Dilogarithm identities for conformal field theories and cluster algebras: simply laced case

TL;DR

This paper proves dilogarithm identities related to conformal field theories and cluster algebras for simply laced Dynkin diagrams, confirming longstanding conjectures using cluster algebra techniques.

Contribution

It provides a complete proof of dilogarithm identities for simply laced types, extending previous partial results and conjectures.

Findings

Proved dilogarithm identities for all simply laced Dynkin diagrams.

Confirmed conjectures by Bazhanov, Kirillov, Reshetikhin, Gliozzi, and Tateo.

Established a cluster algebra framework for Y-systems in this context.

Abstract

The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

Equations

$$L(x)=-\frac{1}{2}{\int }_0^x\left\{\frac{\log (1-y)}{y}+\frac{\log y}{1-y}\right\}dy(0\le x\le 1).$$

$$L(0)=0,L(1)=\frac{{\pi }^2}{6},$$

$$L(x)+L(1-x)=\frac{{\pi }^2}{6},$$

$$L(x)+L(y)+L(1-xy)+L\left(\frac{1-x}{1-xy}\right)+L\left(\frac{1-y}{1-xy}\right)=\frac{{\pi }^2}{2}.$$

$$(Y_m^{(a)}{)}^2=\frac{\prod _{b:b\sim a}(1+Y_m^{(b)})}{(1+Y_{m-1}^{(a)}{}^{-1})(1+Y_{m+1}^{(a)}{}^{-1})},$$

$$\frac{6}{{\pi }^2}\sum _{a\in I}\sum _{m=1}^{\ell -1}L\left(\frac{Y_m^{(a)}}{1+Y_m^{(a)}}\right)=\frac{\ell \dim \mathfrak{g}}{h+\ell }-r,$$

$$\frac{(\ell -1)rh}{h+\ell }.$$

$$Y_m^{(1)}=\frac{{\sin }^2\frac{\pi }{\ell +2}}{\sin \frac{m\pi }{\ell +2}\sin \frac{(m+2)\pi }{\ell +2}},$$

$$\frac{6}{{\pi }^2}\sum _{m=1}^{\ell -1}L\left(\frac{{\sin }^2\frac{\pi }{\ell +2}}{{\sin }^2\frac{(m+1)\pi }{\ell +2}}\right)=\frac{3\ell }{2+\ell }-1.$$

$$Y_{i{i}^{\prime }}(u-1)Y_{i{i}^{\prime }}(u+1)=\frac{\prod _{j:j\sim i}(1+Y_{j{i}^{\prime }}(u))}{\prod _{j^{\prime }:j^{\prime }\sim i^{\prime }}(1+Y_{i{j}^{\prime }}(u{)}^{-1})},$$

$$Y_{i{i}^{\prime }}(u+2(h+h^{\prime }))=Y_{i{i}^{\prime }}(u),$$

$$\frac{6}{{\pi }^2}\sum _{(i,i^{\prime })\in I\times I^{\prime }}\sum _{u=0}^{2(h+h^{\prime })-1}L\left(\frac{Y_{i{i}^{\prime }}(u)}{1+Y_{i{i}^{\prime }}(u)}\right)$$

$$\frac{6}{{\pi }^2}\sum _{(i,i^{\prime })\in I\times I^{\prime }}\sum _{u=0}^{2(h+h^{\prime })-1}L\left(\frac{1}{1+Y_{i{i}^{\prime }}(u)}\right)$$

$$\displaystyle B^{\prime}_{ij}=\begin{cases}-B_{ij}&\mbox{$i=k$ or $j=k$},\\ B_{ij}+\frac{1}{2}(|B_{ik}|B_{kj}+B_{ik}|B_{kj}|)&\mbox{otherwise}.\end{cases}$$

$$y_i^{\prime }=\{\begin{array}{ll}{y}_k{}^{-1}& i=k,\\ y_i{\left(\frac{y_k}{1\oplus y_k}\right)}^{B_{ki}}& i\ne k,B_{ki}\ge 0,\\ y_i(1\oplus y_k{)}^{-B_{ki}}& i\ne k,B_{ki}\le 0.\end{array}$$

$$x_i^{\prime }=\{\begin{array}{ll}{x}_k& i\ne k,\\ \frac{y_k{\prod }_{j:B_{jk}>0}{x}_j^{B_{jk}}+{\prod }_{j:B_{jk}<0}{x}_j^{-B_{jk}}}{(1\oplus y_k)x_k}& i=k.\end{array}$$

$$\displaystyle\mbox{if $C_{ij}<0$, then $(i,j)\in I_{+}\times I_{-}$ or $(i,j)\in I_{-}\times I_{+}$}.$$

$$B_{\mathbf{ij}}=\{\begin{array}{ll}-C_{ij}{\delta }_{i^{\prime }{j}^{\prime }}& \mathbf{i}:(-+),\mathbf{j}:(++)\text{mbox}or\mathbf{i}:(+-),\mathbf{j}:(--),\\ C_{ij}{\delta }_{i^{\prime }{j}^{\prime }}& \mathbf{i}:(++),\mathbf{j}:(-+)\text{mbox}or\mathbf{i}:(--),\mathbf{j}:(+-),\\ -{\delta }_{ij}{C}_{i^{\prime }{j}^{\prime }}^{\prime }& \mathbf{i}:(++),\mathbf{j}:(+-)\text{mbox}or\mathbf{i}:(--),\mathbf{j}:(-+),\\ {\delta }_{ij}{C}_{i^{\prime }{j}^{\prime }}^{\prime }& \mathbf{i}:(+-),\mathbf{j}:(++)\text{mbox}or\mathbf{i}:(-+),\mathbf{j}:(--),\\ 0& \text{mbox}otherwise.\end{array}$$

$$\begin{array}{ccccc}& & {}_{-C}& & \\ & (+-)& \to & (--)& \\ {}_{-C^{\prime }}& \uparrow & & \downarrow & {}_{-C^{\prime }}\\ & (++)& \leftarrow & (-+)& \\ & & {}_{-C}& & \end{array}$$

$$\displaystyle\mbox{if $B_{\mathbf{i}\mathbf{j}}\neq 0$, then $(\mathbf{i},\mathbf{j})\in\mathbf{I}_{+}\times\mathbf{I}_{-}$ or $(\mathbf{i},\mathbf{j})\in\mathbf{I}_{-}\times\mathbf{I}_{+}$}.$$

