††institutetext: 1Physics Department, Ariel University, Ariel 40700,
Israel.††institutetext: 2Department of Particle Physics and Astrophysics,
Weizmann Institute, Rehovot 76100, Israel.††institutetext: 3Saint Petersburg
State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia.

# On different approaches to integrable lattice models

Vladimir Belavin1, Doron Gepner2, J. Ramos Cabezas1, Boris Runov1,3
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Interaction-Round the Face (IRF) models are two-dimensional lattice models of
statistical mechanics defined by an affine Lie algebra and admissibility
conditions depending on a choice of representation of that affine Lie algebra.
Integrable IRF models, i.e., the models the Boltzmann weights of which satisfy
the quantum Yang-Baxter equation, are of particular interest. In this paper,
we investigate trigonometric Boltzmann weights of integrable IRF models. By
using an ansatz proposed by one of the authors in some previous works, the
Boltzmann weights of the restricted IRF models based on the affine Lie
algebras $\mathfrak{su}(2)_{k}$ and $\mathfrak{su}(3)_{k}$ are computed for
fundamental and adjoint representations for some fixed levels $k$. New
solutions for the Boltzmann weights are obtained. We also study the vertex-IRF
correspondence in the context of an unrestricted IRF model based on
$\mathfrak{su}(3)_{k}$ (for general $k$) and discuss how it can be used to
find Boltzmann weights in terms of the quantum $\hat{R}$ matrix when the
adjoint representation defines the admissibility conditions.

##  1 Introduction

Integrable lattice models have various connections with two-dimensional
conformal field theories (CFTs). It is known that at criticality, lattice
models are described by CFTs [1]. The well-studied examples of this
relationship include the Ising model, the Yang-Lee edge singularity, the
tricritical Ising model, the three-state Potts model, the eight-vertex SOS
model, etc., for a review, see [2].

In [3, 4, 5], Jimbo et al. introduced a class of Interaction-Round the Face
(IRF) models that generalize the eight-vertex SOS model [6]. It was discovered
that these IRF models, based on the affine Lie algebras $\mathfrak{su}(n)_{k}$
in a special regime111More precisely, the regime III and one-dimensional
limit, for details see [5]. are associated with minimal models corresponding
to the GKO coset
$\frac{\mathfrak{su}(n)_{k}\oplus\mathfrak{su}(n)_{1}}{\mathfrak{su}(n)_{k+1}}$
[7], in the sense that the characters (branching functions) of this coset
coincide with the one-dimensional configuration sums of the IRF models222See
[8, 9, 10] for recent developments regarding this idea..

In [11], an explicit method for constructing the Boltzmann weights (BWs) of
IRF models based on their relationships with CFTs was proposed. The key idea
was that the BWs can be expressed in terms of the matrix elements of the
braiding matrices of CFTs.

Two other approaches can be used to find BWs of IRF models: the fusion
procedure (see, e.g., [12, 13]) and the Vertex-IRF correspondence. In the
first part of the paper (sections 2,3), we focus on the approach proposed in
[11] in the case of restricted IRF models. Then, in section 4, we discuss the
Vertex-IRF correspondence in the context of unrestricted IRF models. All these
approaches can be combined to analyze more complex IRF models. We discuss this
idea in section 5.

We will define restricted IRF models and explain the key concepts associated
with them in section 2. To present our results, below we give a brief
overview. Our study focuses on two IRF models based on the affine Lie algebras
$\mathfrak{su}(2)_{k}$ and $\mathfrak{su}(3)_{k}$. The models are defined on a
square lattice, where one of the dominant integral representations of the
corresponding algebra sits in each vertex. Two ordered neighboring vertices
$a$ and $b$ are considered admissible if the fusion coefficient $N_{a,h}^{b}$
satisfies the condition

$N_{a,h}^{b}>0\;,$ (1.1)

where $N_{a,h}^{b}$ arises in the tensor product

$a\otimes h=\bigoplus_{i}N_{a,h}^{i}\times[i]\;.$ (1.2)

Here $h$ is some fixed representation, which, together with the algebra
itself, defines the IRF model, and $[i]$ denotes the representations appearing
in this tensor product.

In section 3, we review the ansatz proposed in [11]. In section 3.1, we first
consider a warm-up example applying this ansatz to the model based on
$\mathfrak{su}(2)_{k}$ at level $k=2$ with $h$ corresponding to the
fundamental representation. This model was studied by Jimbo et al., [5]. At
this level, there are eight possible configurations yielding nonzero BWs. We
recover the solutions of [5].

In section 3.2, we study the model based on $\mathfrak{su}(2)_{k}$ at level
$k=4$, with $h$ corresponding to the adjoint representation. This model was
investigated by Tartataglia and Pearce in [14]. At level $k=4$, there are 33
possible configurations of the BWs. Using the ansatz mentioned above [11], we
obtain solutions parameterized by four independent parameters that reproduce
the solutions found in [14].

In section 3.3, we consider the model based on $\mathfrak{su}(3)_{k}$ at level
$k=2$, with $h$ corresponding to the adjoint representation. At this level,
the fusion rules of the algebra are free of multiplicities, and there are 21
possible configurations of the BWs. The solutions we found completely follow
the ansatz [11].

In the case of $\mathfrak{su}(3)_{k}$ algebra at level $k\geq 3$, the
multiplicities in the tensor products are present ($N_{a,h}^{b}\geq 2$) and
the approach [11] is not applicable. In this respect, in section 4, we explore
the vertex-IRF correspondence approach, which is based on the $\hat{R}$ matrix
of quantum algebra $\mathfrak{sl}_{q}(3)$. Using this correspondence, one can
express the BWs of unrestricted IRF models in terms of $\hat{R}$ matrix
elements. In the present paper, we describe how the computation of the quantum
$\hat{R}$ matrix can be performed along the lines of the method outlined in
[15], and provide the necessary ingredients for this method, which are the
generators of $\mathfrak{sl}_{q}(3)$ in the adjoint representation. Because
the computation of the $\hat{R}$ matrix is bulky due to the large size of the
matrix, the explicit derivation of the BWs will be considered in a separate
work [16].

In section 5, we summarize our results and briefly discuss possible
applications to a class of IRF models based on a quotient
$\frac{SU(n)_{k}}{Z_{n}}$ [17]. The case $n=2$ has already been studied in
[18]. For $n>2$, the problem of multiplicity arises, and we expect that the
present research results will help find exact solutions for this class of IRF
models.

##  2 Description of the restricted models

The restricted IRF models are defined on a two-dimensional square lattice as
shown in Figure 1. The fluctuating variables are associated with the lattice
vertices. A face is formed by four nearest neighboring vertices (in Figure 1
is shown a face with vertices $a,b,c,$ and $d$ ).

IRF models can be defined for a general CFT model, with chiral algebra
$\mathcal{O}$. In the case of affine Lie algebras, the corresponding CFTs are
WZW models [19, 2]333In this work, we focus on IRF models based on WZW models
with affine Lie algebras $\mathfrak{su}(2)_{k}$ and $\mathfrak{su}(3)_{k}$..
The primary fields of the CFT model are labeled by the representations of the
algebra $\mathcal{O}$ and the fluctuating variables of the IRF model
correspond to integrable highest-weight representations, characterized by
dominant integral highest-weights (defined below). For affine Lie algebras of
rank $n-1$, we denote the set of fundamental weights as
$\Lambda_{0},\Lambda_{1},\dots,\Lambda_{n-1}$. Consequently, the fluctuating
variables residing on the lattice vertices take values from the set of
dominant integral weights $P_{+}(n,k)$,

$P_{+}(n,k)=\left\\{a=\sum_{i=0}^{n-1}a_{i}\Lambda_{i}=(a_{0},a_{1},...,a_{n-1}),\quad\sum_{i=0}^{n-1}a_{i}=k=\text{level},\quad
a_{i}\in\mathbf{Z},\quad a_{i}\geq 0\right\\},$ (2.1)

where $k$ is a positive integer referred to as the level of weights, serving
as a parameter of the model, and $\mathbf{Z}$ represents the set of integers.
Among all the weights (2.1), we choose two weights, $h$ and $v$, to define the
admissibility conditions in both the horizontal and vertical directions of the
lattice. For a given CFT algebra $\mathcal{O}$ and fields $h$ and $v$, the
corresponding IRF model is denoted as IRF ($\mathcal{O}$, $h$, $v$) following
the notation in [11]. In the present study, we focus on the cases where $h=v$.

