On different approaches to integrable lattice models

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 su(2)k\mathfrak{su}(2)_k and su(3)k\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 su(3)k\mathfrak {su}(3)_k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^\hat{R} matrix when the adjoint representation defines the admissibility conditions.

Full text

Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 57 (2024) 155201 (19pp) https://doi.org/10.1088/1751-8121/ad33dc
On different approaches to integrable
lattice models
Vladimir Belavin1,†, Doron Gepner2,
J Ramos Cabezas1,†,∗and Boris Runov1,3,†
1Physics Department, Ariel University, Ariel 40700, Israel
2Department of Particle Physics and Astrophysics, Weizmann Institute, Rehovot
76100, Israel
3Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg
199034 Russia
E-mail: juanjrcb64@gmail.com,vladimirbe@ariel.ac.il,
doron.gepner@weizmann.ac.il and b.runov@spbu.ru
Received 23 August 2023; revised 6 March 2024
Accepted for publication 14 March 2024
Published 28 March 2024
Abstract
Interaction-round the face (IRF) models are two-dimensional lattice models
of statistical mechanics dened by an afne Lie algebra and admissibility
conditions depending on a choice of representation of that afne 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 afne Lie
algebras su(2)kand su(3)kare computed for fundamental and adjoint repres-
entations for some xed 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 su(3)k(for general k) and discuss how it can
†Publisher’s note. Whilst IOP Publishing adheres to and respects UN resolutions regarding the designations of territ-
ories (available at www.un.org/press/en), the policy of IOP Publishing is to use the afliations provided by its authors
on its published articles.
∗Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution
4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the
title of the work, journal citation and DOI.
© 2024 The Author(s). Published by IOP Publishing Ltd
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
be used to nd Boltzmann weights in terms of the quantum ˆ
Rmatrix when the
adjoint representation denes the admissibility conditions.
Keywords: models, integrable lattice models, lattice, CFT,
interaction-round the face models, approaches
1. Introduction
Integrable lattice models have various connections with two-dimensional conformal eld the-
ories (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 singu-
larity, the tricritical Ising model, the three-state Potts model, the eight-vertex solid-on-solid
(SOS) model, etc for a review, see [2]. The underlying reason for this relationship is the fact
that integrable lattice models implement representations of quantum groups, with Boltzmann
weights (BWs) associated to a vertex or face of the model expressible via quantum R matrix
computed in a particular representation. The same quantum groups also arise in the context of
corresponding CFTs. Additionally, integrable lattice models offer a rich ground for mathem-
atical applications in knot theory, for a comprehensive review, refer to [3].
There are two major classes of integrable lattice models, namely vertex models (such as
6-vertex and 8-vertex models studied by Baxter [4]) and Interaction-Round the Face (IRF)
models (e.g. SOS or RSOS models). The former provide representations of ordinary quantum
groups with BWs of a vertex satisfying Yang-Baxter equation (YBE), while the latter have
BWs obeying star-triangle relation (equivalent to dynamical YBE) and therefore form repres-
entations of dynamical quantum groups introduced in [5] (see [6] and references therein for a
mathematical review). The connection with CFTs becomes particularly profound for a class
of IRF models [7–9] constructed by Jimbo et al Firstly, these models are based on afne Lie
algebras su(n)k, which correspond to the chiral symmetry algebras of Wess–Zumino–Witten
(WZW) models. The solutions for the BWs of these models, as given by elliptic functions
(see rhs of equation (2.6)), depend on the spectral and elliptic parameters uand p, respect-
ively. Secondly, an intriguing connection with CFTs arises from studying the local state prob-
ability Pa, representing the probability of nding the conguration aat the center vertex of
the lattice. Remarkably, it was discovered that in a special regime4,Pais expressed in terms
of one-dimensional conguration sums, precisely matching the branching functions5of the
Goddard–Kent–Olive cosets su(n)k⊕su(n)1
su(n)k+1[13], which describe minimal models in CFTs6.
The profound relationship between IRF models and CFTs makes these models a rich arena
to explore. In [14], a general method was proposed for constructing and solving IRF models
from the data of CFTs. We will elaborate on this method in sections 2and 3. For now, let
us outline its general features. According to [14], one can construct an IRF model based on
some given CFT in the following way: First, the model is built upon the chiral algebra of the
CFT. Second, the uctuating variables situated on the lattice vertices correspond to the primary
elds of the CFT (which, in turn, correspond to representations of the algebra), and third, the
4Specically, in the regime III, that is, the regime 0 <p<1,−πn
2(k+n)<u<0, and the limit p′→0,u→0 and
(p′)u−xed, where p′is a modular transformation of p, namely, if p=e−ϵ, then p′=e−4π2/ϵ. See [9] for more
details on these notions and [10–12] for recent developments regarding this idea.
5The branching functions Ba,b,cof the cosets can be dened as χaχb=∑cχcBa,b,c, were χa, χb, χcare the charac-
ters associated to the representations a,b,cof the algebras su(n)k,su(n)1,su(n)k+1, respectively. See chapter 18 of
[2] for details on these notions.
6The well-known example of these cosets is the case n=2 that corresponds to Virasoro CFT.
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
admissible congurations on the lattice are determined by the fusion rules of the CFT. With
respect to the procedure to solve the IRF model, the key idea of the method [14] is that the