$${\mu }_{+}(B)={\mu }_{-}(B)=-B.$$

$$\begin{array}{rl}\cdots & \stackrel{{\mu }_{-}}{⟷}(B,x(0),y(0))\stackrel{{\mu }_{+}}{⟷}(-B,x(1),y(1))\stackrel{{\mu }_{-}}{⟷}(B,x(2),y(2))\stackrel{{\mu }_{+}}{⟷}\cdots .\end{array}$$

$${\mathbf{P}}_{\pm }:\epsilon (\mathbf{i})(-1{)}^u=\pm ,$$

$$\text{EuScript}{Y}^{\circ }(X_r,X_{r^{\prime }}^{\prime })\simeq \text{EuScript}{Y}^{\circ }(X_r,X_{r^{\prime }}^{\prime }{)}_{+}\times \text{EuScript}{Y}^{\circ }(X_r,X_{r^{\prime }}^{\prime }{)}_{-}.$$

$$\frac{6}{{\pi }^2}\sum _{(\mathbf{i},u)\in S_{+}}L\left(\frac{\phi (y_{\mathbf{i}}(u))}{1+\phi (y_{\mathbf{i}}(u))}\right)$$

$$\frac{6}{{\pi }^2}\sum _{(\mathbf{i},u)\in S_{-}}L\left(\frac{\phi (y_{\mathbf{i}}(u))}{1+\phi (y_{\mathbf{i}}(u))}\right)$$

$$\frac{6}{{\pi }^2}\sum _{(\mathbf{i},u)\in H_{+}}L\left(\frac{\phi (y_{\mathbf{i}}(u))}{1+\phi (y_{\mathbf{i}}(u))}\right)$$

$$\frac{6}{{\pi }^2}\sum _{(\mathbf{i},u)\in H_{-}}L\left(\frac{\phi (y_{\mathbf{i}}(u))}{1+\phi (y_{\mathbf{i}}(u))}\right)$$

$$\prod _{i\in I}{y}_i^{a_i}\oplus \prod _{i\in I}{y}_i^{b_i}=\prod _{i\in I}{y}_i^{\min (a_i,b_i)}.$$

$$N_{+}=h^{\prime }r{r}^{\prime },N_{-}=hr{r}^{\prime }.$$

$$\mathbf{i}\to \mathbf{j}⟺B_{\mathbf{ij}}=1.$$

$$\begin{array}{ccc}0& 0& 0\\ 1& 1& 0\end{array}$$

$$t_{+}=\prod _{i\in I_{+}}{s}_i,t_{-}=\prod _{i\in I_{-}}{s}_i.$$

Full text

1. Introduction

1.1. Dilogarithm identities

Let $L(x)$ be the Rogers dilogarithm function [L, Ki2, Zag2, N]

[TABLE]

It is well known that the following properties hold $(0\leq x,y\leq 1)$.

[TABLE]

In the series of works by Bazhanov, Kirillov, and Reshetikhin [KR1, BR1, KR2, Ki1, BR2] they reached a remarkable conjecture on identities expressing the central charges of conformal field theories in terms of $L(x)$, and partly established it. Let us concentrate on the identities in the simply laced case here.

Let $X_{r}$ be any simply laced Dynkin diagram of finite type with rank $r$, and $I$ be the index set of $X_{r}$. Let $\ell\geq 2$ be any integer. For a family of positive real numbers $\{Y^{(a)}_{m}\mid a\in I;1\leq m\leq\ell-1\}$, consider a system of algebraic relations

[TABLE]

where $b\sim a$ means $b$ is adjacent to $a$ in $X_{r}$, and $Y^{(a)}_{0}{}^{-1}=Y^{(a)}_{\ell}{}^{-1}=0$ if they appear in the right hand sides.

Theorem 1.1** ([NK, Zag2]).**

There exists a unique positive real solution of (1.5).

Conjecture 1.2** (Dilogarithm identities [Ki1, BR2]).**

Suppose that a family of positive real numbers $\{Y^{(a)}_{m}\mid a\in I;1\leq m\leq\ell-1\}$ satisfies (1.5). Then, we have the identities

[TABLE]

where $h$ and $\mathfrak{g}$ are the Coxeter number and the simple Lie algebra of type $X_{r}$, respectively.

Remark 1.3**.**

The conjectures in [Ki1] and [BR2] are not exactly the same. This is the version in [BR2, Eqs. (4.21), (4.22), (5.3)] with the identification of $f^{a}_{j}$ therein with $Y^{(a)}_{j}/(1+Y^{(a)}_{j})$ here. The version in [Ki1] also concerns the construction of the solution of (1.5). We do not touch this issue here, since it is regarded as an independent problem in the framework of the present paper.

For $X_{r}=A_{r}$, Kirillov [Ki1] gave the explicit expression of the solution of (1.5), and proved the corresponding identity (1.6) by the analytic method.

Due to the well-known formula $\dim\mathfrak{g}=r(h+1)$, the right hand side of (1.6) is equal to the number

[TABLE]

It is already remarkable that the left hand side of (1.6) is a rational number. It is much more remarkable that the rational number of the first term in the right hand side of (1.6) is the central charge of the Wess-Zumino-Witten conformal field theory [KZ, GW] of type $X_{r}$ with level $\ell$. (This is one of the reason why the integer $\ell$ is called the level.) The rational number in the right hand side of (1.6) itself is also the central charge of the parafermion conformal field theory of type $X_{r}$ with level $\ell$ [FZ, G]. The identity (1.6) is crucial to establish the connection between conformal field theories and various types of nonconformal integrable models in various limits.

Example 1.4** ([KR1]).**

Consider the case $X_{r}=A_{1}$ and any $\ell$, which is equivalent to the case $X_{r}=A_{\ell-1}$ and $\ell=2$ by the level-rank duality. Then, one has a solution of (1.5)

[TABLE]

and the corresponding identity (1.6) reads

[TABLE]

This identity has been known and studied by various authors in various points of view. To name few, Lewin [L], Richmond-Szekeres [RS], Kirillov-Reshetikhin [KR1], Nahm-Recknagel-Terhoeven [NRT], Dupont-Sah [DS], etc.