Now, we define the admissibility conditions for configurations on the
vertices. Considering the tensor product of the elements of $P_{+}(n,k)$ with
$h$, see (1.2), an ordered pair $(a,b)$ is called admissible if it satisfies
(1.1). Let $(a,b,c,d)$ be the values of the North-West (NW), NE, SE, and SW
corners of a face (e.g., the face shown in Figure 1). The face configuration
$(a,b,c,d)$ is called admissible if the ordered pairs $(a,b)$, $(a,d)$,
$(b,c)$, and $(d,c)$ are all admissible, i.e., if

$N_{a,h}^{b}>0,\quad N_{a,h}^{d}>0,\quad N_{b,h}^{c}>0,\quad N_{d,h}^{c}>0.$
(2.2)

For each face configuration, we assign a Boltzmann weight

$\omega\left(\begin{matrix}a&b\\\ d&c\end{matrix}\bigg{|}u\right).$ (2.3)

Hence, the BWs depend on the configuration $(a,b,c,d)$ and the spectral
parameter $u$. We set the Boltzmann weight to zero if the face configuration
is not admissible. The partition function of the lattice model is given by

$Z=\sum_{\text{configurations}}\prod_{\text{faces}}\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}u\right).$ (2.4)

We wish to define the BWs in such a way that the model will be solvable.
Namely, that the transfer matrices will commute for different values of the
spectral parameter. This is guaranteed by the Yang–Baxter equation (YBE), see,
e.g., [11]

$\sum_{g}\omega\left(\begin{matrix}a&b\\\
f&g\end{matrix}\bigg{|}u+v\right)\omega\left(\begin{matrix}f&g\\\
e&d\end{matrix}\bigg{|}u\right)\omega\left(\begin{matrix}b&c\\\
g&d\end{matrix}\bigg{|}v\right)=\sum_{g}\omega\left(\begin{matrix}a&g\\\
f&e\end{matrix}\bigg{|}v\right)\omega\left(\begin{matrix}a&b\\\
g&c\end{matrix}\bigg{|}u\right)\omega\left(\begin{matrix}g&c\\\
e&d\end{matrix}\bigg{|}u+v\right).$ (2.5)

In the case of trigonometric solutions (considered here), the Boltzmann
weights are parameterized by trigonometric functions. On the other hand, for
elliptic solutions, the BWs are parameterized by elliptic theta functions,
which depend on the elliptic modulus parameter $p$. One can obtain
trigonometric solutions from elliptic solutions by taking the limit
$p\rightarrow 0$.444The elliptic solutions for BWs are given by a ratio of two
elliptic theta functions. Therefore taking this limit $p\rightarrow 0$ results
in a ratio of trigonometric functions. The possibility of going in the
opposite direction is not apparent. Still, the general procedure suggests that
to obtain elliptic solutions from trigonometric ones, it is necessary to make
the following substitution:

$\sin{u}\rightarrow
2p^{\frac{1}{8}}\sin{u}\prod_{n=1}^{\infty}(1-2\cos(2u)p^{n}+p^{2n})(1-p^{n})=\theta_{1}(p,u).$
(2.6)

In the following section, will explain the approach employed to derive the BWs
of the considered models from the Yang-Baxter equation (2.5).

$a$$b$$c$$d$$u$ Figure 1: Two-dimensional lattice.

##  3 CFT approach

In this section, we describe the method proposed in [11] to obtain the
solutions to YBE for the BWs. This method suggests that BWs of the IRF ($\cal
O$, $h,h$) can be computed using the following prescription. We label the $N$
representations appearing in the tensor product $h\otimes h$ as follows

$h\otimes h=\bigoplus_{j=0}^{N-1}N_{h,h}^{j}\times\phi_{j}.$ (3.1)

According to the proposal, the solutions of the YBE are given by

$\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}u\right)\equiv\sum_{j=0}^{N-1}\tilde{P}^{(j)}\left[a,b,c,d\right]\rho_{j}(u).$
(3.2)

Here, $\tilde{P}^{(j)}\left[a,b,c,d\right]$ is a projector defined as

$\tilde{P}^{(j)}\left[a,b,c,d\right]=\langle
a,b,c|P^{(j)}|a,d,c\rangle=\prod_{l\neq
j}\frac{B_{b,d}\left[\begin{matrix}h&h\\\
a&c\end{matrix}\right]-\delta_{b,d}\lambda_{j}}{\lambda_{l}-\lambda_{j}}\;,$
(3.3)

where $B_{b,d}\left[\begin{matrix}h&h\\\ a&c\end{matrix}\right]$ represents
the braiding matrix (for details see, e.g., [11, 20]) associated with the CFT
$\cal O$ and $\lambda_{j}$ are the eigenvalues of the braiding matrix, these
are given by

$\lambda_{j}=\epsilon_{j}e^{i\pi(\Delta_{\phi_{j}}-2\Delta_{h})},$ (3.4)

where $\epsilon_{j}=\pm 1$ depending on whether $\phi_{j}$ appears
symmetrically or anti-symmetrically in (3.1) and
$\Delta_{h},\Delta_{\phi_{j}}$ are conformal dimensions of the primary fields
corresponding to the representations $h,\phi_{j}$ respectively. The conformal
dimension $\Delta_{a}$ of a primary field corresponding to the representation
$a$ is given by the formula

$\Delta_{a}=\frac{\left(a,a+2\rho\right)}{2(k+g)},$ (3.5)

where $g$ represents the dual Coxeter number of the algebra $\cal O$ under
consideration, and $\rho$ is the Weyl vector
$\rho=\sum_{i=0}^{n-1}\Lambda_{i}$. The proposal (3.2) relies significantly on
the functions $\rho_{j}(u)$, which are defined as follows

$\rho_{j}(u)=\frac{\prod_{r=1}^{j}\sin{(\zeta_{r-1}-u)}\prod_{r=j+1}^{N-1}\sin{(\zeta_{r-1}+u)}}{\prod_{r=1}^{N-1}\sin{(\zeta_{r-1})}},$
(3.6)

where the parameters $\zeta_{i}$ referred to as crossing parameters in this
proposal, are determined as follows

$\zeta_{i}=\frac{\pi(\Delta_{\phi_{i+1}}-\Delta_{\phi_{i}})}{2},\quad
i=0,1,...,N-2.$ (3.7)

The projectors $\tilde{P}^{(j)}[a,b,c,d]$ satisfy the property
$\sum_{j=0}^{N-1}P^{(j)}[a,b,c,d]=\mathbf{1}$, where $\mathbf{1}$ denotes the
identity operator. Consequently, the BWs (3.2) satisfy the condition

$\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}0\right)=\delta_{b,d}\;.$ (3.8)

Now, let us explain how we compute the BWs in subsections (3.1, 3.2, 3.3). In
general, the $k$ dependence of the braiding matrix appearing in (3.3) is not
known explicitly, however, it is clear that for concrete values of $k$ and
($a,b,c,d,h$), the braiding matrix is a fixed numerical matrix. Therefore, the
matrix elements of the $\tilde{P}^{(j)}$ projector also have fixed numerical
values, which we need to find. We substitute the ansatz (3.2) into YBE (2.5)
and (3.8), and then find solutions for $\tilde{P}^{(j)}[a,b,c,d]$ for all
possible face configurations. The possible nonzero $\tilde{P}^{(j)}[a,b,c,d]$
values (for a given ($a,b,c,d$)) correspond to the fields $\phi_{j}$, which
belong to the domain:

$(a^{*}\otimes c)\cap(h\otimes h),$ (3.9)

where $a^{*}$ represents the conjugate representation of $a$, this allows us
to reduce the set of unknowns, see [18]. The results of these computations are
presented in the subsequent sub-sections.

It is worth noting that as the level $k$ increases (for nontrivial
representations $h$ and especially for $\mathfrak{su}(n\geq 3)_{k}$), the
number of admissible face configurations, and consequently the number of
equations (2.5) becomes very large. This makes the task of finding solutions
for the BWs more complicated. However, the outlined method shows its
effectiveness in various nontrivial examples. For a general level $k$, to
determine the BWs, it is required to first find expressions in closed form for
the braiding matrix or the projector $\tilde{P}^{(j)}$, as demonstrated for
instance in [21, 18].

###  3.1 Warm-up example: $\mathfrak{su}(2)_{2}$

Let us consider the model based on $su(2)_{2}$ with
$h=\Lambda_{0}+\Lambda_{1}$. At this level $k=2$, we find 8 possible BWs.
These BWs correspond exactly to the trigonometric solutions found by Jimbo et
al., [3, 4]. Let us describe them by labeling the fields as follows

$2\Lambda_{0}+0\Lambda_{1}:=0,\quad 1\Lambda_{0}+1\Lambda_{1}:=1,\quad
0\Lambda_{0}+2\Lambda_{1}:=2.$ (3.10)

The relevant tensor product rules are,

$\centering\begin{array}[]{ll}1\otimes 2=1,\\\ 1\otimes 1=0\oplus 2,&\\\
2\otimes 2=0.&\\\ \end{array}\@add@centering$ (3.11)

Since the tensor product $1\otimes 1$ contains two representations, $N=2$.
Therefore, we have one crossing parameter given by

$\zeta_{0}=\frac{\pi}{4}:=\zeta.$ (3.12)

With these data, we can now search for solutions to the Yang-Baxter equation
(2.5) and the initial condition (3.8) in the form of (3.2). We find the
following solutions

$\begin{split}\begin{array}[]{ll}\omega(0,1,0,1,u)=&\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(0,1,2,1,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(1,0,1,0,u)=&\frac{1}{2}\frac{s(\zeta-u)}{s(\zeta)}+\frac{1}{2}\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(1,0,1,2,u)=&\frac{1}{2}\frac{s(\zeta-u)}{s(\zeta)}-\frac{1}{2}\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(1,2,1,0,u)=&\frac{1}{2}\frac{s(\zeta-u)}{s(\zeta)}-\frac{1}{2}\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(1,2,1,2,u)=&\frac{1}{2}\frac{s(\zeta-u)}{s(\zeta)}+\frac{1}{2}\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(2,1,0,1,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(2,1,2,1,u)=&\frac{s(\zeta+u)}{s(\zeta)}.\\\ \end{array}\end{split}$
(3.13)

Here and in what follows we use the following notations:

$\omega\left(a,b,c,d,u\right):=\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}u\right),$ (3.14) $s(x):=\sin(x).$ (3.15)

The solutions (3.13) coincide with the trigonometric limit of BWs obtained in
[3, 4].