BWs can be expressed in terms of the matrix elements of the braiding matrices of the CFT.
The braiding matrices of a CFT are those objects that allow us to express four-point conformal
blocks in the s-channel in terms of conformal blocks in the u-channel (see footnote 11 for
clarication).
Two other approaches can be used to nd BWs of IRF models: the fusion procedure (ini-
tially formulated in [15] and subsequently applied, e.g. in [16,17]) and the Vertex-IRF cor-
respondence. The fusion procedure is a method that allows the construction of new solutions
to the models from the solutions when the fundamental representation determines the admiss-
ibility conditions on the lattice. On the other hand, as we will explain below, the vertex-IRF
correspondence is a method that allows us to construct solutions to IRF models from the solu-
tions of vertex models. In the rst part of the paper (sections 2and 3), we focus on the approach
proposed in [14] 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 dene restricted IRF models and explain the key concepts associated with them
in section 2. To present our results, we give a brief overview below. Our study focuses on
IRF models based on the afne Lie algebras su(2)kand su(3)k. The models are dened on a
square lattice, as depicted in gure 1. The uctuating variables, which take values from the
set of dominant integral weights at a xed level kof the algebra, are situated on the vertices
of the lattice. To dene admissible congurations on the lattice, we initially select a weight h
from the set of dominant integral weights. Considering two nearest neighboring vertices with
values (a,b), arranged either horizontally from left to right or vertically from top to bottom,
as depicted in gure 1(1). The conguration (a,b) of these vertices is considered admissible if
the fusion coefcient Nb
a,hsatises the condition
Nb
a,h>0,(1.1)
where Nb
a,harises in the tensor product
a⊗h=M
i
Ni
a,h×[i],(1.2)
where [i]denotes the representations appearing in this tensor product. The condition (1.1)
simply implies that the conguration (a,b) is admissible if b∈a⊗h. From the admissibility
condition (1.1), it follows that a conguration of a face (a,b,c,d) as shown in gure 1(2)
is admissible only if each pair of nearest neighboring vertices satises (1.1), i.e. if (2.2) is
satised. It is noteworthy that the algebra itself, together with the values of h,k, completely
dene the model and x the number of possible admissible congurations that a face can take.
The interaction of the model is determined by the BWs ω, which are functions associated with
each particular face conguration and satisfy the YBE (dened below). This work aims to nd
the BWs for admissible7face congurations of the models.
In section 3, we review the ansatz proposed in [14] for nding the BWs of IRF models. In
section 3.1, we rst consider a warm-up example applying this ansatz to the model based on
su(2)kat level k=2 with hcorresponding to the fundamental representation. This model was
studied by Jimbo et al [9]. At this level, there are eight possible admissible face congurations
yielding nonzero BWs. We recover the solutions of [9].
7A BW is assigned zero if the face conguration is not admissible.
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
Figure 1. Two-dimensional lattice on which the models are dened. In (1), a cong-
uration (a,b) of two horizontal and vertical neighboring vertices is shown. In (2), a
conguration (a,b,c,d) of a face is shown.
In section 3.2, we study the model based on su(2)kat level k=4, with hcorresponding
to the adjoint representation. This model was investigated by Tartaglia and Pearce in [18]. At
level k=4, there are 33 possible admissible face congurations for the BWs. Using the ansatz
mentioned above [14], we obtain solutions parameterized by four independent parameters that
reproduce the solutions found in [18].
In section 3.3, we consider the model based on su(3)kat level k=2, with hcorresponding to
the adjoint representation. At this level, the fusion rules of the algebra are free of multiplicities,
and there are 21 possible admissible face congurations for the BWs. The solutions we found
completely follow the ansatz [14].
In the case of su(3)kalgebra at level k⩾3, the multiplicities in the tensor products are
present (Nb
a,h⩾2) and the approach [14] is not applicable. In this respect, in section 4, we
explore the vertex-IRF correspondence approach, which is based on the ˆ
Rmatrix of quantum
algebra slq(3). Using this correspondence, one can express the BWs of unrestricted8IRF mod-
els in terms of ˆ
Rmatrix elements. In the present paper, we describe how the computation of
the quantum ˆ
Rmatrix can be performed along the lines of the method outlined in [19], and
provide the necessary ingredients for this method, which are the generators of slq(3)in the
adjoint representation. Because the computation of the ˆ
Rmatrix is bulky due to the large size
of the matrix, the explicit derivation of the BWs will be considered in a separate work [20].
In section 5, we summarize our results and briey discuss possible applications to a class
of IRF models based on a quotient SU(n)k
Zn[21]. The case n=2 has already been studied in [22].
For n>2, the problem of multiplicity arises, and we expect that the present research results
will help nd exact solutions for this class of IRF models.
2. Description of the restricted models
The restricted IRF models are dened on a two-dimensional square lattice as shown in gure 1.
The uctuating variables are associated with the lattice vertices. A face is formed by four
nearest neighboring vertices (in gure 1(2) is shown a face with vertices a,b,c,and d).
IRF models can be dened for a general CFT model, with chiral algebra O. In the case of
afne Lie algebras, the corresponding CFTs are WZW models [2,23]9. The primary elds of
8Here, ‘unrestricted’, refers that the uctuating variables are not restricted to a xed level k.
9In this work, we focus on IRF models based on WZW models with afne Lie algebras su(2)kand su(3)k.
4

