\vertex [above=1cm of j] (j11) ;
\vertex[left=0.05 of j11] (j111) [label=below left:\(\rho\)];
\vertex[above=1cm of k] (k11);
\node[isosceles triangle,
minimum size =0.25cm] (T2)at (j11){};
\vertex[above=0.8cm of j11] (j22) [label=below right:\(U_i\)];
\vertex[above=0.8cm of k11]  (k22) ;
\vertex  (j2) at (0.45,-3.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-3.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [thick,rounded corners=1mm] (j)  -- (T2);
\draw [thick,rounded corners=1mm] (k)  -- (T3) ;
\draw [thick,rounded corners=1mm] (T2)  -- (j22) -- (beta);
\draw [thick,rounded corners=1mm] (k)  -- (k22) -- (beta);
\diagram*{
(beta) -- [thick,edge label=$U_{k}$] (T4),
(T4)--[thick] (top2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

where $C^{(a,j)b}_{(I)K;\alpha\beta}$ are complex numbers to be determined by the condition of $\Psi^\alpha_{(a,j)b}$ being an $A$-module intertwiner:

\begin{equation}
\Psi^\alpha_{(a,j)b}\circ \rho_{a\otimes j}=\rho_b\circ\left(\operatorname{id}_A\otimes \Psi^\alpha_{(a,j)b}\right)
\end{equation}

Using (<ref>) in the last equation leads to a system of linear equations:

\begin{align}\label{psicongen}
&\forall T\prec A,\forall I\prec M_a, \forall q\prec M_a,\forall S\prec M_b, \nonumber \\
&\forall \gamma\in\{1,\ldots,\tensor{N}{_{ti}}{^q}\}, \forall \beta\in\{1,\ldots,\tensor{N}{_{qj}}{^s}\} \quad \text{the following holds:}\nonumber\\
&\hspace{10mm}\sum_{\zeta=1}^{\mathcal{E}^{M_a}_q}\rho^{a(Q);\gamma}_{(T)(I)}C^{(a,j)b}_{(Q)S;\alpha\beta}=\sum_{\substack{K\prec M_b\\}}\sum_{\delta=1}^{\tensor{N}{_{ij}}{^k}}\sum_{\varepsilon=1}^{\tensor{N}{_{tk}}{^s}}\rho^{b(S);\varepsilon}_{(T)(K)}\mathrm{F}^{(t\,i\,j)s}_{\varepsilon k\delta,\gamma q\beta}C^{(a,j)b}_{(I)K;\alpha\delta}
\end{align}

where   $Q=(q,\zeta)$.

Suppose now that we have obtained these numbers $C^{(a,j)b}_{(I)K;\alpha\beta}$. To evaluate the $\dot{\mathrm{T}}$ matrix elements we introduce the numbers $\bar{C}^{(a,j)b}_{(I)K;\alpha\beta}$ in the same way as the $\bar{A}$ coefficients are introduced in (<ref>) as the coefficients in the decomposition of the dual morphism $\bar{\Psi}^\alpha_{(a,j)b}$ in terms of $\bar{\lambda}^\beta_{(i,j)k}$. The numbers  $\bar{C}$ can be obtained from the normalisation condition

\begin{equation}
\Psi_{(a,j)b}^\alpha\circ \bar{\Psi}^\beta_{(a,j)b}=\delta_{\alpha,\beta}\operatorname{id}_{M_b}
\end{equation}

Subsequently, using the decomposition (<ref>) in both sides of  (<ref>) and composing  with a suitable morphism  we find the following expression

\begin{equation} \small
\dot{\mathrm{T}}^{(x\,a\,k)c}_{\alpha b\beta,\gamma d\delta}=\sum_{\substack{I\prec X_x\\ J\prec M_a}} \, \sum_{\substack{P\prec M_b\\Q\prec M_d}}\sum_{\zeta=1}^{\tensor{N}{_{jk}}{^p}}\sum_{\varepsilon=1}^{\tensor{N}{_{ip}}{^s}}\sum_{\theta=1}^{\tensor{N}{_{ij}}{^q}}\sum_{\xi=1}^{\tensor{N}{_{qk}}{^s}}\sum_{\phi^{\prime\prime}=1}^{\mathcal{E}_s^{M_c}}\mathrm{F}^{(i\,j\,k)s}_{\varepsilon p\zeta,\theta q\xi}C^{(a,k)b}_{(J)P;\beta\zeta}B^{(x,b)c}_{(I,P)S;\alpha\varepsilon}\bar{B}^{(x,a)d}_{(I,J)Q;\gamma\theta}\bar{C}^{(d,k)c}_{(Q)s\phi^{\prime\prime};\delta\xi}
\end{equation}
that holds for all  $S=(s,\phi)\prec M_c$.

In the special case described by

\begin{align}
&U_i\otimes U_j\cong U_k \\
& X_x\otimes_A X_y\cong X_z\\
&X_w\otimes_A M_a\cong M_b \label{speciald}\\
& \operatorname{dim}\operatorname{Hom}(U_i,\dot{M}_a)\in\{0,1\}\\
& M_c\otimes U_i\cong M_c
\end{align}

we find

\begin{equation}\label{tdotboson}
\dot{\mathrm{T}}^{(x\,a\,k)c}_{b,d}=\frac{C^{(a,k)b}_{(j)p}B^{(x,b)c}_{(i,p)s}}{B^{(x,a)d}_{(i,j)q}C^{(d,k)c}_{(q)s}}\mathrm{F}^{(i\,j\,k)s}_{p,q},
\enspace \;   \forall i\prec X_x,\forall j\prec M_a
\end{equation}

Here all possible fusion structures between simple objects are  group-like. For example, while the tensor product $X_w\otimes_A M_a$ is always an $A$-module, the restriction (<ref>) means that it should be irreducible.  Each irreducible defect thus acts on the set of all irreducible modules.

The matrix elements $\mathrm{\dot{T}}^{(x\,a\,k)c}_{b,d}$ are non-zero only if the labels $c,b,d$ are the unique labels that satisfy

\begin{equation}\label{selec3}
M_c\cong \left( X_x\otimes_BM_a\right)\otimes U_k, \quad M_b\cong M_a\otimes U_k \quad \text{and}\quad M_d\cong X_x\otimes_B M_a
\end{equation}

and $s,p,q$ are the unique labels that satisfy

U_s\cong U_i\otimes U_j\otimes U_k,\quad U_p\cong U_j\otimes U_k \quad \text{and} \quad U_q\cong U_i\otimes U_j.

For these reasons, if we work in the special case of the form (<ref>) we will usually omit the labels $c,b,d$ when presenting the non-zero elements $\mathrm{\dot{T}}^{(x\,a\,k)c}_{b,d}$.

§.§ F[A]-matrix

To calculate the  fusing matrix $\mathrm{F}[A]$ we start from the definition (<ref>) and use the decomposition (<ref>) of the  junctions $\Psi$. Suppose now that we have obtained the complex coefficients $C^{(a,j)b}_{(I,j)K;\alpha\beta}$ as a solution of the system (<ref>). Then the entries of the $\mathrm{F}[A]$ matrices are given by the following expression which holds for any $S\prec M_c$

\begin{equation}
\mathrm{F}[A]^{(a\,j\,k)c}_{\alpha p\beta,\gamma b\delta}=\sum_{I\prec M_a}\sum_{Q\prec M_b} \sum_{\varepsilon=1}^{\tensor{N}{_{ip}}{^s}}\sum_{\zeta=1}^{\tensor{N}{_{ij}}{q}}\sum_{\eta=1}^{\tensor{N}{_{qk}}{^s}}\sum_{\phi^{\prime\prime}=1}^{\mathcal{E}_s^{M_c}}\mathrm{F}^{(i\,j\,k)s}_{\varepsilon p\beta,\zeta q\eta}C^{(a,p)c}_{(I)S;\alpha\varepsilon}\overbar{C}^{(b,k)c}_{(Q)s\phi^{\prime\prime};\delta\eta}\overbar{C}^{(a,j)b}_{(I)Q;\gamma\zeta}
\end{equation}

where $S=(s,\phi)$.

In the special case described by (<ref>) we find

\begin{equation}\label{F(1)boson}
\mathrm{F}[A]^{(a\,j\,k)c}_{p,b}=\frac{C^{(a,p)c}_{(i)s}}{C^{(a,j)b}_{(i)q}C^{(b,k)c}_{(q)s}}\mathrm{F}^{(i\,j\,k)s}_{p,q}, \enspace \;  \forall i\prec M_a.
\end{equation}

The matrix elements $\mathrm{F}[A]^{(a\,j\,k)c}_{p,b}$ are non-zero only if the labels $c,p,b$ are the unique labels that satisfy

\begin{equation}
M_c\cong \left(M_a\otimes U_j\right)\otimes U_k,\quad U_p\cong U_j\otimes U_k\quad \text{and} \quad M_b\cong M_a\otimes U_j
\end{equation}

and $s,q$ are the unique labels that satisfy

U_s\cong U_i\otimes U_j\otimes U_k \quad \text{and} \quad U_q\cong U_i\otimes U_j.

Therefore, whenever we work in this special case we are going to omit the labels $s,p,b$ when presenting the non-zero elements  $\mathrm{F}[A]^{(a\,j\,k)s}_{p,b}$. Moreover, in this case where we have no multiplicities for the relevant junction spaces, the boundary OPE coefficients (<ref>) take the form

\begin{equation}\label{OPEcoefbos1}
C_{j,k}^{(a\,b\,c)p}=\mathrm{G}[A]^{(a\,j\,k)c}_{b,p} \, .
\end{equation}

§ FREE BOSON COMPACTIFIED ON A CIRCLE

In this section, we apply the techniques described earlier in this paper in order to evaluate the various fusing matrices introduced in section <ref> for the conformal field theory of a free boson compactified at a radius of rational square.

§.§ Generalities

We will now introduce all the relevant quantities such as the modular tensor category, the symmetric special Frobenius algebras and their representation theory for the free boson theory. The description of these objects will be brief since in this paper we consider them as given data. More details and results are presented in [43, 7, 9].

First, we recall that the chiral symmetry algebra of the free boson $\phi(z)$ is a Heisenberg affine Lie algebra $\hat{\mathfrak{u}}(1)$. For any $N\in \mathbb{Z}_{>0}$ this algebra can be extended by the two vertex operators $W_N^{\pm}(z)=:e^{\pm 2i\sqrt{2N}\phi(z)}:$. The resulting extended chiral algebra, denoted by $\affu(1)_N$, is rational  with $2N$ inequivalent irreducible highest weight representations, labelled $U_0,U_1,\ldots,U_{2N-1}$.

We denote by $\mathcal{U}_N$ the modular tensor category formed by these representations.
A special feature of this category that will simplify calculations is the fact that the dimensions $\tensor{N}{_{ij}}{^k}$ of all coupling spaces $\Hom(U_i\otimes U_j,U_k)$ are either $0$ or $1$. More specifically the fusion, which furnishes the group $\mathbb{Z}_{2N}$ is found to be

\begin{equation}
U_k\otimes U_l\cong U_{[k+l]} \quad\;\text{where} \quad[k+l]=k+l\;\text{mod}\; 2N\;\in\mathcal{I}.
\end{equation}

The dual of $k$ is $\bar{k}=2N-k$ and all simple objects have quantum dimension $\operatorname{dim}(U_k)=1$. Under a suitable choice of bases (<ref>), the braiding and fusing matrices are given by

\begin{equation}\label{Finit}
\mathrm{R}^{(kl)[k+l]}=e^{-\pi ikl/2N},\qquad \mathrm{F}^{(r\,s\,t)[r+s+t]}_{[s+t][r+s]}=(-1)^{r\sigma(s+t)}
\end{equation}

where $\sigma(s+t)=0$ if $s+t<2N$ and $\sigma(s+t)=1$ otherwise. We denote the bases of $\Hom(U_i\otimes U_j,U_k)$ for which those equalities hold by $\lambda_{(i,j)k}$. Notice that the fusing matrices in (<ref>) only depend on the first three upper indices. Hence, in the following we will usually denote them as $\mathrm{F}^{(r\,s\,t)}$ with the understanding that this stands for the only non-zero matrix element of $\mathrm{F}^{(r\,s\,t)}$. Furthermore, it follows from the explicit form (<ref>) that the $\mathrm{F}$-matrices coincide with their inverse

\begin{equation}
\mathrm{F}^{(r\,s\,t)}=\mathrm{G}^{(r\,s\,t)}\,.
\end{equation}

The modular $s$-matrix is found to be

\begin{equation}
s_{k,l}=e^{-\pi ikl/N} \, .
\end{equation}

Now we turn to describing the Frobenius algebras in $\mathcal{U}_N$. For this free boson theory, we will consider haploid special symmetric Frobenius algebras in $\mathcal{U}_N$. The additional property of haploidity corresponds to the uniqueness of the boundary vacuum state.  Every boundary condition of a full CFT gives rise to a special symmetric Frobenius algebra and all such algebra objects are Morita equivalent. It turns out that there is always a haploid representative of each Morita equivalence class. Furthemore, due to the fact that the quantum dimensions of the simple objects of $\mathcal{U}_N$ are all $1$, the algebra object is additionally of the simple current type. Frobenius algebras which have  all of these properties are called Schellekens algebras and they are studied in [9].

Every haploid special symmetric Frobenius algebra in $\mathcal{U}_N$ is isomorphic to one of the algebras $A_r$ defined as follows (see <cit.>). As an object in $\mathcal{U}_N$ we have

\begin{equation}\label{alg1}
A_r=\bigoplus_{a=0}^{r-1}U_{2aN/r} \qquad \text{where}\; r\in\mathbb{Z}_{>0} \;\text{divides} \; N\,.
\end{equation}

For $a\in\mathbb{Z}_r$ we denote by $\iota_a\in\Hom(U_{2aN/r},A_r)$ and $\jmath_a\in\Hom(A_r,U_{2aN/r})$ embedding and restriction morphisms for the subobject $U_{2aN/r}$ of $A_r$ with $\jmath_a\circ \iota_a=\operatorname{id}_{U_{2aN/r}}$. We can choose the multiplication and the unit morphism of $A_r$ to be

\begin{equation}\label{mult1}
m=\sum_{a, b=0}^{r-1} \iota_{a+b} \circ \lambda_{(2 a N / r, 2 b N / r)[2(a+b) N / r]} \circ\left(\jmath_a \otimes \jmath_b\right) \qquad \text { and } \quad \eta=\iota_0\,.
\end{equation}

The compactification radius is related to $r$ and $N$ by

\begin{equation}
\end{equation}

so that different algebra objects correspond to different compactification radii.  A rational  free boson model  is thus specified by the two numbers: $N$ and $r$ which completely characterise the modular tensor category and the Frobenius algebra respectively.

Conversely, the free boson compactified at a radius of the form $R=\sqrt{P/2Q}$ with $P,Q$ coprime positive integers contains in its subset of holomorphic bulk fields the chiral algebra $\affu(1)_{n^2PQ}$ for any choice of $n\in\mathbb{Z}_{>0}$. The corresponding algebra in $\mathcal{U}_{n^2PQ}$ is then $A_{nP}$. Therefore, the following relations hold between the two different aforementioned parametrizations

\begin{equation}
r=nP \qquad \text{and}\qquad N=n^2PQ
\end{equation}

For the left $A_r$-modules one finds that every simple module is isomorphic to an induced module $\operatorname{Ind}_{A_r}(U_k)$, and that the induced modules $\operatorname{Ind}_{A_r}(U_k)$ and $\operatorname{Ind}_{A_r}(U_l)$ are isomorphic if and only if $k\equiv l \;\text{mod}\; 2N/r$. Hence, there are $2N/r$ distinct simple left $A_r$-modules which we denote $M^{(r)}_f,\;f\in \mathcal{J}_{A_r}$ where $\mathcal{J}_{A_r}=\{0,1,\ldots,\frac{2N}{r}-1\}$. Furthermore, there is an additional simplification for this model:  the dimensions of the hom-spaces $\Hom(U_i,A_r\otimes U_k)$ for the subobjects of the induced modules are either $0$ or $1$. The representation numbers (<ref>) are found to be

\begin{equation}\label{rep4}
\rho^{M_f\,[a+i]}_{(a)(i)}=1
\end{equation}

where $a\prec A_{r}$ and $i\prec M_f$.

