#  $\alpha$-induction for bi-unitary connections

Yasuyuki Kawahigashi
Graduate School of Mathematical Sciences
The University of Tokyo, Komaba, Tokyo, 153-8914, Japan
e-mail<EMAIL_ADDRESS>
Kavli IPMU (WPI), the University of Tokyo
5–1–5 Kashiwanoha, Kashiwa, 277-8583, Japan
and
iTHEMS Research Group, RIKEN
2-1 Hirosawa, Wako, Saitama 351-0198,Japan

###### Abstract

The tensor functor called $\alpha$-induction produces a new unitary fusion
category from a Frobenius algebra object, or a $Q$-system, in a braided
unitary fusion category. In the operator algebraic language, it gives
extensions of endomorphism of $N$ to $M$ arising from a subfactor $N\subset M$
of finite index and finite depth which gives a braided fusion category of
endomorphisms of $N$. It is also understood in terms of Ocneanu’s graphical
calculus. We study this $\alpha$-induction for bi-unitary connections, which
give a characterization of finite-dimensional nondegenerate commuting squares
and gives certain $4$-tensors appearing in recent studies of $2$-dimensional
topological order. We show that the resulting $\alpha$-induced bi-unitary
connections are flat if we start with a commutative Frobenius algebra, or a
local $Q$-system. Examples related to chiral conformal field theory and the
Dynkin diagrams are presented.

Dedicated to the memory of Vaughan Jones

##  1 Introduction

A fusion category [13] has recently emerged as a new type of symmetry in a
wide range of topics in mathematics and physics. Theory of operator algebras
gives a nice framework to study this type of new symmetries, as exemplified by
discovery of the Jones polynomial for knots [24] from the Jones theory of
subfactors [23]. We have three operator algebraic realizations of a fusion
category based on endomorphisms of a type III factor [37], [38], bimodules
over type II1 factors [15, Chapter 9], and bi-unitary connections [1, Section
3]. The last approach based on bi-unitary connections recently has renewed
interest because of its relations to 2-dimensional statistical physics [9],
[29].

A certain $4$-tensor, a (finite) family of complex numbers indexed by four
indices, is studied in physics literature such as [9], [41]. This has been
identified with a bi-unitary connection in the subfactor sense in [29], and
further studies [30], [31], [32] have followed in this direction. Also see
[20] for a recent development. Bi-unitary connections and related topics are
also recently studied in [10], [11]. This approach to a fusion category based
on bi-unitary connections has advantage that everything is described with
finite dimensional matrices and thus in principle computable on computers,
while endomorphisms of a type III factor and bimodules over type II1 factors
are infinite dimensional. This computability is one reason physicists are
interested in this approach recently. Our aim in this paper to study
$\alpha$-induction in this framework of bi-unitary connections.

The tensor functor $\alpha$-induction originates in the operator algebraic
studies of chiral conformal field theory [40]. In this context, our basic
object is a conformal net, which is a family of von Neumann algebras
parametrized by intervals contained in the circle $S^{1}$. It has a
representation theory of superselection sectors and this has a structure of
braiding [16], [17]. If a conformal net has a certain finiteness property
called complete rationality, we obtain a modular tensor category of its
representations [34]. See a short review [28] and a longer review [27] for a
general theory of this topic.

In a classical representation theory, if we have a representation of a
subgroup $H\subset G$, we have a method of induction to obtain a
representation of $G$. We have a similar, but more subtle, induction machinery
for representation theory of conformal nets, depending on braiding structure.
Suppose we have an inclusion $A\subset B$ of completely rational conformal
nets and $\lambda$ is a representation of $A$ given as a Doplicher-Haag-
Roberts endomorphism of $A(I)$ for some fixed interval $I$ in $S^{1}$. Then we
can extend this endomorphism to $B(I)$ using the braiding. This was first
defined by Longo-Rehren [40] and further studied by Xu [51] and Böckenhauer-
Evans [4], [5], [6], where this was named as $\alpha$-induction.

In a completely different setting, Ocneanu had a machinery of flat connections
to study subfactors [43], [44], which consists of entries of a large unitary
matrix. Then using a braiding structure of flat connections on the Dynkin
diagrams of type $A$, he introduced a graphical calculus to construct new
fusion categories related to the Goodman-de la Harpe subfactors [18] in [45].
We have unified the two theories of $\alpha$-induction and Ocneanu’s graphical
calculus in a fully general setting of braided fusion categories and proved
various basic properties such as appearance of modular invariants in [7], [8]
in the framework of endomorpshisms of type III factors. In this setting,
$\alpha$-induction is understood as a method of extending an endomorphism of a
factor $N$ to another factor $M\supset N$ using braiding, where we do not
assume anything about conformal field theory. This $\alpha$-induction has been
also understood in the language of bimodules and more abstract braided fusion
category.

Our motivation for this paper is as follows. First, it is nice to have a
concrete realization of $\alpha$-induction in the setting of bi-unitary
connections due to its finite dimensionality and relations to statistical
physics. Secondly, it gives a generalization of Ocneanu’s graphical calculus
in a much more general setting. We also note that the same mathematical
structure as $\alpha$-induction appears in the context of anyon condensation
[2], which gives another reason to study $\alpha$-induction in this setting.
(See [36, Table 1] for this direction.)

Our main result is Theorem 3.6 which shows our bi-unitary connection defined
in Definition 3.5 is a correct description of $\alpha$-induction. We have a
remark that though we use an endomorphism framework [7], [8], we only need
abstract properties of $\alpha$-induction, so we can also use a bimodule
framework or a setting based on more abstract fusion categories for
$\alpha$-induction for our results.

This work was partially supported by JST CREST program JPMJCR18T6 and Grants-
in-Aid for Scientific Research 19H00640 and 19K21832. A part of this work was
done at Microsoft Station Q at Santa Barbara and University of California,
Berkeley. The author thanks for their hospitality.

##  2 Preliminaries on braided fusion categories, subfactors and
$\alpha$-induction

We give our setting for $\alpha$-induction for a subfactor $N\subset M$ and
endomorphisms of $N$ as in [7]. Let $N$ be a type III factor. Throughout this
paper, we assume the following for $\Delta$ which is a finite set of mutually
inequivalent irreducible endomorphisms of $N$ of finite dimension. (See [7,
Definition 2.1].)

###### Assumption 2.1

We have the following for $\Delta$.

(1) The identity automorphism is in $\Delta$.

(2) For any $\lambda\in\Delta$, we have another element $\mu\in\Delta$ which
is equivalent to $\bar{\lambda}$.

(3) For any $\lambda,\mu\in\Delta$, the composition $\lambda\mu$ decomposes
into a direct sum of irreducible endomorphisms each of which is equivalent to
one in $\Delta$.

(4) The set $\Delta$ has a braiding
${\varepsilon}(\lambda,\mu)\in\mathop{\mathrm{Hom}}\nolimits(\lambda\mu,\mu\lambda)$
as in [7, Definition 2.2].

We further make an assumption on connectedness of certain bipartite graphs as
follows, which will be necessary in the following Section. Note that this
assumption depends on a choice of $\mu$, an irreducible endomorphism of $N$ in
$\Delta$. This is assumed throughout the paper, except for Section 6.

###### Assumption 2.2

Consider a bipartite graph defined as follows. Let both even and odd vertex
sets be labeled with the elements in $\Delta$. The number of edges between the
even vertex $\nu_{1}$ and the odd vertex $\nu_{2}$ is given by
$\dim\mathop{\mathrm{Hom}}\nolimits(\nu_{1}\mu,\nu_{2})$. We assume that this
graph is connected.

Endomorphisms of $N$ with finite dimension whose irreducible decompositions
give endomorpshisms equivalent to ones in $\Delta$ produce a fusion category,
where objects are such endomorphisms and morphisms are intertwiners between
endomorphisms. (See [7, Section 2.1] for more details on intertwiners. Also
see [3] for a recent treatment of fusion categories and subfactors. See [13]
for a more abstract and algebraic treatment of fusion categories.) We consider
a subfactor $N\subset M$ with finite index whose canonical endomorphism
$\theta$ decomposes into a sum of endomorphisms equivalent to ones in
$\Delta$. Such a subfactor automatically has a finite depth. For a fixed $N$
and a fusion category of such endomorphisms, an extension $M$ of $N$ is in a
bijective correspondence to a $Q$-system as in [39, Section 6]. In algebraic
literature, this is often called a Frobenius algebra. (In this paper, we only
consider $C^{*}$-tensor categories. Also see [42, Theorem 2.3] for a bimodule
formulation of a $Q$-system.)

