# Periodic Y-systems and Nahm sums: the rank 2 case

Yuma Mizuno

###### Abstract

We classify periodic Y-systems of rank 2 satisfying the symplectic property.
We find that there are six such Y-systems. In all cases, the periodicity
follows from the existence of two reddening sequences associated with the time
evolution of the Y-systems in positive and negative directions, which gives
rise to quantum dilogarithm identities associated with Donaldson-Thomas
invariants. We also consider $q$-series called the Nahm sums associated with
these Y-systems. We see that they are included in Zagier’s list of rank 2 Nahm
sums that are likely to be modular functions. It was recently shown by Wang
that they are indeed modular functions.

##  1 Introduction

###  1.1 Background

The Y-system is a system of algebraic relations satisfied by coefficients of a
cluster algebra, which has the following form:

$Y_{i}(u)Y_{i}(u-r_{i})=\prod_{j\in
I}\prod_{p=1}^{r_{i}-1}Y_{j}(u-p)^{[n_{ij;p}]_{+}}\bigl{(}1+Y_{j}(u-p)\bigr{)}^{-n_{ij;p}}$
(1.1)

where $I$ is a finite index set, $Y_{i}(u)$ for $i\in I$, $u\in\mathbb{Z}$ are
commuting variables, $r_{i}\in\mathbb{Z}_{\geq 1}$, and
$n_{ij;p}\in\mathbb{Z}$. We also use the notation $[n]_{+}\coloneqq\max(0,n)$.
Such equations are first discovered by Zamolodchikov in the study of
thermodynamic Bethe ansats [36], prior to the discovery of cluster algebras by
Fomin and Zelevinsky [7]. The most striking feature of Zamolodchikov’s
Y-systems, as well as their generalizations [22, 30] defined shortly after the
Zamolodchikov’s work, is that they are periodic, which was fully proved by
applying the theory of cluster algebras [9, 10, 15, 16, 21].

A systematic treatment of the Y-systems in the general setting of cluster
algebras, including the Y-systems arising from the thermodynamic Bethe ansatz
as spacial cases, was given by Nakanishi [27]. This approach was further
developed in [24], and it was shown that the algebraic relation (1.1) arises
from a cluster algebra if and only if the data $r_{i},n_{ij;p}$ have a certain
symplectic property. This allows the “axiomatic” study of Y-systems without
explicitly referring to cluster algebras. In this general setting, however,
the Y-system is typically not periodic, and so the study of periodic Y-systems
as a generalization of Zamolodchikov’s Y-systems would be further developed.
In particular, the classification problem for periodic Y-systems is a
challenging open problem (see the last comments in [27, Section 3]).

There are several classification results in the literature. Fomin and
Zelevinsky [10] showed that the classification when $r_{i}=2$, $n_{ij;p}\leq
0$, and $n_{ii;p}=0$ for any $i,j,p$ coincides with the Cartan-Killing
classification. Galashin and Pylyavskyy [12] generalized this result to show
that the classification when $r_{i}=2$ and $n_{ii;p}=0$ for any $i,p$
coincides with the classification of ADE bigraphs of Stembridge [31]. On the
other hand, the situation is more complicated when $r_{i}>2$ for some $i$, and
so far there has been no comprehensive classification results except when
$\lvert I\rvert=1$ where it is not difficult to give a complete classification
thanks to the work by Fordy and Marsh [11] (e.g. see [24, Example 5.6]).

In this paper, we make a first attempt to give a classification result
involving the case $r_{i}>2$ for some $i$. Precisely, we classify the periodic
Y-systems of the form (1.1) with $\lvert I\rvert=2$ satisfying the symplectic
property. We would like to emphasize that we consider general $r_{i},n_{ij;p}$
in the classification. The result is given in the next section.

We also discuss the relation to Nahm’s conjecture on $q$-series [26, 35] in
Section 1.3.

###  1.2 Main result

Let $I$ be a finite set. We denote by $\mathbb{Y}_{0}$ the set of pairs
$(r,n)$ where $r=(r_{i})_{i\in I}$ and $n=(n_{ij;p})_{i,j\in
I,p\in\mathbb{N}}$ are families of integers satisfying $r_{i}\geq 1$ for any
$i$ and

$n_{ij;p}^{\pm}=0$ unless $0<p<r_{i}$ (1.2)

for any $i,j,p$.

######  Definition 1.1.

Let $(r,n)\in\mathbb{Y}_{0}$. Let $\mathbb{P}$ be a semifield, and
$(Y_{i}(u))_{i\in I,u\in\mathbb{Z}}$ be a family of elements in $\mathbb{P}$.
We say that $(Y_{i}(u))$ _satisfies the Y-system_ associated with the pair
$(r,n)$ if the relation (1.1) holds for any $i,u$. The equation (1.1) itself
is called the _Y-system_ associated with $(r,n)$. We also say that
$(Y_{i}(u))$ is a _solution of the Y-system_ if it satisfies the Y-system.

It is useful to think a pair $(r,n)\in\mathbb{Y}_{0}$ as a triple of matrices
with polynomial entries by the map
$(r,n)\mapsto(N_{0}(z),N_{+}(z),N_{-}(z)):\mathbb{Y}_{0}\to(\operatorname{Mat}_{I\times
I}\mathbb{N}[z])^{3}$ defined by

$\displaystyle N_{0}(z)\coloneqq\operatorname{diag}(1+z^{r_{i}})_{i\in
I},\quad
N_{\pm}(z)\coloneqq\biggr{(}\sum_{p\in\mathbb{N}}n_{ij;p}^{\pm}z^{p}\biggl{)}_{i,j\in
I}$ (1.3)

where we set $n_{ij;p}^{\pm}\coloneqq[\pm n_{ij;p}]_{+}$. We also define the
map $(r,n)\mapsto A_{\pm}(z):\mathbb{Y}_{0}\to(\operatorname{Mat}_{I\times
I}\mathbb{Z}[z])^{2}$ by $A_{\pm}(z)\coloneqq N_{0}(z)-N_{\pm}(z)$. Since this
map is injective by the condition (1.2), we will identify $\mathbb{Y}_{0}$
with the image of this map. For example, we will use the term “the Y-system
associated with $A_{\pm}(z)\in\mathbb{Y}_{0}$”.

######  Definition 1.2.

We say that $A_{\pm}(z)\in\mathbb{Y}_{0}$ satisfies the _symplectic property_
if

$\displaystyle
A_{+}(z)A_{-}(z^{-1})^{\mathsf{T}}=A_{-}(z)A_{+}(z^{-1})^{\mathsf{T}},$ (1.4)

where $\mathsf{T}$ is the transpose of a matrix. We denote by $\mathbb{Y}$ the
subset of $\mathbb{Y}_{0}$ consisting of pairs satisfying the symplectic
property.

The pair $A_{\pm}(z)\in\mathbb{Y}_{0}$ satisfies the simplectic property if
and only if the Y-system associated with $A_{\pm}(z)\in\mathbb{Y}_{0}$ is
realized as the exchange relations of coefficients in a cluster algebra [24].
We review this fact in Section 2.1.

$A_{+}(z)$ | $A_{-}(z)$ | $h_{+}$ | $h_{-}$ |
---|---|---|---|---
$\begin{pmatrix}1+z^{2}&-z\\\ -z&1+z^{2}\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}&0\\\ 0&1+z^{2}\end{pmatrix}$ | $3$ | $2$ | (1)
$\begin{pmatrix}1+z^{2}&-z\\\ -z-z^{5}&1+z^{6}\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}&0\\\ -z^{3}&1+z^{6}\end{pmatrix}$ | $8$ | $6$ | (2)
$\begin{pmatrix}1+z^{2}&-z\\\ -z-z^{5}-z^{9}&1+z^{10}\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}&0\\\ -z^{3}-z^{7}&1+z^{10}\end{pmatrix}$ | $18$ | $10$ | (3)
$\begin{pmatrix}1+z^{2}&-z\\\ -z&1+z^{2}\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}-z&0\\\ 0&1+z^{2}-z\end{pmatrix}$ | $3$ | $3$ | (4)
$\begin{pmatrix}1+z^{2}&-z\\\ -z-z^{2}&1+z^{3}\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}-z&0\\\ 0&1+z^{3}\end{pmatrix}$ | $5$ | $3$ | (5)
$\begin{pmatrix}1+z^{2}&-z\\\ -z&1+z^{2}-z\end{pmatrix}$ | $\begin{pmatrix}1+z^{2}&0\\\ 0&1+z^{2}\end{pmatrix}$ | $5$ | $2$ | (6)
Table 1: Finite type classification for Y-systems of rank $2$. The numbers
$h_{\pm}$ are the length of reddening sequences in positive and negative
directions, respectively.

