Classical and Quantum Dilogarithm Identities

Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturall...

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Datum:	2011
Hauptverfasser:	Kashaev, R.M., Nakanishi, T.
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Veröffentlicht:	Інститут математики НАН України 2011
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1479942019-02-17T01:23:27Z Classical and Quantum Dilogarithm Identities Kashaev, R.M. Nakanishi, T. Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method. 2011 Article Classical and Quantum Dilogarithm Identities / R.M. Kashaev, T. Nakanishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 65 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60 DOI: http://dx.doi.org/10.3842/SIGMA.2011.106 http://dspace.nbuv.gov.ua/handle/123456789/147994 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.
format	Article
author	Kashaev, R.M. Nakanishi, T.
spellingShingle	Kashaev, R.M. Nakanishi, T. Classical and Quantum Dilogarithm Identities Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kashaev, R.M. Nakanishi, T.
author_sort	Kashaev, R.M.
title	Classical and Quantum Dilogarithm Identities
title_short	Classical and Quantum Dilogarithm Identities
title_full	Classical and Quantum Dilogarithm Identities
title_fullStr	Classical and Quantum Dilogarithm Identities
title_full_unstemmed	Classical and Quantum Dilogarithm Identities
title_sort	classical and quantum dilogarithm identities
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147994
citation_txt	Classical and Quantum Dilogarithm Identities / R.M. Kashaev, T. Nakanishi // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 65 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kashaevrm classicalandquantumdilogarithmidentities AT nakanishit classicalandquantumdilogarithmidentities
first_indexed	2025-07-11T03:45:23Z
last_indexed	2025-07-11T03:45:23Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 102, 29 pages Classical and Quantum Dilogarithm Identities Rinat M. KASHAEV † and Tomoki NAKANISHI ‡ † Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland E-mail: Rinat.Kashaev@unige.ch ‡ Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan E-mail: nakanisi@math.nagoya-u.ac.jp Received May 03, 2011, in final form October 26, 2011; Published online November 01, 2011 http://dx.doi.org/10.3842/SIGMA.2011.102 Abstract. Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method. Key words: dilogarithm; quantum dilogarithm; cluster algebra 2010 Mathematics Subject Classification: 13F60 1 Introduction 1.1 Pentagon relations The Euler dilogarithm Li2(x) and its variant, the Rogers dilogarithm L(x) have appeared in several branches of mathematics (e.g., [47, 41, 65]). See (2.1) and (2.2) for the definition. The most important property of the functions is the pentagon relation. For L(x), it takes the following form L(x) + L(y) = L ( x(1− y) 1− xy ) + L(xy) + L ( y(1− x) 1− xy ) , 0 ≤ x, y ≤ 1. (1.1) The quantum dilogarithm appears also in several branches of mathematics, e.g., discrete quantum systems [2, 17, 15, 14, 13, 16, 3, 4, 37], hyperbolic geometry and Teichmüller theory [34, 35, 20, 30], quantum topology [33, 1], Donaldson–Thomas invariants [43, 44, 45, 39, 50, 49], string theory [26, 27, 9], representation theory of algebras [56], etc., and it accumulates much attention recently. Actually, there are at least two