1.2. Functional dilogarithm identities

The system (1.5) admits an affinization called the Y-system introduced by Zamolodchikov [Zam], Kuniba-Nakanishi [KN], and Ravanini-Tateo-Valleriani [RTV]. Here, we consider the version in [RTV]. Let $X_{r}$ and $X^{\prime}_{r^{\prime}}$ be a pair of simply laced Dynkin diagrams of finite type. Let $I$ and $I^{\prime}$ be the index sets of $X_{r}$ and $X^{\prime}_{r^{\prime}}$, respectively. For a family of variables $\{Y_{ii^{\prime}}(u)\mid i\in I,i^{\prime}\in I^{\prime},u\in\mathbb{Z}\}$, the Y-system $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$ associated with a pair $(X_{r},X^{\prime}_{r^{\prime}})$ is a system of the algebraic/functional relations

[TABLE]

where $j\sim i$ means $j$ is adjacent to $i$ in $X_{r}$, while $j^{\prime}\sim i^{\prime}$ means $j^{\prime}$ is adjacent to $i^{\prime}$ in $X^{\prime}_{r^{\prime}}$. Two systems $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$ and $\mathbb{Y}(X^{\prime}_{r^{\prime}},X_{r})$ are equivalent to each other by the correspondence $Y_{ii^{\prime}}(u)\leftrightarrow Y_{i^{\prime}i}(u)^{-1}$. This is a generalization of the level-rank duality.

Ravanini-Tateo-Valleriani [RTV] gave the periodicity conjecture, which generalized the one by [Zam] in the case $X_{r}=A_{1}$ or $X^{\prime}_{r^{\prime}}=A_{1}$.

Conjecture 1.5** (Periodicity [RTV]).**

Suppose that a family of positive real numbers $\{Y_{ii^{\prime}}(u)\mid i\in I,i^{\prime}\in I^{\prime},u\in\mathbb{Z}\}$ satisfies $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$. Then, we have the periodicity

[TABLE]

where $h$ and $h^{\prime}$ are the Coxeter numbers of types $X_{r}$ and $X^{\prime}_{r^{\prime}}$, respectively.

Conjecture 1.5 was proved for $(X_{r},X^{\prime}_{r^{\prime}})=(A_{r},A_{1})$ by Gliozzi-Tateo [GT2] and Frenkel-Szenes [FS], for $(X_{r},X^{\prime}_{r^{\prime}})=(\mbox{any},A_{1})$ by Fomin-Zelevinsky [FZ3], and for $(X_{r},X^{\prime}_{r^{\prime}})=(A_{r},A_{r^{\prime}})$ by Volkov [V] and Szenes [S]. More recently, it was proved in full generality by Keller [Kel1, Kel2].

Furthermore, Gliozzi-Tateo [GT1] significantly generalized Conjecture 1.2 as follows:

Conjecture 1.6** (Functional dilogarithm identities [GT1]).**

Suppose that a family of positive real numbers $\{Y_{aa^{\prime}}(u)\mid a\in I,a^{\prime}\in I^{\prime},u\in\mathbb{Z}\}$ satisfies $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$. Then, we have the identities

[TABLE]

Two identities (1.12) and (1.13) are equivalent to each other due to (1.3).

Conjecture 1.6 implies Conjecture 1.2; namely, set $X^{\prime}_{r^{\prime}}=A_{\ell-1}$, and take a constant solution $Y_{ii^{\prime}}=Y_{ii^{\prime}}(u)$ of $\mathbb{Y}(X_{r},A_{\ell-1})$ as a function of $u$. Then, one obtain (1.6) from (1.12) using $h^{\prime}=\ell$, $r^{\prime}=\ell-1$, and (1.7). Conjecture 1.6 was proved for $(X_{r},X^{\prime}_{r^{\prime}})=(A_{r},A_{1})$ by Frenkel-Szenes [FS], and for $(X_{r},X^{\prime}_{r^{\prime}})=(\mbox{any},A_{1})$ by Chapoton [C].

Example 1.7** ([GT1]).**

(i) In the simplest case $(X_{r},X^{\prime}_{r^{\prime}})=(A_{1},A_{1})$, the identity (1.12) is equivalent to (1.3).

(ii) In the next simplest case $(X_{r},X^{\prime}_{r^{\prime}})=(A_{2},A_{1})$, the identity (1.12) is equivalent to the 5-term relation (1.4).

Conjecture 1.6 tells us that what is important in (1.6) is, not the values of $Y^{(a)}_{m}$ themselves, but rather the relations (1.10) they satisfy. We will see below that the algebraic property of the relations (1.10) is efficiently extracted by cluster algebras introduced by Fomin-Zelevinsky [FZ1, FZ2, FZ4].

1.3. Main result

Among several preceding results and methods concerning Conjectures 1.5, and 1.6, we list the ones which are particularly relevant to the present work.

• (i)

Frenkel-Szenes [FS] proved Conjecture 1.6 for $(X_{r},X^{\prime}_{r^{\prime}})=(A_{r},A_{1})$ by showing the constancy property of the left hand side of (1.12).

• (ii)

Caracciolo-Gliozzi-Tateo [CGT] studied the constancy property for any pair $(X_{r},X^{\prime}_{r^{\prime}})$. (But they did not complete the proof of Conjecture 1.6.)

• (iii)

Fomin-Zelevinsky [FZ3] proved Conjecture 1.5 for $(X_{r},X^{\prime}_{r^{\prime}})=(\mbox{any},A_{1})$ by using the ‘cluster algebra like’ formulation of Y-systems and the root systems.

• (iv)

Chapoton [C] proved Conjecture 1.6 for $(X_{r},X^{\prime}_{r^{\prime}})=(\mbox{any},A_{1})$ by combining the constancy property of (i) and the result of (iii).

• (v)

Fomin-Zelevinsky [FZ4] more manifestly integrated the Y-system $\mathbb{Y}(X_{r},A_{1})$ in the framework of cluster algebras with coefficients, where the Y-system is identified with a system of relations among coefficients of the cluster algebras of type $X_{r}$.

• (vi)

Keller [Kel1, Kel2] proved Conjecture 1.5 in full generality for any pair $(X_{r},X^{\prime}_{r^{\prime}})$ of simply laced type by using cluster algebras with coefficients of (v) together with their categorifications by the cluster categories.

