Periodic -Systems and Nahm Sums: The Rank 2 Case
We classify periodic -systems of rank 2 satisfying the symplectic property. We find that there are six such -systems. In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the -systems in positive and negative directions, which give...
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| description | We classify periodic -systems of rank 2 satisfying the symplectic property. We find that there are six such -systems. In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the -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 -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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 094, 17 pages
Periodic Y -Systems and Nahm Sums:
The Rank 2 Case
Yuma MIZUNO
School of Mathematical Sciences, University College Cork,
Western Gateway Building, Western Road, Cork, Ireland
E-mail: ymizuno@ucc.ie
URL: https://yuma-mizuno.github.io/
Received June 04, 2025, in final form October 26, 2025; Published online November 03, 2025
https://doi.org/10.3842/SIGMA.2025.094
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 asso-
ciated 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.
Key words: cluster algebras; Y -systems; Nahm sums
2020 Mathematics Subject Classification: 11F03; 11P84; 13F60
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:
Yi(u)Yi(u− ri) =
∏
j∈I
ri−1∏
p=1
Yj(u− p)[nij;p]+(1 + Yj(u− p))−nij;p , (1.1)
where I is a finite index set, Yi(u) for i ∈ I, u ∈ Z are commuting variables, ri ∈ Z≥1,
and nij;p ∈ Z. We also use the notation [n]+ := max(0, n). Such equations are first discovered by
Zamolodchikov in the study of thermodynamic Bethe ansatz [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 special 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 ri, nij;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]).
mailto:ymizuno@ucc.ie
https://yuma-mizuno.github.io/
https://doi.org/10.3842/SIGMA.2025.094
2 Y. Mizuno
There are several classification results in the literature. Fomin and Zelevinsky [10] showed
that the classification when ri = 2, nij;p ≤ 0, and nii;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 ri = 2 and nii;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 ri > 2 for some i, and so far there has been no comprehensive classification results except
when |I| = 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 ri > 2
for some i. Precisely, we classify the periodic Y -systems of the form (1.1) with |I| = 2 satisfying
the symplectic property. We would like to emphasize that we consider general ri, nij;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 Y0 the set of pairs (r,n) where r = (ri)i∈I and n =
(nij;p)i,j∈I,p∈N are families of integers satisfying ri ≥ 1 for any i and
nij;p = 0 unless 0 < p < ri (1.2)
for any i, j, p.
Definition 1.1. Let (r,n) ∈ Y0. Let P be a semifield, and (Yi(u))i∈I,u∈Z be a family of
elements in P. We say that (Yi(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 (Yi(u)) is a solution of the Y -system if it satisfies the Y -system.
It is useful to think a pair (r,n) ∈ Y0 as a triple of matrices with polynomial entries by the
map (r,n) 7→ (N0(z), N+(z), N−(z)) : Y0 → (MatI×I N[z])3 defined by
N0(z) := diag
(
1 + zri
)
i∈I , N±(z) :=
(∑
p∈N
n±
ij;pz
p
)
i,j∈I
,
where we set n±
ij;p := [±nij;p]+. We also define the map (r,n) 7→ A±(z) : Y0 → (MatI×I Z[z])2
by A±(z) := N0(z)−N±(z). Since this map is injective by the condition (1.2), we will identify Y0
with the image of this map. For example, we will use the term “the Y -system associated
with A±(z) ∈ Y0”.
Definition 1.2. We say that A±(z) ∈ Y0 satisfies the symplectic property if
A+(z)A−
(
z−1
)T
= A−(z)A+
(
z−1
)T
, (1.3)
where T is the transpose of a matrix. We denote by Y the subset of Y0 consisting of pairs
satisfying the symplectic property.
The pair A±(z) ∈ Y0 satisfies the symplectic property if and only if the Y -system associated
with A±(z) ∈ Y0 is realized as the exchange relations of coefficients in a cluster algebra [24].
We review this fact in Section 2.1.
Definition 1.3. We say that a solution of a Y -system is periodic if there is a positive inte-
ger Ω > 0 such that Yi(u+Ω) = Yi(u) for any i, u.
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 3
A+(z) A−(z) h+ h−(
1 + z2 −z
−z 1 + z2
) (
1 + z2 0
0 1 + z2
)
3 2 (1)(
1 + z2 −z
−z − z5 1 + z6
) (
1 + z2 0
−z3 1 + z6
)
8 6 (2)(
1 + z2 −z
−z − z5 − z9 1 + z10
) (
1 + z2 0
−z3 − z7 1 + z10
)
18 10 (3)(
1 + z2 −z
−z 1 + z2
) (
1 + z2 − z 0
0 1 + z2 − z
)
3 3 (4)(
1 + z2 −z
−z − z2 1 + z3
) (
1 + z2 − z 0
0 1 + z3
)
5 3 (5)(
1 + z2 −z
−z 1 + z2 − z
) (
1 + z2 0
0 1 + z2
)
5 2 (6)
Table 1. Finite type classification for Y -systems of rank 2. The numbers h± are the length of reddening
sequences in positive and negative directions, respectively.
