Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems

We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2010
Автори:	Nakanishi, T., Tateo, R.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/146519
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems / T. Nakanishi, R. Tateo // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 36 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-146519
record_format	dspace
spelling	nasplib_isofts_kiev_ua-123456789-1465192025-02-09T10:03:05Z Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems Nakanishi, T. Tateo, R. We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds. It is our great pleasure to thank the anonymous referee who generously pointed out a crucial fact for the subject of the paper. 2010 Article Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems / T. Nakanishi, R. Tateo // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 17B37 DOI:10.3842/SIGMA.2010.085 https://nasplib.isofts.kiev.ua/handle/123456789/146519 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.
format	Article
author	Nakanishi, T. Tateo, R.
spellingShingle	Nakanishi, T. Tateo, R. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Nakanishi, T. Tateo, R.
author_sort	Nakanishi, T.
title	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
title_short	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
title_full	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
title_fullStr	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
title_full_unstemmed	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
title_sort	dilogarithm identities for sine-gordon and reduced sine-gordon y-systems
publisher	Інститут математики НАН України
publishDate	2010
url	https://nasplib.isofts.kiev.ua/handle/123456789/146519
citation_txt	Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems / T. Nakanishi, R. Tateo // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 36 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT nakanishit dilogarithmidentitiesforsinegordonandreducedsinegordonysystems AT tateor dilogarithmidentitiesforsinegordonandreducedsinegordonysystems
first_indexed	2025-11-25T16:02:18Z
last_indexed	2025-11-25T16:02:18Z
_version_	1849778810844610560
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 085, 34 pages Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems Tomoki NAKANISHI † and Roberto TATEO ‡ † Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan E-mail: nakanisi@math.nagoya-u.ac.jp ‡ Dipartimento di Fisica Teorica and INFN, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy E-mail: tateo@to.infn.it Received May 29, 2010, in final form October 16, 2010; Published online October 19, 2010 doi:10.3842/SIGMA.2010.085 Abstract. We study the family of Y-systems and T-systems associated with the sine- Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds. Key words: cluster algebras; quantum groups; integrable models 2010 Mathematics Subject Classification: 13F60; 17B37 1 Introduction The Y-systems and T-systems appeared in the study of two-dimensional integrable S-matrix models and integrable lattice models in 90’s. They play central roles to connect these integrable models with conformal field theories [3] through the method called the thermodynamic Bethe ansatz (TBA) (e.g., [18, 2, 34, 36, 20, 21, 23, 30, 24, 32]). Since the introduction of the cluster algebras by Fomin and Zelevinsky [7, 8], it has been gradually recognized that cluster algebras provide a suitable framework to study the mathema- tical properties of Y and T-systems. As a fruitful outcome, the long standing conjectures of the periodicity of Y-systems [36, 30, 24] (as well as the periodicity of T-systems) and the dilogarithm identities [19, 2, 22, 12] for the class of Y and T-systems associated with the quantum affine algebras have been proved partly by [9, 5], and in full generality by [16, 15, 26, 17, 13, 14], recently. In the above proof, the periodicities of the Y and T-systems are reformulated as the periodici- ties of seeds in the corresponding cluster algebras. It was shown in [13] that the periodicity of seeds in a cluster algebra associated with a skew symmetric matrix reduces to the periodicity of the tropical coefficients (the ‘principal coefficients’ in [10]), which is much simpler than the original problem. In the above examples, the periodicities at the level of the tropical coefficients are realized as combinations of the Coxeter transformations of the A-D-E root systems and their variations. From this point of view, one can regard these periodicities as a natural extension of the periodicities of the Coxeter mutation sequences in the cluster algebras of finite type studied by [8, 9, 10]. Then, it is natural to ask the following question: “Are there any other periodicities of seeds? And, if there are, can we classify them?” The purpose of this paper is to present a new class of (infinitely many) periodicities of seeds in cluster algebras. These cluster algebras correspond to the Y-systems introduced in [32], where their periodicities and the associated dilogarithm identities were also conjectured. More mailto:nakanisi@math.nagoya-u.ac.jp mailto:tateo@to.infn.it http://dx.doi.org/10.3842/SIGMA.2010.085 2 T. Nakanishi and R. Tateo Figure 1. The diagram Xn. precisely, these Y-systems consist of two classes. The first class are called the sine-Gordon (SG) Y-systems and they are associated with the TBA equation for the sine-Gordon model [35]. The second class are called the reduced sine-Gordon (RSG) Y-systems and, as the name suggests, they are associated with the TBA equation for a certain reduction of the SG model [31, 4]. To these cluster algebras, one can apply the method of [13], and prove their periodicities and the associated dilogarithm identities. This is the main result of the paper. To be more precise, we concentrate on the case where the ‘coupling constant’ ξ is a continued fraction with two terms, for simplicity. The result also suggests us a vague perspective to the second question above. Namely, the classification of the periodicities of seeds may be comparable with the classification of the integrable deformations of rational conformal field theories. In particular, those periodicities we have already known should be just a tip of iceberg. The content of the paper is as follows. In Section 2 we introduce the Y-systems and T-systems, then summarize the results of their periodicities and the associated dilogarithm identities in both the SG and the RSG cases. Here, we concentrate on the special case of the choice of the coupling constant ξ in (2.2). In Sections 3 and 4 we give a proof in the SG case. In Sections 5 and 6 we give a proof in the RSG case. In Section 7 we extend the result to a little more general case of ξ in (7.1), i.e., a general continued fraction with two terms. In the derivation of the main result, the properties of the tropical Y-systems in Proposi- tions 4.2 and 6.1 are crucial, and the proofs are provided in detail. The proofs of the rest are rather formal and repetitions of the formerly studied cases [16, 15, 26, 17, 13, 14]. So, instead of repeating similar proofs, we only provide examples which typically representing situations. Note added. After the submission of the paper, the anonymous referee pointed out us that the cluster algebras concerned with the SG and RSG models studied in this paper turn out to be the cluster algebras of type D and A, respectively. Since this is a very important fact, we include it in a new subsection (Section 7.3). This means, somewhat on the contrary to our initial intention and expectation, the periodicities of the mutation sequences studied in this paper are actually the ones inside the cluster algebras of finite type. In particular, the periodicity property itself is an automatic consequence of the finiteness of the number of seeds. However, the determination of their precise periods is still new in the literature; furthermore, it does not crucially change our perspective of the richness of the periodicity phenomena of seeds, which should be uncovered in the future study. 2 Main results 2.1 Results for sine-Gordon Y-systems With an integer n ≥ 4, we associate a diagram Xn in Fig. 1. The diagram Xn should not be regarded as an ordinary Dynkin diagram, since the horizontality and verticality of segments also Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 3 carry information. It appeared in [32] and encodes the structure of the Y-systems which we are going to introduce now. Let In = {1, . . . , n + 1} × Z. Definition 2.1. Fix an integer n ≥ 4. The sine-Gordon (SG) Y-system Yn(SG) is the following system of relations for a family of variables {Yi(u) \| (i, u) ∈ In}, Y1 (u− n + 1) Y1 (u + n− 1) = n−1∏ j=2 (1 + Yj(u− n + j))(1 + Yj(u + n− j))  × (1 + Yn(u))(1 + Yn+1(u)), Y2(u− 1)Y2(u + 1) = 1 + Y1(u) 1 + Y3(u)−1 , Yi(u− 1)Yi(u + 1) = 1∏ j:j∼i (1 + Yj(u)−1) , i = 3, . . . , n + 1, (2.1) where j ∼ i means that j is adjacent to i in Xn. In [32], a more general family of Y-systems was associated with a rational parameter ξ, which is related the coupling constant β of the SG model by (2.15). The system (2.1) corresponds to the special case ξ = n− 1 n = 1 1 + 1 n− 1 , (2.2) namely, F = 2, n1 = 1, and n2 = n in the notation in [32]. The variable u here is related to the variable θ in [32] by u = (2n/π √ −1)θ. Later in Section 2.3 we explain more about the background of (2.1). Definition 2.2. Let Yn(SG) be the semifield (Appendix A(i)) with generators Yi(u) ((i, u) ∈ In) and relations Yn(SG). Let Y◦n(SG) be the multiplicative subgroup of Yn(SG) generated by Yi(u), 1 + Yi(u) ((i, u) ∈ In). (Here we use the symbol + instead of ⊕ for simplicity.) The first main result of the paper is the following two theorems conjectured by [32]. Theorem 2.3 (Periodicity). The following relations hold in Y◦n(SG). (i) Half periodicity: Yi(u+4n−2) = Yω(i)(u), where ω is an involution of the set {1, . . . , n+1} defined by ω(n) = n + 1, ω(n + 1) = n, and ω(i) = i (i = 1, . . . , n− 1). (ii) Full periodicity: Yi(u + 8n− 4) = Yi(u). In our proof of Theorem 2.3 we have a natural interpretation of the half period 4n− 2 = h(Dn) + 2 + h(Dn−1) + 2 in terms of the Coxeter number h(Dn) = 2n− 2 of type Dn. Let L(x) be the Rogers dilogarithm function [25] L(x) = −1 2 ∫ x 0 { log(1− y) y + log y 1− y } dy, 0 ≤ x ≤ 1. It satisfies the following relation L(x) + L(1− x) = π2 6 , 0 ≤ x ≤ 1. (2.3) 4 T. Nakanishi and R. Tateo Theorem 2.4 (Functional dilogarithm identities). Suppose that a family of positive real numbers {Yi(u) \| (i, u) ∈ In} satisfies Yn(SG). Then, we have the identities 6 π2 ∑ (i,u)∈In 0≤u<8n−4 L ( Yi(u) 1 + Yi(u) ) = 8(2n− 1), (2.4) 6 π2 ∑ (i,u)∈In 0≤u<8n−4 L ( 1 1 + Yi(u) ) = 4(n− 1)(2n− 1). (2.5) Two identities (2.4) and (2.5) are equivalent to each other due to (2.3). Using this opportunity, we introduce another system of relations accompanying Yn(SG). Definition 2.5. Fix an integer n ≥ 4. The sine-Gordon (SG) T-system Tn(SG) is the following system of relations for a family of variables {Ti(u) \| (i, u) ∈ In}, T1 (u− n + 1) T1 (u + n− 1) = T2(u) + 1, T2(u− 1)T2(u + 1) = T1(u− n + 2)T1(u + n− 2) + T3(u), Ti(u− 1)Ti(u + 1) = T1(u− n + i)T1(u + n− i) + ∏ j:j∼i Tj(u), i = 3, . . . , n− 1, Tn(u− 1)Tn(u + 1) = T1(u) + Tn−1(u), Tn+1(u− 1)Tn+1(u + 1) = T1(u) + Tn−1(u), (2.6) where j ∼ i means that j is adjacent to i in Xn. There are two connections between Yn(SG) and Tn(SG). The first connection is a formal one. Set d1 = n− 1, di = 1, i = 2, . . . , n− 1, (2.7) and let us write (2.1) in a unified manner as Yi (u− di) Yi (u + di) = ∏ (j,v)∈In (1 + Yj(v))G+(j,v;i,u) ∏ (j,v)∈In (1 + Yj(v)−1)G−(j,v;i,u) . (2.8) Then, (2.6) is written as Ti (u− di) Ti (u + di) = ∏ (j,v)∈In Tj(v)G+(i,u;j,v) + ∏ (j,v)∈In Tj(v)G−(i,u;j,v). (2.9) Note that we took the ‘transpositions’ of G+ and G− in (2.9). The second connection is an algebraic one. Suppose that {Ti(u) \| (i, u) ∈ In} satisfies the T-system Tn(SG). Set Yi(u) = ∏ (j,v)∈In Tj(v)G+(i,u;j,v) ∏ (j,v)∈In Tj(v)G−(i,u;j,v) . Then, {Yi(u) \| (i, u) ∈ In} satisfies the Y-system Yn(SG). One may check the claim directly at this moment using Tn(SG) (with some effort). Alternatively and better, one can automatically obtain it from [10, Proposition 3.9] once we formulate these systems by a cluster algebra in the next section. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 5 Definition 2.6. Let Tn(SG) be the commutative ring over Z with identity element, with gene- rators Ti(u)±1 ((i, u) ∈ In) and relations Tn(SG) together with Ti(u)Ti(u)−1 = 1. Let T◦n(SG) be the subring of Tn(SG) generated by Ti(u) ((i, u) ∈ In). The following theorem can be proved simultaneously with Theorem 2.3. Theorem 2.7 (Periodicity). The following relations hold in T◦n(SG). (i) Half periodicity: Ti(u + 4n− 2) = Tω(i)(u), where ω is the one in Theorem 2.3. (ii) Full periodicity: Ti(u + 8n− 4) = Ti(u). Remark 2.8. Actually, Yn(SG) and Tn(SG) are also considered for n = 3, and they coincide with the Y and T-systems of type B2 with level 2 in [24]. Theorems 2.3, 2.4, and 2.7 remain valid for n = 3 due to [13]. All the results in this subsection will be extended to a more general case (7.1) later in Section 7. 2.2 Results for reduced sine-Gordon Y-systems The SG Y-system in the previous subsection admits a reduction called the reduced SG Y-system. It is obtained from (2.1) by formally setting Yn(u), Yn+1(u) → −1 and Yn−1(u) → ∞ [32]. Let Ĩn = {1, . . . , n− 2} × Z. Definition 2.9. Fix an integer n ≥ 4. The reduced sine-Gordon (RSG) Y-system Yn(RSG) is the following system of relations for a family of variables {Yi(u) \| (i, u) ∈ Ĩn}, Y1 (u− n + 1) Y1 (u + n− 1) = n−2∏ j=2 (1 + Yj(u− n + j))(1 + Yj(u + n− j)) 1 + Yn−2(u)−1 , Y2(u− 1)Y2(u + 1) = 1 + Y1(u) 1 + Y3(u)−1 , Yi(u− 1)Yi(u + 1) = 1∏ j:j∼i (1 + Yj(u)−1) , i = 3, . . . , n− 2, (2.10) where j ∼ i means that j ≤ n − 2 is adjacent to i in Xn. For n = 4, the second relation is replaced with Y2(u− 1)Y2(u + 1) = 1 + Y1(u). Definition 2.10. Let Yn(RSG) be the semifield with generators Yi(u) ((i, u) ∈ Ĩn) and relations Yn(RSG). Let Y◦n(RSG) be the multiplicative subgroup of Yn(RSG) generated by Yi(u), 1+Yi(u) ((i, u) ∈ Ĩn). The second main result of the paper is the following two theorems conjectured by [32]. The first theorem was already proved by [12] for the RSG Y-systems associated with a general rational ξ by using the explicit solution in terms of cross-ratio. Theorem 2.11 (Periodicity [12]). The following relations hold in Y◦n(RSG). Periodicity: Yi(u + 4n− 2) = Yi(u). In our proof of Theorem 2.11 we have a natural interpretation of the period 4n− 2 = 2{h(An−3) + 2 + h(An−4) + 2} in terms of the Coxeter number h(An) = n + 1 of type An. 6 T. Nakanishi and R. Tateo Theorem 2.12 (Functional dilogarithm identities). Suppose that a family of positive real numbers {Yi(u) \| (i, u) ∈ Ĩn} satisfies Yn(RSG). Then, we have the identities 6 π2 ∑ (i,u)∈Ĩn 0≤u<4n−2 L ( Yi(u) 1 + Yi(u) ) = 6(2n− 5), (2.11) 6 π2 ∑ (i,u)∈Ĩn 0≤u<4n−2 L ( 1 1 + Yi(u) ) = 2(2n2 − 11n + 17). (2.12) Two identities (2.11) and (2.12) are equivalent to each other due to (2.3). Again, we introduce the ‘T-system’ accompanying Yn(RSG). Definition 2.13. Fix an integer n ≥ 4. The reduced sine-Gordon (RSG) T-system Tn(RSG) is the following system of relations for a family of variables {Ti(u) \| (i, u) ∈ Ĩn}, T1 (u− n + 1) T1 (u + n− 1) = T2(u) + 1, T2(u− 1)T2(u + 1) = T1(u− n + 2)T1(u + n− 2) + T3(u), Ti(u− 1)Ti(u + 1) = T1(u− n + i)T1(u + n− i) + Ti−1(u)Ti+1(u), i = 3, . . . , n− 3, Tn−2(u− 1)Tn−2(u + 1) = T1(u− 2)T1(u + 2) + T1(u)Tn−3(u). (2.13) For n = 4, the second and the fourth relations are replaced with T2(u− 1)T2(u + 1) = T1(u− 2)T1(u + 2) + T1(u). There are connections between Yn(RSG) and Tn(RSG) parallel to the ones between Yn(SG) and Tn(SG). Definition 2.14. Let Tn(RSG) be the commutative ring over Z with identity element, with generators Ti(u)±1 ((i, u) ∈ Ĩn) and relations Tn(RSG) together with Ti(u)Ti(u)−1 = 1. Let T◦n(RSG) be the subring of Tn(RSG) generated by Ti(u) ((i, u) ∈ Ĩn). The following theorem can be proved simultaneously with Theorem 2.11. Theorem 2.15 (Periodicity). The following relations hold in T◦n(RSG). Periodicity: Ti(u + 4n− 2) = Ti(u). All the results in this subsection will be extended to a more general case (7.1) later in Section 7. 2.3 Background in integrable models To provide the reader with the ‘big picture behind the scene’, we briefly review the origins and the consequences of the Y-systems (Definitions 2.1 and 2.9), the periodicity (Theorems 2.3 and 2.11), and the dilogarithm identities (Theorems 2.4 and 2.12) in the context of integrable models. Mathematically speaking, the whole content in this subsection is completely indepen- dent of the rest of the paper, so that the reader can safely skip it. The study of integrable models of quantum field theory has a long history, with two initially distinct lines of development. One comes through classical statistical mechanics, since any lattice model can be viewed as a regularized Euclidean quantum field theory; the other is the direct study of the models as quantum field theories, either in Euclidean or Minkowski space. On the statistical-mechanical side, the subject can be traced back to Onsager’s solution of the two-dimensional Ising model [27] but, much of the modern approach to these models, owes most to the work of Baxter summarized in his book [1]. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 7 Viewed directly as a problem in quantum field theory, the history begins with the work on the quantum sine-Gordon model, probably the most famous example of an integrable quantum field theory. The Euclidean action of the sine-Gordon model is: ASG = ∫ dx2 ( 1 16π (∂νϕ)2 − 2µ cos(βϕ) ) , (2.14) where 0 < β2 < 1 is related to ξ in (2.2) as ξ = β2 1− β2 , (2.15) and µ fixes the mass scale: m ∝ µ1/(2−2β2), where m is the soliton mass. For ξ < 1 the theory comprises also solitons-antisoliton bound states – the breathers – with masses mn = 2m sin(πnξ/2), n = 1, 2, . . . < 1/ξ. The review by Zamolodchikov and Zamolodchikov [35] covers most of this early work, which focused particularly on characteristics of the models when defined in infinite spatial volumes. The main goal when studying any theory was the exact calculation of its S-matrix, describing the scattering of arbitrary numbers of elementary excitations. All of this work concerned massive quantum field theories, with finite correlation lengths. By contrast, the initial interest in statistical mechanics is often the study of models at criticality, where the correlation length is infinite. Links between the two approaches began to be built with the development of conformal field theory (CFT) [3] which showed how powerful algebraic techniques could be used to solve massless quantum field theories, corresponding to the continuum limits of critical lattice models. Most relevant to the current context was the subsequent discovery by Zamolodchikov [33] that suitable perturbations, of these conformal field theories could lead to models of precisely the sorts which had been studied previously as massive integrable quantum field theories and exact S-matrix models. The Euclidean action of a perturbed CFT is Aµ = ACFT + µ ∫ dx2φ(x), (2.16) where ACFT is the action of the conformal invariant theory and φ is a spinless primary field with conformal dimensions ∆ = ∆̄ < 1. The dimensionful coupling µ measures the deviation from the critical point and introduces an independent mass scale proportional to µ1/(2−2∆). Compa- ring (2.14) with (2.16), we see that the sine-Gordon model corresponds to the perturbation of a CFT with central charge c = 1 – a free massless boson – by the operator φSG = 2 cos(βϕ) with conformal dimension ∆SG = β2. Another interesting family of exact S-matrix models is obtained from the sine-Gordon model, at rational values of β2, through a quantum group restriction [31, 4] of the Hilbert space. Setting β2 = p/q with q > p coprime integers, this infinite family of models corresponds to the minimal conformal field theoriesMp,q perturbed by the operator φ1,3 with ∆RSG = (2p− q)/q. As we will see shortly, it is not the Virasoro central charge c = 1 − 6(p − q)2/pq but the effective central charge ceff = c− 24∆0, with ∆0 the dimension of the field φ0, which generates the ground state of the conformal field theory on a cylinder. For the Mp,q family of models ceff = 1− 6/pq. Links with statistical mechanics grew stronger as finite-size effects began to be explored, using techniques such as the thermodynamic Bethe Ansatz (TBA) [34]. In particular, sets of functional relations, the Y-systems, began to emerge [36] which very closely paralleled mathe- matical structures such as the ‘fusion hierarchies’ (T-systems) found in the developments of Baxter’s pioneering work on integrable lattice models [24]. 