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
the CFT model are labeled by the representations of the algebra Oand the uctuating vari-
ables of the IRF model correspond to integrable highest-weight representations, characterized
by dominant integral highest-weights (dened below). For afne Lie algebras of rank n−1,
we denote the set of fundamental weights as Λ0,Λ1,...,Λn−1. Consequently, the uctuating
variables residing on the lattice vertices take values from the set of dominant integral weights
P+(n,k),
P+(n,k) = (a=
n−1
X
i=0
aiΛi= (a0,a1,...,an−1),
n−1
X
i=0
ai=k=level,ai∈Z,ai⩾0),(2.1)
where kis a positive integer referred to as the level of weights, serving as a parameter of
the model, and Zrepresents the set of integers. Among all the weights (2.1), we choose two
weights, hand v, to dene the admissibility conditions in both the horizontal and vertical
directions of the lattice. For a given CFT algebra Oand elds hand v, the corresponding IRF
model is denoted as IRF (O,h,v) following the notation in [14]. In the present study, we focus
on the cases where h=v.
Now, we dene the admissibility conditions for congurations 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 satises (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 gure 1(2)). The face conguration (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
Nb
a,h>0,Nd
a,h>0,Nc
b,h>0,Nc
d,h>0.(2.2)
For each face conguration, we assign a Boltzmann weight
ωa b
d c
u.(2.3)
Hence, the BWs depend on the conguration (a,b,c,d)and the spectral parameter u. We set
the BW to zero if the face conguration is not admissible. The partition function of the lattice
model is given by
Z=X
congurations Y
faces
ωa b
d c
u.(2.4)
We wish to dene 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 YBE, see, e.g. [14]
X
g
ωa b
f g
u+vωf g
e d
uωb c
g d
v
=X
g
ωa g
f e
vωa b
g c
uωg c
e d
u+v.(2.5)
In the case of trigonometric solutions (considered here), the BWs are parameterized by trigo-
nometric 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
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
trigonometric solutions from elliptic solutions by taking the limit p→010. 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:
sinu→2p1
8sinu
∞
Y
n=11−2cos(2u)pn+p2n(1−pn) = θ1(p,u).(2.6)
In the following section, will explain the approach employed to derive the BWs of the
considered models from the YBE (2.5).
3. CFT approach
In this section, we describe the method proposed in [14] to obtain the solutions to YBE for the
BWs. This method suggests that BWs of the IRF (O,h,h) can be computed using the following
prescription. We label the Nrepresentations appearing in the tensor product h⊗has follows
h⊗h=
N−1
M
j=0
Nj
h,h×ϕj.(3.1)
According to the proposal, the solutions of the YBE are given by
ωa b
d c
u≡
N−1
X
j=0
˜
P(j)[a,b,c,d]ρj(u).(3.2)
Here, ˜
P(j)[a,b,c,d]is a projector dened as
˜
P(j)[a,b,c,d] = ⟨a,b,c|P(j)|a,d,c⟩=Y
l=j
Bb,dh h
a c−δb,dλj
λl−λj
,(3.3)
where Bb,dh h
a crepresents the braiding matrix11 (for details see, e.g. [14,24]) associated
with the CFT Oand λjare the eigenvalues of the braiding matrix, these are given by
λj=ϵjeiπ(∆ϕj−2∆h),(3.4)
where ϵj=±1 depending on whether ϕjappears symmetrically or anti-symmetrically
in (3.1) and ∆h,∆ϕjare conformal dimensions of the primary elds corresponding to the
10 The elliptic solutions for BWs are given by a ratio of two elliptic theta functions. Therefore taking this limit p→0
results in a ratio of trigonometric functions.
11 Diagrammatically, the braiding matrix Bhas the property
where (i,j,k,l)label external primary elds and (n,m) label intermediate primary elds in the four-point correlation
function. The left and right diagrams denote the four-point conformal blocks in s-channel and u-channel, respectively.
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
representations h,ϕjrespectively. The conformal dimension ∆aof a primary eld correspond-
ing to the representation ais given by the formula
∆a=(a,a+2ρ)
2(k+g),(3.5)
where grepresents the dual Coxeter number of the algebra Ounder consideration, and ρis
the Weyl vector ρ=Pn−1
i=0Λi. The proposal (3.2) relies signicantly on the functions ρj(u),
which are dened as follows
ρj(u) = Qj
r=1sin(ζr−1−u)QN−1
r=j+1sin(ζr−1+u)
QN−1
r=1sin(ζr−1),(3.6)
where the parameters ζireferred to as crossing parameters in this proposal, are determined as
follows
ζi=π∆ϕi+1−∆ϕi
2,i=0,1,...,N−2.(3.7)
The projectors ˜
P(j)[a,b,c,d]satisfy the property PN−1
j=0P(j)[a,b,c,d] = 1, where 1denotes the
identity operator. Consequently, the BWs (3.2) satisfy the condition
ωa b
d c
0=δb,d.(3.8)
Now, let us explain how we compute the BWs in sections 3.1–3.3. In general, the kdepend-
ence of the braiding matrix appearing in (3.3) is not known explicitly, however, it is clear
that for concrete values of kand (a,b,c,d,h), the braiding matrix is a xed numerical matrix.
Therefore, the matrix elements of the ˜
P(j)projector also have xed numerical values, which
we need to nd. We substitute the ansatz (3.2) into YBE (2.5) and (3.8), and then nd solu-
tions for ˜
P(j)[a,b,c,d]for all possible face congurations. The possible nonzero ˜
P(j)[a,b,c,d]
values (for a given (a,b,c,d)) correspond to the elds ϕj, which belong to the domain:
(a∗⊗c)∩(h⊗h),(3.9)
where a∗represents the conjugate representation of a, this allows us to reduce the set of
unknowns, see [22]. The results of these computations are presented in the subsequent sub-
sections.
It is worth noting that as the level kincreases (for nontrivial representations hand espe-
cially for su(n⩾3)k), the number of admissible face congurations, and consequently the
number of equations (2.5) becomes very large. This makes the task of nding solutions for the
BWs more complicated. However, the outlined method shows its effectiveness in various non-
trivial examples. For a general level k, to determine the BWs, it is required to rst nd expres-
sions in closed form for the braiding matrix or the projector ˜
P(j), as demonstrated for instance
in [22,25].
3.1. Warm-up example: su(2)2
Let us consider the model based on su(2)2with h= Λ0+ Λ1. At this level k=2, we nd 8
possible BWs. These BWs correspond exactly to the trigonometric solutions found by Jimbo
et al [7,8]. Let us describe them by labeling the elds as follows
2Λ0+0Λ1:= 0,1Λ0+1Λ1:= 1,0Λ0+2Λ1:= 2.(3.10)
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
The relevant tensor product rules are,
1⊗2=1,
1⊗1=0⊕2,
2⊗2=0.
(3.11)
Since the tensor product 1 ⊗1 contains two representations, N=2. Therefore, we have one
crossing parameter given by
ζ0=π
4:= ζ. (3.12)
With these data, we can now search for solutions to the YBE (2.5) and the initial condition (3.8)
in the form of (3.2). We nd the following solutions
ω(0,1,0,1,u) = s(ζ+u)
s(ζ),
ω(0,1,2,1,u) = s(ζ−u)
s(ζ),
ω(1,0,1,0,u) = 1
2
s(ζ−u)
s(ζ)+1
2
s(ζ+u)
s(ζ),
ω(1,0,1,2,u) = 1
2
s(ζ−u)
s(ζ)−1
2
s(ζ+u)
s(ζ),
ω(1,2,1,0,u) = 1
2
s(ζ−u)
s(ζ)−1
2
s(ζ+u)
s(ζ),
ω(1,2,1,2,u) = 1
2
s(ζ−u)
s(ζ)+1
2
s(ζ+u)
s(ζ),
ω(2,1,0,1,u) = s(ζ−u)
s(ζ),
ω(2,1,2,1,u) = s(ζ+u)
s(ζ).(3.13)
Here and in what follows we use the following notations:
ω(a,b,c,d,u) := ωa b
d c
u,(3.14)
s(x) := sin(x).(3.15)
The solutions (3.13) coincide with the trigonometric limit of BWs obtained in [7,8].
3.2. su(2)4in the adjoint representation
In this model, we consider h=2Λ0+2Λ1, which corresponds to the (2,2)-fused model studied
by Tartaglia and Pearce [18]. At this level k=4, the model contains ve elds, which are
labeled as follows
4Λ0+0Λ1:= 0,3Λ0+1Λ1:= 1,2Λ0+2Λ1:= 2,Λ0+3Λ1:= 3,
0Λ0+4Λ1:= 4.(3.16)
8

