# Associativity of fusion products of $C_{1}$-cofinite ${\mathbb{N}}$-gradable
modules of vertex operator algebra

Masahiko Miyamoto 111Partially supported by the Grants-in-Aids for Scientific
Research, No.21K03195, The Ministry of Education, Science and Culture, Japan
Institute of Mathematics, University of Tsukuba Institute of Mathematics,
Academia Sinica

###### Abstract

We prove an associative law of the fusion products $\boxtimes$ of
$C_{1}$-cofinite ${\mathbb{N}}$-gradable modules for a vertex operator algebra
$V$. To be more precise, for $C_{1}$-cofinite ${\mathbb{N}}$-gradable
$V$-modules $A,B,C$ and their fusion products $(A\\!\boxtimes\\!B,{\cal
Y}^{AB})$, $((A\\!\boxtimes\\!B)\\!\boxtimes\\!C,{\cal Y}^{(AB)C})$,
$(B\\!\boxtimes\\!C,{\cal Y}^{BC})$,
$(A\\!\boxtimes\\!(B\\!\boxtimes\\!C),{\cal Y}^{A(BC)})$ with logarithmic
intertwining operators ${\cal Y}^{AB},\ldots,{\cal Y}^{A(BC)}$ satisfying the
universal properties for ${\mathbb{N}}$-gradable modules, we prove that four-
point correlation functions $\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal
Y}^{BC}(u,y)w\rangle$ and $\langle\theta^{\prime},{\cal Y}^{(AB)C}({\cal
Y}^{AB}(v,x-y)u,y)w\rangle$ are locally normally convergent over
$\\{(x,y)\in{\mathbb{C}}^{2}\mid 0\\!<\\!|x\\!-\\!y|\\!<\\!|y|\\!<\\!|x|\\}$.
We then take their respective principal branches

$\widetilde{F}(\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle)$
and $\widetilde{F}(\langle\theta,{\cal Y}^{(AB)C}({\cal
Y}^{AB)}(v,x-y)u,y)w\rangle)$

on ${\cal D}^{2}\\!=\\!\\{(x,y)\in{\mathbb{C}}^{2}\mid
0\\!<\\!|x\\!-\\!y|\\!<\\!|y|\\!<\\!|x|,\mbox{ and
}x,y,x\\!-\\!y\not\in{\mathbb{R}}^{\leq 0}\\}$ and then show that there is an
isomorphism $\phi_{[AB]C}:(A\boxtimes B)\boxtimes C\to A\boxtimes(B\boxtimes
C)$ such that

$\widetilde{F}(\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal
Y}^{BC}(u,y)w\rangle)=\widetilde{F}(\langle\phi_{[AB]C}^{\ast}(\theta),{\cal
Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w)\rangle\vspace{-2mm}$

on ${\cal D}^{2}$ for $\theta\in(A\boxtimes(B\boxtimes C))^{\vee}$, $v\in A$,
$u\in B$, and $w\in C$, where $W^{\vee}$ denotes the contragredient module of
$W$ and $\phi_{[AB]C}^{\ast}$ denotes the dual of $\phi_{[AB]C}$. We also
prove the pentagon identity.

##  1 Introduction

As a counterpart of a tensor product of two modules in the VOA theory, Huang
and Lepowsky[4] have introduced a concept of fusion products
$\boxtimes_{p(x)}$. Huang [3] has also shown that four-point correlation
functions for three $C_{1}$-cofinite ${\mathbb{N}}$-gradable modules satisfy
differential equations with regular singularity and then as an application, he
has proved an associative law of fusion products under some conditions, where
a $V$-module $W$ is called to be ”$C_{1}$-cofinite” if $\dim
V/C_{1}(W)<\infty$ for $C_{1}(W):={\rm span}_{{\mathbb{C}}}\\{\alpha_{-1}w\in
W\mid w\in W,\alpha\in V,{\rm wt}(\alpha)\geq 1\\}$ and a $V$-module $W$ is
called ${\mathbb{N}}$-gradable if $W$ has a decomposition
$W=\oplus_{m\in{\mathbb{N}}}W_{(m)}$ satisfying $\alpha_{k}W_{(m)}\subseteq
W_{(m+{\rm wt}(\alpha)-k-1)}$ for $\alpha\in V$ and $k\in{\mathbb{Z}}$. From
now on, ${\cal N}\\!{\cal C}_{1}$ denotes the set of all $C_{1}$-cofinite
${\mathbb{N}}$-gradable $V$-modules. Before we explain our results, let us
show basic knowledge for $C_{1}$-cofinite ${\mathbb{N}}$-gradable modules. If
$W=\oplus_{m\in{\mathbb{N}}}W_{(m)}\in{\cal N}\\!{\cal C}_{1}$ is
indecomposable, then as we will show in §2, $\dim W_{(m)}<\infty$ for all $m$
and there is $d=d(W)\in{\mathbb{C}}$ such that
$W=\oplus_{m\in{\mathbb{N}}}W_{d+m}$ with $W_{d}\not=0$ and $\dim
W_{d+m}<\infty$, where $W_{s}$ denotes a generalized eigenspace of $L(0)$ with
eigenvalue $s$. In particular, $W^{\vee}=\oplus_{m\in{\mathbb{N}}}{\rm
Hom}(W_{(m)},{\mathbb{C}})\cong\oplus_{d\in{\mathbb{C}}}{\rm
Hom}(W_{d},{\mathbb{C}})$. An element $w\in W_{s}$ is called homogeneous and
denoted by ${\rm wt}(w)=s$. Clearly, $W_{(m)}=W_{d+m}$ defines an
${\mathbb{N}}$-grading on $W$. We always choose this grading for
indecomposable modules $A,B,C$ which come out later. We write ${\rm gr}(w)=m$
for $w\in W_{(m)}$ and call it the grade of $w$.

For $A,B\in{\cal N}\\!{\cal C}_{1}$, as the author has shown in [6], a pair
$(A\boxtimes B,{\cal Y}^{AB})$ of a $V$-module $A\boxtimes B$ and a
logarithmic intertwining operator ${\cal Y}^{AB}\in I\binom{A\boxtimes
B}{A\,\,B}$ satisfying the universal property for ${\mathbb{N}}$-gradable
modules (see §2) is always uniquely well-defined up to isomorphism and
$A\boxtimes B$ is also in ${\cal N}\\!{\cal C}_{1}$ and ${\cal Y}^{AB}$ is
also a logarithmic intertwining operator. Let $A,B,C\in{\cal N}\\!{\cal
C}_{1}$ and assume to be ${\mathbb{N}}$-graded. Since $A\boxtimes B,B\boxtimes
C\in{\cal N}\\!{\cal C}_{1}$, we also have $((A\boxtimes B)\boxtimes C,{\cal
Y}^{(AB)C})$ and $(A\boxtimes(B\boxtimes C),{\cal Y}^{A(BC)})$ and we denote
these logarithmic intertwining operators by

$\begin{array}[]{ll}{\cal Y}^{A(BC)}(v,x)&=\sum_{h=0}^{K_{1}}{\cal
Y}_{h}^{A(BC)}(v,x)\log^{h}(x),\vspace{1mm}\cr{\cal
Y}^{BC}(u,y)&=\sum_{k=0}^{K_{2}}{\cal
Y}^{BC}_{k}(u,y)\log^{k}(y),\vspace{1mm}\cr{\cal
Y}^{AB}(v,x-y)&=\sum_{h=0}^{K_{3}}{\cal Y}^{AB}_{k}(v,x-y)\log^{k}(x-y),\mbox{
and}\vspace{1mm}\cr{\cal Y}^{(AB)C}(\delta,y)&=\sum_{k=0}^{K_{4}}{\cal
Y}_{h}^{(AB)C}(\delta,y)\log^{h}(y),\end{array}$

for $v\in A,u\in B,\delta\in A\boxtimes B$ with formal ${\mathbb{C}}$-power
series ${\cal Y}_{h}^{A(BC)}(v,x)$ of $x$ with coefficients in ${\rm
Hom}(B\boxtimes C,A\boxtimes(B\boxtimes C))$, etc. To simplify the notation,
we use notation $K^{2}=\\{(h,k)\in{\mathbb{N}}^{2}\mid h\leq K_{1},k\leq
K_{2}\\}$, ${K^{\prime}}^{2}=\\{(h,k)\in{\mathbb{N}}^{2}\mid h\leq K_{3},k\leq
K_{4}\\}$ and we often denote $A\boxtimes(B\boxtimes C)$ and $(A\boxtimes
B)\boxtimes C$ by $A(BC)$ and $(AB)C$, respectively. Then for
$\theta\in(A(BC))^{\vee},\theta^{\prime}\in((AB)C)^{\vee}$, we define two
four-point correlation functions and the coefficients of
$\log^{h}(x)\log^{k}(y)$ and $\log^{h}(x-y)\log^{k}(y)$ in them as follows:

$\begin{array}[]{ll}F^{A(BC)}(\theta,v,u,w;x,y)&:=\langle\theta,{\cal
Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle,\vspace{1mm}\cr
F^{(AB)C}(\theta^{\prime},v,u,w;x-y,y)&:=\langle\theta^{\prime},{\cal
Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w\rangle,\vspace{1mm}\cr
F_{h,k}^{A(BC)}(\theta,v,u,w;x,y)&:=\langle\theta,{\cal
Y}_{h}^{A(BC)}(v,x){\cal Y}_{k}^{BC}(u,y)w\rangle,\mbox{ and}\vspace{1mm}\cr
F_{h,k}^{(AB)C}(\theta^{\prime},v,u,w;x-y,y)&:=\langle\theta^{\prime},{\cal
Y}_{h}^{(AB)C}({\cal Y}_{k}^{AB}(v,x-y)u,y)w\rangle.\end{array}$

Set $\Omega=(A(BC))^{\vee}\times A\times B\times C$ (and
$\Omega^{\prime}=((AB)C)^{\vee}\times A\times B\times C$). We use $\vec{\xi}$
(and $\vec{\xi}^{\prime}$) to denote a quadruple $(\theta,v,u,w)\in\Omega$
(and $(\theta^{\prime},v,u,w)\in\Omega^{\prime}$) so that
$F_{h,k}^{A(BC)}(\vec{\xi};x,y)$ denotes $F_{h,k}^{A(BC)}(\theta,v,u,w;x,y)$
and so on. For $\alpha\in V$, we use notation $\alpha_{n}^{[i]}$ to denote the
action of $\alpha_{n}$ on the $i$-th coordinate of $\Omega$, e.g.
$\alpha^{[2]}_{n}\vec{\xi}=(\theta,\alpha_{n}v,u,w)$. For
$\vec{\xi}=(\theta,v,u,w)\in\Omega$, define ${\rm gr}^{234}(\vec{\xi})={\rm
gr}(v)+{\rm gr}(u)+{\rm gr}(w)$ and the total grade ${\rm gr}(\vec{\xi})={\rm
gr}(\theta)+{\rm gr}^{234}(\vec{\xi})$.

The main purpose in this paper is to show the associative law of fusion
products (Theorem 4.1) for $A,B,C\in{\cal N}\\!{\cal C}_{1}$. We may assume
that $A,B,C$ are indecomposable. As a proof, we will construct a surjective
homomorphism $\phi_{[AB]C}:(A\boxtimes B)\boxtimes C\to A\boxtimes(B\boxtimes
C)$ by starting from $F^{A(BC)}(\vec{\xi};x,y)$. The reverse homomorphism is
given by starting from $F^{(AB)C}(\vec{\xi}^{\prime};x,y)$ with the similar
argument. Our proofs are based on Huang’s ideas in [3]. The differences from
[3] are that we will treat formal ${\mathbb{C}}$-power series
$F_{h,k}^{A(BC)}(\vec{\xi};x,y)$ and show that their modified versions satisfy
two Fuchsian systems on discs by fixing one variable
$y=y_{0}\not\in{\mathbb{R}}^{\leq 0}$ or $x=x_{0}\not\in{\mathbb{R}}^{\leq
0}$. Furthermore, in order to expand $F^{A(BC)}(\vec{\xi};x,y_{0})$ in a
${\mathbb{C}}$-formal power series of $(x-y_{0})$ with logarithm functions
$\log^{h}(x-y_{0})$ as a component of a solution of Fuchsian system, we
restrict the variable $x$ into a domain ${\cal
D}^{2}_{(x,y_{0})}\\!=\\!\\{x\in{\mathbb{C}}\mid
0\\!<\\!|x\\!-\\!y_{0}|\\!<\\!|y_{0}|\\!<\\!|x|,\mbox{ and
}x,y_{0},x\\!-\\!y_{0}\not\in{\mathbb{R}}^{\leq 0}\\}$ and take a principal
branch $\widetilde{F}^{A(BC)}(\vec{\xi};x,y_{0})$ of
$F^{A(BC)}(\vec{\xi};x,y_{0})$. A key result we get from our Fuchsian system
is that there is a finite set $\Delta\subseteq{\mathbb{C}}$ such that the
powers of $(x-y_{0})$ in these expansions are all in $\Delta-{\rm gr}(v)-{\rm
gr}(u)+{\mathbb{N}}$ for any $\vec{\xi}$, which supports an existence of an
${\mathbb{N}}$-gradable $V$-module isomorphic to $A\boxtimes B$.

In order to get Fuchsian systems, we consider finite spaces. Let $P_{W}$
denote a complement of $C_{1}(W)$ in $W$ and let $\tilde{P}_{W}$ be a finite
dimensional subspace of $W$ containing $P_{W}$ and spanned by homogeneous
elements. For $N\in{\mathbb{N}}$, we define $W^{\vee}_{(\leq
N)}=\oplus_{m=0}^{N}{\rm Hom}(W_{(m)},{\mathbb{C}})$ and set
$\Omega_{N}=\\{(\theta,v,u,w)\in(A(BC))^{\vee}_{(\leq N)}\times A\times
B\times C\\}$. Choose bases $J_{\tilde{P}_{A}}=\\{v^{i}\mid i\in{\cal
P}_{A}\\},J_{\tilde{P}_{B}}=\\{u^{j}\mid j\in{\cal
P}_{B}\\},J_{\tilde{P}_{C}}=\\{w^{q}\mid q\in{\cal
P}_{C}\\},J_{N}=\\{\theta^{p}\mid p\in{\cal P}_{N}\\}$, of
$\tilde{P}_{A},\tilde{P}_{B},\tilde{P}_{C},(A(BC))^{\vee}_{(\leq N)}$
consisting of homogeneous elements and define ${\cal
J}_{N,\tilde{P}_{A},\tilde{P}_{B},\tilde{P}_{C}}=J_{N}\times
J_{\tilde{P}_{A}}\times J_{\tilde{P}_{B}}\times J_{\tilde{P}_{C}}$, which is a
finite set. We simply denote it by ${\cal J}_{N}$ and denote ${\cal
J}_{N,P_{A},P_{B},P_{C}}$ by ${\cal J}_{N}^{0}$. We always assume ${\cal
J}^{0}_{N}\subseteq{\cal J}_{N}$. We similarly define $\Omega^{\prime}_{N}$
and finite sets ${\cal J}^{\prime}_{N}$ and ${{\cal J}^{\prime}}^{0}_{N}$ for
$((AB)C)^{\vee}$. Set

$\begin{array}[]{ll}G^{A(BC):y}_{h,k}(\vec{\xi};x,y)&:=\langle\theta,{\cal
Y}_{h}^{A(BC)}(v,x){\cal Y}_{k}^{BC}(u,y)w\rangle y^{{\rm
gr}^{234}(\vec{\xi})},\vspace{1mm}\cr
G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)&:=\langle\theta,{\cal
Y}_{h}^{A(BC)}(v,x){\cal Y}_{k}^{BC}(u,y)w\rangle(x-y)^{{\rm
gr}^{234}(\vec{\xi})},\vspace{1mm}\cr
G^{(AB)C:y}_{h,k}(\vec{\xi}^{\prime};x,y)&:=\langle\theta^{\prime},{\cal
Y}_{h}^{(AB)C}({\cal Y}_{k}^{AB}(v,x-y)u,y)w\rangle y^{{\rm
gr}^{234}(\vec{\xi})},\mbox{ and}\vspace{1mm}\cr
G^{(AB)C:x-y}_{h,k}(\vec{\xi}^{\prime};x,y)&:=\langle\theta^{\prime},{\cal
Y}_{h}^{(AB)C}({\cal Y}_{k}^{AB}(v,x-y)u,y)w\rangle(x-y)^{{\rm
gr}^{234}(\vec{\xi})}.\end{array}$

Then we will obtain the following reduction theorem.

Theorem 3.1 for $G^{A(BC)}$ For $\vec{\xi}=(\theta,v,u,w)\in\Omega_{N}$,
$\alpha\in V$, and $x_{0}\not=0\not=y_{0}$,
(1) $G_{h,k}^{A(BC):y}(\vec{\xi};x_{0},y)$ is a linear combination of
$\\{G_{h,k}^{A(BC):y}(\vec{\mu};x_{0},y)\mid\vec{\mu}\in{\cal J}^{0}_{N}\\}$
with coefficients in
${\mathbb{C}}[\iota_{x_{0},y}\\{(x_{0}-y)^{-1}\\}][y]\subseteq{\mathbb{C}}[[y]]$
and
(2) $G_{h,k}^{A(BC):x-y}(\vec{\xi};x,y_{0})$ is a linear combination of
$\\{G_{h,k}^{A(BC):x-y}(\vec{\mu};x,y_{0})\mid\vec{\mu}\in{\cal J}^{0}_{N}\\}$
with coefficients in
${\mathbb{C}}[x,\iota_{y_{0},x-y_{0}}\\{x^{-1}\\}][x-y_{0}]\subseteq{\mathbb{C}}[[x-y_{0}]]$.
Furthermore, for the residue classes of coefficients modulo
${\mathbb{C}}[[x-y_{0}]](x-y_{0})$, we have:

$\begin{array}[]{ll}G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[2]}\vec{\xi};x,y_{0})\equiv\sum_{j=0}^{\infty}G_{h,k}^{A(BC):x-y}(\alpha_{j}^{[3]}\vec{\xi};x,y_{0})&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})},\cr
G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[3]}\vec{\xi};x,y_{0})\equiv\sum_{j=0}^{\infty}(-1)^{j+1}G_{h,k}^{A(BC):x-y}(\alpha_{j}^{[2]}\vec{\xi};x,y_{0})&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})},\cr
G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[4]}\vec{\xi};x,y_{0})\equiv
0&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})}.\end{array}$

Using $L(-1)$-derivative properties:
$F^{A(BC)}(L(-1)^{[3]}\vec{\xi};x,y)=\frac{{\partial}}{{\partial}y}F^{A(BC)}(\vec{\xi};x,y)$
and
$F^{A(BC)}(L(-1)^{[2]}\vec{\xi};x,y)=\frac{{\partial}}{{\partial}x}F^{A(BC)}(\vec{\xi};x,y)$,
we will show that there are
$\lambda^{34,\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)\in{\mathbb{C}}[\iota_{x_{0},y}\\{(x_{0}-y)^{-1}\\}][y]$
and
$\lambda^{23,\vec{\xi},h,k}_{\vec{\mu},p,q}(x,y_{0})\in{\mathbb{C}}[x,\iota_{y_{0},x-y_{0}}\\{x^{-1}\\}][x-y_{0}]$
such that

$\begin{array}[]{l}\frac{{\partial}}{{\partial}y}G^{A(BC):y}_{h,k}(\vec{\xi};x_{0},y)=\frac{1}{y}\sum_{\vec{\mu}\in{\cal
J}^{0}_{N}}\sum_{(p,q)\in
K^{2}}\lambda^{34,\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)G^{A(BC):y}_{p,q}(\vec{\mu};x_{0},y)\mbox{
and}\cr\frac{{\partial}}{{\partial}x}G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})=\frac{1}{x-y_{0}}\sum_{\vec{\mu}\in{\cal
J}^{0}_{N}}\sum_{(p,q)\in
K^{2}}\lambda^{23,\vec{\xi},h,k}_{\vec{\mu},p,q}(x,y_{0})G^{A(BC):x-y}_{p,q}(\vec{\mu};x,y_{0})\end{array}$

as formal ${\mathbb{C}}$-power series. Set $r\\!=\\!|{\cal J}_{N}||K^{2}|$ and
define matrix-valued functions
$\Lambda^{23}(x,y_{0})\\!=\\!\left(\lambda^{23,\vec{\xi},h,k}_{\vec{\mu},p,q}(x,y_{0})\right)_{\\!(\vec{\xi},h,k),(\vec{\mu},p,q)\in{\cal
J}_{N}\times K^{2}}\\!\\!\in\\!M_{r\times r}({\cal
O}({\mathbb{D}}_{|y_{0}|}(y_{0})))$ and
$\Lambda^{34}(x_{0},y)\\!=\\!\left(\lambda^{34,\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)\right)_{(\vec{\xi},h,k),(\vec{\mu},p,q)\\!\\!\in{\cal
J}_{N}\times K^{2}}\\!\in\\!M_{r\times r}({\cal
O}({\mathbb{D}}_{|x_{0}|}(0)))$. Here and hereafter
${\mathbb{D}}_{R}(z_{0})=\\{z\in{\mathbb{C}}\mid|z\\!-\\!z_{0}|\\!<\\!R\\}$
and ${\cal O}(U)$ denotes the set of holomorphic functions on $U$. We note
$\lambda^{23,\vec{\xi},h,k}_{\vec{\mu},p,q}\\!=\\!0\\!=\\!\lambda^{34,\vec{\xi},h,k}_{\vec{\mu},p,q}$
if $\vec{\mu}\not\in{\cal J}^{0}_{N}$. Then we get:

Theorem 3.2 [Differential systems] Fix $x_{0}\not=0$ and $N\in{\mathbb{N}}$.
Then a vector valued function
$G^{y}(x_{0},y):=\left(G^{A(BC):y}_{h,k}(\vec{\xi};x_{0},y)\right)_{\vec{\xi}\in{\cal
J}_{N},(h,k)\in K^{2}}$ satisfies a Fuchsian system

$\frac{d}{dy}G^{y}(x_{0},y)=\frac{\Lambda^{34}(x_{0},y)}{y}G^{y}(x_{0},y)$
$None$

over ${\mathbb{D}}_{|x_{0}|}(0)$ with polar locus $\\{0\\}$. Similarly, for a
fixed $0\not=y_{0}\in{\mathbb{C}}$ and $N\in{\mathbb{N}}$, a vector valued
function
$G^{x-y}(x,y_{0}):=\left(G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})\right)_{\vec{\xi}\in{\cal
J}_{N},(h,k)\in K^{2}}$ satisfies a Fuchsian system

$\frac{d}{dx}G^{x-y}(x,y_{0})=\frac{\Lambda^{23}(x,y_{0})}{x-y_{0}}G^{x-y}(x,y_{0})$
$None$

over ${\mathbb{D}}_{|y_{0}|}(y_{0})$ with polar locus $\\{y_{0}\\}$.
Furthermore, the sets of nonzero eigenvalues (without multiplicities) of the
constant matrices $\Lambda^{34}(x_{0},0)$ of $(D1)$ and
$\Lambda^{23}(y_{0},y_{0})$ of $(D2)$ are determined by only the choices of
the bases $J_{P_{A}}$ and $J_{P_{B}}$ of $P_{A}$ and $P_{B}$.