8 T. Nakanishi and R. Tateo In the TBA approach, the ground state energy E0(R) of a massive integrable quantum field theory confined on an infinitely long cylinder of circumference R is written in terms of dressed single-particle energies εa(θ) (pseudoenergies) as E0(R) = −πceff(r) 6R = − 1 2π N∑ i=1 ∫ ∞ −∞ dθ νi(θ) ln ( 1 + e−εi(θ) ) , (2.17) where θ is the rapidity and r is related to the mass m1 of the lightest excitation in the theory and the circumference R by r = Rm1. The pseudoenergies are the solutions of a set of coupled integral equations known as TBA equations. The latter equations have the general form εi(θ) = νi(θ)− N∑ j=1 ∫ ∞ −∞ dθ φij(θ − θ′) ln ( 1 + e−εj(θ ′) ) . (2.18) When all scattering is diagonal, the integral equations of the TBA follow directly from the mass spectrum {mi} and the two-body S-matrix elements Sij : νi(θ) = Rmi cosh θ, φij(θ) = 1 2π √ −1 d dθ lnSij(θ). In this case, the number of pseudoenergies coincides with the number N of particle types in the original scattering theory. If the scattering is non-diagonal the TBA derivation becomes more complicated but the final result can be still written in the form (2.18), with some of the νi = 0. The pseudoenergies with νi = 0 correspond to fictitious particles transporting zero energy and zero momentum. These new particles are often called ‘magnons’, and can be thought of as constructs introduced to get the counting of states right. The Y functions, the main subject of this paper, are related to the pseudoenergies as Yi(θ) = eεi(θ). The ultraviolet CFT regime corresponds to m1 → 0 or, equivalently, to r → 0. In this special limit E0(R) ∼ −πceff 6R , where ceff = ceff(0) is the effective central charge. It is during the calculation of ceff from equations (2.17) and (2.18) that sum-rules for the Rogers dilogarithm function emerge: ceff = 6 π2 N∑ i=1 [ L ( 1 1 + Yi ) − L ( 1 1 + Υi )] . (2.19) In (2.19), the constants Yi and Υi are the stationary values of Yi(θ) in the limits r → 0 and r →∞, respectively. The TBA equations contain also exact information on the conformal dimension ∆ of the perturbing operator φ. The key idea is to find a set of functional relations satisfied by the Y functions: a Y-system. These relations generally imply a periodicity property for the pseudo- energies under a certain imaginary shift in θ: Yi ( θ + 2π √ −1P ) = Yi(θ). The periodicity phenomenon was first noticed by Al.B. Zamolodchikov in [36] and considerations, based on this periodicity suggest that in the far ultraviolet region ceff(r) will have, apart for a possible irregular anti-bulk term, an expansion in powers of r2/P . This implies for φ either the Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 9 conformal dimension ∆ = 1− 1/P and an expansion for ceff(r) with both even and odd powers of µ, or ∆ = 1− 1/2P and an expansion where only even powers of µ appear. Let us conclude this subsection by demonstrating the validity of the argument for the SG and the RSG models at the simplest case n = 4 in (2.2). This corresponds to ξ = 3/4 and β2 = p/q = 3/7. At this specific value of coupling, the sine-Gordon scattering is non-diagonal with a single soliton-antisoliton bound state. Setting u = 8θ/π √ −1, the corresponding Y- system is Y1(u− 3)Y1(u + 3) = (1 + Y5(u))(1 + Y4(u))(1 + Y3(u− 1)) × (1 + Y3(u + 1))(1 + Y2(u− 2))(1 + Y2(u + 2)), Y2(u− 1)Y2(u + 1) = (1 + Y1(u)) 1 1 + Y3(u)−1 , Y3(u− 1)Y3(u + 1) = 1 1 + Y2(u)−1 1 1 + Y4(u)−1 1 1 + Y5(u)−1 , Y4(u− 1)Y4(u + 1) = 1 1 + Y3(u)−1 , Y5(u− 1)Y5(u + 1) = 1 1 + Y3(u)−1 . (2.20) In (2.20), Y1 corresponds to the breather, Y2 to the soliton-antisoliton pair, Y3, Y4 and Y5 to magnons. From the point of view of the sine-Gordon ground state energy the two magnonic nodes Y4 and Y5 are indistinguishable. Thus, it is the half periodicity property (Theorem 2.3), 14 in u, and P = 7/8 in θ, that is relevant to our purposes. Considering also that the perturbing field φSG is intrinsically ‘self-dual’ with only even powers of µ contributing to the expansion of ceff(r), the final result is ∆ = 1 − 1/2P = 3/7. This coincides with the conformal dimension of φSG. For the dilogarithm sum-rule (2.19) at r → 0, instead of solving the stationary version of the Y-system we can simply use the result (2.5) averaged over the period: 6 π2 n+1∑ i=1 L ( 1 1 + Yi ) = 1 8n− 4 6 π2 ∑ (i,u)∈In 0≤u<8n−4 L ( 1 1 + Yi(u) ) = n− 1, with n = 4. Similarly, in the limit r →∞ both Y1(u) and Y2(u) tend to infinity and decouple. Hence, for the second contribution to the sum-rule (2.19) we can consider the simplified system Υ3(u− 1)Υ3(u + 1) = 1 1 + Υ4(u)−1 1 1 + Υ5(u)−1 , Υ4(u− 1)Υ4(u + 1) = 1 1 + Υ3(u)−1 , Υ5(u− 1)Υ5(u + 1) = 1 1 + Υ3(u)−1 . The latter is a D3 ≡ A3 Y-system. Due to [12], the half-period is 6 in the variable u, and we have 6 π2 5∑ i=3 L ( 1 1 + Υi ) = 1 π2 ∑ (i,u)∈D3 0≤u<6 L ( 1 1 + Υi(u) ) = 2. The result is ceff = 1, as expected. The Y-system for the corresponding RSG model is Y1(u− 3)Y1(u + 3) = (1 + Y2(u− 2))(1 + Y2(u + 2)) 1 1 + Y2(u)−1 , Y2(u− 1)Y2(u + 1) = 1 + Y1(u). 10 T. Nakanishi and R. Tateo We still have P = 7/8 (Theorem 2.11) but now the perturbing operator is ‘anti self-dual’ with respect to the ground state of the conformal field theory on an cylinder. This means that both even and odd powers of µ appear in the expansion of ceff(r). Therefore, ∆ = 1 − 1/P = −1/7, which indeed matches the conformal dimension of φ13 inM3,7. Finally, from the result (2.12) we recover the effective central charge ofM3,7: ceff = 3 7π2 ∑ (i,u)∈Ĩ4 0≤u<14 L ( 1 1 + Yi(u) ) = 5 7 . 3 Cluster algebras for SG Y-systems In this section we identify Tn(SG) and Yn(SG) as relations for cluster variables and coefficients of the cluster algebra associated with a certain quiver Qn(SG). We follow [10] for definitions and conventions concerning cluster algebras with coefficients, which are summarized in Appendix A for the reader’s convenience. 3.1 Parity decompositions of T and Y-systems For (i, u) ∈ In, we set the parity condition P+ by, for even n, P+ : { i + u is even, i = 1, . . . , n, n + u is even, i = n + 1, and, for odd n, P+ :  u is even, i = 1, i + u is even, i = 2, . . . , n, n + u is even, i = n + 1. Let P− be the negation of P+. We write, for example, (i, u) : P+ if (i, u) ∈ In satisfies P+. Let Inε (ε = ±) be the set of all (i, u) : Pε. Define T◦n(SG)ε (ε = ±) to be the subring of T◦n(SG) generated by Ti(u) ((i, u) ∈ Inε). Then, we have T◦n(SG)+ ' T◦n(SG)− by Ti(u) 7→ Ti(u+1) and T◦n(SG) ' T◦n(SG)+ ⊗Z T◦n(SG)−. For (i, u) ∈ In, we introduce another parity condition P′ + by P′ + : { i + u is odd, i = 1, . . . , n, n + u is odd, i = n + 1, We have (i, u) : P′ + ⇐⇒ (i, u± di) : P+, where di is given in (2.7). Let P′ − be the negation of P′ +. Let I ′nε (ε = ±) be the set of all (i, u) : P′ ε. Define Y◦n(SG)ε (ε = ±) to be the subgroup of Y◦n(SG) generated by Yi(u), 1 + Yi(u) ((i, u) ∈ I ′nε). Then, we have Y◦n(SG)+ ' Y◦n(SG)− by Yi(u) 7→ Yi(u+1), 1+Yi(u) 7→ 1+Yi(u+1), and Y◦n(SG) ' Y◦n(SG)+ × Y◦n(SG)−. From now on, we mainly treat the + parts, T◦n(SG)+ and Y◦n(SG)+. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 11 Figure 2. The quiver Qn(SG) for n = 8 (upper) and for n = 7 (lower), where, except for the leftmost vertex of each quiver Qi, all the vertices in the same position in n−1 quivers Q1, . . . , Qn−1 are identified. 3.2 Quiver Qn(SG) Recall that a quiver is an oriented graph, namely, it consists of the vertices and the arrows connecting them. With each n ≥ 4 we associate a quiver Qn(SG) as below. First, as rather general examples, the cases n = 8 and n = 7 are given in Fig. 2, where, except for the leftmost vertex of each quiver Qi, all the vertices in the same position in n− 1 quivers Q1, . . . , Qn−1 are identified. For a general n, the quiver Qn(SG) is defined by naturally extending these examples. Namely, we consider n − 1 quivers Q1, . . . , Qn−1, whose vertices are naturally identified with the vertices of the graph Xn in Fig. 1. (The leftmost vertex corresponds to the vertex 1 in Xn.) The arrows are put in Qi as clearly indicated by the examples in Fig. 2. Note that the pattern of arrows slightly depends on the parity of n. Then, except for the leftmost vertex of each quiver Qi, all the vertices in the same position in n−1 quivers Q1, . . . , Qn−1 are identified. Also we assign the property +/− to each vertex, except for the leftmost one in each Qi, as in Fig. 2. Let us choose the index set I of the vertices of Qn(SG) so that i = (i, 1) ∈ I represents the leftmost vertex in Qi for i = 1, . . . , n−1, and i = (n, i′) ∈ I represents the vertex i′ = 2, . . . , n+1 in any quiver Qi under the natural identification with Xn. Thus, i = 1, . . . , n; and i′ = 1 if i 6= n and i′ = 2, . . . , n + 1 if i = n. Let I+ (resp. I−) denote the set of the vertices i ∈ I with property + (resp. −). We define composite mutations (Appendix A(ii)–(v)), µ+ = ∏ i∈I+ µi, µ− = ∏ i∈I− µi. Note that they do not depend on the order of the product. For a permutation σ of {1, . . . , n−1}, let σ̃ be the permutation of I such that σ̃(i, 1) = (σ(i), 1) for i 6= n and σ̃(n, i′) = (n, i′). Let σ̃(Qn(SG)) denote the quiver induced from Qn(SG) by σ̃. Namely, if there is an arrow i→ j in Qn(SG), then, there is an arrow σ̃(i)→ σ̃(j) in σ̃(Qn(SG)). Lemma 3.1. Let Q(0) := Qn(SG). We have the following periodic sequence of mutations of quivers: Q(0) µ+µ(1,1)←→ Q(1) µ−←→Q(2) µ+µ(2,1)←→ Q(3) µ−←→Q(4) µ+µ(3,1)←→ · · · µ+µ(n−1,1)←→ Q(2n− 3) µ−←→Q(2n− 2) = Q(0), (3.1) 12 T. Nakanishi and R. Tateo Figure 3. The mutation sequence of the quiver Qn(SG) in (3.1) for n = 6. The encircled vertices correspond to the mutation points (i, u) : p+ in the forward direction. where the quiver Q(2p) (p = 1, . . . , n− 2) is given by Q(2p) = σ̃p(Q(0)), σ = ( 1 2 . . . n− 2 n− 1 2 3 . . . n− 1 1 ) . (3.2) Example 3.2. The mutation sequence (3.1) for n = 6 is explicitly given in Fig. 3. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 13 3.3 Embedding maps Let B = Bn(SG) be the skew-symmetric matrix corresponding to the quiver Qn(SG) (Ap- pendix A(iii)). Let A(B, x, y) be the cluster algebra with coefficients in the universal semifield Puniv(y) (Appendix A(vi)). In view of Lemma 3.1 we set x(0) = x, y(0) = y and define clusters x(u) = (xi(u))i∈I (u ∈ Z) and coefficient tuples y(u) = (yi(u))i∈I (u ∈ Z) by the sequence of mutations · · · µ−←→ (B(0), x(0), y(0)) µ+µ(1,1)←→ (B(1), x(1), y(1)) µ−←→ · · · µ−←→ (B(2n− 2), x(2n− 2), y(2n− 2)) µ+µ(1,1)←→ · · · , (3.3) where B(u) is the skew-symmetric matrix corresponding to Q(u). For (i, u) ∈ I× Z, we set the parity condition p+ by p+ : { i ∈ I+ t {(j + 1, 1)}, u ≡ 2j (j = 0, 1, . . . , n− 2), i ∈ I−, u: odd, (3.4) where ≡ is modulo (2n − 2)Z. We define the condition p− by (i, u) : p− ⇐⇒ (i, u − 1) : p+. Plainly speaking, each (i, u) : p+ (resp. (i, u) : p−) is a mutation point of (3.3) in the forward (resp. backward) direction of u. See Fig. 3. Lemma 3.3. Below ≡ means the equivalence modulo (2n− 2)Z. (i) The map g : In+ → {(i, u) ∈ I× Z \| (i, u) : p+}, (i, u− di) 7→ { ((j + 1, 1), u), i = 1; u ≡ 2j (j = 0, 1, . . . , n− 2), ((n, i), u), i = 2, . . . , n + 1 is a bijection. (ii) The map g′ : I ′n+ → {(i, u) ∈ I× Z \| (i, u) : p+}, (i, u) 7→ { ((j + 1, 1), u), i = 1; u ≡ 2j (j = 0, 1, . . . , n− 2), ((n, i), u), i = 2, . . . , n + 1 is a bijection. Based on Lemma 3.3, we introduce alternative notations x̃i(u−di) := xi(u) for (i, u−di) ∈ In+ with (i, u) = g((i, u− di)) and yi(u) := yi(u) for (i, u) ∈ I ′n+ with (i, u) = g′((i, u)), respectively, which turn out to be useful. Let A(B, x) be the cluster algebra with trivial coefficients, where (B, x) is the initial seed and the coefficient semifield is the trivial semifield 1 = {1} (Appendix A(i)). Let π1 : Puniv(y)→ 1, yi 7→ 1 be the projection. Let [xi(u)]1 denote the image of xi(u) by the algebra homomorphism A(B, x, y)→ A(B, x) induced from π1. It is called the trivial evaluation. The following lemma follows from the exchange relation of cluster variables (A.4) and the property of the sequence (3.1) observed in Fig. 3. Lemma 3.4. Let G+ and G− be the ones in (2.8) and (2.9). The family {x̃i(u) \| (i, u) ∈ In+} satisfies a system of relations x̃i (u− di) x̃i (u + di) = yi(u) 1 + yi(u) ∏ (j,v)∈In+ x̃j(v)G+(i,u;j,v) 14 T. Nakanishi and R. Tateo + 1 1 + yi(u) ∏ (j,v)∈In+ x̃j(v)G−(i,u;j,v), (3.5) where (i, u) ∈ I ′n+. In particular, the family {[x̃i(u)]1 \| (i, u) ∈ In+} satisfies the T-system Tn(SG) in A(B, x) by replacing Ti(u) with [x̃i(u)]1. Example 3.5. Consider the case n = 6 in Fig. 3. Let us consider the mutation at the ver- tex (1, 1) in Q(0), to which the variable x̃1(−5) is attached. The next time (1, 1) is mutated is in Q(10), where x̃1(5) is attached. Meanwhile, the only vertex connected to (1, 1) in Q(0) is (6, 2), and the variable attached to (6, 2) in Q(0) is equal to the variable x̃2(0) attached to (6, 2) in Q(1). Taking account of the directions of the arrows, we have the relation x̃1(−5)x̃1(5) = y1(0) 1 + y1(0) x̃2(0) + 1 1 + y1(0) , which agrees with (2.6) and (3.5). Similarly, consider the mutation at the vertex (6, 2) in Q(1), to which the variable x̃2(0) is attached. The next time (6, 2) is mutated is in Q(3), where x̃2(2) is attached. Meanwhile, the vertices connected to (6, 2) in Q(1) are (1, 1), (2, 1), and (6, 3), and the variable attached to them are equal to x̃1(5), x̃1(−3), and x̃3(1), respectively. Taking account of the directions of the arrows, we have the relation x̃2(0)x̃2(2) = y2(1) 1 + y2(1) x̃1(−3)x̃1(5) + 1 1 + y2(1) x̃3(1). As the last example, consider the mutation at the vertex (6, 6) in Q(1), to which the variab- le x̃6(0) is attached. The next time (6, 6) is mutated is in Q(3), where x̃6(2) is attached. Meanwhile, the vertices connected to (6, 6) in Q(1) are (4, 1) and (6, 5), and the variable attached to them are equal to x̃1(1) and x̃5(1), respectively. Taking account of the directions of the arrows, we have the relation x̃6(0)x̃6(2) = y6(1) 1 + y6(1) x̃1(1) + 1 1 + y6(1) x̃5(1). The other cases can be checked in similar manners. Definition 3.6. The T-subalgebra AT (B, x) of A(B, x) associated with the sequence (3.3) is the subring of A(B, x) generated by [xi(u)]1 ((i, u) ∈ I × Z), or equivalently, generated by [x̃i(u)]1 ((i, u) ∈ In+). By Lemma 3.4, we have the following embedding. Theorem 3.7. The ring T◦n(SG)+ is isomorphic to AT (B, x) by the correspondence Ti(u) 7→ [x̃i(u)]1. The coefficient group G(B, y) associated with A(B, x, y) is the multiplicative subgroup of the semifield Puniv(y) generated by all the coefficients y′i of A(B, x, y) together with 1 + y′i. The following lemma follows from the exchange relation of coefficients (A.3) and the property of the sequence (3.1). Lemma 3.8. The family {yi(u) \| (i, u) ∈ I ′n+} satisfies the Y-system Yn(SG) by replacing Yi(u) with yi(u). Example 3.9. Consider the case n = 6 in Fig. 3. Let us consider the mutation at the ver- tex (1, 1) in Q(0), to which the variable y1(0) is attached. The next time (1, 1) is mutated is in Q(10), where y1(10) is attached. Meanwhile, between u = 0 and u = 10, the vertices con- nected to (1, 1) in Q(u) and mutated are (6, 2) at u = 1, 9, (6, 3) at u = 2, 8, (6, 4) at u = 3, 7, Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 15 (6, 5) at u = 4, 6, and (6, 6) and (6, 7) at u = 5, Taking account of the directions of the arrows, we have the relation y1(0)y1(10) = (1 + y2(1))(1 + y2(9))(1 + y3(2))(1 + y3(8))(1 + y4(3)) × (1 + y4(7))(1 + y5(4))(1 + y5(6))(1 + y6(5))(1 + y7(5)), which agrees with (2.1). Similarly, consider the mutation at the vertex (6, 2) in Q(1), to which the variable y2(1) is attached. The next time (6, 2) is mutated is in Q(3), where y2(3) is attached. Meanwhile, between u = 1 and u = 3, the vertices connected to (6, 2) in Q(u) and mutated are (2, 1) and (6, 3) at u = 2. Taking account of the directions of the arrows, we have the relation y2(1)y2(3) = (1 + y1(2)) 1 + y3(2)−1 . As the last example, consider the mutation at the vertex (6, 6) in Q(1), to which the variable y6(1) is attached. The next time (6, 6) is mutated is in Q(3), where y6(3) is attached. Meanwhile, between u = 1 and u = 3, the only vertex connected to (6, 6) in Q(u) and mutated is (6, 5) at u = 2. Taking account of the directions of the arrows, we have the relation y6(1)y6(3) = 1 1 + y5(2)−1 . The other cases can be checked in similar manners. Definition 3.10. The Y-subgroup GY (B, y) of G(B, y) associated with the sequence (3.3) is the subgroup of G(B, y) generated by yi(u) ((i, u) ∈ I × Z) and 1 + yi(u) ((i, u) : p+ or p−), or equivalently, generated by yi(u) and 1 + yi(u) ((i, u) ∈ I ′n+). By Lemma 3.8, we have the following embedding. Theorem 3.11. The group Y◦n(SG)+ is isomorphic to GY (B, y) by the correspondence Yi(u) 7→ yi(u) and 1 + Yi(u) 7→ 1 + yi(u). 4 Proof of Theorems 2.3, 2.4, and 2.7 In this section we prove Theorems 2.3, 2.4, and 2.7 using the method of [26, 13]. Let y = y(0) be the initial coefficient tuple of the cluster algebra A(B, x, y) with B = Bn(SG) in the previous section. Let Ptrop(y) be the tropical semifield for y (Appendix A(i)). Let πT : Puniv(y) → Ptrop(y), yi 7→ yi be the projection. Let [yi(u)]T and [GY (B, y)]T denote the images of yi(u) and GY (B, y) by the multiplicative group homomorphism induced from πT, respectively. They are called the tropical evaluations, and the resulting relations in the group [GY (B, y)]T are called the tropical Y-system. They are first studied in [9] for cluster algebras of finite types. We say a (Laurent) monomial m = ∏ i∈I yki i is positive (resp. negative) if m 6= 1 and ki ≥ 0 (resp. ki ≤ 0) for any i. It is known that every monomial [yi(u)]T is either positive or negative by [10, Proposition 5.6] and [6, Theorem 1.7]. The next ‘tropical mutation rule’ for [yi(u)]T is general and useful. Lemma 4.1. Suppose that y′′ is the coefficient tuple obtained from the mutation of another coefficient tuple y′ at k with mutation matrix B′. Then, for any i 6= k, we have the rule: (i) [y′′i ]T = [y′i]T[y′k]T if one of the following conditions holds: 16 T. Nakanishi and R. Tateo (a) B′ ki > 0, and [y′k]T is positive; (b) B′ ki < 0, and [y′k]T is negative. (ii) [y′′i ]T = [y′i]T if one of the following conditions holds: (a) B′ ki = 0; (b) B′ ki > 0, and [y′k]T is negative; (c) B′ ki < 0, and [y′k]T is positive. Proof. This is an immediate consequence of the exchange relation (A.3) and (A.1). � The following properties of the tropical Y-system are crucial. Proposition 4.2. For [GY (B, y)]T with B = Bn(SG), the following facts hold. (i) For 0 ≤ u < 4n − 2, the monomial [yi(u)]T ((i, u) : p+) is negative if and only if u takes the following values.{ 2n− 2 ≤ u < 4n− 2 for i = (1, 1), . . . , (n− 1, 1), (n, 2), u = 2n− 2, 2n− 1, 4n− 4, 4n− 3 for i = (n, 3), . . . , (n, n + 1). (Note that for each i, u takes only a part of the list due to the condition (i, u) : p+.) (ii) We have [yi(4n− 2)]T = yτ−1(i), where τ is a bijection I→ I defined by (i, 1) 7→ (σ(i), 1), i = 1, . . . , n− 1, (n, i′) 7→  (n, i′), i′ = 2, . . . , n− 1, (n, n + 1), i′ = n, (n, n), i′ = n + 1 and σ is the permutation in (3.2). (iii) The number N− of the negative monomials [yi(u)]T for (i, u) : p+ in the region 0 ≤ u < 4n− 2 is 4n− 2. Proof. (i) Let us factorize [yi(u)]T = [yi(u)]′T[yi(u)]′′T, where [yi(u)]′T is a monomial in y(i,1) (i = 1, . . . , n−1) while [yi(u)]′′T is a monomial in y(n,i′) (i′ = 2, . . . , n+1). One can independently study [yi(u)]′T and [yi(u)]′′T. The claim (i) follows from the following results, which are proved inductively on u by Lemma 4.1 and the results of [9, 10]. (a) [yi(u)]′T part. This part is easier. All the monomials [yi(u)]′T which are not 1 for (i, u) : p+ in the region 0 ≤ u < 4n− 2 are as follows. We have [y(i,1)(2i− 2)]′T = y(i,1) (i = 1, . . . , n− 1), and also [y(1,1)(2n− 2)]′T = y−1 (1,1), [y(n,2)(2n− 1)]′T = y−1 (1,1), [y(2,1)(2n)]′T = y−1 (1,1)y −1 (2,1), [y(n,2)(2n + 1)]′T = y−1 (2,1), [y(3,1)(2n + 2)]′T = y−1 (2,1)y −1 (3,1), [y(n,2)(2n + 3)]′T = y−1 (3,1), (4.1) ... ... [y(n−1,1)(4n− 6)]′T = y−1 (n−2,1)y −1 (n−1,1), [y(n,2)(4n− 5)]′T = y−1 (n−1,1), [y(1,1)(4n− 4)]′T = y−1 (n−1,1). Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 17 (b) [yi(u)]′′T part. Below we list all the monomials [yi(u)]′′T which are not 1 for (i, u) : p+ in the region 0 ≤ u < 4n− 2. We separate the region 0 ≤ u < 4n− 2 into four parts corresponding to the decomposition 4n − 2 = (2n − 2) + 2 + (2n − 4) + 2, where 2n − 2 and 2n − 4 are the Coxeter numbers of Dn and Dn−1. Region I: 0 ≤ u < 2n−2. All the monomials [y(n,i′)(u)]′′T (i′ = 2, . . . , n+1) for ((n, i′), u) : p+ are identified with the positive roots of Dn as in [10, Proposition 10.7]; therefore, they are positive. Here, Dn is identified with the subgraph of Xn consisting of vertices 2, . . . , n + 1. Region II: u = 2n− 2, 2n− 1. We have, for even n, [y(n,i′)(2n− 2)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , n− 1, [y(n,i′)(2n− 1)]′′T = y−1 (n,i′), i′ = 2, 4, . . . , n, n + 1, and, for odd n, [y(n,i′)(2n− 2)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , n− 2, [y(n,n)(2n− 2)]′′T = y−1 (n,n+1), [y(n,n+1)(2n− 2)]′′T = y−1 (n,n), [y(n,i′)(2n− 1)]′′T = y−1 (n,i′), i′ = 2, 4, . . . , n− 1. Region III: 2n ≤ u < 4n−4. All the monomials [y(n,i′)(u)]′′T (i′ = 3, . . . , n+1) for ((n, i′), u) : p+ are identified with the positive roots of Dn−1, therefore, they are positive. Here, Dn−1 is identified with the subgraph of Xn consisting of vertices 3, . . . , n + 1. Region IV: u = 4n− 4, 4n− 3. We have, for even n, [y(n,i′)(4n− 4)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , n− 1, [y(n,i′)(4n− 3)]′′T = y−1 (n,i′), i′ = 2, 4, . . . , n− 2, [y(n,n)(4n− 3)]′′T = y−1 (n,n+1), [y(n,n+1)(4n− 3)]′′T = y−1 (n,n), (4.2) and, for odd n, [y(n,i′)(4n− 4)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , n− 2, [y(n,n)(4n− 4)]′′T = y−1 (n,n+1), [y(n,n+1)(4n− 4)]′′T = y−1 (n,n), [y(n,i′)(4n− 3)]′′T = y−1 (n,i′), i′ = 2, 4, . . . , n− 1. (4.3) Besides, we have the sequences of monomials which appear over Regions III and IV; for even n, [y(2,1)(2n)]′′T = y−1 (n,2)y −1 (n,3), [y(3,1)(2n + 2)]′′T = y−1 (n,4)y −1 (n,5), . . . , [y(n/2,1)(3n− 4)]′′T = y−1 (n,n−2)y −1 (n,n−1), [y(n/2+1,1)(3n− 2)]′′T = y−1 (n,n−1)y −1 (n,n)y −1 (n,n+1), [y(n/2+2,1)(3n)]′′T = y−1 (n,n−3)y −1 (n,n−2), . . . , [y(n−1,1)(4n− 6)]′′T = y−1 (n,3)y −1 (n,4), [y(1,1)(4n− 4)]′′T = y−1 (n,2), (4.4) and, for odd n, the middle three terms are replaced with [y((n−1)/2,1)(3n− 5)]′′T = y−1 (n,n−3)y −1 (n,n−2), [y((n+1)/2,1)(3n− 3)]′′T = y−1 (n,n−1)y −1 (n,n)y −1 (n,n+1), [y(n+3)/2,1)(3n− 1)]′′T = y−1 (n,n−2)y −1 (n,n−1). (4.5) (ii) They follow from (4.2)–(4.5). (iii) By (i), for each i the numbers of the negative monomials [yi(u)]T in the region is 2 for i = (1, 1), 1 for i = (i, 1) (i = 2, . . . , n−1), n for i = (n, 2). and 2 for i = (n, i′) (i′ = 3, . . . , n+1). Summing up, we have N− = 4n− 2. � 18 T. Nakanishi and R. Tateo Now we prove Theorems 2.3 and 2.7. It follows from a very general theorem [13, Theorem 5.1] (based on the work by Plamondon [29, 28]) that the cluster variables xi(u) and coefficients yi(u) have the same periodicity with [yi(u)]T as in Proposition 4.2(ii), namely, xi(4n − 2) = xτ−1(i) and yi(4n− 2) = yτ−1(i). It follows that, under the labelling introduced in Lemma 3.3, we have x̃i(u + 4n− 2) = x̃ω(i)(u), (i, u) : P+, yi(u + 4n− 2) = yω(i)(u), (i, u) : P′ +, (4.6) where ω is the one in Theorem 2.3(i). Then, thanks to the isomorphisms in Theorems 3.7 and 3.11, and also by the isomorphisms T◦n(SG)+ ' T◦n(SG)− and Y◦n(SG)+ ' Y◦n(SG)−, we obtain Theorems 2.3 and 2.7. Since the F -polynomials (Appendix A(vii)) are defined as a certain specialization of cluster variables, they satisfy the same periodicity as the cluster variables. Let F̃i(u) ((i, u) : P+) be the F -polynomial for x̃i(u). Then, from (4.6) we have F̃i(u + 4n− 2) = F̃ω(i)(u), (i, u) : P+. (4.7) Next we prove Theorem 2.4. Let ∧2 Puniv(y) be the quotient of the additive Abelian group Puniv(y)⊗Z Puniv(y) by the subgroup generated by symmetric tensors [11, 5]. Lemma 4.3. In ∧2 Puniv(y), we have∑ (i,u):P′ + 0≤u<4n−2 yi(u) ∧ (1 + yi(u)) = 0. Proof. One can prove it in a similar way as in [26, 13], using [10, Proposition 3.13] and (4.7).� Applying the method of [11, 5, 26], we immediately obtain the following theorem from Lemma 4.3 and Proposition 4.2(iii). Theorem 4.4. For any semifield homomorphism ϕ : Puniv(y) → R+, we have the following identity. 6 π2 ∑ (i,u):P′ + 0≤u<4n−2 L ( ϕ(yi(u)) 1 + ϕ(yi(u)) ) = 4n− 2. This is equivalent to Theorem 2.4. 5 Cluster algebras for RSG Y-systems In this section we identify Tn(RSG) and Yn(RSG) as relations for cluster variables and coeffi- cients of the cluster algebra associated with a certain quiver Qn(RSG). For those things which are quite parallel to the SG case, we skip their precise descriptions just by saying ‘as before’ unless they are not obvious. 5.1 Parity decompositions of T and Y-systems For (i, u) ∈ Ĩn, we set the parity condition P+ by, for even n, P+ : i + u is even, i = 1, . . . , n− 2, Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 19 Figure 4. The quiver Qn(RSG) for n = 8 (upper) and for n = 7 (lower), where, except for the leftmost vertex of each quiver Qi, all the vertices in the same position in n−1 quivers Q1, . . . , Qn−1 are identified. We have an arrow between the leftmost vertices of Q4 and Q5 for n = 8, and Q3 and Q4 for n = 7. and, for odd n, P+ : { u is even, i = 1, i + u is even, i = 2, . . . , n− 2. Define Ĩnε and T◦n(RSG)ε (ε = ±) as before. Then, we have T◦n(RSG)+ ' T◦n(RSG)− by Ti(u) 7→ Ti(u + 1) and T◦n(RSG) ' T◦n(RSG)+ ⊗Z T◦n(RSG)−. For (i, u) ∈ Ĩn, we set the parity condition P′ + by P′ + : i + u is odd, i = 1, . . . , n− 2. We have (i, u) : P′ + ⇐⇒ (i, u± di) : P+, where di is given in (2.7). Define Ĩ ′nε and Y◦n(RSG)ε (ε = ±) as before. Then, we have Y◦n(RSG)+ ' Y◦n(RSG)− by Yi(u) 7→ Yi(u + 1), 1 + Yi(u) 7→ 1 + Yi(u + 1), and Y◦n(RSG) ' Y◦n(RSG)+ × Y◦n(RSG)−. From now on, we mainly treat the + parts, T◦n(RSG)+ and Y◦n(RSG)+. 5.2 Quiver Qn(RSG) With each n ≥ 4 we associate a quiver Qn(RSG) as below. First, the cases n = 8 and n = 7 are given in Fig. 4, where, except for the leftmost vertex of each quiver Qi, all the vertices in the same position in n − 1 quivers Q1,. . . ,Qn−1 are identified. For a general n, the quiver Qn(RSG) is defined by naturally extending these examples; in particular, we have an arrow from the leftmost vertex of Qn/2+1 to the leftmost vertex of Qn/2 for even n, and an arrow from the leftmost vertex of Q(n+1)/2 to the leftmost vertex of Q(n−1)/2 for odd n. Also we assign the property +/− to each vertex, except for the leftmost one in each Qi, as in Fig. 4. Let us choose the index set Ĩ of the vertices of Qn(RSG) naturally obtained from the index set I of the vertices of Qn(SG) by the restriction to the vertices of Qn(RSG). Thus, i = 1, . . . , n; and i′ = 1 if i 6= n and i′ = 2, . . . , n − 2 if i = n. Then, we define Ĩ+, Ĩ−, µ+, µ−, and w̃(Qn(RSG)) as before. 20 T. Nakanishi and R. Tateo Figure 5. The mutation sequence of the quiver Qn(RSG) in (3.1) for n = 6. The encircled vertices correspond to the mutation points (i, u) : p+ in the forward direction. Lemma 5.1. Let Q(0) := Qn(RSG). We have the periodic sequence of mutations of qui- vers (3.1), where the quiver Q(2p) (p = 1, . . . , n− 2) is given by (3.2). Example 5.2. The mutation sequence (3.1) for n = 6 is explicitly given in Fig. 5. Let us explain why we need the ‘extra arrow’ from Q4 to Q3 in Q(0) in this example. Suppose that we do not have it. Then, we will have an extra arrow from Q3 to Q4 in Q(2) due to the mutation at u = 1. This extra arrow will still remain in Q(4), where the vertex (3, 1) will be mutated. This conflicts with Tn(RSG) and Yn(RSG) eventually. The extra arrow is, thus, necessary as a precaution to avoid this confliction. 5.3 Embedding maps Let B = Bn(RSG) be the skew-symmetric matrix corresponding to the quiver Qn(RSG). Let A(B, x, y) be the cluster algebra with coefficients in the universal semifield Puniv(y). In view of Lemma 5.1 we set x(0) = x, y(0) = y and define clusters x(u) = (xi(u))i∈Ĩ (u ∈ Z) and coefficient tuples y(u) = (yi(u))i∈Ĩ (u ∈ Z) by the sequence of mutations (3.3) as before. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 21 For (i, u) ∈ Ĩ × Z, we set the parity condition p+ by the same rule as (3.4), where I± is replaced with Ĩ±. We define the condition p− by (i, u) : p− ⇐⇒ (i, u− 1) : p+. Lemma 5.3. Below ≡ means the equivalence modulo (2n− 2)Z. (i) The map g : Ĩn+ → {(i, u) ∈ Ĩ× Z \| (i, u) : p+}, (i, u− di) 7→ { ((j + 1, 1), u), i = 1; u ≡ 2j (j = 0, 1, . . . , n− 2), ((n, i), u), i = 2, . . . , n− 2 is a bijection. (ii) The map g′ : Ĩ ′n+ → {(i, u) ∈ Ĩ× Z \| (i, u) : p+}, (i, u) 7→ { ((j + 1, 1), u), i = 1; u ≡ 2j (j = 0, 1, . . . , n− 2), ((n, i), u), i = 2, . . . , n− 2 is a bijection. Based on Lemma 5.3, we introduce alternative notations x̃i(u−di) := xi(u) for (i, u−di) ∈ Ĩn+ with (i, u) = g((i, u− di)) and yi(u) := yi(u) for (i, u) ∈ Ĩ ′n+ with (i, u) = g′((i, u)), respectively. Let A(B, x) be the cluster algebra with trivial coefficients. Definition 5.4. The T-subalgebra AT (B, x) of A(B, x) associated with the sequence (3.3) is the subring of A(B, x) generated by [xi(u)]1 ((i, u) ∈ Ĩ × Z), or equivalently, generated by [x̃i(u)]1 ((i, u) ∈ Ĩn+). By the lemma parallel to Lemma 3.4, we have the following embedding. Theorem 5.5. The ring T◦n(RSG)+ is isomorphic to AT (B, x) by the correspondence Ti(u) 7→ [x̃i(u)]1. Example 5.6. Consider the case n = 6 in Fig. 5. Consider the mutation at the vertex (6, 4) in Q(1), to which the variable x̃4(0) is attached. The next time (6, 4) is mutated is in Q(3), where x̃4(2) is attached. Meanwhile, the vertices connected to (6, 4) in Q(1) are (3, 1), (4, 1), (5, 1) and (6, 3), and the variable attached to them are equal to x̃1(−1), x̃1(1), x̃1(3), and x̃3(1), respectively. Taking account of the directions of the arrows, we have the relation x̃4(0)x̃4(2) = y4(1) 1 + y4(1) x̃1(−1)x̃1(3) + 1 1 + y4(1) x̃1(1)x̃3(1), which agree with (2.13). The other cases are the same as in the SG case. Let G(B, y) be the coefficient group associated with A(B, x, y) as before. Definition 5.7. The Y-subgroup GY (B, y) of G(B, y) associated with the sequence (3.3) is the subgroup of G(B, y) generated by yi(u) ((i, u) ∈ Ĩ × Z) and 1 + yi(u) ((i, u) : p+ or p−), or equivalently, generated by yi(u) and 1 + yi(u) ((i, u) ∈ Ĩ ′n+). By the lemma parallel to Lemma 3.8, we have the following embedding. Theorem 5.8. The group Y◦n(RSG)+ is isomorphic to GY (B, y) by the correspondence Yi(u) 7→ yi(u) and 1 + Yi(u) 7→ 1 + yi(u). 