Regarding  $A_r$-$A_r$-bimodules, the simple $A_r$-bimodules are labelled by elements of the abelian group

\begin{equation}
G^{(r,N)}=H^\ast \times_H \operatorname{Pic}(\mathcal{U}_N)
\end{equation}

where $\operatorname{Pic}(\mathcal{U}_N)=\mathbb{Z}_{2N}$ and $H=\mathbb{Z}_r$ is embedded in $\operatorname{Pic}(\mathcal{U}_{N})$ via $\tilde{\imath} (a)=2aN/r$ (see  <cit.>). We will label elements of $H^\ast$ by $x\in\mathbb{Z}_r$ via

\begin{equation}
\psi_x(a)=e^{-2\pi ixa/r}\,.
\end{equation}

The action of $h\in H$ on $k\in\operatorname{Pic}(\mathcal{U}_N)$ is $h.k=k+\tilde{\imath} (h)$. Thus we have

\begin{equation}
\end{equation}

where the product over $\mathbb{Z}_r$ implies the identification $(a,k+2N/r)=(a+N/r,k)$.

Now for each $\psi\in H^\ast$ it turns out <cit.> that we can construct an automorphism of $A_r$ as

\begin{equation}\label{twist1}
t_{\psi}:=\bigoplus_{h\in H}\psi(h)\operatorname{id}_{U_{2hN/r}}\,.
\end{equation}

Then, a simple bimodule corresponding to an element $(\psi,k)\in G^{(r,N)}$ is given by $\alpha_{A_r}^+(U_k)_{t_\psi}$, that is an alpha-induced bimodule for which the right action of $A_r$ is twisted by the automorphism $t_\psi$. More specifically,
the underlying object is $A_r\otimes U_k$ while the left and right actions are

\begin{equation}\label{act2}
\rho^{\alpha_{A_r}^+(U_k)_{t_\psi}}=m\otimes \operatorname{id}_{U_k}, \qquad\Tilde{\rho}^{\alpha_{A_r}^+(U_k)_{t_\psi}}=(m\otimes \operatorname{id}_{U_k})\circ \left(\operatorname{id}_{A_r}\otimes c_{U_k,A_r}\right)\circ\left(\operatorname{id}_{A_r\otimes U_k}\otimes t_\psi\right).
\end{equation}

The representation numbers for the left action are of the same form as (<ref>) while for the right action we have $\forall i\prec A_r\otimes U_k$:

\begin{equation}\label{rightaction0}
\Tilde{\rho}^{M_f\,[i+a]}_{(i)(a)}=\mathrm{R}^{(ka)[k+a]}\tau_\psi(a)
\end{equation}

where $M_f$ denotes a right $A_r$-module with underlying object $A_r\otimes U_k$ and $\tau$ is a number depending on the twist $t_\psi$ and the subobject $a\prec A$. The fusion of two such bimodules is given by the group operation in $G^{(r,N)}$.

It is useful to label the bimodules by another group which is isomorphic to $G^{(r,N)}$, namely

\begin{equation}
\tilde{G}^{(r,N)}=\left(\mathbb{Z}_{r}\times\mathbb{Z}_{\frac{N}{r}}\times \mathbb{Z}\right)\bigg/\langle(1,1,-2)\rangle
\end{equation}

where $\langle(1,1,-2)\rangle$ denotes the subgroup generated by the element $(1,1,-2)$. Notice that every element of $ \tilde{G}^{(r,N)}$ can be written either as $(a,b,0)$ or $(a,b,1)$ for suitable $a,b$, hence the order of the group is $2N$. The simple $A_r$-bimodules labelled by this group will be denoted $B^{(r)}_{(a,b,\rho)}$ for $(a,b,\rho)\in  \tilde{G}^{(r,N)}$. Let us note that for arbitrary $N,r\in\mathbb{Z}_{>0}$ with $r$ dividing $N$ we find that

\begin{equation}\label{defectgroup}
\left(\mathbb{Z}_{r}\times\mathbb{Z}_{\frac{N}{r}}\times \mathbb{Z}\right)\bigg/\langle(1,1,-2)\rangle \cong \mathbb{Z}_{\operatorname{gcd}(r,\frac{N}{r})}\times\mathbb{Z}_{2\operatorname{lcm}(r,\frac{N}{r})}\,.
\end{equation}

The fusion of topological defects corresponds to the tensor product over $A_r$ which is given by the group operation in $\tilde{G}^{(r,N)}$:

\begin{equation}\label{bimodfus}
B_{(a, b, \rho)}^{(r)} \otimes_{A_r} B_{(c, d, \sigma)}^{(r)} \cong B_{(a+c, b+d, \rho+\sigma)}^{(r)}
\end{equation}

while the fusion of a topological defect to a conformal boundary is found to be

\begin{equation}\label{open1}
B_{(a, b, \rho)}^{(r)} \otimes_{A_r} M_f^{(r)} \cong M_{f+2 b+\rho}^{(r)}\,.
\end{equation}

Notice that the free boson theory described here is a simple current theory [9] and in particular has the properties of the special case described in (<ref>). This implies that we can use the simple formulae (<ref>), (<ref>), (<ref>) and (<ref>) for the fusing matrices. Furthermore, the equations (<ref>), (<ref>) and (<ref>) for calculating the coefficients $A^{(x,y)z}_{(I,J)K;\alpha\beta}$ greatly simplify in this special case. The same can be said for the corresponding equations for calculating the numbers $B^{(x,a)b}_{(I,J)K;\alpha\beta}$ and $C^{(a,j)b}_{(I)K;\alpha\beta}$.

In fact, we can derive a general result for the numbers $C^{(a,j)b}_{(I)K;\alpha\beta}$ as well as the related $\mathrm{F}[A]$ matrices  that holds for any choice of parameters $N$ and $r$. These coefficients are required to satisfy (<ref>) which in our free boson theory takes the simple form

\begin{equation}\label{Cconstantsbos}
C^{(a,j)b}_{([t+i])[t+i+j]}=C^{(a,j)b}_{(i)[i+j]},\quad \forall i\prec M_a,\forall t\prec A\,.
\end{equation}

This equation is the result of the explicit form of the Frobenius algebras (<ref>), the triviality of the representation numbers (<ref>) as well as the $\mathrm{F}$ fusing matrices (<ref>). Recall that (<ref>) provides a linear system of equations for each admissible triple $(a,j,b)$ which satisfy $\operatorname{dim}\operatorname{Hom}(M_a\otimes U_j,M_b)\neq 0$. For each such triple, we find that the solution of this system is

\begin{equation}\label{Cconstantsgen}
\tilde{C}^{(a,j)b} & \text{if}\, $U_i$ \;\text{is a subobject of} \; $M_a$  \\
$0$        & \text{otherwise}.
\end{cases*}
\end{equation}

for any $ \tilde{C}^{(a,j)b}\in\mathbb{C}$. In the following, we shall always use the choice of normalisation

\begin{equation}\label{Cconstantsnorm}
\tilde{C}^{(a,j)b}=1,\quad \text{for all admissible triples}\; (a,j,b)\,.
\end{equation}

Using this result in (<ref>), we find that the fusing matrices $\mathrm{F}[A]$ coincide with $\mathrm{F}$ for any choice of $N$ and $r$:

\begin{equation}\label{F[A]boson}
F[A]^{(a\,j\,k)}=\mathrm{F}^{(i\,j\,k)}=(-1)^{i\sigma(j+k)},\quad\forall i\prec M_a\,.
\end{equation}

Let us briefly discuss the boundary operators in these free boson theories which will be important when we consider boundary RG flows. Firstly, we would like to know which boundary operators can live on a single irreducible conformal boundary. In other words, we want to find the non-trivial morphisms

\begin{equation}\label{bosonbc1}
\Psi_{(f,i)f}\in\Hom_{A_r}\left(M_f\otimes U_i,M_f\right), \quad i\neq 0
\end{equation}

where $M_f$ is the left $A_r$-module labelling an irreducible conformal boundary condition. Using the fusion rules for the free boson theory we can see that

\begin{equation}
M_f\otimes U_i\cong M_{f+i\;\text{mod}\; 2N/r},
\end{equation}

hence the hom-space in (<ref>) has non-zero dimension  if

\begin{equation}\label{bosonbc2}
\end{equation}

Secondly, we are interested in the relevant boundary operators, that is, operators with conformal dimension $\Delta<1$ that can trigger an RG flow. In the free boson theory discussed above, the conformal weight of a boundary operator labelled by $i\in\mathcal{I}$ is

\begin{equation}
\Delta(i)= \begin{cases}\frac{1}{4 N} i^2 & \text { for } i \leq N \\ \frac{1}{4 N}(2 N-i)^2 & \text { for } i>N\end{cases}
\end{equation}

from which we conclude that a boundary operator $\Psi_{(f,i)f}$ is relevant if $i$ satisfies

\begin{align}
&i<2\sqrt{N} \qquad \qquad \hspace{-1.5mm} \text { for } i\leq N \nonumber \\
&i>2N-2\sqrt{N} \quad  \text { for } i>N.
\end{align}

The two lines in the above equation correspond to charge conjugation which does not change the dimension. The only self-conjugate boundary operator is  the one with  $i=N$.

Finally, we remind the reader of our convention for the indices in the fusing matrices. Since the fusion rules for the free boson theory are simple, some indices can be omitted. For example, as explained at the end of section <ref> the matrix elements $\mathrm{Y}^{(x\,y\,z)w}_{u,v}$ will be denoted $\mathrm{Y}^{(x\,y\,z)}$. Similar considerations apply to the other  fusing matrices (<ref>), (<ref>) and (<ref>). Namely, we will omit the final upper index as well as both lower indices when presenting the non-zero matrix elements of $\mathrm{T},\dot{\mathrm{T}}$ and $^{(1)}\mathrm{F}$.

We finish with a convention in order to lighten notation which is possible due to the specific structure of the free boson theory having the form summarised in (<ref>). Consider the   matrix elements $\mathrm{Y}^{(x\,y\,z)w}_{u,v}$ as in (<ref>).  They are non-zero only if the labels $w,u,v$ are the unique labels that satisfy

X_w\cong \left( X_x\otimes_BX_y\right)\otimes_C X_z, \quad X_u\cong X_y\otimes_C X_z \quad \text{and}\quad X_v\cong X_x\otimes_B X_y

and $s,p,q$ are the unique labels that satisfy

U_s\cong U_i\otimes U_j\otimes U_k,\quad U_p\cong U_j\otimes U_k \quad \text{and} \quad U_q\cong U_i\otimes U_j

For these reasons,  we will omit the labels $w,u,v$ when presenting the non-zero elements $\mathrm{Y}^{(x\,y\,z)w}_{u,v}$.

§.§ Fusing matrices of the  N=2, r=2 model

In this model, there are four distinct simple objects of the category $\mathcal{U}_2$ denoted $\{U_i,i=0,\ldots,3\}$ and satisfying fusion rules corresponding to the addition in $\mathbb{Z}_4$. The algebra object with $r=2$ is found to be

\begin{equation}
A_2= U_0\oplus U_2
\end{equation}

with the multiplication given by (<ref>) for $N=2=r$.

There are two distinct simple left $A_2$-modules

\begin{equation}
M_0=\left(A_2\otimes U_0,m\otimes \operatorname{id}_{U_0}\right),\qquad\qquad M_1=\left(A_2\otimes U_1,m\otimes \operatorname{id}_{U_1}\right)
\end{equation}

where for the underlying objects we find the following decompositions into simple objects of $\mathcal{U}_2$

\begin{equation}
\dot{M}_0=A_2\otimes U_0 \cong U_0\oplus U_2=A_2 \qquad \text{and}\qquad \dot{M}_1=A_2\otimes U_1 \cong U_1\oplus U_3.
\end{equation}

For the twist automorphisms we have two possible values of $x\in\mathbb{Z}_2$ which lead to

\begin{equation}
\psi_0(a)=1 \qquad \text{and} \qquad \psi_1(a)= e^{-\pi i a} .
\end{equation}

Thus, using (<ref>) we find

\begin{equation}
t_0=\operatorname{id}_{U_0}\oplus \operatorname{id}_{U_2} \qquad \text{and} \qquad  t_1=\operatorname{id}_{U_0}\oplus\left(-\operatorname{id}_{U_2}\right).
\end{equation}

The $t_0$ is trivial so that  there are four distinct simple $A_2$-$A_2$-bimodules with their corresponding labels in $G^{(2,2)}$:

\begin{align}\label{labels}
& X_0:=\alpha^+_{A_2}(U_0)\; \longleftrightarrow\;(0,0) \qquad \qquad X_1:=\alpha^+_{A_2}(U_1) \; \longleftrightarrow \; (0,1) \nonumber \\
& X_2:= \alpha^+_{A_2}(U_0)_{t_1} \;\longleftrightarrow \;(1,0) \qquad \qquad X_3:=\alpha^+_{A_2}(U_1)_{t_1}\; \longleftrightarrow\; (1,1).
\end{align}

These bimodules can be labeled by elements of the group

\begin{equation}
\tilde{G}^{(2,2)} = \left(\mathbb{Z}_{2}\times\mathbb{Z}_{1}\times \mathbb{Z}\right)\bigg/\langle(1,1,-2)\rangle=\left(\mathbb{Z}_{2}\times \mathbb{Z}\right)\bigg/\langle(1,-2)\rangle \cong \mathbb{Z}_4.
\end{equation}

To make contact with the notation introduced in the previous section that uses $\tilde{G}^{(2,2)}$ as the labelling group we note that

\begin{equation}
X_0\equiv B^{(2)}_{(0,0,0)},\qquad X_1\equiv B^{(2)}_{(0,0,1)},\qquad X_2\equiv B^{(2)}_{(1,0,0)},\qquad X_3\equiv B^{(2)}_{(1,0,1)}.
\end{equation}

The fusion of topological defects corresponds to the group operation in  $\mathbb{Z}_4$. The element $X_1$ is the generator, so that

\begin{equation}\label{jun3}
X_x\otimes_{A_2}X_y\cong X_{[x+y]} \qquad \text{where} \; x,y\in\mathbb{Z}_4.
\end{equation}

We note in passing that this  generalises to the case of $N=r$ for which

\begin{equation}
\tilde{G}^{(N,N)}\cong \mathbb{Z}_{2N}.
\end{equation}

This implies that the number of simple topological defects is equal to the number of irreducible representations of the chiral algebra and they both obey the same fusion rules.

The fusion of a topological defect with a conformal boundary can be written compactly as

\begin{equation}
X_x\otimes_{A_2}M_a\cong M_{x+a \;\text{mod}\,2}.
\end{equation}

The possible junctions that can occur in our model are restricted  by these fusion rules. Namely, the fusion of two bimodules  describes the allowed triple defect junctions  (<ref>), while the fusion of a bimodule with a module encodes the allowed open defect junctions  (<ref>).