We also remark that the connection between the dilogarithm and cluster algebras was studied earlier by Fock-Goncharov [FG].

By combining these results, methods, and ideas, we prove Conjecture 1.6.

Theorem 1.8**.**

Conjecture 1.6 is true for any pair $(X_{r},X^{\prime}_{r^{\prime}})$ of simply laced type.

Corollary 1.9**.**

Conjecture 1.2 is true for any $X_{r}$ of simply laced type and any $\ell\geq 2$.

Remark 1.10**.**

The dilogarithm identities of nonsimply laced type by Kirillov [Ki1, Eq. (7)], properly corrected by Kuniba [Ku, Eqs. (A.1a), (A.1c)], are equally important. We stress that they are different from another version of the identities of nonsimply laced type obtained by the folding of simply laced one [C]. Though the situation is more complicated than the simply laced case, a similar approach to the one here is applicable to prove the identities [IIKKN1, IIKKN2].

The organization of the paper is as follows. In section 2 we reformulate Conjecture 1.6 in terms of cluster algebras. In section 3 we study the tropical version of the Y-system in the cluster algebra setting. Proposition 3.2 is a key observation through the paper. In section 4 we prove Theorem 2.8, which is equivalent to Theorem 1.8, by applying the method of [FS] with mixture of the ideas by [CGT, C].

Acknowledgements. It is my great pleasure to thank Atsuo Kuniba for sharing his insight into the dilogarithm identities for many years, and also for useful comments on the manuscript.

2. Reformulation by cluster algebras

As the first step we reformulate Theorem 1.8 in terms of cluster algebras.

2.1. Cluster algebras

Here we collect some basic definitions for cluster algebras [FZ1, FZ2, FZ4] to fix the convention and notation, mainly following [FZ4].

(i) Matrix mutation. An integer matrix $B=(B_{ij})_{i,j\in I}$ is skew-symmetrizable if there is a diagonal matrix $D=\mathrm{diag}(d_{i})_{i\in I}$ with $d_{i}\in\mathbb{N}$ such that $DB$ is skew-symmetric. For a skew-symmetrizable matrix $B$ and $k\in I$, another matrix $B^{\prime}=\mu_{k}(B)$, called the mutation of $B$ at $k$, is defined by

[TABLE]

The matrix $\mu_{k}(B)$ is also skew-symmetrizable.

(ii) Exchange relation of coefficient tuple. A semifield $(\mathbb{P},\oplus)$ is an abelian multiplicative group $\mathbb{P}$ endowed with a binary operation of addition $\oplus$ which is commutative, associative, and distributive with respect to the multiplication in $\mathbb{P}$ [FZ4, HW]. For an $I$-tuple $y=(y_{i})_{i\in I}$, $y_{i}\in\mathbb{P}$ and $k\in I$, another $I$-tuple $y^{\prime}$ is defined by the exchange relation

[TABLE]

(iii) Exchange relation of cluster. Let $\mathbb{QP}$ be the quotient field of the group ring $\mathbb{Z}\mathbb{P}$ of $\mathbb{P}$, and let $\mathbb{QP}(u)$ be the rational function field of algebraically independent variables $u=(u_{i})_{i\in I}$ over $\mathbb{QP}$.

For an $I$-tuple $x=(x_{i})_{i\in I}$ which is a free generating set of $\mathbb{QP}(u)$ and $k\in I$, another $I$-tuple $x^{\prime}$ is defined by the exchange relation

[TABLE]

(iv) Seed mutation. For the above triplet $(B,x,y)$, called a seed, the mutation $\mu_{k}(B,x,y)=(B^{\prime},x^{\prime},y^{\prime})$ at $k$ is defined by combining (i)–(iii).

(v) Cluster algebra. Fix a semifield $\mathbb{P}$ and a seed (initial seed) $(B,x,y)$, where $x=(x_{i})_{i\in I}$ are algebraically independent variables over $\mathbb{Q}\mathbb{P}$. Starting from $(B,x,y)$, iterate mutations and collect all the seeds $(B^{\prime},x^{\prime},y^{\prime})$. We call $y^{\prime}$ and $y^{\prime}_{i}$ a coefficient tuple and a coefficient, respectively. We call $x^{\prime}$ and $x^{\prime}_{i}\in\mathbb{Q}\mathbb{P}(x)$, a cluster and a cluster variable, respectively. The cluster algebra $\mathcal{A}(B,x,y)$ with coefficients in $\mathbb{P}$ is a $\mathbb{Z}\mathbb{P}$-subalgebra of the rational function field $\mathbb{Q}\mathbb{P}(x)$ generated by all the cluster variables.

For further necessary definitions and information for cluster algebras, see [FZ4].

2.2. Matrix $B(X_{r},X^{\prime}_{r^{\prime}})$

For a Cartan matrix $C=(C_{ij})_{i,j\in I}$ of finite type, we say the decomposition $I=I_{+}\sqcup I_{-}$ is bipartite if

[TABLE]

From now on, we assume that $X_{r}$ and $X^{\prime}_{r^{\prime}}$ are a pair of simply laced Dynkin diagrams of finite type and that $C=(C_{ij})_{i,j\in I}$ and $C^{\prime}=(C_{i^{\prime}j^{\prime}})_{i^{\prime},j^{\prime}\in I}$ are the Cartan matrices of $X_{r}$ and $X^{\prime}_{r^{\prime}}$ with fixed bipartite decompositions $I=I_{+}\sqcup I_{-}$ and $I^{\prime}=I^{\prime}_{+}\sqcup I^{\prime}_{-}$, respectively. Set $\mathbf{I}=I\times I^{\prime}$. For $\mathbf{i}=(i,i^{\prime})\in\mathbf{I}$, let us write $\mathbf{i}:(++)$ if $(i,i^{\prime})\in I_{+}\times I^{\prime}_{+}$, etc. Define the matrix $B=B(X_{r},X^{\prime}_{r^{\prime}})=(B_{\mathbf{i}\mathbf{j}})_{\mathbf{i},\mathbf{j}\in\mathbf{I}}$ by

[TABLE]

The rule (2.5) is visualized in the diagram:

[TABLE]

The matrix $B$ corresponds to the square product of alternating quivers by [Kel1].