Since we assume to have a braiding, our fusion category is a braided fusion
category. If a braiding is nondegenerate in the sense of [7, Definition 2.3],
we say that this braided fusion category is a modular tensor category, but we
do not assume this nondegeneracy in this paper. We have a notion of locality,
${\varepsilon}(\theta,\theta)\gamma(v)=\gamma(v)$, for a $Q$-system
corresponding to a subfactor $N\subset M$. (Here $\gamma$ is the canonical
endomorphism of $M$ and an isometry $v\in M$ satisfies $vx=\gamma(x)v$ for
$x\in M$ and $M=Nv$.) The name locality comes from locality of an extension of
a conformal net in the setting of [40, Theorem 4.9] and it was called chiral
locality in [7]. A local $Q$-system is also called a commutative Frobenius
algebra in algebraic literature. We deal with both local and non-local
$Q$-systems in this paper.

The procedure called $\alpha$-induction was defined in [40, Proposition 3.9]
as follows.

$\alpha_{\lambda}^{\pm}=\bar{\iota}^{-1}\cdot\mathrm{Ad}(\varepsilon^{\pm}(\lambda,\theta))\cdot\lambda\cdot\bar{\iota},$

where $\lambda$ is an endomorphism in $\Delta$, $\iota$ is the inclusion map
$N\hookrightarrow M$, $\theta=\bar{\iota}\cdot\iota$ is the dual canonical
endomorphism of $N\subset M$, and $\pm$ stands for a choice of
positive/negative braiding. It is a nontrivial fact that
$\mathrm{Ad}(\varepsilon^{\pm}(\lambda,\theta))\cdot\lambda\cdot\bar{\iota}(x)$
is in the image of $\bar{\iota}$ for $x\in M$. We have
$\alpha_{\lambda}^{\pm}(x)=\lambda(x)$ for $x\in N$ and
$\alpha_{\lambda}^{\pm}(v)=\varepsilon^{\pm}(\lambda,\theta)^{*}v$. See [7,
Section 3] for basic properties of $\alpha$-induction in this setting.

In a very different setting, Ocneanu used Fig. 1 to represent a chiral
generator in the double triangle algebra. (Also see [7, Fig. 47].) It was
identified with the $\alpha$-induction in [7, Theorem 5.3]. (See [7, Section
4] for the double triangle algebra and [7, Fig. 4.1] for a graphical
convention involving small half circles.)

$\displaystyle\sum_{a,b,\mu}$$\lambda$$\mu$$b$$b$$a$$b$$b$$a$ Figure 1: The
chiral generator $p_{\lambda}^{+}$

For various diagrams, we use the convention in [7, Section 3]. In particular,
our convention is as in Fig.2, where $T$ is an isometry in
$\mathop{\mathrm{Hom}}\nolimits(\lambda,\mu\nu)$ as in [7, Fig. 21]. Note that
rotation invariance of this type of diagrams is due to the Frobenius
reciprocity for endomorphisms due to Izumi [21], [22]. Also as in [7, Section
4], we draw a thin wire for an $N$-$N$ morphism, a thick wire for an $N$-$M$
or $M$-$N$ morphism, and a very thick wire for an $M$-$M$ morphism. For a
thick wire, we can tell whether it stands for an $N$-$M$ or $M$-$N$ morphism
from a diagram.

$\mu$$\nu$$\lambda$$\lambda$$\mu$$\nu$$T$$T^{*}$$=\sqrt[4]{\displaystyle\frac{d_{\mu}d_{\nu}}{d_{\lambda}}}T,$$=\sqrt[4]{\displaystyle\frac{d_{\mu}d_{\nu}}{d_{\lambda}}}T^{*}.$
Figure 2: Graphical convention for normalization of an intertwiner

##  3 $\alpha$-induction for bi-unitary connections

We now produce a family of new bi-unitary connections from the setting as in
the previous Section. We simply say a connection for a bi-unitary connection
in this paper.

We first recall the definition of a connection. (See [15, Section 11.3], [1,
Section 3], [30, Section 2]. There is an issue of connectedness of certain
graphs and we follow conventions of [1, Section 3] on this matter.)

We have four finite bipartite graphs
${\mathcal{G}},{\mathcal{G}}^{\prime},{\mathcal{H}},{\mathcal{H}}^{\prime}$.
The vertex set $V_{0}$ is common for even vertices of ${\mathcal{G}}$ and
${\mathcal{H}}$. Similarly, the vertex sets $V_{1},V_{2},V_{3}$ are common for
odd vertices of ${\mathcal{H}}$ and ${\mathcal{G}}^{\prime}$, even vertices of
${\mathcal{G}}^{\prime}$ and ${\mathcal{H}}^{\prime}$, and odd vertices of
${\mathcal{G}}$ and ${\mathcal{H}}^{\prime}$, respectively. The four graphs
satisfy some properties about the Perron-Frobenius eigenvalues and
eigenvectors as in [30, Section 2]. We choose edge
$\xi_{0},\xi_{1},\xi_{2},\xi_{3}$ from the graphs
${\mathcal{H}},{\mathcal{G}}^{\prime},{\mathcal{H}}^{\prime},{\mathcal{G}}$,
respectively so that they make a closed square called a cell. A map $W$ called
a connection assigns a complex number to each of such cells. This complex
value is represented with a diagram in Fig. 3. This map $W$ satisfies axioms
called bi-unitarity as in [30, Definition 2.2]. (See also Fig. 8, 9, 10
below.) The name “bi-unitarirty” means that each number in Fig. 3 is an entry
of a unitary matrix in two ways, one after normalization arising from the
Perron-Frobenius eigenvector entries. Now we require that the graphs
${\mathcal{G}}$ and ${\mathcal{G}}^{\prime}$ to be connected, but we do not
require this for ${\mathcal{H}}$ and ${\mathcal{H}}^{\prime}$. (This
convention is different from the one in [30], and the same as in [1].) We have
an equivalence relation for connections on the same four graphs as in Remark
after [1, Theorem 3]. We call the graphs
${\mathcal{G}},{\mathcal{G}}^{\prime},{\mathcal{H}},{\mathcal{H}}^{\prime}$
the horizontal top graph, the horizontal bottom graph, the vertical left graph
and the vertical right graph of $W$, respectively. We also call the vertices
in $V_{0},V_{1},V_{2},V_{3}$ the upper left vertices, the lower left vertices,
the lower right vertices and the upper right vertices, respectively.

$\xi_{2}$$\xi_{0}$$\xi_{3}$$\xi_{1}$$x_{1}$$x_{2}$$x_{0}$$x_{3}$$W$ Figure 3:
The four graphs for a connection

It is well-known that this bi-unitarity condition is characterized in terms of
a commuting square as we now explain below. A commuting square of finite
dimensional $C^{*}$-algebras $\begin{array}[]{ccc}A&\subset&B\\\ \cap&&\cap\\\
C&\subset&D\end{array}$ with a trace on $D$ is characterized by the following
mutually equivalent conditions. (See [15, Proposition 9.51], for example.)

  1. 1.

The conditional expectation $E_{B}$ from $D$ to $B$ with respect to trace
restricted to $C$ coincides with the conditional expectation $E_{A}$ from $C$
to $A$ with respect to trace.

  2. 2.

The conditional expectation $E_{C}$ from $D$ to $C$ with respect to trace
restricted to $B$ coincides with the conditional expectation $E_{A}$ from $B$
to $A$ with respect to trace.

This notion was originally considered in [46]. We say that this commuting
square is nondegenerate if the span $BC$ is equal to $D$. (Being nondegenerate
is also sometimes said to be symmetric.) A nondegenerate square of finite
dimensional $C^{*}$-algebras is described with a connection, for which we do
not know yet whether it satisfies bi-unitarity, as in [15, Section 11.2]. Then
it has been proved in [49, Theorem 1.10] that the square gives a commuting
square if and only the connection satisfies bi-unitarity. (Also see [15,
Theorem 11.2].) In this sense, having a bi-unitary connection and having a
nondegenerate commuting square of finite dimensional $C^{*}$-algebras are the
same thing.