Now that we have described all the necessary ingredients of this model, we can proceed with the evaluation of the fusing matrices. To this end, we first need to obtain the complex numbers $A^{(x,y)z}_{(i,j)k},B^{(x,a)b}_{(i,j)k}$ and $C^{(a,j)b}_{(i,j)k}$ introduced in sections <ref>, <ref> and <ref> respectively. Since there are  no multiplicity labels  equation (<ref>) simplifies to

\begin{align}\label{conaboson}
%& \quad \text{the following holds:}\nonumber\\
& A^{(x,y)z}_{(i,j)[i+j]}=\sum_{k\prec A_2}\Delta_0^{(k)(\bar{k})}\tilde{\rho}^{x[i+k]}_{(i)(k)}\mathrm{F}^{([i+k]\,\bar{k}\,j)[i+j]}_{[\bar{k}+j],i}G^{(i\,k\,\bar{k})i}_{[i+k],0}A^{(x,y)z}_{([i+k],[\bar{k}+j])[i+j]}\,.
\end{align}

that holds $\forall i\prec X_x,\forall j\prec X_y$.
Furthermore, equation (<ref>) simplifies to

A^(x,y)z_([t+i],j)[t+i+j]=F^(t i j)[t+i+j]_ [i+j], [t+i]A^(x,y)z_(i,j)[i+j]   ∀t≺A_2,∀i≺X_x,∀j≺X_y
while (<ref>) becomes

ρ̃^y[j+t]_(j)(t)F^(i j t)[i+j+t]_ [j+t], [i+j]A^(x,y)z_(i,[j+t])[i+j+t]=ρ^z[i+j+t]_([i+j])(t)A^(x,y)z_(i,j)[i+j]
that holds $\forall t\prec A_2,\forall i\prec X_x,\forall j\prec X_y$.

The functions $\tau_\psi$ introduced in (<ref>) can be derived by using the explicit form of the twists (<ref>):

\begin{align}
&\tau_0(a)=1, \qquad \forall a\prec A_2 \nonumber \\
& \tau_1(a)=\begin{cases}
1,& a=0 \\
-1, & a=2.
\end{cases}
\end{align}

Using (<ref>), (<ref>), (<ref>)  in  (<ref>), (<ref>) and (<ref>) we obtain a  system of linear equations for the complex numbers $A^{(x,y)z}_{(i,j)[i+j]}$. We present the solution of this system in (<ref>).

This notation encodes all the information of the numbers $A^{(x,y)z}_{(i,j)k}$ introduced earlier since in the free boson theories the label $k$ is uniquely determined by the two labels $i$ and $j$. The label $z$ is also uniquely determined by the $A_2$-$A_2$-bimodule fusion rules, therefore in the following we are going to omit both the labels $k$ and $z$  from the numbers $A_{(i,j)k}^{(x,y)z}$.

We can now directly use (<ref>) to find the $\mathrm{Y}$-matrices for this model. To this end, consider the $\mathrm{Y}^{(x\,y\,z)}$ matrix as a 3-cochain $Y: (\mathbb{Z}_4)^3\to \mathbb{C}^\times$. Recall that $\mathrm{Y}$ and $\mathrm{F}$ satisfy the pentagon identities which can be rewritten as $d\mathrm{Y}=0$ and $d\mathrm{F}=0$ where $d$ is the coboundary homomorphism. Hence, $\mathrm{Y}$ and $\mathrm{F}$ are 3-cocycles and then (<ref>) is equivalent to the statement that they are in the same cohomology class:

\begin{equation}\label{coho2}
\mathrm{Y}=\mathrm{F}dA \qquad \text{where}\quad A: \left(\mathbb{Z}_4\right)^2\to \mathbb{C}^\times.
\end{equation}

This implies that $\mathrm{Y}/\mathrm{F}$ must be a coboundary. In  [44] it was shown that for any cyclic group $\mathbb{Z}_n$ the 3-cocycles  $\omega\in H^3(\mathbb{Z}_n,\mathbb{C}^\times)$ can be explicitly written as

\begin{equation}\label{cocycle1}
\omega(a,b,c)=\operatorname{exp}\left[\frac{2\pi il}{n^2}a(b+c-[b+c])\right]
\end{equation}

for some $l\in\mathbb{Z}_n$, where $[b+c]=b+c\;\text{mod}\;n$. In particular we find that the fusing matrices of $\mathcal{U}_2$ given in (<ref>)
are obtained from formula  (<ref>) with $n=4$, $l=2$.

Note that the system of equations (<ref>), (<ref>) and (<ref>) admits a large freedom in choosing the solution.
The general solution we give in   (<ref>) is parameterised by  complex numbers $\tilde{A}^{(x,y)}$. Since  $\mathrm{Y}$ and $\mathrm{F}$ are in the same cohomology class we can choose the solution for $A$ to be a closed 2-cochain so that $\mathrm{Y}^{(x\,y\,z)}=\mathrm{F}^{(x\,y\,z)}$.
We find that this fixes the normalisation constants $\tilde{A}^{(x,y)}$ up to four free parameters. We fix the remaining parameters by imposing an additional

\begin{equation} \label{normalisation1}
\tilde{A}^{(x,x)}=1, \qquad \forall x\in\mathbb{Z}_4.
\end{equation}

that leads to

\begin{equation}\label{newnorm}
\Tilde{A}= \left[
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & -\xi^3 & \xi \\
1 & -\xi & 1 & \xi^3 \\
1 & -\xi^3 & -\xi & 1 \\
\end{array}
\right]
\end{equation}

where $\xi=e^{i\pi/4}$ and the matrix labels run through the simple bimodule labels $x$ and $y$. With this choice we have

\begin{equation}
\mathrm{Y}^{(x\,y\,z)}=\mathrm{F}^{(x\,y\,z)}=(-1)^{x\sigma(y+z)}\, .
\end{equation}

To find the other two fusing matrices $\mathrm{T}$ and $\dot{\mathrm{T}}$ we follow a similar procedure.
The general solution of the equations for the coefficients  $B^{(x,a)b}_{(i,j)k}$ introduced in <ref> is given  in  (<ref>). It is parameterised by the constants
$\Tilde{B}^{(x,a)}$ which we fix by imposing

\begin{equation} \label{normalisation2}
\Tilde{B}^{(x,a)}=1,\quad \forall x\in\mathbb{Z}_4,\forall a\in\{0,1\}.
\end{equation}

Using  (<ref>) we find the $\mathrm{T}$-matrices for this model. With the solutions for the $A$- and $B$-coefficients chosen as above
some of  the  entries of the $\mathrm{T}$-matrices are presented graphically below[ The hollow junctions appearing in these diagrams are the ones introduced in (<ref>).]

\begin{align} %\label{Tvals_pic}
&  \vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.1cm of x1] (x5) [label=below left:\(X_1\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_1\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_1\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=0.8cm of x5] (xx) [label=below left:\(X_1\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_0\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=-i
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.5,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_1\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_1\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.2,1.75) [label=right:\(X_2\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_1\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_1\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}}, \qquad  \vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.1cm of x1] (x5) [label=below left:\(X_3\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_1\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_1\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=0.8cm of x5] (xx) [label=below left:\(X_3\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_0\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=i~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.5,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_1\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_1\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.2,1.75) [label=right:\(X_2\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_3\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_3\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{align}

Note that the defect labelled by the bimodule $X_2$  together with $X_0$ forms a $\mathbb{Z}_2$ subgroup of the defect fusion ring. It has the following $\mathrm{T}$-matrix values:

\begin{equation}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.1cm of x1] (x5) [label=below left:\(X_2\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_a\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_a\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=0.8cm of x5] (xx) [label=below left:\(X_2\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_a\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.5,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{a}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_a\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.05,1.7) [label=right:\(\hspace{-2mm}X_{0}\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_2\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_2\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}} \quad \forall a\in\{0,1\}
\end{equation}

The remaining values for the fusing matrix $\mathrm{T}$ are presented in (<ref>) and (<ref>).

\begin{align*}
&\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (-0.5,3.4) ;
\vertex[below right=0.1cm of x1] (x5) [label=below:\(X_x\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_0\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_0\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.5,1.9) ;
\vertex [below right=0.1cm of y] (y1) [label=below:\(X_x\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=0] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw  (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick] [middlearrow={latex}]   (j) to[out=90,in=0] (y);
\vertex[above=0.65 of j55] [label=right:\(M_{x}\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.5,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_0\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_0\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.25,1.6) [label=right:\(X_{[2x]}\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=0] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_x\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_x\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [middlearrow={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [middlearrow={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}} \quad \forall x\in\{0,1,2,3\}
\end{align*}

We now turn to the $\mathrm{\dot{T}}$-matrices. The general solution to the system of equations (<ref>) for the constants $C^{(a,j)b}_{(i)k}$ is given in (<ref>) . It is parameterised by the coefficients
$\tilde{C}^{(a,j)}$ which we fix by imposing

\begin{equation}\label{standardnormC}
\tilde{C}^{(a,j)} = 1, \quad \forall a\in\{0,1\},\forall j\in\mathbb{Z_4} \, .
\end{equation}

We are ready to use (<ref>) to find the $\mathrm{\dot{T}}$-matrices. The selection rules (<ref>) imply that there are 32 non-zero $\mathrm{\dot{T}}$-matrix elements. When  $x=0$ or $k=0$  the corresponding $\mathrm{\dot{T}}$-matrix element is equal to $1$
which leaves 18 non-zero $\mathrm{\dot{T}}$-matrix elements. Some of these coefficients are presented on the following diagrams which hold $\forall a\in\{0,1\},\forall j\in\{2,3\}$:

\begin{align}
& \vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_2\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_a\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_a\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(X_2\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_a\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= -
\vcenter{\hbox{\hspace{-2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_2\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_a\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_a\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(X_2\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{a}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}},\quad
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8) ;
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_j\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_0\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(X_1\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{j+1}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= -i
\vcenter{\hbox{\hspace{-1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_1\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_0\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(X_1\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{j+1}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{align}

The rest of the non-trivial $\dot{\mathrm{T}}$ matrix elements can be found in (<ref>).

[font=,inner sep=2pt]

[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
(j1) at (0,0);
[above=0.8 of j1] (j2);
[ above =0.9 of j1] (j3);
[small,dot] [ above=2.5cm of j1] (psi)  ;
[right=0.2cm of psi] (psi1) ;
[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_3\)];
[above=3.4cm of j1] (j4) ;
[above=3.5cm of j1] (j5)  ;
[ left=0.15cm of  j4] (j7);
[left=0.15cm of j3] (j8) ;
[above =4cm of j1] (j6);
(0.06,0.8) rectangle (-0.2, 0.9);
[below=0.1cm of j6] (m1) ;
[above=1.6cm of j1] (m3) [label=right:\(M_0\)];
[above=1.1cm of psi] (m2) [label=right:\(M_1\)];
[left=1.2cm of psi] (x) [label=left:\(X_x\)];
[above=0.3 of j1] (jj) [label=right:\(M_{x}\)];
[below=0.25cm of x] (x1);
[above=0.25cm of x] (x2);
[thick]    (j8) to[out=90,in=-90] (x1);
[thick] [middlearrow=latex]   (x1) to (x2);
(j1) –[thick] (j2),
(j3) –[thick] (psi),
(psi) –[thick] (j6)

= -
[font=,inner sep=2pt]

[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
(j1) at (0,0);
[above=2.4 of j1] (j2);
[ above =2.5 of j1] (j3);
[small,dot] [ above=0.9cm of j1] (psi)  ;
[right=0.2cm of psi] (psi1);
[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_3\)];
[above=3.4cm of j1] (j4) ;
[above=3.5cm of j1] (j5)  ;
[ left=0.15cm of  j4] (j7);
[left=0.15cm of j3] (j8);
[above =4cm of j1] (j6);
(0.06,2.4) rectangle (-0.2, 2.5);
[below=0.1cm of j6] (m1) ;
[above=1.6cm of j1] (m3) [label=right:\(M_{x+1}\)];
[above=1.3cm of j2] (m2) [label=right:\(M_1\)];
[above=0.5cm of j4] (xx);
[left=1.2cm of xx] (x);
[below=0.1cm of x] (x100)  [label=left:\(X_x\)];
[above=0.3 of j1] (jj) [label=right:\(M_{x}\)];
[below=0.25cm of x] (x1);
[above=0.15cm of x] (x2);
[left=0.25cm of x2] (x22);
[above=0.6cm of j6] (j7);
[right=0.25cm of j7] (j77);
[left=0.25cm of x2] (x22);
[above=0.05cm of x] (x3);
[left=0.25cm of x3] (x33);
[above=0.1cm of x33] (x333);
[above=0.4cm of x2] (x4);
[thick]    (j8) to[out=90,in=-90] (x1);
[thick] [middlearrow=latex]   (x1) to (x2);
(j1) –[thick] (j2),
(j3) –[thick] (j6)

As explained before the fusing matrices  $\mathrm{F}[A_2]$  coincide with $\mathrm{F}$:

\begin{equation}
\mathrm{F}[A_2]^{(a\,j\,k)}=(-1)^{a\sigma(j+k)}
\end{equation}

where we used the  normalisation (<ref>).

Finally, let us consider the compact open defects introduced in section <ref> and the corresponding associative fusion algebras.  The two smallest algebras  correspond to the two simple $A_2$-modules:

\begin{equation}
\mathcal{A}_{M_0}=\langle \omega_0^{(0)},\omega_2^{(0)}\rangle_{{\mathbb{C}}} \quad \text{and} \quad
\mathcal{A}_{M_1}=\langle \omega_0^{(1)},\omega_2^{(1)}\rangle_{\mathbb{C}}
\end{equation}

where we introduced the generators

\begin{equation} \label{tube_Gen1}
\omega_0^{(0)}=\left(\bar{\tilde{\Omega}}_{(0,0)},\Tilde{\Omega}_{(0,0)}\right),\;\omega_2^{(0)}=\left(\bar{\tilde{\Omega}}_{(2,0)},\Tilde{\Omega}_{(2,0)}\right),\;\omega^{(1)}_0=\left(\bar{\tilde{\Omega}}_{(0,1)},\Tilde{\Omega}_{(0,1)}\right),\;\omega_2^{(1)}=\left(\bar{\tilde{\Omega}}_{(2,1)},\Tilde{\Omega}_{(2,1)}\right)\,.
\end{equation}

Here for brevity we omit the third label of the $\bar{\Tilde{\Omega}}$ and $\Tilde{\Omega}$ since it is uniquely determined by the other two labels.
The product (<ref>) between the compact open defects takes the simple form

\begin{equation}\label{tubeprod1}
\left(\bar{\Tilde{\Omega}}_{(x,a)},\Tilde{\Omega}_{(x,b)}\right)\ast  \left(\bar{\Tilde{\Omega}}_{(y,c)},\Tilde{\Omega}_{(y,d)}\right)=\delta_{a,y+c\;\text{mod}\;2}\,\delta_{b,y+d\;\text{mod}\;2}\overline{\mathrm{T}}^{(x\,y\,c)}\mathrm{T}^{(x\,y\,d)} \left(\bar{\Tilde{\Omega}}_{([x+y],c)},\Tilde{\Omega}_{([x+y],d)}\right).
\end{equation}

Using this we can see that both of these algebras are isomorphic to the group algebra $\mathbb{C}[\mathbb{Z}_2]\cong {\mathbb C}\oplus {\mathbb C}$.