Lemma 2.1**.**

The matrix $B=B(X_{r},X^{\prime}_{r^{\prime}})$ in (2.5) is skew-symmetric and satisfies the following conditions: Let $\mathbf{I}=\mathbf{I}_{+}\sqcup\mathbf{I}_{-}$ with $\mathbf{I}_{+}:=(I_{+}\times I^{\prime}_{+})\sqcup(I_{-}\times I^{\prime}_{-})$ and $\mathbf{I}_{-}:=(I_{+}\times I^{\prime}_{-})\sqcup(I_{-}\times I^{\prime}_{+})$. Then,

[TABLE]

Furthermore, for composed mutations $\mu_{+}=\prod_{\mathbf{i}\in\mathbf{I}_{+}}\mu_{\mathbf{i}}$ and $\mu_{-}=\prod_{\mathbf{i}\in\mathbf{I}_{-}}\mu_{\mathbf{i}}$,

[TABLE]

Proof.

They are easily seen in the quiver picture in [Kel1, Section 8]. ∎

Note that $\mu_{\pm}(B)$ does not depend on the order of the product due to (2.7).

2.3. Cluster algebra and Y-system

For the matrix $B=B(X_{r},X^{\prime}_{r^{\prime}})$ in (2.5), let $\mathcal{A}(B,x,y)$ be the cluster algebra with coefficients in the universal semifield $\mathbb{Q}_{\mathrm{sf}}(y)$, where $(B,x,y)$ is the initial seed [FZ4]. (Here we use the symbol $+$ instead of $\oplus$ in $\mathbb{Q}_{\mathrm{sf}}(y)$, since it is the ordinary addition of subtraction-free expressions of rational functions of $y$.)

To our purpose, it is natural to introduce not only the ‘ring of cluster variables’ but also the ‘group of coefficients’.

Definition 2.2**.**

The coefficient group $\mathcal{G}(B,y)$ associated with $\mathcal{A}(B,x,y)$ is the multiplicative subgroup of the semifield $\mathbb{Q}_{\mathrm{sf}}(y)$ generated by all the coefficients $y_{\mathbf{i}}^{\prime}$ of $\mathcal{A}(B,x,y)$ together with $1+y_{\mathbf{i}}^{\prime}$.

We set $x(0)=x$, $y(0)=y$ and define clusters $x(u)=(x_{\mathbf{i}}(u))_{\mathbf{i}\in\mathbf{I}}$ ($u\in\mathbb{Z}$) and coefficient tuples $y(u)=(y_{\mathbf{i}}(u))_{\mathbf{i}\in\mathbf{I}}$ ($u\in\mathbb{Z}$) by the sequence of mutations

[TABLE]

Definition 2.3**.**

The Y-subgroup ${\mathcal{G}}_{Y}(B,y)$ of ${\mathcal{G}}(B,y)$ associated with the sequence (2.9) is the multiplicative subgroup of ${\mathcal{G}}(B,y)$ generated by $y_{\mathbf{i}}(u)$ and $1+y_{\mathbf{i}}(u)$ ($\mathbf{i}\in\mathbf{I},u\in\mathbb{Z}$).

Let $\varepsilon:\mathbf{I}\rightarrow\{+,-\}$ be the sign function defined by $\varepsilon(\mathbf{i})=\varepsilon$ for $\mathbf{i}\in\mathbf{I}_{\varepsilon}$. For $(\mathbf{i},u)\in\mathbf{I}\times\mathbb{Z}$, we set the ‘parity conditions’ $\mathbf{P}_{+}$ and $\mathbf{P}_{-}$ by

[TABLE]

where we identify $+$ and $-$ with $1$ and $-1$, respectively. We write $(\mathbf{i},u):\mathbf{P}_{\varepsilon}$ if $(\mathbf{i},u)$ satisfies the condition $\mathbf{P}_{\varepsilon}$.

Lemma 2.4** (cf. [KNS, Lemma 6.18]).**

(1) $y_{\mathbf{i}}(u)=y_{\mathbf{i}}(u\pm 1)^{-1}$ for $(\mathbf{i},u):\mathbf{P}_{\pm}$.

(2) The family y_{\pm}=\{y_{\mathbf{i}}(u)\mid\mbox{(\mathbf{i},u):\mathbf{P}_{\pm}}\} satisfies the Y-system $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$ in ${\mathcal{G}}_{Y}(B,y)$ by replacing $Y_{\mathbf{i}}(u)$ in $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$ with $y_{\mathbf{i}}(u)^{\pm 1}$.

Proof.

This follows from the exchange relation (2.2). ∎

Definition 2.5**.**

Let $\EuScript{Y}(X_{r},X^{\prime}_{r^{\prime}})$ be the semifield with generators $Y_{\mathbf{i}}(u)$ ($\mathbf{i}\in\mathbf{I},u\in\mathbb{Z}$) and the relations $\mathbb{Y}(X_{r},X^{\prime}_{r^{\prime}})$. Let $\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})$ be the multiplicative subgroup of $\EuScript{Y}(X_{r},X^{\prime}_{r^{\prime}})$ generated by $Y_{\mathbf{i}}(u)$, $1+Y_{\mathbf{i}}(u)$ ($\mathbf{i}\in\mathbf{I},u\in\mathbb{Z}$). (Here we use the symbol $+$ instead of $\oplus$ for simplicity.)

Define $\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})_{\varepsilon}$ ($\varepsilon=\pm$) to be the subgroup of $\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})$ generated by those $Y_{\mathbf{i}}(u)$, $1+Y_{\mathbf{i}}(u)$ with $(\mathbf{i},u):\mathbf{P}_{\varepsilon}$. Then, we have $\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})_{+}\simeq\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})_{-}$ by $Y_{\mathbf{i}}(u)\mapsto Y_{\mathbf{i}}(u+1)$ and

[TABLE]

Proposition 2.6** (cf. [KNS, Theorem 6.19]).**

The group $\EuScript{Y}^{\circ}(X_{r},X^{\prime}_{r^{\prime}})_{\pm}$ is isomorphic to ${\mathcal{G}}_{Y}(B,y)$ by the correspondence $Y_{\mathbf{i}}(u)\mapsto y_{\mathbf{i}}(u)^{\pm 1}$, $1+Y_{\mathbf{i}}(u)\mapsto 1+y_{\mathbf{i}}(u)^{\pm 1}$ for $(\mathbf{i},u):\mathbf{P}_{\pm}$.