variants of the quantum dilogarithm. The first one Ψq(x), where q is a parameter, is simply called the quantum dilogarithm here. See (3.1) for the definition. The study of the function as ‘quantum exponential’ goes back to [58], but the recognition as ‘quantum dilogarithm’ was made more recently [17, 15]. The following properties explain why it is considered as a quantum analogue of the dilogarithm [17, 15, 36]. (a) Asymptotic behavior: In the limit q → 1−, Ψq(x) ∼ exp ( −Li2(−x) 2 log q ) . (1.2) (b) Pentagon relation: If UV = q2V U , then Ψq(U)Ψq(V ) = Ψq(V )Ψq ( q−1UV ) Ψq(U). (1.3) Moreover, in the limit q → 1−, the relation (1.3) reduces to the relation (1.1). mailto:Rinat.Kashaev@unige.ch mailto:nakanisi@math.nagoya-u.ac.jp http://dx.doi.org/10.3842/SIGMA.2011.102 2 R.M. Kashaev and T. Nakanishi The second variant of the quantum dilogarithm Φb(z), where b is a parameter, was introduced by [14, 13]. Here we call it Faddeev’s quantum dilogarithm (also known as the noncompact quantum dilogarithm). See (4.2) for the definition. The function Φb(z) also satisfies properties parallel to the ones for Ψq(x) [14, 13, 64, 16]. (a) Asymptotic behavior: In the limit b→ 0, Φb ( z 2πb ) ∼ exp ( − Li2(−ez) 2πb2 √ −1 ) . (1.4) (b) Pentagon relation: If [P̂ , Q̂] = (2π √ −1)−1, then Φb(Q̂)Φb(P̂ ) = Φb(P̂ )Φb(P̂ + Q̂)Φb(Q̂). (1.5) Moreover, in the limit b→ 0, the relation (1.5) reduces to the relation (1.1). Despite the appearance of the Euler dilogarithm Li2(x) in (1.2) and (1.4), we have the Rogers dilogarithm L(x) in (1.1) when we take the limits of (1.3) and (1.5). Namely, the limits of (1.3) and (1.5) are not so trivial as termwise limit. The two functions L(x) and Li2(x) differ by logarithms (see (2.3) and (2.4)), and the noncommutativity of U , V and P , Q ‘magically’ turns Li2(x) into L(x). To clarify this phenomenon in a (much) wider situation is the main theme of this paper. 1.2 Classical and quantum dilogarithm identities from cluster algebras In [52], based on cluster algebras by [21, 24], an identity of the Rogers dilogarithm was associated with any period of seeds of a cluster algebra. It looks as follows L∑ t=1 εtL ( ykt(t) εt 1 + ykt(t) εt ) = 0. (1.6) A precise account will be given in Section 2.5. Here we only mention that ε1, . . . , εL is a certain sequence of signs called the tropical sign-sequence. The simplest case of the cluster algebra of type A2 yields the pentagon relation (1.1). Thus, it provides a vast generalization of (1.1). Here we call this family the classical dilogarithm identities. Cluster algebras have the quantum counterparts, called quantum cluster algebras [7, 18]. Here we use the formulation by [18]. Any period of seeds of a classical (nonquantum) cluster algebra is also a period of seeds of the corresponding quantum cluster algebra and vice versa. Recently, in parallel with the classical case, an identity of the quantum dilogarithm Ψq(x) was associated with any period of seeds of a quantum cluster algebra by [39] (see also [56, 50, 49]). Moreover, as a pleasant surprise, we simultaneously obtain at least four variations of quantum dilogarithm identities as follows. 1) Identities in tropical form for Ψq(x). This is the form presented by [39], and it looks as follows Ψq(Y ε1α1)ε1 · · ·Ψq(Y εLαL)εL = 1. (1.7) A precise account will be given in Section 3.4. The simplest case of the quantum cluster algebra of type A2 yields the pentagon relation (1.3). 