We also have similar results for
$\bar{G}^{y}(x_{0},y)=\left(G^{(AB)C:y}_{h,k}(\vec{\xi}^{\prime};x_{0},y)\right)_{\vec{\xi}^{\prime}\in{\cal
J}^{\prime}_{N},(h,k)\in{K^{\prime}}^{2}}$ and
$\bar{G}^{x-y}(x,y_{0})=\left(G^{(AB)C:x-y}_{h,k}(\vec{\xi}^{\prime},x,y_{0})\right)_{\vec{\xi}^{\prime}\in{\cal
J}^{\prime}_{N},(h,k)\in{K^{\prime}}^{2}}$.

As a corollary of $(D1)$ for $G^{A(BC)}$ and $(D2)$ for $G^{(AB)C}$, we will
obtain the following:

Corollary 3.3 $F_{h,k}^{A(BC)}(\vec{\xi};x,y)$ and
$F_{h,k}^{(AB)C}(\vec{\xi};x-y,y)$ are locally normally convergent on
$\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|y|<|x|\\}$ and
$\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|x-y|<|y|\\}$, respectively.

Set ${\cal D}^{2}=\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|x\\!-\\!y|<|y|<|x|\mbox{
and }x,y,x\\!-\\!y\not\in{\mathbb{R}}^{\leq 0}\\}$. Let
$y_{0}\not\in{\mathbb{R}}^{\leq 0}$ and we take a principal branch
$\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})$ of
$G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})$ on a small domain ${\cal
D}^{2}_{(x,y_{0})}\\!=\\!\\{x\in{\mathbb{C}}\mid(x,y_{0})\in{\cal D}^{2}\\}$.
As we will explain in Proposition 2.9 in §2, there is a finite subset
$\Delta^{\prime}=\\{d_{1},...,d_{p}\\}\subseteq{\mathbb{C}}$ which depends
only on the choice of $J_{P_{A}}$ and $J_{P_{B}}$ such that all components of
solutions of the Fuchsian system (D2) with polar locus $\\{y_{0}\\}$ are
written as

$\sum_{d\in\Delta^{\prime}}\sum_{m\in{\mathbb{N}}}\sum_{t=0}^{K}r_{d+m,t}(x-y_{0})^{d+m}\log^{t}(x-y_{0})$

with $r_{d+m,t}\in{\mathbb{C}}$ at
${\mathbb{D}}_{|y_{0}|}(y_{0})=\\{x\in{\mathbb{C}}\mid
0<|x-y_{0}|<|y_{0}|\\}$. Since this is true for every
$y=y_{0}\not\in{\mathbb{R}}^{\leq 0}$ and $\vec{\xi}$, there is
$r^{h,k}_{d+m,t}(\vec{\xi},y)\in{\mathbb{C}}$ and
$K(\vec{\xi})\in{\mathbb{N}}$ such that

$\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)=\sum_{d\in\Delta^{\prime}}\sum_{m\in{\mathbb{N}}}\sum_{t=0}^{K(\vec{\xi})}r^{h,k}_{d+m,t}(\vec{\xi},y)(x-y)^{d+m}\log^{t}(x-y).$

Multiplying it by $(x-y)^{-{\rm gr}^{234}(\vec{\xi})}\log^{h}(x)\log^{k}(y)$
for $h,k$ and taking a sum of them and replacing $\log(x)$ by
$\sum_{j=0}^{\infty}\frac{(-1)^{j}}{j+1}(\frac{x-y}{y})^{j+1}+\log(y)$, we
obtain an expansion of a principal branch $\widetilde{F}(\langle\theta,{\cal
Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle)$ of $F^{A(BC)}(\vec{\xi};x,y)$ on
${\cal D}^{2}$. We denote it by $\widetilde{F}^{A(BC)}(\vec{\xi};x,y)$. Since
$\dim P_{C}\\!<\\!\infty$, setting $\Delta\\!=\Delta^{\prime}!-\\!{\rm
max}\\{{\rm gr}(w)\mid w\in\\!P_{C}\\}$, we can write

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\sum_{t=0}^{K(\vec{\xi})}\sum_{-s-1\in\Delta+{\mathbb{N}}-{\rm
gr}(v)-{\rm gr}(u)}g_{s,t}(\vec{\xi};y)(x-y)^{-s-1}\log^{t}(x-y)$ $None$

for $\vec{\xi}=(\theta,v,u,w)\in\Omega$ with $w\in P_{C}$ on ${\cal D}^{2}$.
Although our argument is depending on the choice ${\cal J}_{N}$, we will show
that $(4.5)$ holds for every $\vec{\xi}\in\Omega$ with the same $\Delta$.

Since $g_{s,t}(\vec{\xi};y)$ is multi-linear on
$\vec{\xi}\in(A(BC))^{\vee}\times A\times B\times C$, we will view it as a map
$A\times B\to{\rm Hom}(C,A(BC))\otimes{\cal
O}({\mathbb{C}}\backslash{\mathbb{R}}^{\leq 0})$, in other words, using formal
operators ${\cal Y}^{3}$ and ${\cal Y}^{4}$, we can rewrite RHS of (4.5) into
$\langle\theta,{\cal Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle$ such that

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle$ $None$

on ${\cal D}^{2}$ for $\vec{\xi}\in\Omega$. In §4, using these notation, we
will prove that ${\cal Y}^{3}$ and ${\cal Y}^{4}$ are logarithmic surjective
intertwining operators, which implies the existence of surjective homomorphism
$(A\boxtimes B)\boxtimes C\to A\boxtimes(B\boxtimes C)$ and then we will get
the main theorem (Theorem 4.1).

We will also prove another axiom of monoidal category called pentagon axiom in
§5.

Acknowledgments. The author wishes thank Toshiyuki Abe, Scott Carnahan, Lam
Ching Hung, Hiroshi Yamauchi and Atsushi Matsuo for their useful advice and
comments. He also wishes to thank Mathematical Institute of Academia Sinica
for hospitality.

##  2 Notation and preliminary results

###  2.1 Differential systems

We quote the following statements from [2]. For the proofs, see [2]. See also
[5].

###### Definition 2.1

A meromorphic system of linear differential equations is a differential
equation of the form $\frac{d}{dz}Y=AY$, where $A\in M_{r\times r}({\cal
M}(U))$ is a square matrix whose coefficients are meromorphic functions over
some complex domain $U\subseteq{\mathbb{C}}$ (open, nonempty).

$\bullet$ A singularity of such a meromorphic system is a point in $U$
corresponding to a pole of (at least one of the coefficients of) $A$.
$\bullet$ For technical reasons, one usually fixes a discrete subset
$\Sigma\subseteq U$, and speaks of meromorphic systems $\frac{d}{dz}Y=AY$ over
$U$ with polar locus $\Sigma$ if the ”true polar locus” (i.e., the set of
poles of $A$) is contained in $\Sigma$.
$\bullet$ A meromorphic system $\frac{d}{dz}Y=\frac{A}{z-z_{0}}Y$ with a
holomorphic matrix function $A$ of size $r\times r$ defined on $U$ is called a
Fuchsian system and its singularity at $z=z_{0}$ is called a Fuchsian
singularity.

###### Definition 2.2

Let $\frac{d}{dz}Y=AY$ be a meromorphic system of rank $r$ over
$U\subseteq{\mathbb{C}}$. Let $U^{\prime}\subseteq U$ be open. A fundamental
solution of $\frac{d}{dz}Y=AY$ over $U^{\prime}$ is a matrix-valued
meromorphic function $\Phi\in M_{r\times r}({\cal M}(U^{\prime}))$ such that

$\frac{d}{dz}\Phi(z)=A(z)\Phi(z)\quad\forall z\in U^{\prime},$

and moreover

$\det(\Phi)(z)\not=0\quad\forall z\in U^{\prime}.$

From now on, let $z_{0}\in U$ and let $A=\sum_{k\geq 0}(z-z_{0})^{k}A_{k}$
with $A_{k}\in M_{r\times r}({\mathbb{C}})$ and we will consider

$\frac{d}{dz}Y=\frac{A}{z-z_{0}}Y,$ $None$

which we will call Fuchsian system. Set
${\mathbb{D}}_{R}(z_{0})=\\{z\in{\mathbb{C}}\mid|z-z_{0}|<R\\}$. If
$A=A_{0}\in M_{r\times r}({\mathbb{C}})$ and $z_{0}\in U$, then

$\frac{d}{dz}Y=\frac{A_{0}}{z-z_{0}}Y$ $None$

is called Euler system. In this case, there is $G\in GL_{r}({\mathbb{C}})$
such that $GA_{0}G^{-1}$ has a Jordan standard form and
$\frac{d}{dz}(GY)=G\frac{d}{dz}Y=\frac{GA_{0}G^{-1}}{z-z_{0}}(GY)$.

###### Lemma 2.3

If an Euler system (2.2) has a Jordan cell $A_{0}=J_{r}(a)$, that is,
$Y={}^{t}(f_{1},\ldots,f_{r})$ satisfies
$(z-z_{0})\frac{d}{dz}f_{i}=af_{i}+f_{i+1}$ for $i=1,...,r$ with $f_{r+1}=0$,
then
$\Phi=\\{(z-z_{0})^{a}\times{}^{t}(\log^{s}(z-z_{0}),...,\log(z-z_{0}),1,0,\cdots,0)\mid
s=0,...,r-1\\}$ is a fundamental solution, where ${}^{t}(\cdots)$ denotes the
transpose of $(\cdots)$.

Therefore, in generally, all components of solutions of Euler system (2.2) are
written as

$f(z)=\sum_{i=1}^{r}\sum_{t=0}^{r}r_{i,t}(z-z_{0})^{d_{i}}\log^{t}(z-z_{0})$
$None$

with $r_{i,t}\in{\mathbb{C}}$ and eigenvalues $d_{i}$ of $A_{0}$. We also note
that if $\frac{d}{dz}Y=\frac{A}{z-z_{0}}Y$, then

$\frac{d}{dz}((z-z_{0})^{d}Y)=\frac{(A+dE_{r})}{z-z_{0}}((z-z_{0})^{d}Y),$
$None$

where $E_{r}$ is the identity matrix of size $r$.

###### Definition 2.4

Let $\frac{d}{dz}Y=AY$ be a meromorphic system over $U$ with polar locus
$\Sigma$.
$\bullet$ A holomorphic gauge transformation over $U^{\prime}\subseteq U$ is
the change of variable $Y=\Delta Z$, yielding a meromorphic system
$\frac{d}{dz}Z=BZ$ over $U^{\prime}$ with polar locus $\Sigma\cap U^{\prime}$,
where $\Delta\in GL_{r}({\cal O}(U^{\prime}))$, i.e. $\Delta$ is a holomorphic
matrix function over $U^{\prime}$ with non-vanishing determinant.
$\bullet$ A meromorphic gauge transformation over $U^{\prime}\subseteq U$ is
the change of variable $Y=\Delta Z$, yielding a meromorphic system
$\frac{d}{dz}Z=BZ$ over $U^{\prime}$ with polar locus $Z\cap U^{\prime}$,
where $\Delta\in GL_{r}({\cal M}(U^{\prime}))$, i.e. $\Delta$ is a meromorphic
matrix function over $U^{\prime}$ whose determinant is not identically zero on
any connected component of $U^{\prime}$, such that moreover,
$\Delta|_{U^{\prime}\backslash\Sigma}\in M_{r\times r}({\cal
O}(U^{\prime}\backslash\Sigma))$ and
$\Delta^{-1}|_{U^{\prime}\backslash\Sigma}\in M_{r\times r}({\cal
O}(U^{\prime}\backslash\Sigma))$.

$\bullet$ As usual, two meromorphic systems over $U$ with polar locus $\Sigma$
are said to be holomorphically (resp. meromorphically) gauge equivalent, if
they are related via a global holomorphic (resp. meromorphic) gauge
transformation.

######  Definition 2.5 (Definition 3.6 in [2])

A Fuchsian system $\frac{d}{dz}Y=\frac{B}{z}Y$ of rank $r$ over
${\mathbb{D}}$, with polar locus $\\{0\\}$, is said to be of Levelt normal
form if $B=\sum_{k\geq 0}B_{k}z^{k}$ such that for any $k\in{\mathbb{N}}$, the
matrix $B_{k}\in M_{r,r}({\mathbb{C}})$ satisfies ${\rm
ad}(B_{0,s})(B_{k})=kB_{k}$, where $B_{0}=B_{0,s}+B_{0,n}$ is the Dunford
decomposition of $B_{0}$ into semisimple part $B_{0,s}$ and nilpotent part
$B_{0,n}$ with $B_{0,s}B_{0,n}=B_{0,n}B_{0,s}$.

###### Lemma 2.6

The eigenvalues of $B(1)=\sum_{k=0}^{\infty}B_{k}$ are equal to the
eigenvalues of $B_{0}$.

[Proof] We may assume that $B_{0,s}$ is a diagonal matrix
$(\lambda_{1},...,\lambda_{r})$ and $B_{0,n}$ is an upper triangular matrix.
Furthermore, we may assume $\Re(\lambda_{1})\geq\cdots\geq\Re(\lambda_{r})$.
Then since ${\rm ad}(B_{0,s})E_{i,j}=(\lambda_{i}-\lambda_{j})E_{i,j}$, ${\rm
ad}(B_{0,s})B_{k}=kB_{k}$ means that $B_{k}$ are strictly upper triangular and
$B(1)=\sum_{k=0}^{\infty}B_{k}$ is an upper triangular matrix with diagonal
entries $\\{\lambda_{1},\ldots,\lambda_{r}\\}$. $\square$

######  Theorem 2.7 (Theorem 3.8 in [2])

Let $\frac{d}{dz}Y=\frac{A}{z}Y$ be a Fuchsian system over ${\mathbb{D}}$.
This system is holomorphically gauge equivalent to some Fuchsian system
$\frac{d}{dz}\widehat{Y}=\frac{B}{z}\widehat{Y}$ of Levelt normal form over
${\mathbb{D}}$.

###### Remark 2.1

Under the setting of Theorem 2.7, the set of eigenvalues of $B(0)$ coincides
with the set of eigenvalues of $A(0)$ counted with multiplicity.

######  Theorem 2.8 (Lemma 3.7 in [2])

Let $B_{0,s}\in M_{r\times r}({\mathbb{C}})$ be semisimple. Then there exists
the unique semisimple matrix $L$ such that $L$ commutes with $B_{0,s}$ and has
only integer eigenvalues and any eigenvalues $\mu$ of $B_{0,s}-L$ satisfies
$\Re(\mu)\in[0,1[$. Moreover, if $\frac{d}{dz}Y=\frac{B}{z}Y$ is a Fuchsian
system of Level normal form with $B(0)=B_{0,s}+B_{0,n}$ as above, then
$B=z^{L}B(1)z^{-L}$.

###### Remark 2.2

As mentioned in the proof of Corollary 3.9 in [2], under the assumption in
Theorem 2.8, $\tilde{Y}:=z^{-L}Y$ satisfies an Euler system
$\frac{d}{dz}\tilde{Y}=\frac{B(1)-L}{z}\tilde{Y}$.

Using the these results, we get the following result.

###### Proposition 2.9

Let $\frac{d}{dz}Z=\frac{B}{z}Z$ be a Fuchsian system with $B\in M_{r\times
r}({\cal O}(U))$ and let $\\{d_{1},\cdots,d_{r}\\}$ be the set of eigenvalues
of the constant matrix $B_{0}$. Choose $\ell_{i}\in{\mathbb{Z}}$ so that
$0\leq d_{i}-\ell_{i}<1$. Then all components of solutions of
$\frac{d}{dz}Z=\frac{B}{z}Z$ are written as

$\sum_{j=1}^{r}\sum_{i=1}^{r}\sum_{m\in{\mathbb{N}}}\sum_{t=0}^{r}r^{j}_{i,m,t}z^{d_{j}-\ell_{j}+\ell_{i}+m}\log^{t}(z)$
$None$

with $r^{j}_{i,m,t}\in{\mathbb{C}}$.

[Proof] If $\frac{d}{dz}Z=\frac{A}{z}Z$ and $\frac{d}{dz}Y=\frac{B}{z}Y$ are
holomorphically equivalent, then all components of solutions of one are linear
combinations of components of solutions of the other with coefficients in
${\mathbb{C}}[[z]]$ and so if all components of solutions of
$\frac{d}{dz}Z=\frac{A}{z}Z$ satisfy (2.5), then those of solutions of
$\frac{d}{dz}Y=\frac{B}{z}Y$ also satisfy (2.5). Therefore, we may assume that
$\frac{d}{dz}Z=\frac{B}{z}Z$ is a Fuchsian system of Levelt normal form over
${\mathbb{D}}$ by Theorem 2.7. Furthermore we may assume that $B_{0,s}={\rm
diag}(d_{1},...,d_{r})$ satisfying $\Re(d_{1})\geq\cdots\geq\Re(d_{r})$ and
$B_{0,n}$ is an upper triangular matrix. In this case, $L$ in Theorem 2.8 is a
diagonal matrix ${\rm diag}(\ell_{1},...,\ell_{r})$. Then by Theorem 2.8 and
Remark 2.1, we have $B=z^{L}B(1)z^{-L}$ and $\tilde{Y}:=z^{-L}Y$ satisfies an
Euler system $\frac{d}{dz}\tilde{Y}=\frac{B(1)-L}{z}\tilde{Y}$. By Lemma 2.6,
$B(1)$ is an upper triangular matrix with diagonal entries
$\\{d_{1},...,d_{r}\\}$ and the eigenvalues of $B(1)-L$ are
$d_{1}-\ell_{1},\ldots,d_{r}-\ell_{r}$. Hence all components of a solution
$\tilde{Y}$ are written as
$\sum_{j=1}^{r}\sum_{t\in{\mathbb{N}}}r^{j}_{t}z^{d_{j}-\ell_{j}}\log^{t}(z)$
by Lemma 2.3. In this case, since $Y=z^{L}\tilde{Y}$, all components of
solutions of $\frac{d}{dz}Y=\frac{B}{z}Y$ are written as
$\sum_{i,j=1}^{r}\sum_{t\in{\mathbb{N}}}r^{j}_{i,t}z^{d_{j}-\ell_{j}+\ell_{i}}\log^{t}(z)$,
as we desired. $\square$

Regarding the radius of convergence of such a solution, we quote the following
theorem.

######  Theorem 2.10 (Theorem 1.6 [2])

Let $A=\sum_{k\in{\mathbb{N}}}A_{k}(z-z_{0})^{k}$ with $A_{k}\in M_{r\times
r}({\mathbb{C}})$ be a matrix power series with radius of convergence $R>0$.
Let $Y=\sum_{k=0}^{\infty}Y_{k}(z-z_{0})^{k}$ with $Y_{k}\in M_{r\times
r}({\mathbb{C}})$ be a formal power series satisfying
$\frac{d}{dz}Y=\frac{A}{z-z_{0}}Y$. Then the radius of convergence of $Y$ is
at least $R$.