A commuting square also naturally appears from a subfactor $N\subset M$ with
finite Jones index as $\begin{array}[]{ccc}M^{\prime}\cap
M_{k}&\subset&M^{\prime}\cap M_{k+1}\\\ \cap&&\cap\\\ N^{\prime}\cap
M_{k}&\subset&N^{\prime}\cap M_{k+1}\end{array}$, where $N\subset M\subset
M_{1}\subset M_{2}\subset\cdots$ is the Jones tower arising from the Jones
basic construction [23]. (Also see [15, Section 9.6] for such a commuting
square.) If the original subfactor $N\subset M$ is hyperfinite, of type II1
and of finite depth, then the above commuting square recovers the original
subfactor $N\subset M$ by Popa’s theorem [47]. In this sense, a connection
encodes complete information about such a subfactor. A connection
corresponding to such a commuting square arising from a subfactor $N\subset M$
satisfies a special extra property called flatness, which was introduced by
Ocneanu [43], [44] and studied in [25]. (See [15, Section 11.4] for more
discussions on flatness.)

We first define the connection $W_{1}(\lambda,\mu)$, but we make a remark on
one issue again. In this paper, we use operator algebraic realization of a
braided fusion category and $\alpha$-induction based on endomorpshisms of a
type III factor as in [7] and [8], such as [7, Fig. 30], but this is simply
because this was the first place where all necessary details were worked out,
and we can equally use other formulations based on bimodules or abstract
fusion categories. Such a choice of formulation does not cause any change in
our results here.

###### Definition 3.1

Let $\nu_{1},\nu_{2},\nu_{3},\nu_{4}\in\Delta$. Consider the diagram in Fig.
4. By composing isometries
$T_{1}\in\mathop{\mathrm{Hom}}\nolimits(\nu_{1}\mu,\nu_{2})$,
$T_{2}\in\mathop{\mathrm{Hom}}\nolimits(\lambda\nu_{2},\nu_{4})$,
$T_{3}\in\mathop{\mathrm{Hom}}\nolimits(\lambda\nu_{1},\nu_{3})$, and
$T_{4}\in\mathop{\mathrm{Hom}}\nolimits(\nu_{3}\mu,\nu_{4})$, we obtain a
complex number
$T_{4}T_{3}\lambda(T_{1}^{*})T_{2}^{*}\in\mathop{\mathrm{Hom}}\nolimits(\nu_{4},\nu_{4})$.
We define the connection $W_{1}(\lambda,\mu)$ by this number and draw Fig. 5.

We note that the left vertical bipartite graph in this Definition is defined
as follows. One set of vertices is given by $\Delta$ and the other is the
same. The number of edges between the vertices $\nu_{1}$ and $\nu_{3}$ is
given by $\dim\mathop{\mathrm{Hom}}\nolimits(\lambda\nu_{1},\nu_{3})$. The
other graphs are defined similarly, and this remark applies to all the
Definitions of the connections in this Section.

$\nu_{3}\mu$$\nu_{4}$$\lambda\nu_{1}\mu$$\lambda\nu_{2}$$\lambda(T_{1})$$T_{4}$$T_{3}$$T_{2}$
Figure 4: The diagram for the connection $W_{1}(\lambda,\mu)$ in Fig. 5
$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$$T_{4}$$T_{1}$$T_{3}$$T_{2}$$W_{1}(\lambda,\mu)$
Figure 5: The standard diagram for a connection $W_{1}(\lambda,\mu)$

We often drop labels for the connection and/or intertwiners as long as no
confusion arises, and simply draw a diagram in Fig. 6.

$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$ Figure 6: The standard diagram for a
connection $W_{1}(\lambda,\mu)$
$\nu_{3}$$\nu_{2}$$\lambda$$\nu_{1}$$\mu$$\bar{\nu}_{4}$ Figure 7: The diagram
for the connection $W_{1}(\lambda,\mu)$ in Fig. 5

We draw a diagram as in Fig. 7 which represents a complex number in the
standard convention as in [7, Section 3]. (Note that in Fig. 7, we drop
orientations of wires which go from the top to the bottom.) This complex
number is equal to the one represented by the composition of isometries as in
Fig. 4 multiplied by $\sqrt{d_{\lambda}d_{\mu}d_{\nu_{1}}d_{\nu_{4}}}$ by the
standard convention in Fig. 2.

Then unitarity of the connection $W_{1}(\lambda,\mu)$ is represented as in
Fig. 8, 9. Here the bars above the right squares denote the complex
conjugates.

$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$$T_{4}$$T_{1}$$T_{3}$$T_{2}$$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu^{\prime}_{2}$$T_{4}$$T^{\prime}_{1}$$T_{3}$$T^{\prime}_{2}$$\displaystyle\sum_{\nu_{3},T_{3},T_{4}}$$=\delta_{T_{1},T^{\prime}_{1}}\delta_{T_{2},T^{\prime}_{2}}$
Figure 8: Unitarity (1) for the connection $W_{1}(\lambda,\mu)$
$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$$T_{4}$$T_{1}$$T_{3}$$T_{2}$$\nu^{\prime}_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$$T^{\prime}_{4}$$T_{1}$$T^{\prime}_{3}$$T_{2}$$\displaystyle\sum_{\nu_{2},T_{1},T_{2}}$$=\delta_{T_{3},T^{\prime}_{3}}\delta_{T_{4},T^{\prime}_{4}}$
Figure 9: Unitarity (2) for the connection $W_{1}(\lambda,\mu)$

Crossing symmetry for the connection $W_{1}(\lambda,\mu)$ is given as in Fig.
10, where $\tilde{T}_{1}$ and $\tilde{T}_{4}$ stand for the Frobenius duals of
$T_{1}$ and $T_{4}$, respectively. This is a well-known relation arising from
the tetrahedral symmetry of the $6j$-symbols as in [15, Definition 12.15], but
we include a simple argument for this, because we need a similar argument
later for $W_{3}(\lambda)$ as in Fig. 21.

$\nu_{4}$$\nu_{3}$$\nu_{2}$$\nu_{1}$$\tilde{T}_{4}$$\tilde{T}_{1}$$T_{2}$$T_{3}$$\nu_{3}$$\nu_{4}$$\nu_{1}$$\nu_{2}$$T_{4}$$T_{1}$$T_{3}$$T_{2}$$=\displaystyle\sqrt{\frac{d_{\nu_{1}}d_{\nu_{4}}}{d_{\nu_{2}}d_{\nu_{3}}}}$
Figure 10: Crossing symmetry for the connection $W_{1}(\lambda,\mu)$
$\nu_{3}$$\nu_{2}$$\lambda$$\nu_{1}$$\mu$$\bar{\nu}_{4}$ Figure 11: A vertical
reflection of Fig. 7

We first make a vertical reflection of the diagram in Fig. 7 to obtain Fig.
11. We then redraw Fig. 11 to obtain Fig. 12, which represents the complex
number given by the diagram in Fig. 13 multiplied by
$\sqrt{d_{\lambda}d_{\nu_{2}}d_{\mu}d_{\nu_{3}}}$.

$\nu_{4}$$\nu_{1}$$\lambda$$\nu_{2}$$\bar{\mu}$$\bar{\nu}_{3}$ Figure 12:
Redrawing of Fig. 11
$\nu_{4}$$\nu_{3}$$\nu_{2}$$\nu_{1}$$\tilde{T}_{4}$$\tilde{T}_{1}$$T_{2}$$T_{1}$$W_{1}(\lambda,\bar{\mu})$
Figure 13: The diagram for a connection $W_{1}(\lambda,\bar{\mu})$

This shows we have crossing symmetry, Fig. 10. This, together with unitarity
of the left hand side of Fig. 10, shows bi-unitarity of the connection
$W_{1}(\lambda,\mu)$.

We now prove that the fusion category of endomorphisms of $N$ arising from
$\Delta$ and the one arising from the connections $W_{1}(\lambda,\mu)$ for
various $\lambda$ and fixed $\mu$ are equivalent. This is where we need
Assumption 2.2. Then we have the following theorem.