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
The relevant tensor product rules that are used in the admissibility conditions and when xing
the set of unknowns ˜
P(j), are the following
2⊗1=1⊕3,1⊗1=0⊕2,
2⊗2=0⊕2⊕4,1⊗3=2⊕4,
2⊗3=1⊕3,3⊗3=0⊕2,
2⊗4=2,4⊗4=0.
(3.17)
Since N=3, there are two crossing parameters
ζ0=π(∆2−∆0)
2=π
3:= ζ,
ζ1=π(∆4−∆2)
2=2ζ.
(3.18)
With this data, we solve the YBE with the ansatz (3.2), taking into account (3.8) and nd
solutions which are parameterized by four parameters s1,s2,s3,s4. In this case, there are 33
admissible congurations on the faces. The solutions for the BWs are
ω(0,2,0,2,u) = s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ),
ω(0,2,2,2,u) = s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(0,2,4,2,u) = s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ),
ω(1,1,1,1,u) = s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(1,1,3,1,u) = s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(1,1,1,3,u) = s(ζ+u)s(2ζ+u)
4s1s(ζ)s(2ζ)−s(ζ−u)s(2ζ+u)
4s1s(ζ)s(2ζ),
ω(1,1,3,3,u) = s2s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ)−s2s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ),
ω(1,3,1,1,u) = s1s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ)−s1s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(1,3,3,1,u) = s(ζ−u)s(2ζ+u)
4s2s(ζ)s(2ζ)−s(ζ−u)s(2ζ−u)
4s2s(ζ)s(2ζ),
ω(1,3,1,3,u) = s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(1,3,3,3,u) = s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(2,0,2,0,u) = s(ζ−u)s(2ζ−u)
4s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
4s(ζ)s(2ζ),
ω(2,0,2,2,u) = s(ζ−u)s(2ζ−u)
16s3s4s(ζ)s(2ζ)−s(ζ+u)s(2ζ+u)
16s3s4s(ζ)s(2ζ),
ω(2,0,2,4,u) = −s(ζ−u)s(2ζ−u)
8s4s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
4s4s(ζ)s(2ζ)−s(ζ+u)s(2ζ+u)
8s4s(ζ)s(2ζ),
9