Let us now turn to the tube algebra $\mathcal{A}_{M^U}$. It is the complex algebra with a basis consisting of   all of the
sixteen elementary compact open defects. In addition to the generators given in (<ref>) we also have

\begin{align}\label{tube1case1}
& \omega_{1}^{(0)}= \left(\bar{\tilde{\Omega}}_{(1,1)},\Tilde{\Omega}_{(1,1)}\right),\; \omega_{3}^{(0)}=\left(\bar{\tilde{\Omega}}_{(3,1)},\Tilde{\Omega}_{(3,1)}\right) ,\;\omega_{1}^{(1)}=\left(\bar{\tilde{\Omega}}_{(1,0)},\Tilde{\Omega}_{(1,0)}\right),\;\omega_{3}^{(1)}=\left(\bar{\tilde{\Omega}}_{(3,0)},\Tilde{\Omega}_{(3,0)}\right), \nonumber \\
& \gamma^{(1)}_0=\left(\bar{\tilde{\Omega}}_{(0,0)},\Tilde{\Omega}_{(0,1)}\right) ,\;\gamma^{(1)}_2=\left(\bar{\tilde{\Omega}}_{(2,0)},\Tilde{\Omega}_{(2,1)}\right),\;\gamma^{(0)}_0=\left(\bar{\tilde{\Omega}}_{(0,1)},\Tilde{\Omega}_{(0,0)}\right),\;\gamma^{(0)}_2=\left(\bar{\tilde{\Omega}}_{(2,1)},\Tilde{\Omega}_{(2,0)}\right),\nonumber \\
& \gamma_{1}^{(1)}=\left(\bar{\tilde{\Omega}}_{(1,1)},\Tilde{\Omega}_{(1,0)}\right),\;\gamma_{3}^{(1)}=\left(\bar{\tilde{\Omega}}_{(3,1)},\Tilde{\Omega}_{(3,0)}\right),\; \gamma_{1}^{(0)}= \left(\bar{\tilde{\Omega}}_{(1,0)},\Tilde{\Omega}_{(1,1)}\right),\;\gamma_{3}^{(0)}= \left(\bar{\tilde{\Omega}}_{(3,0)},\Tilde{\Omega}_{(3,1)}\right).
\end{align}

The rationale behind the notation is as follows. All compact defects denoted by $\omega_{i}^{(j)}$ have the same boundary condition on the outside (of the attached arc) while those  denoted by $\gamma_{i}^{(j)}$ have different boundary conditions on the outside. The bottom index is the index of the defect while the top index gives the boundary condition which on the diagrams is below the defect arc. It is clear from the notational rule that
the algebra splits into two ideals
𝒜_M^U = ω⊕γwhere
ω= ⟨ω_0^(0), ω_0^(1), ω_2^(0), ω_2^(1),  ω_1^(0), ω_3^(0),  ω_1^(1), ω_3^(1)⟩_ℂ   ,
γ=⟨γ^(1)_0, γ^(1)_2, γ^(0)_0, γ^(0)_2,  γ_1^(1), γ_3^(1),
γ_1^(0), γ_3^(0) ⟩_ℂ   .
Each pair of boundary  fields:  $(\Psi_{(0,0)0}, \Psi_{(1,0)1})$, $  (\Psi_{(0,2)0}, \Psi_{(1,2)1})$ spans a two-dimensional representation of the
subalgebra $\omega$. The kernels of these representations are two-sided ideals which are easy to find using the $\dot {\mathrm T}$ matrices.
They give a decomposition $\omega = \omega_{+}\oplus \omega_{-} $ with
ω_+ = ⟨ω_0^(0) + ω_2^(0) , ω_0^(1)+ ω_2^(1),  ω_1^(0)+ ω_3^(0),  ω_1^(1)+  ω_3^(1)⟩_ℂ
ω_- = ⟨ω_0^(0) - ω_2^(0) , ω_0^(1)- ω_2^(1),  ω_1^(0)- ω_3^(0),  ω_1^(1)-  ω_3^(1)⟩_ℂ .
Each of the subalgebras $\omega_{+}$ and $\omega_{-}$ is isomorphic to the matrix algebra $\mathrm{Mat}_{2}({\mathbb C})$. Similarly the second ideal
$\gamma$ has two two-dimensional representations spanned by the pairs of boundary fields  $(\Psi_{(0,1)1}, \Psi_{(1,1)0})$,
$(\Psi_{(0,3)1}, \Psi_{(1,3)0})$. Using these representations we find a decomposition
$\gamma = \gamma_{+}\oplus\gamma_{-} $ with
γ_+ = ⟨γ^(1)_0+ γ^(1)_2, γ^(0)_0- γ^(0)_2,  γ_1^(1)- γ_3^(1),
γ_1^(0)+ γ_3^(0) ⟩_ℂ   ,
γ_- = ⟨γ^(1)_0- γ^(1)_2, γ^(0)_0+ γ^(0)_2,  γ_1^(1)+ γ_3^(1),
γ_1^(0)- γ_3^(0) ⟩_ℂ
where each of $\gamma_{\pm}$ is isomorphic to $\mathrm{Mat}_{2}({\mathbb C})$. The above gives an explicit decomposition of the tube algebra
into a direct sum of 4 matrix algebras
𝒜_M^U = ω_+⊕ω_-⊕γ_+ ⊕γ_- ≅Mat_2(ℂ)⊕Mat_2(ℂ)⊕Mat_2(ℂ)⊕Mat_2(ℂ)
as expected from semi-simplicity.

§.§ Fusing matrices of the  N=4, r=2 model

In this model, there are eight distinct simple objects of the category $\mathcal{U}_4$ denoted $\{U_i,\,i=0,\ldots 7\}$ and satisfying fusion rules corresponding to the addition in $\mathbb{Z}_8$. The algebra object with $r=2$ is found to be

\begin{equation}
\tilde{A}_2= U_0\oplus U_4.
\end{equation}

This algebra has four simple left modules $\{M_a,\,a=0,1,2, 3\}$ with $\dot{M}_a= \tilde{A}_2\otimes U_a$. The labelling group for the $ \tilde{A}_2$-$ \tilde{A}_2$-bimodules is

\begin{equation}
\tilde{G}^{(2,4)}=\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times \mathbb{Z}\right)\bigg/\langle(1,1,-2)\rangle\cong \mathbb{Z}_2\times \mathbb{Z}_4.
\end{equation}

We notice that again the number of simple topological defects is equal to the number of irreducible representations of the chiral algebra. However, in this model the fusion rules for the two aforementioned objects do not coincide, unlike in the $N=2=r$ case. We denote these bimodules as $X_x$ where the labels $x$ correspond to the labels $(a,b,\rho)$ according to[Here, the labels $x$ are introduced only for notational convenience and they do not satisfy the $\mathbb{Z}_8$ fusion rules.]

\begin{align}
&X_0\equiv B_{(0,0,0)}^{(2)},\quad X_1\equiv B_{(0,0,1)}^{(2)} ,\quad X_2\equiv B_{(0,1,0)}^{(2)} ,\quad X_3\equiv B_{(0,1,1)}^{(2)} \nonumber \\
& X_4\equiv B_{(1,0,0)}^{(2)},\quad X_5\equiv B_{(1,0,1)}^{(2)} ,\quad X_6\equiv B_{(1,1,0)}^{(2)} ,\quad X_7\equiv B_{(1,1,1)}^{(2)}.
\end{align}

The fusion of these bimodules is given by  (<ref>) which for this model coincides with the group operation in $\mathbb{Z}_2\times \mathbb{Z}_4$. The fusion of a bimodule  with a module $M_a$ takes the form

\begin{equation}
X_x\otimes_{\tilde{A}_2}M_a\cong M_{a+x\;\text{mod} \;4}\,.
\end{equation}

We can now proceed in the same way as in <ref> to obtain the complex numbers $A^{(x,y)}_{(i,j)},B^{(x,a)}_{(i,j)}$ and $C^{(a,j)}_{(i)}$ which we present in Appendix <ref>. Using  (<ref>) we obtain

\begin{equation}
\mathrm{Y}^{(x\,y\,z)}=1 \quad \text{if any} \; x,y,z \;\text{is} \; 0.
\end{equation}

The remaining values of the $\mathrm{Y}$ fusing matrices are given in (<ref>).

Furthermore, using (<ref>) we find the  values for the $\mathrm{T}$ fusing matrix. We present some of its entries graphically:

\begin{align}
&\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.45cm of x1] (x5) [label=below:\(\hspace{-2mm}B^{(2)}_{(a,b,\rho)}\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{2+2b+\rho}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_{3}\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=1cm of x5] (xx) [label=below left:\(B^{(2)}_{(0,1,1)}\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_{3+2b+\rho}\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=-i~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.7,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{2+2b+\rho}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_3\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.25,1.3) [label=right:\(\hspace{-8mm}B^{(2)}_{(a,b+1,\rho+1)}\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1.2,2.8) ;
\vertex (k2) at (-1.2,3.4)  [label=left:\(B^{(2)}_{(0,1,1)}\)];
\vertex (k3) at (-0.2,2.8) ;
\vertex (k4) at (-0.2,3.4) [label=right:\(B^{(2)}_{(a,b,\rho)}\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{align}

$\forall(a,b,\rho)\in \tilde{G}^{(2,4)} \mysetminus \{(0,0,0),(1,0,0)\}$. The defect labelled by the bimodule $X_4$ which together with $X_0$ form a $\mathbb{Z}_2$ subgroup of the fusion has the following $\mathrm{T}$-matrix values:

\begin{align}
&\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.45cm of x1] (x5) [label=below left:\(X_4\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_a\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_{a}\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=1cm of x5] (xx) [label=below left:\(X_4\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_a\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.5,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{a}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_a\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.25,1.7) [label=right:\(X_{0}\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_4\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_4\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}} \quad \forall a\in\{0,1,2,3\}
\end{align}

The remaining values are presented in (<ref>) and (<ref>).

\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (-0.5,3.4) ;
\vertex[below right=0.1cm of x1] (x5) [label=below:\(B^{(2)}_{(a,1,\rho)}\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{\rho}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_2\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  ;
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.5,1.9) ;
\vertex [below right=0.1cm of y] (y1) [label=below:\(B^{(2)}_{(1,0,0)}\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) ;
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=0] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw  (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick] [middlearrow={latex}]   (j) to[out=90,in=0] (y);
\vertex[above=0.65 of j55] [label=right:\(M_{\rho}\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~=-~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [empty dot] (x1) at (-0.7,2) {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_{\rho}\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_2\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] ;
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.25,1.6) [label=right:\(B^{(2)}_{(a,b+1,\rho+1)}\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=0] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1.2,2.8) ;
\vertex (k2) at (-1.2,3.4)  [label=left:\(B^{(2)}_{(0,1,1)}\)];
\vertex (k3) at (-0.2,2.8) ;
\vertex (k4) at (-0.2,3.4) [label=right:\(B^{(2)}_{(a,1,\rho)}\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7);
\draw [middlearrow={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [middlearrow={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}} \qquad \forall a,\rho\in\{0,1\}
\end{equation*}

Next, we use (<ref>) to find the  $\dot{\mathrm{T}}$ fusing matrix. According to the selection rules, there are $256$ non-zero  $\dot{\mathrm{T}}$ matrix elements. Those with $x=0$ or $k=0$ are equal to $1$.  Among the pairs of open defects and boundary operators we find an anticommuting pair:
$B^{(2)}_{(1,0,0)}$, $\Psi_{(4,a)a}$ inserted on the same conformal boundary condition:

\begin{align}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_4\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_a\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_a\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(B^{(2)}_{(1,0,0)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_a\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= -
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_4\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_a\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_a\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(B^{(2)}_{(1,0,0)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{a}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}} \quad \forall a\in\{0,1,2,3\}
\end{align}

Some of the other  values of $\dot{\mathrm{T}}$ are given below

\begin{align}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_{j+1}\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_1\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(B^{(2)}_{(0,0,1)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{j+2}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= -i
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_2\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_1\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(B^{(2)}_{(0,0,1)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{j+2}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}} \quad \forall j\in\{3,4,5,6\} \nonumber \\[1.5ex]
& \vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_{3+j}\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_3\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(B^{(2)}_{(1,0,1)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_j\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= -
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_{0}\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_3\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(B^{(2)}_{(1,0,1)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_{j}\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}\; \quad \forall j\in\{5,6,7\}
\end{align}

while the rest are given in (<ref>). Here we have used the normalisation given by (<ref>) and (<ref>) for all the possible pairs of labels.

The fusing matrix $\mathrm{F}[ \tilde{A}_2]$ is obtained from (<ref>):

\begin{equation}
\mathrm{F}[ \tilde{A}_2]^{(a\,j\,k)}=(-1)^{a\sigma(j+k)}\, .
\end{equation}

Finally, the tube algebra $\mathcal{A}_{M^U}$ with $M^U=M_0\oplus M_1\oplus M_2\oplus M_3$   is found to be a $128$-dimensional associative complex algebra that splits into a direct sum of 8 full matrix algebras $\mathrm{Mat}_{4}({\mathbb C})$  that correspond to the 8 boundary representations spanned by
the boundary fields with a fixed object $U_{j}$, $j=0,\dots , 7$.

§.§ Boundary RG flows

In this section we discuss how our calculations can be used to impose constraints on boundary RG flows in the free boson theory.
All conformal boundary conditions for the compact free boson theory  have been classified [45], [46], [47].
For a generic radius the only conformal boundary conditions with finite boundary entropy are  the Dirichlet and Neumann ones while for
the radius being a rational number times the self-dual radius there are exceptional boundary conditions which preserve only Virasoro algebra.
The Dirichlet boundary conditions can be pictured as D0 branes located at points on the circle while the Neumann ones correspond to D1 branes with Wilson lines (or alternatively as D0 branes localised on the T-dual circle).
In the rational case the exceptional boundary conditions appear for the theories with $N$ being a square.  Since they
break the extended chiral algebra we do not see them in the TFT formalism.
We first discuss the simplest case when the conformal boundary condition in the UV is a single irreducible boundary condition $M_{a}$ in a rational theory.
As we already discussed in section <ref> the non-trivial relevant boundary operators on $M_{a}$ are $\Psi_{(a,j)a}$ with
j= 0 mod 2N/r

j<2√(N)  j>2(N-√(N))   .
We are interested in RG flows triggered by perturbing $M_{a}$ by a boundary action perturbation of the form

ΔS =   ∫ dt ( λΨ_(a,j)a(t) +   λ^∗Ψ_(a,j)a^†(t) )

where $\lambda$ is a complex coupling constant with the asterisk standing for complex conjugation and $\Psi_{(a,j)a}^{\dagger}$ is the Hermitian conjugate field that is proportional to $\Psi_{(a,2N-j)a}$.

Since we preserve unitarity the strongest constraint on such flows comes from the $g$-theorem [48], [49], [50].
The $g$-factor $g_{N}$ for a Neumann boundary condition and $g_{D}$ for a Dirichlet one are
g_N = √(R)   ,     g_D = 1/√(2R)
where $R$ is the radius of the free boson normalised so that the self-dual radius is $R_{\rm s.d.} =\frac{1}{\sqrt{2}}$.
We observe that when $R\le \sqrt{2}$ we have $2 g_{D} \ge g_{N}$.
Since in the UV our flows start from a single Neumann boundary condition (as all the boundary conditions preserving the extended chiral algebra are Neumann) we see that when $r\le 2\sqrt{N}$ the only possible end point of the flow is a single Dirichlet boundary condition.
Each Dirichlet boundary condition has the same spectrum so if we just observe the flow of the spectrum putting the same perturbed boundary
condition on both ends of an interval we cannot distinguish the different Dirichlet boundary conditions at the end point. If, however, we put
different perturbed boundary conditions on both ends – those corresponding to different couplings $\lambda_{1}$ and $\lambda_{2}$,
we can discern from the resulting spectrum the relative positions of the D0 branes. Information about these relative positions can be
obtained from topological defects.