###### Theorem 3.2

Under Assumptions 2.1 and 2.2, the fusion category arising from the
connections $W_{1}(\lambda,\mu)$ for all $\lambda\in\Delta$ is equivalent to
the one arising from endomorphisms $\lambda\in\Delta$ of $N$.

  * Proof.

By the description of the intertwiners between open string bimodules in the
proof of [1, Theorem 3], we have a natural injective linear map from
$\mathop{\mathrm{Hom}}\nolimits(\lambda_{1}\lambda_{2},\lambda_{3})$ to
$\mathop{\mathrm{Hom}}\nolimits(W_{1}(\lambda_{1},\mu))W_{1}(\lambda_{2},\mu)),W_{1}(\lambda_{3},\mu))$
for $\lambda_{1},\lambda_{2},\lambda_{3}\in\Delta$. The morphism space
$\mathop{\mathrm{Hom}}\nolimits(W_{1}(\lambda_{1},\mu))W_{1}(\lambda_{2},\mu)),W_{1}(\lambda_{3},\mu))$
is described with the higher relative commutants of the corresponding
subfactor by the proof of [1, Theorem 3]. These higher relative commutants are
described with $C_{k,-1}$ for the subfactor $M_{0}\subset M_{1}$ in the proof
of [48, Theorem 3.3]. They are described with the intertwiner spaces of the
$N$-$N$ morphisms as in the proof of [48, Theorem 3.3]. This means that the
dimensions of the two intertwiner spaces
$\mathop{\mathrm{Hom}}\nolimits(\lambda_{1}\lambda_{2},\lambda_{3})$ and
$\mathop{\mathrm{Hom}}\nolimits(W_{1}(\lambda_{1},\mu))W_{1}(\lambda_{2},\mu)),W_{1}(\lambda_{3},\mu))$
are the same, and thus the above natural linear map is surjective. The
compositions of intertwiners in the two fusion categories are also compatible
with this identification, so they are equivalent.  $\square$

Then we see that the structure of the braided fusion category of these
endomorphisms of $N$ passes to that of these connections.

We next introduce two types of connections. The first is the easier one
$W_{2}(\mu)$.

###### Definition 3.3

Let $\nu_{1},\nu_{2}\in\Delta$ and $a_{1},a_{2}$ be irreducible $M$-$N$
morphisms arising from $\Delta$ and the subfactor $N\subset M$. Fig. 14
represents the complex number
$T_{4}T_{3}\iota(T_{1})^{*}T_{2}^{*}\in\mathop{\mathrm{Hom}}\nolimits(a_{2},a_{2})$
for isometries $T_{1}\in\mathop{\mathrm{Hom}}\nolimits(\nu_{1}\mu,\nu_{2})$,
$T_{2}\in\mathop{\mathrm{Hom}}\nolimits(\iota\nu_{2},a_{2})$,
$T_{3}\in\mathop{\mathrm{Hom}}\nolimits(\iota\nu_{1},a_{1})$,
$T_{4}\in\mathop{\mathrm{Hom}}\nolimits(a_{1}\nu_{2},a_{2})$, where $a_{1}$
and $a_{2}$ are $M$-$N$ morphisms. We define the connection $W_{2}(\mu)$ by
this number, and use the diagram as in Fig. 15 to represent this connection.

Note that Remark after Definition 3.1 applies here again. For example, the
left vertical bipartite graph in this Definition is defined as follows. One
set of vertices is given by $\Delta$ and the other is given by the
representatives of the irreducible $M$-$N$ morphisms. The number of edges
between the vertices $\nu_{1}\in\Delta$ and an $M$-$N$ morphism $a_{1}$ is
given by $\dim\mathop{\mathrm{Hom}}\nolimits(\iota\nu_{1},a_{1})$.

$a_{1}\mu$$a_{2}$$\iota\nu_{1}\mu$$\iota\nu_{2}$$\iota(T_{1})$$T_{4}$$T_{3}$$T_{2}$
Figure 14: The diagram for the connection $W_{2}(\mu)$
$a_{1}$$a_{2}$$\nu_{1}$$\nu_{2}$ Figure 15: The standard diagram for a
connection $W_{2}(\mu)$

We next introduce the other connection, $W_{3}(\lambda)$.

###### Definition 3.4

Let $\nu_{1},\nu_{2}\in\Delta$ and $a_{1},a_{2}$ be irreducible $M$-$N$
morphisms arising from $\Delta$ and the subfactor $N\subset M$. Fig. 16
represents the complex number
$T_{4}\alpha_{\lambda}^{+}(T_{3}){\mathcal{E}}^{+}(\lambda,\iota)^{*}\iota(T_{1})^{*}T_{2}^{*}\in\mathop{\mathrm{Hom}}\nolimits(a_{2},a_{2})$
for isometries
$T_{1}\in\mathop{\mathrm{Hom}}\nolimits(\lambda\nu_{1},\nu_{2})$,
$T_{2}\in\mathop{\mathrm{Hom}}\nolimits(\iota\nu_{2},\nu_{4})$,
$T_{3}\in\mathop{\mathrm{Hom}}\nolimits(\iota\nu_{1},a_{1})$,
$T_{4}\in\mathop{\mathrm{Hom}}\nolimits(\alpha_{\lambda}^{+}a_{1},a_{2})$,
which is represented with the connection diagram in Fig. 17. We define the
connection $W_{3}(\lambda)$ by this number. Here ${\mathcal{E}}^{\pm}$ is
defined on page 455 below (14) in [7] and we recall Fig. 18, which is taken
from [7, Fig. 30].

$\alpha^{+}_{\lambda}a_{1}$$\alpha^{+}_{\lambda}\iota\nu_{1}$$a_{2}$$\iota\lambda\nu_{1}$$\iota\nu_{2}$$\iota(T_{1})$$T_{4}$${\mathcal{E}}^{+}(\lambda,\iota)^{*}$$\alpha_{\lambda}^{+}(T_{3})$$T_{2}$
Figure 16: The diagram for the connection $W_{3}(\lambda)$
$a_{1}$$a_{2}$$\nu_{1}$$\nu_{2}$ Figure 17: The standard diagram for a
connection $W_{3}(\lambda)$ $\alpha_{\lambda}^{+}$$\iota$$\iota$$\lambda$
Figure 18: The braiding operator ${\mathcal{E}}^{+}(\lambda,\iota)^{*}$
$a_{2}$$\nu_{2}$$\iota$$\lambda$$\nu_{1}$$\alpha_{\lambda}^{+}$$a_{1}$ Figure
19: The diagram for the connection $W_{3}(\lambda)$ in Fig. 17

The complex number represented by the diagram in Fig. 19 is equal to the one
represented by the connection diagram in Fig. 17 multiplied by
$\sqrt{d_{\iota}d_{\lambda}d_{\nu_{1}}d_{a_{2}}}$ by the standard convention
in Fig. 2. Note that we drop orientations of wires which go from the top to
the bottom in Fig. 19.

We make a vertical reflection of Fig. 19 to get Fig. 20. We again drop
orientations of wires which go from the top to the bottom. Note that the
complex number values given by these two diagrams are complex conjugate to
each other.

$a_{2}$$a_{1}$$\alpha^{+}_{\lambda}$$\iota$$\lambda$$\nu_{1}$$\nu_{2}$ Figure
20: A vertical reflection of Fig. 19

By redrawing Fig. 20, we have Fig. 21.

$a_{1}$$\nu_{1}$$\iota$$\bar{\lambda}$$\nu_{2}$$\alpha_{\bar{\lambda}}^{+}$$a_{2}$
Figure 21: Redrawing of Fig. 20

The complex number represented by the diagram in Fig. 21 is equal to the one
represented by the connection diagram in Fig. 22 multiplied by
$\sqrt{d_{\lambda}d_{\iota}d_{\nu_{2}}d_{a_{1}}}$ This means that the value of
the connection diagram in Fig.22 for $W_{3}(\bar{\lambda})$ is equal to the
complex conjugate of the value of the connection diagram in Fig.17 for
$W_{3}(\lambda)$ multiplied by
$\displaystyle\sqrt{\frac{d_{\nu_{1}}d_{a_{2}}}{d_{\nu_{2}}d_{a_{1}}}}$. This
proves bi-unitarity of $W_{3}(\lambda)$, as Fig. 10 gave bi-unitarity of
$W_{1}(\lambda,\mu)$.