######  Definition 1.3.

We say that a solution of a Y-system is _periodic_ if there is a positive
integer $\Omega>0$ such that $Y_{i}(u+\Omega)=Y_{i}(u)$ for any $i,u$.

######  Definition 1.4.

We say that a pair $A_{\pm}(z)\in\mathbb{Y}$ is of _finite type_ if any
solution (in any semifield) of the Y-system associated with this pair is
periodic. In this case, we also say that Y-system itself is _periodic_.

The purpose of this paper is to classify periodic Y-systems of rank $2$.
Before stating the result, we give a few remarks. We say that
$A_{\pm}(z)\in\mathbb{Y}_{I}$ is _decomposable_ if it is a direct sum of some
$A^{\prime}_{\pm}(z)\in\mathbb{Y}_{I^{\prime}}$ and
$A^{\prime\prime}_{\pm}(z)\in\mathbb{Y}_{I^{\prime\prime}}$ with nonempty
$I^{\prime}$ and $I^{\prime\prime}$. We say that $A_{\pm}(z)\in\mathbb{Y}_{I}$
is _indecomposable_ if it is not decomposable. It is enough to consider
indecomposable pairs in the classification. We also note that $A_{\pm}(z)$ is
of finite type if and only if $A_{\pm}(z)^{\mathrm{op}}\coloneqq A_{\mp}(z)$
is of finite type by the correspondence between solutions $Y_{i}(u)\mapsto
Y_{i}(u)^{-1}$. The main results are summarized as follows:

######  Theorem 1.5.

Suppose that $I=\\{1,2\\}$.

  1. (1)

Any pair $A_{\pm}(z)\in\mathbb{Y}$ in Table 1 is of finite type.

  2. (2)

Any indecomposable pair $A_{\pm}(z)\in\mathbb{Y}$ of finite type is reduced to
exactly one pair in Table 1 by permuting the indices, changing sign, and
changing slices (see Section 3.1), if necessary.

The claim (1) can be proved by concrete calculation in a suitable universal
algebra since $A_{\pm}(z)$ in Table 1 is concrete. We, however, give another
proof involving cluster algebras. We give a quiver and a sequence of mutations
for each $A_{\pm}(z)$ in Table 1 that yields the Y-system as the exchange
relation of coefficients in the cluster algebra. See Table 3 for quivers and
mutations. We can verify that some iteration of this sequence of mutations, as
well as its inverse, is a reddening sequence (Theorem 2.8). Thanks to the deep
results in the theory of cluster algebras, this property is enough to imply
the periodicity (Proposition 2.7). The number $h_{\pm}$ in Table 1 are the
length of reddening sequences in positive and negative directions,
respectively. This verification of the periodicity is interesting not only
because it is computationally more efficient, but also because it leads to
nontrivial dilogarithm identities associated with Donaldson-Thomas invariants
(Corollary 2.10).

The claim (2) is proved in Section 3.2 by the following steps:

  1. Step 1.

We recall the result in [24] that asserts that $A_{\pm}(1)$ satisfies a
certain positivity, which in particular implies that
$\operatorname{tr}A_{\pm}(1)$ and $\det A_{\pm}(1)$ are positive. This allows
us to significantly reduce the candidates for finite type $A_{\pm}(z)$.

  2. Step 2.

For a fixed $A_{+}(1)$ in the candidates obtained in Step 1, we search for
$A_{-}(1)$ satisfying the symplectic property (1.4) at $z=1$.

  3. Step 3.

During the search in Step 2, we discard the pair $A_{\pm}(1)$ that cannot be
endowed with the parameter $z$ (Lemma 3.3 and 3.4).

  4. Step 4.

At this point, we have six candidates up to a permutation of the indices and a
change of sign. For each $A_{\pm}(1)$ in the six candidates, we try to endow
with the parameter $z$. It turns out that this is possible for all the six
candidates. We give all possible $A_{\pm}(z)$ in Lemma 3.5–3.8.

  5. Step 5.

We finally check that each remaining candidate reduces to one of $A_{\pm}(z)$
in Table 1 by change of slices.

######  Remark 1.6.

Most of the Y-systems obtained from Table 1 are already known in the
literature. $\mathrm{(1)}^{\mathrm{op}}$ and $\mathrm{(6)}^{\mathrm{op}}$ are
Zamolodchikov’s Y-system of type $A_{2}$ [36] and $T_{2}$ (“tadpole”) [30],
respectively. $\mathrm{(2)}^{\mathrm{op}}$ is the reduced sine-Gordon Y-system
associated with the continued fraction $3/4=[1,3]=1/(1+1/3)$, and
$\mathrm{(5)}^{\mathrm{op}}$ with $z$ replaced by $z^{2}$ is the reduced sine-
Gordon Y-system associated with $3/5=[1,1,2]=1/(1+1/(1+1/2))$ [32] . (4) is
the “half” of the Y-system associated with the pair $(A_{2},A_{2})$ [30]. (3)
appears to be new:

$\begin{split}Y_{1}(u)Y_{1}(u-2)&=\frac{1}{1+Y_{2}(u-1)^{-1}}\\\
Y_{2}(u)Y_{2}(u-10)&=\frac{\bigl{(}1+Y_{1}(u-3)\bigr{)}\bigl{(}1+Y_{1}(u-7)\bigr{)}}{\bigl{(}1+Y_{1}(u-1)^{-1}\bigr{)}\bigl{(}1+Y_{1}(u-5)^{-1}\bigr{)}\bigl{(}1+Y_{1}(u-9)^{-1}\bigr{)}}\end{split}$

although it is implicitly given in the author’s previous work [24, Table 2].

######  Remark 1.7.

The pair $A_{\pm}(z)\in\mathbb{Y}$ is called the _T-datum_ in [24] since it
describes the T-systems, which is a companion to the Y-systems. We do not use
this term since we only consider the Y-systems in this paper. Moreover, the
definition of the T-datum in [24] allows to have a non-diagonal $N_{0}$ and
have a nontrivial symmetrizer $D$, which is more general than the definition
in this paper. See also Section 1.4 for the Y-systems involving nontrivial
symmetrizers.

######  Remark 1.8.

There is another expression of the Y-system using a pair of matrices
$A_{\pm}(z)$ directly. Let $A_{\pm}(z)\in\mathbb{Y}_{0}$, and define
$a_{ij;p}\in\mathbb{Z}$ by

$A_{\pm}(z)=\biggl{(}\sum_{p\in\mathbb{N}}a_{ij;p}^{\pm}z^{p}\biggr{)}_{i,j\in
I}.$

Let $(P_{i}^{\pm}(u))_{i\in I,u\in\mathbb{Z}}$ be a family of elements in a
multiplicative abelian group $\mathbb{P}$. We say that $(P_{i}^{\pm}(u))$
_satisfies the multiplicative Y-system_ associated with $A_{\pm}(z)$ if

$\displaystyle\prod_{j\in
I}\prod_{p\in\mathbb{N}}P_{j}^{+}(u-p)^{a_{ij;p}^{+}}=\prod_{j\in
I}\prod_{p\in\mathbb{N}}P_{j}^{-}(u-p)^{a_{ij;p}^{-}}$ (1.5)

for any $i,u$ (schematically, “$A_{+}(z)\cdot\log P^{+}=A_{-}(z)\cdot\log
P^{-}$” under the action $z:u\mapsto u-1$). The solution $(P_{i}^{\pm}(u))$ is
called _normalized_ if $\mathbb{P}$ is endowed with a semifield structure, and

$P_{i}^{+}(u)+P_{i}^{-}(u)=1$

for any $i,u$. We have a one-to-one correspondence between solutions of the
Y-system (1.1) and normalized solutions of the multiplicative Y-system (1.5).
The correspondence is given by

$Y_{i}(u)\mapsto\frac{P_{i}^{+}(u)}{P_{i}^{-}(u)},\quad
P_{i}^{+}(u)\mapsto\frac{Y_{i}(u)}{1+Y_{i}(u)},\quad
P_{i}^{-}(u)\mapsto\frac{1}{1+Y_{i}(u)}.$

In the setting of cluster algebras, this correspondence is nothing but the
normalization of the coefficients described by Fomin and Zelevinsky [7,
Section 5].