Definition 1.4. We say that a pair A±(z) ∈ 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±(z) ∈ YI is decomposable if it is a direct sum
of some A′
±(z) ∈ YI′ and A′′
±(z) ∈ YI′′ with nonempty I ′ and I ′′. We say that A±(z) ∈ YI
is indecomposable if it is not decomposable. It is enough to consider indecomposable pairs in
the classification. We also note that A±(z) is of finite type if and only if A±(z)
op := A∓(z) is
of finite type by the correspondence between solutions Yi(u) 7→ Yi(u)
−1. The main results are
summarized as follows.
Theorem 1.5. Suppose that I = {1, 2}.
(1) Any pair A±(z) ∈ Y in Table 1 is of finite type.
(2) Any indecomposable pair A±(z) ∈ 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±(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±(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 (see Theorem 2.8). Thanks to the deep results in the theory of cluster algebras, this
property is enough to imply the periodicity (see Proposition 2.7). The numbers h± 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 (see Corollary 2.10).
The claim (2) is proved in Section 3.2 by the following steps:
4 Y. Mizuno
Step 1. We recall the result in [24] that asserts that A±(1) satisfies a certain positivity, which
in particular implies that trA±(1) and detA±(1) are positive. This allows us to significantly
reduce the candidates for finite type A±(z).
Step 2. For a fixed A+(1) in the candidates obtained in Step 1, we search for A−(1) satisfying
the symplectic property (1.3) at z = 1.
Step 3. During the search in Step 2, we discard the pair A±(1) that cannot be endowed with
the parameter z (see Lemmas 3.4 and 3.5).
Step 4. At this point, we have six candidates up to a permutation of the indices and a change of
sign. For each A±(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±(z) in Lemmas 3.6–3.9.
Step 5. We finally check that each remaining candidate reduces to one of A±(z) in Table 1 by
change of slices.
Remark 1.6. In fact, we can see that six pairs in Table 1 are not related to each other by
permuting the indices, changing sign, and changing slices. It can be verified by comparing the
mutation sequences of quivers associated with these pairs, which are given in Table 3. See
Remark 3.2.
Remark 1.7. Most of the Y -systems obtained from Table 1 are already known in the litera-
ture. (1)op and (6)op are Zamolodchikov’s Y -system of type A2 [36] and T2 (“tadpole”) [30],
respectively. (2)op is the reduced sine-Gordon Y -system associated with the continued frac-
tion 3/4 = [1, 3] = 1/(1 + 1/3), and (5)op with z replaced by z2 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 (A2, A2) [30]. (3) appears to be new
Y1(u)Y1(u− 2) =
1
1 + Y2(u− 1)−1
,
Y2(u)Y2(u− 10) =
(1 + Y1(u− 3))(1 + Y1(u− 7))(
1 + Y1(u− 1)−1
)(
1 + Y1(u− 5)−1
)(
1 + Y1(u− 9)−1
)
although it is implicitly given in the author’s previous work [24, Table 2].
Remark 1.8. The pair A±(z) ∈ 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 N0 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 1.9. There is another expression of the Y -system using a pair of matrices A±(z)
directly. Let A±(z) ∈ Y0, and define aij;p ∈ Z by
A±(z) =
(∑
p∈N
a±ij;pz
p
)
i,j∈I
.
Let
(
P±
i (u)
)
i∈I,u∈Z be a family of elements in a multiplicative abelian group P. We say that(
P±
i (u)
)
satisfies the multiplicative Y -system associated with A±(z) if∏
j∈I
∏
p∈N
P+
j (u− p)a
+
ij;p =
∏
j∈I
∏
p∈N
P−
j (u− p)a
−
ij;p (1.4)
for any i, u (schematically, “A+(z)·logP+ = A−(z)·logP−” under the action z : u 7→ u−1). The
solution
(
P±
i (u)
)
is called normalized if P is endowed with a semifield structure, and P+
i (u) +
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 5
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 (1.4). The correspondence
is given by
Yi(u) 7→
P+
i (u)
P−
i (u)
, P+
i (u) 7→ Yi(u)
1 + Yi(u)
, P−
i (u) 7→ 1
1 + Yi(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) =
∞∑
n=0
qn
2
(q)n
, H(q) =
∞∑
n=0
qn
2+n
(q)n
, (1.5)
where (q)n := (1 − q)
(
1 − q2
)
· · · (1 − qn) is the q-Pochhammer symbol. The famous Rogers–
Ramanujan identities express these q-series as the following infinite products:
G(q) =
∏
n≡±1 mod 5
1
1− qn
, H(q) =
∏
n≡±2 mod 5
1
1− qn
.