###  3.2 $\mathfrak{su}(2)_{4}$ in the adjoint representation

In this model, we consider $h=2\Lambda_{0}+2\Lambda_{1}$, which corresponds to
the (2,2)-fused model studied by Tartaglia and Pearce [14]. At this level
$k=4$, the model contains five fields, which are labeled as follows

$4\Lambda_{0}+0\Lambda_{1}:=0,\quad 3\Lambda_{0}+1\Lambda_{1}:=1,\quad
2\Lambda_{0}+2\Lambda_{1}:=2,\quad\Lambda_{0}+3\Lambda_{1}:=3,\quad
0\Lambda_{0}+4\Lambda_{1}:=4.$ (3.16)

The relevant tensor product rules that are used in the admissibility
conditions and when fixing the set of unknowns $\tilde{P}^{(j)}$, are the
following

$\begin{array}[]{ll}2\otimes 1=1\oplus 3,&1\otimes 1=0\oplus 2,\\\ 2\otimes
2=0\oplus 2\oplus 4,&1\otimes 3=2\oplus 4,\\\ 2\otimes 3=1\oplus 3,&3\otimes
3=0\oplus 2,\\\ 2\otimes 4=2,&4\otimes 4=0.\end{array}$ (3.17)

Since $N=3$, there are two crossing parameters

$\begin{split}&\zeta_{0}=\frac{\pi(\Delta_{2}-\Delta_{0})}{2}=\frac{\pi}{3}:=\zeta,\\\
&\zeta_{1}=\frac{\pi(\Delta_{4}-\Delta_{2})}{2}=2\zeta.\end{split}$ (3.18)

With this data, we solve the YBE with the ansatz (3.2), taking into account
(3.8) and find solutions which are parameterized by four parameters
$s_{1},s_{2},s_{3},s_{4}$. In this case, there are 33 admissible
configurations on the faces. The solutions for the BWs are

$\begin{array}[]{ll}\omega(0,2,0,2,u)=&\frac{s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(0,2,2,2,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(0,2,4,2,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)},\\\
\omega(1,1,1,1,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(1,1,3,1,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(1,1,1,3,u)=&\frac{s(\zeta+u)s(2\zeta+u)}{4s_{1}s(\zeta)s(2\zeta)}-\frac{s(\zeta-u)s(2\zeta+u)}{4s_{1}s(\zeta)s(2\zeta)},\\\
\omega(1,1,3,3,u)=&\frac{s_{2}s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{s_{2}s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)},\\\
\omega(1,3,1,1,u)=&\frac{s_{1}s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{s_{1}s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(1,3,3,1,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{4s_{2}s(\zeta)s(2\zeta)}-\frac{s(\zeta-u)s(2\zeta-u)}{4s_{2}s(\zeta)s(2\zeta)},\\\
\omega(1,3,1,3,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(1,3,3,3,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(2,0,2,0,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{4s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{4s(\zeta)s(2\zeta)},\\\
\omega(2,0,2,2,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{16s_{3}s_{4}s(\zeta)s(2\zeta)}-\frac{s(\zeta+u)s(2\zeta+u)}{16s_{3}s_{4}s(\zeta)s(2\zeta)},\\\
\omega(2,0,2,4,u)=&-\frac{s(\zeta-u)s(2\zeta-u)}{8s_{4}s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{4s_{4}s(\zeta)s(2\zeta)}-\frac{s(\zeta+u)s(2\zeta+u)}{8s_{4}s(\zeta)s(2\zeta)},\\\
\omega(2,2,2,0,u)=&\frac{2s_{3}s_{4}s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)}-\frac{2s_{3}s_{4}s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(2,2,0,2,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(2,2,2,2,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(2,2,4,2,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(2,2,2,4,u)=&\frac{s_{3}s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{s_{3}s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)},\\\
\omega(2,4,2,0,u)=&\frac{s_{4}s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{s_{4}s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}-\frac{s_{4}s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(2,4,2,2,u)=&\frac{s(\zeta+u)s(2\zeta+u)}{8s_{3}s(\zeta)s(2\zeta)}-\frac{s(\zeta-u)s(2\zeta-u)}{8s_{3}s(\zeta)s(2\zeta)},\\\
\omega(2,4,2,4,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{4s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{4s(\zeta)s(2\zeta)},\\\
\omega(3,1,1,1,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(3,1,3,1,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(3,1,1,3,u)=&\frac{s_{2}s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{s_{2}s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)},\\\
\omega(3,1,3,3,u)=&\frac{4s_{1}s_{2}^{2}s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}-\frac{4s_{1}s_{2}^{2}s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(3,3,1,1,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{4s_{2}s(\zeta)s(2\zeta)}-\frac{s(\zeta-u)s(2\zeta-u)}{4s_{2}s(\zeta)s(2\zeta)},\\\
\omega(3,3,3,1,u)=&\frac{s(\zeta+u)s(2\zeta+u)}{16s_{1}s_{2}^{2}s(\zeta)s(2\zeta)}-\frac{s(\zeta-u)s(2\zeta+u)}{16s_{1}s_{2}^{2}s(\zeta)s(2\zeta)},\\\
\end{array}$

$\begin{array}[]{ll}\omega(3,3,1,3,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(3,3,3,3,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)}+\frac{s(\zeta+u)s(2\zeta+u)}{2s(\zeta)s(2\zeta)},\\\
\omega(4,2,0,2,u)=&\frac{s(\zeta-u)s(2\zeta-u)}{s(\zeta)s(2\zeta)},\\\
\omega(4,2,2,2,u)=&\frac{s(\zeta-u)s(2\zeta+u)}{s(\zeta)s(2\zeta)},\\\
\omega(4,2,4,2,u)=&\frac{s(\zeta+u)s(2\zeta+u)}{s(\zeta)s(2\zeta)}.\end{array}$
(3.19)

We find that in the particular case

$s_{1}=-\frac{3}{2},\quad s_{2}=-\frac{1}{2},\quad s_{3}=-\frac{1}{4},\quad
s_{4}=-\frac{1}{2},$ (3.20)

our solutions reproduce exactly the Tartaglia and Pearce solutions.

###  3.3 $\mathfrak{su}(3)_{2}$ in the adjoint representation

In this subsection, we present the 21 BWs of the model based on $su(3)_{2}$
with $h=0\Lambda_{0}+\Lambda_{1}+\Lambda_{2}=(0,1,1)$. In this case, there are
six fields, which we enumerate as follows

$(2,0,0):=0,\quad(1,1,0):=1,\quad(1,0,1):=2,\quad(0,2,0):=3,\quad(0,0,2):=4,\quad(0,1,1):=5.$
(3.21)

The relevant tensor product decompositions for solving this model are

$\begin{array}[]{ll}5\otimes 5=0\oplus 5,&5\otimes 0=5,\\\ 5\otimes 1=1\oplus
4,&2\otimes 4=5,\\\ 5\otimes 2=2\oplus 3,&1\otimes 3=5,\\\ 5\otimes
3=2,&1\otimes 2=0\oplus 5,\\\ 5\otimes 4=1,&3\otimes 4=0,\\\ &4\otimes
3=0.\end{array}$ (3.22)

At this level, the tensor product $5\otimes 5$ contains only two
representations, indicating that we have $N=2$ and therefore a single crossing
parameter denoted as $\zeta_{0}$. According to the formula (3.7), this
parameter is expected to be $\frac{3\pi}{10}$. However, our calculation shows
that in order to obtain solutions to (2.5), the parameter is actually given by

$\zeta_{0}=\frac{\pi}{5}:=\zeta.$ (3.23)

It is worth noting the important fact that when computing the crossing
parameter, one should be aware of the phenomenon called “pseudo-conformal
field theory” [11]. Suppose that the theory is a WZW theory. Then the
conformal dimensions are (3.5). Actually, we can describe a related pseudo-
conformal theory where

$\bar{\Delta}_{a}=r\Delta_{a}\quad\text{ modulo $1$},$ (3.24)

where $r$ is any integer strange to $k+g$. This is a different conformal field
theory with the same fusion rules. Thus, the crossing parameters $\zeta_{i}$
become $\zeta_{i}r$ modulo one. In our particular example,
$\zeta_{0}=3\pi/10$. But multiplying by $r=4$ we get $\zeta_{0}=\pi/5$, so
these parameters are consistent with one another. Note that one has to change
also the braiding matrices, basically taking $\sin(x)$ to $\sin(rx)$, where
$x$ is the appropriate solution, and similarly for the elliptic solution.