###  1.3 Relation to Nahm sums

Consider the $q$-series defined by

$G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q)_{n}},\quad
H(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(q)_{n}},$ (1.6)

where $(q)_{n}\coloneqq(1-q)(1-q^{2})\cdots(1-q^{n})$ is the $q$-Pochhammer
symbol. The famous Rogers-Ramanujan identities express these $q$-series as the
following infinite products:

$G(q)=\prod_{n\equiv\pm 1\bmod 5}\frac{1}{1-q^{n}},\quad
H(q)=\prod_{n\equiv\pm 2\bmod 5}\frac{1}{1-q^{n}}.$

These expressions in particular implies that $q^{-1/60}G(q)$ and
$q^{11/60}H(q)$ are modular functions on some finite index subgroup of
$\operatorname{SL}(2,\mathbb{Z})$. In fact, it is a rare case that an infinite
sum of the form (1.6) is modular. It is known that the $q$-series

$\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}an^{2}+bn+c}}{(q)_{n}}$ (1.7)

with $a,b,c\in\mathbb{Q}$ is modular only if $a=1/2$, $1$, or $2$ [35].

Nahm [26] considered higher rank generalization of (1.7), which we call the
_Nahm sum_. Let $I$ be a finite set, and suppose that $A\in\mathbb{Q}^{I\times
I}$ is a symmetric positive definite matrix, $B\in\mathbb{Q}^{I}$ is a vector,
and $C\in\mathbb{Q}$ is a scalar. The Nahm sum is the $q$-series defined by

$f_{A,B,C}(q)\coloneqq\sum_{n\in\mathbb{N}^{I}}\frac{q^{\frac{1}{2}n^{\mathsf{T}}An+n^{\mathsf{T}}B+C}}{\prod_{a}(q)_{n_{i}}}.$

When $\lvert I\rvert\geq 2$, it is not well understood when $f_{A,B,C}(q)$ is
modular. Nahm gave a conjecture providing a criterion on the modularity of
$f_{A,B,C}(q)$ in terms of torsion elements in the Bloch group [26, 35]. See
[3, 33] for the development of this conjecture.

Nahm used Zamolodchikov’s periodicity to provide an evidence of the
conjecture. In fact, there is a natural way to give a candidate of modular
Nahm sums from finite type $A_{\pm}(z)\in\mathbb{Y}$ in general. Precisely,
the matrix $K\coloneqq A_{+}(1)^{-1}A_{-}(1)$ is always symmetric and positive
definite for finite type $A_{\pm}(z)\in\mathbb{Y}$, and it is conjectured that
it gives a modular Nahm sum $f_{K,0,C}(q)$ for some $C$ [24]. (This
construction is essentially the same as that in [18], except that they did not
prove that $K$ is symmetric and positive definite. A special case can also be
found in [23].) We note that the symplectic property (1.4) at $z=1$ plays an
important role here since it implies that $K$ is symmetric. On the other hand,
the positive definiteness is related to the periodicity of the Y-system.

$A_{\pm}(z)$ | $K$ | $-24C$ | RR |  | $A_{\pm}(z)$ | $K$ | $-24C$ | RR
---|---|---|---|---|---|---|---|---
(1) | $\begin{pmatrix}4/3&2/3\\\ 2/3&4/3\end{pmatrix}$ | $\dfrac{4}{5}$ | [6] |  | $\mathrm{(1)}^{\mathrm{op}}$ | $\begin{pmatrix}1&-1/2\\\ -1/2&1\end{pmatrix}$ | $\dfrac{6}{5}$ | [33]
(2) | $\begin{pmatrix}3/2&1\\\ 1&2\end{pmatrix}$ | $\dfrac{5}{7}$ | [34] |  | $\mathrm{(2)}^{\mathrm{op}}$ | $\begin{pmatrix}1&-1/2\\\ -1/2&3/4\end{pmatrix}$ | $\dfrac{9}{7}$ | [34]
(3), (6) | $\begin{pmatrix}2&2\\\ 2&4\end{pmatrix}$ | $\dfrac{4}{7}$ | [1] |  |  $\mathrm{(3)}^{\mathrm{op}}$, $\mathrm{(6)}^{\mathrm{op}}$ | $\begin{pmatrix}1&-1/2\\\ -1/2&1/2\end{pmatrix}$ | $\dfrac{10}{7}$ | [34]
(4) | $\begin{pmatrix}2/3&1/3\\\ 1/3&2/3\end{pmatrix}$ | $1$ | [35] |  | $\mathrm{(4)}^{\mathrm{op}}$ | $\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix}$ | $1$ | [35]
(5) | $\begin{pmatrix}1&1\\\ 1&2\end{pmatrix}$ | $\dfrac{3}{4}$ | [34] |  | $\mathrm{(5)}^{\mathrm{op}}$ | $\begin{pmatrix}2&-1\\\ -1&1\end{pmatrix}$ | $\dfrac{5}{4}$ | [4]
Table 2: The list of the matrix $K=A_{+}(1)^{-1}A_{-}(1)$. The Nahm sum
$f_{K,0,C}(q)$ is modular, which can be proved by using Rogers-Ramanujan type
identities (RR for short) given in the references in the table.

Based on our classification, we find that:

######  Theorem 1.9.

Suppose that $I=\\{1,2\\}$. The Nahm sum $f_{K,0,C}(q)$ is modular for any
finite type $A_{\pm}(z)\in\mathbb{Y}$, where $C$ is given in Table 2.

In fact, every $K$ from finite type $A_{\pm}(z)$ is included in the Zagier’s
list [35, Table 2] for rank $2$ candidates of modular Nahm sums. There are
Rogers-Ramanujan type identities that enable us to write each Nahm sum in the
list in terms of theta functions. The proof of the desired identities was
partially given in [1, 4, 6, 33, 35], and was recently completed by Wang [34]
except for one candidate that does not appears in our construction from
Y-systems. See Table 2.

######  Remark 1.10.

We can define the refinement $f_{A_{\pm}(z)}^{(s)}(q)$ of the Nahm sum
$f_{K,0,0}(q)$, which is parametrized by $s\in H$ for an abelian group $H$ of
order $\det A_{+}(1)$ such that it reduces to the original one by taking
summation [24, Definition 5.12]:

$f_{K,0,0}(q)=\sum_{s\in H}f_{A_{\pm}(z)}^{(s)}(q).$

It is conjectured that each $f_{A_{\pm}(z)}^{(s)}(q)$ is already modular after
multiplying $q^{C}$ for some $C$. We note that the symplectic property (1.4)
at $z=1$ again plays an important role in the definition of the refinement. We
will discuss this refinement for rank 2 case in more detail elsewhere. We
remark that similar refinement also appears in the context of $3$-dimensional
quantum topology [13, Section 6.3].

###  1.4 Remarks on higher rank and skew-symmetrizable case

We have seen that the following properties hold for rank $2$ case:

  1. (P1)

We have reddening sequences in both positive and negative directions.

  2. (P2)

The map $A_{\pm}(z)\mapsto A_{+}(1)^{-1}A_{-}(1)$ gives modular Nahm sums.

We expect that the properties 1 and 2 also hold for any finite type
$A_{\pm}(z)\in\mathbb{Y}$ of general rank. The followings are some known
examples:

  * •

For the Y-system associated with the untwisted quantum affine algebras
$U_{q}(X_{r}^{(1)})$ with level $\ell$ restriction [22], 1 holds with
$h_{+}=t\ell$ and $h_{-}=t\cdot(\text{dual Coxeter number of $X_{r}$})$ where
$t=1$, $2$, or $3$ is the multiplicity in the Dynkin diagram of $X_{r}$ [15,
16], and 2 holds under the assumption [14, Conjecture 5.3] by the result of
Kac and Peterson [17].