###  2.2 Intertwining operators among $C_{1}$-cofinite modules

Let $A,B,C$ be $V$-modules. Let $I\binom{C}{A\,\,B}$ denote the set of
logarithmic intertwining operators of type $\binom{C}{A\,\,B}$. We introduce a
generalized concept of (logarithmic) intertwining operators. For a nonempty
domain $U\subseteq{\mathbb{C}}\backslash\\{0\\}$, we define a local
intertwining operator ${\cal Y}$ of type $\binom{C}{A\,\,B}$ on $U$ as a
linear map $A\otimes B\to C\otimes{\cal O}(U)$ , we denote it by
$\langle\theta,{\cal Y}(v,z)u\rangle$ for $v\in A,u\in B,\theta\in C^{\vee}$
and $z\in U$, satisfying
${\rm(I\,\,1)}$ [commutativity] $\langle\theta,\alpha_{m}{\cal Y}(v,z)u-{\cal
Y}(v,z)\alpha_{m}u\rangle=\sum_{j=0}^{\infty}\binom{m}{j}\langle\theta,{\cal
Y}(\alpha_{j}v,z)u\rangle z^{m-j}$,
${\rm(I\,\,2)}$ [associativity]
$\langle\theta,{\cal
Y}(\alpha_{m}v,z)u\rangle=\sum_{j=0}^{\infty}\binom{m}{j}(-1)^{j}\langle\theta,\\{\alpha_{m-j}z^{j}{\cal
Y}(v,z)u-(-1)^{m}{\cal Y}(v,z)\alpha_{j}z^{m-j}u\\}\rangle$, and
${\rm(I\,\,3)}$ [$L(-1)$-derivative property] $\langle\theta,{\cal
Y}(L(-1)v,z)u\rangle=\frac{d}{dz}\langle\theta,{\cal Y}(v,z)u\rangle$
for $\alpha\in V$ and $m\in{\mathbb{Z}}$, where $\langle\theta,\alpha_{m}{\cal
Y}(v,z)u\rangle$ denotes $\langle(\alpha_{m})^{\ast}\theta,{\cal
Y}(v,z)u\rangle$ with a contragredient operator $(\alpha_{m})^{\ast}$ of
$\alpha_{m}$.

An aim in this subsection is to show that every local intertwining operator
for $A,B,C\in{\cal N}{\cal C}_{1}$ on a domain $U$ is a branch of a
logarithmic intertwining operator on $U$.

Let $W\in{\cal N}\\!{\cal C}_{1}$. Since $C_{1}(W)={\rm
Span}_{{\mathbb{C}}}\\{\alpha_{-1}(P_{W}+C_{1}(W))\mid\alpha\in V_{\geq
1}\\}$, we have:

###### Lemma 2.11

$W={\rm Span}_{{\mathbb{C}}}\\{\alpha^{1}_{-1}\cdots\alpha^{k}_{-1}w\mid w\in
P_{W},\alpha^{j}\in V_{\geq 1},k\geq 0\\}$. In particular, $W$ is a finitely
generated $V$-module.

Therefore, there are finitely many $d_{1},...,d_{t}\in{\mathbb{C}}$ such that
$W=\oplus_{j=1}^{t}\oplus_{n\in{\mathbb{N}}}W_{d_{j}+n}$ and $\dim
W_{s}<\infty$ for $s\in{\mathbb{C}}$, where $W_{s}$ denotes a generalized
eigenspace in $W$ of $L(0)$ with eigenvalue $s$. In particular, if $W$ is
indecomposable, then $W=\oplus_{m\in{\mathbb{N}}}W_{d+m}$ for some
$d\in{\mathbb{C}}$ and $W_{d}\not=0$. We use the notation ${\rm wt}(w)=s$ for
$w\in W_{s}$ and ${\rm wt}(\alpha_{n})={\rm wt}(\alpha)-n-1$ for $\alpha\in
V$. Define ${\rm wt}()1\in{\rm End}_{{\mathbb{C}}}(W)$ by $({\rm wt}()1)w={\rm
wt}(w)w$ for $w\in W$. Since $W_{(m)}\subseteq{\rm
Span}_{{\mathbb{C}}}\\{\alpha^{1}_{-1}\cdots\alpha^{k}_{-1}w\mid w\in
P_{W},\alpha^{j}\in V_{\geq 1},\sum_{j=1}^{k}{\rm wt}(\alpha^{j})\leq m\\}$,
we obtain:

###### Lemma 2.12

For $m\in{\mathbb{N}}$, there is $\mu(m)\in{\mathbb{N}}$ such that $\dim
W_{(m)}\leq\mu(m)\times\dim P_{W}$. Moreover, the restricted dual
$W^{\vee}=\oplus_{m\in{\mathbb{N}}}{\rm Hom}(W_{(m)},{\mathbb{C}})$ of an
${\mathbb{N}}$-graded module $W=\oplus W_{(m)}\in{\cal N}{\cal C}_{1}$ is
isomorphic to the contragredient module $\oplus_{s\in{\mathbb{C}}}{\rm
Hom}(W_{s},{\mathbb{C}})$ of $W=\oplus_{s\in{\mathbb{C}}}W_{s}$. In
particular, $W^{\vee}$ does not depend on the choice of an
${\mathbb{N}}$-grading on $W$.

Since ${\rm wt}(\alpha_{n}w)={\rm wt}(w)+{\rm wt}(\alpha)-n-1$ and
$L(0)\alpha_{n}w=({\rm wt}(\alpha)-n-1)\alpha_{n}w+\alpha_{n}L(0)w=({\rm
wt}(\alpha_{n}w)\alpha_{n}w+\alpha_{n}(L(0)-{\rm wt}(w))w$, we get:

###### Lemma 2.13

$L(0)-{\rm wt}()1$ is a $V$-homomorphism. If $W\in{\cal N}\\!{\cal C}_{1}$,
then there is $K\in{\mathbb{N}}$ such that $(L(0)-{\rm wt}()1)^{K}W=0$ and
$(L(0)-{\rm wt}()1)^{K}W^{\vee}=0$.

[Proof] As we have just proved, $L(0)\\!-\\!{\rm wt}()1$ is a
$V$-homomorphism. Since $\dim P_{W}\\!<\\!\infty$, there is
$n\\!\in\\!{\mathbb{N}}$ such that
$P_{W}\\!\subseteq\\!\oplus_{i=0}^{n}W_{(i)}$, which is $L(0)\\!-\\!{\rm
wt}()1$-invariant and $\dim\oplus_{i=0}^{n}W_{(i)}\\!<\\!\infty$. Hence
$(L(0)\\!-\\!{\rm wt}()1)^{K}(\oplus_{i=0}^{n}W_{(i)})=0$ for some $K$. Since
$W$ is generated from $P_{W}$ by the actions of $V$, we have $(L(0)-{\rm
wt}()1)^{K}W=0$. From $\langle(L(0)-{\rm wt}()1)^{K}W^{\vee},W\rangle=\langle
W^{\vee},(L(0)\\!-\\!{\rm wt}()1)^{K}W\rangle=0$, we also obtain
$(L(0)\\!-\\!{\rm wt}()1)^{K}W^{\vee}=0$. $\square$

###### Lemma 2.14

Let $A,B,C\in{\cal N}{\cal C}_{1}$ and let ${\cal Y}$ be a local intertwining
operator of type $\binom{C}{A\,\,B}$ on a domain $U$ with a form
$\langle\theta,{\cal
Y}(v,z)u\rangle=\sum_{h=0}^{K(\theta,v,u)}\langle\theta,{\cal
Y}_{h}(v,z)u\rangle\log^{h}(z)$ with ${\mathbb{C}}$-formal power series ${\cal
Y}_{h}(v,z)u$ and $K(\theta,v,u)\in{\mathbb{N}}$. Then there is
$K\in{\mathbb{N}}$ such that $K(\theta,v,u)\leq K$.

[Proof] Since

$\sum_{h=0}^{K(\theta,v,u)}\langle\theta,{\cal
Y}_{h}(L(-1)v,z)u\rangle\log^{h}(z)=\frac{{\partial}}{{\partial}z}\\{\sum_{h=0}^{K(\theta,v,u)}\langle\theta,{\cal
Y}_{h}(v,z)u\rangle\log^{h}(z)\\},$

we have ${\cal Y}_{k+1}(v,z)u={\cal
Y}_{k}(L(-1)v,z)u-\frac{{\partial}}{{\partial}z}({\cal
Y}_{k}(v,z)u)\times\frac{z}{k+1}$. We also get

$\begin{array}[]{l}\langle\theta,{\rm wt}()1({\cal
Y}_{k}(v,x)u)\rangle=\langle\theta,{\rm
wt}()1(\sum_{m}v_{m,k}uz^{-m-1})\rangle\cr\mbox{}\qquad=({\rm gr}(u)+{\rm
gr}(v))\langle\theta,\sum_{m}v_{m,k}uz^{-m-1}\rangle+\langle\theta,\sum_{m}(-m-1)v_{m,k}uz^{-m-1}\rangle\cr\mbox{}\qquad=({\rm
gr}(v)+{\rm gr}(u))\langle\theta,{\cal
Y}_{k}(v,z)u\rangle+\frac{{\partial}}{{\partial}z}\langle\theta,{\cal
Y}_{k}(v,z)u\rangle z.\end{array}$

Using the notation $\Xi=L(0)\\!-\\!{\rm wt}()1$ and
$(X+Y+Z)^{\ell}=\sum_{i,j,p\in{\mathbb{N}}}\lambda^{\ell}_{i,j,p}X^{i}Y^{j}Z^{p}$,
we obtain

$\begin{array}[]{l}\langle\theta,(k+1){\cal
Y}_{k+1}(v,z)u\rangle\cr\mbox{}\qquad=\langle\theta,(L(0)\\!-\\!{\rm
wt}()1){\cal Y}_{k}(v,x)u+{\cal Y}_{k}(v,x)(L(0)\\!-\\!{\rm wt}()1)u+{\cal
Y}_{k}((L(0)\\!-\\!{\rm
wt}()1)v,x)u\rangle\cr\mbox{}\qquad=\langle\theta,\Xi{\cal
Y}_{k}(v,x)u\rangle+\langle\theta,{\cal Y}_{k}(v,x)\Xi
u\rangle+\langle\theta,{\cal Y}_{k}(\Xi
v,x)u\rangle\cr\mbox{}\qquad=\cdots=\sum_{\\{i,j,p\in{\mathbb{N}}\mid
i+j+p=\ell\leq k\\}}\lambda^{\ell}_{i,j,p}\langle\theta,\Xi^{i}({\cal
Y}_{k+1-\ell}(\Xi^{j}v,z)\Xi^{p}u\rangle.\end{array}$

Since there is $\bar{K}\in{\mathbb{N}}$ such that
$(\Xi)^{\bar{K}}A=(\Xi)^{\bar{K}}B=(\Xi)^{\bar{K}}C=0$ by Lemma 2.13, we have
$\langle\theta,{\cal Y}_{k}(v,z)u\rangle=0$ for $k\geq 3\bar{K}$, that is, we
can take $K(\theta,v,u)\leq 3\bar{K}$ for any $\theta,v,u$. $\square$

We will use the following theorem after the proof of Theorem 4.1.

###### Theorem 2.15

Let $A,B,C\in{\cal N}\\!{\cal C}_{1}$ and $N\in{\mathbb{N}}$. For a local
intertwining operator ${\cal Y}$ of type $\binom{C}{A\,\,B}$ over $U$
satisfying ${\rm(I\,\,1)}\sim{\rm(I\,\,3)}$, there is a finite subset
$\Delta_{N}$ of ${\mathbb{C}}$ and $K\in{\mathbb{N}}$ such that
$\langle\theta,{\cal Y}(v,z)u\rangle$ has the following expression:

$\langle\theta,{\cal
Y}(v,z)u\rangle=\sum_{t=0}^{K}\sum_{-s-1\in\Delta_{N}-{\rm gr}(v)-{\rm
gr}(u)+{\rm gr}(\theta)}\langle\theta,v_{s,t}u\rangle z^{-s-1}\log^{t}(z)$

for any $\theta\in C^{\vee}_{(\leq N)},v\in A,u\in B$, where $v_{s,t}\in{\rm
Hom}(B,C)$ for $v\in V$ and we take a principal branch of $z^{-s}$ and
$\log(z)$. Furthermore, $\widetilde{{\cal
Y}}(v,z)=\sum_{t=0}^{K}\sum_{s\in{\mathbb{C}}}v_{s,t}z^{-1}\log^{t}(z)\in
I\binom{C}{A\,\,B}$. Namely, ${\cal Y}$ is a branch of a logarithmic
intertwining operator $\widetilde{{\cal Y}}$ on $U$.

[Proof] Set $S(\theta,v,u;z)=\langle\theta,{\cal Y}(v,z)u\rangle$. Choose
$N\in{\mathbb{N}}$ so that ${\rm gr}(\theta)\leq N$. Let
$Q_{N}=\\{\theta^{p}:p\in{\cal Q}_{N}\\}$, $J_{A}=\\{v^{i}\mid i\in{\cal
P}_{A}\\}$, $J_{B}=\\{u^{j}\mid j\in{\cal P}_{B}\\}$ be bases of
$C^{\vee}_{(\leq N)}$, $P_{A}$, $P_{B}$, respectively. From ${\rm(I\,\,1)}$
and ${\rm(I\,\,2)}$, for $\alpha\in V_{\geq 1}$, we have

$\begin{array}[]{l}S(\theta,\alpha_{-1}v,u;z)=\sum_{j=0}^{\infty}S((\alpha_{-1-j})^{\ast}\theta,v,u;z)\rangle
z^{j}+\sum_{j=0}^{\infty}S(\theta,v,\alpha_{j}u;z)z^{-1-j}\quad\mbox{ and}\cr
S(\theta,v,\alpha_{-1}u;z)=S((\alpha_{-1})^{\ast}\theta,v,u;z)-\sum_{j=0}^{\infty}(-1)^{j}S(\theta,\alpha_{j}v,u;z)z^{-1-j}.\end{array}$

Define ${\rm gr}^{-}(\theta,v,u)={\rm gr}(v)+{\rm gr}(u)-{\rm gr}(\theta)$ and
the total grade ${\rm gr}^{+}(\theta,v,u)={\rm gr}(v)+{\rm gr}(u)+{\rm
gr}(\theta)$ and set $T(\theta,v,u;z)=F(\theta,v,u;z)z^{{\rm
gr}^{-}(\theta,v,u)}$. Then we obtain

$\begin{array}[]{l}T(\theta,\alpha_{-1}v,u;z)=\sum_{j=0}^{\infty}T((\alpha_{-1-j})^{\ast}\theta,v,u;z)+\sum_{j=0}^{\infty}T(\theta,v,\alpha_{j}u;z)\quad\mbox{
and}\cr
T(\theta,v,\alpha_{-1}u;z)=T((\alpha_{-1})^{\ast}\theta,v,u;z)-\sum_{j=0}^{\infty}(-1)^{j}T(\theta,\alpha_{j}v,u;z).\end{array}$
$None$

###### Comment 1

(1) The coefficients are all constants.
(2) The total grades of triples in RHS of (2.6) are less than the total grade
${\rm gr}(\theta)+{\rm gr}(v)+{\rm gr}(u)+{\rm gr}(\alpha)$ of LHS of (2.6).
Hence for $(\theta,v,u)$ with ${\rm gr}(\theta)\leq N$, by iterating the
expansion (2.6) as long as we have elements of $C_{1}(A)$ or $C_{1}(B)$ in the
second and third coordinates, we finally obtain that $T(\theta,v,u;z)$ is a
linear combination of $\\{T(\theta^{p},v^{i},u^{j};z)\mid(p,i,j)\in{\cal
Q}_{N}\times{\cal P}_{A}\times{\cal P}_{B}\\}$ with constant coefficients.

By the direct calculation, we also have

$\begin{array}[]{l}\frac{d}{dz}T(\theta,v,u;z)=\frac{d}{dz}\\{S(\theta,v,u;z)z^{{\rm
gr}^{-}(\theta,v,u)}\\}\cr\mbox{}\qquad=S(\theta,L(-1)v,u;z)z^{{\rm
gr}^{-}(\theta,v,u)}+({\rm gr}^{-}(\theta,v,u))S(\theta,v,u;z)z^{{\rm
gr}^{-}(\theta,v,u)-1}\cr\mbox{}\qquad=T(\theta,L(-1)v,u;z)z^{-1}+({\rm
gr}^{-}(\theta,v,u))G(\theta,v,u;z)z^{-1}.\end{array}$

Therefore, for a vector valued function
$Z=(T(\theta^{p},v^{i},u^{j};z))_{(p,i,j)\in{\cal Q}_{N}\times{\cal
P}_{A}\times{\cal P}_{B}}$ of size $s$, there is a matrix $A_{0}\in M_{s\times
s}({\mathbb{C}})$ such that

$\frac{d}{dz}Z=\frac{A_{0}}{z}Z,$ $None$

where $s=|{\cal Q}_{N}|\times|{\cal P}_{A}|\times|{\cal P}_{B}|$, that is, $Z$
satisfies an Euler system (2.7) and $T(\theta,v,u;z)$ is a component of a
solution of (2.7) for $(\theta,v,u)\in Q_{N}\times J_{A}\times J_{B}$. By
Lemma 2.3, all components of solutions $T(\theta,v,u;z)$ of (2.7) on
${\mathbb{C}}\backslash\\{0\\}$ are linear combinations of
$\\{z^{s}\log^{t}(z)\mid s\in{\mathbb{C}},t\in{\mathbb{N}}\\}$. Namely, for
each $N,v,u$ with $N\geq{\rm gr}(\theta)$, there is a finite set
$\Delta_{N}\subseteq{\mathbb{C}}$ and $K(N,v,u)\in{\mathbb{N}}$ such that

$\langle\theta,{\cal Y}(v,z)u\rangle=\sum_{s_{j}\in\Delta_{N}-{\rm gr}(v)-{\rm
gr}(u)+{\rm
gr}(\theta)}\sum_{t=0}^{K(N,v,u)}\langle\theta,v_{-s_{j}-1,t}u\rangle
z^{s_{j}}\log^{t}(z)$ $None$

with $v_{-s_{j}-1,t}\in{\rm Hom}(B,C)$. Since $A$ and $B$ are generated from
$P_{A}$ and $P_{B}$ as $V$-modules, respectively, we know that Eq.(2.8) holds
for any $\theta\in C^{\vee},v\in A,u\in B$ and $z\in U$ by using (2.6).
Although our argument depends on the choice of $N\geq{\rm gr}(\theta)$, we
note that there is $K$ such that $K(N,v,u)\leq K$ by Lemma 2.4. Therefore it
is easy to check that $\widetilde{{\cal
Y}}(v,z)=\sum_{t=0}^{K}\sum_{s\in{\mathbb{C}}}v_{s,t}z^{-s-1}\log^{t}(z)$ is a
logarithmic intertwining operator of type $\binom{C}{A\,\,B}$ and ${\cal Y}$
is a branch of it on $U$. $\square$

$[$Fusion product$]$ In this paper, for ${\mathbb{N}}$-gradable $V$-modules
$A$ and $B$, a fusion product $(A\boxtimes B,{\cal Y}^{AB})$ is defined to be
a pair of an ${\mathbb{N}}$-gradable $V$-module $A\boxtimes B$ and ${\cal
Y}^{AB}\in I\binom{A\boxtimes B}{A\,\,B}$ satisfying the universal property
for ${\mathbb{N}}$-gradable modules, that is, for any ${\mathbb{N}}$-gradable
$V$-module $C$ and ${\cal Y}\in I\binom{C}{A\,\,B}$, there is the unique
isomorphism $\phi:A\boxtimes B\to C$ such that $\phi({\cal
Y}^{AB}(v,z)u)={\cal Y}(v,z)u$ for $v\in A$ and $u\in B$.