The solutions we derived depend on three parameters: $s_{1},s_{2},s_{3}$. They
can be expressed as follows

$\begin{split}\begin{array}[]{ll}\omega(0,5,0,5,u)=&\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(0,5,5,5,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(1,1,1,1,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta+u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta-u)}{2s(\zeta)},\\\
\omega(1,1,4,1,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(1,1,1,4,u)=&\frac{s_{1}s(\zeta+u)}{s(\zeta)}-\frac{s_{1}s(\zeta-u)}{s(\zeta)},\\\
\omega(1,4,1,1,u)=&\frac{\left(\sqrt{5}-2\right)s(\zeta+u)}{s_{1}s(\zeta)}+\frac{\left(2-\sqrt{5}\right)s(\zeta-u)}{s_{1}s(\zeta)},\\\
\omega(1,4,1,4,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta-u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta+u)}{2s(\zeta)},\\\
\omega(2,2,2,2,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta+u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta-u)}{2s(\zeta)},\\\
\omega(2,2,3,2,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(2,2,2,3,u)=&\frac{s_{2}s(\zeta+u)}{s(\zeta)}-\frac{s_{2}s(\zeta-u)}{s(\zeta)},\\\
\omega(2,3,2,2,u)=&\frac{\left(\sqrt{5}-2\right)s(\zeta+u)}{s_{2}s(\zeta)}+\frac{\left(2-\sqrt{5}\right)s(\zeta-u)}{s_{2}s(\zeta)},\\\
\omega(2,3,2,3,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta-u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta+u)}{2s(\zeta)},\\\
\omega(3,2,2,2,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(3,2,3,2,u)=&\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(4,1,1,1,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(4,1,4,1,u)=&\frac{s(\zeta+u)}{s(\zeta)},\\\
\omega(5,0,5,0,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta-u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta+u)}{2s(\zeta)},\\\
\omega(5,0,5,5,u)=&\frac{s_{3}s(\zeta+u)}{s(\zeta)}-\frac{s_{3}s(\zeta-u)}{s(\zeta)},\\\
\omega(5,5,5,0,u)=&\frac{\left(\sqrt{5}-2\right)s(\zeta+u)}{s_{3}s(\zeta)}+\frac{\left(2-\sqrt{5}\right)s(\zeta-u)}{s_{3}s(\zeta)},\\\
\omega(5,5,0,5,u)=&\frac{s(\zeta-u)}{s(\zeta)},\\\
\omega(5,5,5,5,u)=&\frac{\left(\sqrt{5}-1\right)s(\zeta+u)}{2s(\zeta)}+\frac{\left(3-\sqrt{5}\right)s(\zeta-u)}{2s(\zeta)}.\\\
\end{array}\end{split}$ (3.25)

By imposing some specific conditions on the parameters $s_{1},s_{2},s_{3}$,
these solutions satisfy the symmetry

$\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}u\right)=\omega\left(\begin{matrix}a&d\\\
b&c\end{matrix}\bigg{|}u\right).$ (3.26)

This is an example of an IRF model based on $\mathfrak{su}(3)_{k}$ with $h$ in
the adjoint representation. For $k>2$ there are multiplicities in tensor
products, we will return to this question in section 5.

##  4 Vertex-IRF correspondence approach for unrestricted models

This section focuses on the Vertex-IRF correspondence approach for
unrestricted IRF models. Unlike restricted models, the fluctuating variables
in unrestricted models are not limited by the level $k$. Specifically, the
fluctuating variables located at the vertices of the lattice can assume values
from the set $P$ of integral weights of the algebra. Here we are interested in
a model based on $\mathfrak{su}(3)_{k}$, particularly when the adjoint
representation defines the admissibility conditions. For the affine Lie
algebra $\mathfrak{su}(3)_{k}$ the set $P$ can be expressed as follows

$P=\left\\{a=\sum_{i=0}^{2}a_{i}\Lambda_{i}=(a_{0},a_{1},a_{2}),\quad
a_{i}\in\mathbf{Z}\right\\}.$ (4.1)

We introduce the weights of the adjoint representation of the corresponding
finite algebra $\mathfrak{su}(3)$ as follows

$\begin{array}[]{cccc}e_{1}=(1,1),&e_{2}=(-1,2),&e_{3}=(2,-1),&e_{4}=(0,0),\\\
e_{5}=(0,0),&e_{6}=(1,-2),&e_{7}=(-2,1),&e_{8}=(-1,-1).\end{array}$ (4.2)

Their corresponding affine extensions (at level zero) are

$\begin{array}[]{cccc}\hat{e}_{1}=(-2,1,1),&\hat{e}_{2}=(-1,-1,2),&\hat{e}_{3}=(-1,2,-1),&\hat{e}_{4}=(0,0,0)_{1},\\\
\hat{e}_{5}=(0,0,0)_{2},&\hat{e}_{6}=(1,1,-2,),&\hat{e}_{7}=(1,-2,1),&\hat{e}_{8}=(2,-1,-1).\end{array}$
(4.3)

For the unrestricted IRF model, a pair $(a,b)$ is termed admissible if

$b=a+\hat{e}_{i},\quad\text{for some $i=1,2,...,8.$}$ (4.4)

A face configuration $(a,b,c,d)$ is admissible if the pairs $(a,b)$, $(b,c)$,
$(a,d)$, $(d,c)$ are all admissible in the sense (4.4).

According to [22, 23, 24], the BWs of unrestricted IRF models can be defined
in terms of the matrix elements of the so-called quantum $\hat{R}$ matrix. We
call this approach “vertex-IRF correspondence” because the matrix $\hat{R}$
directly determines the BWs of vertex-type models. As mentioned earlier, our
study here involves two main objectives: (1) determining the BWs of the
unrestricted IRF model based on $\mathfrak{su}(3)_{k}$ when the adjoint
representation defines the admissibility conditions, and (2) extending the
approach of Section 3 to a situation where multiplicities occur by obtaining
the exact expressions for the functions $\rho_{j}(u)$ and $\tilde{P}^{(j)}$.
Here, we explain the Vertex-IRF correspondence and the method outlined in [15]
for computing the $\hat{R}$ matrix elements. Since the computation of the
$\hat{R}$ matrix elements in the adjoint representation is a demanding problem
in its own right, we plan to address it separately in [16]. By definition, the
quantum $\hat{R}$ matrix satisfies the Yang-Baxter equation in the form

$(I\otimes\hat{R}(u))(\hat{R}(u+v)\otimes
I)(I\otimes\hat{R}(v))=(\hat{R}(v)\otimes
I)(I\otimes\hat{R}(u+v))(\hat{R}(u)\otimes I)\;.$ (4.5)

Both sides of the equation (4.5) act on the tensor product of three vector
spaces: $V_{1}\otimes V_{2}\otimes V_{3}$. $I$ is the identity operator and
$\hat{R}$ is defined as follows

$\hat{R}=P\cdot R\;,$ (4.6)

where $P$ denotes the transposition $Px\otimes y=y\otimes x$, and $R$, also
called quantum $R$ matrix,555It is also common to write the YBE (4.5) in the
form $R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)$, where
$R_{ij}(u)$ is an operator acting as $R$ on the $i$th and $j$th components and
as identity on the other component. acts on the tensor product of two vector
spaces in terms of its matrix elements as follows

$R(u)(e_{i}\otimes e_{j})=\sum_{m,n}R_{i,j}^{m,n}(u)\quad(e_{m}\otimes
e_{n}).$ (4.7)

We can rewrite (4.5) in terms of $R$ for a given sextuplet
($i_{1},i_{2},i_{3},f_{1},f_{2},f_{3}$), yielding

$\sum_{s_{1},s_{2},s_{3}}R_{s_{2},s_{3}}^{f_{1},f_{2}}(u)R_{i_{1},s_{1}}^{s_{2},f_{3}}(u+v)R_{i_{2},i_{3}}^{s_{3},s_{1}}(v)=\sum_{s_{1},s_{2},s_{3}}R_{s_{2},s_{3}}^{f_{2},f_{3}}(v)R_{s_{1},i_{3}}^{f_{1},s_{3}}(u+v)R_{i_{1},i_{2}}^{s_{1},s_{2}}(u)\;.$
(4.8)

In the context of IRF models, we are interested in the case
$V_{1}=V_{2}=V_{3}=V$ and $V$ is the vector space of the adjoint
representation of $su(3)$. However, in this case, obtaining the matrix $R$ is
a nontrivial task due to its size. Since the adjoint representation contains
eight states, the matrix $R$ becomes a $64\times 64$ matrix. We explain how
one can proceed to compute this matrix $R$, but before let us discuss its
relations with the BWs of IRF models. The $R$ matrix has the property

$R_{i,j}^{k,l}=0\quad\text{if $e_{i}+e_{j}\neq e_{k}+e_{l}$.}$ (4.9)

Following the Vertex-IRF correspondence (see, e.g., [22, 23]), one can compute
the BWs of unrestricted IRF models in terms of the matrix elements of
$\hat{R}$ as follows

$\omega\left(\begin{matrix}a&b\\\
d&c\end{matrix}\bigg{|}u\right)=\begin{cases}\hat{R}_{i,j}^{k,l}(u)\quad\text{if
$b-a=\hat{e}_{k}$, $c-b=\hat{e}_{l}$, $d-a=\hat{e}_{i}$,
$c-d=\hat{e}_{j}$},\\\ 0\quad\text{otherwise}.\end{cases}$ (4.10)

This relation can be graphically represented as

$a$$b$$c$$d$$u$$=$$e_{k}$$e_{l}$$e_{j}$$e_{i}$$u$ (4.11)

The lhs square in the equation represents the BWs of the IRF666In IRF models,
when multiplicities arise in the tensor product rules, additional labels are
required on the edges to characterize the BWs effectively. We explain this in
Section 5. model, while the rhs square corresponds to the BWs of the Vertex
model.

