2i),$
(81) $\displaystyle(L(\ell,j_{1}))^{*}\times
L(\ell,j_{2})=L(\ell,j_{2})\times(L(\ell,j_{1}))^{*}=\left\\{\begin{array}[]{ll}L(\ell,-j_{1}+j_{2}),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\ (L(\ell,j_{1}-j_{2}))^{*},&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$ (82)
$\displaystyle(L(\ell,j_{1}))^{*}\times(L(\ell,j_{2}))^{*}=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}(L(\ell,j_{1}+j_{2}-2i))^{*}.$

###### Proof.

(78) was proved in [DLM97]. We use the similar method to prove (81) and (82).

We prove the (81). For any admissible weight $-j$, let $\mathbb{C}v_{-j}$ be
the one-dimensional module for Lie algebra $\mathbb{C}h$ such that
$hv_{-j}=-jv_{-j}$. Then $\mathbb{C}v_{-j}$ is the lowest weight space of
$L(\ell,-j)$. By Frenkel-Zhu’s Theorem, we need calculate the
$A(L(\ell,0))$-module
$A(L(\ell,-j_{1}))\otimes_{A(L(\ell,0))}\mathbb{C}v_{j_{2}}.$ Note
$ev_{j_{2}}=0$. We get

$A(L(\ell,-j_{1}))\otimes_{A(L(\ell,0))}\mathbb{C}v_{j_{2}}\cong\mathbb{C}[x,z]/J$

where $J$ is the subspace of $\mathbb{C}[x,z]$ spanned by

$\\{x-j_{2},z\\}.$

Thus, $\mathbb{C}[x,z]/J\cong v_{-j_{1}}\otimes v^{\prime}_{j_{2}}.$ And
$x*(v_{-j_{1}}\otimes v^{\prime}_{j_{2}})=j_{2}-j_{1}$, as required.

For (82), let $\mathbb{C}v_{-j}$ be the one-dimensional module for Lie algebra
$\mathbb{C}h$ such that $hv_{-j}=-jv_{-j}$. Using Proposition 4.4 we get

$A(L(\ell,j_{1}))\otimes_{A(L(\ell,0))}\mathbb{C}v_{-j_{2}}\cong\mathbb{C}[x,z]/J$

where $J$ is the subspace of $\mathbb{C}[x,z]$ spanned by

$\\{x+j_{2},\mathbb{C}[x,z]z^{n_{1}},f^{\prime}_{j_{1},i}(-j_{2},1)\mathbb{C}[x]z^{i},i=0,1,...,n_{1}-1\\}.$

If $j_{2}$ does not satisfy the relation $0\leq k_{2}-1\leq q-k_{1}$, then

$f^{\prime}_{j_{1},i}(-j_{2},1)=\prod_{r=0}^{p-n_{1}-1}\prod_{s=0}^{q-k_{1}}(-j_{2}+r+i-st)\neq
0$

for $0\leq i\leq n_{1}-1.$ Thus
$A(L(\ell,-j_{1}))\otimes_{A(L(\ell,0))}\mathbb{C}v_{j_{2}}=0$ so that all the
corresponding fusion rules are zero.

Suppose $0\leq k_{2}-1\leq q-k_{1}.$ As before $\mathbb{C}[x]z^{i}=0$ in
$\mathbb{C}[x,z]/J$ if $f^{\prime}_{j_{1},i}(j_{2},1)\neq 0.$ Noticing
$f^{\prime}_{j_{1},i}(j_{2},1)=0$ if and only if $-j_{2}+r+i-st=0$ for some
$0\leq r\leq p-n_{1}-1$ and $0\leq s\leq q-k_{1}$. This implies that
$r+i=n_{2}-1$. Thus $n_{1}+n_{2}-p\leq i\leq n_{2}-1.$ Therefore

$\max\\{0,n_{1}+n_{2}-p\\}\leq i\leq\min\\{n_{1}-1,n_{2}-1\\}.$

If $n_{1}+n_{2}-p\leq i\leq n_{2}-1$, then $\mathbb{C}[x]z^{i}$ is not zero in
$\mathbb{C}[x,z]/J$. Thus

$\mathbb{C}[x,z]/J\cong\oplus_{\max\\{0,n_{1}+n_{2}-p\\}\leq
i\leq\min\\{n_{1}-1,n_{2}-1\\}}\mathbb{C}y^{i}.$

From (60) we get $x*z^{i}=(-j_{2}-j_{1}+2i)z^{i}$, as required.

∎

###  4.5. Fusion rules among twisted modules

We follow the same notations as in previous subsection. Let
$\hat{\sigma}^{\prime}=e^{-\frac{\pi ih_{(0)}}{2}}$. By using the similar
arguments as in Section 3, one has

######  Theorem 4.6.

  * •

We have the following isomorphism

$A(L(\ell,j)^{*})\cong
A_{\hat{\sigma}^{\prime},\hat{\sigma}^{\prime}}(L(\ell,j)^{*}),$

via $\Delta_{\frac{1}{2}}(1)$.

  * •

We also have

(83)
$\displaystyle(L(\ell,j_{1}))^{*}\times\sigma^{\frac{1}{2}}((L(\ell,j_{2}))^{*})=\sum_{i=\max\\{0,n_{1}+n_{2}-p\\}}^{\min\\{n_{1}-1,n_{2}-1\\}}\sigma^{\frac{1}{2}}((L(\ell,j_{1}+j_{2}-2i))^{*}).$
(86)
$\displaystyle(L(\ell,j_{1}))^{*}\times\sigma^{\frac{1}{2}}(L(\ell,j_{2}))=\left\\{\begin{array}[]{ll}\sigma^{\frac{1}{2}}(L(\ell,-j_{1}+j_{2})),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\
\sigma^{\frac{1}{2}}((L(\ell,j_{1}-j_{2}))^{*}),&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$ (89) $\displaystyle
L(\ell,j_{2})\times\sigma^{-\frac{1}{2}}((L(\ell,j_{1}))^{*})=\left\\{\begin{array}[]{ll}\sigma^{-\frac{1}{2}}(L(\ell,-j_{1}+j_{2})),&\hbox{if
$n_{2}-n_{1}\geq 0$;}\\\
\sigma^{-\frac{1}{2}}((L(\ell,j_{1}-j_{2}))^{*}),&\hbox{if
$n_{2}-n_{1}<0$.}\end{array}\right.$

##  5\. Twisted modules from spectral flow and their MLDEs

In [Li23], we have the following result

######  Theorem 5.1.

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

In this section and the next, we shall provide examples for this Theorem,
i.e., twisted modules coming from spectral flow. We also discuss their
applications in physics.