In summary, the Laurent monomials in $Y_{\mathbf{i}}(u)$ and $1+Y_{\mathbf{i}}(u)$ with $(\mathbf{i},u):\mathbf{P}_{+}$ are embedded in the coefficient group ${\mathcal{G}}(B,y)$.

2.4. Reformulation of Theorem 1.8

We recall the periodicity theorem, originally conjectured by [RTV]:

Theorem 2.7** ([Kel1, Kel2]).**

In ${\mathcal{G}}(B,y)$, the following relations hold:

(i) Periodicity: $y_{\mathbf{i}}(u+2(h+h^{\prime}))=y_{\mathbf{i}}(u)$.

(ii) Half periodicity: $y_{ii^{\prime}}(u+(h+h^{\prime}))=y_{\omega(i)\omega^{\prime}(i^{\prime})}(u)$, where $\omega$ (resp. $\omega^{\prime}$) is the Dynkin automorphism of $X_{r}$ (resp. $X^{\prime}_{r^{\prime}}$) for types $A_{r}$, $D_{r}$ ($r:odd$), or $E_{6}$, and the identity otherwise.

Proof.

(i). This is due to [Kel1, Theorem 8.2] or [Kel2, Theorem 2.3]. (ii). This is obtained by combining [Kel1, Theorem 7.13] and the proof of [IIKNS, Theorem 4.27]. ∎

Let $\mathbb{R}_{+}$ be the semifield of the positive real numbers by the usual multiplication and addition. By Proposition 2.6, Theorem 1.8 is equivalent to the following one:

Theorem 2.8**.**

Let $y_{\mathbf{i}}(u)\in\mathbb{Q}_{\mathrm{sf}}(y)$ $(\mathbf{i}\in\mathbf{I}$, $u\in\mathbb{Z})$ be as above. Let $\varphi:\mathbb{Q}_{\mathrm{sf}}(y)\rightarrow\mathbb{R}_{+}$ be any semifield homomorphism. Then, the following identities hold:

[TABLE]

where S_{\pm}=\{(\mathbf{i},u)\mid\mathbf{i}\in I,0\leq u\leq 2(h+h^{\prime})-1,\mbox{(\mathbf{i},u):\mathbf{P}_{\pm}}\}. Also,

[TABLE]

where H_{\pm}=\{(\mathbf{i},u)\mid\mathbf{i}\in I,0\leq u\leq(h+h^{\prime})-1,\mbox{(\mathbf{i},u):\mathbf{P}_{\pm}}\}.

The identities (2.12) and (2.13) are equivalent to each other by Lemma 2.4. The identities (2.14) and (2.15) follow from (2.12) and (2.13) by the half periodicity in Theorem 2.7.

We are going to prove Theorem 2.8.

3. Tropical Y-system

Let us have an interlude to establish a property of the tropical Y-system [FZ4] associated with the cluster algebra $\mathcal{A}(B,x,y)$ for $B=B(X_{r},X^{\prime}_{r^{\prime}})$.

Let $B=(B_{ij})_{i,j\in I}$ be a general skew-symmetrizable matrix. Let $y$ be the initial coefficient tuple of the cluster algebra $\mathcal{A}(B,x,y)$ with coefficients in the universal semifield $\mathbb{Q}_{\mathrm{sf}}(y)$. The tropical semifield $\mathrm{Trop}(y)$ is an abelian multiplicative group freely generated by the elements $y_{i}$ ($i\in I$) with the addition $\oplus$

[TABLE]

The image of $f\in\mathbb{Q}_{\mathrm{sf}}(y)$ by the natural projection $\mathbb{Q}_{\mathrm{sf}}(y)\rightarrow\mathrm{Trop}(y)$ is denoted by $[f]_{\mathrm{T}}$ and called the tropical evaluation of $f$ [FZ4].

We say a (Laurent) monomial in $y=(y_{i})_{\i\in I}$ is positive if its exponents are all nonnegative and at least one of them is positive. A negative monomial is defined similarly.

Lemma 3.1**.**

Suppose that $y^{\prime\prime}$ is the coefficient tuple obtained from the mutation of another coefficient tuple $y^{\prime}$ at $k$. Then, for any $i\neq k$, $[y^{\prime\prime}_{i}]_{\mathrm{T}}=[y^{\prime}_{i}]_{\mathrm{T}}$ if one of the following conditions holds.

(i) $B_{ki}=0$.

(ii) $B_{ki}>0$, and $[y^{\prime}_{k}]_{\mathrm{T}}$ is negative.

(iii) $B_{ki}<0$, and $[y^{\prime}_{k}]_{\mathrm{T}}$ is positive.

Proof.

This is an immediate consequence of the exchange relation (2.2). ∎

Now we claim a key proposition in our proof of Theorem 2.8.

Proposition 3.2**.**

For the cluster algebra $\mathcal{A}(B,x,y)$ for $B=B(X_{r},X^{\prime}_{r^{\prime}})$, the following properties hold.

(i) The tropical evaluation $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ of $y_{\mathbf{i}}(u)$ $(\mathbf{i}\in\mathbf{I},u\in\mathbb{Z})$ is a positive or negative monomial in $y=y(0)$.

(ii) For $0\leq u\leq h^{\prime}-1$ and $(\mathbf{i},u):\mathbf{P}_{+}$, $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ is a positive monomial. For $-h\leq u\leq-1$ and $(\mathbf{i},u):\mathbf{P}_{+}$, $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ is a negative monomial.

(iii) Let $N_{+}$ (resp. $N_{-}$) be the number of the positive (resp. negative) monomials $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ $((\mathbf{i},u)\in S_{+})$, where $S_{+}$ is the domain in Theorem 2.8. Then,

[TABLE]

The properties (i) and (iii) follow from (ii) by the half periodicity in Theorem 2.7. In the case $X^{\prime}_{r^{\prime}}=A_{1}$ or $X_{r}=A_{1}$, i.e., the ‘level 2 case’ or its level-rank dual in the original context, Proposition 3.2 reduces to the known one for the cluster algebra of finite type [FZ4, Proposition 10.7].

Before giving a proof, it is instructive to observe some examples.