2) Identities in universal form for Ψq(x). This is the form presented by [63, 62], and it looks as follows Ψq(YkL(L)εL)εL · · ·Ψq(Yk1(1)ε1)ε1 = 1. (1.8) Classical and Quantum Dilogarithm Identities 3 A precise account will be given in Section 3.5. The simplest case of type A2 yields the new variation of the pentagon relation for Ψq(x) recently found by [63] with a suitable identification of variables. In general, they are obtained from the identities in tropical form (1.7) by the ‘shuffle method’ due to A.Yu. Volkov [62]. 3) Identities in tropical form for Φb(z). This is the counterpart of the form (1.7), and it looks as follows Φb(ε1α1D̂)ε1 · · ·Φb(εLαLD̂)εL = 1. (1.9) A precise account will be given in Section 4.5. The simplest case of type A2 yields the pentagon relation (1.5). 4) Identities in local form for Φb(z). This is the form presented by [20, 30]. In general they are specified not only by a period of seeds but also by any choice of sign-sequence. The case of tropical sign-sequence is important for our purpose, and in that case it looks as follows Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗ = 1. (1.10) A precise account will be given in Section 4.6. We call these identities (1.7)–(1.10) together the quantum dilogarithm identities. With these classical and the corresponding quantum dilogarithm identities, it is natural to ask how the latter reduce to the former in the limit q → 1 or b → 0. In this paper we address this question. More precisely, we demonstrate how in the limit b → 0 the classical dilogarithm identities (1.6) emerge as the leading term in the asymptotic expansion from the quantum dilogarithm identities in the form (1.10), that is, the local form with tropical sign-sequence. To do it, we apply the saddle point method (see, e.g., [59, p. 95]), also known as the stationary phase method, à la [15]. In particular, we show transparently how the aforementioned logarithmic gap between the Euler and Rogers dilogarithms is filled. See Section 5.5 for the bottom line. Three remarks follow. First, the variables of quantum cluster algebras admit a natural quantum-mechanical formulation, where the limit b → 0 corresponds to the limit ~ → 0 of the Planck constant ~. See (4.7) and (4.10). Furthermore, the classical dilogarithm identities appear as the leading terms of the quantum dilogarithm identities for the asymptotic expansion in ~. Therefore, following the standard terminology of quantum mechanics, we call the limiting procedure the semiclassical limit. Second, even though our treatment of the saddle point method here is standard in quantum mechanics, we admit and stress that we did not pursue the complete, functional-analytic rig- orousness. Namely, the validity of the method in total and specific details, for example, the uniqueness of the solution of the saddle point equations, the specification of the integration contour through the saddle point, etc., are not argued. Our objective here is not to prove the classical dilogarithm identities by this method, but to make a direct bridge between the classical and the quantum dilogarithm identities. Third, there is actually the fifth form of quantum dilogarithm identities, namely, the identities in local form for Ψq(x) with tropical sign-sequence. This is the counterpart of the form (1.10), and it looks as follows Ψq(Ŷk1(1)ε1)ε1ρ∗k1,ε1 · · ·Ψq(ŶkL(L)εL)εLρ∗kL,εLν ∗ = 1. (1.11) One can also obtain the classical dilogarithm identities (1.6) from (1.11) in the semiclassical limit. However, the relevant differential operators are not self-adjoint. Therefore, the semiclassical limit is more natural for Φb(z) from the quantum-mechanical point of view. For the completeness, we also present it in Appendix A. 4 R.M. Kashaev and T. Nakanishi In summary, our result establishes the following scheme periods of quantum cluster algebras // quantum dilogarithm identities semiclassical limit �� periods of classical cluster algebras // �� OO classical dilogarithm identities The organization of the paper is the following. In Section 2 we present the classical di- logarithm identities obtained from periods of cluster algebras. In Section 3 we present the quantum dilogarithm identities for the quantum dilogarithm Ψq(x). In Section 4 we present the quantum dilogarithm identities for the Faddeev’s quantum dilogarithm Φb(z). In Section 5 we demonstrate how Rogers dilogarithm identities naturally emerge from the quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method. This is the main part of the paper. In Appendix A, we present the quantum dilogarithm identities in local form for Ψq(x). Then, we derive the classical dilogarithm identities from them in the semiclassical limit. 2 Classical dilogarithm identities In this section we present the classical dilogarithm identities obtained from periods of cluster algebras following [52]. 2.1 Euler and Rogers dilogarithms Let Li2(x) and L(x) be the Euler and Rogers dilogarithm functions, respectively [47], Li2(x) = − ∫ x 0 { log(1− y) y } dy, x ≤ 1, (2.1) L(x) = −1 2 ∫ x 0 { log(1− y) y + log y 1− y } dy, 0 ≤ x ≤ 1. (2.2) Two functions are related as follows L(x) = Li2(x) + 1 2 log x log(1− x), 0 ≤ x ≤ 1, (2.3) −L ( x 1 + x ) = Li2(−x) + 1 2 log x log(1 + x), 0 ≤ x. (2.4) The function L(x) satisfies the property (1.1) and also the following ones L(0) = 0, L(1) = π2 6 , L(x) + L(1− x) = π2 6 , 0 ≤ x ≤ 1. (2.5) 2.2 y-variables in cluster algebras In this subsection we recall some definitions and properties of the cluster algebras with coeffi- cients [21, 22], following the convention of [24] with slight change of notation and terminology. Here, we concentrate on the ‘coefficients’ or ‘y-variables’, since we do not explicitly use the ‘cluster variables’ or ‘x-variables’. Classical and Quantum Dilogarithm Identities 5 Let I be a finite set, and fix the initial y-seed (B, y), which is a pair of a skew-symmetric (integer) matrix B = (bij)i,j∈I and an I-tuple of commutative variables y = (yi)i∈I . Let Puniv(y) be the universal semifield of y, which consists of all nonzero rational functions of y having subtraction-free expressions. It is a semifield, i.e., the Abelian multiplicative group with addition (but not with subtraction), by the ordinary multiplication and addition of rational functions. Let (B′, y′) be any pair of a skew-symmetric matrix B′ = (b′ij)i,j∈I and an I-tuple y′ = (y′i)i∈I with y′i ∈ Puniv(y). For each k ∈ I, we define another pair (B′′, y′′) = µk(B ′, y′) of a skew- symmetric matrix B′′ = (b′′ij)i,j∈I and an I-tuple y′′ = (y′′i )i∈I with y′′i ∈ Puniv(y), called the mutation of (B′, y′) at k, by the following rule: (i) Mutation of matrix: b′′ij =  −b′ij , i = k or j = k, b′ij + [−b′ik]+b′kj + b′ik[b ′ kj ]+ = b′ij + [b′ik]+b ′ kj + b′ik[−b′kj ]+, otherwise. (2.6) (ii) Exchange relation of y-variables: y′′i =  y′k −1, i = k, y′iy ′ k [b′ki]+(1 + y′k) −b′ki = y′iy ′ k [−b′ki]+(1 + y′k −1)−b ′ ki , i 6= k. (2.7) Here, [a]+ = a for a ≥ 0 and 0 for a < 0. Starting from the initial y-seed (B, y), repeat the mutations. Each resulting pair (B′, y′) is called a y-seed of (B, y). Remark 2.1. The convention of [24] adopted here is related with the convention of [18, 20, 39] by exchanging the matrix B′ with its transposition. 