It follows from the fusion rule (<ref>) that the only rational topological defects which can end topologically on $M_{b}$ are
The fusion  of these defects generates a subring isomorphic to the group ring of  ${\mathbb Z}_{r}$.
In the language of [15] each boundary condition $M_{b}$ is strongly symmetric with respect to the action of this subring.
By looking at the action of the defects $B^{(r)}_{(a,0,0)}$ on the $U(1)$ bulk primaries we can identify them with rotations of the circle
by the angle $\frac{2\pi a}{r}$ that act on the D0 branes by shifting their positions by the same angle.
For any boundary field $ \Psi_{(b,j)b}$ passing a junction with $B^{(r)}_{(a,0,0)}$ through it results in multiplying the field by a

\begin{equation}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_b\)];
\vertex[below=0.4cm of j6] (m2) [label=right:\(M_b\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(B^{(r)}_{(a,0,0)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_b\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=1.45cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
= \xi_{a,b,j}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.25cm of psi1] (psi2) [label=\(\,\;\Psi_j\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8);
\vertex [above =4cm of j1] (j6);
\draw  (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_b\)];
\vertex[above=1.3cm of j2] (m2) [label=right:\(M_b\)];
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(B^{(r)}_{(a,0,0)}\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_b\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

which we can call a charge. The mixed pentagon equation (<ref>) implies that these charges are all $r$-th roots of unity and that
$\xi_{a,b,j}=(\xi_{1,b,j})^{a}$ so that it suffices to
know the charges  $\xi_{1,b,j}$. Our calculations  give the values
ξ_1,b,j=e^-πi j/N   .

We further observe that a general relation
ξ_1,b,2N-j = ξ_1,b,j^*
takes place. It is easy to derive this from the consistency condition between the
$\dot{\mathrm{T}}$-matrix and the boundary OPE discussed in  section <ref>. This means that passing the  defect
junction of $M_{b}$ with $B^{(r)}_{(a,0,0)}$ through the perturbation (<ref>) results in rotating the complex coupling

λ→λ(ξ_1,b,j)^a   .
General considerations of [29] imply that there exists a topological junction with $B^{(r)}_{(a,0,0)}$ between the endpoints
of the boundary RG flows for which the couplings differ by the rotation (<ref>). In the situation when the end points are D0 branes
they must be spaced accordingly on the circle. When $g$-theorem allows superpositions of two or more D0 branes as the end-points
they must be again related by the corresponding rotations.

A more interesting situation arises when the starting point of the flow is described by a superposition of irreducible  modules $M_{a}$.
We are going to look at a single example that arises for the $r=N=2$ rational free boson theory we discussed in section <ref>.
Let us consider the direct sum $M=M_{0}\oplus M_{1}$. The relevant boundary fields are $\Psi_{(0,j)1}$, $\Psi_{(1,j)0}$ for $j=1,3$ and
$\Psi_{(0,2)0}$, $\Psi_{(1,2)1}$ so that there is a two-dimensional space of fields for each object $U_{j}$, $j=0,1,2,3$.
We introduce coordinates in the corresponding subspaces as

[  α

β] = αΨ_(0,j)1 + βΨ_(1,j)0   , j=1, 3   ,

[  α

β] = αΨ_(0,2)0 + βΨ_(1,2)1    .
The elementary topological defects are $X_{j}$, $j=0,1,2,3$.  Their  fusion ring is isomorphic to ${\mathbb Z}_{4}$. Each  defect can be
attached to $M$. Moreover the space of junctions for each $X_{j}$ is two-dimensional. We introduce coordinates
in each such subspace for fixed $j$ according to

Ω_j^[γ, δ] ≡γ  Ω_(j,1)  j+1  mod  2 + δ   Ω_(j,0)  j  mod  2   ,     γ, δ∈ℂ   .
Using our results from section <ref> it is straightforward to calculate the matrices that give the action of junctions in the two-dimensional subspaces
of boundary fields with fixed $i$. We denote the matrices that give the passing of  $\Omega_{j}^{[\gamma, \delta]}$ through the subspace corresponding to
$U_{i}$  as $\hat X_{j}^{(i)}$. Explicitly we obtain

X̂_1^(1) = [0   -iγ/δ

δ/γ    0]   ,   X̂_2^(1) =
[γ/δ   0

0   -δ/γ]   ,   X̂_3^(1) =  [0   iγ/δ

δ/γ    0]    ,

X̂_1^(2) = [0   -i

-i    0]   ,   X̂_2^(2) =
[-1   0

0   -1]   ,   X̂_3^(2) =  [0   i

i    0]    ,

X̂_1^(3) = [0   -γ/δ

-iδ/γ    0]   ,   X̂_2^(3) =
[-γ/δ   0

0   δ/γ]   ,   X̂_3^(3) =  [0   -γ/δ

iδ/γ    0]
where we assume that $\gamma \ne 0$, $\delta \ne 0$.

The fields with $j=1,2,3$ are all relevant. We can draw consequences from the topological junctions for the RG flows provided we construct junctions
that preserve the subspace of Hermitian boundary fields in the vector space spanned by the relevant fields.
To describe this subspace we need to know the boundary two-point functions. The complete two-point functions contain the OPE coefficient at the identity field and a factor from the two-point conformal blocks. The two point functions on a circle with insertions at the opposite points are
⟨Ψ_(0,1)1(1)  Ψ_(1,3)0(-1) ⟩= C_1,3^(010)0 B_13 ⟨1 ⟩_0 = C_3,1^(101)0B_31 ⟨1 ⟩_1   ,

⟨Ψ_(0,3)1(1)  Ψ_(1,1)0(-1) ⟩= C_3,1^(010)0 B_31 ⟨1 ⟩_0 = C_1,3^(101)0B_13 ⟨1 ⟩_1     ,

⟨Ψ_(1,2)1 (1) Ψ_(1,2)1(-1) ⟩= C_2,2^(111)0 B_22 ⟨1⟩_1   ,

⟨Ψ_(0,2)0(1)  Ψ_(0,2)0(-1) ⟩= C_2,2^(000)0 B_22  ⟨1⟩_0
where $C_{i,j}^{(abc)k}$ are the boundary OPE coefficients (<ref>), $\langle 1 \rangle_{a}$ are the disk partition functions for the boundary labelled by $a$, and
$B_{ij}$ are the normalisation constants for two-point conformal blocks. The latter  are defined as follows. Let
V_ij^k : ℋ_j →ℋ_k
be the vertex operators corresponding to the primaries of the extended chiral algebra. The two-point conformal block can be written as
⟨0| V_k̅ k^0(z_2) V_k0^k(z_2) |0⟩= (z_2-z_1)^-2Δ_k B_k̅ k   .
Note that in (<ref>), (<ref>) there are two expressions for each two point function. Those equalities should  hold due to the continuity of the two-point functions [51], [52] .
Using the braiding matrix (<ref>) and the conformal dimension $\Delta_{1}=\Delta_{3}=\frac{1}{8} $ we obtain the relation $B_{13}=-B_{31}$.
Furthermore as noted in (<ref>)  the values of the boundary OPE coefficients are given by the $\mathrm{F}$-matrix elements. In particular we find
C_1,3^(010)0=-C_3,1^(101)0= 1   ,   C_3,1^(010)0 = -  C_1,3^(101)0 = 1   ,   C_2,2^(000)0 = - C_2,2^(111)0= 1   .
Without loss of generality we assume that our fields are normalised so that $B_{13}=1$ and $B_{22}=1$.  The Hermitian conjugation is then given by
Ψ_(0,1)1^† = Ψ_(1,3)0   ,     Ψ_(1,1)0^† = -Ψ_(0,3)1   ,

Ψ_(0,2)0^† = Ψ_(0,2)0   ,     Ψ_(1,2)1^† = -  Ψ_(1,2)1   .
Next we introduce a basis of Hermitian operators

\begin{align}
v_{1}&= \Psi_{(0,1)1} + \Psi_{(1,3)0} \, ,   &v_{2}&=  \Psi_{(1,1)0} - \Psi_{(0,3)1} \, , \nonumber \\
v_{3}&= i( \Psi_{(0,1)1} - \Psi_{(1,3)0})  \, ,   &v_{4}& = i( \Psi_{(1,1)0} + \Psi_{(0,3)1}) \, , \nonumber \\
v_{5}&=  \Psi_{(0,2)0}  \, ,  & v_6 &=  i \Psi_{(1,2)1} \, .
\end{align}

There are two  real subspaces
W^(1,3) = ⟨v_1, v_2, v_3, v_4 ⟩_ℝ  ,     W^(2)=⟨v_5, v_6  ⟩_ℝ  .
Choosing $\gamma = i\delta$ in (<ref>) for each junction[In principle one can choose $\gamma$ and $\delta$ separately for each $j$. Moreover the choice we give is not the only solution. Here however we are not after an exhaustive classification but after a proof of concept example.] with $X_{j}$ we find from (<ref>),  (<ref>),  (<ref>)  that on the subspace $W^{(1,3)}$ the corresponding junctions are represented by the matrices

X̂_1^(1,3) = [
0   1   0   0

0   0   1   0

0   0   0   1

-1   0   0   0
]   ,   X̂_2^(1,3)= [
0   0   -1   0

0   0   0   -1

1   0   0   0

0   1   0   0
]   ,   X̂_3^(1,3)= [
0   -1   0   0

0   0   1   0

0   0   0   -1

-1   0   0   0
]   .
We see that the matrix elements are all real and hence the chosen junctions preserve the real subspace.
Consider now a boundary perturbation given by
ΔS = ∑_i=1^4 λ^i ∫ dt   v_i(t)
where $\lambda_{i}$ are four real coupling constants which form a vector space with a fixed basis.
The junctions we consider  act on this vector space by the  transpose of the matrices given in  (<ref>). Each orbit of this action is finite. The neighbouring points on the orbit give
two deformations for which the IR fixed point boundary conditions must admit a topological junction with the corresponding defect $X_{j}$.

Similarly, in the second real subspace $W^{(2)}$ each junction (<ref>) with any $\gamma \ne 0$, $\delta \ne 0$ acts by
the matrices

X̃_1^(2) = [
0   1

-1   0
]   ,     X̃_2^(2)
= [
-1   0

0   -1
]   ,     X̃_3^(2) = [
0   -1

1   0
]   .
These matrices are real so that the subspace  $W_2$ is preserved by these junctions. Considering the deformations
ΔS =  λ^5 ∫ dt   v_5(t) +  λ^6 ∫ dt   v_6(t)  ,       λ^5 ∈ℝ   , λ^6 ∈ℝ
the junctions act on the couplings vector $(\lambda^{5}, \lambda^{6} )$ by the transpose of the matrices in (<ref>).
The corresponding orbits have only two elements so that
the IR fixed points at the ends of the corresponding flows must have topological junctions with the corresponding defects.

Notice that there are no real eigenvectors for the above matrices in $W^{(1,3)}$ and $W^{(2)}$ so that we do not obtain any constraints on individual flows but rather a collection of  relative constraints between them. It would be interesting to explore systematically what all of these constraints say about the possible set of IR end points.

§ FUTURE DIRECTIONS

We conclude by mentioning some open problems.
In this paper  have worked out the details of the linear algebra problems that allow to compute the fusing matrices $\mathrm{Y}$, $\mathrm{T}$,
$\dot{\mathrm{T}}$ from the RCFT representation theoretic data related to a pair  $({\mathcal C}, A)$. For concrete models like the free boson or the WZW models with non-trivial modular invariants it would be desirable to obtain general analytic expressions for these quantities. We hope this can be done in the future. Although a different approach is needed for this kind of task our method can be always used to check the results.

In section <ref> we outlined possible applications of  the fusing matrices we discussed to boundary RG flows. One can approach these applications more systematically and try to classify all pairs of junctions with boundary conditions and their Hermitian boundary field eigenvectors or subspaces. We hope to address such classification tasks for direct sums of boundary conditions in concrete models in future work. It would be  also interesting to explore
how restrictive these constraints are in concrete examples of multi-coupling families of flows.

Another interesting direction would be to understand better  the associative fusion algebras of compact open defects we discussed in section <ref>.
Their applications to boundary and bulk RG flows should be further explored.

We are grateful to Ingo Runkel for stimulating discussions, in particular for explaining the
construction of associators relevant to the fusing matrices we consider in this paper.

§ ASSOCIATORS

The structure of a modular tensor category [53, 54, 55, 37] naturally arises from conformal field theory as the representation category of certain rational conformal vertex algebras [56, 57, 61, 62, 58] which are the mathematical structure encoding physical information of a 2-dimensional chiral RCFT such as its chiral algebra and primary fields. More precisely, the symmetries of a chiral RCFT include conformal symmetries that can be encoded in an conformal vertex algebra $\mathfrak{V}_{\text{Vir}}$ which furnishes in particular a representation of the Virasoro Lie algebra. But, most models have additional symmetries where the  chiral algebra is a larger conformal vertex algebra $\mathfrak{V}\supseteq \mathfrak{V}_{\text{Vir}}$ [59]. The spaces of fields of the chiral RCFT are then given by  $\mathfrak{V}$-modules and the category of interest is the representation category $\mathcal{C}:=\operatorname{Rep} (\mathfrak{V})$.

For reference   a tensor category is defined as follows

A tensor category $\mathcal{C}=(\operatorname{Obj}(\mathcal{C}),\otimes,\mathbf{1},\alpha,\lambda,\rho)$ is a category $\mathcal{C}$, a bifunctor $\otimes:\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, an object $\mathbf{1}\in\operatorname{Obj}(\mathcal{C})$ called the tensor unit, and three natural isomorphisms $\alpha,\lambda,\rho$. Explicitly, the natural isomorphism

\begin{equation}\label{aa1}
\alpha=\alpha_{U,V,W}:U\otimes(V\otimes W)\to(U\otimes V)\otimes W
\end{equation}

called the associator is such that the following pentagon diagram commutes

\begin{equation}\label{pentag1}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=\(U\otimes(V\otimes(W\otimes Z))\)];
\vertex [right=4.5cm of j1] (j2) [label=\((U\otimes V)\otimes (W\otimes Z)\)];
\vertex[right=4.5cm of j2] (j3) [label=\( ((U\otimes V)\otimes W)\otimes Z\)];
\draw [-stealth] (1.7,0.25)--(2.8,0.25);
\draw [-stealth] (6.2,0.25)--(7.3,0.25);
\vertex at (2.2,0.3) [label=\footnotesize\(\alpha\)];
\vertex at (6.7,0.3) [label=\footnotesize\(\alpha\)];
\vertex[below=1.5cm of j1] (j4) [label=\(U\otimes((V\otimes W)\otimes Z)\)];
\vertex [below =1.5cm of j3] (j5) [label=\((U\otimes(V\otimes W))\otimes Z\)];
\draw [-stealth] (1.7,-1.25)--(7.3,-1.25);
\draw [-stealth] (0.25,-0.1)--(0.25,-0.8);
\draw [-stealth] (8.8,-0.1)--(8.8,-0.8);
\vertex at (0.85,-0.65) [label=\footnotesize\(\operatorname{id}_U\!\otimes\alpha\)];
\vertex at (9.4,-0.65) [label=\footnotesize\(\alpha\otimes \operatorname{id}_Z\)];
\vertex at (4,-1.2) [label=\footnotesize\(\alpha\)];
\diagram*{

\end{feynman}
\end{tikzpicture}}}
\end{equation}

Furthermore, the natural isomorphisms

\begin{equation}
\lambda_U:\mathbf{1}\otimes U\to U,\qquad \rho_U:U\otimes \mathbf{1}\to U
\end{equation}

called the left and right unitor respectively, are such that the following triangle diagram commutes

\begin{equation}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=\(U\otimes(\mathbf{1}\otimes W)\)];
\vertex [right=5cm of j1] (j2) [label=\((U\otimes\mathbf{1})\otimes W\)];
\vertex[below right=3.5cm of j1] (j3) [label=\(U\otimes W\)];
\draw [-stealth] (1.2,0.25)--node[above]{\footnotesize$\alpha$}(3.9,0.25);
\draw [-stealth] (0,-0.2)--node[below left]{\footnotesize$\operatorname{id}_U\otimes\lambda_W$}(2.2,-1.9);
\draw [-stealth] (5,-0.2)--node[below right]{\footnotesize$\rho_U\otimes\operatorname{id}_W$}(2.7,-1.9);
\diagram*{

\end{feynman}
\end{tikzpicture}}}
\end{equation}

Based on coherence theorems <cit.>, we will assume without loss of generality that our tensor category is strict which means that all the isomorphisms $\alpha,\lambda,\rho$ are identities.