Firstly, let us review some useful facts about modular forms and modular
differential operators. See any standard reference, or [KNS13] for further
details. The ordinary Eisenstein series are modular forms for the full modular
group $\Gamma$ of weight $2k$ with $k\geq 2$. We define our Eisenstein series,
following the notation of [BR18],

(90)
$\mathbbm{E}_{k}(\tau)=-\frac{B_{2k}}{2k!}+\frac{2}{(2k-1)!}\sum_{n=1}^{\infty}\frac{n^{2k-1}q^{n}}{1-q^{n}}$

where $B_{2k}$ is the $2k$’th Bernoulli number. The ring of modular forms for
the full modular group $\Gamma$ is freely generated by $\mathbbm{E}_{4}(\tau)$
and $\mathbbm{E}_{6}(\tau)$, so we have,

(91)
$\bigoplus_{k=0}^{\infty}M_{k}(\Gamma,\mathbb{C})=\mathbb{C}[\mathbbm{E}_{4}(\tau),\mathbbm{E}_{6}(\tau)]$

We also make use of a class of twisted Eisenstein series that are modular
forms for certain congruence subgroups of $\Gamma$,

(92) $\mathbbm{E}_{k}\begin{bmatrix}\varphi\\\
\vartheta\end{bmatrix}(\tau)\equiv-\frac{B_{k}(\lambda)}{k!}+\frac{1}{(k-1)!}\sum_{r\geq
0}{}^{\prime}\frac{(r+\lambda)^{k-1}\vartheta^{-1}q^{r+\lambda}}{1-\vartheta^{-1}q^{r+\lambda}}+\frac{(-1)^{k}}{(k-1)!}\sum_{r\geq
1}\frac{(r-\lambda)^{k-1}\vartheta q^{r-\lambda}}{1-\vartheta q^{r-\lambda}}$

where $\varphi=e^{2\pi i\lambda}$ with $\lambda\in[0,1)$ and now $B_{k}(x)$ is
the $k$’th Bernoulli polynomial. The prime in the first summation means that
the $r=0$ term should be omitted when $\varphi=\vartheta=1$. The spaces of
modular forms for $\Gamma(2)$, $\Gamma^{0}(2)$ all admit a simple description
in terms of theta functions. For example,

(93)
$M_{2k}(\Gamma^{0}(2))=\text{span}_{\mathbb{C}}\left\\{\bar{\Theta}_{r,s}(\tau)|r+s=k\right\\}$

where the $\bar{\Theta}_{r,s}$ takes the following form,

(94)
$\bar{\Theta}_{r,s}(\tau):=\theta_{2}(\tau)^{4r}\theta_{3}(\tau)^{4s}+\theta_{2}(\tau)^{4s}\theta_{3}(\tau)^{4r},\qquad
r\leq s.$

We define $k$’th order modular differential operators $D^{(k)}_{q}$ as

(95)
$D^{(k)}_{q}\chi(q):=\partial_{(2k-2)}\circ\dots\circ\partial_{(2)}\circ\partial_{(0)}\chi(q),$

then modular linear differential operators that are holomorphic and monic have
the following generic form,

(96) $\mathcal{D}^{(k)}_{q}\equiv
D^{(k)}_{q}+\sum_{r=1}^{k}f_{r}(q)D^{(k-r)}_{q},\quad f_{r}(q)\in
M_{2k}(\tilde{\Gamma},\mathbb{C})$

where $\tilde{\Gamma}$ denote any congruence subgroup of $\Gamma$.

###  5.1. $A^{(1)}_{1}$ at boundary admissible levels $k=-2+\frac{2}{u}$

In this section, we give some specific MLDEs of irreducible
$\mathbb{Z}_{2}$-twisted modules for $A^{(1)}_{1}$ at boundary admissible
level $k=-2+\frac{2}{u}$. Following from Proposition 2.9 and (63), we have:

######  Theorem 5.2.

All irreducible $\mathbb{Z}_{2}$-twisted modules of $L_{k}(\mathfrak{sl}_{2})$
at admissible level in category $\mathcal{O}$ can be obtained by using
$\ell=-\frac{1}{2}$ spectral flow on the untwisted modules in category
$\mathcal{O}$. In particular, for boundary admissible level, all of those
irreducible twisted modules are ordinary modules. we find that the $q$-series
characters satisfy the following relation,

(97)
$\mathrm{ch}[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,j}))](q)=\mathrm{ch}[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,u-1-j}))](q)$

Furthermore, the number of independence $q$ series characters is
$\frac{u+1}{2}$.

Now let us give some concrete examples for small values of $u$.

######  Example 5.3.

Let us consider $\hat{\mathfrak{g}}=A^{(1)}_{1}$ at level $k=-\frac{4}{3}$,
the number of independence $q$-series characters is two, we denote these two
characters of twisted modules as
$\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,0}))\right]$ and
$\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,1}))\right]$. They are
annihilated by a second-order $\Gamma^{0}(2)$-MLDE which we display here:

(98)
$\displaystyle\left(D^{(2)}_{q}-\frac{1}{96}\bar{\Theta}_{1,1}(\tau)\right)\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,i}))\right](q)=0,\quad
i=0,1.$

Since the modular form $M_{2k}(\Gamma^{0}(2))$ spanned by the $\Theta_{r,s}$
which can be rewritten in terms of twisted Eisenstein series, we can rewrite
above $\mathrm{MLDE}$ (98) as

(99)
$\displaystyle\left(D^{(2)}_{q}+\frac{4}{3}\mathbbm{E}_{4}\begin{bmatrix}-1\\\
1\end{bmatrix}+\frac{28}{3}\mathbbm{E}_{4}\begin{bmatrix}1\\\
-1\end{bmatrix}+\frac{28}{3}\mathbbm{E}_{4}\begin{bmatrix}-1\\\
-1\end{bmatrix}\right)\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,i}))\right](q)=0,\quad
i=0,1.$

Actually, the twisted module character
$\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,1}))\right]\big{|}_{y=1}$
has two different physical interpretations. In [CGS17], the authors computed
the defect Schur indices of the $(A_{1},A_{3})$ Argyres-Douglas theory,

(100)
$\displaystyle\mathcal{I}_{\mathbb{S}}(q,x)=(q)^{2}_{\infty}\sum^{\infty}_{\begin{subarray}{c}\ell_{1},...,\ell_{3},\\\
k_{1},...,k_{3}=0\end{subarray}}\frac{(-1)^{\sum_{i=1}^{3}(k_{i}+\ell_{i})}q^{\frac{1}{2}\sum_{i=1}^{3}(k_{i}+\ell_{i})+\ell_{2}(\ell_{1}+\ell_{3})}}{\prod_{i=1}^{3}(q)_{k_{i}}(q)_{\ell_{i}}}(x)^{\ell_{1}-k_{1}}\left(q^{\frac{\ell_{1}-k_{1}}{2}}+q^{\frac{k_{1}-\ell_{1}}{2}}-q^{\frac{\ell_{1}+k_{1}}{2}}\right)\delta_{k_{2},\ell_{2}}\delta_{k_{1}+k_{3},\ell_{1}+\ell_{3}}.$

The corresponding VOA of the $(A_{1},A_{3})$ AD theory is just
$L_{-\frac{4}{3}}(A_{1}^{(1)})$. One can check that the above surface defect
index agrees with the twisted module character
$\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,1}))\right]\big{|}_{y=1}$
with $x$ replaced by $\mathbf{z}^{2}$,

(101)
$\displaystyle\mathcal{I}_{\mathbb{S}}(q,\mathbf{z}^{2})=\text{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,1}))\right]\big{|}_{y=1}.$

In [FS18], the authors compute the lens space index of the $(A_{1},A_{3})$ AD
theory. For example, we have checked that
$\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,1}))\right]\big{|}_{y=1}$
agrees with the lens space index
$\mathcal{I}^{\text{Mac}}_{(A_{1},D_{3})}\big{|}_{t=1}$ up to an overall
factor by identify their ”twisting parameter” with the spectral flow
parameter. One advantage of our expression is that operator spectrum of the 4d
theory is much easier to read off, and modular properties are also apparent.

######  Example 5.4.

Let us consider $\hat{\mathfrak{g}}=A^{(1)}_{1}$ at level $k=-\frac{8}{5}$,
the number of independent $q$-series characters is three, we denote these
three characters of twisted modules as
$\mathrm{ch}[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,j}))](q),j=0,1,2$. They
satisfy a third-order $\Gamma^{0}(2)$-MLDE.