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
ω(2,2,2,0,u) = 2s3s4s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ)−2s3s4s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ),
ω(2,2,0,2,u) = s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(2,2,2,2,u) = s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(2,2,4,2,u) = s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(2,2,2,4,u) = s3s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ)−s3s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ),
ω(2,4,2,0,u) = s4s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ)−s4s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)−s4s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(2,4,2,2,u) = s(ζ+u)s(2ζ+u)
8s3s(ζ)s(2ζ)−s(ζ−u)s(2ζ−u)
8s3s(ζ)s(2ζ),
ω(2,4,2,4,u) = s(ζ−u)s(2ζ−u)
4s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
4s(ζ)s(2ζ),
ω(3,1,1,1,u) = s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(3,1,3,1,u) = s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(3,1,1,3,u) = s2s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ)−s2s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ),
ω(3,1,3,3,u) = 4s1s2
2s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ)−4s1s2
2s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(3,3,1,1,u) = s(ζ−u)s(2ζ+u)
4s2s(ζ)s(2ζ)−s(ζ−u)s(2ζ−u)
4s2s(ζ)s(2ζ),
ω(3,3,3,1,u) = s(ζ+u)s(2ζ+u)
16s1s2
2s(ζ)s(2ζ)−s(ζ−u)s(2ζ+u)
16s1s2
2s(ζ)s(2ζ),
ω(3,3,1,3,u) = s(ζ−u)s(2ζ−u)
2s(ζ)s(2ζ)+s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(3,3,3,3,u) = s(ζ−u)s(2ζ+u)
2s(ζ)s(2ζ)+s(ζ+u)s(2ζ+u)
2s(ζ)s(2ζ),
ω(4,2,0,2,u) = s(ζ−u)s(2ζ−u)
s(ζ)s(2ζ),
ω(4,2,2,2,u) = s(ζ−u)s(2ζ+u)
s(ζ)s(2ζ),
ω(4,2,4,2,u) = s(ζ+u)s(2ζ+u)
s(ζ)s(2ζ).(3.19)
We nd that in the particular case
s1=−3
2,s2=−1
2,s3=−1
4,s4=−1
2,(3.20)
our solutions reproduce exactly the Tartaglia and Pearce solutions.
10

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
3.3. su(3)2in the adjoint representation
In this subsection, we present the 21 BWs of the model based on su(3)2with h=0Λ0+ Λ1+
Λ2= (0,1,1). In this case, there are six elds, which we enumerate as follows
(2,0,0) := 0,(1,1,0) := 1,(1,0,1) := 2,(0,2,0) := 3,(0,0,2) := 4,
(0,1,1) := 5.(3.21)
The relevant tensor product decompositions for solving this model are
5⊗5=0⊕5,5⊗0=5,
5⊗1=1⊕4,2⊗4=5,
5⊗2=2⊕3,1⊗3=5,
5⊗3=2,1⊗2=0⊕5,
5⊗4=1,3⊗4=0,
4⊗3=0.
(3.22)
At this level, the tensor product 5 ⊗5 contains only two representations, indicating that we have
N=2 and therefore a single crossing parameter denoted as ζ0. According to the formula (3.7),
this parameter is expected to be 3π
10 . However, our calculation shows that in order to obtain
solutions to (2.5), the parameter is actually given by
ζ0=π
5:= ζ. (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 eld theory’ [14]. 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
¯
∆a=r∆amodulo1,(3.24)
where ris any integer strange to k+g. This is a different CFT with the same fusion rules. Thus,
the crossing parameters ζibecome ζirmodulo one. In our particular example, ζ0=3π/10.
But multiplying by r=4 we get ζ0=π/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
xis the appropriate solution, and similarly for the elliptic solution.
The solutions we derived depend on three parameters: s1,s2,s3. They can be expressed as
follows
ω(0,5,0,5,u) = s(ζ+u)
s(ζ),
ω(0,5,5,5,u) = s(ζ−u)
s(ζ),
ω(1,1,1,1,u) = √5−1s(ζ+u)
2s(ζ)+3−√5s(ζ−u)
2s(ζ),
ω(1,1,4,1,u) = s(ζ−u)
s(ζ),
ω(1,1,1,4,u) = s1s(ζ+u)
s(ζ)−s1s(ζ−u)
s(ζ),
ω(1,4,1,1,u) = √5−2s(ζ+u)
s1s(ζ)+2−√5s(ζ−u)
s1s(ζ),
11