Let us now go through the various additional properties that our tensor category of interest $\mathcal{C}$  has that qualifies it as a modular tensor category.

* It is abelian, $\mathbb{C}$-linear, and semisimple.  An object $U$ of our abelian tensor category is called simple if $\operatorname{End}(U)=\mathbb{C}\operatorname{id}_U$. Then, a semisimple category has the property that every object is the direct sum of finitely many simple objects.

* It is a ribbon category. This equips $\mathcal{C}$ with a notion of (right) duality which associates to every $U\in\operatorname{Obj}(\mathcal{C})$ another object $U^\vee\in\operatorname{Obj}(\mathcal{C})$ and so-called evaluation and co-evaluation morphisms

\begin{equation}
d_U\in \operatorname{Hom}(U^\vee\otimes U,\mathbf{1}),\hspace{15mm} b_U\in \operatorname{Hom}(\mathbf{1},U\otimes U^\vee)
\end{equation}

Moreover, this ribbon category is equipped with braiding which consists of a family of isomorphisms $c_{U,V}\in\operatorname{Hom}(U\otimes V,V\otimes U)$, one for each pair $U,V\in\operatorname{Obj}(\mathcal{C})$. Finally, the last ingredient is the twist, a family of isomorphisms $\theta_U\in\operatorname{End}(U)$.

* It is a sovereign and spherical category. In a ribbon category there is automatically also a left duality which is constructed from the right duality, the braiding and the twist. It is defined on objects by $^\vee U:= U^\vee$ while the left duality morphisms are defined as

\begin{align}
\Tilde{b}_U&=\left(\operatorname{id}_{^\vee U}\otimes \theta_U\right)\circ c_{U,^\vee U}\circ b_U \in\operatorname{Hom}(\mathbf{1},^\vee U\otimes U) \nonumber \\ \Tilde{d}_U&=d_U\circ c_{^\vee U,U} \circ \left(\theta_U\otimes \operatorname{id}_{^\vee U}\right)\in \operatorname{Hom}(U\otimes {^\vee U},\mathbf{1})
\end{align}

One can check that the left duality coincides with the right duality not only on objects, but also on morphisms; categories with this property are called sovereign. We can now define left and right traces of endomorphisms $f\in\operatorname{Hom}(U,U)$ as

\begin{equation}
\operatorname{tr}_L(f)=d_U\circ\left(\operatorname{id}_{U^\vee}\otimes f\right)\circ \Tilde{b}_U,\qquad  \operatorname{tr}_R(f)=\Tilde{d}_U\circ\left(f\otimes \operatorname{id}_{U^\vee}\right)\circ b_U
\end{equation}

In our case, the two traces coincide, thus we have a so-called spherical category. The quantum dimension of an object in $\mathcal{C}$ is defined as

\begin{equation}
\operatorname{dim}(U)=\operatorname{tr}(\operatorname{id}_U)
\end{equation}

* It is a modular tensor category. That is, the number of isomorphism classes of simple objects is finite and the braiding is maximally non-degenerate: the numerical matrix $s$ with entries

\begin{equation}
s_{i, j}:=\operatorname{tr}\left(c_{U_i, U_j}\circ c_{U_j,U_i}\right)
\end{equation}

is invertible. Here we denote by $\{U_i\,|\,i\in\mathcal{I}\}$ a finite set of representatives of isomorphism classes of simple objects. We will usually take $U_0:=\mathbf{1}$ as the representative for the class of the tensor unit. The tensor product of objects induces the structure of a commutative and associative ring over $\mathbb{Z}$ on the set of isomorphism classes, called the Grothendieck ring $K_0(\mathcal{C})$ of $\mathcal{C}$. A distinguished basis of this ring is given by the isomorphism classes of the simple objects $U_i$ with $i\in\mathcal{I}$. In this basis the structure constants are the non-negative integers $\operatorname{dim}\operatorname{Hom}(U_i\otimes U_j,U_k)$.

Suppose that we have fixed bases in the hom-spaces of $\mathcal{C}$ as in (<ref>). A crucial piece of information involving the braiding morphisms $c_{U_i,U_j}$ of $\mathcal{C}$ is encoded in the braiding matrices which are defined by

\begin{equation}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [right=1cm of j1] (i1) [label=below:\(U_j\)];
\vertex [above=2cm of j1] (j2);
\vertex [above=0.7cm of j1] (j4);
\vertex [above=1.5cm of j1] (j5);
\vertex [above=1.5cm of i1] (i5);
\vertex[above=2cm of i1] (i2) ;
\vertex[above=0.7 cm of i1] (i4);
\vertex[above left=0.5cm of i4] (i6);
\vertex [above left=0.2 cm of i6] (i7);
\draw [thick,rounded corners=1mm] (j4)--(i5)--(i2);
\draw [thick,rounded corners=1mm] (i1)--(i4) -- (i6);
\draw [thick,rounded corners=1mm] (i7)--(j5) -- (j2);
\vertex [right=0.5cm of j2] (k1);
\vertex [small,orange, dot] [above=0.5cm  of k1] (k2) [label=right:\( \alpha\)] {};
\draw [thick,rounded corners=1mm] (j2)--(k2);
\draw [thick,rounded corners=1mm] (i2)--(k2);
\vertex[above=0.6cm of k2] (k3) [label=above:\(U_k\)];
\diagram*{
(j1)--[thick] (j4),
(k3)--[thick] (k2)
\end{feynman}
\end{tikzpicture}}}
~=\;\mathlarger{\sum}_\beta \mathrm{R}^{(i\,j)k}_{\alpha \beta}~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_i\)];
\vertex [right=1cm of i1] (j1) [label=below:\(U_j\)];
\vertex [above=0.5cm of i1] (i2);
\vertex [above =0.5cm of j1] (j2);
\vertex  [right=0.5cm of i1] (k1) ;
\vertex [small,orange, dot][above =1.5cm of k1] (k2)[label=right:\(\beta\)] {};
\vertex [above=1.5cm of k2] (k3) [label=above:\(U_k\)];
\draw [thick,rounded corners=1mm] (i1)--(i2)--(k2);
\draw [thick,rounded corners=1mm] (j1)--(j2)--(k2);
\diagram*{
(k2)--[thick] (k3)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

Let us now provide more details about the structure of a Frobenius algebra in a tensor category.

* A Frobenius algebra  in a tensor category $\mathcal{C}$ is a quintuple $(A,m,\eta,\Delta,\varepsilon)$ such that $(A,m,\eta)$ is an algebra, $(A,\Delta,\varepsilon)$ is a coalgebra and the Frobenius property is satisfied:

\begin{equation}
\left(\mathrm{id}_A \otimes m\right) \circ\left(\Delta \otimes \mathrm{id}_A\right)=\Delta \circ m=\left(m \otimes \mathrm{id}_A\right) \circ\left(\mathrm{id}_A \otimes \Delta\right)
\end{equation}

* A special algebra in a tensor category $\mathcal{C}$ is an object that is both an algebra and a coalgebra such that

\begin{equation}
\varepsilon\circ \eta=\beta_1 \operatorname{id}_{\mathbf{1}} \qquad \text{and} \qquad m\circ \Delta =\beta_A\operatorname{id}_A
\end{equation}

for non-zero complex numbers $\beta_1,\beta_A$. In the following it will be assumed without loss of generality that the coproduct is normalised such that $\beta_A=1$.

* A symmetric algebra in a sovereign tensor category $\mathcal{C}$ is an algebra object $(A,m,\eta)$ together with a morphism $\varepsilon\in\Hom(A,\mathbf{1})$ such that the two morphisms $\Phi_1,\Phi_2\in\Hom(A,A^\vee)$ defined as

\begin{equation}
\Phi_1:=\left[(\varepsilon \circ m) \otimes \operatorname{id}_{A^{\vee}}\right] \circ\left(\operatorname{id}_A \otimes b_A\right) \quad \text { and } \quad \Phi_2:=\left[\operatorname{id}_{A^\vee} \otimes(\varepsilon \circ m)\right] \circ\left(\tilde{b}_A \otimes \operatorname{id}_A\right)
\end{equation}

are equal.

* A haploid  algebra in a tensor category is an algebra object for which $\operatorname{dim} \Hom(\mathbf{1},A)=1$.

The main result that connects this structure with conformal field theory is theorem 3.6 in [7].

A module category is defined as

Let $\mathcal{C}$ be a tensor category. A right module category over $\mathcal{C}$ is a category $\mathcal{M}$ equipped with a module product bifunctor $\otimes: \mathcal{M}\times\mathcal{C} \to M$ and a natural isomorphism with components

\begin{equation}
m_{M,U,V}: M\otimes (U\otimes V)\to (M\otimes U)\otimes V, \qquad U,V\in\operatorname{Obj}(\mathcal{C}),\, M\in\operatorname{Obj}(\mathcal{M})
\end{equation}

called module associativity constraint such that the functor $M\mapsto M \otimes \mathbf{1}$ is an autoequivalence and the following module pentagon diagram commutes

\begin{equation}\label{modpent1}
\vcenter{\hbox{\hspace{-3mm}\begin{tikzpicture}[inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=\(M\otimes(U\otimes(V\otimes W))\)];
\vertex [right=6cm of j1] (j2) [label=\((M\otimes U)\otimes (V\otimes W)\)];
\vertex[right=6cm of j2] (j3) [label=\( ((M\otimes U)\otimes V)\otimes W\)];
\draw [-stealth] (1.75,0.25)--(4.3,0.25);
\draw [-stealth] (7.8,0.25)--(10.3,0.25);
\vertex at (2.9,0.3) [label=\footnotesize\(m_{M,U,V\otimes W}\)];
\vertex at (8.9,0.3) [label=\footnotesize\(m_{M\otimes U,V,W}\)];
\vertex[below=2cm of j1] (j4) [label=\(M\otimes((U\otimes V)\otimes W)\)];
\vertex [below =2cm of j3] (j5) [label=\((M\otimes(U\otimes V))\otimes W\)];
\draw [-stealth] (1.75,-1.7)--(10.3,-1.7);
\draw [-stealth] (0.25,-0.1)--(0.25,-1.4);
\draw [-stealth] (11.6,-0.1)--(11.6,-1.4);
\vertex at (1.4,-0.9) [label=\footnotesize\(\operatorname{id}_M\!\otimes \alpha_{U,V,W}\)];
\vertex at (10.2,-0.9) [label=\footnotesize\(m_{M,U,V}\otimes \operatorname{id}_W\)];
\vertex at (5.2,-1.6) [label=\footnotesize\(m_{M,U\otimes V,W}\)];
\diagram*{

\end{feynman}
\end{tikzpicture}}}
\end{equation}

where $\alpha$ is the associator of $\mathcal{C}$ introduced in (<ref>). There is also a similar definition for a left module category.

Consider a left $A$-module $(\dot{M},\rho^M)$. From the properties of $\mathcal{C}$ we obtain the following completeness relation involving the morphisms introduced in (<ref>)

\begin{equation}
\sum_{i\in\mathcal{I}}\sum_\alpha b^M_{i,\alpha}\circ b_M^{i,\alpha}=\operatorname{id}_{\dot{M}}
\end{equation}

§.§ Associator of TEXT

Here we prove the pentagon identity for the associator (<ref>) which can be written as

\begin{equation}\label{a1}   \check{\alpha}_{X\otimes_AY,Z,W}\circ\check{\alpha}_{X,Y,Z\otimes_AW}=\left(\check{\alpha}_{X,Y,Z}\otimes_A\operatorname{id}_W\right)\circ \check{\alpha}_{X,Y\otimes_AZ,W}\circ\left(\id_X\otimes_A\check{\alpha}_{Y,Z,W}\right)
\end{equation}

We will proceed by calculating both sides of the last equation and showing that they are identical. Using (<ref>) we find the left-hand side of (<ref>) to have the following graphical form