22 T. Nakanishi and R. Tateo Example 5.9. Consider the case n = 6 in Fig. 5. Let us consider the mutation at the vertex (1, 1) in Q(0), to which the variable y1(0) is attached. The next time (1, 1) is mutated is in Q(10), where y1(10) is attached. Meanwhile, between u = 0 and u = 10, the vertices connected to (1, 1) in Q(u) and mutated are (6, 2) at u = 1, 9, (6, 3) at u = 2, 8, and (6, 4) at u = 3, 5, 7. Taking account of the directions of the arrows, we have the relation y1(0)y1(10) = (1 + y2(1))(1 + y2(9))(1 + y3(2))(1 + y3(8))(1 + y4(3))(1 + y4(7)) 1 + y4(5)−1 , which agrees with (2.10). The other cases are the same as in the SG case. 6 Proof of Theorems 2.11, 2.12, and 2.15 In this section we prove Theorems 2.11, 2.12, and 2.15 by the same method as in Section 4. Let Bn(RSG) be the one in the previous section. The following properties of the tropical Y-system are crucial. Proposition 6.1. For [GY (B, y)]T with B = Bn(RSG), the following facts hold. (i) For 0 ≤ u < 4n − 2, the monomial [yi(u)]T ((i, u) : p+) is negative if and only if u takes the following values. 2n− 2 ≤ u < 4n− 2 for i = (1, 1), . . . , (n− 1, 1), u = n− 2, n− 1; 2n− 2 ≤ u < 4n− 2 for i = (n, 2), u = n− 2, n− 1, 2n− 2, 2n− 1, 3n− 3, 3n− 2, 4n− 4, 4n− 3 for i = (n, 3), . . . , (n, n− 2). (Note that for each i, u takes only a part of the list due to the condition (i, u) : p+.) (ii) We have [yi(4n− 2)]T = yτ−1(i), where τ is a bijection I→ I defined by (i, 1) 7→ (σ(i), 1), i = 1, . . . , n− 1, (n, i′) 7→ (n, i′), i′ = 2, . . . , n− 2, and σ is the permutation in (3.2). (iii) The number N− of the negative monomials [yi(u)]T for (i, u) : p+ in the region 0 ≤ u < 4n− 2 is 6n− 15. Proof. (i) Let us factorize [yi(u)]T = [yi(u)]′T[yi(u)]′′T, where [yi(u)]′T is a monomial in y(i,1) (i = 1, . . . , n− 1) while [yi(u)]′′T is a monomial in y(n,i′) (i′ = 2, . . . , n− 2). The claim (i) follows from the following results. (a) [yi(u)]′T part. This part is exactly the same as (4.1) in the Bn(SG) case. (b) [yi(u)]′′T part. Below we list all the monomials [yi(u)]′′T which are not 1 for (i, u) : p+ in the region 0 ≤ u < 4n−2. We separate the region 0 ≤ u < 4n−2 into eight parts corresponding to the decomposition 4n− 2 = (n− 2) + 2 + (n− 2) + 2 + (n− 3) + 2 + (n− 3) + 2, where n− 2 and n− 3 are the Coxeter numbers of An−3 and An−4. Region I: 0 ≤ u < n− 2. All the monomials [y(n,i′)(u)]′′T (i′ = 2, . . . , n− 2) for ((n, i′), u) : p+ are identified with the positive roots of An−3; therefore, they are positive. Here, An−3 is identified with the subgraph of Xn consisting of vertices 2, . . . , n− 2. Region II: u = n− 2, n− 1. We have, for even n, [y(n,i′)(n− 2)]′′T = y−1 (n,n−i′), i′ = 3, 5, . . . , n− 3, Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 23 [y(n,i′)(n− 1)]′′T = y−1 (n,n−i′), i′ = 2, 4, . . . , n− 2, and, for odd n, [y(n,i′)(n− 2)]′′T = y−1 (n,n−i′), i′ = 2, 4, . . . , n− 3, [y(n,i′)(n− 1)]′′T = y−1 (n,n−i′), i′ = 3, 5, . . . , n− 2. Region III: n ≤ u < 2n− 2. Once again, all the monomials [y(n,i′)(u)]′′T (i′ = 2, . . . , n− 2) for ((n, i′), u) : p+ are identified with the positive roots of An−3; therefore, they are positive. Region IV: n = 2n− 2, 2n− 1. We have [y(n,i′)(2n− 2)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , [y(n,i′)(2n− 1)]′′T = y−1 (n,i′), i′ = 2, 4, . . . . Region V: 2n ≤ u < 3n− 3. All the monomials [y(n,i′)(u)]′′T (i′ = 3, . . . , n− 2) for ((n, i′), u) : p+ are identified with the positive roots of An−4; therefore, they are positive. Here, An−4 is identified with the subgraph of Xn consisting of vertices 3, . . . , n− 2. Region VI: u = 3n− 3, 3n− 2. We have, for even n, [y(n,i′)(3n− 3)]′′T = y−1 (n,n+1−i′), i′ = 4, 6, . . . , n− 2, [y(n,i′)(3n− 2)]′′T = y−1 (n,n+1−i′), i′ = 3, 5, . . . , n− 3, and, for odd n, [y(n,i′)(3n− 3)]′′T = y−1 (n,n+1−i′), i′ = 3, 5, . . . , n− 2, [y(n,i′)(3n− 2)]′′T = y−1 (n,n+1−i′), i′ = 4, 6, . . . , n− 3. Region VII: 3n−1 ≤ u < 4n−4. Once again, all the monomials [y(n,i′)(u)]′′T (i′ = 3, . . . , n−2) for ((n, i′), u) : p+ are identified with the positive roots of An−4; therefore, they are positive. Region VIII: u = 4n− 4, 4n− 3. We have [y(n,i′)(4n− 4)]′′T = y−1 (n,i′), i′ = 3, 5, . . . , [y(n,i′)(4n− 3)]′′T = y−1 (n,i′), i′ = 2, 4, . . . . (6.1) Besides, we have the sequences of monomials which appear over Regions V–VIII; for even n, [y(2,1)(2n)]′′T = y−1 (n,2)y −1 (n,3), y(3,1)(2n + 2)]′′T = y−1 (n,4)y −1 (n,5), . . . , [y(n/2,1)(3n− 4)]′′T = y−1 (n,n−2), [y(n/2+2,1)(3n)]′′T = y−1 (n,n−3)y −1 (n,n−2), . . . , [y(n−1,1)(4n− 6)]′′T = y−1 (n,3)y −1 (n,4), [y(1,1)(4n− 4)]′′T = y−1 (n,2), (6.2) and, for odd n, the middle two terms are replaced with [y((n−1)/2,1)(3n− 5)]′′T = y−1 (n,n−3)y −1 (n,n−2), [y(n+3)/2,1)(3n− 1)]′′T = y−1 (n,n−2). (6.3) (ii) They follow from (6.1)–(6.3). (iii) By (i), for each i the numbers of the negative monomials [yi(u)]T in the region is 2 for i = (1, 1), 1 for i = (i, 1) (i = 2, . . . , n − 1), n + 1 for i = (n, 2). and 4 for i = (n, i′) (i′ = 3, . . . , n− 2). Summing up, we have N− = 6n− 15. � Theorems 2.11, 2.15 and 2.12 follow from Proposition 6.1 as before. 24 T. Nakanishi and R. Tateo Figure 6. The diagram Xm,n. 7 Further extension Seeing that the cluster algebraic setting perfectly works for the case (2.2), it is rather natural to expect that the method in this paper is, at least in principle, applicable for a general rational ξ. On the other hand, working out in full generality seems a complicated task, and we do not pursue it in this paper. However, repeating our method to a little more general case ξ = n− 1 mn−m + 1 = 1 m + 1 n− 1 (7.1) is not difficult. Since the proof is mostly the repetition of the previous case m = 1, we concen- trate on exhibiting the results. 7.1 Further result for SG Y-systems With a pair of integers m ≥ 1 and n ≥ 4, we associate a diagram Xm,n in Fig. 6, following [32]. Let Im,n = {1, . . . ,m + n} × Z. Definition 7.1 ([32]). Fix a pair of integers m ≥ 1 and n ≥ 4. The sine-Gordon (SG) Y-system Ym,n(SG) is the following system of relations for a family of variables {Yi(u) \| (i, u) ∈ Im,n}, Yi(u− n + 1)Yi(u + n− 1) = ∏ j:j∼i (1 + Yj(u)), i = 1, . . . ,m− 1, Ym (u− n + 1) Ym (u + n− 1) = (1 + Ym−1(u)) × ( m+n−2∏ j=m+1 (1 + Yj(u−m− n + 1 + j))(1 + Yj(u + m + n− 1− j)) ) × (1 + Ym+n−1(u))(1 + Ym+n(u)), Ym+1(u− 1)Ym+1(u + 1) = 1 + Ym(u) 1 + Ym+2(u)−1 , Yi(u− 1)Yi(u + 1) = 1∏ j:j∼i (1 + Yj(u)−1) , i = m + 2, . . . ,m + n, where j ∼ i means that j is adjacent to i in Xm,n. Definition 7.2. Fix a pair of integers m ≥ 1 and n ≥ 4. The sine-Gordon (SG) T-system Tm,n(SG) is the following system of relations for a family of variables {Ti(u) \| (i, u) ∈ Im,n}, Ti(u− n + 1)Ti(u + n− 1) = ∏ j:j∼i Tj(u) + 1, i = 1, . . . ,m, Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 25 Tm+1(u− 1)Tm+1(u + 1) = Tm(u− n + 2)Tm(u + n− 2) + Tm+2(u), Ti(u− 1)Ti(u + 1) = Tm(u−m− n + 1 + i) × Tm(u + m + n− 1− i) + ∏ j:j∼i Tj(u), i = m + 2, . . . ,m + n− 2, Tm+n−1(u− 1)Tm+n−1(u + 1) = Tm(u) + Tm+n−2(u), Tm+n(u− 1)Tm+n(u + 1) = Tm(u) + Tm+n−2(u), where j ∼ i means that j is adjacent to i in Xm,n. We define the associated multiplicative Abelian group Y◦m,n(SG) and the commutative ring T◦m,n(SG) as before. Theorem 7.3. (i) (Conjectured by [32].) The following relations hold in Y◦m,n(SG). (a) Suppose that both m and n are even. (In other words, mn−m+n is even.) Then, we have the periodicity: Yi(u + 2(mn−m + n)) = Yi(u). (There is no half periodicity.) (b) Suppose that at least one of m or n is odd. (In other words, mn − m + n is odd.) Then, we have the half periodicity: Yi(u + 2(mn−m + n)) = Yω(i)(u), where ω is an involution of the set {1, . . . ,m + n} defined by ω(m + n − 1) = m + n, ω(m + n) = m+n−1, and ω(i) = i (i = 1, . . . ,m+n−2). Therefore, we have the full periodicity: Yi(u + 4(mn−m + n)) = Yi(u). (ii) The same periodicity holds in T◦m,n(SG) by replacing Yi(u) in (i) with Ti(u). (iii) (Conjectured by [32].) Suppose that a family of positive real numbers {Yi(u) \| (i, u) ∈ In} satisfies Ym,n(SG). Then, we have the identities 6 π2 ∑ (i,u)∈Im,n 0≤u<4(mn−m+n) L ( Yi(u) 1 + Yi(u) ) = 4(m + 1)(mn−m + n), 6 π2 ∑ (i,u)∈Im,n 0≤u<4(mn−m+n) L ( 1 1 + Yi(u) ) = 4(n− 1)(mn−m + n). In our proof of Theorem 7.3 we have a natural interpretation of the full/half period 2(mn−m + n) = h(Dn) + 2 + m(h(Dn−1) + 2). Remark 7.4. Actually, Ym,n(SG) and Tm,n(SG) are also considered for n = 3, and they coincide with the Y and T-systems of type Bm+1 with level 2 in [24]. Theorem 7.3 remains valid for n = 3 due to [13]. The cluster algebraic formulation of Ym,n(SG) and Tm,n(SG) is done by the quiver Qm,n(SG) defined as follows. As a rather general example, the case (m,n) = (4, 7) is given in Fig. 7, where, all the vertices with • in the same position in the quivers Q1, . . . , Q6 are identified. Note that the arrows between the vertices with ◦ in Q1, Q2, Q3 and Q4, Q5, Q6 are opposite. For a general n, the quiver Qn(SG) is defined by naturally extending this example. Namely, add m− 1 vertices to the left of the leftmost vertex in each quiver Qi of Qn(SG). We assign the property +/− to each vertex as in Fig. 7 such that the leftmost vertex of each Qi has the property − for even m and + for odd m. Then, we put the arrows between the vertices with ◦ such that, for even n, 26 T. Nakanishi and R. Tateo Figure 7. The quiver Qm,n(SG) for (m,n) = (4, 7), where, all the vertices with • in the same position in the quivers Q1, . . . , Q6 are identified. Note that the arrows between the vertices with ◦ in Q1, Q2, Q3 and Q4, Q5, Q6 are opposite. each vertex with