(102)
$\left[D^{(3)}_{q}-\left(\frac{7}{450}\bar{\Theta}_{0,2}(\tau)+\frac{31}{1800}\bar{\Theta}_{1,1}(\tau)\right)D^{(1)}_{q}-\frac{1}{400}\bar{\Theta}_{1,2}(\tau)\right]\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\lambda_{k,j}))\right](q)=0,\quad
j=0,1,2.$

######  Example 5.5.

Let us consider $\hat{\mathfrak{g}}=A^{(1)}_{1}$ at level $k=-\frac{12}{7}$,
the number of independent $q$-series characters is four, we denote these three
characters of twisted modules as
$\mathrm{ch}[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,j}))],j=0,1,\cdots,3$. They
satisfy a fourth-order $\Gamma^{0}(2)$-MLDE.

(103)
$\displaystyle[D^{(4)}_{q}-\left(\frac{1}{18}\bar{\Theta}_{0,2}(\tau)+\frac{17}{1008}\bar{\Theta}_{1,1}(\tau)\right)D^{(2)}_{q}$
$\displaystyle+\left(\frac{50}{9261}\bar{\Theta}_{0,3}(\tau)-\frac{883}{49392}\bar{\Theta}_{1,2}(\tau)\right)D^{(1)}_{q}+\left(\frac{9}{10976}\bar{\Theta}_{1,3}(\tau)-\frac{225}{175616}\bar{\Theta}_{2,2}(\tau)\right)]\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(L(\Lambda_{k,j}))\right](q)=0.$

###  5.2. $A^{(1)}_{1}$ at admissible level $k=-\frac{1}{2}$

For $A^{(1)}_{1}$ at level $k=-\frac{1}{2}$, we can write down characters of
admissible highest weight modules following [Rid09],

(104)
$\displaystyle\text{ch}[\mathcal{L}_{0}]=\frac{\mathbf{y}^{-\frac{1}{2}}}{2}\left[\frac{\eta(\tau)}{\theta_{4}(\mathbf{z};q)}+\frac{\eta(\tau)}{\theta_{3}(\mathbf{z};q)}\right],\quad\text{ch}[\mathcal{D}^{+}_{-1/2}]=\frac{\mathbf{y}^{-\frac{1}{2}}}{2}\left[\frac{-i\eta(\tau)}{\theta_{1}(\mathbf{z};q)}+\frac{\eta(\tau)}{\theta_{2}(\mathbf{z};q)}\right]$
(105)
$\displaystyle\text{ch}[\mathcal{L}_{1}]=\frac{\mathbf{y}^{-\frac{1}{2}}}{2}\left[\frac{\eta(\tau)}{\theta_{4}(\mathbf{z};q)}-\frac{\eta(\tau)}{\theta_{3}(\mathbf{z};q)}\right],\quad\text{ch}[\mathcal{D}^{+}_{-3/2}]=\frac{\mathbf{y}^{-\frac{1}{2}}}{2}\left[\frac{-i\eta(\tau)}{\theta_{1}(\mathbf{z};q)}-\frac{\eta(\tau)}{\theta_{2}(\mathbf{z};q)}\right]$

The characters $\text{ch}\left[\sigma^{-\frac{1}{2}}(M)\right]$ of associated
$\mathbb{Z}_{2}$-twisted modules from $\ell=-\frac{1}{2}$ spectral flow are,

(106)
$\displaystyle\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(\mathcal{L}_{0})\right](\mathbf{y},\mathbf{z},q)=\frac{\mathbf{y}^{-\frac{1}{2}}\mathbf{z}^{\frac{1}{4}}q^{-\frac{1}{32}}}{2}\left[\frac{\eta(q\tau)}{\theta_{4}(\mathbf{z}q^{-\frac{1}{4}};q)}+\frac{\eta(\tau)}{\theta_{3}(\mathbf{z}q^{-\frac{1}{4}};q)}\right]$
$\displaystyle\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(\mathcal{L}_{1})\right](\mathbf{y},\mathbf{z},q)=\frac{\mathbf{y}^{-\frac{1}{2}}\mathbf{z}^{\frac{1}{4}}q^{-\frac{1}{32}}}{2}\left[\frac{\eta(\tau)}{\theta_{4}(\mathbf{z}q^{-\frac{1}{4}};q)}-\frac{\eta(\tau)}{\theta_{3}(\mathbf{z}q^{-\frac{1}{4}};q)}\right]$
$\displaystyle\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(\mathcal{D}^{+}_{-1/2})\right](\mathbf{y},\mathbf{z},q)=\frac{\mathbf{y}^{-\frac{1}{2}}\mathbf{z}^{\frac{1}{4}}q^{-\frac{1}{32}}}{2}\left[\frac{-i\eta(\tau)}{\theta_{1}(\mathbf{z}q^{-\frac{1}{4}};q)}+\frac{\eta(\tau)}{\theta_{2}(\mathbf{z}q^{-\frac{1}{4}};q)}\right]$
$\displaystyle\mathrm{ch}\left[\sigma^{-\frac{1}{2}}(\mathcal{D}^{+}_{-3/2})\right](\mathbf{y},\mathbf{z},q)=\frac{\mathbf{y}^{-\frac{1}{2}}\mathbf{z}^{\frac{1}{4}}q^{-\frac{1}{32}}}{2}\left[\frac{-i\eta(\tau)}{\theta_{1}(\mathbf{z}q^{-\frac{1}{4}};q)}-\frac{\eta(\tau)}{\theta_{2}(\mathbf{z}q^{-\frac{1}{4}};q)}\right]$

For the character $\mathrm{ch}\left[\mathcal{L}_{0}\right]$ of vacuum module
of $L_{-1/2}(\mathfrak{sl}_{2})$, it is a solution of a third-order MLDE under
full $SL(2,\mathbb{Z})$ group,

(107)
$\left[D^{(3)}_{q}-\frac{235}{4}\mathbbm{E}_{4}(\tau)D^{(1)}_{q}-\frac{455}{8}\mathbbm{E}_{6}(\tau)\right]\mathrm{ch}[\mathcal{L}_{0}](q)=0$

There are two independent well-defined $q$-series characters of
$\mathbb{Z}_{2}$-twisted modules denoted by
$\mathrm{ch}[\sigma^{-\frac{1}{2}}(\mathcal{L}_{0})](q)$ and
$\mathrm{ch}[\sigma^{-\frac{1}{2}}(\mathcal{L}_{1})](q)$. They satisfy a
second-order $\Gamma^{0}(2)$ MLDE.

(108)
$\left[D^{(2)}_{q}-\frac{5}{48}\bar{\Theta}_{0,1}(\tau)D^{(1)}_{q}+\left(\frac{25}{9216}\bar{\Theta}_{0,2}(\tau)-\frac{41}{9216}\bar{\Theta}_{1,1}(\tau)\right)\right]\mathrm{ch}[\sigma^{-\frac{1}{2}}(\mathcal{L}_{i})](q)=0,\quad
i=0,1.$

###  5.3. $A^{(1)}_{2}$ at boundary admissible level $k=-\frac{3}{2}$

Consider the boundary principal admissible weight modules of $A^{(1)}_{2}$ at
the boundary admissible level $k=-\frac{3}{2}$. There are four irreducible
admissible highest weight modules of affine Lie algebra $A^{(1)}_{2}$, they
are exactly the complete list of irreducible weak
$L_{-3/2}(\mathfrak{sl}_{3})$ modules from category $\mathcal{O}$ [Ara16,
Per08]. Their characters can be written in terms of Jacobi theta function
[KW88, KW17].