Example 3.3**.**

Let $X_{r}=A_{1}$, $I_{+}=\{1\}$, $I_{-}=\emptyset$, and $X^{\prime}_{r^{\prime}}=A_{2}$, $I^{\prime}_{+}=\{1\}$, $I^{\prime}_{-}=\{2\}$. We have $h=2$ and $h^{\prime}=3$. We visualize the mutation matrix $B$ by a quiver in the correspondence

[TABLE]

Set $y_{1}:=y_{11}$ and $y_{2}:=y_{12}$. Then, $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ for $0\leq u\leq 5$ is given as follows:

[TABLE]

Here, the framed variables are all the elements in the domain $H_{+}$. Certainly, we have $N_{+}/2=3$ and $N_{-}/2=2$, which agree with (3.2). Moreover, we observe that the positive monomials occur consecutively for $0\leq u\leq 2$. This is a consequence of [FZ4, Proposition 10.7]. In fact, they correspond to the positive roots $\alpha_{1}$, $\alpha_{1}+\alpha_{2}$, $\alpha_{2}$ of $A_{2}$. For the later use we abbreviate the above diagram as follows:

[TABLE]

Example 3.4**.**

Let $X_{r}=A_{3}$, $I_{+}=\{1,3\}$, $I_{-}=\{2\}$, and $X^{\prime}_{r^{\prime}}=A_{1}$, $I^{\prime}_{+}=\{1\}$, $I^{\prime}_{-}=\emptyset$. We have $h=4$ and $h^{\prime}=2$. We consider mutations for $-6\leq u\leq 0$ by moving in the reverse direction of $u$. The result is abbreviated in the diagram:

[TABLE]

Here, $110$, for example, represents the monomial $y_{11}y_{21}$. The framed variables correspond to all the elements in the domain $H_{+}$ modulo half period $h+h^{\prime}=6$. Certainly, we have $N_{+}/2=3$ and $N_{-}/2=6$, which agree with (3.2). We observe that the negative monomials occur consecutively for $-4\leq u\leq-1$. Again, this is a consequence of [FZ4, Proposition 10.7], and they correspond to the positive roots of $A_{3}$.

Now we are ready to proceed to ‘higher level’. The next example looks a toy example, but completely clarifies why Proposition 3.2 holds.

Example 3.5**.**

Let $X_{r}=A_{3}$, $I_{+}=\{1,3\}$, $I_{-}=\{2\}$, and $X^{\prime}_{r^{\prime}}=A_{2}$, $I^{\prime}_{+}=\{1\}$, $I^{\prime}_{-}=\{2\}$. We have $h=4$ and $h^{\prime}=3$. We consider mutations for $-4\leq u\leq 3$ by moving in the both directions of $u$. The result is shown in Figure 1.

The framed variables corresponds to all the elements in the domain $H_{+}$ modulo half period $h+h^{\prime}=7$, and

[TABLE]

for example, represents the monomial $y_{11}y_{12}$. Certainly, we have $N_{+}/2=9$ and $N_{-}/2=12$, which agree with (3.2). We observe that the positive monomials occur consecutively for $0\leq u\leq 2$, while the negative monomials do so for $-4\leq u\leq-1$.

Let us look at Figure 1 more closely, then we find a remarkable factorization property of the tropical Y-system, which does not occur in the nontropical Y-system. First, see the region $0\leq u\leq 2$. Then, the contents in the left and the right columns for $y(u)$ mutate exactly in the same pattern as in Example 3.3. So does the middle column with the other choice of bipartite decomposition of $I^{\prime}$. In particular, there is no interaction in the horizontal direction. This is because the property (iii) in Lemma 3.1 is satisfied at any mutation point $\mathbf{k}$ and any $\mathbf{i}$ horizontally adjacent to $\mathbf{k}$. On the other hand, in the region $-4\leq u\leq-1$, the contents in the lower row mutate exactly in the same pattern as in Example 3.4. So does the upper row with the other choice of bipartite decomposition of $I$. Now, there is no interaction in the vertical direction. Again, this is because the property (iii) in Lemma 3.1 is satisfied at any mutation point $\mathbf{k}$ and any $\mathbf{i}$ vertically adjacent to $\mathbf{k}$. (Note that $(\mathbf{k},u):\mathbf{P}_{-}$ for mutations in the reverse direction of $u$.)

Proof of Proposition 3.2.

It is enough to prove (ii). We just repeat the argument in Example 3.5 in a general manner. Let us recall the definition of the integer vector $\mathbf{d}(i,u)$ in [FZ4, Definition 10.2] (in our notation). Let $s_{1},\dots,s_{r}$ be the simple reflections of the Weyl group of type $X_{r}$, and let

[TABLE]

Define the piecewise-linear analogue $\tau_{\pm}$ of $t_{\pm}$ acting on the set $\Phi_{\geq-1}$ of all the positive roots and the negative simple roots of $X_{r}$ by

[TABLE]

Then, $\mathbf{d}(i,u)$ is defined by

[TABLE]

It is known by [FZ4, Proposition 9.3] that $\mathbf{d}(i,u)$ is a positive root of $X_{r}$ for $1\leq u\leq h$. We naturally identify $\mathbf{d}(i,u)=\sum_{k\in I}d(i,u)_{k}\alpha_{k}$ with the integer vector $(d(i,u)_{k})_{k\in I}$.

1. The case $0\leq u\leq h^{\prime}-1$. Let $\mathbf{d}(i^{\prime},u)=(d(i^{\prime},u)_{k^{\prime}})_{k^{\prime}\in I^{\prime}}$ be the integer vector as above for $X^{\prime}_{r^{\prime}}$ with $I^{\prime}=I^{\prime}_{+}\sqcup I^{\prime}_{-}$. Let $\mathbf{\tilde{d}}(i^{\prime},u)$ be the same vector but for the opposite choice of the bipartite decomposition of $I^{\prime}$. Forget temporarily all the arrows in the horizontal direction. Then, one has

[TABLE]

by applying [FZ4, Proposition 10.7] in our convention. This remains to be true even in the presence of the horizontal arrows (factorization property), because of Lemma 3.1 and the positivity of vectors $\mathbf{d}(i^{\prime},u)$ and $\mathbf{\tilde{d}}(i^{\prime},u)$.