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
ω(1,4,1,4,u) = √5−1s(ζ−u)
2s(ζ)+3−√5s(ζ+u)
2s(ζ),
ω(2,2,2,2,u) = √5−1s(ζ+u)
2s(ζ)+3−√5s(ζ−u)
2s(ζ),
ω(2,2,3,2,u) = s(ζ−u)
s(ζ),
ω(2,2,2,3,u) = s2s(ζ+u)
s(ζ)−s2s(ζ−u)
s(ζ),
ω(2,3,2,2,u) = √5−2s(ζ+u)
s2s(ζ)+2−√5s(ζ−u)
s2s(ζ),
ω(2,3,2,3,u) = √5−1s(ζ−u)
2s(ζ)+3−√5s(ζ+u)
2s(ζ),
ω(3,2,2,2,u) = s(ζ−u)
s(ζ),
ω(3,2,3,2,u) = s(ζ+u)
s(ζ),
ω(4,1,1,1,u) = s(ζ−u)
s(ζ),
ω(4,1,4,1,u) = s(ζ+u)
s(ζ),
ω(5,0,5,0,u) = √5−1s(ζ−u)
2s(ζ)+3−√5s(ζ+u)
2s(ζ),
ω(5,0,5,5,u) = s3s(ζ+u)
s(ζ)−s3s(ζ−u)
s(ζ),
ω(5,5,5,0,u) = √5−2s(ζ+u)
s3s(ζ)+2−√5s(ζ−u)
s3s(ζ),
ω(5,5,0,5,u) = s(ζ−u)
s(ζ),
ω(5,5,5,5,u) = √5−1s(ζ+u)
2s(ζ)+3−√5s(ζ−u)
2s(ζ).(3.25)
By imposing some specic conditions on the parameters s1,s2,s3, these solutions satisfy the
symmetry
ωa b
d c
u=ωa d
b c
u.(3.26)
This is an example of an IRF model based on su(3)kwith hin the adjoint representation. For
k>2 there are multiplicities in tensor products, we will return to this question in section 5.
12

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
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 uctuating variables in unrestricted models are not limited by the
level k. Specically, the uctuating variables located at the vertices of the lattice can assume
values from the set Pof integral weights of the algebra. Here we are interested in a model based
on su(3)k, particularly when the adjoint representation denes the admissibility conditions. For
the afne Lie algebra su(3)kthe set Pcan be expressed as follows
P=(a=
2
X
i=0
aiΛi= (a0,a1,a2),ai∈Z).(4.1)
We introduce the weights of the adjoint representation of the corresponding nite algebra su(3)
as follows
e1= (1,1),e2= (−1,2),e3= (2,−1),e4= (0,0),
e5= (0,0),e6= (1,−2),e7= (−2,1),e8= (−1,−1).(4.2)
Their corresponding afne extensions (at level zero) are
ˆ
e1= (−2,1,1),ˆ
e2= (−1,−1,2),ˆ
e3= (−1,2,−1),ˆ
e4= (0,0,0)1,
ˆ
e5= (0,0,0)2,ˆ
e6= (1,1,−2,),ˆ
e7= (1,−2,1),ˆ
e8= (2,−1,−1).(4.3)
For the unrestricted IRF model, a pair (a,b) is termed admissible if
b=a+ˆ
ei,for somei=1,2,...,8.(4.4)
A face conguration (a,b,c,d)is admissible if the pairs (a,b), (b,c), (a,d), (d,c) are all admiss-
ible in the sense (4.4).
According to [26–28], the BWs of unrestricted IRF models can be dened in terms of the
matrix elements of the so-called quantum ˆ
Rmatrix. We call this approach ‘vertex-IRF corres-
pondence’ because the matrix ˆ
Rdirectly determines the BWs of vertex-type models. As men-
tioned earlier, our study here involves two main objectives: (1) determining the BWs of the
unrestricted IRF model based on su(3)kwhen the adjoint representation denes the admissib-
ility conditions, and (2) extending the approach of section 3to a situation where multiplicities
occur by obtaining the exact expressions for the functions ρj(u)and ˜
P(j). Here, we explain
the Vertex-IRF correspondence and the method outlined in [19] for computing the ˆ
Rmatrix
elements. Since the computation of the ˆ
Rmatrix elements in the adjoint representation is a
demanding problem in its own right, we plan to address it separately in [20]. By denition, the
quantum ˆ
Rmatrix satises the YBE in the form
I⊗ˆ
R(u)ˆ
R(u+v)⊗II⊗ˆ
R(v)=ˆ
R(v)⊗II⊗ˆ
R(u+v)ˆ
R(u)⊗I.(4.5)
Both sides of the equation (4.5) act on the tensor product of three vector spaces: V1⊗V2⊗
V3.Iis the identity operator and ˆ
Ris dened as follows
ˆ
R=P·R,(4.6)
13