###  2.3 Borcherds Identities for four-point correlation functions

From the commutativity ${\rm(I\,\,1)}$ and associativity ${\rm(I\,\,2)}$, we
have the following identities:

######  Lemma 2.16 (Borcherds identities)

Let ${\cal Y}^{1}\in I\binom{A(BC)}{A\,\,B\boxtimes C}$, ${\cal Y}^{2}\in
I\binom{B\boxtimes C}{B\,\,C}$, ${\cal Y}^{3}\in I\binom{A\boxtimes
B}{A\,\,B}$, ${\cal Y}^{4}\in I\binom{(AB)C}{A\boxtimes B\,\,C}$. For
$\theta\in(A(BC))^{\vee},\theta^{\prime}\in((AB)C)^{\vee},v\in A,u\in B$, and
$w\in C$ we set $F^{12}(\vec{\xi};x,y)\\!=\\!\langle\theta,{\cal
Y}^{1}(v,x){\cal Y}^{2}(u,y)w\rangle$ and
$F^{34}(\vec{\xi}^{\prime};x,y)\\!=\\!\langle\theta^{\prime},{\cal
Y}^{3}({\cal Y}^{4}(v,x\\!-y)u,y)w\rangle$. Then for $\alpha\in V$ and
$n\in{\mathbb{Z}}$, as formal ${\mathbb{C}}$-power series with logarithm
functions, we have:

$\begin{array}[]{l}(1A)\,\,F^{12}({(\alpha_{n})^{\ast}}^{[1]}\vec{\xi};x,y)=\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha^{[2]}_{j}\vec{\xi};x,y){x^{n-j}}\cr\mbox{}\quad\quad+\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha^{[3]}_{j}\vec{\xi};x,y)y^{n-j}+F^{12}(\alpha_{n}^{[4]}\vec{\xi};x,y).\cr(1B)\,\,F^{34}({(\alpha_{n})^{\ast}}^{[1]}\vec{\xi}^{\prime};x,y)=\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[2]}\vec{\xi}^{\prime};x,y)\iota_{y,x-y}\\{x^{n-j}\\}\cr\mbox{}\quad\quad+\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[3]}\vec{\xi}^{\prime};x,y)y^{n-j}+F^{34}(\alpha_{n}^{[4]}\vec{\xi}^{\prime};x,y).\end{array}$
$\begin{array}[]{l}(2A)\,\,F^{12}(\alpha_{n}^{[2]}\vec{\xi};x,y)=\sum_{j=0}^{\infty}\binom{n}{j}F^{12}({(\alpha_{n-j})^{\ast}}^{[1]}\vec{\xi};x,y){(-x)^{j}}\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{i}F^{12}(\alpha_{i}^{[3]}\vec{\xi};x,y)\iota_{x,y}\\{(-x+y)^{n-j}\\}-\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha_{j}^{[4]}\vec{\xi};x,y){(-x)^{n-j}}.\cr(2B)\,\,F^{34}(\alpha_{n}^{[2]}\vec{\xi}^{\prime};x,y)=\\!\sum_{j=0}^{\infty}\binom{n}{j}F^{34}({(\alpha_{n-j})^{\ast}}^{[1]}\vec{\xi}^{\prime};x,y)\iota_{y,x-y}\\{(-x)^{j}\\}\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[3]}\vec{\xi}^{\prime};x,y){(-x\\!+\\!y)^{n-j}}-\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[4]}\vec{\xi}^{\prime};x,y)\iota_{y,x-y}\\{(-x)^{n-j}\\}.\end{array}$
$\begin{array}[]{l}(3A)\,\,F^{12}(\alpha_{n}^{[3]}\vec{\xi};x,y)=\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}F^{12}({(\alpha_{n-j})^{\ast}}^{[1]}\vec{\xi};x,y){y^{j}}\cr\mbox{}\quad\quad-\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha_{j}^{[2]}\vec{\xi};x,y)\iota_{x,y}\\{(x-y)^{n-j}\\}-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}F^{12}(\alpha_{i}^{[4]}\vec{\xi};x,y){y^{n-j}}.\cr(3B)\,\,F^{34}(\alpha_{n}^{[3]}\vec{\xi}^{\prime};x,y)\\!=\\!\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}F^{34}({(\alpha_{n-j})^{\ast}}^{[1]}\vec{\xi}^{\prime};x,y){y^{j}}\cr\mbox{}\quad\quad-\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[2]}\vec{\xi}^{\prime};x,y){(x-y)^{n-j}}-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}F^{34}(\alpha_{j}^{[4]}\vec{\xi}^{\prime};x,y){y^{n-j}}.\end{array}$
$\begin{array}[]{l}(4A)\,\,F^{12}(\alpha_{n}^{[4]}\vec{\xi};x,y)=F^{12}({(\alpha_{n})^{\ast}}^{[1]}\vec{\xi};x,y)-\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha_{j}^{[2]}\vec{\xi};x,y){x^{n-j}}\cr\mbox{}\quad\quad-\sum_{j=0}^{\infty}\binom{n}{j}F^{12}(\alpha_{j}^{[3]}\vec{\xi};x,y){y^{n-j}}.\cr(4B)\,\,F^{34}(\alpha_{n}^{[4]}\vec{\xi}^{\prime};x,y)=F^{34}({(\alpha_{n})^{\ast}}^{[1]}\vec{\xi}^{\prime};x,y)-\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[2]}\vec{\xi}^{\prime};x,y)\iota_{y,x-y}\\{x^{n-j}\\}\cr\mbox{}\quad\quad-\sum_{j=0}^{\infty}\binom{n}{j}F^{34}(\alpha_{j}^{[3]}\vec{\xi}^{\prime};x,y){y^{n-j}}.\end{array}$

Here $(\alpha_{m})^{\ast}$ denotes its adjoint operator of $\alpha_{m}$ and
$\iota_{x,y}\\{(x-y)^{s}\\}:=\sum_{j=0}^{\infty}\binom{s}{j}x^{s-j}(-y)^{j}$
and
$\iota_{y,x-y}\\{x^{s}\\}:=\iota_{y,x-y}\\{(y+(x-y))^{s}\\}=\sum_{j=0}^{\infty}\binom{s}{j}y^{s-j}(x-y)^{j}$.

###### Comment 2

(1) The coefficients are in ${\mathbb{C}}[x^{\pm 1},y^{\pm
1},\tau_{x,y}\\{(x-y)^{\pm 1}\\},\tau_{y,x-y}\\{x^{\pm 1}\\}]$. Hence the same
equations also hold for coefficients of logarithm functions.
(2) Then for $n\in{\mathbb{Z}}$, $m,\ell=1,2,3,4$, and $j\geq 0$, the
coefficients of $F^{12}(\alpha^{[\ell]}_{j}\vec{\xi};x,y)$ in RHS of $(mA)$
for $F^{12}(\alpha^{[m]}_{n}\vec{\xi};x,y)$ are equal to the corresponding
coefficients of $F^{34}(\alpha^{[\ell]}_{j}\vec{\xi};x-y,y)$ in RHS of $(mB)$
for $F^{34}(\alpha^{[m]}_{n}\vec{\xi};x-y,y)$ excepting the expansions by
$\iota_{x,y}$ or $\iota_{x-y,y}$. Hence the corresponding coefficients in (mA)
and (mB) are same on a domain $\\{(x,y)\in{\mathbb{C}}^{2}\mid
0<|x-y|<|y|<|x|\\}$.

[Proof] First, we prepare the following equations:

$\begin{array}[]{lr}\sum_{\ell=0}^{\infty}\sum_{j=0}^{\infty}\binom{n}{j}\binom{j}{\ell}x^{n-j}y^{j-\ell}Z^{\ell}=\sum_{\ell=0}^{\infty}\binom{n}{i}\iota_{x,y}\\{(x+y)^{n-\ell}\\}Z^{\ell}&(2.9)\cr\sum_{i,j\in{\mathbb{N}}}\binom{n-j}{i}\binom{n}{j}(-1)^{i+j}(x-y)^{j}y^{i}Z^{i+j}=\sum_{\ell=0}^{\infty}\binom{n}{\ell}(-1)^{\ell}\iota_{y,x-y}\\{x^{\ell}\\}Z^{\ell}&(2.10)\end{array}$

and we will use the equations between the coefficients of $Z^{\ell}$ of LHS
and RHS.

By commutativity ${\rm(I\,\,1)}$ and associativity ${\rm(I\,\,2)}$, we have:

$\begin{array}[]{rl}(1A)&\langle(\alpha_{n})^{\ast}\theta,{\cal
Y}^{1}(v,x){\cal Y}^{2}(u,y)w\rangle=\langle\theta,\alpha_{n}{\cal
Y}^{1}(v,x){\cal Y}^{2}(u,y)w\rangle\hfill\cr=&\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)(\alpha_{n}w)\rangle+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{1}(\alpha_{j}v,x){\cal
Y}^{2}(u,y)w\rangle{x^{n-j}}\cr&+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(\alpha_{j}u,y)w\rangle{y^{n-j}}.\cr(1B)&\langle\alpha_{a}\theta^{\prime},{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle=\langle\theta^{\prime},\alpha_{n}{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle\hfill\cr=&\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u)y)\alpha_{n}w\rangle+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}(\alpha_{j}{\cal
Y}^{4}(v,x-y)u),y)w\rangle\cr=&\langle\theta^{\prime},{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)\alpha_{n}w\rangle+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle{y^{n-j}}\cr&+\sum_{j=0}^{\infty}\binom{n}{j}\sum_{i=0}^{\infty}\binom{j}{i}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}((\alpha_{i}v),x-y)u),y)w\rangle{(x-y)^{j-i}y^{n-j}}\cr=&\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u)y)\alpha_{n}w\rangle+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle{y^{n-j}}\cr&+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{i}v,x-y)u,y)w\rangle{\iota_{y,x-y}\\{x^{n-j}\\}}\mbox{ by
$(2.9)$}.\end{array}$ $\begin{array}[]{rl}(2A)&\langle\theta,{\cal
Y}^{1}(\alpha_{n}v,x){\cal
Y}^{2}(u,y)w\rangle=\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle\theta,\alpha_{n-j}{\cal
Y}^{1}(v,x){\cal
Y}(u,y)w\rangle{x^{j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta,{\cal
Y}^{1}(v,x)\alpha_{j}{\cal Y}^{2}(u,y)w\rangle
x^{n-j}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)w\rangle{x^{j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)(\alpha_{j}w)\rangle{x^{n-j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\sum_{i=0}^{\infty}\binom{j}{i}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(\alpha_{i}u,w)\rangle{x^{n-j}y^{j-i}}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)w\rangle{x^{j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{n+j}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)(\alpha_{j}w)\rangle{x^{n-j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{n+j}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(\alpha_{j}u,w)\rangle{\iota_{x,y}\\{(x-y)^{n-j}\\}}\mbox{ by
$(2.9)$}.\cr(2B)&\langle\theta^{\prime},{\cal Y}^{3}({\cal
Y}^{4}(\alpha_{n}v,x-y)u,y)w\rangle\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle\theta^{\prime},{\cal
Y}^{3}(\alpha_{n-j}{\cal
Y}^{4}(v,x-y)u,y)w\rangle(x-y)^{j}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle(x-y)^{n-j}\cr=&\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\binom{n-j}{i}\binom{n}{j}(-1)^{i+j}\langle\theta^{\prime},\alpha_{n-i-j}{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle{(x-y)^{j}y^{i}}\cr&-\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\binom{n}{j}\binom{n-j}{i}(-1)^{i+n}\langle\theta^{\prime},{\cal
Y}_{h}^{3}({\cal
Y}_{k}^{4}(v,x-y)u,y)\alpha_{i}w\rangle{(x-y)^{j}y^{n-i-j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle{(x-y)^{n-j}}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta^{\prime},{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle{\iota_{y,x-y}\\{x^{j}\\}}\mbox{ by
$(2.10)$}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{n+j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)\alpha_{j}w\rangle{\iota_{y,x-y}\\{x^{n-j}\\}}\mbox{ by
$(2.9)$}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta^{\prime},{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle{(x-y)^{n-j}}.\end{array}$
$\begin{array}[]{rl}(3A)&\langle\theta,{\cal Y}^{1}(v,x){\cal
Y}^{2}(\alpha_{n}u,y)w\rangle=\sum_{i=0}^{\infty}\binom{n}{i}(-1)^{i}\langle\theta,{\cal
Y}^{1}(v,x)\alpha_{n-i}{\cal Y}^{2}(u,y)w\rangle
y^{j}\cr&+\sum_{i=0}^{\infty}\binom{n}{i}(-1)^{i+n+1}\langle\theta,{\cal
Y}^{1}(v,x){\cal Y}^{2}(u,y)\alpha_{i}w\rangle
y^{n-i}\cr=&\sum_{i=0}^{\infty}\binom{n}{i}(-1)^{i}\langle\theta,\alpha_{n-i}{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)w\rangle{y^{i}}\cr&-\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{n}{i}(-1)^{i}\binom{n-i}{j}\langle\theta,{\cal
Y}^{1}(\alpha_{j}v,x){\cal
Y}^{2}(u,y)w\rangle{x^{n-i-j}y^{i}}\cr&-\sum_{i=0}^{\infty}\binom{n}{i}(-1)^{i+n}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)(\alpha_{i}w)\rangle{y^{n-i}}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)w\rangle{y^{j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{1}(\alpha_{j}v,x){\cal
Y}^{2}(u,y)w\rangle{\iota_{x,y}\\{(x-y)^{n-j}\\}}\mbox{ by
$(2.9)$}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)\alpha_{i}w\rangle{y^{n-j}}.\cr(3B)&\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x\\!-\\!y)(\alpha_{n}u),y)w\rangle=\langle\theta^{\prime},{\cal
Y}^{3}(\alpha_{n}{\cal
Y}^{4}(v,x-y)u,y)w\rangle\cr&-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{j}v,x-y)u,y)w\rangle(x-y)^{n-j}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\theta^{\prime},\alpha_{n-j}{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle
y^{j}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta^{\prime},{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)\alpha_{j}w)\rangle
y^{n-j}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{j}v,u,y)w\rangle(x-y)^{n-j}\cr=&\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x\\!-\\!y)u,y)w\rangle{y^{j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x\\!-\\!y)u,y)\alpha_{j}w\rangle{y^{n-j}}\cr&-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{j}v,x\\!-\\!y)u,y)w\rangle{(x-y)^{n-j}}.\end{array}$
$\begin{array}[]{rl}(4A)&\langle\theta,{\cal Y}^{1}(v,x){\cal
Y}^{2}(u,y)(\alpha_{n}w)\rangle=\langle(\alpha_{n})^{\ast}\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(u,y)w\rangle\cr&-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{1}(v,x){\cal
Y}^{2}(\alpha_{j}u,y)w\rangle{y^{n-j}}\\!-\\!\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,{\cal
Y}^{1}(\alpha_{j}v,x){\cal
Y}^{2}(u,y)w\rangle{x^{n-j}}.\cr(4B)&\langle\theta^{\prime},{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)(\alpha_{n}w)\rangle\cr=&\langle\theta^{\prime},\alpha_{n}{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}(\alpha_{j}({\cal
Y}^{4}(v,x-y)u,y)w\rangle\cr=&\langle(\alpha_{n})^{\ast}\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u),y)w\rangle\\!-\\!\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)(\alpha_{j}u)),y)w\rangle{y^{n-j}}\cr&\\!\\!-\\!\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\binom{n}{j}\binom{j}{i}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{i}v,x-y)u,y)w\rangle{(x-y)^{j-i}y^{n-j}}\cr=&\langle(\alpha_{n})^{\ast}\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u),y)w\rangle-\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle{y^{n-j}}\cr&\\!\\!-\\!\sum_{i=0}^{\infty}\binom{n}{i}\langle\theta^{\prime},{\cal
Y}^{3}({\cal
Y}^{4}(\alpha_{i}v,x-y)u,y)w\rangle{\iota_{y,x-y}\\{x^{n-i}\\}}\mbox{ by
$(2.9)$}.\hfill\square\\\ \end{array}$

Let us explain a role of $L(-1)$-derivative properties ${\rm(I\,\,3)}$.

###### Remark 2.3

From $L(-1)$-derivative property given by $L(-1)^{[3]}$, we have:

$\begin{array}[]{rl}\frac{{\partial}}{{\partial}y}(\sum
F^{A(BC)}_{h,k}(\vec{\xi};x,y)\log^{h}(x)\log^{k}(y))&=\frac{{\partial}}{{\partial}y}F^{A(BC)}(\vec{\xi};x,y)=F^{A(BC)}(L(-1)^{[3]}\vec{\xi};x,y)\cr=&\sum_{h,k}F_{h,k}^{A(BC)}(L(-1)^{[3]}\vec{\xi};x,y)\log^{h}(x)\log^{k}(y).\end{array}\vspace{-2mm}$

By comparing the coefficients of $\log^{h}(x)\log^{k}(y)$, we get

$F^{A(BC)}_{h,k}(L(-1)^{[3]}\vec{\xi};x,y)=\frac{{\partial}}{{\partial}y}F^{A(BC)}_{h,k}(\vec{\xi};x,y)+\frac{k+1}{y}F^{A(BC)}_{h,k+1}(\vec{\xi};x,y).$
$None$

Similarly, by considering $\frac{{\partial}}{{\partial}x}$ and $L(-1)^{[2]}$,
we obtain:

$F^{A(BC)}_{h+1,k}(\vec{\xi};x,y)=\frac{x}{h+1}\\{F^{A(BC)}_{h,k}(L(-1)^{[2]}\vec{\xi};x,y)-\frac{{\partial}}{{\partial}x}F^{A(BC)}_{h,k}(\vec{\xi};x,y)\\}.$
$None$

In particular, we can recover all $F^{A(BC)}_{h,k}$ from $F^{A(BC)}_{0,0}$ by
$\frac{{\partial}}{{\partial}x}$ and $\frac{{\partial}}{{\partial}y}$.

##  3 The proof of theorems 3.1, 3.2 and Corollary 3.3

###  3.1 Theorem 3.1 (Reduction to small weight subspaces)

We recall Lemma 2.11. See Table 1 for Borcherds identities $(2A)\sim(4A)$ for
$n=-1$.

Table 1: Borcherds identity

Borcherds identities | D:coefficient | q | $D\times(x-y)^{q}\mbox{ and }D\times y^{q}$
---|---|---|---
$(2A)$ $F(\alpha_{-1}^{[2]}\vec{\xi})\times$ | 1 | 0 | 1
$\mbox{}\quad=\sum_{j=0}^{\infty}F(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi})\times$ | $x^{j}$ | ${\rm wt}(\alpha)$ | no negative powers
$\mbox{}\qquad+\sum_{j=0}^{\infty}F(\alpha_{j}^{[3]}\vec{\xi})\times$ | $\iota_{x,y}\\{(x-y)^{-j-1}\\}$ | j+1 |  $1$ and $(x-y)^{-j-1}y^{j+1}$
$\mbox{}\qquad+\sum_{j=0}^{\infty}F(\alpha_{j}^{[4]}\vec{\xi})\times$ | ${x^{-j-1}}.$ | j+1 |  $x^{-j-1}(x\\!-\\!y)^{j+1}$ and $(y/x)^{j+1}$
$(3A)$ $F(\alpha_{-1}^{[3]}\vec{\xi})\times$ | 1 | 0 | 1
$\mbox{}\quad=\sum_{j=0}^{\infty}F(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi})\times$ | ${y^{j}}$ | ${\rm wt}(\alpha)$ | no negative powers
$\mbox{}\qquad-\sum_{j=0}^{\infty}(-1)^{j}F(\alpha_{j}^{[2]}\vec{\xi})\times$ | ${\iota_{x,y}\\{(x-y)^{-j-1}\\}}$ | $j+1$ |  $1$ and $(x-y)^{-j-1}y^{j+1}$
$\mbox{}\qquad+\sum_{j=0}^{\infty}F(\alpha_{j}^{[4]}\vec{\xi})\times$ | ${y^{-j-1}}.$ | $j+1$ |  $y^{-j-1}(x-y)^{j+1}$ and $1$
$(4A)$ $F(\alpha_{-1}^{[4]}\vec{\xi})\times$ | 1 | 0 | 1
$\mbox{}\quad=F(((\alpha_{-1})^{\ast})^{[1]}\vec{\xi})\times$ | 1 | ${\rm wt}(\alpha)$ | no negative powers
$\mbox{}\qquad-\sum_{j=0}^{\infty}(-1)^{j}F(\alpha_{j}^{[3]}\vec{\xi})\times$ | $y^{-j-1}$ | j+1 |  $(x-y)^{j+1}y^{-j-1}$ and $1$
$\mbox{}\qquad-\sum_{j=0}^{\infty}(-1)^{j}F(\alpha_{j}^{[2]}\vec{\xi})\times$ | ${x^{-j-1}}.$ | j+1 |  $(x-y)^{j+1}x^{-j-1}$ and $(y/x)^{j+1}$

Here the first columns are Borcherds identities for $n=-1$ and $F(\vec{\xi})$
denotes
---
$F^{A(BC)}_{h,k}(\vec{\xi};x,y)$ and $q={\rm
gr}^{234}(\alpha_{-1}^{[m]}\vec{\xi})-{\rm
gr}^{234}(\alpha_{j}^{[i]}\vec{\xi})$ for a quadruple
$\alpha_{j}^{[i]}\vec{\xi}$ in (mA).

###### Comment 3

Since ${\rm gr}((\alpha_{-1-j})^{\ast}\theta)<{\rm gr}(\theta)$, we have
$(\alpha_{-1-j})^{\ast}\theta\in(A(BC))^{\vee}_{(<N)}$ for $\alpha\in V_{\geq
1}$ and $j\in{\mathbb{N}}$. Since ${\rm gr}(\alpha_{j}w)={\rm
gr}(\alpha_{-1}w)-j-1$ for $j\in{\mathbb{N}}$, the total grades of quadruples
of each term in RHS are less than the total grade of quadruples in LHS, that
is, ${\rm gr}(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi}),{\rm
gr}(\alpha_{j}^{[k]}\vec{\xi})<{\rm gr}(\alpha_{-1}^{[i]}\vec{\xi})$.
Therefore by iterating these reductions as long as we have an element of
$C_{1}(A),C_{1}(B)$ or $C_{1}(C)$ in the second, third or fourth coordinates
of quadruples, we finally get an expression of
$F_{h,k}^{A(BC)}(\vec{\xi};x,y)$ as a linear combination of

$\\{F_{h,k}^{A(BC)}(\vec{\mu};x,y):\vec{\mu}\in{\cal J}^{0}_{N}\\}$

with coefficients in
${\mathbb{C}}[x,x^{-1},y,y^{-1},\iota_{x,y}\\{(x-y)^{-1}\\}]$ for any
$\vec{\xi}\in\Omega_{N}$.