These expressions in particular implies that q−1/60G(q) and q11/60H(q) are modular functions
on some finite index subgroup of SL(2,Z). In fact, it is a rare case that an infinite sum of the
form (1.5) is modular. It is known that the q-series
∞∑
n=0
q
1
2
an2+bn+c
(q)n
(1.6)
with a, b, c ∈ Q is modular only if a = 1/2, 1, or 2 [35].
Nahm [26] considered higher rank generalization of (1.6), which we call the Nahm sum. Let I
be a finite set, and suppose that A ∈ QI×I is a symmetric positive definite matrix, B ∈ QI is
a vector, and C ∈ Q is a scalar. The Nahm sum is the q-series defined by
fA,B,C(q) :=
∑
n∈NI
q
1
2
nTAn+nTB+C∏
a(q)ni
.
When |I| ≥ 2, it is not well understood when fA,B,C(q) is modular. Nahm gave a conjecture
providing a criterion on the modularity of fA,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±(z) ∈ Y
in general. Precisely, the matrix K := A+(1)
−1A−(1) is always symmetric and positive definite
for finite type A±(z) ∈ Y, and it is conjectured that it gives a modular Nahm sum fK,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 (1.3) 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.
Based on our classification, we find the following statement.
6 Y. Mizuno
A±(z) K −24C RR A±(z) K −24C RR
(1)
(
4/3 2/3
2/3 4/3
)
4
5
[6] (1)op
(
1 −1/2
−1/2 1
)
6
5
[33]
(2)
(
3/2 1
1 2
)
5
7
[34] (2)op
(
1 −1/2
−1/2 3/4
)
9
7
[34]
(3), (6)
(
2 2
2 4
)
4
7
[1] (3)op, (6)op
(
1 −1/2
−1/2 1/2
)
10
7
[34]
(4)
(
2/3 1/3
1/3 2/3
)
1 [35] (4)op
(
2 −1
−1 2
)
1 [35]
(5)
(
1 1
1 2
)
3
4
[34] (5)op
(
2 −1
−1 1
)
5
4
[4]
Table 2. The list of the matrix K = A+(1)
−1A−(1). The Nahm sum fK,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.
Theorem 1.10. Suppose that I = {1, 2}. The Nahm sum fK,0,C(q) is modular for any finite
type A±(z) ∈ Y, where C is given in Table 2.
In fact, every K from finite type A±(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.
Example 1.11. For a matrix K =
(
2 2
2 4
)
, we have
fK,(0,0),−1/42(q) =
θ7,1(τ)
η(τ)
, fK,(0,1),5/42(q) =
θ7,2(τ)
η(τ)
, fK,(1,2),17/42(q) =
θ7,3(τ)
η(τ)
,
by the Andrews–Gordon identities [1] and the Jacobi’s triple product identity, where
η(τ) := q1/24
∞∏
n=1
(1− qn), θ7,j(τ) :=
∑
n≡2j−1 mod 14
(−1)[n/14]qn
2/56,
with q = e2πiτ . The Dedekind eta function η(τ) and the theta functions θ7,j(τ) are modular
forms of weight 1/2, and their modular transformation formulas imply that the vector-valued
function f = (fK,0,−1/42fK,(0,1),5/42fK,(1,2),17/42)
T is a vector-valued modular function satisfying
f(−1/τ) =
2√
7
sin 3π
7 sin 2π
7 sin π
7
sin 2π
7 − sin π
7 − sin 3π
7
sin π
7 − sin 3π
7 sin 2π
7
f(τ).
Remark 1.12. We can define the refinement f
(s)
A±(z)(q) of the Nahm sum fK,0,0(q), which is
parametrized by s ∈ H for an abelian groupH of order detA+(1) such that it reduces to the orig-
inal one by taking summation [24, Definition 5.12]
fK,0,0(q) =
∑
s∈H
f
(s)
A±(z)(q).
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 7
It is conjectured that each f
(s)
A±(z)(q) is already modular after multiplying qC for some C. We note
that the symplectic property (1.3) 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:
(P1) We have reddening sequences in both positive and negative directions.
(P2) The map A±(z) 7→ A+(1)
−1A−(1) gives modular Nahm sums.