\begin{equation}\label{lhs1}
\vcenter{\hbox{\hspace{2mm}\scalebox{0.8}{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (x1) at (0,0) [label=below:\(X\)];
\vertex [right=0.7cm of x1] (y1) [label=below:\(Y\)];
\vertex[right=0.7cm of y1] (z1) [label=below:\(Z\)];
\vertex[right=0.7cm of z1] (w1) [label=below:\(W\)];
\vertex[above=0.5cm of x1] (x2) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)] ;
\vertex[above=1cm of w1] (w2);
\vertex[above=0.5cm of y1] (y2);
\vertex[above=0.5cm of z1] (z2);
\vertex[above=0.5cm of w1] (w22);
\vertex[left=0.25cm of x2)] (x3);
\vertex [right=0.25cm of w2] (w3);
\draw [fill=yellow] (x3) rectangle (w3);
\vertex[above=0.5cm of x1] (x2) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)] ;
\vertex[above=2cm of x2] (x4);
\vertex[above=1cm of y2] (y3);
\vertex[left=0.25cm of y3] (y4);
\vertex[above=1cm of z2] (z3);
\vertex[above=1cm of w22] (w4);
\vertex[above=0.5cm of w4] (w5);
\vertex[right=0.25cm of w5] (w6);
\draw [fill=yellow] (y4) rectangle (w6);
\vertex[left=0.25cm of y3] (y4) [label=above right:\(e_{Y,Z\otimes_AW}\)];
\vertex[above=0.5cm of x2] (x22);
\vertex[above=0.5cm of y2] (y22);
\vertex[above=0.5cm of z2] (z22);
\vertex[above=1.5cm of y22] (y5);
\vertex[above=2.5cm of z22] (z5);
\vertex[above=3cm of w22] (w7);
\vertex[above=0.5cm of y5] (y55);
\vertex[right=0.25cm of y55] (y6);
\vertex[left=0.25cm of x4] (x5);
\draw [fill=yellow] (x5) rectangle (y6);
\vertex[left=0.25cm of x4] (x5) [label=above right:\(r_{X,Y}\)];
\vertex[above=0.5cm of x4] (x44);
\vertex[above=0.5cm of y3] (y33);
\vertex[above=0.5cm of z3] (z33);
\vertex[above=0.5cm of x44] (x444);
\vertex[above=0.5cm of y33] (y333);
\vertex[above=0.5cm of y55] (y555);
\vertex[left=0.25cm of x444] (x6);
\vertex[above=0.5cm of w7] (w8);
\vertex[right=0.25cm of w8] (w88);
\draw [fill=yellow] (x6) rectangle (w88);
\vertex[left=0.25cm of x444] (x6) [label=above right:\(P_{X\otimes_AY,Z\otimes_AW}\)];
\vertex[above=0.5cm of x444] (xx1);
\vertex[above=0.5cm of y555] (yy1);
\vertex[above=0.5cm of z5] (zz1);
\vertex[above=0.5cm of w7] (ww1);
\vertex[above=0.5cm of zz1] (zz2);
\vertex[above=0.5cm of ww1] (ww2);
\vertex[above=0.5cm of ww2] (ww3);
\vertex[left=0.25cm of zz2] (zz3);
\vertex[right=0.25cm of ww3] (ww4);
\draw [fill=yellow] (zz3) rectangle (ww4);
\vertex[left=0.25cm of zz2] (zz3) [label=above right:\(e_{Z,W}\)];
\vertex[above=1.5cm of xx1] (xx2);
\vertex[above=1.5cm of yy1] (yy2);
\vertex[above=0.5cm of zz2] (zz4);
\vertex[above=1.5cm of ww3] (ww5);
\vertex[left=0.25cm of xx2] (xx3);
\vertex[above=1cm of zz4] (zz5);
\vertex[above=0.5cm of zz4] (zz55);
\vertex[right=0.25cm of zz5] (zz6);
\draw [fill=yellow] (zz6) rectangle (xx3);
\vertex[left=0.25cm of xx2] (xx3) [label=above right:\(r_{X\otimes_AY,Z}\)];
\vertex[above=0.5cm of xx2] (xx4);
\vertex[above=0.5cm of yy2] (yy3);
\vertex[above=0.5cm of xx4] (xx5);
\vertex[above=0.5cm of yy3] (yy4);
\vertex[above=0.5cm of zz5] (zz7);
\vertex[left=0.25cm of xx5] (xx6);
\vertex[above=0.5cm of ww5] (ww6);
\vertex[right=0.25cm of ww6] (ww7);
\draw [fill=yellow] (ww7) rectangle (xx6);
\vertex[left=0.25cm of xx5] (xx6) [label=above right:\(r_{(X\otimes_AY)\otimes_AZ,W}\)];
\vertex[above=0.5cm of xx5] (xx7);
\vertex[above=0.5cm of yy4] (yy5);
\vertex[above=0.5cm of zz7] (zz8);
\vertex[above=0.5cm of xx7] (xx8) [label=above:\(X\)];
\vertex[above=0.5cm of yy5] (yy6) [label=above:\(Y\)];
\vertex[above=0.5cm of zz8] (zz9) [label=above:\(Z\)];
\vertex[above=0.5cm of ww6] (ww8) [label=above:\(W\)];
\diagram*{
(x1)--[thick] (x2),
(y1)--[thick] (y2),
(z1)--[thick] (z2),
(w1)--[thick] (w22),
(x22)--[thick] (x4),
(y22)--[thick] (y3),
(z22)--[thick] (z3),
(w4)--[thick] (w2),
(y33)--[thick] (y5),
(z5)--[thick] (z33),
(w7)--[thick] (w5),
(x44)--[thick] (x444),
(y55)--[thick] (y555),
(zz1)--[thick] (zz2),
(ww1)--[thick] (ww2),
(xx1)--[thick] (xx2),
(yy1)--[thick] (yy2),
(ww5)--[thick] (ww3),
(zz55)--[thick] (zz4),
(xx5)--[thick] (xx4),
(yy4)--[thick] (yy3),
(zz7)--[thick] (zz5),
(xx7)--[thick] (xx8),
(yy5)--[thick] (yy6),
(zz8)--[thick] (zz9),
(ww6)--[thick] (ww8)
\end{feynman}
\end{tikzpicture}}}}
\end{equation}

It will prove useful to manipulate the three morphisms in the middle of this expression. To this end, we can use the fact that $r_{X,Y}$ and $e_{Z,W}$ are bimodule intertwiners so they satisfy in particular

\begin{align}\label{a3}
&e_{Z,W}\circ \rho_{Z\otimes_AW}=\rho_{Z\otimes W}\circ\left(\id_A\otimes e_{Z,W}\right) \qquad \text{and} \nonumber \\
& r_{X,Y}\circ \Tilde{\rho}_{X\otimes Y}= \Tilde{\rho}_{X\otimes_AY}\circ\left(r_{X,Y}\otimes\id_A\right)
\end{align}

Now using the definition of the projectors (<ref>) and (<ref>) we can show that for any three $A\text{-}A$ bimodules $X,Y,Z$ we have

\begin{align}\label{a4}
& P_{X\otimes_AY,Z}\circ(r_{X,Y}\otimes\id_Z)=(r_{X,Y}\otimes\id_Z)\circ P_{X\otimes Y,Z} \qquad \text{and} \nonumber \\
&(\id_X\otimes e_{Y,Z})\circ P_{X,Y\otimes_AZ}=P_{X,Y\otimes Z}\circ (\id_X\otimes e_{Y,Z})
\end{align}

Using (<ref>) into (<ref>) leads to

\begin{equation}\label{a5}
\vcenter{\hbox{\hspace{2mm}\scalebox{0.8}{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (x1) at (0,0) [label=below:\(X\)];
\vertex [right=0.7cm of x1] (y1) [label=below:\(Y\)];
\vertex[right=0.7cm of y1] (z1) [label=below:\(Z\)];
\vertex[right=0.7cm of z1] (w1) [label=below:\(W\)];
\vertex[above=0.5cm of x1] (x2) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)] ;
\vertex[above=1cm of w1] (w2);
\vertex[above=0.5cm of y1] (y2);
\vertex[above=0.5cm of z1] (z2);
\vertex[above=0.5cm of w1] (w22);
\vertex[left=0.25cm of x2)] (x3);
\vertex [right=0.25cm of w2] (w3);
\draw [fill=yellow] (x3) rectangle (w3);
\vertex[above=0.5cm of x1] (x2) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)] ;
\vertex[above=2cm of x2] (x4);
\vertex[above=1cm of y2] (y3);
\vertex[left=0.25cm of y3] (y4);
\vertex[above=1cm of z2] (z3);
\vertex[above=1cm of w22] (w4);
\vertex[above=0.5cm of w4] (w5);
\vertex[right=0.25cm of w5] (w6);
\draw [fill=yellow] (y4) rectangle (w6);
\vertex[left=0.25cm of y3] (y4) [label=above right:\(e_{Y,Z\otimes_AW}\)];
\vertex[above=0.5cm of x2] (x22);
\vertex[above=0.5cm of y2] (y22);
\vertex[above=0.5cm of z2] (z22);
\vertex[above=1.5cm of y22] (y5);
\vertex[above=2.5cm of z22] (z5);
\vertex[above=3cm of w22] (w7);
\vertex[above=0.5cm of y5] (y55);
\vertex[right=0.25cm of y55] (y6);
\vertex[left=0.25cm of x4] (x5);
\vertex[above=0.5cm of x4] (x44);
\vertex[above=0.5cm of y3] (y33);
\vertex[above=0.5cm of z3] (z33);
\vertex[above=0.5cm of z33] (zzz1);
\vertex[above=0.5cm of w5] (www1);
\vertex[above=0.5cm of www1] (www2);
\vertex[left=0.25cm of zzz1] (zzz2);
\vertex[right=0.25cm of www2] (www3);
\draw [fill=yellow] (zzz2) rectangle (www3);
\vertex[left=0.25cm of zzz1] (zzz2) [label=above right:\(e_{Z,W}\)];
\vertex[above=0.5cm of zzz1] (zzz3);
\vertex[above=0.5cm of x44] (x444);
\vertex[above=0.5cm of y33] (y333);
\vertex[above=0.5cm of y55] (y555);
\vertex[left=0.25cm of x444] (x6);
\vertex[above=0.5cm of w7] (w8);
\vertex[right=0.25cm of w8] (w88);
\draw [fill=yellow] (x6) rectangle (w88);
\vertex[left=0.25cm of x444] (x6) [label=above right:\(P_{X\otimes Y,Z\otimes W}\)];
\vertex[above=0.5cm of x444] (xx1);
\vertex[above=0.5cm of y555] (yy1);
\vertex[above=0.5cm of z5] (zz1);
\vertex[above=0.5cm of w7] (ww1);
\vertex[above=0.5cm of xx1] (xxx1);
\vertex[above=0.5cm of yy1] (yyy1);
\vertex[above=0.5cm of yyy1] (yyy2);
\vertex[right=0.25cm of yyy2] (yyy3);
\vertex[left=0.25cm of xxx1] (xxx2);
\draw [fill=yellow] (yyy3) rectangle (xxx2);
\vertex[left=0.25cm of xxx1] (xxx2) [label=above right:\(r_{X,Y}\)];
\vertex[above=0.5cm of xxx1] (xxx3);
\vertex[above=0.5cm of zz1] (zz2);
\vertex[above=0.5cm of ww1] (ww2);
\vertex[above=0.5cm of ww2] (ww3);
\vertex[left=0.25cm of zz2] (zz3);
\vertex[right=0.25cm of ww3] (ww4);
\vertex[above=1.5cm of xx1] (xx2);
\vertex[above=1.5cm of yy1] (yy2);
\vertex[above=0.5cm of zz2] (zz4);
\vertex[above=1.5cm of ww3] (ww5);
\vertex[left=0.25cm of xx2] (xx3);
\vertex[above=1cm of zz4] (zz5);
\vertex[above=0.5cm of zz4] (zz55);
\vertex[right=0.25cm of zz5] (zz6);
\draw [fill=yellow] (zz6) rectangle (xx3);
\vertex[left=0.25cm of xx2] (xx3) [label=above right:\(r_{X\otimes_AY,Z}\)];
\vertex[above=0.5cm of xx2] (xx4);
\vertex[above=0.5cm of yy2] (yy3);
\vertex[above=0.5cm of xx4] (xx5);
\vertex[above=0.5cm of yy3] (yy4);
\vertex[above=0.5cm of zz5] (zz7);
\vertex[left=0.25cm of xx5] (xx6);
\vertex[above=0.5cm of ww5] (ww6);
\vertex[right=0.25cm of ww6] (ww7);
\draw [fill=yellow] (ww7) rectangle (xx6);
\vertex[left=0.25cm of xx5] (xx6) [label=above right:\(r_{(X\otimes_AY)\otimes_AZ,W}\)];
\vertex[above=0.5cm of xx5] (xx7);
\vertex[above=0.5cm of yy4] (yy5);
\vertex[above=0.5cm of zz7] (zz8);
\vertex[above=0.5cm of xx7] (xx8) [label=above:\(X\)];
\vertex[above=0.5cm of yy5] (yy6) [label=above:\(Y\)];
\vertex[above=0.5cm of zz8] (zz9) [label=above:\(Z\)];
\vertex[above=0.5cm of ww6] (ww8) [label=above:\(W\)];
\diagram*{
(x1)--[thick] (x2),
(y1)--[thick] (y2),
(z1)--[thick] (z2),
(w1)--[thick] (w22),
(x22)--[thick] (x4),
(y22)--[thick] (y3),
(z22)--[thick] (z3),
(w4)--[thick] (w2),
(y33)--[thick] (y5),
(zzz1)--[thick] (z33),
(www1)--[thick] (w5),
(x44)--[thick] (x444),
(y55)--[thick] (y555),
(zz1)--[thick] (zz2),
(ww1)--[thick] (ww2),
(xx1)--[thick] (xxx1),
(yy1)--[thick] (yyy1),
(ww5)--[thick] (ww3),
(zz55)--[thick] (zz4),
(xx5)--[thick] (xx4),
(yy4)--[thick] (yy3),
(zz7)--[thick] (zz5),
(xx7)--[thick] (xx8),
(yy5)--[thick] (yy6),
(zz8)--[thick] (zz9),
(ww6)--[thick] (ww8),
(x4)--[thick] (x44),
(y55)--[thick] (y5),
(zz2)--[thick] (zz4),
(ww2)--[thick] (ww3),
(xx2)--[thick] (xxx3),
(yyy2)--[thick] (yy2),
(www2)--[thick] (w7),
(z5)--[thick] (zzz3)
\end{feynman}
\end{tikzpicture}}}}
\end{equation}