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
where Pdenotes the transposition Px ⊗y=y⊗x, and R, also called quantum Rmatrix12, acts
on the tensor product of two vector spaces in terms of its matrix elements as follows
R(u)(ei⊗ej) = X
m,n
Rm,n
i,j(u) (em⊗en).(4.7)
We can rewrite (4.5) in terms of Rfor a given sextuplet (i1,i2,i3,f1,f2,f3), yielding
X
s1,s2,s3
Rf1,f2
s2,s3(u)Rs2,f3
i1,s1(u+v)Rs3,s1
i2,i3(v) = X
s1,s2,s3
Rf2,f3
s2,s3(v)Rf1,s3
s1,i3(u+v)Rs1,s2
i1,i2(u).(4.8)
In the context of IRF models, we are interested in the case V1=V2=V3=Vand Vis the
vector space of the adjoint representation of su(3). However, in this case, obtaining the matrix
Ris a nontrivial task due to its size. Since the adjoint representation contains eight states, the
matrix Rbecomes a 64 ×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 Rmatrix has the
property
Rk,l
i,j=0 ifei+ej=ek+el.(4.9)
Following the Vertex-IRF correspondence (see, e.g. [26,27]), one can compute the BWs of
unrestricted IRF models in terms of the matrix elements of ˆ
Ras follows
ωa b
d c
u=(ˆ
Rk,l
i,j(u)ifb−a=ˆ
ek,c−b=ˆ
el,d−a=ˆ
ei,c−d=ˆ
ej,
0 otherwise.(4.10)
This relation can be graphically represented as
(4.11)
The lhs square in the equation represents the BWs of the IRF13 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 [19]. This scheme
relies on certain fundamental elements. First, we introduce the slq(3)quantum algebra and its
generators in the adjoint representation. Two simple roots of the algebra are denoted as αa
(a=1,2). Three generators, namely Ha,Ea, and Fa, 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
12 It is also common to write the YBE (4.5) in the form R12(u)R13 (u+v)R23(v) = R23(v)R13 (u+v)R12(u), where
Rij(u)is an operator acting as Ron the ith and jth components and as identity on the other component.
13 In 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.
14

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
H1=













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













,H2=













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













,
(4.12)
the raising and lowering generators are
E1=














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
001000 q2+1
q0
000000 0 q2+1
q
0 0 0 0 0 0 0 0














,E2=














0 0 0 0 0 1 0 0
0 0 q2+1
q0 0010
0 0 0 q2+1
q0 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














,
(4.13)
F1=













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 q
q2+100010













,F2=













0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
001000 q
q2+10
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













.
(4.14)
They dene the slq(3)quantum algebra
ka=qHa/2,
kaEa=qEaka,kaEb=q−1/2Ebka,
kaFa=q−1Faka,kaFb=q1/2Fbka,
[k1,k2]=[E1,F2]=[E2,F1] = 0,
[Ea,Fa] = k2
a−k−2
a
q−q−1,
E2
aEb−q+q−1EaEbEb+EbE2
a=0,
F2
aFb−q+q−1FaFbFa+FbF2
a=0,(4.15)
15