2.3 Tropical y-variables Let Ptrop(y) be the tropical semifield of y = (yi)i∈I , which is the Abelian multiplicative group freely generated by y endowed with the addition ⊕∏ i∈I yaii ⊕ ∏ i∈I ybii = ∏ i∈I y min(ai,bi) i . There is a canonical surjective semifield homomorphism πT (the tropical evaluation) from Puniv(y) to Ptrop(y) defined by πT(yi) = yi and πT(α) = 1 (α ∈ Q+). For any y-variable y′i of a y-seed (B′, y′) of (B, y), let us write [y′i] := πT(y′i) for simplicity. We call [y′i]’s the tropi- cal y-variables (the principal coefficients in [24]). They satisfy the exchange relation (2.7) by replacing y′i and + with [y′i] and ⊕. We say that a Laurent monomial [y′i] is positive (resp. negative) if it is not 1 and all the exponents are nonnegative (resp. nonpositive). Proposition 2.2 (Sign-coherence [11, 55, 48]). For any y-seed (B′, y′) of (B, y), the Laurent monomial [y′i] in y is either positive or negative. Based on Proposition 2.2, for any y-seed (B′, y′) of (B, y), let ε(y′i) be 1 (resp. −1) if [y′i] is positive (resp. negative). We call it the tropical sign of y′i by identifying ±1 with the signs ±. Using the tropical sign ε(y′i), the tropical exchange relation is written as follows: [y′′i ] = { [y′k] −1, i = k, [y′i][y ′ k] [ε(y′k)b ′ ki]+ , i 6= k. (2.8) 6 R.M. Kashaev and T. Nakanishi 2.4 Periodicity of y-seeds For any I-sequence (k1, k2, . . . , kL), set (B(1), y(1)) := (B, y), and consider the sequence of mutations of y-seeds of (B, y), (B(1), y(1)) µk1←→ (B(2), y(2)) µk2←→ · · · µkL←→ (B(L+ 1), y(L+ 1)). (2.9) Definition 2.3. Let ν : I → I be any bijection. We say that an I-sequence (k1, k2, . . . , kL) is a ν-period of (B, y) if the following holds bν(i)ν(j)(L+ 1) = bij(1), yν(i)(L+ 1) = yi(1), i, j ∈ I. (2.10) See [23, 38, 31, 32, 53] for various examples of periodicity. Remarkably, the periodicity of y-seeds reduces to the periodicity of tropical y-variables, which is much simpler. Proposition 2.4 ([31, 55]). The condition (2.10) holds if and only if [yν(i)(L+ 1)] = [yi(1)], i ∈ I. For Ĩ ⊃ I and a skew-symmetric matrix B̃ = (b̃ij)i,j∈Ĩ , we say that B̃ is an Ĩ-extension of B if b̃ij = bij for any i, j ∈ I. Example 2.5. For any skew-symmetric matrix B with index set I, which may be degenerate, let I ′ = {i′ \| i ∈ I} be a copy of I and let Ĩ = I t I ′. Define the skew-symmetric matrix B̃ = (b̃ij)i,j∈Ĩ by b̃ij =  bij , i, j ∈ I, 1, j ∈ I, i = j′, −1, i ∈ I, j = i′, 0, otherwise. Then, B̃ is an Ĩ-extension of B; furthermore, B̃ is nondegenerate. The matrix B̃ is called the principal extension of B. Proposition 2.6 (Extension Theorem [52]). Suppose that an I-sequence (k1, . . . , kL) is a ν- period of (B, y). Then, for any Ĩ-extension B̃ of B, (k1, . . . , kL) is also a ν-period of (B̃, ỹ). In Proposition 2.6 the periodicity of the ‘external’ variables ỹi (i ∈ Ĩ \ I) is nontrivial. 