(109)
$\displaystyle\text{ch}\left[\mathcal{L}\left(-\frac{3}{2}\Lambda_{0}\right)\right](\mathbf{y},\mathbf{z}_{1},\mathbf{z}_{2},q)=\mathbf{y}^{-\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}\frac{\theta_{1}\left(\mathbf{z}_{1};q^{2}\right)\theta_{1}\left(\mathbf{z}_{2};q^{2}\right)\theta_{1}\left(\mathbf{z}_{1}\mathbf{z}_{2};q^{2}\right)}{\theta_{1}\left(\mathbf{z}_{1};q\right)\theta_{1}\left(\mathbf{z}_{2};q\right)\theta_{1}\left(\mathbf{z}_{1}\mathbf{z}_{2};q\right)}$
$\displaystyle\text{ch}\left[\mathcal{L}\left(-\frac{3}{2}\Lambda_{1}\right)\right](\mathbf{y},\mathbf{z}_{1},\mathbf{z}_{2},q)=-\mathbf{y}^{-\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}\frac{\theta_{1}(\mathbf{z}_{2};q^{2})\theta_{4}(\mathbf{z}_{1};q^{2})\theta_{4}(\mathbf{z}_{1}\mathbf{z}_{2};q^{2})}{\theta_{1}(\mathbf{z}_{1};q)\theta_{1}(\mathbf{z}_{2};q)\theta_{1}(\mathbf{z}_{1}\mathbf{z}_{2};q)}$
$\displaystyle\text{ch}\left[\mathcal{L}\left(-\frac{3}{2}\Lambda_{2}\right)\right](\mathbf{y},\mathbf{z}_{1},\mathbf{z}_{2},q)=-\mathbf{y}^{-\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}\frac{\theta_{1}(\mathbf{z}_{1};q^{2})\theta_{4}(\mathbf{z}_{1}\mathbf{z}_{2};q^{2})\theta_{4}(\mathbf{z}_{2};q^{2})}{\theta_{1}(\mathbf{z}_{1};q)\theta_{1}(\mathbf{z}_{2};q)\theta_{1}(\mathbf{z}_{1}\mathbf{z}_{2};q)}$
$\displaystyle\text{ch}\left[\mathcal{L}\left(-\frac{\rho}{2}\right)\right](\mathbf{y},\mathbf{z}_{1},\mathbf{z}_{2},q)=\mathbf{y}^{-\frac{3}{2}}(\mathbf{z}_{1}\mathbf{z}_{2})^{\frac{3}{2}}q^{\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}\frac{\theta_{1}(\mathbf{z}_{2}^{-1}q^{-1};q^{2})\theta_{1}(\mathbf{z}_{1}^{-1}q^{-1};q^{2})\theta_{1}(\mathbf{z}_{1}^{-1}\mathbf{z}_{2}^{-1}q^{-2};q^{2})}{\theta_{1}(\mathbf{z}_{1};q)\theta_{1}(\mathbf{z}_{2};q)\theta_{1}(\mathbf{z}_{1}\mathbf{z}_{2};q)}$

where $\Lambda_{i}$ denote the fundamental weights of affine Lie algebra
$A^{(1)}_{2}$ and $\rho$ is the affine Weyl vector. Letting
$z=\sum_{i=1}^{2}\mathfrak{z}_{i}\bar{\Lambda}_{i}$, where $\bar{\Lambda}_{i}$
are the fundamental weights of finite Lie algebra
$\mathfrak{g}=\mathfrak{sl}_{3}$, we define $\mathbf{z}_{i}=e^{2\pi
i\mathfrak{z}_{i}}$ which appeared in Jacobi theta function.

Now, we consider the action of spectral flow on these irreducible highest
weight modules. Firstly, we consider the spectral flow along
$\frac{1}{2}\bar{\Lambda}^{\vee}_{1}$ direction, and the character becomes,

(110)
$\text{ch}\left[\sigma^{\frac{1}{2}\Lambda_{1}^{\vee}}\left(\mathcal{L}\left(-\frac{3}{2}\Lambda_{0}\right)\right)\right](\mathbf{y},\mathbf{z}_{1},\mathbf{z}_{2},q)=(\mathbf{y}\mathbf{z}_{1}^{\frac{1}{3}}\mathbf{z}_{2}^{\frac{1}{6}}q^{\frac{1}{12}})^{-\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}\frac{\theta_{1}\left(\mathbf{z}_{1}q^{\frac{1}{2}};q^{2}\right)\theta_{1}\left(\mathbf{z}_{2};q^{2}\right)\theta_{1}\left(\mathbf{z}_{1}\mathbf{z}_{2}q^{\frac{1}{2}};q^{2}\right)}{\theta_{1}\left(\mathbf{z}_{1}q^{\frac{1}{2}};q\right)\theta_{1}\left(\mathbf{z}_{2};q\right)\theta_{1}\left(\mathbf{z}_{1}\mathbf{z}_{2}q^{\frac{1}{2}};q\right)}.$

This spectral flowed module is an ordinary module. It satisfies a second-order
$\Gamma^{0}(2)$-modular linear differential equation,

(111)
$\left(D^{(2)}_{q}-\frac{5}{576}\bar{\Theta}_{0,2}(\tau)-\frac{11}{576}\bar{\Theta}_{1,1}(\tau)\right)\text{ch}\left[\sigma^{\frac{1}{2}\Lambda_{1}^{\vee}}\left(\mathcal{L}\left(-\frac{3}{2}\Lambda_{0}\right)\right)\right](q)=0.$

Secondly, consider the spectral flow of the character
$\text{ch}\left[\mathcal{L}\left(-\frac{\rho}{2}\right)\right]$ along
$\frac{1}{3}(\bar{\Lambda}^{\vee}_{1}+\bar{\Lambda}^{\vee}_{2})$ direction,

(112)
$\displaystyle\text{ch}\left[\sigma^{\frac{1}{3}\overline{\Lambda}_{1}^{\vee}+\frac{1}{3}\overline{\Lambda}_{2}^{\vee}}\left(\mathcal{L}\left(-\frac{\rho}{2}\right)\right)\right](y,\mathbf{z}_{1},\mathbf{z}_{2},q)$
$\displaystyle=(\mathbf{y}\mathbf{z}_{1}^{\frac{1}{3}}\mathbf{z}_{2}^{\frac{1}{3}}q^{\frac{1}{9}})^{-\frac{3}{2}}(\mathbf{z}_{1}\mathbf{z}_{2}q^{\frac{3}{2}})^{\frac{3}{2}}q^{\frac{3}{2}}\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{-1}$
$\displaystyle\times\frac{\theta_{1}((\mathbf{z}_{2}q^{\frac{1}{3}})^{-1}q^{-1};q^{2})\theta_{1}((\mathbf{z}_{1}q^{\frac{1}{3}})^{-1}q^{-1};q^{2})\theta_{1}((\mathbf{z}_{1}\mathbf{z}_{2}q^{-\frac{2}{3}})^{-1}q^{-2};q^{2})}{\theta_{1}(\mathbf{z}_{1}q^{\frac{1}{3}};q)\theta_{1}(\mathbf{z}_{2}q^{\frac{1}{3}};q)\theta_{1}(\mathbf{z}_{1}\mathbf{z}_{2}q^{\frac{2}{3}};q)}$

This spectral flowed character matches with the lens space index of
$(A_{1},D_{4})$ AD theory [FS18] with suitable change of variables.

###  5.4. Bershadsky-Polyakov Algebra $\mathrm{BP}^{k}$ with $k=-\frac{9}{4}$

In previous examples, we consider the spectral flowed modules of affine Lie
algebra $\hat{\mathfrak{g}}$. Now we consider an example of the affine
$W$-algebra [BS95, FF90a, FF90b, KRW03], the
$W^{k}(\mathfrak{sl}_{3},f_{\mathrm{min}})$ which agrees with the
$\mathrm{BP}^{k}$-algebra defined in [Ber91]. First, let us review the
definition of $\text{BP}^{k}$-algebra.