  * •

For the Y-system associated with a pair of finite type simply laced Dynkin
type $(X_{r},X^{\prime}_{r^{\prime}})$ [30], 1 holds with
$h_{+}=(\text{Coxeter number of $X_{r}$})$ and $h_{-}=(\text{Coxeter number of
$X^{\prime}_{r^{\prime}}$})$ [19, 21].

  * •

For the (reduced) sine-Gordon Y-system associated with the continued fraction
$p/q=[n_{F},\dots,n_{1}]=1/(n_{F}+1/(\cdots+1/n_{1}))$ [29, 32], 1 appears to
hold with $h_{+}=2p$ and $h_{-}=2q$.

  * •

For the Y-system associated with an admissible ADE bigraph $(\Gamma,\Delta)$
[12], 1 appears to hold with $h_{+}=(\text{Coxeter number of $\Gamma$})$ and
$h_{-}=(\text{Coxeter number of $\Delta$})$.

Moreover, we can consider Y-systems associated with skew-symmetrizable cluster
algebras rather than skew-symmetric ones discussed in this paper. In this
case, the symplectic property (1.4) becomes

$A_{+}(z)DA_{-}(z^{-1})^{\mathsf{T}}=A_{-}(z)DA_{+}(z^{-1})^{\mathsf{T}},$

where $D$ is a diagonal matrix called symmetrizer [24]. We also expect that
the properties 1 and 2 also hold for skew-symmetrizable case. See [24,
Definition 5.12] for the definition of the Nahm sum in skew-symmetrizable
case.

######  Acknowledgment.

This work is supported by JSPS KAKENHI Grant Number JP21J00050.

##  2 Y-systems and cluster algebras

###  2.1 Preliminaries on cluster algebras

In this paper, a _semifield_ is a multiplicative abelian group equipped with
an addition that is commutative, associative, and distributive with respect to
the multiplication.

######  Definition 2.1.

Let $I$ be a set. The set of all nonzero rational functions in the variables
$\boldsymbol{y}=(y_{i})_{i\in I}$ with natural number coefficients is a
semifield with respect to the usual addition and multiplication. This
semifield is called the _universal semifield_ , and denoted by
$\mathbb{Q}_{>0}(\boldsymbol{y})$. We have a canonical bijection
$\operatorname{Hom}_{\mathrm{semifield}}(\mathbb{Q}_{>0}(\boldsymbol{y}),\mathbb{P})\cong\operatorname{Hom}_{\mathrm{set}}(I,\mathbb{P})$
for any set $I$ and semifield $\mathbb{P}$.

######  Definition 2.2.

Let $I$ be a set. The _tropical semifield_
$\operatorname{Trop}(\boldsymbol{y})$ is the multiplicative free abelian group
generated by the variables $\boldsymbol{y}=(y_{i})_{i\in I}$ equipped with the
addition defined by

$\prod_{i}y_{i}^{a_{i}}+\prod_{i}y_{i}^{b_{i}}=\prod_{i}y_{i}^{\min(a_{i},b_{i})}.$

Let $I$ be a finite set and $\mathbb{P}$ be a semifield. A _Y-seed_ is a pair
$(B,\boldsymbol{y})$ where $B=(B_{ij})_{i,j\in I}$ is a skew-symmetric integer
matrix and $\boldsymbol{y}=(y_{i})_{i\in I}$ is a tuple of elements in
$\mathbb{P}$. We sometimes represent $B$ as the quiver whose signed adjacency
matrix is $B$. For a Y-seed $(B,\boldsymbol{y})$ and $k\in I$, the _mutation_
in direction $k$ transforms $(B,\boldsymbol{y})$ into the new Y-seed
$\mu_{k}(B,\boldsymbol{y})=(B^{\prime},\boldsymbol{y}^{\prime})$ given by

$\displaystyle B^{\prime}_{ij}$
$\displaystyle\coloneqq\begin{cases}-B_{ij}&\text{if $i=k$ or $j=k$},\\\
B_{ij}+[-B_{ik}]_{+}B_{kj}+B_{ik}[B_{kj}]_{+}&\text{otherwise},\end{cases}$
(2.1) $\displaystyle y^{\prime}_{i}$
$\displaystyle\coloneqq\begin{cases}y_{k}&\text{if $i=k$},\\\
y_{i}y_{k}^{[B_{ki}]_{+}}(1+y_{k})^{-B_{ki}}&\text{otherwise}.\end{cases}$
(2.2)

A mutation is involutive, that is,
$\mu_{k}(B,\boldsymbol{y})=(B^{\prime},\boldsymbol{y}^{\prime})$ implies
$(B,\boldsymbol{y})=\mu_{k}(B^{\prime},\boldsymbol{y}^{\prime})$. We have the
commutativity

$\mu_{i}\mu_{j}=\mu_{j}\mu_{i}\quad\text{if $B_{ij}=0$},$ (2.3)

which allows us to write $\mu_{\boldsymbol{i}}$ for a set
$\boldsymbol{i}\subseteq I$ such that $B_{ij}=0$ for any
$i,j\in\boldsymbol{i}$ to mean the successive mutations along arbitrarily
chosen order on $\boldsymbol{i}$.

For a Y-seed $(B,\boldsymbol{y})$ and a bijection $\nu:I\to I$, we define a
new Y-seed $\nu(B,\boldsymbol{y})=(B^{\prime},\boldsymbol{y}^{\prime})$ by
$B^{\prime}_{\nu(i)\nu(j)}\coloneqq B_{ij}$ and $y^{\prime}_{\nu(i)}\coloneqq
y_{i}$.

###  2.2 Solving Y-systems by cluster algebras

Let $A_{\pm}(z)\in\mathbb{Y}$. We will construct a solution of the Y-system
associated with $A_{\pm}(z)$ based on [24, Section 3.3]. We first define a
subset $R\subseteq I\times\mathbb{Z}$ by

$\displaystyle R\coloneqq\\{(i,u)\in I\times\mathbb{Z}\mid 0\leq u<r_{i}\\},$
(2.4)

and define a skew-symmetric $R\times R$ integer matrix $B$ by

$B_{(i,p)(j,q)}=-n_{ij;p-q}+n_{ji;q-p}+\sum_{k\in
I}\sum_{v=0}^{\min(p,q)}\bigl{(}n_{ik;p-v}^{+}{n}_{jk;q-v}^{-}-n_{ik;p-v}^{-}{n}_{jk;q-v}^{+}\bigr{)},$
(2.5)

where we understand $n_{ij;p}=0$ if $p<0$. We then define
$\boldsymbol{i}\coloneqq\\{(i,u)\mid u=0\\}\subseteq R$. We also define a
bijection $\nu:R\to R$ by

$\displaystyle\nu(i,p)=\begin{cases}(i,p-1)&\text{if $p>0$,}\\\
(i,r_{i})&\text{if $p=0$.}\end{cases}$ (2.6)

Then the symplectic property (1.4) ensures that
$\nu(\mu_{\boldsymbol{i}}(B))=B$ [24, Lemma 3.16]. We finally define a
sequence of Y-seeds

$\displaystyle\cdots\to(B,\boldsymbol{y}(-1))\to(B,\boldsymbol{y}(0))\to(B,\boldsymbol{y}(1))\to\cdots$
(2.7)

in $\mathbb{Q}_{>0}(\boldsymbol{y})$ by
$\boldsymbol{y}(0)\coloneqq\boldsymbol{y}$ and
$(B,\boldsymbol{y}(u+1))=\nu(\mu_{\boldsymbol{i}}(B,\boldsymbol{y}(u)))$. The
sequence (2.7) gives a solution of the Y-system:

######  Lemma 2.3.

[24, Theorem 3.13] $(y_{i,0}(u))_{i\in I,u\in\mathbb{Z}}$ satisfies the
Y-system associated with $A_{\pm}(z)$.

This solution is universal in the following sense.