$a_{2}$$a_{1}$$\nu_{2}$$\nu_{1}$ Figure 22: The standard diagram for a
connection $W_{3}(\bar{\lambda})$

We next introduce the connection $W_{4}(\alpha^{+}_{\lambda},\mu)$, which is
the $\alpha^{+}$-induced connection.

###### Definition 3.5

Let $a_{1},a_{2},a_{3},a_{4}$ be irreducible $M$-$N$ morphisms arising from
$\Delta$ and the subfactor $N\subset M$. Consider the diagram in Fig. 23. By
composing isometries $T_{1}\in\mathop{\mathrm{Hom}}\nolimits(a_{1}\mu,a_{2})$,
$T_{2}\in\mathop{\mathrm{Hom}}\nolimits(\alpha_{\lambda}^{+}a_{2},a_{4})$,
$T_{3}\in\mathop{\mathrm{Hom}}\nolimits(\alpha_{\lambda}^{+}a_{1},a_{3})$, and
$T_{4}\in\mathop{\mathrm{Hom}}\nolimits(a_{3}\mu,a_{4})$, we obtain a complex
number
$T_{4}T_{3}\alpha_{\lambda}^{+}(T_{1}^{*})T_{2}^{*}\in\mathop{\mathrm{Hom}}\nolimits(a_{4},a_{4})$.
We define the connection $W_{4}(\alpha^{+}_{\lambda},\mu)$ by this number and
represent this as in Fig. 24.

$a_{3}\mu$$a_{4}$$\alpha_{\lambda}^{+}a_{1}\mu$$\alpha_{\lambda}^{+}a_{2}$$\alpha_{\lambda}^{+}(T_{1})$$T_{4}$$T_{3}$$T_{2}$
Figure 23: The diagram for the connection $W_{4}(\alpha_{\lambda}^{+},\mu)$ in
Fig. 5 $a_{3}$$a_{4}$$a_{1}$$a_{2}$ Figure 24: The standard diagram for a
connection $W_{4}(\alpha^{+}_{\lambda},\mu)$

By a very similar argument to the one for bi-unitarity of
$W_{1}(\lambda,\mu)$, we obtain bi-unitarity of
$W_{4}(\alpha^{+}_{\lambda},\mu)$.

We then have the intertwining Yang-Baxter equation for $W_{1}(\lambda,\mu)$,
$W_{3}(\lambda)$, $W_{3}(\mu)$, and $W_{4}(\alpha^{+}_{\lambda},\mu)$ as in
Fig. 25, which was given in [26, Axiom 7]. The meaning of this diagram is as
follows. On the both hand sides of the identity, the six isometries are fixed
for the six boundary edges of the hexagons in the same way. The left hand side
means the summation of the product of the three connection values
$W_{1}(\lambda,\mu)$, $W_{3}(\lambda)$, and $W_{2}(\mu)$ over all possible
choices of the three isometries corresponding to the three internal edges of
the hexagon. The right hand side means the summation of the product of the
three connection values $W_{3}(\lambda)$, $W_{2}(\mu)$, and
$W_{4}(\alpha^{+}_{\lambda},\mu)$ over all possible choices of the three
isometries corresponding to the three internal edges of the hexagon. The both
hand sides are equal because they are both equal to the composition of the six
(co-)isometries corresponding to the six boundary edges of the hexagons.

$=$$W_{2}(\mu)$$W_{1}(\lambda,\mu)$$W_{3}(\lambda)$$W_{3}(\lambda)$$W_{4}(\alpha^{+}_{\lambda},\mu)$$W_{2}(\mu)$
Figure 25: The intertwining Yang-Baxter equation for $W_{1}(\lambda,\mu)$,
$W_{3}(\lambda)$, $W_{3}(\mu)$, and $W_{4}(\alpha^{+}_{\lambda},\mu)$

We have seen $\alpha$-induction applied to $N\subset M$ produces bi-unitary
connections. From this construction and the description of intertwiner spaces
for connections as in [1, Theorem 3] again, we have the following theorem
under Assumption 2.2 with the same arguments as in the proof of Theorem 3.2.

###### Theorem 3.6

Under Assumptions 2.1 and 2.2 for a fixed $\mu$, the fusion category arising
from the connections $W_{4}(\alpha_{\lambda}^{\pm},\mu)$ for all
$\lambda\in\Delta$ is equivalent to the one arising from endomorphisms
$\alpha_{\lambda}^{\pm}$ of $M$ given by $\alpha$-induction.

Note that it is nontrivial that we have only finite many irreducible
endomorphisms of $M$ up to equivalence when we consider those arising from
$\alpha_{\lambda}^{\pm}$. This has been proved in [7, Theorem 5.10].

##  4 Rewriting $W_{4}(\alpha^{+}_{\lambda},\mu)$ and switching of positive
and negative braiding

Though we have defined the $\alpha$-induced connection
$W_{4}(\alpha^{+}_{\lambda},\mu)$ as in Fig. 23, we need to know full
information about $\alpha^{+}_{\lambda}$ to compute this value. Since we are
now in the process of defining the new connection
$W_{4}(\alpha^{+}_{\lambda},\mu)$ before knowing $\alpha^{+}_{\lambda}$, we
certainly hope to compute this number without using the information about
$\alpha^{+}_{\lambda}$. We show in this Section that this is possible.

Fig. 26 represents the complex number

$T_{4}T^{\prime}_{3}{\mathcal{E}}^{+}(\lambda,a_{1})\alpha_{\lambda}^{+}(T_{1})^{*}{\mathcal{E}}^{+}(\lambda,a_{2})^{*}{T^{\prime}_{2}}^{*}\in\mathop{\mathrm{Hom}}\nolimits(a_{4},a_{4})$

for isometries $T_{1}\in\mathop{\mathrm{Hom}}\nolimits(a_{1}\mu,a_{2})$,
$T^{\prime}_{2}\in\mathop{\mathrm{Hom}}\nolimits(a_{2}\lambda,a_{4})$,
$T^{\prime}_{3}\in\mathop{\mathrm{Hom}}\nolimits(a_{1}\lambda,a_{3})$,
$T_{4}\in\mathop{\mathrm{Hom}}\nolimits(a_{3}\mu,a_{4})$, which is represented
with the connection diagram in Fig. 24. This composition corresponds to the
diagram in Fig. 27 up to normalization constant, and we can redraw this as in
Fig. 28. Note that we drop orientations of wires which go from the top to the
bottom in Fig. 27, 28. The complex number represented by the diagram in Fig.
28 is equal to the one represented by the connection diagram in Fig. 26
multiplied by $\sqrt{d_{\lambda}d_{\mu}d_{a_{1}}d_{a_{4}}}$. Since Fig. 28
does not involve $\alpha_{\lambda}^{+}$, this diagram gives
$W_{4}(\alpha_{\lambda}^{+},\mu)$, up to normalization constant, in terms of
$N$-$N$ and $M$-$N$ morphisms and intertwiners including the braiding
operators. This is what we asked for at the beginning of this Section.

$a_{3}\mu$$a_{1}\lambda\mu$$\alpha_{\lambda}^{+}a_{1}\mu$$a_{4}$$a_{2}\lambda$$\alpha_{\lambda}^{+}a_{2}$$\alpha_{\lambda}^{+}(T_{1})$$T_{4}$${\mathcal{E}}^{+}(\lambda,a_{1})$$T^{\prime}_{3}$${\mathcal{E}}^{+}(\lambda,a_{2})$$T^{\prime}_{2}$
Figure 26: The diagram for the connection $W_{4}(\alpha^{+}_{\lambda},\mu)$
$\lambda$$a_{2}$$\alpha_{\lambda}^{+}$$a_{1}$$\lambda$$\mu$$a_{3}$$a_{4}$
Figure 27: The diagram for the connection in Fig. 26
$a_{4}$$a_{2}$$a_{1}$$\mu$$\lambda$$a_{3}$ Figure 28: Redrawing of Fig. 27

We next study what the effect of switching the positive and negative braiding
is. Vertical reflection of Fig. 28 gives Fig. 29. The complex numbers given by
these two Figures are mutually complex conjugate. By comparing these two
Figures, we have the following Proposition.