######  Definition 5.6.

[FKR21] Given $k\in\mathbb{C}$, $k\neq-3$, the level-k universal Bershadsky-
Polyakov algebra $\text{BP}\,^{k}$ is the vertex operator algebra with vacuum
$\mathbf{1}$ that is strongly and freely generated by fields $J(z)$,
$G^{+}(z)$, $G^{-}(z)$ and $L(z)$ satisfying the complicated operator product
expansions. The conformal weights of the generating fields $J(z)$, $G^{+}(z)$,
$G^{-}(z)$ and $L(z)$ are $1$, $\frac{3}{2}$, $\frac{3}{2}$ and $2$
respectively, the central charge is,

(113) $c^{\mathrm{BP}}_{u,v}=-\frac{(2k+3)(3k+1)}{k+3}$

The action of the spectral flow automorphism $\sigma^{\ell}$,
$\ell\in\mathbb{Z}$ of the vertex algebra $\text{BP}\,^{k}$ on the modes of
the generating field $J(z)$, $G^{+}(z)$, $G^{-}(z)$ and $L(z)$ is

(114) $\displaystyle\sigma^{l}(J_{n})$
$\displaystyle=J_{n}-\frac{2k+3}{3}l\delta_{n,0}\mathbf{1},$
$\displaystyle\sigma^{l}(G^{+}_{r})$ $\displaystyle=G^{+}_{r-l},$
$\displaystyle\sigma^{l}(G^{-}_{r})$ $\displaystyle=G^{-}_{r+l},$
$\displaystyle\sigma^{l}(L_{n})$
$\displaystyle=L_{n}-lJ_{n}+\frac{2k+3}{6}l^{2}\delta_{n,0}\mathbf{1}.$

When $\ell$ is a half-integer, $\sigma^{l}$ exchanges twisted and untwisted
mode algebras [FKR21].

If we consider the $|\lambda,\Delta\rangle\in M$ is a state of weight
$\lambda$ and conformal dimension $\Delta$ for any module $M$, then the state
$(\sigma^{\ell})^{*}|\lambda,\Delta\rangle\in(\sigma^{\ell})^{*}(M)$
satisfied,

(115) $\displaystyle
h_{0}(\sigma^{\ell})^{*}|\lambda,\Delta\rangle=(\lambda+\ell\frac{2k+3}{3})(\sigma^{\ell})^{*}|\lambda,\Delta\rangle$
$\displaystyle
L_{0}(\sigma^{\ell})^{*}|\lambda,\Delta\rangle=(\Delta+\ell\lambda+\frac{2k+3}{6}\ell^{2})(\sigma^{\ell})^{*}|\lambda,\Delta\rangle$

For a $\mathrm{BP}^{k}$-algebra module $M$, the character has been defined in
[FKR21],

(116)
$\text{ch}[M](\theta|\zeta|\tau)=\mathbf{y}^{\kappa}\text{tr}_{M}\left(\mathbf{z}^{J_{0}}q^{L_{0}-c^{\mathrm{BP}}_{u,v}/24}\right),\quad\kappa=\frac{2k+3}{6}$

where $\mathbf{y}=e^{2\pi i\theta}$, $\mathbf{z}=e^{2\pi i\zeta}$ and
$q=e^{2\pi i\tau}$. The character of spectral flowed module $\sigma^{\ell}(M)$
for $\ell\in\frac{1}{2}\mathbb{Z}$ given by Lemma 4.3 in [FKR21],

(117)
$\mathrm{ch}[\sigma^{\ell}(M)]\left(\theta|\zeta|\tau\right)=\mathrm{ch}[M]\left(\theta+2\ell\zeta+\ell^{2}\tau|\zeta+\ell\tau|\tau\right)$

We consider the special case which the level $k=-\frac{9}{4}$ and
$c^{\mathrm{BP}}_{u,v}=-\frac{23}{2}$, the highest weight vector
$|\lambda,\Delta\rangle=|\frac{1}{4},-\frac{3}{8}\rangle$, and spectral flow
parameter $\ell=\frac{1}{2}$. After spectral flow, the weight and conformal
dimension become,

(118)
$\lambda^{{}^{\prime}}=\frac{1}{4}+\frac{1}{2}\times(-\frac{1}{2})=0,\qquad\Delta^{\prime}=-\frac{3}{8}+\frac{1}{2}\times\frac{1}{4}-\frac{1}{4}\times\frac{1}{4}=-\frac{5}{16}$

According to [CGS17], the character of spectral flowed module can be written
as follows,

(119)
$\text{ch}[\sigma^{\frac{1}{2}}(M_{|\frac{1}{4},-\frac{3}{8}\rangle})](q)=q^{\frac{1}{6}}\left(1+4q+10q^{2}+24q^{3}+51q^{4}+100q^{5}+\mathcal{O}(q^{6})\right)$

This character satisfies a third-order MLDE under full $SL(2,\mathbb{Z})$
group,

(120)
$\left[D^{(3)}_{q}-25\mathbbm{E}_{4}(\tau)D^{(1)}_{q}-175\mathbbm{E}_{6}(\tau)\right]\text{ch}[\sigma^{\frac{1}{2}}(M_{|\frac{1}{4},-\frac{3}{8}\rangle})](q)=0$

##  6\. $\mathfrak{d}_{4}$ with non-admissible level $k=-2$

In this section, we propose a relation between simple modules and spectral
flowed modules of the non-admissible affine vertex algebra
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$. We also find an ordinary
module
$\sigma^{\frac{1}{2}\bar{\Lambda}_{2}}(\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2}))$,
whose character satisfies a second-order $\Gamma^{0}(2)$ MLDE.

Firstly, recall the result for the simple modules of
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$.

######  Theorem 6.1.

[Per13] The set
$\\{\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}),\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{1}),\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{3}),\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{4}),\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2})\\}$
provides a complete list of irreducible weak
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$-modules from the Category
$\mathcal{O}.$ $\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$ is the unique
irreducible ordinary module for
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$. And every ordinary
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$-module is completely
reducible.

We consider the spectral flow automorphsim that act on this vacuum character.
The fundamental weights $\bar{\Lambda}_{i}$ of finite Lie algebra
$\mathfrak{d}_{4}$ can be written in terms of the linear combination of simple
roots $\alpha_{i}$.

(121)
$\displaystyle\bar{\Lambda}_{1}=\alpha_{1}+\alpha_{2}+\frac{1}{2}\alpha_{3}+\frac{1}{2}\alpha_{4},\quad\bar{\Lambda}_{2}=\alpha_{1}+2\alpha_{2}+\alpha_{3}+\alpha_{4}$
$\displaystyle\bar{\Lambda}_{3}=\frac{1}{2}\alpha_{1}+\alpha_{2}+\alpha_{3}+\frac{1}{2}\alpha_{4},\quad\bar{\Lambda}_{4}=\frac{1}{2}\alpha_{1}+\alpha_{2}+\frac{1}{2}\alpha_{3}+\alpha_{4}$

Since the $\mathfrak{d}_{4}$ is simple-laced, then
$\alpha_{i}=\alpha^{\vee}_{i}$ and
$\bar{\Lambda}_{i}=\bar{\Lambda}^{\vee}_{i}$, we will use roots (weights) or
coroots (coweights) without distinction. The highest root of finite Lie
algebra $\mathfrak{d}_{4}$ is $\theta=\bar{\Lambda}_{2}$, therefore the marks
and comarks of affine Lie algebra $\hat{\mathfrak{d}}_{4}$ are
$(a_{i})=(a_{i}^{\vee})=(1,1,2,1,1)$, the level of affine weight
$\Lambda=\sum_{i=0}^{4}\lambda_{i}\Lambda_{i}$ is then given by,

(122) $k=\lambda_{0}+\lambda_{1}+2\lambda_{2}+\lambda_{3}+\lambda_{4}$

######  Theorem 6.2.

The $\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{i})$ with $i=1,3,4$ is the same
as spectral flowed module
$\sigma^{\bar{\Lambda}_{i}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))$,
respectively.