######  Lemma 2.4.

[24, Theorem 3.19] Suppose that a family $(Y_{i}(u))_{i\in I,u\in\mathbb{Z}}$
satisfies the Y-system associated with $A_{\pm}(z)$. Define a semifield
homomorphism $f:\mathbb{Q}_{>0}(\boldsymbol{y})\to\mathbb{P}$ by

$\displaystyle f(y_{i,p})\coloneqq Y_{i}(p)\prod_{j\in
I}\prod_{q=0}^{p}Y_{j}(p-q)^{-[n_{ij;q}]_{+}}\bigl{(}1+Y_{j}(p-q)\bigr{)}^{n_{ij;q}}.$
(2.8)

Then $f(y_{i,0}(u))=Y_{i}(u)$ for any $i,u$.

######  Corollary 2.5.

$A_{\pm}(z)\in\mathbb{Y}$ is of finite type if and only if there are different
integers $u,v$ such that $\boldsymbol{y}(u)=\boldsymbol{y}(v)$ in (2.7).

###  2.3 Periodicity and reddening sequences

Similarly to (2.7), we define a sequence of Y-seeds

$\displaystyle\cdots\to(B,\boldsymbol{y}(-1))\to(B,\boldsymbol{y}(0))\to(B,\boldsymbol{y}(1))\to\cdots$
(2.9)

by the same formulas but now in $\operatorname{Trop}(\boldsymbol{y})$ rather
than $\mathbb{Q}(\boldsymbol{y})$.

######  Definition 2.6.

We say that the Y-system associated with $A_{\pm}(z)\in\mathbb{Y}$ is
_positive (resp. negative) reddening_ if there is a positive integer $u$ such
that all the exponents in $y_{i}(u)$ (resp. $y_{i}(-u)$) in (2.9) are
nonpositive for any $i$. We denote by $h_{+}$ (resp. $h_{-}$) the least such
positive integer $u$.

Equivalently, the Y-system is positive (resp. negative) reddening if and only
if all the entries in the C-matrix associated with the sequence of mutations
$(B,\boldsymbol{y}(0))\to(B,\boldsymbol{y}(u))$ (resp.
$(B,\boldsymbol{y}(0))\to(B,\boldsymbol{y}(-u))$) are nonpositive for some
$u>0$.

######  Proposition 2.7.

Suppose that the Y-system associated with $A_{\pm}(z)$ is positive and
negative reddening. Then $A_{\pm}(z)$ is of finite type.

###### Proof.

We verify the equivalent condition in Corollary 2.5. By [2, Proposition 2.10],
there are bijections $\sigma,\sigma^{\prime}:R\to R$ such that
$y_{i}(h_{+})=y_{\sigma(i)}^{-1}$ and
$y_{i}(-h_{-})=y_{\sigma^{\prime}(i)}^{-1}$ for any $i$ (in other words, the
C-matrices associated with them are the minus of permutation matrices). Now
the claim follows from the separation formula for $y$-variables [10,
Proposition 3.13] and the result on C-matrices shown by Cao, Huang, and Li [5,
Theorem 2.5]. See also [28, Theorem 5.2] for the corresponding statement
dealing with permutations that is actually suitable here. ∎

Quiver $B$ | $A_{\pm}(z)$
---|---
$(1,0)$$(2,1)$$(1,1)$$(2,0)$ | (1)
$(1,0)$$(2,1)$$(2,3)$$(2,5)$$(1,1)$$(2,0)$$(2,2)$$(2,4)$ | (2)
$(1,0)$$(2,1)$$(2,3)$$(2,5)$$(2,7)$$(2,9)$$(1,1)$$(2,0)$$(2,2)$$(2,4)$$(2,6)$$(2,8)$ | (3)
$(1,0)$$(2,1)$$(2,0)$$(1,1)$ | (4)
$(2,0)$$(1,1)$$(1,0)$$(2,2)$$(2,1)$ | (5)
$(1,0)$$(2,1)$$(2,0)$$(1,1)$ | (6)
Table 3: Quivers associated with $A_{\pm}(z)$ in Table 1. Each quiver is
preserved by the mutation at $({*},0)$ followed by the permutation
$(i,p)\mapsto(i,p-1)$ (the second argument is considered modulo $r_{i}$),
which yields Y-system. For (1)–(3), this operation interchanges the connected
components (see Section 3.1).

######  Theorem 2.8.

The Y-system associated with each $A_{\pm}(z)$ in Table 1 is positive and
negative reddening.

###### Proof.

The quiver $B$ associated with $A_{\pm}(z)$ is given in Table 3. We can verify
the assertion by concrete calculation on the quiver. The numbers $h_{\pm}$ are
given in Table 1. ∎

Theorem 1.5 (1) now follows from Proposition 2.7 and Theorem 2.8.

######  Remark 2.9.

A connected component of each quiver in Table 3 has the following cluster
type:

(1) $A_{2}$ | (2) $A_{4}$ | (3) $E_{6}$ | (4) $D_{4}$ | (5) $A_{5}$ | (6) $A_{4}$
---|---|---|---|---|---

These are of finite type in the sense of [8], which also implies Theorem 1.5
(1). We remark, however, that this observation is somewhat misleading since
the quiver associated with a periodic Y-system of general rank is typically of
infinite type. It might be better to think that the appearance of only finite
type quivers happens “by chance” due to the smallness of $2$, the rank of
Y-systems considered in this paper.

Theorem 2.8 also gives quantum dilogarithm identifies associated with
Donaldson-Thomas invariants. For any reddening sequence $\boldsymbol{i}$
starting from a quiver $B$, we can define a quantity
$\mathbb{E}(\boldsymbol{i})$ by using the quantum dilogarithm. We refer to
[20, Remark 6.6] as the definition. This quantity coincides with Kontsevich-
Soibelman’s refined Donaldson-Thomas invariant associated with $B$ [20, 25].
In particular, $\mathbb{E}(\boldsymbol{i})$ does not depend on
$\boldsymbol{i}$, which gives the quantum dilogarithm identifies. In our case,
we have:

######  Corollary 2.10.

For each $A_{\pm}(z)$ in Table 1, we have

$\mathbb{E}(\mu^{h_{+}})=\mathbb{E}(\mu^{-h_{-}}),$

where $\mu\coloneqq\nu\circ\mu_{\boldsymbol{i}}$ is the sequence of mutations
(together with the permutation)
$(B,\boldsymbol{y}(0))\to(B,\boldsymbol{y}(1))$ in (2.7).

For example, the pair (1) in Table 1 yields the famous pentagon identity of
the quantum dilogarithm.

##  3 Classification

###  3.1 Change of slices

We need to introduce an appropriate equivalence relation on the set
$\mathbb{Y}$, which identifies essentially the same Y-systems. Before we get
into the definition, we will see a typical example. Consider the following
Y-system:

$\begin{split}&Y_{1}(u)Y_{1}(u-2)=(1+Y_{2}(u-1)^{-1})^{-1}\\\
&Y_{2}(u)Y_{2}(u-2)=(1+Y_{1}(u-1)^{-1})^{-1}\end{split}$ (3.1)

which corresponds to $A_{\pm}(z)\in\mathbb{Y}$ given by (1) in Table 1. This
system of equations are defined on the set $[1,2]\times\mathbb{Z}$, but
actually can be defined on each component of the following disjoint union:

$[1,2]\times\mathbb{Z}=\bigsqcup_{k=0}^{1}\\{(i,u)\mid i-u\equiv k\bmod 2\\}.$

We informally call the algebraic relation defined on each subset the _slice_
of the whole Y-system. If $(Y_{i}(u))$ is a solution of the Y-system for
$i-u\equiv 0\bmod 2$, then $(Y_{i}(u+1))$ is a solution of the Y-system for
$i-u\equiv 1\bmod 2$. Thus it is enough to consider only one slice when
considering solutions. Now we consider another Y-system:

$\begin{split}&Y^{\prime}_{1}(u)Y^{\prime}_{1}(u-3)=(1+Y^{\prime}_{2}(u-2)^{-1})^{-1}\\\
&Y^{\prime}_{2}(u)Y^{\prime}_{2}(u-3)=(1+Y^{\prime}_{1}(u-1)^{-1})^{-1}.\end{split}$
(3.2)

which corresponds to $A^{\prime}_{\pm}(z)\in\mathbb{Y}$ given by

$\displaystyle A^{\prime}_{+}(z)\coloneqq\begin{pmatrix}1+z^{3}&-z^{2}\\\
-z&1+z^{3}\end{pmatrix},\quad
A^{\prime}_{-}(z)\coloneqq\begin{pmatrix}1+z^{3}&0\\\ 0&1+z^{3}\end{pmatrix}.$

The Y-system (3.2) is decomposed into three slices:

$[1,2]\times\mathbb{Z}=\bigsqcup_{k=0}^{2}\\{(i,u)\mid i-u\equiv k\bmod 3\\}.$

We see that for any solution of (3.1) for $i-u\equiv 0\bmod 2$,

$Y^{\prime}_{1}(u)\coloneqq Y_{1}\left(\frac{2}{3}u-\frac{1}{3}\right),\quad
Y^{\prime}_{2}(u)\coloneqq Y_{2}\left(\frac{2}{3}u\right)$

is a solution of (3.2) for $i-u\equiv 2\bmod 3$. We also obtain solutions for
the other two slices by shifting $u$. Conversely, any solution of (3.1) is
obtained from a solution of (3.2). Therefore, it is enough to consider one of
the Y-systems (3.1) and (3.2). In particular, $A_{\pm}(z)$ is of finite type
if and only if $A^{\prime}_{\pm}(z)$ is.