To compute the matrix $R$, one can employ the scheme given by Jimbo [15]. This
scheme relies on certain fundamental elements. First, we introduce the
$sl_{q}(3)$ quantum algebra and its generators in the adjoint representation.
Two simple roots of the algebra are denoted as $\alpha_{a}$ ($a=1,2$). Three
generators, namely $H_{a}$, $E_{a}$, and $F_{a}$, are associated with each
simple root. In the adjoint representation, we found these generators can be
written in the following manner. The Cartan generators are given by

$H_{1}=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\ 0&-1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&2&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-2\\\ \end{array}\right),\quad
H_{2}=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\ 0&2&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&-2&0&0&0&0\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ \end{array}\right),$ (4.12)

the raising and lowering generators are

$E_{1}=\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&\frac{q^{2}+1}{q}&0\\\ 0&0&0&0&0&0&0&\frac{q^{2}+1}{q}\\\
0&0&0&0&0&0&0&0\\\ \end{array}\right),\quad
E_{2}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\
0&0&\frac{q^{2}+1}{q}&0&0&0&1&0\\\ 0&0&0&\frac{q^{2}+1}{q}&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ \end{array}\right),$ (4.13)

$F_{1}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&\frac{q}{q^{2}+1}&0&0&0&1&0\\\ \end{array}\right),\quad
F_{2}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&\frac{q}{q^{2}+1}&0\\\ 0&0&0&0&0&0&0&1\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ \end{array}\right).$
(4.14)

They define the $sl_{q}(3)$ quantum algebra

$\begin{split}&k_{a}=q^{H_{a}/2},\\\ &k_{a}E_{a}=qE_{a}k_{a},\quad
k_{a}E_{b}=q^{-1/2}E_{b}k_{a},\\\ &k_{a}F_{a}=q^{-1}F_{a}k_{a},\quad
k_{a}F_{b}=q^{1/2}F_{b}k_{a},\\\
&[k_{1},k_{2}]=[E_{1},F_{2}]=[E_{2},F_{1}]=0,\\\
&[E_{a},F_{a}]=\frac{k_{a}^{2}-k_{a}^{-2}}{q-q^{-1}},\\\
&E_{a}^{2}E_{b}-(q+q^{-1})E_{a}E_{b}E_{b}+E_{b}E_{a}^{2}=0,\\\
&F_{a}^{2}F_{b}-(q+q^{-1})F_{a}F_{b}F_{a}+F_{b}F_{a}^{2}=0,\end{split}$ (4.15)

where $a,b=1,2$ and $a\neq b$. The states of the adjoint representation (4.2)
in the Euclidean basis can be written as follows

$\begin{array}[]{ccc}e_{1}=(1,0,0,0,0,0,0,0)^{T},&e_{2}=(0,1,0,0,0,0,0,0)^{T},&e_{3}=(0,0,0,0,0,1,0,0)^{T},\\\
e_{4}=(0,0,1,0,0,0,0,0)^{T},&e_{5}=(0,0,0,0,0,0,1,0)^{T},&e_{6}=(0,0,0,1,0,0,0,0)^{T},\\\
e_{7}=(0,0,0,0,0,0,0,1)^{T},&e_{8}=(0,0,0,0,1,0,0,0)^{T}.\end{array}$ (4.16)

According to Jimbo’s scheme, one needs the following two elements

$k_{0}=q^{(H_{1}+H_{2})/2},\quad
E_{0}=q^{(H_{1}-H_{2})/3}\left(F_{2}F_{1}-q^{-1}F_{1}F_{2}\right).$ (4.17)

In [15] it was shown that the quantum $R$ matrix (4.7, 4.8) satisfies the
following system of linear equations (the solution of which is unique up to an
overall factor)

$\begin{split}&R(u)\left(E_{a}\otimes k_{a}^{-1}+k_{a}\otimes
E_{a}\right)=\left(E_{a}\otimes k_{a}+k_{a}^{-1}\otimes E_{a}\right)R(u),\\\
&R(u)\left(F_{a}\otimes k_{a}^{-1}+k_{a}\otimes
F_{a}\right)=\left(F_{a}\otimes k_{a}+k_{a}^{-1}\otimes F_{a}\right)R(u),\\\
&[R(u),H_{a}\otimes I+I\otimes H_{a}]=0,\\\ &\end{split}$ (4.18)

and an important fact is that

$R(u)\left(e^{u}E_{0}\otimes k_{0}+k_{0}^{-1}\otimes
E_{0}\right)=\left(e^{u}E_{0}\otimes k_{0}^{-1}+k_{0}\otimes
E_{0}\right)R(u).$ (4.19)

From (4.19), it is clear that besides the spectral parameter $u$, the $R$
matrix also depends on the parameter $q$ of the quantum algebra. To compute
the $R$ matrix, one needs to put the above generators in (4.18), (4.19) and
solve these equations for the matrix elements $R_{i,j}^{k,l}$. The results of
this computation will be reported in [16].

##  5 Conclusions and discussion

In this work, we have investigated restricted IRF models based on the affine
Lie algebras $\mathfrak{su}(2)_{k}$ and $\mathfrak{su}(3)_{k}$ for various
levels $k$ and representations $h$ that determine the admissibility conditions
of the models. Specifically, we have examined the models
$\mathfrak{su}(2)_{2}$ with $h$ corresponding to the fundamental
representation, $\mathfrak{su}(2)_{4}$ with $h$ corresponding to the adjoint
representation, and $\mathfrak{su}(3)_{2}$ with $h$ also in the adjoint
representation. For these models, we have employed a previously proposed
approach based on the relationship between CFTs and IRF models, described in
Section 3. This approach allows determining the BWs of restricted IRF models
in terms of the braiding matrix of the associated CFT and certain
trigonometric functions $\rho_{j}(u)$ that depend on the spectral parameter
$u$ and the crossing parameters $\zeta_{j}$ associated with the conformal
dimensions of the fields (representations) appearing in the tensor product
$h\otimes h$. By incorporating this ansatz into the Yang-Baxter equation
(2.5), we have obtained solutions for the BWs of the abovementioned models.
Our solutions for $\mathfrak{su}(2)_{2}$ and $\mathfrak{su}(2)_{4}$ reproduce
known results [4, 14], while our solutions for $\mathfrak{su}(3)_{2}$ are new.

The work is focused on particular fixed $k$ levels, however, it is worth
noting that the approach can be extended to general levels. To this end, one
must first determine the corresponding braiding matrix [21, 18]. By
successfully solving these models, we have demonstrated the efficiency of the
approach in addressing new IRF models.

In the second part, we explored the approach based on the Vertex-IRF
correspondence. The final objective in this direction is to determine the BWs
of the unrestricted model based on $\mathfrak{su}(3)_{k}$ in terms of the
quantum $\hat{R}$ matrix. We discussed the method proposed in [15] for finding
$\hat{R}$ and provided all the elements needed to address this problem.

The motivation for studying the Vertex-IRF correspondence is combining this
approach with the CFT approach. Notice that although the Vertex-IRF
correspondence provides BWs for unrestricted models, the specific set of BWs
for the restricted models can be obtained as a subset of unrestricted BWs.
This is because these BWs from this subset satisfy YBE between themselves, as
discussed, e.g., in [25, 26]. Since the CFT approach is not applicable when
multiplicities are present, we plan to investigate how this method generalizes
to such cases. Below we describe some details regarding this matter. For
$k\geq 5$, the tensor product of the adjoint representation (excluding the
$\Lambda_{0}$ component) $h=\Lambda_{1}+\Lambda_{2}=(1,1)$ yields:

$h\otimes
h=(0,0)\oplus(1,1)_{1}\oplus(1,1)_{2}\oplus(0,3)\oplus(3,0)\oplus(2,2).$ (5.1)

Here, $(1,1)_{1}$ and $(1,1)_{2}$ correspond to the adjoint representation
that appears twice (this multiplicity corresponds to the multiplicity of the
null weights $e_{4},e_{5}$ in (4.2)). Notice that to define a face
configuration in these cases completely, it becomes necessary to introduce
additional labels on the edges. For example, Figure 2 illustrates two
configurations where the fields at the southeast vertex are the same, but the
edges connected to this vertex may have different labels. The multiplicity of
$(1,1)$ representation leads to a complication, as it results in two identical
values for (3.4), which is not consistent with the ansatz (3.3).

$\alpha$$(0,0)$$(1,1)$$\beta_{1}$$(1,1)_{1}$$(1,1)$$\gamma_{1}$$\lambda$$u$$(0,0)$$(1,1)$$(1,1)_{2}$$(1,1)$$\alpha$$\beta_{2}$$\gamma_{2}$$\lambda$$u$
Figure 2:

Studying the generalization of the CFT approach is a crucial step in further
investigations of IRF models which cannot be solved solely by applying the
fusion procedure [13]. Interesting examples are the models based on the
algebra of the quotient group $\frac{SU(n)_{k}}{Z_{n}}$. These quotients were
investigated in the context of non-diagonal modular invariant CFTs [27]. Their
tensor product rules were computed in [17]. These theories exhibit novel
features not present in the original $\mathfrak{su}(n)_{k}$ models. For
instance, only representations that are $Z_{n}$-invariant are present, and
there are so-called “fixed-point representations”, which play a significant
role.

To illustrate these features, let us consider an example from [17], the
quotient $\frac{SU(3)_{6}}{Z_{3}}$. The tensor product of two adjoint
representations, in this case, is given by

$(1,1)\otimes(1,1)=(0,0)\oplus 2(1,1)\oplus
2(3,3)\oplus(2,2)\oplus(2,2)^{\prime}\oplus(2,2)^{\prime\prime},$ (5.2)

on rhs, the first three representations are also present in (5.1), but in this
theory, the representations are identified up to the external automorphism
$\sigma$, defined as $\sigma(a_{1},a_{2})=(k-(a_{1}+a_{2}),a_{1})$.
Consequently, the representations $(3,3)$, $(0,3)$, and $(3,0)$ are
identified. The fixed-point representations are those invariant under the
automorphism $\sigma$, and for this level, they are $(2,2)$, $(2,2)^{\prime}$,
and $(2,2)^{\prime\prime}$. These fixed-point representations are treated as
three distinct fields (representations). In the original
$\mathfrak{su}(3)_{k}$ models, these fixed-point representations are absent,
and the question is how to incorporate them in the fusion procedure. Thus,
finding the BWs of IRF models based on these quotients remains an open
question. In [18], the case of $\frac{SU(2)_{k}}{Z_{2}}$ was solved by
combining the CFT approach and known solutions of the $\mathfrak{su}(2)_{k}$
model. The strategy involved the approach 3 to find the BWs containing fixed-
point representations, and for other BWs, the solutions of
$\mathfrak{su}(2)_{k}$ were used. However, if we want to apply this strategy
to other cases (such as $\frac{SU(3)_{k}}{Z_{3}}$), we encounter the following
difficulty. The tensor product rule of fixed-point representations in these
cases (see section 2.1 of [17]) involves multiplicity, unlike the case of
$\frac{SU(2)_{k}}{Z_{2}}$. Consequently, this prevents us from employing the
approach 3. We leave this problem as an open question for future research.