$a_{4}$$a_{3}$$a_{1}$$\lambda$$\mu$$a_{2}$ Figure 29: Vertical reflection of
Fig. 28

###### Proposition 4.1

We have the identity as in Fig. 30.

$a_{3}$$a_{1}$$a_{4}$$a_{2}$$W_{4}(\alpha_{\lambda}^{+},\mu)$$=$$W_{4}(\alpha_{\mu}^{-},\lambda)$$a_{2}$$a_{1}$$a_{4}$$a_{3}$
Figure 30: Complex conjugate and opposite braiding

We also show that we can rewrite $W_{3}(\lambda)$ into the form without using
$\alpha_{\lambda}^{+}$ as follows.

$\nu_{2}$$\iota$$\lambda$$\nu_{1}$$a_{1}$$\alpha_{\lambda}^{+}$$\lambda$$a_{2}$
Figure 31: Redrawing of $W_{3}(\lambda)$
$\nu_{2}$$\iota$$\lambda$$\nu_{1}$$a_{1}$$a_{2}$ Figure 32: Further redrawing
of $W_{3}(\lambda)$

##  5 Locality of the $Q$-system and flatness of
$W_{4}(\alpha_{\lambda}^{+},\mu)$

In the case of Ocneanu’s construction [45], he obtained flat connections only
for $A_{n}$, $D_{2n}$, $E_{6}$ and $E_{8}$ as we see in the next Section in
detail. (Note that Assumption 2.2 does not hold for these examples. We will
treat this issue in the following Section. Also see [15, Theorem 11.24].) It
has been observed that these cases exactly correspond to the $Q$-systems with
locality ${\varepsilon}(\theta,\theta)\gamma(v)=\gamma(v)$ as in [8, Section
5] (where this property was called chiral locality). So we expect some
relations between locality of the $Q$-system and flatness of the corresponding
$\alpha^{\pm}$-induced connection in general. We show that this is indeed the
case in this Section.

We now assume locality. Note that we then have irreducibility of
$\iota:N\hookrightarrow M$ by [4, Corollary 3.6]. (Having a $Q$-system with
locality in a general modular tensor category is enough in [4] rather than a
conformal net.)

We prove flatness of the connection $W_{4}(\alpha_{\lambda}^{+},\mu)$ in the
sense of Fig. 33, which is taken from [15, Fig. 11.21]. We first need a lemma.

$*$$*$$*$$*$$=1$$\rho$$\rho$$\sigma$$\sigma$ Figure 33: Flatness as in [15,
Fig. 11.21]

###### Lemma 5.1

Suppose $\lambda,\mu$ are irreducible endomorphisms of $N$ contained in
$\theta$. We then have the identity as in Fig. 34.

$\iota$$\iota$$=$$\lambda$$\mu$$\lambda$$\mu$ Figure 34: A consequence of
locality

  * Proof.

Locality gives Fig. 35, where the triple vertices on the both hand sides
represent $\gamma(v)$. We then have the identity in Fig. 36, where small black
circles represent co-isometries in
$\mathop{\mathrm{Hom}}\nolimits(\theta,\lambda)$ and
$\mathop{\mathrm{Hom}}\nolimits(\theta,\mu)$. We next have the identity in
Fig. 37 by a property of braiding and then the identity in Fig. 38 by
rewriting $\theta$ with $\bar{\iota}\iota$. Pulling the thick wires straight
and rotating this diagram for $90$ degrees gives the desired conclusion.
$\square$

$=$$\theta$$\theta$$\theta$$\theta$$\theta$$\theta$ Figure 35: Locality of a
$Q$-system
$=$$\theta$$\theta$$\theta$$\theta$$\theta$$\theta$$\lambda$$\mu$$\lambda$$\mu$
Figure 36: A consequence of Fig. 35
$=$$\theta$$\theta$$\theta$$\theta$$\theta$$\theta$$\lambda$$\mu$$\lambda$$\mu$
Figure 37: A consequence of Fig. 36
$=$$\bar{\iota}$$\iota$$\iota$$\lambda$$\mu$$\bar{\iota}$$\iota$$\iota$$\lambda$$\mu$
Figure 38: A consequence of Fig. 37

###### Theorem 5.2

Under Assumption 2.1, the connection $W_{4}(\alpha_{\lambda}^{+},\mu)$ is
flat.

  * Proof.

We prove the identity in Fig. 33. Both horizontal and vertical sizes of the
large diagram in Fig. 33, are supposed to be even, but we can take both of
them to be $1$ in our current setting, so we first give a proof for this case.

Now our $*$ vertex in Fig. 33 corresponds to $\iota$. Because all the four
corner vertices in Fig. 33 are now $\iota$, we set
$a_{1}=a_{2}=a_{3}=a_{4}=\iota$ in Fig. 28. We then have Fig. 39 and the
complex value this diagram represents is equal to the one given by the
partition function in Fig. 33 multiplied by
$\sqrt{d_{\lambda}d_{\mu}}d_{\iota}$.

By Lemma 5.1, the value Fig. 39 represents is equal to the one Fig. 40
represents. The latter is equal to $\sqrt{d_{\lambda}d_{\mu}}d_{\iota}$, so
the complex value given by the partition function in Fig. 33 is $1$.

When the horizontal and vertical sizes of the large diagram in Fig. 33 are
arbitrary, we replace $\lambda$ and $\mu$ in the above arguments by
$\bar{\lambda}\lambda\cdots\lambda$ and $\mu\bar{\mu}\cdots\bar{\mu}$. Then
the same arguments give the value $1$ and we are done.  $\square$

$\iota$$\iota$$\iota$$\mu$$\lambda$$\iota$ Figure 39: A diagram for the
connection $W_{4}(\alpha_{\lambda}^{+},\mu)$ for
$a_{1}=a_{2}=a_{3}=a_{4}=\iota$ $\iota$$\iota$$\iota$$\iota$$\mu$$\lambda$
Figure 40: Redrawing of Fig. 39 with Lemma 5.1

##  6 Examples

Ocneanu considered connections on $A$-$D$-$E$ Dynkin diagrams in [45]. We
revisit this topic from our viewpoint now and would like to apply the results
in Section 3. Here we have a nontrivial issue since Assumption 2.2 does not
hold now for any choice of $\mu$.

Let $\Delta$ be the set of endomorphisms of a type III factor $N$
corresponding to the Wess-Zumino-Witten model $SU(2)_{k}$, where $k$ is a
positive integer called a level. (See [12, Subsection 16.2.3], for example.)
We label the irreducible objects of the modular tensor category with
$0,1,2,\dots,k$, using the Dynkin diagram of type $A_{k+1}$ as in Fig. 41,
where the label $0$ denotes the vacuum representation, that is, the identity
automorphism of $N$. Such a system of endomorphisms with braiding has been
constructed from a conformal net by Wassermann [50] and this braiding is
nondegenerate by [34, Corollary 37] and [52, Theorem 4.1].

$0$$1$$2$$3$$k-1$$k$ Figure 41: Irreducible objects for the $SU(2)_{k}$ WZW-
model

All $Q$-systems on $\Delta$ for all $k$ have been classified in [33, Section
2.5] for the local case and in [35, Section 2] for the general case. They are
classified with a pair consisting one of the $A$-$D$-$E$ Dynkin diagrams and
its vertex. We explain this from a viewpoint of the Goodman-de la Harpe-Jones
subfactor in [18, Section 4.5].

Let ${\mathcal{G}}$ be one of the $A$-$D$-$E$ Dynkin diagrams and choose the
vertex with the smallest entry of the Perron-Frobenius eigenvector entry. We
then have the Goodman-de la Harpe-Jones subfactor as in [18, Section 4.5] or
[15, Section 11.6], and the corresponding $Q$-system. If ${\mathcal{G}}$ is of
type $A$, then this $Q$-system has index $1$ and is trivial. If
${\mathcal{G}}$ is of type $D$, then the $Q$-system has index $2$ and
corresponds to a crossed product by $\mathbb{Z}/2\mathbb{Z}$. If
${\mathcal{G}}$ is $E_{6}$ or $E_{8}$, then the $Q$-systems correspond to the
subfactors arising from conformal embeddings $SU(2)_{10}\subset S(5)_{1}$ and
$SU(2)_{28}\subset(\mathrm{G}_{2})_{1}$ by [8, Proposition A.3]. Also see [8,
Appendix] for the case of $E_{7}$.