We expect that the properties (P1) and (P2) also hold for any finite type A±(z) ∈ Y of general
rank. The following are some known examples:
� For the Y -system associated with the untwisted quantum affine algebras Uq
(
X
(1)
r
)
with
level ℓ restriction [22], (P1) holds with h+ = tℓ and h− = t · (dual Coxeter number of Xr)
where t = 1, 2, or 3 is the multiplicity in the Dynkin diagram of Xr [15, 16], and (P2)
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 (Xr, X
′
r′)
[30], (P1) holds with h+ = (Coxeter number of Xr) and h− = (Coxeter number of X ′
r′)
[19, 21].
� For the (reduced) sine-Gordon Y -system associated with the continued fraction p/q =
[nF , . . . , n1] = 1/(nF + 1/(· · · + 1/n1)) [29, 32], (P1) appears to hold with h+ = 2p and
h− = 2q.
� For the Y -system associated with an admissible ADE bigraph (Γ,∆) [12], (P1) appears to
hold with h+ = (Coxeter number of Γ) and h− = (Coxeter number of ∆).
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 prop-
erty (1.3) becomes
A+(z)DA−
(
z−1
)T
= A−(z)DA+
(
z−1
)T
,
where D is a diagonal matrix called symmetrizer [24]. We also expect that the properties (P1)
and (P2) also hold for skew-symmetrizable case. See [24, Definition 5.12] for the definition of
the Nahm sum in skew-symmetrizable case.
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
y = (yi)i∈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 Q>0(y). We
have a canonical bijection Homsemifield(Q>0(y),P) ∼= Homset(I,P) for any set I and semifield P.
8 Y. Mizuno
Definition 2.2. Let I be a set. The tropical semifield Trop(y) is the multiplicative free abelian
group generated by the variables y = (yi)i∈I equipped with the addition defined by∏
i
yaii +
∏
i
ybii =
∏
i
y
min(ai,bi)
i .
Let I be a finite set and P be a semifield. A Y-seed is a pair (B,y) where B = (Bij)i,j∈I
is a skew-symmetric integer matrix and y = (yi)i∈I is a tuple of elements in P. We sometimes
represent B as the quiver whose signed adjacency matrix is B. For a Y-seed (B,y) and k ∈ I,
the mutation in direction k transforms (B,y) into the new Y-seed µk(B,y) = (B′,y′) given by
B′
ij :=
{
−Bij if i = k or j = k,
Bij + [−Bik]+Bkj +Bik[Bkj ]+ otherwise,
y′i :=
{
yk if i = k,
yiy
[Bki]+
k (1 + yk)
−Bki otherwise.
A mutation is involutive, that is, µk(B,y) = (B′,y′) implies (B,y) = µk(B
′,y′). We have the
commutativity
µiµj = µjµi if Bij = 0, (2.1)
which allows us to write µi for a set i ⊆ I such that Bij = 0 for any i, j ∈ i to mean the
successive mutations along arbitrarily chosen order on i.
For a Y-seed (B,y) and a bijection ν : I → I, we define a new Y-seed ν(B,y) = (B′,y′)
by B′
ν(i)ν(j)
:= Bij and y′ν(i) := yi.
2.2 Solving Y -systems by cluster algebras
Let A±(z) ∈ Y. We will construct a solution of the Y -system associated with A±(z) based
on [24, Section 3.3]. We first define a subset R ⊆ I × Z by R := {(i, u) ∈ I × Z | 0 ≤ u < ri},
and define a skew-symmetric R×R integer matrix B by
B(i,p)(j,q) = −nij;p−q + nji;q−p +
∑
k∈I
min(p,q)∑
v=0
(
n+
ik;p−vn
−
jk;q−v − n−
ik;p−vn
+
jk;q−v
)
, (2.2)
where we understand nij;p = 0 if p < 0. We then define i := {(i, u) | u = 0} ⊆ R. We also define
a bijection ν : R → R by
ν(i, p) =
{
(i, p− 1) if p > 0,
(i, ri) if p = 0.
Then the symplectic property (1.3) ensures that ν(µi(B)) = B [24, Lemma 3.16]. We finally
define a sequence of Y-seeds
· · · → (B,y(−1)) → (B,y(0)) → (B,y(1)) → · · · (2.3)
in Q>0(y) by y(0) := y and (B,y(u+1)) = ν(µi(B,y(u))). The sequence (2.3) gives a solution
of the Y -system.
Lemma 2.3 ([24, Theorem 3.13]). (yi,0(u))i∈I,u∈Z satisfies the Y -system associated with A±(z).
This solution is universal in the following sense.
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 9
Lemma 2.4 ([24, Theorem 3.19]). Suppose that a family (Yi(u))i∈I,u∈Z satisfies the Y -system
associated with A±(z). Define a semifield homomorphism f : Q>0(y) → P by
f(yi,p) := Yi(p)
∏
j∈I
p∏
q=0
Yj(p− q)−[nij;q ]+(1 + Yj(p− q))nij;q .