2. The case $-h\leq u\leq-1$. We consider mutations in the reverse direction of $u$ at points $(\mathbf{i},u):\mathbf{P}_{-}$. Let $\mathbf{d}(i,u)$ be the corresponding vector for $X_{r}$ with $I=I_{+}\sqcup I_{-}$, and $\mathbf{\tilde{d}}(i,u)$ be the one for the opposite choice of the bipartite decomposition of $I$. Then, repeating the same argument, and also using Lemma 2.4 (1), we obtain

[TABLE]

This completes the proof of Proposition 3.2. ∎

4. Proof of Theorem 2.8

We prove Theorem 2.8 by applying the method of [FS] with mixture of ideas by [CGT, C].

The proof is divided into two steps. First, we prove that the left hand side of (2.12) is independent of a semifield homomorphism $\varphi$ (constancy or rigidity property). Next, we evaluate its value at the $0/\infty$* limit*.

4.1. Constancy property

Let $A$ be a multiplicative abelian group. The group $A\otimes_{\mathbb{Z}}A$ is the additive abelian group generated by $g\otimes h$ ($g,h\in A$) with relations

[TABLE]

As a consequence, we also have the following relations

[TABLE]

Let $S^{2}A$ be the subgroup of $A\otimes_{\mathbb{Z}}A$ generated by $f\otimes f$ ($f\in A$), and $\bigwedge^{2}A$ be the quotient of $A\otimes_{\mathbb{Z}}A$ by $S^{2}A$. In $\bigwedge^{2}A$ we use $\wedge$ instead of $\otimes$.

According to a very general theorem by [FS, Proposition 1] (see also [B, Zag1]), the constancy property of the left hand side of (2.12) follows from the following fact.

Proposition 4.1**.**

In $\bigwedge^{2}\mathbb{Q}_{\mathrm{sf}}(y)$, we have

[TABLE]

Motivated by [CGT, C], we use the $F$-polynomials of [FZ4] to prove Proposition 4.1.

The $F$-polynomial $F_{\mathbf{i}}(u)\in\mathbb{Q}_{\mathrm{sf}}(y)$ at $(\mathbf{i},u)$ ($\mathbf{i}\in\mathbf{I}$, $u\in\mathbb{Z}$) is defined by the specialization of $[x_{\mathbf{i}}(u)]_{\mathrm{T}}$ at $x_{\mathbf{j}}=x_{\mathbf{j}}(0)=1$ ($\mathbf{j}\in\mathbf{I}$). (Caution: Do not make the tropical evaluation for the addition in $\mathbb{Q}\mathbb{P}(x)$!) It is represented as a polynomial in $y$ with integer coefficients due to the Laurent phenomenon [FZ4, Proposition 3.6].

For our matrix $B=B(X_{r},X^{\prime}_{r^{\prime}})$, it is convenient to define the incidence matrices $M=(M_{\mathbf{i}\mathbf{j}})_{\mathbf{i},\mathbf{j}\in\mathbf{I}}$ and $M^{\prime}=(M^{\prime}_{\mathbf{i}\mathbf{j}})_{\mathbf{i},\mathbf{j}\in\mathbf{I}}$ as

[TABLE]

Note that they are symmetric matrices.

Lemma 4.2**.**

(i) For $(\mathbf{i},u):\mathbf{P}_{+}$, the following relations hold in $\mathbb{Q}_{\mathrm{sf}}(y)$.

[TABLE]

(ii) Periodicity: $F_{\mathbf{i}}(u+2(h+h^{\prime}))=F_{\mathbf{i}}(u).$

(iii) Each polynomial $F_{\mathbf{i}}(u)$ has constant term 1.

Proof.

(i). The first two relations follow from (2.3). The last one is due to [FZ4, Proposition 3.13]. (ii). This was shown by [Kel1, Kel2]. (iii). For $u=0$, this is true by $F_{\mathbf{i}}(0)=1$. Then, the claim is shown by induction on $u$, by using (4.6), (4.7), and Proposition 3.2 (i) (cf. [FZ4, Proposition 5.6]). ∎

Remark 4.3**.**

Lemma 4.2 (iii) is also true by [DWZ, Theorem 1.7]. Then, Proposition 3.2 (i) is a consequence of Lemma 4.2 (iii) due to [FZ4, Proposition 5.6].

By (4.7) and (4.8), we also have, for $(\mathbf{i},u):\mathbf{P}_{+}$,

[TABLE]

Now, we put (4.8) and (4.9) into (4.4), and expand it.

Firstly,

[TABLE]

since each monomial $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ is either positive or negative by Proposition 3.2.

Secondly, the contributions from the terms involving only $F_{\mathbf{i}}(u)$’s vanish due to the symmetry argument of [CGT, Section 3]. For example,

[TABLE]

by changing the variables twice; therefore, it vanishes.

Thirdly, the contribution from the remaining five terms cancel due to the Y-system (1.10):

[TABLE]

This completes the proof of Proposition 4.1.

Remark 4.4**.**

Eq. (4.4) was studied in [CGT, Eq. (3.32)] by using the parametrization of $y_{\mathbf{i}}(u)$ by the T-system. However, it is known that this does not give a ‘general’ solution of the Y-system [IIKNS, remark after Proposition 3.8]. Therefore, we use $F$-polynomials instead of the T-system.

4.2. Evaluation in the $0/\infty$ limit

Following [C], we evaluate the value of the left hand side of (2.12) in the limit such that each $\varphi(y_{\mathbf{i}}(u))$ goes to either zero or infinity (the $0/\infty$* limit*). Then, the value is equal to the number of the variables which go to infinity due to (1.2).

Thanks to Proposition 3.2, we already have such a limit at hand. Take the one parameter family of semifield homomorphisms $\varphi_{t}:\mathbb{Q}_{\mathrm{sf}}(y)\rightarrow\mathbb{R}_{+}$ ($0<t<1$) defined by $\varphi_{t}(y_{\mathbf{i}})=t$ for any $\mathbf{i}\in I$. Then, in the limit $t\rightarrow 0$, $\varphi_{t}(y_{\mathbf{i}}(u))$ is 0 if $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ is positive and $\infty$ if $[y_{\mathbf{i}}(u)]_{\mathrm{T}}$ is negative, due to (4.8) and Lemma 4.2 (iii). Therefore, the value of the left hand side of (2.12) is $N_{-}=hrr^{\prime}$ by Proposition 3.2.

This completes the proof of Theorem 2.8.