###### Comment 4

Since Borcherds identities for $F^{A(BC)}_{h,k}$ and $F^{(AB)C}_{h,k}$ have
the same coefficients except the way of expansions, we can get Borcherds
identities $(2B)\sim(4B)$ by viewing $F(\vec{\xi})$ as
$F^{(AB)C}_{h,k}(\vec{\xi}^{\prime};x,y)$ and expanding all coefficients by
$\iota_{y,x-y}$.

###### Comment 5

In the case of $D\times(x-y)^{q}$, negative powers appear only on $x$ or $y$.
In the case of $D\times y^{q}$, negative powers occur only on $(x-y)$ or $x$.
We hence consider

$\begin{array}[]{ll}G_{h,k}^{A(BC):y}(\vec{\xi};x_{0},y)&:=F_{h,k}^{A(BC)}(\vec{\xi};x_{0},y)y^{{\rm
gr}^{234}(\vec{\xi})},\cr
G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})&:=F^{A(BC)}_{h,k}(\vec{\xi};x,y_{0})(x-y_{0})^{{\rm
gr}^{234}(\vec{\xi})},\cr
G_{h,k}^{(AB)C:y}(\vec{\xi}^{\prime},w;x_{0},y)&:=F_{h,k}^{(AB)C}(\vec{\xi}^{\prime};x_{0},y)y^{{\rm
gr}^{234}(\vec{\xi})},\mbox{ and}\cr
G^{(AB)C:x-y}_{h,k}(\vec{\xi}^{\prime};x,y_{0})&:=F^{(AB)C}_{h,k}(\vec{\xi}^{\prime};x,y_{0})(x-y_{0})^{{\rm
gr}^{234}(\vec{\xi})}.\end{array}$ $None$

Then we have:

###### Theorem 3.1

For $\vec{\xi}=(\theta,v,u,w)$ with $\theta\in J_{N}$, $\alpha\in V$, and
$x_{0}\not=0\not=y_{0}$,
(1) $G_{h,k}^{A(BC):y}(\vec{\xi};x_{0},y)$ is a linear combination of
$\\{G_{h,k}^{A(BC):y}(\vec{\delta};x_{0},y)\mid\vec{\mu}\in{\cal
J}^{0}_{N}\\}$ with coefficients in
${\mathbb{C}}[\iota_{x_{0},y}\\{(x_{0}-y)^{-1}\\}][y]\subseteq{\mathbb{C}}[[y]]$
and
(2) $G_{h,k}^{A(BC):x-y}(\vec{\xi};x,y_{0})$ is a linear combination of
$\\{G_{h,k}^{A(BC):x-y}(\vec{\mu};x,y_{0})\mid\vec{\mu}\in{\cal J}^{0}_{N}\\}$
with coefficients in
${\mathbb{C}}[x,\iota_{y_{0},x-y_{0}}\\{x^{-1}\\}][x-y_{0}]\subseteq{\mathbb{C}}[[x-y_{0}]]$.
Furthermore, for the residue classes of coefficients modulo
${\mathbb{C}}[[x-y_{0}]](x-y_{0})$, we have:

$\begin{array}[]{ll}G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[2]}\vec{\xi};x,y_{0})\equiv\sum_{j=0}^{\infty}G_{h,k}^{A(BC):x-y}(\alpha_{j}^{[3]}\vec{\xi};x,y_{0})&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})},\cr
G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[3]}\vec{\xi};x,y_{0})\equiv\sum_{j=0}^{\infty}(-1)^{j+1}G_{h,k}^{A(BC):x-y}(\alpha_{j}^{[2]}\vec{\xi};x,y_{0})&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})},\cr
G_{h,k}^{A(BC):x-y}(\alpha_{-1}^{[4]}\vec{\xi};x,y_{0})\equiv
0&\pmod{{\mathbb{C}}[[x\\!-\\!y_{0}]](x\\!-\\!y_{0})},\cr
G_{h,k}^{A(BC):y}(\alpha_{-1}^{[2]}\vec{\xi};x_{0},y)\equiv
0&\pmod{{\mathbb{C}}[[y]]y},\cr
G_{h,k}^{A(BC):y}(\alpha_{-1}^{[3]}\vec{\xi};x_{0},y)\equiv\sum_{j=0}^{\infty}G_{h,k}^{A(BC):y}(\alpha_{j}^{[4]}\vec{\xi};x_{0},y)&\pmod{{\mathbb{C}}[[y]]y},\cr
G_{h,k}^{A(BC):y}(\alpha_{-1}^{[4]}\vec{\xi};x_{0},y)\equiv-\sum_{j=0}^{\infty}(-1)^{j}G_{h,k}^{A(BC):y}(\alpha_{j}^{[3]}\vec{\xi};x_{0},y)&\pmod{{\mathbb{C}}[[y]]y}.\end{array}$

We also obtain the same results for
$G_{h,k}^{(AB)C:x-y}(\vec{\xi}^{\prime};x-y_{0},y_{0})$ and
$G_{h,k}^{(AB)C:y}(\vec{\xi}^{\prime};x_{0}-y,y)$ by replacing
$A(BC),\vec{\xi},\vec{\mu},{\cal J}_{N},{\cal J}_{N}^{0},\iota_{x,y}$ by
$(AB)C,\vec{\xi}^{\prime},\vec{\mu}^{\prime},{\cal J}^{\prime}_{N},{{\cal
J}^{\prime}}_{N}^{0},\iota_{y,x-y}$, respectively.

[Proof] We will show the proofs for $G^{A(BC):*}_{h,k}(\vec{\xi};x,y)$. To
simplify the notation, we denote $G^{A(BC):*}_{h,k}(\vec{\xi};x,y)$ by
$G^{*}(\vec{\xi};x,y)$ for $\ast=y,x-y$. From Table 1, we have

$\begin{array}[]{l}G^{y}(\alpha_{-1}^{[2]}\vec{\xi};x_{0},y)=F(\alpha_{-1}^{[2]}\vec{\xi};x_{0},y)y^{p+{\rm
wt}(\alpha)}=\sum_{j=0}^{\infty}G^{y}(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi};x_{0},y)x_{0}^{j}y^{{\rm
wt}(\alpha)}\cr\mbox{}\quad+\sum_{j=0}^{\infty}G^{y}(\alpha_{j}^{[3]}\vec{\xi};x_{0},y)x_{0}\iota_{x_{0},y}\\{(x_{0}-y)^{-j-1}\\}y^{j+1}+\sum_{j=0}^{\infty}G^{y}(\alpha_{j}^{[4]}\vec{\xi};x_{0},y)x_{0}^{-j-1}y^{j+1},\cr
G^{y}(\alpha_{-1}^{[3]}\vec{\xi};x_{0},y)=F(\alpha_{-1}^{[3]}\vec{\xi};x_{0},y)y^{p+{\rm
wt}(\alpha)}=\sum_{j=0}^{\infty}G^{y}(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi};x_{0},y)y^{j+{\rm
wt}(\alpha)}\cr\mbox{}\quad-\sum_{j=0}^{\infty}(-1)^{j}G^{y}(\alpha_{j}^{[2]}\vec{\xi};x_{0},y)\iota_{x_{0},y}\\{(x_{0}\\!-\\!y)^{-j-1}\\}y^{j+1}+\sum_{j=0}^{\infty}G^{y}(\alpha_{j}^{[4]}\vec{\xi};x_{0},y),\cr
G^{y}(\alpha_{-1}^{[4]}\vec{\xi};x_{0},y)=F(\alpha_{-1}^{[4]}\vec{\xi};x_{0},y)y^{p+{\rm
wt}(\alpha)}=G^{y}(((\alpha_{-1})^{\ast})^{[1]}\vec{\xi};x_{0},y)y^{{\rm
wt}(\alpha)}\cr\mbox{}\quad-\sum_{j=0}^{\infty}(-1)^{j}G^{y}(\alpha_{j}^{[3]}\vec{\xi};x_{0},y)-\sum_{j=0}^{\infty}(-1)^{j}G^{y}(\alpha_{j}^{[2]}\vec{\xi};x_{0},y)x_{0}^{-j-1}y^{j+1}.\end{array}$

For $y=y_{0}\not=0$ and a variable $x$, we have:

$\begin{array}[]{l}G^{x-y}(\alpha_{-1}^{[2]}\vec{\xi};x,y_{0})=\sum_{j=0}^{\infty}G(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi};x,y_{0})x^{j}(x-y_{0})^{{\rm
wt}(\alpha)}\cr\mbox{}\quad+\sum_{j=0}^{\infty}G(\alpha_{j}^{[3]}\vec{\xi};x,y_{0})\iota_{y_{0},x-y_{0}}\\{y_{0}+(x-y_{0})\\}\cr\mbox{}\quad+\sum_{j=0}^{\infty}G(\alpha_{j}^{[4]}\vec{\xi};x,y_{0})\iota_{y_{0},x-y_{0}}\\{(y_{0}+(x-y_{0}))^{-j-1}\\}(x-y_{0})^{j+1},\cr
G^{x-y}(\alpha_{-1}^{[3]}\vec{\xi};x,y_{0})=\sum_{j=0}^{\infty}G^{x-y}(((\alpha_{-1-j})^{\ast})^{[1]}\vec{\xi};x,y_{0})y_{0}^{j}(x-y_{0})^{{\rm
wt}(\alpha)}\cr\mbox{}\quad-\sum_{j=0}^{\infty}(-1)^{j}G^{x-y}(\alpha_{j}^{[2]}\vec{\xi};x,y_{0})+\sum_{j=0}^{\infty}G^{x-y}(\alpha_{j}^{[4]}\vec{\xi};x,y_{0})y_{0}^{-j-1}(x-y_{0})^{j+1},\cr
G^{x-y}(\alpha_{-1}^{[4]}\vec{\xi};x,y_{0})=G^{x-y}(((\alpha_{-1})^{\ast})^{[1]}\vec{\xi};x,y_{0})(x-y_{0})^{{\rm
wt}(\alpha)}\cr\mbox{}\quad-\sum_{j=0}^{\infty}(-1)^{j}G^{x-y}(\alpha_{j}^{[3]}\vec{\xi};x,y_{0})y_{0}^{-j-1}(x-y_{0})^{j+1}\cr\mbox{}\quad-\sum_{j=0}^{\infty}(-1)^{j}G^{x-y}(\alpha_{j}^{[2]}\vec{\xi};x,y_{0})\iota_{y_{0},x-y_{0}}\\{(y_{0}+(x-y_{0}))^{-j-1}(x-y_{0})^{j+1}.\end{array}$

In particular, for the residue classes of coefficients modulo
${\mathbb{C}}[[x-y_{0}]](x-y_{0})$, it is easy to check the statements. Since
these processes don’t depend on the choice of $G^{A(BC)}$ nor $G^{(AB)C}$, we
get the same expressions for $G^{(AB)C:y}(\vec{\xi}^{\prime};x_{0},y)$ and
$G^{(AB)C:x-y}(\vec{\xi}^{\prime};x,y_{0})$ except the notation and expansions
by $\iota_{\ast,\ast}$.

This completes the proof of Theorem 3.1.

###  3.2 Proof of Theorem 3.2 (Differential systems)

We will prove the Fuchsian systems (D1) and (D2) for $G^{A(BC):y}_{h,k}$ and
$G^{A(BC):x-y}_{h,k}$. The proofs for the other cases are similar. From the
$L(-1)$-derivative property of intertwining operators, as
${\mathbb{C}}$-formal power series with logarithmic terms, for
$\vec{\xi}=(\theta,v,u,w)\in\Omega$ and
$\vec{\xi}^{\prime}=(\theta^{\prime},v,u,w)\in\Omega^{\prime}$, we have

$\begin{array}[]{l}\frac{{\partial}}{{\partial}y}F^{A(BC)}(\vec{\xi};x,y)=F^{A(BC)}(L(-1)^{[3]}\vec{\xi};x,y),\vspace{2mm}\cr\frac{{\partial}}{{\partial}x}F^{A(BC)}(\vec{\xi};x,y)=F^{A(BC)}(L(-1)^{[2]}\vec{\xi};x,y),\mbox{
and}\vspace{2mm}\cr\frac{{\partial}}{{\partial}(x-y)}F^{(AB)C}(\vec{\xi}^{\prime};x-y,y)=F^{(AB)C}(L(-1)^{[2]}\vec{\xi}^{\prime};x-y,y).\end{array}$

From (2.11) in Remark 2.3, we get

$\begin{array}[]{l}\frac{{\partial}}{{\partial}y}\\{G_{h,k}^{A(BC):y}(\vec{\xi};x,y)y\\}=\frac{{\partial}}{{\partial}y}\\{F_{h,k}^{A(BC)}(\vec{\xi};x,y)y^{{\rm
gr}^{234}(\vec{\xi})+1}\\}\cr\mbox{}\quad=\\{\frac{{\partial}}{{\partial}y}F_{h,k}^{A(BC)}(\vec{\xi};x,y)\\}y^{{\rm
gr}^{234}(\vec{\xi})+1}+({\rm
gr}^{234}(\vec{\xi})+1)F_{h,k}^{A(BC)}(\vec{\xi};x,y)y^{{\rm
gr}^{234}(\vec{\xi})}\cr\mbox{}\quad=G_{h,k}^{A(BC):y}(L(-1)^{[3]}\vec{\xi};x,y)\\!-\\!(k\\!+\\!1)G_{h,k+1}^{A(BC):y}(\vec{\xi};x,y)\\}y\\!+\\!({\rm
gr}^{234}(\vec{\xi})\\!+\\!1)G_{h,k}^{A(BC):y}(\vec{\xi};x,y)\cr\mbox{}\quad=\sum_{\vec{\xi}\in{\cal
J}^{0}_{N}}s_{\vec{\xi}}(x,y)G_{h,k}^{A(BC):y}(\vec{\xi};x,y),\end{array}$

where
$s_{\vec{\xi}}(x,y)\in{\mathbb{C}}[x,x^{-1},\iota_{x,,y}\\{(x-y)^{-1}\\}][y]$.
On the other hand, we obtain:

$\frac{{\partial}}{{\partial}y}\\{G_{h,k}^{A(BC):y}(\vec{\xi};x,y)y\\}=\frac{{\partial}}{{\partial}y}\\{G_{h,k}^{A(BC):y}(\vec{\xi};x,y)\\}y+G_{h,k}^{A(BC):y}(\vec{\xi};x,y).$

Combining it with the above, and then using Theorem 3.1, we get

$\begin{array}[]{rl}\frac{d}{dy}\\{G_{h,k}^{A(BC):y}(\vec{\xi};x_{0},y)\\}=&y^{-1}\frac{d}{dy}\\{G^{A(BC):y}_{h,k}(\vec{\xi};x_{0},y)y\\}-y^{-1}G^{A(BC):y}_{h,k}(\vec{\xi};x_{0},y)\cr=&\frac{1}{y}\sum_{\vec{\mu}\in{\cal
J}^{0}_{N}}\sum_{(p,q)\in
K^{2}}\lambda^{34:\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)G^{A(BC):y}_{p,q}(\vec{\mu};x_{0},y)\end{array}$

for some
$\lambda^{34:\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)\in{\mathbb{C}}[x_{0},x_{0}^{-1},\iota_{x_{0},y}\\{(x_{0}-y)^{-1}\\}][y]\subseteq{\mathbb{C}}[[y]]$.
Define

$\begin{array}[]{l}G^{y}(x_{0},y)=(G^{A(BC):y}_{h,k}(\vec{\xi};x_{0},y))_{(\vec{\xi},h,k)\in{\cal
J}_{N}\times K^{2}}\mbox{ and
}\cr\Lambda^{34}(x_{0},y)=\left(\lambda^{34:\vec{\xi},h,k}_{\vec{\mu},p,q}(x_{0},y)\right)_{(\vec{\xi},h,k),(\vec{\mu},p,q)\in{\cal
J}_{N}\times K^{2}},\end{array}$

then
$\frac{d}{dy}G^{y}(x_{0},y)=\frac{1}{y}\Lambda^{34}(x_{0},y)G^{y}(x_{0},y)$,
which is a Fuchsian system over ${\mathbb{D}}_{|x_{0}|}(x_{0})$. We note
$\lambda^{34,\vec{\xi},h,k}_{\vec{\mu},p,q}=0$ if $\vec{\mu}\not\in{\cal
J}^{0}_{N}$. Similarly, from Remark 2.3 (2.12), we have:

$\begin{array}[]{l}(x-y_{0})\frac{{\partial}}{{\partial}(x-y_{0})}G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})=G^{A(BC):x-y}_{h,k}(L(-1)^{[2]}\vec{\xi};x,y_{0})\cr\mbox{}\qquad-(h+1)G^{A(BC):x-y}_{h+1,k}(\vec{\xi};x,y_{0})x^{-1}(x-y_{0})+{\rm
gr}^{234}(\vec{\xi})G^{A(BC):x-y}(\vec{\xi};x,y_{0}).\end{array}$ $None$

Therefore, there are
$\lambda^{23:\vec{\xi},h,k}_{\vec{\mu},p,q}(x,y_{0})\in{\mathbb{C}}[x,\iota_{y_{0},x-y_{0}}\\{x^{-1}\\},y_{0}^{-1}][x-y_{0}]\subseteq{\mathbb{C}}[[x-y_{0}]]$
such that

$\frac{{\partial}}{{\partial}x}G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})=\frac{1}{x-y_{0}}\sum_{\vec{\mu}\in{\cal
J}_{N}}\sum_{(p,q)\in
K^{2}}\lambda^{23:\vec{\xi},h,k}_{\vec{\mu},p,q}(x,y_{0})G^{A(BC):x-y}_{p,q}(\vec{\mu};x,y_{0}).$

Thus
$G^{x-y}(x,y_{0}):=(G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0}))_{(\vec{\xi},p,q)\in{\cal
J}_{N}\times K^{2}}$ satisfies a Fuchsian system

$\frac{d}{dx}G^{x-y}(x,y_{0})=\frac{\Lambda^{23:x-y}(x,y_{0})}{x-y_{0}}G^{x-y}(x,y_{0}),$

where
$\Lambda^{23:x,y}(x,y_{0})=\left(\lambda^{23:\vec{\xi},h,k}_{\vec{\mu},p,q}\right)_{(\vec{\xi},h,k),(\vec{\mu},p,q)\in{\cal
J}_{N}\times K^{2}}$ and $\lambda^{23,\vec{\xi},h,k}_{\vec{\mu},p,q}=0$ if
$\vec{\mu}\not\in{\cal J}^{0}_{N}$.

We next investigate the set of eigenvalues of constant matrices. Since
$\lambda^{23:\vec{\xi},h,k}_{\vec{\mu},p,q}=0$ for $\vec{\mu}\not\in{\cal
J}_{N}^{0}$, the set of nonzero eigenvalues of the constant matrix
$\Lambda^{23;x-y}(y_{0},y_{0})$ for ${\cal J}_{N}$ is equal to that for ${\cal
J}^{0}_{N}$. So we may assume ${\cal J}_{N}={\cal J}_{N}^{0}$. From (3.2), we
know

$\begin{array}[]{l}(x-y_{0})\frac{{\partial}}{{\partial}(x-y_{0})}G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})\cr\equiv
G^{A(BC):x-y}_{h,k}(L(-1)^{[2]}\vec{\xi};x,y_{0})\\!+\\!{\rm
gr}^{234}(\vec{\xi})G_{h,k}^{A(BC):x-y}(\vec{\xi};x,y_{0})\pmod{{\mathbb{C}}[[x-y_{0}]](x-y_{0})}.\end{array}$

Furthermore, on each step of the reduction of
$G^{A(BC):x-y}_{h,k}(L(-1)^{[2]}\vec{\xi};x,y_{0})$ into a linear combination
of $\\{G^{A(BC):x-y}_{h,k}(\vec{\mu};x,y_{0})\mid\vec{\mu}\in{\cal
J}^{0}_{N}\\}$, the coefficients at the first term with
${(\alpha_{j})^{\ast}}^{[1]}\vec{\xi}$ and at the 4th term with
$\alpha_{j}^{[4]}\vec{\xi}$ are contained in
${\mathbb{C}}[[x-y_{0}]](x-y_{0})$ by Theorem 3.1. In other words, modulo
${\mathbb{C}}[[x-y_{0}]](x-y_{0})$, the process of these reductions are
independent from the choice of $(\theta,w)$ and we don’t change the subscripts
$h,k$. Therefore, for
$\vec{\xi}=(\theta,v,u,w),\vec{\mu}=(\tilde{\theta},\tilde{v},\tilde{u},\tilde{w})\in{\cal
J}_{N}$, there are
$\lambda^{(v,u)}_{(\tilde{v},\tilde{u})}(h,k;x,y_{0})\in{\mathbb{C}}$ such
that

$\lambda^{23,(\theta,v,u,w),h,k}_{(\tilde{\theta},\tilde{v},\tilde{u},\tilde{w}),p,q}(x,y_{0})\equiv\delta_{h,p}\delta_{k,q}\delta_{\theta,\tilde{\theta}}\delta_{w,\tilde{w}}\lambda^{(v,u)}_{(\tilde{v},\tilde{u})}(h,k;x,y_{0})\pmod{{\mathbb{C}}[[x-y_{0}]](x-y_{0})},$
$None$

which means that the set of eigenvalues (without counting multiplicities) of
the constant matrix $\Lambda^{23,x-y}(y_{0},y_{0})$ is determined by only the
choice of $J_{P_{A}}$ and $J_{P_{B}}$.