###### Proof.

The powers of $\tau_{i}$ acts as follows,

(123)
$\tau_{i}^{\ell}(h^{j}_{n})=h^{j}_{n}-\ell(\alpha_{i},\alpha_{i})\delta_{n,0}K$

We can compute the action of this automorphsim along $\bar{\Lambda}_{1}$ on
$h^{i}_{0}$,

(124)
$\tau_{1}\tau_{2}\tau_{3}^{\frac{1}{2}}\tau_{4}^{\frac{1}{2}}(h^{1}_{0})=h^{1}_{0}+K,\quad\tau_{1}\tau_{2}\tau_{3}^{\frac{1}{2}}\tau_{4}^{\frac{1}{2}}(h^{2}_{0})=h^{2}_{0}$
(125)
$\tau_{1}\tau_{2}\tau_{3}^{\frac{1}{2}}\tau_{4}^{\frac{1}{2}}(h^{3}_{0})=h^{0}_{3},\quad\tau_{1}\tau_{2}\tau_{3}^{\frac{1}{2}}\tau_{4}^{\frac{1}{2}}(h^{4}_{0})=h^{4}_{0}$

That means the spectral flow automorphsim change highest weight
$-2\Lambda_{0}$ into another highest weight $-2\Lambda_{1}$. One can get
highest weight $-2\Lambda_{3}$ and $-2\Lambda_{4}$ in the same way. According
to cite [DLM96], the $\bar{\Lambda}_{i}$ direction preserves the irreduciblity
of the module $\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{i})$111We would like
to thank Tomoyuki Arakawa and Kazuya Kawasetsu for pointing out this fact.,
this completes the proof. ∎

Actually, the closed form expressions of characters of all simple modules of
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$ are conjectured from the
SCFT/VOA correspondence as [PP22, ZPW22]

(126)
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})]=\mathcal{I}_{0,4}$
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{1})]=\mathcal{I}_{0,4}-2R_{1}$
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2})]=-2\mathcal{I}_{0,4}+2R_{1}+2R_{2}$
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{3})]=\mathcal{I}_{0,4}-R_{1}-R_{2}-R_{3}-R_{4}$
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{4})]=\mathcal{I}_{0,4}-R_{1}-R_{2}-R_{3}+R_{4}$

where $\mathcal{I}_{0,4}$ is the Schur index of an $SU(2)$ gauge theory with
four hypermultiplets, and the $R_{j}$ functions are

(127)
$R_{j}(\widetilde{\mathbf{m}}_{i},\tau)=\frac{i}{2}\frac{\theta_{1}(2\widetilde{\mathbf{m}}_{j})}{\eta(\tau)}\prod_{l\neq
j}\frac{\eta(\tau)}{\theta_{1}(\widetilde{\mathbf{m}}_{j}+\widetilde{\mathbf{m}}_{l})}\frac{\eta(\tau)}{\theta_{1}(\widetilde{\mathbf{m}}_{j}-\widetilde{\mathbf{m}}_{l})},\quad
j=1,2,3,4.$

Here $\widetilde{m}_{i}=e^{2\pi i\mathbf{\widetilde{m}}_{i}}$ are related to
the Cartan element $\widetilde{z}_{i}=e^{2\pi i\mathbf{\widetilde{z}}_{i}}$ of
$\mathfrak{d}_{4}$ as follows,

(128)
$\displaystyle\widetilde{z}_{1}=\frac{\widetilde{m}_{1}}{\widetilde{m}_{2}},\qquad\widetilde{z}_{2}=\frac{\widetilde{m}_{2}}{\widetilde{m}_{3}},\qquad\widetilde{z}_{3}=\widetilde{m}_{3}\widetilde{m}_{4},\qquad\widetilde{z}_{4}=\frac{\widetilde{m}_{3}}{\widetilde{m}_{4}}$

For our purpose, we give
$\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})]$,
$\mathrm{ch}[\sigma^{\bar{\Lambda}_{1}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))]$,
$\mathrm{ch}[\sigma^{\bar{\Lambda}_{3}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))]$,
$\mathrm{ch}[\sigma^{\bar{\Lambda}_{4}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))]$
explicitly,

(129)
$\displaystyle\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})](q;,\widetilde{z}_{1},\widetilde{z}_{2},\widetilde{z}_{3},\widetilde{z}_{4})$
$\displaystyle=\frac{1}{2}\frac{\eta(\tau)^{2}}{\theta_{1}(\tilde{z}_{1}\tilde{z}_{2}^{2}\tilde{z}_{3}\tilde{z}_{4};q)\theta_{1}(\tilde{z}_{1};q)\theta_{1}(\tilde{z}_{4};q)\theta_{1}(\tilde{z}_{3};q)}$
$\displaystyle\times\sum_{\vec{\alpha}=\pm}\left(\prod_{i=1}^{4}\alpha_{i}\right)E_{2}\begin{bmatrix}1\\\
\left(\widetilde{z}_{1}^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{1}}\left(\widetilde{z}_{1}^{\frac{1}{2}}\right)^{\alpha_{2}}\left(\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{3}}\left(\widetilde{z}_{3}^{\frac{1}{2}}\right)^{\alpha_{4}}\end{bmatrix}$
$\displaystyle\mathrm{ch}[\sigma^{\bar{\Lambda}_{1}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))](q;,\widetilde{z}_{1},\widetilde{z}_{2},\widetilde{z}_{3},\widetilde{z}_{4})$
$\displaystyle=\frac{1}{2}\left(y\widetilde{z}_{1}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}\widetilde{z}_{4}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{-2}\frac{\eta(\tau)^{2}}{\theta_{1}(\widetilde{z}_{1}q\widetilde{z}_{2}^{2}\widetilde{z}_{3}\widetilde{z}_{4};q)\theta_{1}(\widetilde{z}_{1}q;q)\theta_{1}(\widetilde{z}_{4};q)\theta_{1}(\widetilde{z}_{3};q)}$
$\displaystyle\times\sum_{\vec{\alpha}=\pm}\left(\prod_{i=1}^{4}\alpha_{i}\right)E_{2}\begin{bmatrix}1\\\
\left(\widetilde{z}_{1}^{\frac{1}{2}}q^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{1}}\left(\widetilde{z}_{1}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{\alpha_{2}}\left(\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{3}}\left(\widetilde{z}_{3}^{\frac{1}{2}}\right)^{\alpha_{4}}\end{bmatrix}$
$\displaystyle\mathrm{ch}[\sigma^{\bar{\Lambda}_{3}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))](q;,\widetilde{z}_{1},\widetilde{z}_{2},\widetilde{z}_{3},\widetilde{z}_{4})$
$\displaystyle=\frac{1}{2}\left(y\widetilde{z}_{1}^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}\widetilde{z}_{4}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{-2}\frac{\eta(\tau)^{2}}{\theta_{1}(\widetilde{z}_{1}\widetilde{z}_{2}^{2}\widetilde{z}_{3}q\widetilde{z}_{4};q)\theta_{1}(\widetilde{z}_{1};q)\theta_{1}(\widetilde{z}_{4};q)\theta_{1}(\widetilde{z}_{3}q;q)}$
$\displaystyle\times\sum_{\vec{\alpha}=\pm}\left(\prod_{i=1}^{4}\alpha_{i}\right)E_{2}\begin{bmatrix}1\\\
\left(\widetilde{z}_{1}^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}q^{\frac{1}{2}}\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{1}}\left(\widetilde{z}_{1}^{\frac{1}{2}}\right)^{\alpha_{2}}\left(\widetilde{z}_{4}^{\frac{1}{2}}\right)^{\alpha_{3}}\left(\widetilde{z}_{3}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{\alpha_{4}}\end{bmatrix}$
$\displaystyle\mathrm{ch}[\sigma^{\bar{\Lambda}_{4}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))](q;,\widetilde{z}_{1},\widetilde{z}_{2},\widetilde{z}_{3},\widetilde{z}_{4})$
$\displaystyle=\frac{1}{2}\left(y\widetilde{z}_{1}^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}\widetilde{z}_{4}q^{\frac{1}{2}}\right)^{-2}\frac{\eta(\tau)^{2}}{\theta_{1}(\widetilde{z}_{1}\widetilde{z}_{2}^{2}\widetilde{z}_{3}\widetilde{z}_{4}q;q)\theta_{1}(\widetilde{z}_{1};q)\theta_{1}(\widetilde{z}_{4}q;q)\theta_{1}(\widetilde{z}_{3};q)}$
$\displaystyle\times\sum_{\vec{\alpha}=\pm}\left(\prod_{i=1}^{4}\alpha_{i}\right)E_{2}\begin{bmatrix}1\\\
\left(\widetilde{z}_{1}^{\frac{1}{2}}\widetilde{z}_{2}\widetilde{z}_{3}^{\frac{1}{2}}\widetilde{z}_{4}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{\alpha_{1}}\left(\widetilde{z}_{1}^{\frac{1}{2}}\right)^{\alpha_{2}}\left(\widetilde{z}_{4}^{\frac{1}{2}}q^{\frac{1}{2}}\right)^{\alpha_{3}}\left(\widetilde{z}_{3}^{\frac{1}{2}}\right)^{\alpha_{4}}\end{bmatrix}$