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
where a,b=1,2 and a=b. The states of the adjoint representation (4.2) in the Euclidean basis
can be written as follows
e1= (1,0,0,0,0,0,0,0)T,e2= (0,1,0,0,0,0,0,0)T,e3= (0,0,0,0,0,1,0,0)T,
e4= (0,0,1,0,0,0,0,0)T,e5= (0,0,0,0,0,0,1,0)T,e6= (0,0,0,1,0,0,0,0)T,
e7= (0,0,0,0,0,0,0,1)T,e8= (0,0,0,0,1,0,0,0)T.
(4.16)
According to Jimbo’s scheme, one needs the following two elements
k0=q(H1+H2)/2,E0=q(H1−H2)/3F2F1−q−1F1F2.(4.17)
In [19] it was shown that the quantum Rmatrix (4.7) and (4.8) satises the following system
of linear equations (the solution of which is unique up to an overall factor)
R(u)Ea⊗k−1
a+ka⊗Ea=Ea⊗ka+k−1
a⊗EaR(u),
R(u)Fa⊗k−1
a+ka⊗Fa=Fa⊗ka+k−1
a⊗FaR(u),
[R(u),Ha⊗I+I⊗Ha] = 0,
(4.18)
and an important fact is that
R(u)euE0⊗k0+k−1
0⊗E0=euE0⊗k−1
0+k0⊗E0R(u).(4.19)
From (4.18), it is clear that besides the spectral parameter u, the Rmatrix also depends on
the parameter qof the quantum algebra. To compute the Rmatrix, one needs to put the above
generators in (4.17) and (4.18) and solve these equations for the matrix elements Rk,l
i,j. The
results of this computation will be reported in [20].
5. Conclusions and discussion
In this work, we have investigated restricted IRF models based on the afne Lie algebras su(2)k
and su(3)kfor various levels kand representations hthat determine the admissibility conditions
of the models. Specically, we have examined the models su(2)2with hcorresponding to
the fundamental representation, su(2)4with hcorresponding to the adjoint representation,
and su(3)2with halso 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 ρj(u)
that depend on the spectral parameter uand the crossing parameters ζjassociated with the
conformal dimensions of the elds (representations) appearing in the tensor product h⊗h. By
incorporating this ansatz into the YBE (2.5), we have obtained solutions for the BWs of the
abovementioned models. Our solutions for su(2)2and su(2)4reproduce known results [8,18],
while our solutions for su(3)2are new.
The work is focused on particular xed klevels, however, it is worth noting that the approach
can be extended to general levels. To this end, one must rst determine the corresponding
braiding matrix [22,25]. By successfully solving these models, we have demonstrated the
efciency of the approach in addressing new IRF models.
In the second part, we explored the approach based on the vertex-IRF correspondence.
The nal objective in this direction is to determine the BWs of the unrestricted model based
on su(3)kin terms of the quantum ˆ
Rmatrix. We discussed the method proposed in [19] for
nding ˆ
Rand 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
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J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
Figure 2. An example of two congurations that may differ only in the labels on their
edges.
for unrestricted models, the specic 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 [29,30]. 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⩾5, the tensor product of the
adjoint representation (excluding the Λ0component) h= Λ1+ Λ2= (1,1)yields:
h⊗h= (0,0)⊕(1,1)1⊕(1,1)2⊕(0,3)⊕(3,0)⊕(2,2).(5.1)
Here, (1,1)1and (1,1)2correspond to the adjoint representation that appears twice (this mul-
tiplicity corresponds to the multiplicity of the null weights e4,e5in (4.2)). Notice that to dene
a face conguration in these cases completely, it becomes necessary to introduce additional
labels on the edges. For example, gure 2illustrates two congurations where the elds 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).
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 [17]. Interesting
examples are the models based on the algebra of the quotient group SU(n)k
Zn. These quotients
were investigated in the context of non-diagonal modular invariant CFTs [31]. Their tensor
product rules were computed in [21]. These theories exhibit novel features not present in the
original su(n)kmodels. For instance, only representations that are Zn-invariant are present, and
there are so-called ‘xed-point representations’, which play a signicant role.
To illustrate these features, let us consider an example from [21], the quotient SU(3)6
Z3. The
tensor product of two adjoint representations, in this case, is given by
(1,1)⊗(1,1)=(0,0)⊕2(1,1)⊕2(3,3)⊕(2,2)⊕(2,2)′⊕(2,2)′′ ,(5.2)
on rhs, the rst three representations are also present in (5.1), but in this theory, the repres-
entations are identied up to the external automorphism σ, dened as σ(a1,a2)=(k−(a1+
a2),a1). Consequently, the representations (3,3), (0,3), and (3, 0) are identied. The xed-
point representations are those invariant under the automorphism σ, and for this level, they are
(2, 2), (2,2)′, and (2,2)′ ′ . These xed-point representations are treated as three distinct elds
(representations). In the original su(3)kmodels, these xed-point representations are absent,
and the question is how to incorporate them in the fusion procedure. Thus, nding the BWs
of IRF models based on these quotients remains an open question. In [22], the case of SU(2)k
Z2
was solved by combining the CFT approach and known solutions of the su(2)kmodel. The
strategy involved the approach 3to nd the BWs containing xed-point representations, and
for other BWs, the solutions of su(2)kwere used. However, if we want to apply this strategy to
other cases (such as SU(3)k
Z3), we encounter the following difculty. The tensor product rule of
xed-point representations in these cases (see section 2.1 of [21]) involves multiplicity, unlike
17

J. Phys. A: Math. Theor. 57 (2024) 155201 V Belavin et al
the case of SU(2)k
Z2. Consequently, this prevents us from employing the approach section 3. We
leave this problem as an open question for future research.
Data availability statement
All data that support the ndings of this study are included within the article (and any
supplementary les).
Acknowledgments
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).
ORCID iDs
Doron Gepner https://orcid.org/0000-0002-3296-5873
J Ramos Cabezas https://orcid.org/0000-0003-0477-7178
Boris Runov https://orcid.org/0000-0002-3473-4534
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