Now we work in the general setting. The idea is that each slice corresponds to
each connected component of the quiver associated with the matrix $B$ defined
by (2.5). Let $A_{\pm}(z)\in\mathbb{Y}$, and assume that it is indecomposable.
By [24, Proposition 3.24], we have a decomposition of the matrix $B$ and its
index set $R$:

$\displaystyle B=\bigoplus_{u=0}^{t-1}B(u),\quad R=\bigsqcup_{u=0}^{t-1}R(u)$

such that each $B(u)$ is indecomposable and we have a cyclic sequence of
mutations

$\displaystyle
B(0)\xrightarrow{\nu|_{R(0)}\circ\mu_{\boldsymbol{i}(0)}}B(1)\longrightarrow\cdots\longrightarrow
B(t-1)\xrightarrow{\nu|_{R(t-1)}\circ\mu_{\boldsymbol{i}(t-1)}}B(0)$ (3.3)

where $\boldsymbol{i}(u)\coloneqq\boldsymbol{i}\cap R(u)$. We say that two
pairs $A_{\pm}(z)$ and $A^{\prime}_{\pm}(z)$ are _related by change of slices_
if they yield the same cyclic sequence (3.3) up to a change of indices and the
commutativity of mutations (2.3). (This commutativity is already implicitly
used to justify the notation $\mu_{\boldsymbol{i}(u)}$ as stated below (2.3).)

######  Example 3.1.

The pairs $A_{\pm}(z)$ and $A^{\prime}_{\pm}(z)$ associated with (3.1) and
(3.2), respectively, are related by change of slices. Indeed, we see that the
sequence (3.3) for (3.1) is

$\leavevmode\hbox to71.57pt{\vbox
to16.67pt{\pgfpicture\makeatletter\hbox{\hskip 14.44412pt\lower-8.33301pt\hbox
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$(1,0)$}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
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}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\xrightarrow{\nu\circ\mu_{(0,0)}}\leavevmode\hbox
to71.57pt{\vbox to16.67pt{\pgfpicture\makeatletter\hbox{\hskip
14.44412pt\lower-8.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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whereas the sequence (3.3) for (3.2) is

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These are the same sequence up to a change of indices.

###  3.2 Proof of the classification

In this section, we will prove Theorem 1.5 (2). We first recall the following
result.

######  Lemma 3.2 ([24, Theorem 5.5]).

Let $A_{\pm}(z)\in\mathbb{Y}$. Assume that $A_{\pm}(z)$ is of finite type.
Then there is a vector $v\in\mathbb{R}^{I}$ such that $v>0$, $vA_{+}(1)>0$,
and $vA_{-}(1)>0$. In particular, $\operatorname{tr}A_{\pm}(1)>0$ and $\det
A_{\pm}(1)>0$.

By Lemma 3.2, $A_{+}(1)$ and $A_{-}(1)$ are equal to one of the following
matrices:

$\displaystyle\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix},\begin{pmatrix}2&-1\\\
-2&2\end{pmatrix},\begin{pmatrix}2&-1\\\
-3&2\end{pmatrix},\begin{pmatrix}2&-1\\\
-1&1\end{pmatrix},\begin{pmatrix}2&0\\\
-n&2\end{pmatrix},\begin{pmatrix}2&0\\\
-n&1\end{pmatrix},\begin{pmatrix}1&0\\\ -n&1\end{pmatrix}$

up to a permutation of the indices. We give several lemmas about impossible
pairs. Before giving lemmas, we note that

$n_{ij;p}^{+}=0\quad\text{or}\quad n_{ij;p}^{-}=0$ (3.4)

for any $i,j,p$.

######  Lemma 3.3.

It is impossible that $A_{\pm}(z)\in\mathbb{Y}$ has the following forms:

  1. (1)

$A_{+}(1)=\begin{pmatrix}2&-a\\\ *&*\end{pmatrix}$,
$A_{-}(1)=\begin{pmatrix}2&-b\\\ *&*\end{pmatrix}$ for odd $a,b$.

  2. (2)

$A_{+}(1)=\begin{pmatrix}2&-a\\\ *&*\end{pmatrix}$,
$A_{-}(1)=\begin{pmatrix}1&-b\\\ *&*\end{pmatrix}$ for odd $a,b$.

  3. (3)

$A_{+}(1)=\begin{pmatrix}1&-1\\\ *&*\end{pmatrix}$,
$A_{-}(1)=\begin{pmatrix}1&-1\\\ *&*\end{pmatrix}$.

  4. (4)

$A_{+}(1)=\begin{pmatrix}1&0\\\ *&*\end{pmatrix}$,
$A_{-}(1)=\begin{pmatrix}1&*\\\ *&*\end{pmatrix}$.

###### Proof.

For (2), we can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r}&-f(z)\\\ *&*\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}1+z^{r}-z^{a}&-g(z)\\\ *&*\end{pmatrix}.$

By the symplectic property (1.4), we have

$z^{a}+z^{a-r}+f(z)g(z^{-1})=z^{-a}+z^{r-a}+g(z)f(z^{-1}).$ (3.5)

Since $0<a$ and $a-r<0$ by (1.2), the sum of the coefficients of the terms in
$f(z)g(z^{-1})$ with positive exponents is equal to that with negative
exponents. Since $f(1)g(1)$ (=$ab$) is odd, $f(z)g(z^{-1})$ should contain the
constant term $z^{0}$, which contradicts (3.4). The proof for (1) is similar.

For (3), we can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r}-z^{a}&-z^{b}\\\
*&*\end{pmatrix},\quad A_{-}(z)=\begin{pmatrix}1+z^{r}-z^{c}&-z^{d}\\\
*&*\end{pmatrix}$

with $0<a,b,c,d<r$. Without loss of generality, we can assume $a<c$. By (1.4),
we have

$z^{-c}+z^{r-c}+z^{a}+z^{a-r}+z^{c-a}+z^{d-b}=z^{c}+z^{c-r}+z^{-a}+z^{r-a}+z^{a-c}+z^{b-d}$

Since $c-a>0$, we see that $c-a$ is equal to $c$, $r-a$, or $b-d$. However,
the first two cases are impossible by (1.2). Thus $c-a=b-d$, which implies
that

$z^{-c}+z^{r-c}+z^{a}+z^{a-r}=z^{c}+z^{c-r}+z^{-a}+z^{r-a}.$

Since $a>0$, we see that $a$ is equal to $c$ or $r-a$. However, $a=c$ is
impossible by (3.4). Thus $a=r-a$, which implies that

$z^{-c}+z^{r-c}=z^{c}+z^{c-r}.$

Since $c>0$, we see that $c=r-c$. However, this implies that $a=r/2=c$, which
is impossible by (3.4).