Acknowledgment: J.R. expresses gratitude to the organizers of the Workshop on
Integrability 2023 held at the University of Amsterdam, where this work was
presented. The work of B.R. was supported by the Ministry of Science and
Higher Education of the Russian Federation (agreement no. 075–15–2022–287).

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lemmatheorem lemma propositiontheorem proposition corollarytheorem corollary
definitiontheorem definition conjecturetheorem conjecture remarktheorem remark
claimtheorem claim thebibliography

# A faster direct sampling algorithm for equilateral closed polygons

Jason Cantarella  Mathematics Department, University of Georgia, Athens, GA,
USA  Henrik Schumacher  Mathematics Department, University of Georgia, Athens,
GA, USA  Clayton Shonkwiler  Department of Mathematics, Colorado State
University, Fort Collins, CO, USA

###### Abstract

We present a faster direct sampling algorithm for random equilateral closed
polygons in three-dimensional space. This method improves on the moment
polytope sampling algorithm of Cantarella, Duplantier, Shonkwiler, and Uehara
[4] and has (expected) time per sample quadratic in the number of edges in the
polygon.

closed random walk, random polygon, random knot, polymer models

Equilateral polygons in $\mathbb{R}^{3}$—that is, polygonal walks in 3-space
forming closed loops and consisting of unit-length steps—provide a standard,
if highly simplified, model of ring polymers under “$\theta$-conditions” (see,
e.g., the survey [22], which gives a number of applications of these models in
physics and biology). The closure condition imposes subtle global correlations
between edge directions, which means it is not obvious how to generate random
equilateral polygons. Indeed, algorithms have been proposed for at least 4
decades [1, 5, 13, 18, 20, 25, 26, 7, 8, 9, 19], though most are numerically
unstable or have not been proved to sample from the correct probability
distribution.

In previous work [4], we introduced the _action-angle method_ , which is a
numerically stable and provably correct algorithm for generating random
equilateral $n$-gons in $\mathbb{R}^{3}$ based on rejection sampling the
hypercube. The action-angle method is the fastest extant method: it produces
samples in expected time $\Theta(n^{5/2})$.

The purpose of this paper is to give a $\sqrt{n}$ speedup for the action-angle
method, yielding an algorithm which produces random equilateral $n$-gons in
expected time $\Theta(n^{2})$. It is based on rejection sampling the same
subset of the hypercube as in the action-angle method; the speedup comes from
progressively checking the defining inequalities as we generate coordinates
rather than checking all the inequalities in a batch. Hence, we call this
algorithm the _progressive action-angle method_. The main challenge is to
prove that the progressive action-angle method really gives a $\sqrt{n}$
speedup over the action-angle method, which we do in A faster direct sampling
algorithm for equilateral closed polygons, A faster direct sampling algorithm
for equilateral closed polygons, and A faster direct sampling algorithm for
equilateral closed polygons by reducing to the computation of the volume of a
certain convex polytope.

We begin by establishing notation. For an $n$-gon in $\mathbb{R}^{3}$, let
$v_{1},\dots,v_{n}\in\mathbb{R}^{3}$ be the coordinates of its vertices, and
let $e_{1},\dots,e_{n}$ be the edge vectors, meaning that
$e_{i}=v_{i+1}-v_{i}$ for $i=1,\dots,n-1$ and $e_{n}=v_{1}-v_{n}$. We will
assume throughout that our polygons are equilateral, so that $|e_{i}|=1$ for
all $i$; equivalently, $e_{1},\dots,e_{n}\in S^{2}$, the unit sphere in
$\mathbb{R}^{3}$. The space $\operatorname{Pol}(n)$ consists of sets of edge
vectors in $(S^{2})^{n}$ which obey the closure condition
$\sum_{i=1}^{n}e_{i}=0$. One can show that the set

$\displaystyle\operatorname{Pol}(n)^{\times}\coloneqq\big{\\{}\vec{e}\in(S^{2})^{n}:\text{$\textstyle\sum_{i=1}^{n}e_{i}=0$
and for all $i\neq j$: $e_{i}\neq e_{j}$ }\big{\\}}$

is a $(2n-3)$-dimensional submanifold of $(S^{2})^{n}$ and that the
$(2n-3)$-dimensional Hausdorff measure of
$\operatorname{Pol}(n)\setminus\operatorname{Pol}(n)^{\times}$ vanishes. In
this sense $\operatorname{Pol}(n)$ is almost everywhere a submanifold of
$(S^{2})^{n}$. We may give it the submanifold metric and corresponding volume;
it is equivalent to take the $(2n-3)$-dimensional Hausdorff measure on
$\operatorname{Pol}(n)$ with respect to the metric on $(S^{2})^{n}$.

Since we are interested in shapes of polygons, we focus on the quotient space
${\widehat{\operatorname{Pol}}(n)=\operatorname{Pol}(n)/\operatorname{SO}(3)}$.
This space has a Riemannian metric—defined by the condition that the quotient
map $\operatorname{Pol}(n)\to\widehat{\operatorname{Pol}}(n)$ is a Riemannian
submersion—and hence a natural probability measure after normalizing the
Riemannian volume form.

\begin{overpic}[width=93.95122pt]{fanTriangulationPentagon}
\put(45.0,98.0){$v_{1}$} \put(-9.0,57.0){$v_{2}$} \put(17.0,-5.0){$v_{3}$}
\put(77.0,-5.0){$v_{4}$} \put(100.0,57.0){$v_{5}$}
\end{overpic}\begin{overpic}[width=133.69994pt]{triangles_constructed}
\put(29.5,22.5){$d_{1}$} \put(57.5,24.0){$d_{2}$}
\end{overpic}\begin{overpic}[width=79.49744pt]{dihedrals_set}
\put(72.0,15.0){$\theta_{1}$} \put(82.0,84.0){$\theta_{2}$}
\end{overpic}\begin{overpic}[width=79.49744pt]{finalpoly_edited} \end{overpic}

Figure 1: Constructing an equilateral pentagon from diagonals and dihedrals.
The far left shows the fan triangulation of an abstract pentagon. Given
diagonal lengths $d_{1}$ and $d_{2}$ of the pentagon which obey the triangle
inequalities, build the three triangles in the triangulation from their side
lengths (middle left). Given dihedral angles $\theta_{1}$ and $\theta_{2}$,
embed these triangles as a piecewise-linear surface in space (middle right).
The far right shows the final space polygon, which is the (solid) boundary of
this triangulated surface.††margin:  1 fig:assembly

Now we introduce some new coordinates on this space. Connecting the vertices
$v_{3},\dots,v_{n-1}$ to $v_{1}$, as in Figure 1 (far left), produces a
collection of $n-3$ triangles. The shape of the triangulated surface
determined by these triangles (and hence also its boundary, which is the
$n$-gon) is completely determined by the lengths $d_{i}$ of the diagonals
joining $v_{1}$ and $v_{i+2}$ and the dihedral angles between triangles
meeting at each diagonal. Hence, we can reconstruct the surface (and hence the
polygon) up to orientation from the data
$d_{1},\dots,d_{n-3},\theta_{1},\dots,\theta_{n-3}$, and so these give a
system of coordinates for $\widehat{\operatorname{Pol}}(n)$.

Indeed, as we have shown [5], these coordinates are natural from the
symplectic geometry point of view: in that context, they are called _action-
angle coordinates_. Note that, while the dihedral angles can be chosen
completely independently, the diagonal lengths cannot: they must obey the
system of triangle inequalities

$0\leq d_{1}\leq 2\qquad\begin{matrix}1\leq d_{i}+d_{i+1}\\\ -1\leq
d_{i+1}-d_{i}\leq 1\end{matrix}\qquad 0\leq d_{n-3}\leq
2.\marginpar{\raggedright\tiny\ref{eq:fan
polytope}eq:fanpolytope}\ignorespaces\@add@raggedright$ (1)

Let $\mathcal{P}_{n}\subset[-1,1]^{n-3}$ be the polytope defined by the
inequalities (1). If $T^{n-3}=(S^{1})^{n-3}$ is the $(n-3)$-dimensional torus
realized as the product of unit circles, then the action-angle coordinates are
defined on $\mathcal{P}_{n-3}\times T^{n-3}$, and we have previously shown
that the standard probability measure on this space—that is, the one coming
from the product of Lebesgue measure on $\mathcal{P}_{n}$ and the standard
product measure on $T^{n-3}$—is measure-theoretically equivalent to
$\widehat{\operatorname{Pol}}(n)$:

######  Theorem 1 (Cantarella–Shonkwiler [5]).

††margin:  1 thm:measures

The reconstruction map $\alpha:\mathcal{P}_{n}\times
T^{n-3}\to\widehat{\operatorname{Pol}}(n)$ defining action-angle coordinates
(i.e., the procedure illustrated in Figure 1) is measure-preserving.

Therefore, to sample points in $\widehat{\operatorname{Pol}}(n)$ (that is,
equilateral $n$-gons), it suffices to sample $\vec{d}$ from Lebesgue measure
on $\mathcal{P}_{n}$ and $\vec{\theta}$ uniformly from $T^{n-3}$. Of course,
the only challenge is to produce the sample $\vec{d}\in\mathcal{P}_{n}$.