###  3.3 Corollary 3.3 (Local normal convergence)

Let $S^{1}$ be an indecomposable direct summand of $B\boxtimes C$ and $T^{1}$
an indecomposable direct summand of $A\boxtimes S^{1}$. We note that
$A\boxtimes(B\boxtimes C)$ is a direct sum of such $T^{1}$’s. We may choose
$\theta\in(T^{1})^{\vee}$. Set $p_{1}=d(T^{1})-d(A)-d(S^{1})-1$ and
$p_{2}=d(S^{1})-d(B)-d(C)+1$. Then there are
$\lambda^{h,k}_{m}(\vec{\xi})\in{\mathbb{C}}$ such that

$G^{A(BC):y}_{h,k}(\vec{\xi};x,y)=\langle\theta,{\cal Y}^{1}_{h}(v,x){\cal
Y}^{2}_{k}(u,y)w\rangle y^{{\rm
gr}^{234}(\vec{\xi})}=\sum_{m=0}^{\infty}\lambda^{h,k}_{m}(\vec{\xi})x^{p_{1}}y^{p_{2}}(y/x)^{m},$

where ${\cal Y}^{1}_{h}(v,z)\delta=\pi_{T^{1}}({\cal
Y}_{h}^{A(BC)}(v,z)\delta)$ and ${\cal Y}^{2}_{k}(u,z)w=\pi_{S^{1}}({\cal
Y}_{k}^{BC}(u,z)w)$ for $\delta\in S^{1}$ and $\pi_{P}$ denotes the projection
to $P$.

Therefore, in order to prove its convergence on a domain
$\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|y|<|x|\\}$, it is enough to check the case
where $x=1$ and $0<|y|<1$. In other words, it is enough to treat a one-
variable vector valued function
$Z=\left(G^{A(BC):y}_{h,k}(\vec{\xi};1,y)\right)_{\vec{\xi}\in{\cal
J}_{N},(h,k)\in K^{2}}$ satisfying a Fuchsian system
$\frac{d}{dy}Z=\frac{\left(\lambda^{\vec{\xi},h,k}_{\vec{\mu},p,q}(1,y)\right)}{y}Z$
with polar locus $\\{0\\}$, where
$\lambda^{\vec{\xi},h,k}_{\vec{\mu},p,q}(1,y)\in{\mathbb{C}}[(1-y)^{-1},y]$ by
Theorem 3.1. Since $(1-y)^{-1}$ and $y$ have radius of convergence at least
$1$, so does
$\lambda^{\vec{\xi},h,k}_{\vec{\mu},p,q}(1,y)-\delta_{(\vec{\xi},h,k),(\vec{\mu},p,q)}$.
Therefore, from Theorem 2.10, the radius of convergence of
$G^{A(BC):y}_{h,k}(\vec{\xi};1,y)$ (and also that of
$F^{A(BC)}_{h,k}(\vec{\xi};x,y)$) are at least $1$ for $\vec{\xi}\in\Omega$.
In a similar way, by using $(D2)$ for $G^{(AB)C}(\vec{\xi}^{\prime};x,y_{0})$,
we can show that $F^{(AB)C}(\vec{\xi}^{\prime};x,y)$ is locally normally
convergent on a domain $\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|x-y|<|y|\\}$.

This completes the proof of Corollary 3.3.

##  4 Associativity

Set ${\cal D}^{2}=\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|x-y|<|y|<|x|,\mbox{ and
}x,y,x-y\not\in{\mathbb{R}}^{\leq 0}\\}$. In this section, we will prove the
following main theorem.

###### Theorem 4.1

On ${\cal D}^{2}$, we choose a principal branch
$\widetilde{F}(\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle)$
of $\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle$ by taking
the values of $\log(x),\log(y)$ which satisfy
$-\pi<\Im(\log(x)),\Im(\log(y))<\pi$ and viewing $x^{d_{1}}y^{d_{2}}$ as
$e^{d_{1}\log(x)+d_{2}\log(y)}$. Similarly
$\widetilde{F}(\langle\theta^{\prime},{\cal Y}^{(AB)C}({\cal
Y}^{AB}(v,x-y)u,y)w\rangle)$ is a branch of $\langle\theta^{\prime},{\cal
Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w\rangle$ by taking the values of
$\log(y),\log(x-y)$ which satisfy $-\pi<\Im(\log(y)),\Im(\log(x-y))<\pi$ and
considering $y^{d}_{1}(x-y)^{d_{2}}=e^{d_{1}\log(y)+d_{2}\log(x-y)}$. Then
there is an isomorphism $\phi_{[AB]C}:(A\boxtimes B)\boxtimes C\to
A\boxtimes(B\boxtimes C)$ such that

$\widetilde{F}(\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal
Y}^{BC}(y,u)w\rangle)=\widetilde{F}(\langle\phi_{[AB]C}^{\ast}(\theta),{\cal
Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w)\rangle\vspace{-2mm}$

on ${\cal D}^{2}$ for any $\theta\in((A\boxtimes B)\boxtimes C)^{\vee}$, $v\in
A,u\in B,w\in C$, where $\phi^{\ast}_{[AB]C}$ denotes the dual of
$\phi_{[AB]C}$.

###  4.1 Equation (4.9)

We may still assume $\theta\in(T^{1})^{\vee}$. Hence there are
$p_{1},p_{2}\in{\mathbb{C}}$ such that

$G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)=\langle\theta,{\cal
Y}^{A(BC)}_{h}(v,x){\cal Y}^{BC}_{k}(u,y)w\rangle(x-y)^{{\rm
gr}^{234}(\vec{\xi})}\in{\mathbb{C}}[[y/x]]x^{p_{1}}y^{p_{2}}$

is absolutely convergent on $\\{(x,y)\in{\mathbb{C}}^{2}\mid 0<|y|<|x|\\}$. We
also recall the definition ${\cal J}_{N}$ which is the set of quadruples of
basis of $(A(BC))^{\vee}_{N},\tilde{P}_{A},\tilde{P}_{B}$, and
$\tilde{P}_{C}$. We choose a branch
$\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)$ of
$G^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)$ on ${\cal D}^{2}$ by taking values of
$\log(x),\log(y)$ which satisfy $-\pi<\Im(\log(x)),\Im(\log(y))<\pi$ and
viewing $x^{p_{1}+m}y^{p_{2}+n}$ as
$e^{(p_{1}+m)\log(x)+(p_{2}+n)\log(y)}=e^{p_{1}\log(x)+p_{2}\log(y)}x^{m}y^{n}$
for $m,n\in{\mathbb{Z}}$. Then by Theorem 3.2, for
$y_{0}\not\in{\mathbb{R}}^{\leq 0}$, a vector valued function

$G^{x-y}(x,y_{0})=\left(\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})\right)_{\vec{\xi}\in{\cal
J}_{N},(h,k)\in K^{2}}$

satisfies a Fuchsian system (D2) for $\vec{\xi}\in{\cal J}_{N}$. Furthermore,
the set of nonzero eigenvalues of the constant matrix
$\Lambda^{23:x-y}(y_{0},y_{0})$ depends only on the choice of bases
$J_{P_{A}}$ and $J_{P_{B}}$. Therefore, by Proposition 2.9, there is a finite
subset $\Delta^{\prime}\subseteq{\mathbb{C}}$ such that all components of
solutions of $(D2)$ are written as

$\sum_{t=0}^{K}\sum_{d\in\Delta^{\prime}}\sum_{m\in{\mathbb{N}}}r_{d,m,t}(x-y_{0})^{d+m}\log^{t}(x-y_{0})$

with $r_{d,m,t}\in{\mathbb{C}}$. We note that we have also taken a branch of
$(x-y_{0})^{d+m}\log^{j}(x-y_{0})$ on ${\cal D}^{2}_{(x,y_{0})}$ by the same
way. Since this holds for $\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y_{0})$
with $\vec{\xi}\in{\cal J}_{N}$ and $y=y_{0}\not\in{\mathbb{R}}^{\leq 0}$,
there are $r^{h,k}_{s,t}(\vec{\xi};y)\in{\mathbb{C}}$ and
$K(\vec{\xi})\in{\mathbb{N}}$ such that we can write

$\widetilde{G}^{A(BC):x-y}_{h,k}(\vec{\xi};x,y)=\sum_{s\in\Delta^{\prime}+{\mathbb{N}}}\sum_{t=0}^{K(\vec{\xi})}r^{h,k}_{s,t}(\vec{\xi};y)(x-y)^{s}\log^{t}(x-y)$
$None$

on ${\cal D}^{2}$. Multiplying (4.1) by $\log^{h}(x)\log^{k}(y)(x-y)^{-{\rm
gr}^{234}(\vec{\xi})}$ for all $(h,k)\in K^{2}$ and taking the sum of all of
them, we obtain an equality:

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\sum_{h,k}\sum_{s\in\Delta^{\prime}+{\mathbb{N}}}\sum_{j=0}^{K(\xi)}r^{h,k}_{d,j}(\vec{\xi};y)\log^{k}(y)\log^{h}(x)(x-y)^{s-{\rm
gr}^{234}(\vec{\xi})}\log^{j}(x-y)$ $None$

for a branch $\widetilde{F}^{A(BC)}(\vec{\xi};x,y)$ of
$F^{A(BC)}(\vec{\xi};x,y)$ on ${\cal D}^{2}$. About the derivations, we use
$\frac{{\partial}x}{{\partial}(x-y)}=1$ and
$\frac{{\partial}y}{{\partial}(x-y)}=0$. Since
$\frac{{\partial}\log(x)}{{\partial}x}=\frac{1}{x}=\frac{1}{y+(x-y)}=\frac{1}{y}\sum_{j=0}^{\infty}(-1)^{j}(\frac{x-y}{y})^{j}$
and $\frac{{\partial}\log(x)}{{\partial}y}=0$, we define

$p(x,y):=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{j+1}\left(\frac{x-y}{y}\right)^{j+1}+\log(y),$
$None$

which satisfies the same property with $\log(x)$ in a neighborhood of
$y\not\in{\mathbb{R}}^{\leq 0}$. We replace $\log(x)$ in (4.2) by $p(x,y)$.
Since the powers of $(x-y)$ in $p(x,y)$ are non-negative, we can write

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\sum_{t=0}^{K(\vec{\xi})}\sum_{s\in{\rm
gr}^{234}(\vec{\xi})-\Delta^{\prime}-{\mathbb{N}}-1}g_{s,t}(\vec{\xi};y)(x-y)^{-s-1}\log^{t}(x-y)$
$None$

with $g_{s,t}(\vec{\xi};y)\in{\mathbb{C}}$ for $\vec{\xi}\in{\cal J}_{N}$.
Since $\dim P_{C}<\infty$, using $\Delta=\\{d-{\rm max}\\{{\rm gr}(w):w\in
P_{C}\\}\mid d\in\Delta^{\prime}\\}$, we can rewrite (4.4) into

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\sum_{t=0}^{K(\vec{\xi})}\sum_{s\in{\rm
gr}(v)+{\rm
gr}(u)-\Delta-{\mathbb{N}}-1}g_{s,t}(\vec{\xi};y)(x-y)^{-s-1}\log^{t}(x-y)$
$None$

for $\vec{\xi}=(\theta,v,u,w)\in\Omega$ with $w\in P_{C}$.

###### Lemma 4.2

$(4.5)$ holds for all $\vec{\xi}\in\Omega$.

[Proof] Suppose false and let $\vec{\xi}=(\theta,v,u,w^{\prime})$ be a
counterexample. Subject to counterexample, we choose $w^{\prime}$ with the
smallest grade. Since $(4.5)$ holds for elements with $w$ in $P_{C}$, we may
assume $w^{\prime}=\alpha_{-1}w$ with $\alpha\in V_{\geq 0}$ and
$g_{s,t}(\theta,v,u,\alpha_{-1}w;y)\not=0$ for some $s\not\in{\rm wt}(v)+{\rm
wt}(u)-1-\Delta-{\mathbb{N}}$. In this case, by Borcherds identities, we
obtain

$\begin{array}[]{l}\widetilde{F}^{A(BC)}(\alpha_{-1}^{[4]}\vec{\xi};x,y)=\widetilde{F}^{A(BC)}({(\alpha_{-1})^{\ast}}^{[1]}\vec{\xi};x,y)\\!-\\!\sum_{j=0}^{\infty}(-1)^{j}\widetilde{F}^{A(BC)}(\alpha_{j}^{[2]}\vec{\xi};x,y)x^{-j-1}\cr\mbox{}\,\,-\sum_{j=0}^{\infty}(-1)^{j}\widetilde{F}^{A(BC)}(\alpha_{j}^{[3]}\vec{\xi};x,y)y^{-j-1}\cr=\widetilde{F}^{A(BC)}({(\alpha_{-1})^{\ast}}^{[1]}\vec{\xi};x,y)-\sum_{j=0}^{\infty}\binom{-1}{j}\widetilde{F}^{A(BC)}(\alpha_{j}^{[2]}\vec{\xi};x,y)x^{-j-1}\cr\mbox{}\,\,-\sum_{j=0}^{\infty}\binom{-1}{j}y^{-j-1}\\{\sum_{i\in{\mathbb{N}}}\binom{j}{i}(-1)^{i}\widetilde{F}^{A(BC)}({(\alpha_{j-i}^{\ast}}^{[1]}\vec{\xi};x,y)y^{i}\cr\mbox{}\,\,-\sum_{i\in{\mathbb{N}}}\binom{j}{i}\widetilde{F}^{A(BC)}(\alpha_{i}^{[2]}\vec{\xi};x,y)(x-y)^{j-i}\\!-\\!\sum_{i\in{\mathbb{N}}}\binom{j}{i}(-1)^{i+j+1}\widetilde{F}^{A(BC)}(\alpha_{i}^{[4]}\vec{\xi};x,y)y^{j-i}\\}\cr=\widetilde{F}^{A(BC)}({(\alpha_{-1})^{\ast}}^{[1]}\vec{\xi};x,y)-\sum_{j,i\in{\mathbb{N}}}\binom{j}{i}(-1)^{i+j}y^{i-j-1}\widetilde{F}^{A(BC)}({(\alpha_{j-i})^{\ast}}^{[1]}\vec{\xi};x,y)\cr\mbox{}\,\,+\sum_{j=0}^{\infty}\sum_{i\in{\mathbb{N}}}\binom{j}{i}(-1)^{i+1}\widetilde{F}^{A(BC)}(\alpha_{i}^{[4]}\vec{\xi};x,y)y^{-1},\end{array}$

by (2.9). However, since ${\rm gr}(\alpha_{j}w)<{\rm gr}(\alpha_{-1}w)={\rm
gr}(w^{\prime})$ for $j\geq 0$ and the 2nd and 3rd components of
$(\alpha_{j})^{[i]}\vec{\xi}$ in RHS of the last equation are $v$ and $u$,
none of the terms in RHS of the last equation have nonzero coefficients of
$(x-y)^{-s-1}\log^{t}(x-y)$ by the minimality of ${\rm gr}(w^{\prime})$, which
is a contradiction. $\square$

Since $g_{s,t}(\vec{\xi};y)$ is multi-linear on
$\vec{\xi}\in\Omega_{N}=(A(BC))^{\vee}_{(\leq N)}\times A\times B\times C$, we
split $\vec{\xi}=(\theta,v,u,w)$ into $\\{v,u\\}$ and $\\{\theta,w\\}$. In
other words, using a formal operator $\widetilde{{\cal Y}}^{3}_{s,t}\in{\rm
Hom}(A\otimes B,{\rm Hom}(C,A(BC)\otimes
O({\mathbb{C}}\backslash{\mathbb{R}}^{\leq 0})))$, we can write

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\sum_{t=0}^{K(\vec{\xi})}\sum_{s\in{\rm
gr}(v)+{\rm gr}(u)-\Delta-{\mathbb{N}}-1}\langle\theta,\widetilde{{\cal
Y}}^{3}_{s,t}(v,u;y)w\rangle(x-y)^{-s-1}\log^{t}(x-y)$ $None$

for each $(x,y)\in{\cal D}^{2}$. Namely, $\langle\theta,\widetilde{{\cal
Y}}^{3}_{s,t}(v,u;y)w\rangle={\rm
Cf}_{s,t}(\tilde{F}^{A(BC)}(\vec{\xi};x,y);y)$, where
${\rm Cf}_{s,t}(f(x,y);y)$ denotes a coefficient $\lambda_{s,t}(y)$ of
$(x-y)^{-s-1}\log^{t}(x-y)$ in
$f(x,y)=\sum_{t\in{\mathbb{N}}}\sum_{s\in{\mathbb{C}}}\lambda_{s,t}(y)(x-y)^{-s-1}\log^{t}(x-y)$.

We introduce a formal vector space $A\otimes_{s,t}B$, which is isomorphic to
$A\otimes B$ as a vector space and its isomorphism is given by $v\otimes
v\mapsto v\otimes_{s,t}u$ for $s,t$. Set
$A\otimes_{\infty}B=\oplus_{(s,t)\in{\mathbb{C}}\times{\mathbb{N}}}(A\otimes_{s,t}B)$
and define a formal operator $\widetilde{{\cal Y}}^{4}$ of type
$\binom{A\otimes_{\infty}B}{A\,\,\,\,B}$ by $\widetilde{{\cal
Y}}^{4}(v,x-y)u=\sum_{s\in{\mathbb{C}}}\sum_{t\in{\mathbb{N}}}v\otimes_{s,t}u(x-y)^{-s-1}\log^{t}(x-y)$.
We also define another formal operator $\widetilde{{\cal
Y}}^{3}:(A\otimes_{\infty}B)\times C\to A(BC)\otimes{\cal O}({\cal
D}^{2}_{(x,y)})$ by $\langle\theta,\widetilde{{\cal
Y}}^{3}(v\otimes_{s,t}u,y)w\rangle=\langle\theta,\widetilde{{\cal
Y}}^{3}_{s,t}(v,u;y)w\rangle(x-y)^{-s-1}\log^{t}(x-y)$. Then we can rewrite
Eq. (4.6) into

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\langle\theta,\widetilde{{\cal
Y}}^{3}(\widetilde{{\cal Y}}^{4}(v,x-y)u,y)w\rangle.$ $None$

As it is well-known, $V[z,z^{-1}]$ has a Lie algebra structure by

$[\alpha z^{m},\beta
z^{n}]=\sum_{j\in{\mathbb{N}}}\binom{m}{j}(\alpha_{j}\beta)z^{n+m-j}.$

###### Lemma 4.3

Define the action of $V[z,z^{-1}]$ on $A\otimes_{\infty}B$ by $(\alpha
z^{m})(v\otimes_{s,t}u)=v\otimes_{s,t}\alpha_{m}u+\sum_{j=0}^{\infty}\binom{m}{j}(\alpha_{j}v\otimes_{s+m-j,t}u)\mbox{
for }\alpha z^{m}\in V[z,z^{-1}]$. Then $A\otimes_{\infty}B$ is a
$V[z,z^{-1}]$-module.

[Proof] To simplify the notation, we will omit the index $t$ from
$v\otimes_{s,t}u$. We also denote the action of $\alpha z^{m}$ by
$\alpha_{[m]}$. Then for $\alpha,\beta\in V$, we obtain

$\begin{array}[]{l}\alpha_{[m]}\beta_{[n]}(v\otimes_{s}u)=\alpha_{[m]}(\sum_{j=0}^{\infty}\binom{n}{j}(\beta_{j}v)\otimes_{n+s-j}u)+\alpha_{[m]}(v\otimes_{s}(\beta_{n}u))\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}(\alpha_{i}\beta_{j}v)\otimes_{m+n+s-i-j}u+\sum_{j=0}^{\infty}\binom{n}{j}(\beta_{j}v)\otimes_{n+s-j}(\alpha_{m}u)\cr\mbox{}\qquad+\sum_{i=0}^{\infty}\binom{m}{i}(\alpha_{i}v)\otimes_{m+s-i}(\beta_{n}u)+v\otimes_{s}(\alpha_{m}\beta_{n}u).\end{array}$

We hence get:

$\begin{array}[]{l}[\alpha_{[m]},\beta_{[n]}](v\otimes_{s}u)\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}(\alpha_{i}\beta_{j}v)\otimes_{m+n+s-i-j}u+\sum_{j=0}^{\infty}\binom{n}{j}(\beta_{j}v)\otimes_{n+s-j}(\alpha_{m}u)\cr\mbox{}\qquad+\sum_{i=0}^{\infty}\binom{m}{i}(\alpha_{i}v)\otimes_{m+s-i}(\beta_{n}u)+v\otimes_{s}(\alpha_{m}\beta_{n}u)\cr\mbox{}\qquad-\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\binom{n}{j}\binom{m}{i}(\beta_{j}\alpha_{i}v)\otimes_{m+n+s-i-j}u-\sum_{i=0}^{\infty}\binom{m}{i}(\alpha_{i}v)\otimes_{m+s-i}(\beta_{n}u)\cr\mbox{}\qquad-\sum_{j=0}^{\infty}\binom{n}{j}(\beta_{j}v)\otimes_{n+s-j}(\alpha_{m}u)-v\otimes_{s}(\beta_{n}\alpha_{m}u)\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}([\alpha_{i},\beta_{j}]v)\otimes_{m+n+s-i-j}u+v\otimes_{s}([\alpha_{m},\beta_{n}]u)\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}([\alpha_{i},\beta_{j}]v)\otimes_{m+n+s-i-j}u+v\otimes_{s}([\alpha_{m},\beta_{n}]u)\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}(\sum_{p=0}^{\infty}\binom{i}{p}(\alpha_{p}\beta)_{i+j-p}v)\otimes_{m+n+s-i-j}u+v\otimes_{s}([\alpha_{m},\beta_{n}]u).\end{array}$
$None$

On the other hand, we have:

$\begin{array}[]{l}\sum_{p=0}^{\infty}\binom{m}{p}(\alpha_{p}\beta)_{[m+n-p]}(v\otimes_{s}u)\cr\mbox{}\quad=v\otimes_{s}(\sum_{p=0}^{\infty}\binom{m}{p}(\alpha_{p}\beta)_{m+n-p}u)+\sum_{p=0}^{\infty}\binom{m}{p}\sum_{q=0}^{\infty}\binom{m+n-p}{q}((\alpha_{p}\beta)_{q}v)\otimes_{s+m+n-p-q}u\cr\mbox{}\quad=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{m}{i}\binom{n}{j}(\sum_{p=0}^{\infty}\binom{i}{p}(\alpha_{p}\beta)_{i+j-p}v)\otimes_{m+n+s-i-j}u+[v\otimes_{s}([\alpha_{m},\beta_{n}]u),\end{array}$

which coincides with the last terms in (4.8). Therefore $A\otimes_{\infty}B$
is a $V[z,z^{-1}]$-module. $\square$

$A\otimes_{\infty}B$ is not a $V$-module. We choose a better one.