We next choose $\lambda=\mu=1$ in the setting of Section 3 and consider the
connection $W_{4}(\alpha_{\lambda}^{\pm},\mu)$. Since the irreducible $N$-$M$
morphisms are labeled with the vertices of ${\mathcal{G}}$ by the arguments in
[15, Section 11.6], all the four vertex sets for
$W_{4}(\alpha_{\lambda}^{\pm},\mu)$ are also labeled with the vertices of
${\mathcal{G}}$. Then the requirements for the Perron-Frobenius eigenvalues
and the Perron-Frobenius eigenvector entries force all the four graphs of
$W_{4}(\alpha_{\lambda}^{\pm},\mu)$ to be ${\mathcal{G}}$, but Assumption 2.2
does not hold, since the horizontal top and bottom graphs are never connected,
due to a $\mathbb{Z}/2\mathbb{Z}$-grading on the vertices of the Dynkin
diagrams. This issue is resolved as follows.

The connection $W_{4}(\alpha_{\lambda}^{\pm},\mu)$ splits into two connections
on mutually disjoint graphs, both of which are isomorphic to ${\mathcal{G}}$.
Then both of the two connections must be of the following form as given in
[44]. (Also see [15, Fig. 11.32].) Let $n$ be its Coxeter number of
${\mathcal{G}}$ and set
$\displaystyle\varepsilon=\sqrt{-1}\exp\frac{\pi\sqrt{-1}}{2(n+1)}$. We write
$\mu_{x}$ for the Perron-Frobenius eigenvector entry for a vertex $x$. Then
our connection is given as follows. (We can replace ${\varepsilon}$ with
$\bar{\varepsilon}$. By the arguments in [15, Section 11.5], these two choices
give the only possible connections on ${\mathcal{G}}$. If ${\mathcal{G}}$ is
of type $A$, they give equivalent connections. If ${\mathcal{G}}$ is of type
$D$ or $E$, they are not mutually equivalent. Here we allow only vertical
gauges while horizontal gauges are also allowed in [15, Section 11.5], but
this does not cause problems since our graph ${\mathcal{G}}$ is a tree. See
Remark after [1, Theorem 3].)

$W$$l$$j$$m$$k$$\displaystyle=\delta_{kl}\varepsilon+\sqrt{\frac{\mu_{k}\mu_{l}}{\mu_{j}\mu_{m}}}\delta_{jm}\bar{\varepsilon}$
Figure 42: A connection on the Dynkin diagram

The horizontal top and bottom graphs for $W_{4}(\alpha_{j}^{\pm},1)$ always
have exactly two connected components. This is still valid after we make
irreducible decomposition of such connections. Let $W$ be a pair of two such
connections $W_{a}$ and $W_{b}$ arising from irreducible decomposition of
$W_{4}(\alpha_{j}^{\pm},1)$ for some $j$ and $W^{\prime}$ be a pair of two
such connections $W^{\prime}_{a}$ and $W^{\prime}_{b}$ from
$W_{4}(\alpha_{k}^{\pm},1)$ for some $k$. We can compose $W_{a}$ with exactly
one of $W^{\prime}_{a}$ and $W^{\prime}_{b}$, and compose $W_{b}$ with the
other one, since they have the matching horizontal bottom and top graphs. For
the fusion rules and intertwiner spaces of this composition product, we can
use either of the two compositions and have the same results, by the same
arguments to the ones in the proof of Theorem 3.6. In this way, we obtain a
fusion category of connections which is equivalent to the one of $M$-$M$
morphisms arising from $\alpha^{\pm}$-induction. We now obtain the diagrams of
decompositions of connections which are the same as in [6, Fig. 2, 5, 8, 9]
and [8, Fig. 40, 42]. These were originally found by Ocneanu for such
connections. We also know that the results in the previous Section on flatness
apply to these cases, though Assumption 2.2 does not hold now. For this type
of computations, we do not need exact information of the connections, and
simply having the graphs involved is often sufficient. See [19] for such
computations.

We now discuss the issue of complex conjugate connections. For
$W_{4}(\alpha_{1}^{\pm},1)$, the connection Fig. 42 is symmetric in $j$ and
$m$, so the effect of switching positive and negative braiding in Proposition
4.1 now amounts to taking simply complex conjugate connections. That is,
$W_{4}(\alpha_{1}^{+},1)$ and $W_{4}(\alpha_{1}^{-},1)$ are mutually complex
conjugate. Now consider the case of ${\mathcal{G}}=E_{6}$. The connections
$W_{4}(\alpha_{2}^{+},1)$ and $W_{4}(\alpha_{2}^{-},1)$ are also mutually
complex conjugate. Both of the connections $W_{4}(\alpha_{3}^{+},1)$ and
$W_{4}(\alpha_{3}^{-},1)$ decomposes into two irreducible connections each.
Since irreducible decomposition of a complex conjugate connection gives
complex conjugate connections of those appearing in the irreducible
decomposition in general, the complex conjugates of the two irreducible
connections arising from $W_{4}(\alpha_{3}^{+},1)$ appear in the irreducible
decomposition of $W_{4}(\alpha_{3}^{-},1)$. In this way, we see that switching
the positive and negative braiding for the $\alpha$-induced connections
amounts to taking complex conjugate connections. The same argument also works
for $E_{7}$ and $E_{8}$. This was also observed by Ocneanu [45], but is
special to the $A$-$D$-$E$ Dynkin diagrams. The decomposition rules for
$E_{7}$ as in [8, Fig. 42] follows from our computations in [14] showing that
the principal graph of the subfactor arising from the $E_{7}$ connection is
$D_{10}$.

The modular tensor categories corresponding to the Wess-Zumino-Witten model
$SU(N)_{k}$ also have $\mathbb{Z}/N\mathbb{Z}$-grading, and a similar method
to the above gives how to handle this issue.

We add a remark on the effect of Assumption 2.2. Even without this Assumption,
our Definitions 3.1, 3.3, 3.4, 3.5 on our new connections make sense. The only
problem is that if horizontal graphs are disconnected, we cannot compose two
connections in a usual way in Theorem 3.6, and other results such as Theorem
5.2 are not affected. Another way of handling the issue in Assumption 2.2 is
to make $\Delta$ smaller. For example, if we choose $\lambda=\mu=2$ in the
numbering of irreducible objects for the fusion category $SU(2)_{k}$ as in
Fig. 41 and consider only irreducible objects numbered with even integers,
then everything in the above Sections works fine.

##  7 Triple sequence of string algebras and another interpretation of the
$\alpha$-induction in terms of bi-unitary connections

Let $*$ be the vertex corresponding to the identity automorphism of $N$ in
$\Delta$. We construct a triple sequence $\\{A_{jkl}\\}_{jkl}$ as in [26,
Section 2]. This is a triple sequence version of the standard construction of
the double sequence of string algebras in [15, Section 11.3]. A new property
we need for compatibility of identification is the Intertwining Yang-Baxter
Equations as in [26, Axiom 7], and we now have this identity as in Fig. 25.

For the commuting squares

$\displaystyle\begin{array}[]{ccc}A_{2j,2k,2l}&\subset&A_{2j,2k+1,2l}\\\
\cap&&\cap\\\ A_{2j+1,2k,2l}&\subset&A_{2j+1,2k+1,2l}\end{array},$
$\displaystyle\begin{array}[]{ccc}A_{2j,2k,2l+1}&\subset&A_{2j,2k+1,2l+1}\\\
\cap&&\cap\\\ A_{2j+1,2k,2l+1}&\subset&A_{2j+1,2k+1,2l+1}\end{array},$
$\displaystyle\begin{array}[]{ccc}A_{2j,2k,2l}&\subset&A_{2j,2k+1,2l}\\\
\cap&&\cap\\\ A_{2j,2k,2l+1}&\subset&A_{2j,2k+1,2l}\end{array},$
$\displaystyle\begin{array}[]{ccc}A_{2j+1,2k,2l}&\subset&A_{2j+1,2k+1,2l}\\\
\cap&&\cap\\\ A_{2j+1,2k,2l+1}&\subset&A_{2j+1,2k+1,2l}\end{array},$
$\displaystyle\begin{array}[]{ccc}A_{2j,2k,2l}&\subset&A_{2j+1,2k,2l}\\\
\cap&&\cap\\\ A_{2j,2k,2l+1}&\subset&A_{2j+1,2k,2l+1}\end{array},$
$\displaystyle\begin{array}[]{ccc}A_{2j,2k+1,2l}&\subset&A_{2j+1,2k+1,2l}\\\
\cap&&\cap\\\ A_{2j,2k+1,2l+1}&\subset&A_{2j+1,2k+1,2l+1}\end{array},$

we use the connections $W_{1}(\lambda,\mu)$,
$W_{4}(\alpha_{\lambda}^{+},\mu)$, $W_{2}(\mu)$, $W_{2}(\mu)$,
$W_{3}(\lambda)$, and $W_{3}(\lambda)$, respectively. We then have the
following identification between the string algebras and the endomorphism
spaces.