Using the definition (<ref>), the right-hand side of (<ref>) reads

\begin{equation} \label{a6}
\vcenter{\hbox{\hspace{2mm}\scalebox{0.8}{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (x1) at (0,0) [label=below:\(X\)];
\vertex [right=0.7cm of x1] (y1) [label=below:\(Y\)];
\vertex [right=0.7cm of y1] (z1) [label=below:\(Z\)];
\vertex [right=0.7cm of z1] (w1) [label=below:\(W\)];
\vertex[above=1cm of w1] (w2);
\vertex [above=0.5cm of w1] (w11);
\vertex[above=0.5cm of x1] (x2);
\vertex[above=0.5cm of y1] (y2);
\vertex[above=0.5cm of z1] (z2);
\vertex[left=0.25cm of x2] (x22);
\vertex[right=0.25cm of w2] (w22);
\draw [fill=yellow] (x22) rectangle (w22);
\vertex[left=0.25cm of x2] (x22) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)];
\vertex[above=0.5cm of x2] (x3); %end of 1st box
\vertex[above=0.5cm of y2] (y3);
\vertex[above=0.5cm of z2] (z3);
\vertex[above=0.5cm of y3] (y4);
\vertex[above=0.5cm of z3] (z4);
\vertex[above=0.5cm of w2] (w3);
\vertex[above=0.5cm of w3] (w33);
\vertex[right=0.25cm of w33] (w333);
\vertex[left=0.25cm of y4] (y44);
\draw [fill=yellow] (y44) rectangle (w333);
\vertex[left=0.25cm of y4] (y44) [label=above right:\(e_{Y,Z\otimes_AW}\)];
\vertex[above=0.5cm of y4] (y5); %end of 2nd box
\vertex[above=0.5cm of z4] (z5);
\vertex[above=0.5cm of z5] (z6);
\vertex[above=0.5cm of w33] (w4);
\vertex[above=0.5cm of w4] (w5); %end of 3rd box
\vertex[right=0.25cm of w5] (w55);
\vertex[left=0.25cm of z6] (z66);
\draw [fill=yellow] (z66) rectangle (w55);
\vertex[left=0.25cm of z6] (z66) [label=above right:\(e_{Z,W}\)];
\vertex[above=0.5cm of z6] (z7); %end of 3rd box
\vertex[above=0.5cm of z7] (z8);
\vertex[above=1.5cm of y5] (y6);
\vertex[above=0.5cm of z8] (z9); %end of 4th box
\vertex[right=0.25cm of z9] (z99);
\vertex[left=0.25cm of y6] (y66);
\draw [fill=yellow] (y66) rectangle (z99);
\vertex[left=0.25cm of y6] (y66) [label=above right:\(r_{Y,Z}\)];
\vertex[above=1.5cm of w5] (w6); %start of 5th box
\vertex[above=0.5cm of w6] (w7);
\vertex[right=0.25cm of w7] (w77);
\vertex[above=0.5cm of y6] (y7); %end of 4th box
\vertex[above=0.5cm of y7] (y8); %start of 5th box
\vertex[left=0.25cm of y8] (y88);
\draw [fill=yellow] (y88) rectangle (w77);
\vertex[left=0.25cm of y8] (y88) [label=above right:\(r_{Y\otimes_AZ,W}\)];
\vertex[above=0.5cm of z9] (z10);
\vertex[above=4.5cm of x3] (x4);
\vertex[left=0.25cm of x4] (x44);
\vertex[above=0.5cm of y8] (yy1); %end of 5th box
\vertex[above=0.5cm of z10] (zz1);
\vertex[above=0.5cm of yy1] (yy2); %start of 6th box
\vertex[above=0.5cm of zz1] (zz2);
\vertex[above=0.5cm of w7] (ww1);
\vertex[above=0.5cm of ww1] (ww2); %end of 6th box
\vertex[right=0.25cm of ww2] (ww22);
\draw [fill=yellow] (ww22) rectangle (x44);
\vertex[left=0.25cm of x4] (x44) [label=above right:\(P_{X,(Y\otimes_AZ)\otimes_AW}\)];
\vertex[above=0.5cm of x4] (x5); %end of 6th box
\vertex[above=0.5cm of yy2] (yy3);
\vertex[above=0.5cm of zz2] (zz3);
\vertex[above=0.5cm of yy3] (yy4); %start of 7th box
\vertex[above=0.5cm of zz3] (zz4);
\vertex[above=0.5cm of ww2] (ww3);
\vertex[left=0.25cm of yy4] (yy44);
\vertex[above=0.5cm of ww3] (ww4);
\vertex[right=0.25cm of ww4] (ww44);
\draw [fill=yellow] (ww44) rectangle (yy44);
\vertex[left=0.25cm of yy4] (yy44) [label=above right:\(e_{Y\otimes_AZ,W}\)];
\vertex[above=0.5cm of yy4] (yy5); %end of 7th box
\vertex[above=0.5cm of zz4] (zz5); %end of 7th box
\vertex[above=1.5cm of x5] (x6); %start of 8th
\vertex[above=0.5cm of yy5] (yy6); % start of 8th
\vertex[above=0.5cm of zz5] (zz6); % start of 8th
\vertex[above=0.5cm of zz6] (zz7);
\vertex[right=0.25cm of zz7] (zz77);
\vertex[left=0.25cm of x6] (x66);
\draw [fill=yellow] (x66) rectangle (zz77);
\vertex[left=0.25cm of x6] (x66)[label=above right:\(r_{X,Y\otimes_AZ}\)];
\vertex[above=0.5cm of x6] (x7); %end of 8th
\vertex[above=0.5cm of yy6] (yy7); %end of 8th
\vertex[above=0.5cm of zz6] (zz7); %end of 8th
\vertex[above=1.5cm of ww4] (ww5); %start of 9th
\vertex[above=0.5cm of ww5] (ww6); %end of 9th
\vertex[right=0.25cm of ww6] (ww66);
\vertex[above=0.5cm of x7] (x8); %start of 9th
\vertex[left=0.25cm of x8] (x88);
\vertex[above=0.5cm of yy7] (yy8); %start of 9th
\vertex[above=0.5cm of zz7] (zz8); %start of 9th
\draw [fill=yellow] (ww66) rectangle (x88);
\vertex[left=0.25cm of x8] (x88)[label=above right:\(P_{X\otimes_A(Y\otimes_AZ),W}\)];
\vertex[above=0.5cm of x8] (xx1); %end of 9th
\vertex[above=0.5cm of yy8] (yy9); %end of 9th
\vertex[above=0.5cm of zz8] (zz9); %end of 9th
\vertex[above=0.5cm of xx1] (xx2); %start of 10th
\vertex[above=0.5cm of yy9] (yyy1); %start of 10th
\vertex[above=0.5cm of zz9] (zzz1); %start of 10th
\vertex[left=0.25cm of xx2] (xx22);
\vertex[above=0.5cm of zzz1] (zzz2); %end of 10th
\vertex[right=0.25cm of zzz2] (zzz22);
\draw [fill=yellow] (zzz22) rectangle (xx22);
\vertex[left=0.25cm of xx2] (xx22)[label=above right:\(e_{X,Y\otimes_AZ}\)];
\vertex[above=0.5cm of xx2] (xx3); %end of 10th
\vertex[above=0.5cm of yyy1] (yyy2); %end of 10th
\vertex[above=0.5cm of yyy2] (yyy3); %start of 11th
\vertex[above=0.5cm of zzz2] (zzz3); %start of 11th
\vertex[above=0.5cm of zzz3]  (zzz4); %end of 11th
\vertex[left=0.25cm of yyy3] (yyy33);
\vertex[right=0.25cm of zzz4] (zzz44);
\draw [fill=yellow] (zzz44) rectangle (yyy33);
\vertex[left=0.25cm of yyy3] (yyy33)[label=above right:\(e_{Y,Z}\)];
\vertex[above=0.5cm of yyy3] (yyy4); %end of 11th
\vertex[above=1.5cm of xx3] (xx4); %start of 12th
\vertex[above=0.5cm of yyy4] (yyy5); %start of 12th
\vertex[above=0.5cm of yyy5] (yyy6); %end of 12th
\vertex[left=0.25cm of xx4] (xx44);
\vertex[right=0.25cm of yyy6] (yyy66);
\draw [fill=yellow] (xx44) rectangle (yyy66);
\vertex[left=0.25cm of xx4] (xx44)[label=above right:\(r_{X,Y}\)];
\vertex[above=0.5cm of xx4] (xx5); %end of 12th
\vertex[above=1.5cm of zzz4] (zzz5); %start of 13th
\vertex[above=0.5cm of xx5] (xx6); %start of 13th
\vertex[above=0.5cm of yyy6] (yyy7); %start of 13th
\vertex[above=0.5cm of zzz5] (zzz6); %end of 13th
\vertex[left=0.25cm of xx6] (xx66);
\vertex[right=0.25cm of zzz6] (zzz66);
\draw [fill=yellow] (zzz66) rectangle (xx66);
\vertex[left=0.25cm of xx6] (xx66)[label=above right:\(r_{X\otimes_AY,Z}\)];
\vertex[above=0.5cm of xx6] (xx7); %end of 13th
\vertex[above=0.5cm of yyy7] (yyy8); %end of 13th
\vertex[above=0.5cm of xx7] (xx8); %start of 14th
\vertex[above=0.5cm of yyy8] (yyy9); %start of 14th
\vertex[above=0.5cm of zzz6] (zzz7); %start of 14th
\vertex[above=4.5cm of ww6] (ww7); %start of 14th
\vertex[above=0.5cm of ww7] (ww8); %end of 14th
\vertex[left=0.25cm of xx8] (xx88);
\vertex[right=0.25cm of ww8] (ww88);
\draw [fill=yellow] (ww88) rectangle (xx88);
\vertex[left=0.25cm of xx8] (xx88)[label=above right:\(r_{(X\otimes_AY)\otimes_AZ,W}\)];
\vertex[above=0.5cm of xx8] (xx9);
\vertex[above=0.5cm of yyy9] (yyy0);
\vertex[above=0.5cm of zzz7] (zzz8);
\vertex[above=0.5cm of xx9] (xx0) [label=above:\(X\)];
\vertex[above=0.5cm of yyy0] (y0) [label=above:\(y\)];
\vertex[above=0.5cm of zzz8] (z0) [label=above:\(Z\)];
\vertex[above=0.5cm of ww8] (w0)  [label=above:\(W\)];
\diagram*{
(x1)--[thick] (x2),
(y1)--[thick] (y2),
(z1)--[thick] (z2),
(w1)--[thick] (w11),
(y3)--[thick] (y4),
(z3)--[thick] (z4),
(w2)--[thick] (w3),
(z5)--[thick] (z6),
(w33)--[thick] (w4),
(y5)--[thick] (y6),
(z7)--[thick] (z8),
(z9)--[thick] (z10),
(y7)--[thick] (y8),
(w5)--[thick] (w6),
(x3)--[thick] (x4),
(yy1)--[thick] (yy2),
(zz1)--[thick] (zz2),
(w7)--[thick] (ww1),
(ww2)--[thick] (ww3),
(yy3)--[thick] (yy4),
(zz3)--[thick] (zz4),
(x5)--[thick] (x6),
(yy5)--[thick] (yy6),
(zz5)--[thick] (zz6),
(x7)--[thick] (x8),
(yy7)--[thick] (yy8),
(zz7)--[thick] (zz8),
(ww4)--[thick] (ww5),
(xx1)--[thick] (xx2),
(yy9)--[thick] (yyy1),
(zz9)--[thick] (zzz1),
(yyy2)--[thick] (yyy3),
(zzz2)--[thick] (zzz3),
(xx3)--[thick] (xx4),
(yyy4)--[thick] (yyy5),
(xx5)--[thick] (xx6),
(yyy6)--[thick] (yyy7),
(zzz4)--[thick] (zzz5),
(xx7)--[thick] (xx8),
(yyy8)--[thick] (yyy9),
(zzz6)--[thick] (zzz7),
(ww6)--[thick] (ww7),
(ww8)--[thick] (w0),
(xx9)--[thick] (xx0),
(yyy0)--[thick] (y0),
(zzz8)--[thick] (z0)
\end{feynman}
\end{tikzpicture}}}}
\end{equation}

To manipulate this expression we proceed in a similar way as the one described in (<ref>) and (<ref>), namely we  use the fact that $e_{X,Y\otimes_AZ}$ and $e_{Y\otimes_AZ,W}$ are bimodule intertwiners. Combining this with the explicit form of the relevant projectors we can show that

\begin{align}\label{a7}
& \left(e_{X,Y\otimes_AZ}\otimes\id_W\right)\circ P_{X\otimes_A(Y\otimes_AZ),W} =P_{X\otimes(Y\otimes_AZ),W}\circ\left(e_{X,Y\otimes_AZ}\otimes\id_W\right) \qquad \text{and}\nonumber \\
& \left(\id_X\otimes e_{Y\otimes_AZ,W}\right)\circ P_{X,(Y\otimes_AZ)\otimes_AW}=P_{X,(Y\otimes_AZ)\otimes W}\circ \left(\id_X\otimes e_{Y\otimes_AZ,W}\right)
\end{align}

Next we  use (<ref>) and also notice that for any three bimodules $X,Y,Z$ we have

\begin{equation}\label{a8}
P_{X\otimes Y,Z}=\id_X\otimes P_{Y,Z} \qquad \text{and} \qquad P_{X,Y\otimes Z}=P_{X,Y}\otimes \id_Z
\end{equation}

which follows from the explicit form of the projector (<ref>)  and from the definition of the right and left action of $A$ on the bimodule $X\otimes Y$ and $Y\otimes Z$ respectively, that is, $\Tilde{\rho}_{X\otimes Y}=\id_X\otimes \tilde{\rho}^Y$ and $\rho_{Y\otimes Z}=\rho^Y\otimes\id_Z$. Finally, we  use the fact that for any $A\text{-}A$-bimodules $X,Y$ the projector $P_{X,Y}$ is an idempotent. Combining all the above, we  put (<ref>) into the following form

\begin{equation}\label{a9}
\vcenter{\hbox{\hspace{2mm}\scalebox{0.8}{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (x1) at (0,0) [label=below:\(X\)];
\vertex [right=0.7cm of x1] (y1) [label=below:\(Y\)];
\vertex [right=0.7cm of y1] (z1) [label=below:\(Z\)];
\vertex [right=0.7cm of z1] (w1) [label=below:\(W\)];
\vertex[above=1cm of w1] (w2);
\vertex [above=0.5cm of w1] (w11);
\vertex[above=0.5cm of x1] (x2);
\vertex[above=0.5cm of y1] (y2);
\vertex[above=0.5cm of z1] (z2);
\vertex[left=0.25cm of x2] (x22);
\vertex[right=0.25cm of w2] (w22);
\draw [fill=yellow] (x22) rectangle (w22);
\vertex[left=0.25cm of x2] (x22) [label=above right:\(e_{X,Y\otimes_A(Z\otimes_AW)}\)];
\vertex[above=0.5cm of x2] (x3); %end of 1st box
\vertex[above=0.5cm of y2] (y3);
\vertex[above=0.5cm of z2] (z3);
\vertex[above=0.5cm of y3] (y4);
\vertex[above=0.5cm of z3] (z4);
\vertex[above=0.5cm of w2] (w3);
\vertex[above=0.5cm of w3] (w33);
\vertex[right=0.25cm of w33] (w333);
\vertex[left=0.25cm of y4] (y44);
\draw [fill=yellow] (y44) rectangle (w333);
\vertex[left=0.25cm of y4] (y44) [label=above right:\(e_{Y,Z\otimes_AW}\)];
\vertex[above=0.5cm of y4] (y5); %end of 2nd box
\vertex[above=0.5cm of z4] (z5);
\vertex[above=0.5cm of z5] (z6);
\vertex[above=0.5cm of w33] (w4);
\vertex[above=0.5cm of w4] (w5); %end of 3rd box
\vertex[right=0.25cm of w5] (w55);
\vertex[left=0.25cm of z6] (z66);
\draw [fill=yellow] (z66) rectangle (w55);
\vertex[left=0.25cm of z6] (z66) [label=above right:\(e_{Z,W}\)];
\vertex[above=0.5cm of z6] (z7); %end of 3rd box
\vertex[above=0.5cm of z7] (z8);
\vertex[above=1.5cm of y5] (y6);
\vertex[above=0.5cm of z8] (z9); %end of 4th box
\vertex[right=0.25cm of z9] (z99);
\vertex[left=0.25cm of y6] (y66);
\draw [fill=yellow] (y66) rectangle (z99);
\vertex[left=0.25cm of y6] (y66) [label=above right:\(r_{Y,Z}\)];
\vertex[above=0.5cm of y6] (y7); %end of 4th
\vertex[above=0.5cm of y7] (y8); %start of 5th
\vertex[above=0.5cm of z9] (zz1); %start of 5th
\vertex[above=1.5cm of w5] (w6); %start of 5th
\vertex[above=0.5cm of w6] (w7); %end of 5th
\vertex[left=0.25cm of y8] (y88);
\vertex[right=0.25cm of w7] (w77);
\draw [fill=yellow] (w77) rectangle (y88);
\vertex[left=0.25cm of y8] (y88)[label=above right:\(P_{Y\otimes_AZ,W}\)];
\vertex[above=0.5cm of y8] (y9); %end of 5th
\vertex[above=0.5cm of zz1] (zz2); %end of 5th
\vertex[above=4.5cm of x3] (x4); %start of 6th
\vertex[above=0.5cm of y9] (yy1); %start of 6th
\vertex[above=0.5cm of zz2] (zz3); %start of 6th
\vertex[above=0.5cm of zz3] (zz4); %end of 6th
\vertex[left=0.25cm of x4] (x44);
\vertex[right=0.25cm of zz4] (zz44);
\draw [fill=yellow] (zz44) rectangle (x44);
\vertex[left=0.25cm of x4] (x44)[label=above right:\(P_{X,Y\otimes_AZ}\)];
\vertex[above=0.5cm of x4] (x5); %end of 6th
\vertex[above=0.5cm of yy1] (yy2); %end of 6th
\vertex[above=0.5cm of yy2] (yy3); %start of 7th
\vertex[above=0.5cm of zz4] (zz5); %start of 7th
\vertex[above=0.5cm of zz5] (zz6); %end of 7th
\vertex[left=0.25cm of yy3] (yy33);
\vertex[right=0.25cm of zz6] (zz66);
\draw [fill=yellow] (zz66) rectangle (yy33);
\vertex[left=0.25cm of yy3] (yy33)[label=above right:\(e_{Y,Z}\)];
\vertex[above=0.5cm of yy3] (yy4); %end of 7th
\vertex[above=1.5cm of x5] (x6); %start of 8th
\vertex[above=0.5cm of yy4] (yy5); %start of 8th
\vertex[above=0.5cm of yy5] (yy6); %end of 8th
\vertex[left=0.25cm of x6] (x66);
\vertex[right=0.25cm of  yy6] (yy66);
\draw [fill=yellow] (yy66) rectangle (x66);
\vertex[left=0.25cm of x6] (x66)[label=above right:\(r_{X,Y}\)];
\vertex[above=0.5cm of x6] (x7); %end of 8th
\vertex[above=0.5cm of x7] (x8); %start of 9th
\vertex[above=0.5cm of yy6] (yy7); %start of 9th