In [4], we showed how to do this efficiently. The key observation is that the
consecutive differences $s_{i}\coloneqq d_{i+1}-d_{i}$ lie in the hypercube
$[-1,1]^{n-3}$. Therefore, we can generate points in $\mathcal{P}_{n}$ by
rejection sampling: generate proposed differences $(s_{0},\dots,s_{n-4})$
uniformly from $[-1,1]^{n-3}$, and simply check whether the proposed diagonal
lengths $(d_{1},\dots,d_{n-3})$ given by $d_{i+1}=d_{i}+s_{i}$ with
$d_{0}=|v_{2}-v_{1}|=1$ satisfy (1). This is surprisingly efficient:

######  Theorem 2 (Cantarella–Duplantier–Shonkwiler–Uehara [4]).

††margin:  2 thm:rejection sampling probability

The probability that a random point $(s_{0},\dots,s_{n-4})\in[-1,1]^{n-3}$
produces a valid collection of diagonal lengths
$(d_{1},\dots,d_{n-3})\in\mathcal{P}_{n}$ is asymptotically equivalent to
$\frac{6\sqrt{6}}{\sqrt{\pi}}\frac{1}{n^{3/2}}$ as $n\to\infty$.

In the above and throughought the rest of the paper, we say that $g(n)$ and
$h(n)$ are _asymptotically equivalent_ , denoted $g(n)\sim h(n)$, if
$\lim_{n\to\infty}\frac{g(n)}{h(n)}=1$.

Since the time it takes to generate points in $[-1,1]^{n-3}$ is linear in $n$,
rejection sampling the hypercube yields a valid point in $\mathcal{P}_{n}$ in
expected time $\Theta(n^{5/2})$. The steps of generating dihedral angles and
assembling the $n$-gon from $(d_{1},\dots,d_{n-3})$ and
$(\theta_{1},\dots,\theta_{n-3})$ do not affect the time bound since they are
both linear in $n$. Therefore, this gives a numerically stable algorithm for
generating random equilateral $n$-gons in expected time $\Theta(n^{5/2})$
which we called the _action-angle method_.

Recently, one of us (Schumacher) created a new implementation of the action-
angle sampler which, in practice, seemed to scale quadratically in $n$ rather
than as $n^{5/2}$. While we were initially worried that there was a mistake in
our analysis of the action-angle method, it turns out that some clever
programming led to a $\sqrt{n}$ speedup: our goal now is to explain and
justify this.

By fixing $d_{0}=1$ and generating proposed consecutive differences $s_{i}$
uniformly from $[-1,1]^{n-3}$, the inequalities $0\leq d_{1}\leq 2$ and
$-1\leq d_{i+1}-d_{i}\leq 1$ for $i=1,\dots,n-4$ are automatically satisfied.
Of course, the final inequality $0\leq d_{n-3}\leq 2$ can only be checked at
the very end, but the inequalities $1\leq d_{i}+d_{i+1}$ can be checked one at
a time as each $s_{i}$ is generated, and we can abort and start over as soon
as one of these inequalities fails. Naively, one might expect to have to
check, on average, some constant fraction of the inequalities. This would
speed up the algorithm by a constant factor, but not change the complexity
bound. However, as we will show below, the expected number of coordinates that
get generated before failure is actually $\Theta(\\!\sqrt{n})$, yielding an
overall time bound of $\Theta(n^{2})$. Algorithm 1 summarizes our algorithm,
which we call the _progressive action-angle method_. A reference C
implementation of this algorithm is included in plCurve [2], where it is now
the default algorithm for producing random equilateral polygons in
$\mathbb{R}^{3}$.

procedure ProgressiveActionAngleMethod($n$)$\triangleright$ Generate closed
equilateral $n$-gon

$d_{0}\leftarrow 1$

$i\leftarrow 0$

repeat

repeat

$s_{i}\leftarrow$UniformRandom($[-1,1]$)

$d_{i+1}\leftarrow d_{i}\,+s_{i}$

if $d_{i}+d_{i+1}<1$ then

$i\leftarrow 0$

else

$i\leftarrow i+1$

end if

until $i=n-3$

until $0\leq d_{n-3}\leq 2$

Sample $n-3$ i.i.d. dihedral angles $\theta_{i}$ uniformly from $[0,2\pi)$.

Reconstruct $P$ from diagonals $d_{1},\dots,d_{n-3}$ and dihedrals
$\theta_{1},\dots,\theta_{n-3}$.

end procedure

Algorithm 1 Progressive action-angle method††margin:  1 alg:PAAM

Let $\mathcal{I}(n)$ be the expected number of iterations in the inner loop of
the progressive action-angle method: that is, $\mathcal{I}(n)$ is the number
of coordinates $s_{i}$ that we expect to generate before failing one of the
$d_{i}+d_{i+1}\geq 1$ inequalities. Since we know from Theorem 2 that the
overall acceptance probability is
$\sim\frac{6\sqrt{6}}{\sqrt{\pi}}\frac{1}{n^{3/2}}$, the expected number of
iterations of the outermost loop is $\Theta(n^{3/2})$. Multiplying the time
per iteration by the number of iterations yields
$\Theta(n^{3/2}\mathcal{I}(n))$ for the expected time to produce a valid list
of diagonals. The postprocessing steps of generating dihedral angles and
assembling the polygon are both linear in $n$, so do not affect the overall
time bound. Hence, our goal is to understand how $\mathcal{I}(n)$ scales.

######  Definition .

††margin:  A faster direct sampling algorithm for equilateral closed polygons
def:pi

Let $p(k)$ be the probability that the proposed diagonal lengths generated by
a random point in $\vec{s}\in[-1,1]^{n-3}$ satisfy each of the first $k$
inequalities $d_{0}+d_{1}\geq 1,\dots,d_{k-1}+d_{k}\geq 1$. In other words,
$p(k)$ (thought of as a function of $k$) is the survival function for the
distribution of the index of the first inequality which fails. For ease of
notation, we also declare $p(0)\coloneqq 1$.

By a standard integration by parts argument (see, for example, [11, Exercise
1.7.2]), the expected value of a non-negative random variate is the integral
of the survival function, so we see that

$\marginpar{\raggedright\tiny\ref{eq:fn}eq:fn}\ignorespaces\mathcal{I}(n)=\sum_{k=0}^{n-3}p(k).\@add@raggedright$
(2)

Since our goal is to understand the asymptotics of $\mathcal{I}(n)$, we will
proceed in the following steps:

  * •

Find an exact expression for $p(k)$ (A faster direct sampling algorithm for
equilateral closed polygons).

  * •

Find an asymptotic approximation for $p(k)$ (A faster direct sampling
algorithm for equilateral closed polygons).

  * •

Plug this into (2) to get the desired asymptotic expression for
$\mathcal{I}(n)$ (A faster direct sampling algorithm for equilateral closed
polygons).

This will give us the desired complexity bound $\Theta(n^{3/2}\mathcal{I}(n))$
for the progressive action-angle method (Theorem 3).

######  Proposition .

††margin:  A faster direct sampling algorithm for equilateral closed polygons
prop:pk

For any nonnegative integer $k$,

$p(k)=\frac{2}{\pi}\int_{0}^{\infty}\operatorname{sinc}^{k+1}(t)\,\mathrm{d}t.$

###### Proof.

For each $k$, let $\mathcal{S}_{k}\subset[-1,1]^{k}$ be the subset of points
$\vec{s}$ satisfying the in the definition of $p(k)$. As a simple calculation
confirms, the affine transformation from $\vec{s}$ to $\vec{d}$ given by
$d_{i}=1+\sum_{j=0}^{i-1}s_{j}$ is volume-preserving. Let $\mathcal{Q}_{k}$ be
the image of $\mathcal{S}_{k}$ under this map; i.e., $\mathcal{Q}_{k}$ is the
polytope of $(d_{1},\dots,d_{k})$ satisfying the inequalities $d_{0}+d_{1}\geq
1,\dots,d_{k-1}+d_{k}\geq 1$ (again, recall that $d_{0}=1$, which implies in
particular that all the $d_{i}$ are nonnegative). Since
$p(k)=\frac{\operatorname{Vol}(\mathcal{S}_{k})}{\operatorname{Vol}([-1,1]^{k})}=\frac{\operatorname{Vol}(\mathcal{S}_{k})}{2^{k}}$,
to prove the proposition it suffices to prove that

$\operatorname{Vol}(\mathcal{Q}_{k})=2^{k}\frac{2}{\pi}\int_{0}^{\infty}\operatorname{sinc}^{k+1}(t)\,\mathrm{d}t.$

Notice that the defining inequalities for $\mathcal{Q}_{k}$ are precisely the
inequalities (1) _except_ the last inequality $0\leq d_{k}\leq 2$. This
suggests that these inequalities may just be the diagonal lengths of an
equilateral polygonal path in $\mathbb{R}^{3}$ which is not required to close
up.