2.5 Classical dilogarithm identities Let (k1, . . . , kL) be a ν-period of (B, y). For the mutation sequence (2.9), let N+ and N− be the numbers of the positive and negative monomials among [yk1(1)], . . . , [ykL(L)], respectively, so that N+ +N− = L. The following is a generalization of the identities [28, 29, 25, 10, 51, 31, 32, 53] originated from the central charge identities in conformal field theory [40, 42, 5, 46]. Theorem 2.7 (Classical dilogarithm identities [52]). The following identities hold 6 π2 L∑ t=1 L ( ykt(t) 1 + ykt(t) ) = N−, (2.11) 6 π2 L∑ t=1 L ( 1 1 + ykt(t) ) = N+, (2.12) where the initial variables yi (i ∈ I) arbitrarily take values in positive real numbers. Classical and Quantum Dilogarithm Identities 7 Two identities (2.11) and (2.12) are equivalent due to (2.5). Remark 2.8. In [52, Theorems 4.3 & 6.4], Proposition 2.6 and Theorem 2.7 are stated only for ν = id. However, the proofs therein are also applicable to a general ν. We introduce the sign-sequence (ε1, . . . , εL) so that εt is the tropical sign of ykt(t). We call it the tropical sign-sequence of (2.9). Using (2.5), one can also rewrite (2.11) and (2.12) in the following way. Theorem 2.9. For the tropical sign-sequence (ε1, . . . , εL), L∑ t=1 εtL ( ykt(t) εt 1 + ykt(t) εt ) = 0. (2.13) 2.6 Example of type A1 Consider the simplest case, I = {1} and B = (0). Let (k1, k2) = (1, 1), and consider the sequence of mutations of y-seeds of (B, y), (B(1), y(1)) µ1←→ (B(2), y(2)) µ1←→ (B(3), y(3)). Then, y1(1) = y1, y1(2) = y−11 , y1(3) = y1. Thus, (k1, k2) is a ν-period with ν = id, which is nothing but the involution property of the mutation. Also [y1(1)] = y1, [y1(2)] = y−11 , [y1(3)] = y1 and ε1 = 1, ε2 = −1. The classical dilogarithm identity (2.13) is L ( y1 1 + y1 ) − L ( y1 1 + y1 ) = 0, which is trivial. 2.7 Example of type A2 Consider the simplest nontrivial case B = ( 0 −1 1 0 ) , which is also represented by the quiver of type A2 e e� 1 2 8 R.M. Kashaev and T. Nakanishi Let (k1, . . . , k5) = (1, 2, 1, 2, 1), and consider the sequence of mutations of y-seeds of (B, y), (B(1), y(1)) µ1←→ (B(2), y(2)) µ2←→ · · · µ1←→ (B(6), y(6)). Then,{ y1(1) = y1, y2(1) = y2, { y1(2) = y−11 , y2(2) = y2(1 + y1), { y1(3) = y−11 (1 + y2 + y1y2), y2(3) = y−12 (1 + y1) −1,{ y1(4) = y1(1 + y2 + y1y2) −1, y2(4) = y−11 y−12 (1 + y2), { y1(5) = y−12 , y2(5) = y1y2(1 + y2) −1, { y1(6) = y2, y2(6) = y1. Thus, (k1, . . . , k5) is a ν-period, where ν = (12) is the permutation of 1 and 2. Also [y1(1)] = y1, [y2(2)] = y2, [y1(3)] = y−11 , [y2(4)] = y−11 y−12 , [y1(5)] = y−12 , and ε1 = ε2 = 1, ε3 = ε4 = ε5 = −1. The classical dilogarithm identity (2.13) is L ( y1 1 + y1 ) + L ( y2(1 + y1) 1 + y2 + y1y2 ) − L ( y1 (1 + y1)(1 + y2) ) − L ( y1y2 1 + y2 + y1y2 ) − L ( y2 1 + y2 ) = 0. By identifying x = y1/(1 + y1), y = y2(1 + y1)/(1 + y2 + y1y2), it coincides with the pentagon relation (1.1). 3 Quantum dilogarithm identities for Ψq(x) In this section we present the quantum dilogarithm identities for Ψq(x). The content heavily relies on [18, 20, 39]. 