property (◦,−) is a source for Q1, . . . , Qn/2 and a sink for Qn/2+1, . . . , Qn−1; and, for odd n, each vertex with property (◦,−) is a source for Q1, . . . , Q(n−1)/2 and a sink for Q(n+1)/2+1, . . . , Qn−1. Let us choose the index set I of the vertices of Qm,n(SG) so that, under the natural iden- tification with Xm,n, i = (i, i′) ∈ I (i = 1, . . . , n− 1; i′ = 1, . . . ,m) represents the vertex with ◦ in Qi, and i = (n, i′) ∈ I (i′ = m + 1, . . . ,m + n) represents the vertex with • in any quiver Qi. Let I•+ (resp. I•−) denote the set of the vertices with property (•,+) (resp. (•,−)). Let I◦+,i (resp. I◦−,i) denote the set of the vertices with property (◦,+) (resp. (◦,−)) in the ith quiver Qi. We define composite mutations, µ•+ = ∏ i∈I•+ µi, µ•− = ∏ i∈I•− µi, µ◦+,i = ∏ i∈I◦+,i µi, µ◦−,i = ∏ i∈I◦−,i µi. (7.2) Lemma 7.5. Let Q(0) := Qm,n(SG). We have the following periodic sequence of mutations of quivers: for even n, Q(0) µ•+µ◦+,1←→ Q(1) µ•−µ◦−,n/2+1←→ Q(2) µ•+µ◦+,2←→ Q(3) µ•−µ◦−,n/2+2←→ · · · µ•+µ◦ +,n/2←→ Q(n− 1) µ•−µ◦−,1←→ Q(n) µ•+µ◦ +,n/2+1←→ Q(n + 1) µ•−µ◦−,2←→ · · · µ•−µ◦−,n/2−1←→ Q(2n− 4) µ•+µ◦+,n−1←→ Q(2n− 3) µ•−µ◦−,n/2←→ Q(2n− 2) = Q(0), (7.3) and, for odd n, Q(0) µ•+µ◦+,1µ◦−,(n+1)/2←→ Q(1) µ•−←→Q(2) µ•+µ◦+,2µ◦−,(n+3)/2←→ Q(3) µ•−←→ · · · µ•−←→Q(n− 1) µ•+µ◦ +,(n+1)/2 µ◦−,1←→ Q(n) µ•−←→Q(n + 1) µ•+µ◦ +,(n+3)/2 µ◦−,2←→ · · · µ•−←→Q(2n− 4) µ•+µ◦+,n−1µ◦−,(n−1)/2←→ Q(2n− 3) µ•−←→Q(2n− 2) = Q(0). (7.4) For (i, u) ∈ I×Z, write (i, u) : p+ if (i, u) is the forward mutation points in (7.3) for even n and in (7.4) for odd n, modulo (2n− 2)Z for u. Then, one can repeat and extend all the arguments for m = 1 in Sections 3 and 4, prove the following proposition, and obtain Theorem 7.3. Proposition 7.6. For [GY (B, y)]T with B = Bm,n(SG), the following facts hold. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 27 (i) For 0 ≤ u < 2(mn −m + n), the monomial [yi(u)]T ((i, u) : p+) is negative if and only if u takes the values 2n − 2 ≤ u < 2(mn − m + n) for i = (i, i′) (i = 1, . . . , n − 1; i′ = 1, . . . ,m) and for i = (n, m + 1), and u = 2k(n− 1), 2k(n− 1) + 1 (k = 1, . . . ,m + 1) for i = (n, m + 2), . . . , (n, m + n). (ii) We have [yi(2(mn −m + n))]T = yτ−1(i), where τ is a bijection I → I defined as follows. If both m and n are even, (i, i′) 7→ (σ(i), i′), i = 1, . . . , n− 1, i′ = 1, . . . ,m, (n, i′) 7→ (n, i′), i′ = m + 1, . . . ,m + n (7.5) and σ is the permutation in (3.2). If at least one of m or n is odd, we modify τ in (7.5) by (n, m + n− 1) 7→ (n, m + n) and (n, m + n) 7→ (n, m + n− 1). (The rest are the same as in (7.5).) (iii) The number N− of the negative monomials [yi(u)]T for (i, u) : p+ in the region 0 ≤ u < 2(mn−m + n) is (m + 1)(mn−m + n). 7.2 Further result for RSG Y-system The RSG case is quite similar. Let Ĩm,n = {1, . . . ,m + n− 3} × Z. Definition 7.7 ([32]). Fix a pair of integers m ≥ 1 and n ≥ 4. The reduced sine-Gordon (RSG) Y-system Ym,n(RSG) is the following system of relations for a family of variables {Yi(u) \| (i, u) ∈ Ĩm,n}, Yi(u− n + 1)Yi(u + n− 1) = ∏ j:j∼i (1 + Yj(u)), i = 1, . . . ,m− 1, Ym (u− n + 1) Ym (u + n− 1) = (1 + Ym−1(u)) × ( m+n−3∏ j=m+1 (1 + Yj(u−m− n + 1 + j))(1 + Yj(u + m + n− 1− j)) ) × (1 + Ym+n−3(u)−1)−1, Ym+1(u− 1)Ym+1(u + 1) = 1 + Ym(u) 1 + Ym+2(u)−1 , Yi(u− 1)Yi(u + 1) = 1∏ j:j∼i (1 + Yj(u)−1) , i = m + 2, . . . ,m + n− 3, where j ∼ i means that j ≤ m + n− 3 is adjacent to i in Xm,n. Definition 7.8. Fix a pair of integers m ≥ 1 and n ≥ 4. The reduced sine-Gordon (RSG) T- system Tm,n(RSG) is the following system of relations for a family of variables {Ti(u) \| (i, u) ∈ Ĩm,n}, Ti(u− n + 1)Ti(u + n− 1) = ∏ j:j∼i Tj(u) + 1, i = 1, . . . ,m, Tm+1(u− 1)Tm+1(u + 1) = Tm(u− n + 2)Tm(u + n− 2) + Tm+2(u), Ti(u− 1)Ti(u + 1) = Tm(u−m− n + 1 + i) × Tm(u + m + n− 1− i) + ∏ j:j∼i Tj(u), i = m + 2, . . . ,m + n− 4, Tm+n−3(u− 1)Tm+n−3(u + 1) = Tm(u− 2)Tm(u + 2) + Tm(u)Tm+n−4(u), where j ∼ i means that j ≤ m + n− 3 is adjacent to i in Xm,n. 28 T. Nakanishi and R. Tateo Figure 8. The quiver Qm,n(RSG) for (m,n) = (4, 7), where, all the vertices with • in the same position in the quivers Q1, . . . , Q6 are identified. We have an arrow between the fourth vertices (from the left) of Q3 and Q4. We define the associated multiplicative Abelian group Y◦m,n(RSG) and the commutative ring T◦m,n(RSG) as before. Theorem 7.9. (i) (Conjectured by [32].) The following relations hold in Y◦m,n(RSG). Periodicity: Yi(u + 2(mn−m + n)) = Yi(u). (ii) The following relations hold in T◦m,n(RSG). Periodicity: Ti(u + 2(mn−m + n)) = Ti(u). (iii) (Conjectured by [32].) Suppose that a family of positive real numbers {Yi(u) \| (i, u) ∈ In} satisfies Ym,n(RSG). Then, we have the identities 6 π2 ∑ (i,u)∈Im,n 0≤u<2(mn−m+n) L ( Yi(u) 1 + Yi(u) ) = 2 ( nm2 −m2 + 3mn− 8m + 2n− 6 ) , 6 π2 ∑ (i,u)∈Im,n 0≤u<2(mn−m+n) L ( 1 1 + Yi(u) ) = 2 ( n2m− 6nm + 11m + n2 − 5n + 6 ) . In our proof of Theorem 7.3 we have a natural interpretation of the period 2(mn−m + n) = 2{h(An−3) + 2 + m(h(An−4) + 2)}. The cluster algebraic formulation of Ym,n(RSG) and Tm,n(RSG) is done by the quiver Qm,n(RSG) defined as follows. Add m − 1 vertices to the left of the leftmost vertex in each quiver Qi of Qn(RSG) and put the arrows between them exactly in the same way as in the quiver Qm,n(SG). As an example, the case (m,n) = (4, 7) is given in Fig. 8. We also assign the properties +/− and ◦/• to each vertex in a similar way to Qm,n(SG) as in Fig. 8. Let us choose the index set Ĩ of the vertices of Qm,n(RSG) naturally obtained from the index set I of the vertices of Qm,n(SG) by the restriction to the vertices of Qm,n(RSG). Thus, i = (i, i′) ∈ Ĩ (i = 1, . . . , n−1; i′ = 1, . . . ,m) represents the vertex with ◦ in Qi, and i = (n, i′) ∈ Ĩ (i′ = m + 1, . . . ,m + n− 3) represents the vertex with • in any quiver Qi. Then, we define Ĩ•+, Ĩ•−, Ĩ◦+,i, Ĩ◦−,i and the corresponding mutations by (7.2). Lemma 7.10. Let Q(0) := Qm,n(RSG). We have the periodic sequence of mutations of qui- vers (7.3) for even n and quivers (7.4) for odd n. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 29 For (i, u) ∈ I×Z, write (i, u) : p+ if (i, u) is the forward mutation points in (7.3) for even n and in (7.4) for odd n, modulo (2n− 2)Z for u. Then, one can repeat and extend all the arguments for m = 1 in Sections 5 and 6, prove the following proposition, and obtain Theorem 7.9. Proposition 7.11. For [GY (B, y)]T with B = Bm,n(RSG), the following facts hold. (i) For 0 ≤ u < 2(mn −m + n), the monomial [yi(u)]T ((i, u) : p+) is negative if and only if u takes the values 2n − 2 ≤ u < 2(mn − m + n) for i = (i, i′) (i = 1, . . . , n − 1; i′ = 1, . . . ,m), u = n − 2, n − 1, 2n − 2 ≤ u < 2(mn − m + n) for i = (n, m + 1), and u = n−2, n−1, k(n−1), k(n−1)+1 (k = 2, . . . , 2m+2) for i = (n, m+2), . . . , (n, m+n−3). (ii) We have [yi(2(mn−m + n))]T = yτ−1(i), where τ is a bijection Ĩ→ Ĩ defined by (i, i′) 7→ (σ(i), i′), i = 1, . . . , n− 1, i′ = 1, . . . ,m, (n, i′) 7→ (n, i′), i′ = m + 1, . . . ,m + n− 3 and σ is the permutation in (3.2). (iii) The number N− of the negative monomials [yi(u)]T for (i, u) : p+ in the region 0 ≤ u < 2(mn−m + n) is nm2 −m2 + 3mn− 8m + 2n− 6. 7.3 Remarks by referee The content of this subsection is based on the communication from the referee of this paper. Recall that two quivers are said to be mutation-equivalent if one is obtained from the other by a sequence of mutations. The following important observation was made by the referee. Fact 1. The quiver Qm,n(SG) in Fig. 7 (Fig. 2 for m = 1) is mutation-equivalent to the quiver of type Dmn−m+n, i.e., its underlying graph is the Dynkin diagram of type Dmn−m+n. Similarly, the quiver Qm,n(RSG) in Fig. 8 (Fig. 4 for m = 1) is mutation-equivalent to the quiver of type Amn−m+n−3. In other words, the cluster algebra A(B, x, y) with B = Bm,n(SG) (resp. B = Bm,n(RSG)) is the cluster algebra of type Dmn−m+n (resp. type Amn−m+n−3). For example, for (m,n) = (1, 7), this is seen as follows: Here, we do mutations at the encircled vertices. For m ≥ 2, the situation is a little more complicated and the examples for (m,n) = (4, 7) are given in Figs. 9 and 10. The referee made a further observation on the periods in Theorems 7.3 and 7.9. Fact 2. (a) Suppose that mn −m + n is even. Then, the period 2(mn −m + n) in Theo- rems 7.3(a) coincides with h(Dmn−m+n)+2, which is the period of the Coxeter mutation sequence for the cluster algebra of type Dmn−m+n studied by Fomin–Zelevinsky in [8, 9, 10]. Similarly, suppose that mn−m + n is odd. Then, the period 4(mn−m + n) in Theorems 7.3(b) coincides with 2(h(Dmn−m+n) + 2), which is the period of the Coxeter mutation sequence for the cluster algebra of type Dmn−m+n in [8, 9, 10]. 30 T. Nakanishi and R. Tateo Figure 9. Sequence of mutations from Q4,7(SG) to the quiver of type D31. (b) The period 2(mn−m + n) in Theorems 7.9 coincides with 2(h(Amn−m+n−3) + 2), which is the period of the Coxeter mutation sequence for the cluster algebra of type Amn−m+n−3 in [8, 9, 10]. Facts 1 and 2 suggest that, even though the mutation sequences studied in this paper and the ones studied in [8, 9, 10] are seemingly different, they may be related, or may be treated in a unified way. We leave this interesting question as an open problem. A Cluster algebras Here we collect basic definitions for cluster algebras to fix the convention and notation, mainly following [10]. For further necessary definitions and information for cluster algebras, see [10]. Let I be a finite index set throughout the appendix. Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 31 Figure 10. Sequence of mutations from Q4,7(RSG) to the quiver of type A28. (i) Semifield. A semifield (P,⊕, ·) is an Abelian multiplicative group endowed with a binary operation of addition ⊕ which is commutative, associative, and distributive with respect to the multiplication in P. The following three examples are important. (a) Trivial semifield. The trivial semifield 1 = {1} consists of the multiplicative identity element 1 with 1⊕ 1 = 1. (b) Universal semifield. For an I-tuple of variables y = (yi)i∈I , the universal semifield Puniv(y) consists of all the rational functions of the form P (y)/Q(y) (subtraction-free rational expressions), where P (y) and Q(y) are the nonzero polynomials in yi’s with nonnegative inte- ger coefficients. The multiplication and the addition are given by the usual ones of rational functions. (c) Tropical semifield. For an I-tuple of variables y = (yi)i∈I , the tropical semifield Ptrop(y) is the Abelian multiplicative group freely generated by the variables yi’s endowed with the addition ⊕∏ i yai i ⊕ ∏ i ybi i = ∏ i y min(ai,bi) i . (A.1) (ii) Mutations of matrix and quiver. An integer matrix B = (Bij)i,j∈I is skew-symmetri- zable if there is a diagonal matrix D = diag(di)i∈I with di ∈ N such that DB is skew-symmetric. For a skew-symmetrizable matrix B and k ∈ I, another matrix B′ = µk(B), called the mutation of B at k, is defined by B′ ij = { −Bij , i = k or j = k, Bij + 1 2(\|Bik\|Bkj + Bik\|Bkj \|), otherwise. (A.2) The matrix µk(B) is also skew-symmetrizable. It is standard to represent a skew-symmetric (integer) matrix B = (Bij)i,j∈I by a quiver Q without loops or 2-cycles. The set of the vertices of Q is given by I, and we put Bij arrows from i 32 T. Nakanishi and R. Tateo to j if Bij > 0. The mutation Q′ = µk(Q) of quiver Q is given by the following rule: For each pair of an incoming arrow i → k and an outgoing arrow k → j in Q, add a new arrow i → j. Then, remove a maximal set of pairwise disjoint 2-cycles. Finally, reverse all arrows incident with k. (iii) Exchange relation of coeff icient tuple. Let P be a given semifield. For an I-tuple y = (yi)i∈I , yi ∈ P and k ∈ I, another I-tuple y′ is defined by the exchange relation y′i =  yk −1, i = k, yi ( yk 1⊕ yk )Bki , i 6= k, Bki ≥ 0, yi(1⊕ yk)−Bki , i 6= k, Bki ≤ 0. (A.3) (iv) Exchange relation of cluster. Let QP be the quotient field of the group ring ZP of P, and let QP(z) be the rational function field of algebraically independent variables, say, z = (zi)i∈I over QP. For an I-tuple x = (xi)i∈I which is a free generating set of QP(z) and k ∈ I, another I-tuple x′ is defined by the exchange relation x′i =  xk, i 6= k, yk ∏ j:Bjk>0 x Bjk j + ∏ j:Bjk<0 x −Bjk j (1⊕ yk)xk , i = k. (A.4) (v) Seed mutation. For the above triplet (B, x, y) in (ii)–(iv), which is called a seed, the mutation µk(B, x, y) = (B′, x′, y′) at k is defined by combining (A.2), (A.3), and (A.4). (vi) Cluster algebra. Fix a semifield P and a seed (initial seed) (B, x, y), where x = (xi)i∈I are algebraically independent variables over QP, and y = (yi)i∈I are elements in P. Starting from (B, x, y), iterate mutations and collect all the seeds (B′, x′, y′). We call y′ and y′i a coefficient tuple and a coefficient, respectively. We call x′ and x′i ∈ QP(x), a cluster and a cluster variable, respectively. The cluster algebra A(B, x, y) with coefficients in P is a ZP-subalgebra of the rational function field QP(x) generated by all the cluster variables. (vii) F -polynomial. The cluster algebra A(B, x, y) with coefficients in the tropical semi- field Ptrop(y) is called the cluster algebra with the tropical coefficients (the principal coefficients in [10]). There, each cluster variable x′i is an element in Z[x±1, y]. The F -polynomial F ′ i (y) ∈ Z[y] (for x′i) is defined as the specialization of x′i with xi = 1 (i ∈ I). Acknowledgements It is our great pleasure to thank the anonymous referee who generously pointed out a crucial fact for the subject of the paper. References [1] Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982. [2] Bazhanov V.V., Reshetikhin N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477–1492. [3] Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quan- tum field theory, Nuclear Phys. B 241 (1984), 333–380. [4] Bernard D., LeClair A., Residual quantum symmetries of the restricted sine-Gordon theories, Nuclear Phys. B 340 (1990), 721–751. [5] Chapoton F., Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. London Math. Soc. 37 (2005), 755–760. http://stacks.iop.org/0305-4470/23/1477 http://dx.doi.org/10.1016/0550-3213(84)90052-X http://dx.doi.org/10.1016/0550-3213(90)90466-Q http://dx.doi.org/10.1016/0550-3213(90)90466-Q http://dx.doi.org/10.1112/S0024609305004510 http://dx.doi.org/10.1112/S0024609305004510 Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems 33 [6] Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations. II. Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790, arXiv:0904.0676. [7] Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529, math.RT/0104151. [8] Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63–121, math.RA/0208229. [9] Fomin S., Zelevinsky A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977–1018, hep-th/0111053. [10] Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112–164, math.RT/0602259. [11] Frenkel E., Szenes A., Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), 677–693, hep-th/9506215. [12] Gliozzi F., Tateo R., ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995), 677–693, hep-th/9411203. [13] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras. I. Type Br, arXiv:1001.1880. [14] Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras. II. Types Cr, F4, and G2, arXiv:1001.1881. [15] Inoue R., Iyama O., Kuniba A., Nakanishi T., Suzuki J., Periodicities of T and Y-systems, Nagoya Math. J. 197 (2010), 59–174, arXiv:0812.0667. [16] Keller B., Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories, Editors T. Holm, P. Jørgensen and R. Rouquier, Lecture Note Series, Vol. 375, London Mathematical Society, Cambridge University Press, 2010, 76–160, arXiv:0807.1960. [17] Keller B., The periodicity conjecture for pairs of Dynkin diagrams, arXiv:1001.1531. [18] Kirillov A.N., Reshetikhin N.Y., Exact solution of the Heisenberg XXZ model of spin s, J. Soviet Math. 35 (1986), 2627–2643. [19] Kirillov A.N., Reshetikhin N.Y., Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math. 52 (1990), 3156–3164. [20] Klassen T.R., Melzer E., Purely elastic scattering theories and their ultraviolet limits, Nuclear Phys. B 338 (1990), 485–528. [21] Klümper A., Pearce P.A., Conformal weights of RSOS lattice models and their fusion hierarchies, Phys. A 183 (1992), 304–350. [22] Kuniba A., Thermodynamics of the Uq(X (1) r ) Bethe ansatz system with q a root of unity, Nuclear Phys. B 389 (1993), 209–244. [23] Kuniba A., Nakanishi T., Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), 3487–3494, hep-th/9206034. [24] Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215–5266, hep-th/9309137. [25] Lewin L., Polylogarithms and associated functions, North-Holland, Amsterdam, 1981. [26] Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J., to appear, arXiv:0909.5480. [27] Onsager L., Crystal statistics. I. A two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944), 117–149. [28] Plamondon P., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, arXiv:1004.0830. [29] Plamondon P., Cluster characters for cluster categories with infinite-dimensional morphism spaces, arXiv:1002.4956. [30] Ravanini R., Tateo R., Valleriani A., Dynkin TBA’s, Internat. J. Modern Phys. A 8 (1993), 1707–1727, hep-th/9207040. [31] Smirnov F.A., Reductions of the sine-Gordon model as a perturbation of minimal models of conformal field theory, Nuclear Phys. B 337 (1990), 156–180. http://dx.doi.org/10.1090/S0894-0347-10-00662-4 http://arxiv.org/abs/0904.0676 http://dx.doi.org/10.1090/S0894-0347-01-00385-X http://arxiv.org/abs/math.RT/0104151 http://dx.doi.org/10.1007/s00222-003-0302-y http://arxiv.org/abs/math.RA/0208229 http://dx.doi.org/10.4007/annals.2003.158.977 http://arxiv.org/abs/hep-th/0111053 http://dx.doi.org/10.1112/S0010437X06002521 http://arxiv.org/abs/math.RT/0602259 http://arxiv.org/abs/hep-th/9506215 http://dx.doi.org/10.1016/0370-2693(95)00125-5 http://arxiv.org/abs/hep-th/9411203 http://arxiv.org/abs/1001.1880 http://arxiv.org/abs/1001.1881 http://arxiv.org/abs/0812.0667 http://arxiv.org/abs/0807.1960 http://arxiv.org/abs/1001.1531 http://dx.doi.org/10.1016/0550-3213(90)90643-R http://dx.doi.org/10.1016/0378-4371(92)90149-K http://dx.doi.org/10.1016/0550-3213(93)90291-V http://dx.doi.org/10.1142/S0217732392002895 http://dx.doi.org/10.1142/S0217732392002895 http://arxiv.org/abs/hep-th/9206034 http://dx.doi.org/10.1142/S0217751X94002119 http://arxiv.org/abs/hep-th/9309137 http://arxiv.org/abs/0909.5480 http://dx.doi.org/10.1103/PhysRev.65.117 http://arxiv.org/abs/1004.0830 http://arxiv.org/abs/1002.4956 http://dx.doi.org/10.1142/S0217751X93000709 http://arxiv.org/abs/hep-th/9207040 http://dx.doi.org/10.1016/0550-3213(90)90255-C 34 T. Nakanishi and R. Tateo [32] Tateo R., New functional dilogarithm identities and sine-Gordon Y-systems, Phys. Lett. B 355 (1995), 157–164, hep-th/9505022. [33] Zamolodchikov A.B., Integrable field theory from conformal field theory, in Integrable Systems in Quantum Field Theory and Statistical Mechanics, Adv. Stud. Pure Math., Vol. 19, Academic Press, Boston, MA, 1989, 641–674. [34] Zamolodchikov A.B., Thermodynamic Bethe ansatz in relativistic models: scaling 3-state Potts and Lee– Yang models, Nuclear Phys. B 342 (1990), 695–720. [35] Zamolodchikov A.B., Zamolodchikov A.B., Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Physics 120 (1979), 253–291. [36] Zamolodchikov A.B., On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theo- ries, Phys. Lett. B 253 (1991), 391–394. http://dx.doi.org/10.1016/0370-2693(95)00751-6 http://arxiv.org/abs/hep-th/9505022 http://dx.doi.org/10.1016/0550-3213(90)90333-9 http://dx.doi.org/10.1016/0003-4916(79)90391-9 http://dx.doi.org/10.1016/0370-2693(91)91737-G 1 Introduction 2 Main results 2.1 Results for sine-Gordon Y-systems 2.2 Results for reduced sine-Gordon Y-systems 2.3 Background in integrable models 3 Cluster algebras for SG Y-systems 3.1 Parity decompositions of T and Y-systems 3.2 Quiver Qn(SG) 3.3 Embedding maps 4 Proof of Theorems 2.3, 2.4, and 2.7 5 Cluster algebras for RSG Y-systems 5.1 Parity decompositions of T and Y-systems 5.2 Quiver Qn(RSG) 5.3 Embedding maps 6 Proof of Theorems 2.11, 2.12, and 2.15 7 Further extension 7.1 Further result for SG Y-systems 7.2 Further result for RSG Y-system 7.3 Remarks by referee A Cluster algebras References

Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems

Репозитарії

Схожі ресурси