Then f(yi,0(u)) = Yi(u) for any i, u.
Corollary 2.5. A±(z) ∈ Y is of finite type if and only if there are different integers u, v such
that y(u) = y(v) in (2.3).
2.3 Periodicity and reddening sequences
Similarly to (2.3), we define a sequence of Y-seeds
· · · → (B,y(−1)) → (B,y(0)) → (B,y(1)) → · · · (2.4)
by the same formulas but now in Trop(y) rather than Q(y).
Definition 2.6. We say that the Y -system associated with A±(z) ∈ Y is positive (resp. negative)
reddening if there is a positive integer u such that all the exponents in yi(u) (resp. yi(−u)) in (2.4)
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,y(0)) → (B,y(u)) (resp. (B,y(0)) →
(B,y(−u))) are nonpositive for some u > 0.
Proposition 2.7. Suppose that the Y -system associated with A±(z) is positive and negative
reddening. Then A±(z) is of finite type.
Proof. We verify the equivalent condition in Corollary 2.5. By [2, Proposition 2.10], there
are bijections σ, σ′ : R → R such that yi(h+) = y−1
σ(i) and yi(−h−) = y−1
σ′(i) 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. ■
Theorem 2.8. The Y -system associated with each A±(z) in Table 1 is positive and negative
reddening.
Proof. The quiver B associated with A±(z) is given in Table 3. We can verify the assertion by
concrete calculation on the quiver. The numbers h± 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) A2, (2) A4, (3) E6, (4) D4, (5) A5, (6) A4.
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.
10 Y. Mizuno
Quiver B A±(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±(z) in Table 1. Each quiver is preserved by the mutation at (∗, 0)
followed by the permutation (i, p) 7→ (i, p − 1) (the second argument is considered modulo ri), which
yields Y -system. For (1)–(3), this operation interchanges the connected components (see Section 3.1).
Theorem 2.8 also gives quantum dilogarithm identities associated with Donaldson–Thomas
invariants. For any reddening sequence i starting from a quiver B, we can define a quan-
tity E(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 as-
sociated with B [20, 25]. In particular, E(i) does not depend on i, which gives the quantum
dilogarithm identities. In our case, we have the following.
Corollary 2.10. For each A±(z) in Table 1, we have
E
(
µh+
)
= E
(
µ−h−
)
,
where µ := ν ◦ µi is the sequence of mutations (together with the permutation) (B,y(0)) →
(B,y(1)) in (2.3).
For example, the pair (1) in Table 1 yields the famous pentagon identity of the quantum
dilogarithm.
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 11
3 Classification
3.1 Change of slices
We need to introduce an appropriate equivalence relation on the set 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
Y1(u)Y1(u− 2) =
(
1 + Y2(u− 1)−1
)−1
, Y2(u)Y2(u− 2) =
(
1 + Y1(u− 1)−1
)−1
, (3.1)
which corresponds to A±(z) ∈ Y given by (1) in Table 1. This system of equations are defined
on the set [1, 2] × Z, but actually can be defined on each component of the following disjoint
union
[1, 2]× Z =
1⊔
k=0
{(i, u) | i− u ≡ k mod 2}.
We informally call the algebraic relation defined on each subset the slice of the whole Y -system.
If (Yi(u)) is a solution of the Y -system for i− u ≡ 0 mod 2, then (Yi(u+ 1)) is a solution of the
Y -system for i − u ≡ 1 mod 2. Thus it is enough to consider only one slice when considering
solutions. Now we consider another Y -system
Y ′
1(u)Y
′
1(u− 3) =
(
1 + Y ′
2(u− 2)−1
)−1
, Y ′
2(u)Y
′
2(u− 3) =
(
1 + Y ′
1(u− 1)−1
)−1
. (3.2)
which corresponds to A′
±(z) ∈ Y given by
A′
+(z) :=
(
1 + z3 −z2
−z 1 + z3
)
, A′
−(z) :=
(
1 + z3 0
0 1 + z3
)
.
The Y -system (3.2) is decomposed into three slices
[1, 2]× Z =
2⊔
k=0
{(i, u) | i− u ≡ k mod 3}.
We see that for any solution of (3.1) for i− u ≡ 0 mod 2,
Y ′
1(u) := Y1
(
2
3
u− 1
3
)
, Y ′
2(u) := Y2
(
2
3
u
)
is a solution of (3.2) for i − u ≡ 2 mod 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±(z) is of finite type if
and only if A′
±(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 (2.2). Let A±(z) ∈ 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
B =
t−1⊕
u=0
B(u), R =
t−1⊔
u=0
R(u)
such that each B(u) is indecomposable and we have a cyclic sequence of mutations
B(0)
ν|R(0)◦µi(0)−−−−−−−→ B(1) −→ · · · −→ B(t− 1)
ν|R(t−1)◦µi(t−1)−−−−−−−−−−→ B(0), (3.3)
12 Y. Mizuno
where i(u) := i ∩ R(u). We say that two pairs A±(z) and A′
±(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 (2.1). (This commutativity is already implicitly used to justify the notation µi(u) as
stated below (2.1).)