###### Definition 4.4

Let $\Gamma$ be the kernel of $\widetilde{{\cal Y}}^{3}$, that is,

$\Gamma=\\{\gamma\in A\otimes_{\infty}B\mid\langle\theta,\widetilde{{\cal
Y}}^{3}(\gamma,y)w\rangle=0\mbox{ for all }\theta\in(A(BC))^{\vee},w\in
C\mbox{ and }y\not\in{\mathbb{R}}^{\leq 0}\\}.$

In other words, $\sum_{i=1}^{n}v^{i}\otimes_{s_{i},t_{i}}u^{i}\in\Gamma$ if
and only if $\sum_{i=1}^{n}{\rm
Cf}_{s_{i},t_{i}}(\widetilde{F}^{A(BC)}(\theta,v^{i},u^{i},w;x,y);y)=0$ for
any $\theta\in(A(BC))^{\vee}$ and $w\in C$. Clearly, $\Gamma$ is a subspace of
$A\otimes_{\infty}B$. Define
$A\tilde{\boxtimes}B=(A\otimes_{\infty}B)/\Gamma$. We denote
$v\otimes_{s,t}u+\Gamma$ in $A\tilde{\boxtimes}B$ by $[v\otimes_{s,t}u]$ or
$[v_{s,t}u]$.

###### Lemma 4.5

$\widetilde{{\cal Y}}^{3}$ satisfies associativity ${\rm(I\,\,2)}$ and
$\Gamma$ is $V[z,z^{-1}]$-invariant.

[Proof] By the definition of the action of $V[z,z^{-1}]$, we get:

$\begin{array}[]{l}\langle\theta,\widetilde{{\cal
Y}}^{3}(\alpha_{[n]}(v\otimes_{s,t}u),y)w\rangle\cr\mbox{}\quad=\langle\theta,\widetilde{{\cal
Y}}^{3}(v\otimes_{s,t}\alpha_{n}u),y)w\rangle+\sum_{i=0}^{\infty}\binom{n}{i}\langle\theta,\widetilde{{\cal
Y}}^{3}(\alpha_{i}v\otimes_{s+n-i,t}u,y)w\rangle\cr\mbox{}\quad={\rm
Cf}_{s,t}(\widetilde{F}^{A(BC)}(\theta,v,\alpha_{n}u,w;x,y);y)\\!+\\!\sum_{i=0}^{\infty}\binom{n}{i}{\rm
Cf}_{s+n-i,t}(\widetilde{F}^{A(BC)}(\theta,\alpha_{j}v,u,w;x,y);y).\end{array}$

Since $\widetilde{F}^{A(BC)}$ satisfies Borcherds identity (3B), we also have:

$\begin{array}[]{l}{\rm
Cf}_{s,t}(\widetilde{F}^{A(BC)}(\theta,v,\alpha_{n}u,w;x,y);y)={\rm
Cf}_{s,t}\left(\left(\sum_{j=0}^{\infty}(-1)^{j}\widetilde{F}^{A(BC)}(((\alpha_{n-j})^{\ast})^{[1]}\vec{\xi};x,y)y^{j}\right.\right.\cr\left.\left.\mbox{}\quad\\!-\\!\sum_{j=0}^{\infty}\binom{n}{j}\widetilde{F}^{A(BC)}(\alpha_{j}^{[2]}\vec{\xi};x,y)(x\\!-\\!y)^{n-j}\\!-\\!\sum_{j=0}^{\infty}\binom{n}{j}\widetilde{F}^{A(BC)}(\alpha_{j}^{[4]}\vec{\xi};x,y)(-y)^{n-j}\right);y\right)\cr\mbox{}=\sum_{j=0}^{\infty}(-1)^{j}{\rm
Cf}_{s,t}(\widetilde{F}^{A(BC)}(((\alpha_{n-j})^{\ast})^{[1]}\vec{\xi};x,y);y)y^{j}\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{j}{\rm
Cf}_{s+n-j,t}(\widetilde{F}^{A(BC)}(\alpha_{j}^{[2]}\vec{\xi};x,y);y)\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}{\rm
Cf}_{s,t}(\widetilde{F}^{A(BC)}(\alpha_{j}^{[4]}\vec{\xi};x,y);y)y^{n-j}.\end{array}$

Therefore, we obtain ${\rm(I\,\,2)}$ for $\widetilde{{\cal Y}}^{3}$, that is,

$\begin{array}[]{l}\langle\theta,\widetilde{{\cal
Y}}^{3}(\alpha_{[n]}(v\otimes_{s,t}u),y)w\rangle=\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta,\widetilde{{\cal
Y}}^{3}(v\otimes_{s,t}u,y)w\rangle
y^{j}\cr\mbox{}\qquad-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta,\widetilde{{\cal
Y}}^{3}(v\otimes_{s,t}u,y)\alpha_{j}w\rangle y^{n-j}.\end{array}$

We next show that $\Gamma$ is $V[z,z^{-1}]$-invariant. If
$\gamma=\sum_{i=1}^{r}v^{i}\otimes_{s_{i},t_{i}}u^{i}\in\Gamma$, then we have

$\begin{array}[]{l}\langle\theta,\widetilde{{\cal
Y}}^{3}(\alpha_{n}\gamma,y)w\rangle\cr=\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle(\alpha_{n-j})^{\ast}\theta,\widetilde{{\cal
Y}}^{3}(\gamma,y)w\rangle
y^{j}-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j+n}\langle\theta,\widetilde{{\cal
Y}}^{3}(\gamma,y)\alpha_{j}w\rangle y^{n-j}=0\end{array}$

for any $\theta$ and $w$, which means $\alpha_{n}\gamma\in\Gamma$ for any
$\alpha z^{n}=\alpha_{n}$. Hence $\Gamma$ is $V[z,z^{-1}]$-invariant.
$\square$

Clearly, we can define formal operators ${\cal Y}^{3}$ of type
$\binom{A(BC)}{A\tilde{\boxtimes}B\,\,C}$ and ${\cal Y}^{4}$ of type
$\binom{A\tilde{\boxtimes}B}{A\,\,B}$ by ${\cal Y}^{4}(v,z)u=[\widetilde{{\cal
Y}}^{4}(v,z)u]=\sum_{s,t}[v\otimes_{s,t}u]z^{-s-1}\log^{t}(z)$ and
$\langle\theta,{\cal
Y}^{3}([v_{s,t}u],z)w\rangle=\langle\theta,\widetilde{{\cal
Y}}^{3}(v\otimes_{s,t}u,z)w\rangle$ for $\vec{\xi}\in\Omega$. Then we have

$\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle$ $None$

for $\vec{\xi}\in\Omega$. We use $F^{34}(\vec{\xi};x,y)$ to denote
$\langle\theta,{\cal Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle$.

###### Theorem 4.6

$\langle\theta,{\cal Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle$ satisfies the
Borcherds identities $(1B)\sim(4B)$.

[Proof] As we mentioned in Comment 2, the coefficients of
$F^{A(BC)}(\vec{\mu};x,y)$ in the expansion of
$F^{A(BC)}(\alpha^{[i]}_{a}\vec{\xi};x,y)$ by Borcherds identities are all
${\mathbb{C}}[x^{\pm 1},y^{\pm 1},(x-y)^{\pm 1}]$, that is, integer powers of
$x,y$ and $x\\!-\\!y$. Hence branches $\widetilde{F}^{A(BC)}(\vec{\xi};x,y)$
of them on ${\cal D}^{2}$ also satisfy the Borcherds identities
$(1A)\sim(4A)$. By Eq. (4.9), $F^{34}(\vec{\xi};x,y)$ also satisfies the
Borcherds identities $(1A)\sim(4A)$ on ${\cal D}^{2}$. Since the corresponding
coefficients in $(1A)\sim(4A)$ and $(1B)\sim(4B)$ are same on ${\cal D}^{2}$,
we have the desired identities. $\square$

###### Proposition 4.7

${\cal Y}^{3}$ and ${\cal Y}^{4}$ satisfy $L(-1)$-derivative properties.

[Proof] We note that we are considering the derivations satisfying
$\frac{{\partial}}{{\partial}(x-y)}y=0=\frac{{\partial}}{{\partial}y}(x-y)$
and
$\frac{{\partial}}{{\partial}y}y=1=\frac{{\partial}}{{\partial}(x-y)}(x-y)$.
By the choice of $p(x,y)$ in (4.4), we get

$\begin{array}[]{l}\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(L(-1)v,x-y)u,y)w\rangle=\widetilde{F}^{A(BC)}(L(-1)^{[2]}\vec{\xi};x,y)=\frac{d}{dx}\widetilde{F}^{A(BC)}(\vec{\xi};x,y)\cr\mbox{}\qquad=\frac{d}{d(x-y)}\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\frac{{\partial}}{{\partial}(x-y)}\langle\theta,{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle.\end{array}$

We fix $v\in A$. Since the above holds for every $\theta\in(A(BC))^{\vee},u\in
B,w\in C$, the above implies $L(-1)$-derivative property of ${\cal Y}^{4}$:
${\cal Y}^{4}(L(-1)v,x-y)=\frac{d}{d(x-y)}{\cal Y}^{4}(v,x-y)$. Using
$F^{34}(L(-1)^{[3]}\vec{\xi};x,y)=\frac{d}{dy}\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=\frac{{\partial}}{{\partial}y}F^{34}(\vec{\xi};x,y)$,
we obtain

$\begin{array}[]{l}\langle\theta,{\cal Y}^{3}(L(-1){\cal
Y}^{4}(v,x-y)u,y)w\rangle\cr\mbox{}\qquad=\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(L(-1)v,x-y)u,y)w\rangle+\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)L(-1)u,y)w\rangle\cr\mbox{}\qquad=\frac{{\partial}}{{\partial}x}\widetilde{F}^{A(BC)}(\vec{\xi};x,y)+\frac{{\partial}}{{\partial}y}\widetilde{F}^{A(BC)}(\vec{\xi};x,y)=(\frac{{\partial}}{{\partial}x}+\frac{{\partial}}{{\partial}y})\widetilde{F}^{A(BC)}(\vec{\xi};x,y).\end{array}$

Since $(\frac{{\partial}}{{\partial}y}+\frac{{\partial}}{{\partial}x})(x-y)=0$
and $(\frac{{\partial}}{{\partial}y}+\frac{{\partial}}{{\partial}x})y=1$, we
also get

$\begin{array}[]{l}\langle\theta,{\cal Y}^{3}(L(-1){\cal
Y}^{4}(v,x-y)u,y)w\rangle=\frac{{\partial}}{{\partial}y}{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle\end{array}$

under the assumption $\frac{{\partial}(x-y)}{{\partial}y}=0$, which proves
$L(-1)$-derivative property of ${\cal Y}^{3}$. $\square$

As a corollary of the above and the existence of a finite set $\Delta$ in
(4.5), we have:

###### Corollary 4.8

$L(-1)[v_{s,t}u]=-s[v_{s-1,t}u]+(t+1)[v_{s-1,t+1}u]$ and ${\rm
wt}([v_{s,t}u])={\rm wt}(v)+{\rm wt}(u)-s-1$. The weights of elements in
$A\tilde{\boxtimes}B$ are contained in $\Delta+{\mathbb{N}}$ and so
$A\tilde{\boxtimes}B$ is ${\mathbb{N}}$-gradable. In particular,
$\alpha_{n}[v_{s,t}u]=0$ for $n>\\!>0$ for $\alpha\in V$.

Although Borcherds identities are given from ${\rm(I\,\,1)}$ and
${\rm(I\,\,2)}$, we will show the reverse.

###### Proposition 4.9

${\cal Y}^{3}$ and ${\cal Y}^{4}$ satisfy ${\rm(I\,\,1)}$ and ${\rm(I\,\,2)}$.

[Proof] We have already proved ${\rm(I\,\,2)}$ for ${\cal Y}^{3}$. By the
action of $V$ on $A\otimes_{\infty}B$, we have:

$\begin{array}[]{l}\langle\theta,{\cal Y}^{3}(\alpha_{[n]}{\cal
Y}^{4}(v,x-y)u,y)w\rangle=\langle\theta,\widetilde{{\cal
Y}}^{3}(\alpha_{n}\widetilde{{\cal
Y}}^{4}(v,x-y)u,y)w\rangle\cr\mbox{}\quad=\langle\theta,\widetilde{{\cal
Y}}^{3}(\widetilde{{\cal
Y}}^{4}(v,x-y)\alpha_{n}u,y)w\rangle+\sum_{j=0}^{\infty}\binom{n}{j}\langle\theta,\widetilde{{\cal
Y}}^{3}(\widetilde{{\cal
Y}}^{4}(\alpha_{j}v,x-y)u,y)w\rangle(x-y)^{n-j}\end{array}$

for all $\theta$ and $w$, which implies ${\rm(I\,\,1)}$ for ${\cal Y}^{4}$.
From Borcherds identity (2B), we obtain:

$\begin{array}[]{l}\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(\alpha_{n}v,x-y)u,y)w\rangle=\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle\theta,\alpha_{n-j}{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)w\rangle
x^{j}\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{n+j}\langle\theta,{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)u,y)\alpha_{j}w\rangle
x^{n-j}\cr\mbox{}\quad-\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{n+j}\langle\theta,{\cal
Y}^{3}({\cal Y}^{4}(v,x-y)\alpha_{j}u,y)w\rangle(x-y)^{n-j}.\end{array}$

On the other hand, from the associativity ${\rm(I\,\,2)}$ for ${\cal Y}^{3}$,
we have

$\begin{array}[]{l}\sum_{j=0}^{\infty}\binom{n}{j}(-1)^{j}\langle\theta,{\cal
Y}^{3}(\alpha_{n-j}{\cal
Y}^{4}(v,x-y)u,y)w\rangle(x-y)^{j}\cr\mbox{}\quad=\sum_{i,j\in{\mathbb{N}}}\binom{n}{j}\binom{n-j}{i}(-1)^{i+j}\langle\theta,\alpha_{n-i-j}{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle(x-y)^{j}y^{i}\cr\mbox{}\qquad-\sum_{i,j\in{\mathbb{N}}}\binom{n}{j}(-1)^{i+n}\binom{n-j}{i}\langle\theta,{\cal
Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)\alpha_{j}w\rangle(x-y)^{j}y^{n-j-i}.\end{array}$

Since
$\sum_{j=0}^{\ell}\binom{n}{j}\binom{n-j}{\ell-j}(x-y)^{j}y^{\ell-j}=\binom{n}{\ell}x^{\ell}$
for $\ell\in{\mathbb{N}}$ and
$\sum_{j}\binom{n}{j}\binom{n-j}{i}(-1)^{i+n}(x-y)^{j}y^{n-j-i}=\binom{n}{i}(-1)^{n-i}x^{n^{i}}$
by (2.9), we obtain ${\rm(I\,\,2)}$ for ${\cal Y}^{4}$. Commutativity
${\rm(I\,\,1)}$ for ${\cal Y}^{3}$ is a direct consequence of Borcherds
identity $(B1)$ and the definition of action of $V$ on $A\tilde{\boxtimes}B$.
$\square$

###  4.2 Recovery of $V$-modules and intertwining operators

###### Theorem 4.10

$A\tilde{\boxtimes}B$ is an ${\mathbb{N}}$-gradable $V$-module.

[Proof] We have proved that $A\tilde{\boxtimes}B$ is an
${\mathbb{N}}$-gradable $V[z,z^{-1}]$-module in Corollary 4.8. In particular,
the action of $V$ on $A\tilde{\boxtimes}B$ satisfies commutativity. Therefore,
the remaining thing is to prove associativity of the action of $V$. In
particular, it is enough to show

$\begin{array}[]{l}\langle\theta,{\cal
Y}^{3}(\alpha_{-1}\beta)_{[-1]}\gamma,y)w\rangle\cr\mbox{}\quad=\langle\theta,{\cal
Y}^{3}(\alpha_{[-1]}\beta_{[-1]}\gamma,y)w\rangle+\sum_{j=0}^{\infty}\langle\theta,(\alpha_{[-2-j]}\beta_{[j]}+\beta_{[-2-j]}\alpha_{[j]})\gamma,y)w\rangle\end{array}$
$None$

for $\gamma\in A\tilde{\boxtimes}B,w\in C,\theta\in(A(BC))^{\vee}$. We will
develop the action of $V$ into a triple $(A(BC))^{\vee}\times
A\tilde{\boxtimes}B\times C$ by using ${\rm(I\,\,1)}$ and ${\rm(I\,\,2)}$. The
rule is simple, that is, we shift $\alpha_{<0}$ and $\beta_{<0}$ to the left
and $\alpha_{\geq 0}$ and $\beta_{\geq 0}$ to the right. Finally, the actions
of $V$ on $A\tilde{\boxtimes}B$ and $C$ are given by $\alpha_{\geq 0}$ or
$\beta_{\geq 0}$. We use notation $P(X(\alpha,\beta))$ to denote
$X(\alpha,\beta)+X(\beta,\alpha)$ for any form $X$ of $\alpha_{i}$ and
$\beta_{j}$. For example, $P(\langle\theta,{\cal
Y}^{3}(\alpha_{[-2-j]}\beta_{[j]})\gamma,y)w\rangle)$ denotes
$\langle\theta,{\cal
Y}^{3}(\alpha_{[-2-j]}\beta_{[j]}+\beta_{[-2-j]}\alpha_{[j]})\gamma,y)w\rangle$.
Then by ${\rm(I\,\,2)}$ and associativity of the action of $V$ on $A(BC)$ and
$C$, we have:

$\begin{array}[]{l}{\rm LHS}\mbox{ of
(4.10)}=\sum_{j\in{\mathbb{N}}}\langle\theta,(\alpha_{-1}\beta)_{-1-j}{\cal
Y}^{3}(\gamma,y)w\rangle y^{j}+\sum_{j\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)(\alpha_{-1}\beta)_{j}w\rangle
y^{-1-j}\cr\mbox{}=\sum_{i,j\in{\mathbb{N}}}\langle\theta,\\{\alpha_{-1-j}\beta_{-1-j+i}+\beta_{-2-j-i}\alpha_{i}\\}{\cal
Y}^{3}(\gamma,y)w\rangle
y^{j}\cr\mbox{}\,\,+\sum_{i,j\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\\{\alpha_{-1-i}\beta_{j+i}+\beta_{-1+j-i}\alpha_{i}\\}w\rangle
y^{-1-j}\cr\mbox{}=\sum_{\ell,i\in{\mathbb{N}}}\langle\theta,\alpha_{-1-i}\beta_{-1-\ell}{\cal
Y}^{3}(\gamma,y)w\rangle
y^{\ell+i}+\sum_{\ell,j\in{\mathbb{N}}}\langle\theta,\alpha_{-2-j-\ell}\beta_{\ell}{\cal
Y}^{3}(\gamma,y)w\rangle
y^{j}\cr\mbox{}\,\,+\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-2-i-j}\alpha_{i}{\cal
Y}^{3}(\gamma,y)w\rangle y^{j}+\sum_{i,j\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\alpha_{-1-i}\beta_{i+j}w\rangle
y^{-1-j}\cr\mbox{}\,\,+\sum_{i,j\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\beta_{-1-\ell}\alpha_{\ell+j}w\rangle
y^{-1-j}+\sum_{\ell,i\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\beta_{\ell}\alpha_{i}w\rangle
y^{-2-\ell-i}\cr\mbox{}=\sum_{\ell,i\in{\mathbb{N}}}\langle\theta,\alpha_{-1-i}\beta_{-1-\ell}{\cal
Y}^{3}(\gamma,y)w\rangle
y^{\ell+i}+P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-2-i-j}[\alpha_{i},{\cal
Y}^{3}(\gamma,y)]w\rangle
y^{j})\cr\mbox{}\,\,+\\!P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-2-i-j}{\cal
Y}^{3}(\gamma,y)\alpha_{i}w\rangle
y^{j})\\!-\\!P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,[\alpha_{-1-i},{\cal
Y}^{3}(\gamma,y)]\beta_{i+j}w\rangle
y^{-1-j})\cr\mbox{}\,\,+\\!P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,\alpha_{-1-i}{\cal
Y}^{3}(\gamma,y)\beta_{i+j}w\rangle
y^{-1-j})+\sum_{\ell,i\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\beta_{\ell}\alpha_{i}w\rangle y^{-2-\ell-i}.\end{array}$