More precisely, let

$\widehat{\operatorname{APol}}(k)\coloneqq\Big{\\{}(e_{1},\dots,e_{k+1})\in(S^{2})^{k+1}:z_{1}+\dots+z_{k+1}=0\Big{\\}}/\operatorname{SO}(2),$

where $e_{i}=(x_{i},y_{i},z_{i})$, so that the defining condition says that
the path starts and ends in the $x$-$y$-plane, and $\operatorname{SO}(2)$ acts
by simultaneously rotating all edges around the $z$-axis; this is the diagonal
subgroup of the $T^{k+1}=\operatorname{SO}(2)^{k+1}$ action on $(S^{2})^{k+1}$
which rotates edges around the $z$-axis. $\widehat{\operatorname{APol}}(k)$ is
the space of _abelian polygons_ introduced by Hausmann and Knutson [12], and
we will see that it admits two different effective, Hamiltonian $T^{k}$
actions, as shown in Figure 2.

\begin{overpic}[height=113.82832pt]{apol-torus-action}
\put(34.0,23.0){$v_{1}$} \put(50.0,55.0){$v_{2}$} \put(80.0,40.0){$v_{3}$}
\put(60.0,10.0){$v_{4}$}
\end{overpic}\begin{overpic}[height=113.82832pt]{diags-torus-action}
\put(15.0,25.0){$v_{1}$} \put(32.0,80.0){$v_{2}$} \put(92.0,67.0){$v_{3}$}
\put(72.0,5.0){$v_{4}$} \end{overpic}

.

Figure 2: Here we see both torus actions on the space
$\widehat{\operatorname{APol}}(2)$. On the left, we may rotate each of the
three edges (independently) around the $z$-axis, sweeping out three cones.
Then we identify configurations which are the same under the diagonal circle
action rotating the entire configuration around the $z$-axis (indicated by the
dark circle in the $xy$-plane). On the right, we may rotate the first two
edges around the diagonal joining vertices $v_{1}$ and $v_{3}$ or rotate the
entire polygon around the diagonal joining $v_{1}$ and $v_{4}$. In each case,
this is a Hamiltonian $2$-torus action on
$\widehat{\operatorname{APol}}(2)$.††margin:  2 fig:actions

The first is the residual $T^{k+1}/\operatorname{SO}(2)\simeq T^{k}$ action
above. The moment map for this action simply records the $z$-coordinates of
the edges; since the defining equation $z_{1}+\dots+z_{k+1}=0$ implies that
$z_{k+1}$ is determined by the remaining $z_{i}$’s, the last coordinate can be
dropped and we can think of the moment map as recording the vector
$(z_{1},\dots,z_{k})$. Let $\mathcal{H}_{k}$ be the image of this map; that
is, the moment polytope for this torus action. Of course, $-1\leq z_{i}\leq 1$
for all $i$, and, since $-1\leq z_{k+1}\leq 1$, we see that the defining
inequalities of $\mathcal{H}_{k}$ are

$-1\leq z_{i}\leq 1\text{ for all }i=1,\dots,k\quad\text{and}\quad-1\leq
z_{1}+\dots+z_{k}\leq 1.$

In other words, $\mathcal{H}_{k}$ is the central slab of the hypercube
$[-1,1]^{k}$ of points whose coordinates sum to between $-1$ and $1$. This is
a well-studied polytope, and its volume has been known at least since Pólya
[23] to be

$\operatorname{Vol}(\mathcal{H}_{k})=2^{k}\frac{2}{\pi}\int_{0}^{\infty}\operatorname{sinc}^{k+1}(t)\,\mathrm{d}t,$

where $\operatorname{sinc}(t)=\frac{\sin(t)}{t}$ for $t\neq 0$ and
$\operatorname{sinc}(0)=1$ is the sinc function (see also Borwein, Borwein,
and Mares [3] for generalizations of the above formula).

On the other hand, we get a $T^{k}$ action on
$\widehat{\operatorname{APol}}(k)$ which is analogous to the bending flows on
$\widehat{\operatorname{Pol}}(n)$ described above and in more detail in [5].
Specifically, the $i$-th $\operatorname{SO}(2)$ factor acts by rotating the
first $i+1$ edges of the polygonal arm around the $i$-th diagonal, which is
the axis through $v_{1}$ and $v_{i+1}$. Just as in the case of
$\widehat{\operatorname{Pol}}(n)$, the moment map records the lengths
$d_{1},\dots,d_{k}$ of the diagonals, and the image of the moment map is
precisely $\mathcal{Q}_{k}$.

But now we’ve realized $\widehat{\operatorname{APol}}(k)$ as a toric
symplectic manifold in 2 ways, with moment polytopes $\mathcal{H}_{k}$ and
$\mathcal{Q}_{k}$. Since the Duistermaat–Heckman theorem [10] implies that the
volume of $\widehat{\operatorname{APol}}(k)$ must be the product of the volume
$(2\pi)^{k}$ of the torus $T^{k}$ and the volume of either of its moment
polytopes, it follows that

$\operatorname{Vol}(\mathcal{Q}_{k})=\operatorname{Vol}(\mathcal{H}_{k})=2^{k}\frac{2}{\pi}\int_{0}^{\infty}\operatorname{sinc}^{k+1}(t)\,\mathrm{d}t,$

as desired. ∎

We have the following estimate for $p(k)$, which goes back at least to Laplace
[16, pp. 172–173] (see also [17]):

######  Proposition .

††margin:  A faster direct sampling algorithm for equilateral closed polygons
prop:sinc-integral-estimate

$p(k)=\sqrt{\frac{6}{\pi k}}+O\left(\frac{1}{k^{3/2}}\right).$

Now we can easily get the desired asymptotic expression for $\mathcal{I}(n)$:

######  Proposition .

††margin:  A faster direct sampling algorithm for equilateral closed polygons
prop:sum-asymptotics

$\mathcal{I}(n)\sim\sqrt{\frac{24n}{\pi}}.$

###### Proof.

By A faster direct sampling algorithm for equilateral closed polygons,
$p(k)=\sqrt{\frac{6}{\pi k}}+O\left(\frac{1}{k^{3/2}}\right)$, so, from (2),

$\mathcal{I}(n)=\sum_{k=0}^{n-3}p(k)=1+\sum_{k=1}^{n-3}\left[\sqrt{\frac{6}{\pi
k}}+O\left(\frac{1}{k^{3/2}}\right)\right]=1+\sum_{k=1}^{n-3}\sqrt{\frac{6}{\pi
k}}+\sum_{k=1}^{n-3}O\left(\frac{1}{k^{3/2}}\right).$

The first sum is asymptotic to $\sqrt{\frac{24n}{\pi}}$ by the integral test.
The rest of this expression has order $O(1)$ since the second sum is a partial
sum of a convergent series, and the result follows. ∎

Therefore, $\Theta(n^{3/2}\mathcal{I}(n))=\Theta(n^{2})$ and we have proved a
sharp complexity bound on the progressive action-angle method:

Figure 3: This plot compares average time per sample for random equilateral
$n$-gons with $n=2^{2},2^{3},\dots,2^{12}$ using the action-angle method (AAM)
and the progressive action-angle method (PAAM). The time needed to generate
samples scales as predicted by Theorem 2 and Theorem 3.††margin:  3 fig:timing

######  Theorem 3.

††margin:  3 thm:main bound

The progressive action-angle method generates uniform random samples of
closed, equilateral $n$-gons in expected time $\Theta(n^{2})$.

This agrees with the behavior we see in practice; see Figure 3.

## Discussion

††margin:  Discussion sec:discussion

Since it requires listing $n$ edges (or vertices), the time needed to generate
random equilateral $n$-gons must be at least linear in $n$. Theorem 3 shows
that the optimal bound is no worse than quadratic in $n$. It would be
interesting to see if even the quadratic bound can be improved.

In the proof of A faster direct sampling algorithm for equilateral closed
polygons, we showed that
$\operatorname{Vol}(\mathcal{Q}_{k})=\operatorname{Vol}(\mathcal{H}_{k})$ by
showing that $\mathcal{Q}_{k}$ and $\mathcal{H}_{k}$ are moment polytopes for
different toric structures on the manifold $\widehat{\operatorname{APol}}(k)$.
In fact, some additional analysis indicates that if one could generate
polygons in $\widehat{\operatorname{APol}}(n-3)$ directly, their diagonals
would  _automatically_ obey $0\leq d_{1}\leq 2$, $1\leq d_{i}+d_{i+1}$, and
$-1\leq d_{i+1}-d_{i}\leq 1$ for $i\in 1,\dots,n-2$, but still have
probability only $\sim\frac{6}{n}$ of satisfying the final inequality $0\leq
d_{n-3}\leq 2$ and lying in the moment polytope $\mathcal{P}_{n}$. This
(hypothetical) algorithm would be faster by a constant factor, but still
quadratic in $n$. This may indicate that a new idea is required, but it also
seems plausible that the moment polytope $\mathcal{P}_{n}$ could be
transformed more cleverly to occupy a larger fraction of an even smaller
polytope.

We note that the equivalence
$\operatorname{Vol}(\mathcal{Q}_{k})=\operatorname{Vol}(\mathcal{H}_{k})$
seems potentially interesting as a statement in combinatorics. We could not
find a more elementary proof, which leads us to ask: Are there other pairs of
polytopes which can be identified as moment polytopes for the same toric
symplectic manifold which are difficult to prove equivalent otherwise?

As in the case of the action-angle method [24], the progressive action-angle
method can easily be modified to give an algorithm for sampling so-called
_unit-norm tight frames_ in $\mathbb{C}^{2}$. Unit-norm tight frames in
$\mathbb{C}^{d}$ are of considerable interest for applications in signal
processing, compressed sensing, and quantum information (see [6, 14, 15, 27]
for introductions to this area). It would be very interesting to generalize
the approach from this paper to the $d>2$ setting.

## Acknowledgments

We are very grateful for the On-Line Encyclopedia of Integer Sequences [21],
without which progress on this paper would have been much slower. Thanks to
Bertrand Duplantier for helping us think asymptotically and to the National
Science Foundation (DMS–2107700 to Shonkwiler) and the Simons Foundation
(#524120 to Cantarella, #709150 to Shonkwiler) for their support.

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