then, one can show that these expressions satisfy the following relations

(130)
$\mathrm{ch}[\sigma^{\bar{\Lambda}_{i}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))]=\mathrm{ch}\left[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{i})\right],\quad
i=1,3,4.$

It means that we prove the closed form expressions of characters of some
simple modules of $\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$ in terms of
$R_{j}$ function. We can also prove another relation between characters of
$\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})$ modules and the partition
function of the curved $\beta\gamma$ system [ELS20]. The partition function of
the curved $\beta\gamma$ system on complex Grassmannian $\text{Gr}(2,4)$ is
given by,

(131)
$Z_{\mathfrak{d}_{4}}(t,\mathbf{m}^{\mathfrak{a}_{3}}_{3},\tau)=\frac{i\eta(\tau)\theta_{1}(2\sigma,\tau)}{\prod_{\omega\in\rho}\theta_{1}(\sigma+(\mathbf{m}^{\mathfrak{a}_{3}},\omega),\tau)},$

where $t=e^{2\pi i\sigma}$ and $q=e^{2\pi i\tau}$ as usual. The product in the
denominator is over the weights in the representation $\rho$ of
$\mathfrak{a}_{3}$ with highest weight $\bar{\Lambda}_{\mathfrak{a}_{3}}$. The
authors of [ELS20] found that this partition function is also given by

(132)
$Z_{\mathfrak{d}_{4}}(t,\mathbf{m}^{\mathfrak{a}_{3}}_{3},\tau)=\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})](\mathbf{m}^{\mathfrak{d}_{4}},\tau)-\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{4})](\mathbf{m}^{\mathfrak{d}_{4}},\tau)$

with the following identifications of parameters

(133)
$\mathbf{m}^{\mathfrak{d}_{4}}_{i}=\mathbf{m}^{\mathfrak{a}_{3}}_{i},\quad\text{for}\,\,i=1,2,3,\quad\mathbf{m}^{\mathfrak{d}_{4}}_{4}=\sigma-\frac{\mathbf{m}^{\mathfrak{a}_{3}}_{1}}{2}-\mathbf{m}^{\mathfrak{a}_{3}}_{2}-\frac{\mathbf{m}^{\mathfrak{a}_{3}}_{3}}{2}.$

Using the expression of
$\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})]$, one also sees that

(134)
$Z_{\mathfrak{d}_{4}}(t,\mathbf{m}^{\mathfrak{a}_{3}}_{3},\tau)=\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0})](\mathbf{m}^{\mathfrak{d}_{4}},\tau)-\text{ch}[\sigma^{\bar{\Lambda}_{4}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))](\mathbf{m}^{\mathfrak{d}_{4}},\tau).$

Therefore we have

(135)
$\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{4})](\mathbf{m}^{\mathfrak{d}_{4}},\tau)=\text{ch}[\sigma^{\bar{\Lambda}_{4}}(\mathcal{L}_{\mathfrak{d}_{4}}(-2\Lambda_{0}))](\mathbf{m}^{\mathfrak{d}_{4}},\tau).$

Triality gives similar results for $i=1$ and $2$.

Now, let us consider the character of the spectral flowed module of
$\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2})$ along
$-\frac{1}{2}\bar{\Lambda}_{2}$ direction,

(136)
$\text{ch}[\sigma^{-\frac{1}{2}\bar{\Lambda}_{2}}(\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2}))]=\left(y\tilde{z}_{1}^{\frac{1}{2}}\tilde{z}_{2}\tilde{z}_{3}^{\frac{1}{2}}\tilde{z}_{4}^{\frac{1}{2}}q^{\frac{1}{4}}\right)^{-2}\text{ch}[\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2})]\left(q;\tilde{z}_{1},\tilde{z}_{2}q^{-\frac{1}{2}},\tilde{z}_{3},\tilde{z}_{4}\right).$

$\text{ch}[\sigma^{-\frac{1}{2}\bar{\Lambda}_{2}}(\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2}))]$
is the character of an ordinary module as it converges under the limit
$\tilde{z}_{i}\rightarrow 1$. It equals to the defect index
$\mathcal{I}^{\text{defet}}_{0,4}(k=1)$ of [ZPW22], which satisfies the
following second-order $\Gamma^{0}(2)$ MLDE

(137)
$\left(D_{q}^{(2)}+\frac{1}{144}\bar{\Theta}_{0,2}(\tau)-\frac{37}{288}\bar{\Theta}_{1,1}(\tau)\right)\text{ch}[\sigma^{\frac{1}{2}\bar{\Lambda}_{2}}(\mathcal{L}_{\mathfrak{d}_{4}}(-\Lambda_{2}))]\big{|}_{\tilde{z}_{i}\rightarrow
1}=0,$

without $D_{q}^{(1)}$ term.

##  Appendix A Theta functions

We first summarize some basic facts about affine Lie of type $A^{(1)}_{1}$.