Example 3.1. The pairs A±(z) and A′
±(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
(1, 0) (2, 1)
ν◦µ(0,0)−−−−−→ (1, 1) (2, 0)
ν◦µ(1,0)−−−−−→ (1, 0) (2, 1),
whereas the sequence (3.3) for (3.2) is
(1, 0) (2, 1)
ν′◦µ(0,0)−−−−−→ (1, 2) (2, 0)
ν′◦µ(1,0)−−−−−→ (1, 1) (2, 2)
ν′−→ (1, 0) (2, 1).
These are the same sequence up to a change of indices.
Remark 3.2. The mutation sequences associated with the quivers in Table 3 are not related
to each other by a change of indices and the commutativity of mutations (2.1).
3.2 Proof of the classification
In this section, we will prove Theorem 1.5 (2). We first recall the following result.
Lemma 3.3 ([24, Theorem 5.5]). Let A±(z) ∈ Y. Assume that A±(z) is of finite type. Then
there is a vector v ∈ RI such that v > 0, vA+(1) > 0, and vA−(1) > 0. In particular,
trA±(1) > 0 and detA±(1) > 0.
Since the off-diagonal entries of A±(1) are nonpositive integers by the definition of A±(1),
the condition detA±(1) > 0 in Lemma 3.3 implies that the product of diagonal entries of A±(1)
is positive. By combining the condition trA±(1) > 0 in Lemma 3.3, we see that the diagonal
entries of A±(1) are positive. By the definition of A±(1), the diagonal entries of A±(1) are
less than or equal to 2, and hence they are equal to 1 or 2. Consequently, we see that A+(1)
and A−(1) are equal to one of the following matrices:(
2 −1
−1 2
)
,
(
2 −1
−2 2
)
,
(
2 −1
−3 2
)
,
(
2 −1
−1 1
)
,
(
2 0
−n 2
)
,(
2 0
−n 1
)
,
(
1 0
−n 1
)
up to a permutation of the indices. We give several lemmas about impossible pairs. Before
giving lemmas, we note that
n+
ij;p = 0 or n−
ij;p = 0 (3.4)
for any i, j, p.
Lemma 3.4. It is impossible that A±(z) ∈ Y has the following forms:
(1) A+(1) =
(
2 −a
∗ ∗
)
, A−(1) =
(
2 −b
∗ ∗
)
for odd a, b.
(2) A+(1) =
(
2 −a
∗ ∗
)
, A−(1) =
(
1 −b
∗ ∗
)
for odd a, b.
(3) A+(1) =
(
1 −1
∗ ∗
)
, A−(1) =
(
1 −1
∗ ∗
)
.
(4) A+(1) =
(
1 0
∗ ∗
)
, A−(1) =
(
1 ∗
∗ ∗
)
.
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 13
Proof. For (2), we can set
A+(z) =
(
1 + zr −f(z)
∗ ∗
)
, A−(z) =
(
1 + zr − za −g(z)
∗ ∗
)
for some f, g ∈ N[z]. By the symplectic property (1.3), we have
za + za−r + f(z)g
(
z−1
)
= z−a + zr−a + g(z)f
(
z−1
)
.
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 z0, which contradicts (3.4). The proof for (1) is
similar.
For (3), we can set
A+(z) =
(
1 + zr − za −zb
∗ ∗
)
, A−(z) =
(
1 + zr − zc −zd
∗ ∗
)
with 0 < a, b, c, d < r. Without loss of generality, we can assume a < c. By (1.3), we have
z−c + zr−c + za + za−r + zc−a + zd−b = zc + zc−r + z−a + zr−a + za−c + zb−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 + zr−c + za + za−r = zc + zc−r + z−a + zr−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+zr−c = zc+zc−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
A+(z) =
(
1 + zr − za 0
∗ ∗
)
, A−(z) =
(
1 + zr − zb ∗
∗ ∗
)
.
By (1.3), we have
za + za−r + z−b + zr−b + zb−a = z−a + zr−a + zb + zb−r + za−b.
Comparing the number of the terms with positive and negative exponents, we should have a = b.
This is impossible by (3.4). ■
Lemma 3.5. It is impossible that indecomposable A±(z) ∈ Y has the form A+(1) =
(
∗ 0
∗ ∗
)
,
A−(1) =
(
∗ 0
∗ ∗
)
.