For (4), we can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r}-z^{a}&0\\\
*&*\end{pmatrix},\quad A_{-}(z)=\begin{pmatrix}1+z^{r}-z^{b}&*\\\
*&*\end{pmatrix}.$

By (1.4), we have

$\displaystyle
z^{a}+z^{a-r}+z^{-b}+z^{r-b}+z^{b-a}=z^{-a}+z^{r-a}+z^{b}+z^{b-r}+z^{a-b}.$

Comparing the number of the terms with positive and negative exponents, we
should have $a=b$. This is impossible by (3.4). ∎

######  Lemma 3.4.

It is impossible that indecomposable $A_{\pm}(z)\in\mathbb{Y}$ has the form

$A_{+}(1)=\begin{pmatrix}*&0\\\ *&*\end{pmatrix},\quad
A_{-}(1)=\begin{pmatrix}*&0\\\ *&*\end{pmatrix}.$

###### Proof.

We can set

$\displaystyle A_{\pm}(z)=\begin{pmatrix}1+z^{r_{1}}-f_{\pm}(z)&0\\\
-g_{\pm}(z)&1+z^{r_{2}}-h_{\pm}(z)\end{pmatrix}.$

Since $g_{+}(z)\neq 0$ or $g_{-}(z)\neq 0$, we can pick the least integer $c$
among the exponents in $g_{+}(z)$ and $g_{-}(z)$. Without loss of generality,
we can assume $g_{+}(1)$ contains the term $z^{c}$. By (1.4), we have

$\displaystyle
f_{+}(z)g_{-}(z^{-1})+(1+z^{r_{1}})g_{+}(z^{-1})=f_{-}(z)g_{+}(z^{-1})+(1+z^{r_{1}})g_{-}(z^{-1}).$
(3.6)

The left-hand side in (3.6) contains the term $z^{r_{1}-c}$, but any exponent
in the right-hand side is strictly smaller that $r_{1}-c$ by (1.2) and (3.4),
which is a contradiction. ∎

We now search for possible pairs $A_{\pm}(1)$ case by case using the
symplectic property (1.4) at $z=1$ together with Lemma 3.3 and 3.4:

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix}$. The possibilities
for $A_{-}(1)$ are:

$\begin{pmatrix}2&0\\\ 0&2\end{pmatrix},\ \begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix}$. The possibilities
for $A_{-}(1)$ are:

$\begin{pmatrix}2&0\\\ 0&2\end{pmatrix},\ \begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix}$. The possibilities
for $A_{-}(1)$ are:

$\begin{pmatrix}2&0\\\ -1&2\end{pmatrix},\ \begin{pmatrix}1&0\\\
0&2\end{pmatrix}.$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&-1\\\ -3&2\end{pmatrix}$. The possibilities
for $A_{-}(1)$ are:

$\begin{pmatrix}2&0\\\ -2&2\end{pmatrix}.$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&-1\\\ -1&1\end{pmatrix}$. The possibilities
for $A_{-}(1)$ are:

$\begin{pmatrix}2&0\\\ 0&2\end{pmatrix},\ \begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&0\\\ -n&2\end{pmatrix}$. The possibilities
for $A_{\pm}(1)$ are:

$\left(\begin{pmatrix}2&0\\\ 0&2\end{pmatrix},\ \begin{pmatrix}2&-1\\\
-1&2\end{pmatrix}\right),\ \left(\begin{pmatrix}2&0\\\ -1&2\end{pmatrix},\
\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix}\right),\ \left(\begin{pmatrix}2&0\\\
0&2\end{pmatrix},\ \begin{pmatrix}1&-1\\\ -1&2\end{pmatrix}\right).$

  * •

Case: $A_{+}(1)=\begin{pmatrix}2&0\\\ -n&1\end{pmatrix}$. The possibilities
for $A_{\pm}(1)$ are:

$\left(\begin{pmatrix}2&0\\\ 0&1\end{pmatrix},\ \begin{pmatrix}2&-2\\\
-1&2\end{pmatrix}\right).$

  * •

Case: $A_{+}(1)=\begin{pmatrix}1&0\\\ -n&1\end{pmatrix}$. The possibilities
for $A_{\pm}(1)$ are:

$\left(\begin{pmatrix}2&0\\\ 0&1\end{pmatrix},\ \begin{pmatrix}2&-2\\\
-1&2\end{pmatrix}\right).$

In summary, the remaining possible pairs, up to a permutation of the indices
and an change of sign, are given in the following table:

(3.15)

We now start searching for possible $A_{\pm}(z)$.

######  Lemma 3.5.

Let $n\geq 1$. Suppose that

$\displaystyle A_{+}(1)=\begin{pmatrix}2&-1\\\ -n&2\end{pmatrix},\quad
A_{-}(1)=\begin{pmatrix}2&0\\\ -(n-1)&2\end{pmatrix}.$

Then

$\displaystyle A_{+}(z)=\begin{pmatrix}[2]_{r}&-z^{-a}\\\
-z^{r-a}[n]_{2r}&[2]_{(2n-1)r}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}[2]_{r}&0\\\
-z^{2r-a}[n-1]_{2r}&[2]_{(2n-1)r}\end{pmatrix}$

for some $r,a$, where $[n]_{r}$ is the _$z$ -integer_ defined by

$[n]_{r}\coloneqq\frac{1-z^{rn}}{1-z^{r}}.$ (3.16)

###### Proof.

We can set

$\displaystyle A_{+}(z)=\begin{pmatrix}[2]_{r_{1}}&-z^{a}\\\
-\sum_{i=1}^{n}z^{b_{i}}&[2]_{r_{2}}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}[2]_{r_{1}}&0\\\
-\sum_{i=1}^{n-1}z^{c_{i}}&[2]_{r_{2}}\end{pmatrix}.$

Without loss of generality, we can assume that

$\displaystyle b_{1}\leq b_{2}\leq\dots\leq b_{n},\quad c_{1}\leq
c_{2}\leq\dots\leq c_{n-1}.$

By the symplectic property (1.4), we have

$\displaystyle\sum_{i=1}^{n-1}(z^{-c_{i}}+z^{r_{1}-c_{i}})+z^{a}+z^{a-r_{2}}=\sum_{i=1}^{n}(z^{-b_{i}}+z^{r_{1}-b_{i}}).$

Comparing the degree by using the conditions (1.2) and (3.4), we obtain the
system of linear equations

$\displaystyle a=r_{1}-b_{1},\quad a-r_{2}=-b_{n},\quad
r_{1}=c_{i}-b_{i}=b_{i+1}-c_{i}\quad(i=1,\dots,n-1),$

which implies that

$\displaystyle r_{2}=(2n-1)r_{1},\quad b_{i}=(2i-1)r_{1}-a,\quad
c_{i}=2ir_{1}-a.$

∎

######  Lemma 3.6.

Suppose that

$\displaystyle A_{+}(1)=\begin{pmatrix}2&-1\\\ -1&2\end{pmatrix},\quad
A_{-}(1)=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$

Then

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{2r}&-z^{a}\\\
-z^{2r-a}&1+z^{2r}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}1+z^{2r}-z^{r}&0\\\ 0&1+z^{2r}-z^{r}\end{pmatrix}$

for some $r,a$.

###### Proof.

We can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r_{1}}&-z^{a}\\\
-z^{b}&1+z^{r_{2}}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}1+z^{r_{1}}-z^{c}&0\\\
0&1+z^{r_{2}}-z^{d}\end{pmatrix}.$

By (1.4), we have $r_{1}=r_{2}=a+b=2c=2d$. ∎

######  Lemma 3.7.

Suppose that

$\displaystyle A_{+}(1)=\begin{pmatrix}2&-1\\\ -2&2\end{pmatrix},\quad
A_{-}(1)=\begin{pmatrix}1&0\\\ 0&2\end{pmatrix}.$

Then

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{2r}&-z^{a}\\\
-z^{2r-a}-z^{3r-a}&1+z^{3r}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}1+z^{2r}-z^{r}&0\\\ 0&1+z^{2r}\end{pmatrix}$

for some $r,a$.