Since there are many subscripts, in order to count the coefficients of terms
in summations, we use monomials. For example, we denote
$\langle\theta,\beta_{a}{\cal Y}^{3}([v_{s}u],y)\alpha_{b}w\rangle y^{c}$ by
$X^{a}Y^{b}Z^{c}$. Then since $i,j\in{\mathbb{N}}$, keeping the non-negative
powers of $Y$, the following equation

$\begin{array}[]{l}\sum_{i,j\in{\mathbb{N}}}X^{-2-i-j}Y^{i}Z^{j}+\sum_{i,j\in{\mathbb{N}}}X^{-1-i}Y^{i+j}Z^{-1-j}\cr\mbox{}\quad=\iota_{X,Y}\\{(X-Y)^{-1}\\}\iota_{X,Z}\\{(X-Z)^{-1}\\}+\iota_{X,Y}\\{(X-Y)^{-1}\\}\iota_{Z,Y}\\{(Z-Y)^{-1}\\}\cr\mbox{}\quad=\iota_{X,Z}\\{(X-Z)^{-1}\\}\iota_{X,Y}\\{(X-Y)^{-1}\\}\iota_{Z,Y}\\{(Z-Y)^{-1}\\}(X-Y)\cr\mbox{}\quad=\iota_{X,Z}\\{(X-Z)^{-1}\\}\iota_{Z,Y}\\{(Z-Y)^{-1}\\}\cr\mbox{}\quad=\sum_{i,j\in{\mathbb{N}}}X^{-1-i}Y^{j}Z^{i-j-1}\end{array}$

implies

$\begin{array}[]{l}\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-2-i-j}{\cal
Y}^{3}(\gamma,y)\alpha_{i}w\rangle
y^{j}+\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-1-i}{\cal
Y}^{3}(\gamma,y)\alpha_{i+j}w\rangle
y^{-1-j}\cr\mbox{}\qquad=\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-1-i}{\cal
Y}^{3}(\gamma,y)\alpha_{j}w\rangle y^{-1-j+i}.\end{array}$

Furthermore, from

$\begin{array}[]{l}\langle\theta,{\cal
Y}^{3}(\alpha_{[-1]}\beta_{[-1]}[v_{s}u],y)w\rangle\cr\mbox{}\quad=\sum_{i,j\in{\mathbb{N}}}\langle\theta,\alpha_{-1-j}\beta_{-1-i}{\cal
Y}^{3}(\gamma,y)w\rangle
y^{i+j}+\sum_{i,j\in{\mathbb{N}}}\langle\theta,\alpha_{-1-j}{\cal
Y}^{3}(\gamma,y)\beta_{i}w\rangle
y^{j-i-1}\cr\mbox{}\qquad+\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-1-i}{\cal
Y}^{3}(\gamma,y)\alpha_{j}w\rangle
y^{-j-1+i}+\sum_{i,j\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\gamma,y)\beta_{j}\alpha_{i}w\rangle y^{-2-i-j},\end{array}$

we get

$\begin{array}[]{l}{\rm LHS}\mbox{ of (4.10)}-\langle\theta,{\cal
Y}^{3}(\alpha_{[-1]}\beta_{[-1]}[v_{s}u],y)w\rangle\cr=P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,\beta_{-2-i-j}[\alpha_{i},{\cal
Y}^{3}(\gamma,y)]w\rangle
y^{j})-P(\sum_{i,j\in{\mathbb{N}}}\langle\theta,[\alpha_{-1-i},{\cal
Y}^{3}(\gamma,y)]\beta_{i+j}w\rangle
y^{-1-j})\cr=P(\sum_{i,j,k\in{\mathbb{N}}}\binom{i}{k}\langle\theta,\beta_{-2-i-j}{\cal
Y}^{3}(\alpha_{[k]}\gamma,y)w\rangle
y^{j+i-k})\cr\mbox{}\qquad-P(\sum_{i,j,k\in{\mathbb{N}}}\binom{-1-i}{k}\langle\theta,{\cal
Y}^{3}(\alpha_{[k]}\gamma,y)\beta_{i+j}w\rangle y^{-2-j-i-k}).\end{array}$

On the other hand, we obtain

$\begin{array}[]{l}{\rm RHS}\mbox{ of (4.10)}-\langle\theta,{\cal
Y}^{3}(\alpha_{[-1]}\beta_{[-1]}\gamma,y)w\rangle=P(\sum_{i\in{\mathbb{N}}}\langle\theta,{\cal
Y}^{3}(\alpha_{[-2-i]}\beta_{[i]}\gamma,y)w)\cr\mbox{}\quad=P(\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j}\langle\theta,\alpha_{-2-i-j}{\cal
Y}^{3}(\beta_{[i]}\gamma,y)w\rangle
y^{j})\cr\mbox{}\qquad-P(\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j-2-i}\langle\theta,{\cal
Y}^{3}(\beta_{[i]}\gamma,y)\alpha_{j}w\rangle y^{-2-i-j}).\end{array}$

which coincides with the above, since we obtain

$\begin{array}[]{l}P(\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j}\langle\theta,\alpha_{-2-i-j}{\cal
Y}^{3}(\beta_{[i]}\gamma,y)w\rangle
y^{j})\cr\mbox{}\qquad=P(\sum_{i,j,k\in{\mathbb{N}}}\binom{i}{k}\langle\theta,\beta_{-2-i-j}{\cal
Y}^{3}(\alpha_{[k]}\gamma,y)w\rangle y^{j+i-k})\end{array}$

from

$\begin{array}[]{l}\sum_{i,j,k\in{\mathbb{N}}}\binom{i}{k}X^{-2-i-j}Y^{k}Z^{j+i-k}=\sum_{i,j\in{\mathbb{N}}}X^{-2-i-j}Z^{j}(Z+Y)^{i}\cr\mbox{}\quad=X^{-1-i}(Z+Y)^{i}\iota_{X,Z}\\{(X-Z)^{-1}\\}=\iota_{X,Z,Y}\\{(X-Z-Y)^{-1}(X-Z)^{-1}\\}\cr\mbox{}\quad=\sum_{i\in{\mathbb{N}}}\iota_{X,Z}\\{(X-Z)^{-2-i}\\}Y^{i}=\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j}X^{-2-i-j}Y^{i}Z^{j},\end{array}$

where we denote $\langle\theta,\alpha_{a}{\cal
Y}^{3}(\beta_{[b]}\gamma,y)w\rangle y^{c}$ by $X^{a}Y^{b}Z^{c}$ and
$b\in{\mathbb{N}}$ and we also have

$\begin{array}[]{l}P(\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j-2-i}\langle\theta,{\cal
Y}^{3}(\beta_{[i]}\gamma,y)\alpha_{j}w\rangle
y^{-2-i-j})\cr\mbox{}\qquad=P(\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{j-2-i}\langle\theta,{\cal
Y}^{3}(\beta_{[i]}\gamma,y)\alpha_{j}w\rangle y^{-2-i-j})\end{array}$

from

$\begin{array}[]{l}\sum_{i,j,k\in{\mathbb{N}}}\binom{-1-i}{k}X^{k}Y^{i+j}Z^{-2-i-j-k}=\sum_{i\in{\mathbb{N}}}(Z+X)^{-1-i}\sum_{j\in{\mathbb{N}}}Y^{i}Z^{-1-j}Y^{j}\cr\mbox{}\quad=\iota_{Z-Y,X}\\{(Z-Y+X)^{-1}\\}\iota_{Z,Y}\\{(Z-Y)^{-1}\\}=\sum_{i\in{\mathbb{N}}}(-1)^{i}\iota_{Z,Y}\\{(Z-Y)^{-2-i}\\}X^{i}\cr\mbox{}\quad=\sum_{i,j\in{\mathbb{N}}}\binom{-2-i}{j}(-1)^{-2-i+j}X^{i}Y^{j}Z^{-2-i+j},\end{array}$

where we replace $\langle\theta,{\cal
Y}^{3}(\beta_{[a]}\gamma,y)\alpha_{b}w\rangle y^{c}$ with $X^{a}Y^{b}Z^{c}$
and $a,b\in{\mathbb{N}}$.

This completes the proof of Theorem 4.10. $\square$

We now start the proof of Theorem 4.1. Since $A\tilde{\boxtimes}B$ is an
${\mathbb{N}}$-gradable modules, by Theorem 2.15, ${\cal Y}^{3}$ and ${\cal
Y}^{4}$ are branches of some logarithmic intertwining operators of type
$\binom{A(BC)}{A\tilde{\boxtimes}B\,\,C}$ and
$\binom{A\tilde{\boxtimes}B}{A\,\,B}$ on ${\cal D}^{2}$. By the construction
of ${\cal Y}^{3},{\cal Y}^{4}$, for any $\theta\in(A(BC))^{\vee}$, there are
$v\in A,u\in B,w\in C$ such that $\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle\not=0$, that is, the set of coefficients of
$(x-y)^{r}y^{s}\log^{h}(x-y)\log^{k}(y)$ in $\\{{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\mid v\in A,u\in B,w\in C\\}$ spans $A(BC)$. From the
universal properties of fusion products $(A\boxtimes B,{\cal Y}^{AB})$ and
$((A\tilde{\boxtimes}B)\boxtimes C,{\cal Y}^{(A\tilde{\boxtimes}B)\boxtimes
C})$ and $((A\boxtimes B)\boxtimes C,{\cal Y}^{(AB)C})$, there is a surjective
$V$-homomorphism $\phi_{[AB]C}:(A\boxtimes B)\boxtimes C\to
A\boxtimes(B\boxtimes C)$ such that

$\begin{array}[]{rl}\langle\theta,\phi_{[AB]C}({\cal Y}^{(AB)C}({\cal
Y}^{AB}(v,x-y)u,y)w\rangle=&\langle\theta,{\cal Y}^{3}({\cal
Y}^{4}(v,x-y)u,y)w\rangle\cr=&\langle\theta,{\cal Y}^{A(BC)}(v,x){\cal
Y}^{BC}(u,y)w\rangle\end{array}$

on ${\cal D}^{2}$ for any $v\in A,u\in B,w\in C$ and
$\theta\in(A(BC))^{\vee}$. Similarly, by starting from
$\langle\theta^{\prime},{\cal Y}^{(AB)C}({\cal Y}^{(AB)}(v,x-y)u,y)w\rangle$,
we also have a surjective homomorphism $\phi_{A[BC]}:(A\boxtimes B)\boxtimes
C\to A\boxtimes(B\boxtimes C)$ satisfying

$\langle\theta^{\prime},\phi_{A[BC]}({\cal Y}^{A(BC)}(v,x){\cal
Y}^{BC}(u,y)w)\rangle=\langle\theta^{\prime},{\cal Y}^{(AB)C}({\cal
Y}^{AB}(v,x-y)u,y)w\rangle$

on ${\cal D}^{2}$ for all $v,u,w,\theta^{\prime}$. From the construction, it
is easy to see that $\phi_{A[BC]}\circ\phi_{[AB]C}=1$ and
$\phi_{[AB]C}\circ\phi_{A[BC]}=1$.

This completes the proof of the associativity law of fusion products.
$\square$

###### Corollary 4.11

If $V^{\vee}\cong V$ and $A$ and $A^{\vee}$ are both $C_{1}$-cofinite
${\mathbb{N}}$-gradable modules, then for any non-zero $C_{1}$-cofinite
${\mathbb{N}}$-gradable $V$-module $B$, $A\boxtimes B$ and $B\boxtimes A$ are
not zero.

[Proof] There is a surjective intertwining operator of type
$\binom{V^{\vee}}{A\,\,A^{\vee}}$ which comes from the $V$-module structure on
$A$ by skew-symmetry and duality. Hence $B\cong B\boxtimes V$ is a
homomorphism image of $B\boxtimes(A\boxtimes A^{\vee})\cong(B\boxtimes
A)\boxtimes A^{\vee}$. Therefore, $(B\boxtimes A)\boxtimes A^{\vee}\not=0$ and
$B\boxtimes A\not=0$. Similarly, we get $A\boxtimes B\not=0$.

##  5 Pentagon axiom

######  Theorem 5.1 (Pentagon axiom)

For $C_{1}$-cofinite ${\mathbb{N}}$-gradable modules $A,B,C,D$, by using the
above isomorphisms $\phi_{[\ast]\ast}$, we have the following commutative
diagram.

$\begin{array}[]{c}((A\boxtimes B)\boxtimes C)\boxtimes
D\xrightarrow{\phi_{([AB]C)}\times 1_{D}}(A\boxtimes(B\boxtimes C))\boxtimes
D\xrightarrow{\phi_{[A(BC)]D}}A\boxtimes((B\boxtimes C)\boxtimes
D)\cr\mbox{}\hskip 71.13188pt\searrow{}^{\phi_{[(AB)C]D}}\mbox{}\qquad\hskip
128.0374pt\swarrow{{}_{1_{A}\times\phi_{[BC]D}}}\cr(A\boxtimes
B)\boxtimes(C\boxtimes
D)\xrightarrow{\phi_{[AB](CD)}}A\boxtimes(B\boxtimes(C\boxtimes
D))\end{array}$

[Proof]

We consider a domain ${\cal
D}^{3}\\!=\\!\\{(x,y,z)\\!\in\\!{\mathbb{C}}^{3}\mid|x|\\!>\\!|y|\\!>\\!|z|\\!>\\!|x\\!-\\!z|\\!>\\!|y\\!-\\!z|\\!>\\!|x\\!-\\!y|\\!>\\!0,\mbox{
and }x,y,z,x\\!-\\!y,y\\!-\\!z,x\\!-\\!z\not\in{\mathbb{R}}^{\leq 0}\\}$. We
note $(7,6,4)\in{\cal D}^{3}\not=\emptyset$. Simplify the notation, we omit
the notation $\boxtimes$. Let $\theta\in(A(B(CD)))^{\vee},v\in A,u\in B,w\in
C,d\in D$ and let $\vec{\xi}$ denote a quadruple $(v,u,w,d)\in A\times B\times
C\times D$. We fix all intertwining operators for fusion products, say ${\cal
Y}^{AB},{\cal Y}^{BC},{\cal Y}^{CD},{\cal Y}^{B(CD)},{\cal Y}^{(BC)D},{\cal
Y}^{A(B(CD))},\ldots$, and consider their five-point correlation functions:

$\begin{array}[]{l}F^{A(B(CD))}(\theta^{1},\vec{\xi};x,y,z)=\langle\theta^{1},{\cal
Y}^{A(B(CD))}(v,x){\cal Y}^{B(CD)}(u,y){\cal Y}^{CD}(w,z)d\rangle,\cr
F^{A((BC)D)}(\theta^{2},\vec{\xi};x,y\\!-\\!z,z)=\langle\theta^{2},{\cal
Y}^{A((BC)D)}(v,x){\cal Y}^{(BC)D}({\cal Y}^{BC}(u,y\\!-\\!z)c,z)d\rangle,\cr
F^{(A(BC))D}(\theta^{3},\vec{\xi};x\\!-\\!z,y\\!-\\!z,z)=\langle\theta^{3},{\cal
Y}^{(A(BC))D}({\cal Y}^{(A(BC)}(v,x\\!-\\!z){\cal
Y}^{BC}(u,y-z)w,z)d\rangle,\cr
F^{((AB)C)D}(\theta^{4},\vec{\xi};x\\!-\\!y,y\\!-\\!z,z)\\!=\\!\langle\theta^{4},{\cal
Y}^{((AB)C)D}({\cal Y}^{(AB)C}({\cal
Y}^{AB}(v,x\\!-\\!y)u,y\\!-\\!z)w,z)d\rangle,\\!\mbox{ and}\cr
F^{(AB)(CD)}(\theta^{5},\vec{\xi};x\\!-\\!y,y,z)=\langle\theta^{5},{\cal
Y}^{(AB)(CD)}({\cal Y}^{AB}(v,x\\!-\\!y)u,y){\cal
Y}^{CD}(w,z)d\rangle\end{array}$ $None$

for
$\theta^{1}\in(A(B(CD)))^{\vee},\theta^{2}\in(A((BC)D))^{\vee},\theta^{3}\in((A(BC))D)^{\vee},\theta^{4}\in(((AB)C)D)^{\vee},\theta^{5}\in((AB)(CD))^{\vee}$.
By the same way as we did for four-point correlation functions, it is not
difficult to see that these five-point correlation functions are all locally
normal convergent on
$\\{(x,y,z)\\!\in\\!{\mathbb{C}}^{3}\mid|x|\\!>\\!|y|\\!>\\!|z|\\!>\\!|x\\!-\\!z|\\!>\\!|y\\!-\\!z|\\!>\\!|x\\!-\\!y|\\!>\\!0\\}$.
We next choose their principle branches

$\widetilde{F}^{A(B(CD))}(\theta^{1},\vec{\xi};x,y,z),\widetilde{F}^{A((BC)D)}(\theta^{2},\vec{\xi};x,y-z,z),\ldots,\widetilde{F}^{(AB)(CD)}(\theta^{5},\vec{\xi};x-y,y,z)$

on ${\cal D}^{3}$ by taking the values of
$\log(x),\log(y),\log(z),\log(x-y),\log(x-z),\log(y-z)$ which satisfy
$-\pi<\Im(\log(x)),\Im(\log(y)),\Im(\log(z)),\Im(\log(x-y)),\Im(\log(x-z)),\Im(\log(y-z))<\pi$
and viewing $x^{a},y^{b},z^{c},(x-y)^{d},(x-z)^{e},(y-z)^{f}$ as
$e^{a\log(x)}$, $e^{b\log(y)}$, $e^{c\log(z)}$, $e^{d\log(x-y)}$,
$e^{e\log(x-z)}$, $e^{f\log(y-z)}$, respectively.

From our construction of isomorphisms $\phi_{[\ast,\ast]\ast}$, by Theorem
4.1, we have equations:

$\begin{array}[]{l}\widetilde{F}^{A(B(CD))}(\theta^{1},\vec{\xi};x,y,z)=\widetilde{F}^{A((BC)D)}((1_{A}\times\phi_{[BC]D}^{\ast})(\theta^{1}),\vec{\xi};x,y-z,z),\cr\widetilde{F}^{A((BC)D)}(\theta^{2},\vec{\xi};x,y-z,z)=\widetilde{F}^{(A(BC))D}(\phi_{[A(BC)]D}^{\ast}(\theta^{2}),\vec{\xi};x-z,y-z,z),\cr\widetilde{F}^{(A(BC))D}(\theta^{3},\vec{\xi};x-z,y-z,z)=\widetilde{F}^{((AB)C)D}((\phi_{[AB]C}^{\ast}\times
1_{D})(\theta^{3}),\vec{\xi};x-y,y-z,z),\cr\widetilde{F}^{A(B(CD))}(\theta^{1},\vec{\xi};x,y,z)=\widetilde{F}^{(AB)(CD)}(\phi_{[AB](CD)}^{\ast}(\theta^{1}),\vec{\xi};x-y,y,z),\mbox{
and}\cr\widetilde{F}^{(AB)(CD))}(\theta^{5},\vec{\xi};x-y,y,z)=\widetilde{F}^{((AB)C)D}(\phi_{[(AB)C]D}^{\ast}(\theta^{5}),\vec{\xi};x-y,y,z),\end{array}$

on ${\cal D}^{3}$ at least, where $\phi^{\ast}:U^{\vee}\to W^{\vee}$ denotes
the dual of $\phi:W\to U$. We hence have

$\begin{array}[]{l}\widetilde{F}^{((AB)C)D}((\phi_{[AB]C}^{\ast}\otimes
1_{D})(\phi_{[A(BC)]D}^{\ast}((1_{A}\otimes\phi_{[BC]D}^{\ast})(\theta^{1}))),\vec{\xi};x-y,y-z,z)\cr\mbox{}\quad=\widetilde{F}^{A(B(CD))}(\theta^{1},\vec{\xi};x,y,z)=\widetilde{F}^{((AB)C)D}(\phi_{[(AB)C]D}^{\ast}(\phi_{[AB](CD)}^{\ast}(\theta^{1})),\vec{\xi};x-y,y-z,z)\end{array}$

for all $\theta^{1}\in(A(B(CD)))^{\vee}$ and $\vec{\xi}$. As a consequence, we
obtain $1_{A}\otimes\phi_{[BC]D}\circ\phi_{[A(BC)]D}\circ\phi_{[AB]C}\otimes
1_{D}=\phi_{[AB](CD)}\circ\phi_{[(AB)C]D}$, which implies the commutativity of
the diagram.

This completes the proof of Theorem 5.1.

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