Cartan matrix | $\begin{pmatrix}2&-2\\\ -2&2\end{pmatrix}$
---|---
Simple roots | $\\{\alpha_{0},\alpha_{1}\\}$
$\mathfrak{h}^{*}$ | $\text{Span}_{\mathbb{C}}\\{\alpha_{0},\alpha_{1},\Lambda_{0}\\}$
Bilinear form on $\mathfrak{h}^{*}$ | $\langle\alpha_{0},\alpha_{0}\rangle=2$
| $\langle\alpha_{1},\alpha_{1}\rangle=2$
| $\langle\alpha_{1},\alpha_{0}\rangle=\langle\alpha_{0},\alpha_{1}\rangle=-2$
| $\langle\alpha_{0},\Lambda_{0}\rangle=1$
|
$\langle\Lambda_{0},\Lambda_{0}\rangle=\langle\alpha_{1},\Lambda_{0}\rangle=0$
basic imaginary root | $\delta=\alpha_{0}+\alpha_{1}$
$\mathfrak{h}$ | $\text{Span}_{\mathbb{C}}\\{\alpha_{0}^{\vee}=h_{1},\alpha_{2}^{\vee}=h_{2},d\\}$
Central element | $c=\alpha_{0}^{\vee}+\alpha_{1}^{\vee}$
Fundamental weights | $\\{\Lambda_{0},\Lambda_{1}\\}$
|  $\langle\Lambda_{i},\alpha_{j}^{\vee}\rangle=\delta_{i,j}$,
$\Lambda_{i}(d)=0$, $i=0,1$
Lattice $M$ | $\mathbb{Z}h_{1}$
Lattice $M^{*}$ | $\frac{1}{2}\mathbb{Z}\alpha_{1}$
Integral forms $P$ | $\\{\lambda\in\mathfrak{h}^{*}|\langle\lambda(\alpha_{i}^{\vee})\in\mathbb{Z},i=0,1\\}$
Positive integral forms $P_{+}$ | $\\{\lambda\in\mathfrak{h}^{*}|\langle\lambda(\alpha_{i}^{\vee})\in\mathbb{Z}_{\geq 0},i=0,1\\}$
Weyl group |  $t(M)\rtimes\overline{W}$, $t(M)=\\{t_{m}|m\in M\\}$, $\overline{W}=\\{s_{1}\\}$
Lacing number | 1
Coxeter dual number | 2

One can use classical theta functions to define Jacobi theta functions of
degree two:

(138)
$\displaystyle\begin{split}&\theta_{3}(\mathbf{z};q)\equiv\vartheta_{00}(\tau,z)=\Theta_{2,2}(\tau,z)+\Theta_{0,2}(\tau,z)=\displaystyle\sum_{n\in\mathbb{Z}}\mathbf{z}^{2(n+\frac{1}{2})}q^{2(n+\frac{1}{2})^{2}}+\mathbf{z}^{2n}q^{2n^{2}}=\sum_{n\in\mathbb{Z}}\mathbf{z}^{n}q^{\frac{n^{2}}{2}},\\\
&\theta_{4}(\mathbf{z};q)\equiv\vartheta_{01}(\tau,z)=-\Theta_{2,2}(\tau,z)+\Theta_{0,2}(\tau,z)=\displaystyle\sum_{n\in\mathbb{Z}}-\mathbf{z}^{2(n+\frac{1}{2})}q^{2(n+\frac{1}{2})^{2}}+\mathbf{z}^{2n}q^{2n^{2}}=\sum_{n\in\mathbb{Z}}(-1)^{n}\mathbf{z}^{n}q^{\frac{n^{2}}{2}},\\\
&\theta_{2}(\mathbf{z};q)\equiv\vartheta_{10}(\tau,z)=\Theta_{1,2}(\tau,z)+\Theta_{-1,2}(\tau,z)=\displaystyle\sum_{n\in\mathbb{Z}}\mathbf{z}^{2(n+\frac{1}{4})}q^{2(n+\frac{1}{4})^{2}}+\mathbf{z}^{2(n-\frac{1}{4})}q^{2(n-\frac{1}{4})^{2}}=\sum_{n\in\mathbb{Z}}\mathbf{z}^{(n+\frac{1}{2})}q^{\frac{(n+\frac{1}{2})^{2}}{2}},\\\
&\theta_{1}(\mathbf{z};q)\equiv-\vartheta_{11}(\tau,z)=i\Theta_{1,2}(\tau,z)-i\Theta_{-1,2}(\tau,z)=\displaystyle\sum_{n\in\mathbb{Z}}i\mathbf{z}^{2(n+\frac{1}{4})}q^{2(n+\frac{1}{4})^{2}}-i\mathbf{z}^{2(n-\frac{1}{4})}q^{2(n-\frac{1}{4})^{2}}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\sum_{n\in\mathbb{Z}}e^{\pi
i(n+\frac{1}{2})}\mathbf{z}^{(n+\frac{1}{2})}q^{\frac{(n+\frac{1}{2})^{2}}{2}}.\end{split}$

Using $\eta(q)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$, one can gets
the following infinite product identities

(139)
$\displaystyle\begin{split}&\displaystyle\prod_{n=1}^{\infty}(1+\mathbf{z}q^{n-\frac{1}{2}})(1+\mathbf{z}^{-1}q^{n-\frac{1}{2}})=q^{\frac{1}{24}}\frac{\vartheta_{00}(\tau,z)}{\eta(q)},\\\
&\displaystyle\prod_{n=1}^{\infty}(1-\mathbf{z}q^{n-\frac{1}{2}})(1-\mathbf{z}^{-1}q^{n-\frac{1}{2}})=q^{\frac{1}{24}}\frac{\vartheta_{01}(\tau,z)}{\eta(q)},\\\
&\displaystyle\prod^{\infty}_{n=1}(1+\mathbf{z}q^{n})(1+\mathbf{z}^{-1}q^{n-1})=q^{-\frac{1}{12}}\mathbf{z}^{-\frac{1}{2}}\frac{\vartheta_{10}(\tau,z)}{\eta(q)},\\\
&\displaystyle\prod^{\infty}_{n=1}(1-\mathbf{z}q^{n})(1-\mathbf{z}^{-1}q^{n-1})=-iq^{-\frac{1}{12}}\mathbf{z}^{-\frac{1}{2}}\frac{\vartheta_{11}(\tau,z)}{\eta(q)},\end{split}$

where $q$ is as always $e^{2\pi i\tau}.$

##  Appendix B Proof in Proposition 4.4

The calculation of (76):

$\displaystyle\;\;\;T_{-}^{a}T_{0}^{b}T_{+}^{d}P(E_{1}(n,k))$
$\displaystyle=T_{-}^{a}T_{0}^{b}T_{+}^{d}\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}G_{-r-st}\right)T_{+}^{n}$
$\displaystyle=T_{-}^{a}\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}G_{-r-st-d}\right)T_{0}^{b}T_{+}^{d+n}$
$\displaystyle=T_{-}^{a}\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}(T_{-}T_{+}-(-r-st-d)T_{0}+(-r-st-d)(-r-st-d+1))\right)T_{0}^{b}T_{+}^{d+n}$
$\displaystyle=T_{-}^{a}\left(\prod_{r=1}^{n}\prod_{s=1}^{k-1}(r+st+d)(T_{0}+r+st+d-1)\right)T_{0}^{b}T_{+}^{n+d}\;\;\;\mod
T_{-}U(L_{0}).$

The calculation of (77):

$\displaystyle\;\;\;T_{-}^{a}T_{0}^{b}T_{+}^{d}P(E_{2}(n,k))$
$\displaystyle=T_{-}^{a}T_{0}^{b}T_{+}^{m}\left(\prod_{i=1}^{p-n}G_{-i}\right)\prod_{r=0}^{p-n-1}\prod_{s=1}^{q-k}G_{r+st-p+n}$
$\displaystyle=T_{-}^{a}T_{0}^{b}T_{+}^{m}\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}G_{r+st-p+n}$
$\displaystyle=T_{-}^{a}\prod_{r=0}^{p-n-1}\prod_{s=0}^{q-k}G_{r+st-p+n-m}T_{0}^{b}T_{+}^{m}$
$\displaystyle=T_{-}^{a}\prod_{r=1}^{p-n}\prod_{s=0}^{q-k}G_{st-
m-r}T_{0}^{b}T_{+}^{m}$
$\displaystyle=T_{-}^{a}\prod_{r=1}^{p-n}\prod_{s=0}^{q-k}(st-m-r)(-T_{0}+st-
m-r+1)T_{0}^{b}T_{+}^{m}\;\;\;\mod T_{-}U(L_{0}).$

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