Proof. We can set
A±(z) =
(
1 + zr1 − f±(z) 0
−g±(z) 1 + zr2 − h±(z)
)
.
Since g+(z) ̸= 0 or g−(z) ̸= 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+(z) contains the term zc. By (1.3),
we have
f+(z)g−
(
z−1
)
+ (1 + zr1)g+
(
z−1
)
= f−(z)g+
(
z−1
)
+ (1 + zr1)g−
(
z−1
)
. (3.5)
The left-hand side in (3.5) contains the term zr1−c, but any exponent in the right-hand side is
strictly smaller that r1 − c by (1.2) and (3.4), which is a contradiction. ■
14 Y. Mizuno
We now search for possible pairs A±(1) case by case using the symplectic property (1.3)
at z = 1 together with Lemmas 3.4 and 3.5:
� Case: A+(1) =
(
2 −1
−1 2
)
. The possibilities for A−(1) are
(
2 0
0 2
)
,
(
1 0
0 1
)
.
� Case: A+(1) =
(
2 −1
−1 2
)
. The possibilities for A−(1) are
(
2 0
0 2
)
,
(
1 0
0 1
)
.
� Case: A+(1) =
(
2 −1
−2 2
)
. The possibilities for A−(1) are
(
2 0
−1 2
)
,
(
1 0
0 2
)
.
� Case: A+(1) =
(
2 −1
−3 2
)
. The possibilities for A−(1) are
(
2 0
−2 2
)
.
� Case: A+(1) =
(
2 −1
−1 1
)
. The possibilities for A−(1) are
(
2 0
0 2
)
.
� Case: A+(1) =
(
2 0
−n 2
)
. The possibilities for A±(1) are
((
2 0
0 2
)
,
(
2 −1
−1 2
))
,
((
2 0
−1 2
)
,
(
2 −1
−2 2
))
,((
2 0
0 2
)
,
(
1 −1
−1 2
))
.
� Case: A+(1) =
(
2 0
−n 1
)
. The possibilities for A±(1) are
((
2 0
0 1
)
,
(
2 −2
−1 2
))
.
� Case: A+(1) =
(
1 0
−n 1
)
. The possibilities for A±(1) are
((
2 0
0 1
)
,
(
2 −2
−1 2
))
.
In summary, the remaining possible pairs, up to a permutation of the indices and a change
of sign, are given in the following table:
A+(1) A−(1)(
2 −1
−1 2
) (
2 0
0 2
)
(
2 −1
−2 2
) (
2 0
−1 2
)
(
2 −1
−3 2
) (
2 0
−2 2
)
A+(1) A−(1)(
2 −1
−1 2
) (
1 0
0 1
)
(
2 −1
−2 2
) (
1 0
0 2
)
(
2 −1
−1 1
) (
2 0
0 2
)
.
We now start searching for possible A±(z).
Lemma 3.6. Let n ≥ 1. Suppose that A+(1) =
(
2 −1
−n 2
)
, A−(1) =
( 2 0
−(n−1) 2
)
. Then
A+(z) =
(
[2]r −za
−zr−a[n]2r [2](2n−1)r
)
, A−(z) =
(
[2]r 0
−z2r−a[n− 1]2r [2](2n−1)r
)
for some r, a, where [n]r is the z-integer defined by [n]r :=
1−zrn
1−zr .
Proof. We can set
A+(z) =
[2]r1 −za
−
n∑
i=1
zbi [2]r2
, A−(z) =
[2]r1 0
−
n−1∑
i=1
zci [2]r2
.
Without loss of generality, we can assume that b1 ≤ b2 ≤ · · · ≤ bn, c1 ≤ c2 ≤ · · · ≤ cn−1. By the
symplectic property (1.3), we have
n−1∑
i=1
(
z−ci + zr1−ci
)
+ za + za−r2 =
n∑
i=1
(
z−bi + zr1−bi
)
.
Comparing the degree by using the conditions (1.2) and (3.4), we obtain the system of linear
equations a = r1 − b1, a − r2 = −bn, r1 = ci − bi = bi+1 − ci (i = 1, . . . , n − 1), which implies
that r2 = (2n− 1)r1, bi = (2i− 1)r1 − a, ci = 2ir1 − a. ■
Periodic Y -Systems and Nahm Sums: The Rank 2 Case 15
Lemma 3.7. Suppose that A+(1) =
(
2 −1
−1 2
)
, A−(1) =
(
1 0
0 1
)
. Then
A+(z) =
(
1 + z2r −za
−z2r−a 1 + z2r
)
, A−(z) =
(
1 + z2r − zr 0
0 1 + z2r − zr
)
for some r, a.