###### Proof.

We can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r_{1}}&-z^{a}\\\
-z^{b_{1}}-z^{b_{2}}&1+z^{r_{2}}\end{pmatrix},\quad
A_{-}(z)=\begin{pmatrix}1+z^{r_{1}}-z^{c}&0\\\ 0&1+z^{r_{2}}\end{pmatrix}.$

Without loss of generality, we can assume $b_{1}\leq b_{2}$. By (1.4), we have
$r_{1}=2c$, $r_{2}=3c$, $b_{1}=2c-a$, and $b_{2}=3c-a$. ∎

######  Lemma 3.8.

Suppose that

$\displaystyle A_{+}(1)=\begin{pmatrix}2&-1\\\ -1&1\end{pmatrix},\quad
A_{-}(1)=\begin{pmatrix}2&0\\\ 0&2\end{pmatrix}.$

Then

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{2r}&-z^{a}\\\
-z^{2r-a}&1+z^{2r}\end{pmatrix},\quad A_{-}(z)=\begin{pmatrix}1+z^{2r}&0\\\
0&1+z^{2r}\end{pmatrix}$

for some $r,a$.

###### Proof.

We can set

$\displaystyle A_{+}(z)=\begin{pmatrix}1+z^{r_{1}}&-z^{a}\\\
-z^{b}&1+z^{r_{2}}\end{pmatrix},\quad A_{-}(z)=\begin{pmatrix}1+z^{r_{1}}&0\\\
0&1+z^{r_{2}}\end{pmatrix}.$

By (1.4), we have $r_{1}=r_{2}=a+b=2c$. ∎

###### Proof of Theorem 1.5 (2).

The remaining possibilities for finite type $A_{\pm}(z)\in\mathbb{Y}$, up to a
permutation of indices and change of sign, are the six families of the pairs
given in Lemma 3.5–3.8, which contain the parameters $r,a$. We can verify that
these six families belong to $\mathbb{Y}$, and they can be reduced to the
pairs in Table 1 by change of slices. ∎

## References

  * [1] George E. Andrews.  An analytic generalization of the Rogers-Ramanujan identities for odd moduli.  Proc. Nat. Acad. Sci. U.S.A., 71:4082–4085, 1974.
  * [2] Thomas Brüstle, Grégoire Dupont, and Matthieu Pérotin.  On maximal green sequences.  Int. Math. Res. Not. IMRN, (16):4547–4586, 2014.
  * [3] Frank Calegari, Stavros Garoufalidis, and Don Zagier.  Bloch groups, algebraic K-theory, units, and Nahm’s conjecture.  arXiv preprint arXiv:1712.04887, 2017.
  * [4] Corina Calinescu, Antun Milas, and Michael Penn.  Vertex algebraic structure of principal subspaces of basic $A_{2n}^{(2)}$-modules.  J. Pure Appl. Algebra, 220(5):1752–1784, 2016.
  * [5] Peigen Cao, Min Huang, and Fang Li.  A conjecture on $C$-matrices of cluster algebras.  Nagoya Math. J., 238:37–46, 2020.
  * [6] Ivan Cherednik and Boris Feigin.  Rogers-Ramanujan type identities and Nil-DAHA.  Adv. Math., 248:1050–1088, 2013.
  * [7] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. I. Foundations.  J. Amer. Math. Soc., 15(2):497–529, 2002.
  * [8] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. II. Finite type classification.  Invent. Math., 154(1):63–121, 2003.
  * [9] Sergey Fomin and Andrei Zelevinsky.  $Y$-systems and generalized associahedra.  Ann. of Math. (2), 158(3):977–1018, 2003.
  * [10] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. IV. Coefficients.  Compos. Math., 143(1):112–164, 2007.
  * [11] Allan P. Fordy and Robert J. Marsh.  Cluster mutation-periodic quivers and associated Laurent sequences.  J. Algebraic Combin., 34(1):19–66, 2011.
  * [12] Pavel Galashin and Pavlo Pylyavskyy.  The classification of Zamolodchikov periodic quivers.  Amer. J. Math., 141(2):447–484, 2019.
  * [13] Stavros Garoufalidis and Don Zagier.  Knots, perturbative series and quantum modularity.  arXiv preprint arXiv:2111.06645, 2021.
  * [14] Goro Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi, and Zengo Tsuboi.  Paths, crystals and fermionic formulae.  In MathPhys odyssey, 2001, volume 23 of Prog. Math. Phys., pages 205–272. Birkhäuser Boston, Boston, MA, 2002.
  * [15] Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi.  Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras I: type $B_{r}$.  Publ. Res. Inst. Math. Sci., 49(1):1–42, 2013.
  * [16] Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi.  Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras II: types $C_{r}$, $F_{4}$, and $G_{2}$.  Publ. Res. Inst. Math. Sci., 49(1):43–85, 2013.
  * [17] Victor G. Kac and Dale H. Peterson.  Infinite-dimensional Lie algebras, theta functions and modular forms.  Adv. in Math., 53(2):125–264, 1984.
  * [18] Akishi Kato and Yuji Terashima.  Quiver mutation loops and partition $q$-series.  Comm. Math. Phys., 336(2):811–830, 2015.
  * [19] Bernhard Keller.  On cluster theory and quantum dilogarithm identities.  In Representations of algebras and related topics, EMS Ser. Congr. Rep., pages 85–116. Eur. Math. Soc., Zürich, 2011.
  * [20] Bernhard Keller.  Cluster algebras and derived categories.  In Derived categories in algebraic geometry, EMS Ser. Congr. Rep., pages 123–183. Eur. Math. Soc., Zürich, 2012.
  * [21] Bernhard Keller.  The periodicity conjecture for pairs of Dynkin diagrams.  Ann. of Math. (2), 177(1):111–170, 2013.
  * [22] A. Kuniba and T. Nakanishi.  Spectra in conformal field theories from the Rogers dilogarithm.  Modern Phys. Lett. A, 7(37):3487–3494, 1992.
  * [23] Chul-hee Lee.  Nahm’s conjecture and $Y$-systems.  Commun. Number Theory Phys., 7(1):1–14, 2013.
  * [24] Yuma Mizuno.  Difference equations arising from cluster algebras.  J. Algebraic Combin., 54(1):295–351, 2021.
  * [25] Kentaro Nagao.  Quantum dilogarithm identities.  In Infinite analysis 2010—Developments in quantum integrable systems, RIMS Kôkyûroku Bessatsu, B28, pages 165–170. Res. Inst. Math. Sci. (RIMS), Kyoto, 2011.
  * [26] Werner Nahm.  Conformal field theory and torsion elements of the Bloch group.  In Frontiers in number theory, physics, and geometry. II, pages 67–132. Springer, Berlin, 2007.
  * [27] Tomoki Nakanishi.  Periodicities in cluster algebras and dilogarithm identities.  In Representations of algebras and related topics, EMS Ser. Congr. Rep., pages 407–443. Eur. Math. Soc., Zürich, 2011.
  * [28] Tomoki Nakanishi.  Synchronicity phenomenon in cluster patterns.  J. Lond. Math. Soc., II. Ser., 103(3):1120–1152, 2021.
  * [29] Tomoki Nakanishi and Salvatore Stella.  Wonder of sine-Gordon $Y$-systems.  Trans. Amer. Math. Soc., 368(10):6835–6886, 2016.
  * [30] F. Ravanini, A. Valleriani, and R. Tateo.  Dynkin TBAs.  Internat. J. Modern Phys. A, 8(10):1707–1727, 1993.
  * [31] John R. Stembridge.  Admissible $W$-graphs and commuting Cartan matrices.  Adv. in Appl. Math., 44(3):203–224, 2010.
  * [32] R. Tateo.  New functional dilogarithm identities and sine-Gordon $Y$-systems.  Phys. Lett. B, 355(1-2):157–164, 1995.
  * [33] Masha Vlasenko and Sander Zwegers.  Nahm’s conjecture: asymptotic computations and counterexamples.  Commun. Number Theory Phys., 5(3):617–642, 2011.
  * [34] Liuquan Wang.  Identities on Zagier’s rank two examples for Nahm’s conjecture.  arXiv preprint arXiv:2210.10748, 2022.
  * [35] Don Zagier.  The dilogarithm function.  In Frontiers in number theory, physics, and geometry. II, pages 3–65. Springer, Berlin, 2007.
  * [36] Al. B. Zamolodchikov.  On the thermodynamic Bethe ansatz equations for reflectionless $ADE$ scattering theories.  Phys. Lett. B, 253(3-4):391–394, 1991.

Department of Mathematics and Informatics, Faculty of Science, Chiba
University, Chiba 263-8522, Japan.

E-mail address, Y. Mizuno<EMAIL_ADDRESS>