$\displaystyle A_{2j,2k,2l}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits((\bar{\lambda}\lambda)^{j}(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}),$
$\displaystyle A_{2j,2k+1,2l}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits((\bar{\lambda}\lambda)^{j}(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}\mu),$
$\displaystyle A_{2j+1,2k,2l}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits(\lambda(\bar{\lambda}\lambda)^{j}(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}),$
$\displaystyle A_{2j+1,2k+1,2l}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits(\lambda(\bar{\lambda}\lambda)^{j}(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}\mu),$
$\displaystyle A_{2j,2k,2l+1}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits((\alpha_{\bar{\lambda}}^{+}\alpha_{\lambda}^{+})^{j}\iota(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}),$
$\displaystyle A_{2j,2k+1,2l+1}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits((\alpha_{\bar{\lambda}}^{+}\alpha_{\lambda}^{+})^{j}\iota(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}\mu),$
$\displaystyle A_{2j+1,2k,2l+1}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits(\alpha_{\lambda}^{+}(\alpha_{\bar{\lambda}}^{+}\alpha_{\lambda}^{+})^{j}\iota(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}),$
$\displaystyle A_{2j+1,2k+1,2l+1}$
$\displaystyle=\mathop{\mathrm{End}}\nolimits(\alpha_{\lambda}^{+}(\alpha_{\bar{\lambda}}^{+}\alpha_{\lambda}^{+})^{j}\iota(\bar{\iota}\iota)^{l}(\mu\bar{\mu})^{k}\mu),$

Note that for inclusions such as $A_{2j,2k,2l}\subset A_{2j,2k,2l+1}$, we use
unitary equivalences $\alpha^{+}_{\lambda}\iota\cong\iota\lambda$,
$\alpha^{+}_{\bar{\lambda}}\iota\cong\iota\bar{\lambda}$,
$\lambda\bar{\iota}\cong\bar{\iota}\alpha_{\lambda}^{+}$, and
$\bar{\lambda}\bar{\iota}\cong\bar{\iota}\alpha_{\bar{\lambda}}^{+}$. These
are compatible with ways of identification of strings with the connections.
(The only nontrivial identification is done with $W_{3}(\lambda)$, where we
use unitary equivalence of $\iota\lambda$ and $\alpha_{\lambda}\iota$ as in
Fig.16.)

By taking unions over $k$ and making the GNS-completions with respect to the
compatible trace, we have a commuting square of hyperfinite type II1 factors
as in [26, Assumption 1.1]. We see that our triple sequence of string algebras
arise from this commuting square as in [26, Section 3]. See [26, Section 5]
for the case of the Dynkin diagrams of type $A$-$D$-$E$.

Our construction of $\alpha$-induced connections from $N\subset M$ uses
information on all the $N$-$M$ morphisms and their intertwiners. This is also
true for Ocneanu’s chiral generator picture in Fig. 1 in the double triangle
algebra. This is theoretically fine, but we would like to have a method to
obtain the $\alpha$-induced connections purely in terms of connections. We
discuss such a method at the end of this paper.

In the original setting of the set $\Delta$ of irreducible endomorphisms of
$N$, we fix $\mu\in\Delta$ and suppose we have a family of connections
$W_{1}(\lambda,\mu)$ for $\lambda\in\Delta$ and the horizontal top and bottom
graphs for all of them are the same finite bipartite graph ${\mathcal{G}}$.
Let $*$ be the vertex of ${\mathcal{G}}$ corresponding to the identity
automorphism in $\Delta$. Our positive braiding gives equivalence of the two
composite connections $W_{1}(\lambda_{1},\mu)\cdot W_{1}(\lambda_{2},\mu)$ and
$W_{2}(\lambda_{1},\mu)\cdot W_{1}(\lambda_{1},\mu)$. This is given by Fig.
43, where $S$ and $T$ give unitary matrices giving vertical gauge choices
arising from a positive braiding between $\lambda_{1}$ and $\lambda_{2}$.

$W_{1}(\lambda_{1},\mu)$$W_{1}(\lambda_{2},\mu)$$S$$T$$=$$W_{1}(\lambda_{2},\mu)$$W_{1}(\lambda_{1},\mu)$
Figure 43: Braiding ${\varepsilon}^{+}(\lambda_{1},\lambda_{2})$

Suppose we have a flat connection $W_{0}$ with respect to $*$ (in the sense of
[15, Definition 11.16]) with the horizontal top and bottom graphs being finite
bipartite graphs ${\mathcal{G}}$ and ${\mathcal{H}}$ such that the composition
$W_{0}\cdot\bar{W}_{0}$ decomposes into a direct sum of irreducible
connections each of which is equivalent to one of $W_{1}(\lambda,\mu)$. This
is a connection version of our setting in Section 2 and our connection $W_{0}$
must be of the form $W_{2}(\mu)$ for some $\iota$.

Construct a double sequence of string algebras $\\{B_{kl}\\}_{kl}$ as in [15,
Section 11.3] from the connection $W_{0}$. That is, the commuting square
$\begin{array}[]{ccc}B_{2l,2k}&\subset&B_{2l,2k+1}\\\ \cap&&\cap\\\
B_{2l+1,2k}&\subset&B_{2l+1,2k+1}\end{array}$ is described with $W_{0}$. We
further construct a double sequence of string algebras $\\{C_{kl}\\}_{kl}$ so
that $\begin{array}[]{ccc}B_{2l,2k}&\subset&B_{2l,2k+1}\\\ \cap&&\cap\\\
C_{2l,2k}&\subset&C_{2l,2k+1}\end{array}$ is described with
$W_{1}(\lambda,\mu)$ and $\begin{array}[]{ccc}C_{2l,2k}&\subset&C_{2l,2k+1}\\\
\cap&&\cap\\\ C_{2l+1,2k}&\subset&C_{2l+1,2k+1}\end{array}$ is described with
$W_{0}$ . Because of the braiding between $W_{1}(\lambda,\mu)$ and
$W_{0}\cdot\bar{W}_{0}$, we have compatibility between

$\begin{array}[]{ccc}B_{2l,k}&\subset&B_{2l,k+1}\\\ \cap&&\cap\\\
C_{2l,k}&\subset&C_{2l,k+1}\\\ \cap&&\cap\\\
C_{2l+2,k}&\subset&C_{2l+2,k+1}\end{array}$

and

$\begin{array}[]{ccc}B_{2l,k}&\subset&B_{2l,k+1}\\\ \cap&&\cap\\\
B_{2l+2,k}&\subset&B_{2l+2,k+1}\\\ \cap&&\cap\\\
C_{2l+2,k}&\subset&C_{2l+2,k+1}\end{array}$

as in the case of the Intertwining Yang-Baxter equation.

Consider an element in $B_{2l+1,k}$. It is embedded into $B_{2l+2,k}$ and then
into $C_{2l+2,k}$. It is not a priori clear whether the image in $C_{2l+2,k}$
or not, but actually the image is in $C_{2l+1,k}$ because we are in the
setting of the triple sequence of string algebras introduced in this Section.
This gives an embedding of $B_{2l+1,k}$ into $C_{2l+1,k}$. Then the results in
Section 3 mean that the connection describing the commuting square
$\begin{array}[]{ccc}B_{2l+1,2k}&\subset&B_{2l+1,2k+1}\\\ \cap&&\cap\\\
C_{2l+1,2k}&\subset&C_{2l+1,2k+1}\end{array}$ is
$W_{4}(\alpha_{\lambda}^{+},\mu)$.

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