Proof. We can set
A+(z) =
(
1 + zr1 −za
−zb 1 + zr2
)
, A−(z) =
(
1 + zr1 − zc 0
0 1 + zr2 − zd
)
.
By (1.3), we have r1 = r2 = a+ b = 2c = 2d. ■
Lemma 3.8. Suppose that A+(1) =
(
2 −1
−2 2
)
, A−(1) =
(
1 0
0 2
)
. Then
A+(z) =
(
1 + z2r −za
−z2r−a − z3r−a 1 + z3r
)
, A−(z) =
(
1 + z2r − zr 0
0 1 + z2r
)
for some r, a.
Proof. We can set
A+(z) =
(
1 + zr1 −za
−zb1 − zb2 1 + zr2
)
, A−(z) =
(
1 + zr1 − zc 0
0 1 + zr2
)
.
Without loss of generality, we can assume b1 ≤ b2. By (1.3), we have r1 = 2c, r2 = 3c, b1 = 2c−a,
and b2 = 3c− a. ■
Lemma 3.9. Suppose that A+(1) =
(
2 −1
−1 1
)
, A−(1) =
(
2 0
0 2
)
. Then
A+(z) =
(
1 + z2r −za
−z2r−a 1 + z2r − zr
)
, A−(z) =
(
1 + z2r 0
0 1 + z2r
)
for some r, a.
Proof. We can set
A+(z) =
(
1 + zr1 −za
−zb 1 + zr2 − zc
)
, A−(z) =
(
1 + zr1 0
0 1 + zr2
)
.
By (1.3), we have r1 = r2 = a+ b = 2c. ■
Proof of Theorem 1.5 (2). The remaining possibilities for finite type A±(z) ∈ Y, up to a per-
mutation of indices and change of sign, are the six families of the pairs given in Lemmas 3.6–3.9,
which contain the parameters r, a. We can verify that these six families belong to Y if only if
� 1 < r and 0 < a < r for Lemma 3.6,
� 1 < r and 0 < a < 2r for Lemmas 3.7, 3.8, and 3.9,
where the only if part follows from (1.2). With these conditions, the families for n = 1, 2, 3 in
Lemma 3.6 can be reduced to the pairs (1), (2), and (3) in Table 1, respectively, by change of
slices, and the families in Lemmas 3.7, 3.8, and 3.9 can be reduced to the pairs (4), (5), and (6)
in Table 1, respectively, by change of slices. ■
We note that the pairs in Table 1 correspond to the value (r, a) = (2, 1) in Lemmas 3.6–3.9.
16 Y. Mizuno
Acknowledgements
The authors would like to thank the anonymous referees for their constructive and helpful
suggestions. This work is supported by JSPS KAKENHI Grant Number JP21J00050.
References
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1 Introduction
1.1 Background
1.2 Main result
1.3 Relation to Nahm sums
1.4 Remarks on higher rank and skew-symmetrizable case
2 Y-systems and cluster algebras
2.1 Preliminaries on cluster algebras
2.2 Solving Y-systems by cluster algebras
2.3 Periodicity and reddening sequences
3 Classification
3.1 Change of slices
3.2 Proof of the classification
References
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| id | nasplib_isofts_kiev_ua-123456789-214189 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:00:50Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mizuno, Yuma 2026-02-20T07:58:52Z 2025 Periodic -Systems and Nahm Sums: The Rank 2 Case. Yuma Mizuno. SIGMA 21 (2025), 094, 17 pages 1815-0659 2020 Mathematics Subject Classification: 11F03; 11P84; 13F60 arXiv:2301.13239 https://nasplib.isofts.kiev.ua/handle/123456789/214189 https://doi.org/10.3842/SIGMA.2025.094 We classify periodic -systems of rank 2 satisfying the symplectic property. We find that there are six such -systems. In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the -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 -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. The authors would like to thank the anonymous referees for their constructive and helpful suggestions. This work is supported by JSPS KAKENHI Grant Number JP21J00050. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Periodic -Systems and Nahm Sums: The Rank 2 Case Article published earlier |
| spellingShingle | Periodic -Systems and Nahm Sums: The Rank 2 Case Mizuno, Yuma |
| title | Periodic -Systems and Nahm Sums: The Rank 2 Case |
| title_full | Periodic -Systems and Nahm Sums: The Rank 2 Case |
| title_fullStr | Periodic -Systems and Nahm Sums: The Rank 2 Case |
| title_full_unstemmed | Periodic -Systems and Nahm Sums: The Rank 2 Case |
| title_short | Periodic -Systems and Nahm Sums: The Rank 2 Case |
| title_sort | periodic -systems and nahm sums: the rank 2 case |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214189 |
| work_keys_str_mv | AT mizunoyuma periodicsystemsandnahmsumstherank2case |