3.1 Quantum dilogarithm Following [17, 15], define the quantum dilogarithm Ψq(x), for \|q\| < 1 and x ∈ C, by Ψq(x) = ∞∑ n=0 (−qx)n (q2; q2)n = 1 (−qx; q2)∞ , (a; q)n = n−1∏ k=0 ( 1− aqk ) . (3.1) We have the properties (1.2) and (1.3), and also the following recursion relations Ψq ( q±2x ) = ( 1 + q±1x )±1 Ψq(x). (3.2) 3.2 Quantum y-variables So far, two kinds of quantum cluster algebras are known in the literature. The first one was introduced earlier by [7], where the x-variables are noncommutative and the y-variables are noncommutative but restricted to the tropical one. The second one was introduced by [18, 20], where the y-variables are noncommutative and the universal one but x-variables are commuta- tive. For the relation between them, see [20, Section 2.7] and also [60]. Here we use the second one by [18, 20], and concentrate on the quantum y-variables only. Classical and Quantum Dilogarithm Identities 9 Let I be a finite set, and q be an indeterminate. We start from the initial quantum y-seed (B, Y ), which is a pair of a skew-symmetric (integer) matrix B = (bij)i,j∈I and an I-tuple of noncommutative variables Y = (Yi)i∈I with YiYj = q2bjiYjYi. (3.3) Accordingly, let T(B,Y) be the associated quantum torus, which is the Q(q)-algebra generated by the noncommutative variables Yα (α ∈ ZI) with the relations q〈α,β〉YαYβ = Yα+β, 〈α, β〉 = −〈β, α〉 = tαBβ. Thus, we have YαYβ = q2〈β,α〉YβYα. Set Yi := Yei for the standard unit vector ei (i ∈ I). Then, by identifying Yi with Yi, we recover (3.3). Following [39], let A(B,Y) be the associated quantum affine space, which is the Q(q)-subal- gebra of T(B,Y) generated by Yα’s with α ∈ (Z≥0)I . Let Â(B,Y) be the completion of A(B,Y), which consists of the noncommutative formal power series of Yi’s. The complete quantum torus T̂(B,Y) is the localization of Â(B,Y) at Yα’s with α ∈ (Z≥0)I . Let Frac(A(B,Y)) be the noncommutative fraction field of the algebra A(B,Y), which is viewed as a subskewfield of T̂(B,Y) [7]. Let (B′, Y ′) be any pair of a skew-symmetric matrix B′ = (b′ij)i,j∈I and an I-tuple Y ′ = (Y ′i )i∈I with Y ′i ∈ Frac(A(B,Y)) satisfying the relations (3.3) where everything is primed. For each k ∈ I, we define another same kind of pair (B′′, Y ′′) = µk(B ′, Y ′), called the mutation of (B′, Y ′) at k, where B′′ = (b′′ij)i,j∈I , which is the same as (2.6), and Y ′′ = (Y ′′i )i∈I , Y ′′ i ∈ Frac(A(B,Y)) is given by the following rule [18, 20]: Exchange relation of quantum y-variables Y ′′i =  Y ′k −1, i = k, qb ′ ik[b ′ ki]+Y ′i Y ′ k [b′ki]+ \|b′ki\|∏ m=1 ( 1 + q−sgn(b ′ ki)(2m−1)Y ′k )−sgn(b′ki) = qb ′ ik[−b ′ ki]+Y ′i Y ′ k [−b′ki]+ \|b′ki\|∏ m=1 ( 1 + qsgn(b ′ ki)(2m−1)Y ′k −1)−sgn(b′ki), i 6= k. (3.4) Formally setting q = 1, it reduces to (2.7). Now, starting from the quantum initial y-seed (B, Y ), repeat the mutations. Each resulting pair (B′, Y ′) is called a quantum y-seed of (B, Y ). 3.3 Decomposition of mutations Let (B′, Y ′) and (B′′, Y ′′) be a pair of quantum y-seeds of (B, Y ) such that (B′′, Y ′′) = µk(B ′, Y ′). Following [18], we decompose the mutation (3.4) into two parts, namely, the monomial part and the automorphism part. (a) Monomial part. Define the isomorphisms τk,ε for each ε = ±1 by τk,ε : Frac(A(B′′,Y′′))→ Frac(A(B′,Y′)), Y′′i 7→ { Y′−1k , i = k, Y′ei+[εb′ki]+ek , i 6= k. (3.5) The dependence of the map τk,ε on its source (B′′, Y ′′) and target (B′, Y ′) is suppressed for the notational simplicity and should be understood in the context. One can check that they are indeed homomorphisms using (2.6); furthermore, they are isomorphisms because the inverses 10 R.M. Kashaev and T. Nakanishi are given by τk,−ε with b′ki being replaced by b′′ki = −b′ki in (3.5). Compare with the exchange relation of tropical y-variables (2.8). Also note that, in A(B′,Y′), Y′ei+[εb′ki]+ek = qb ′ ik[εb ′ ki]+Y′iY ′ k [εb′ki]+ . (3.6) (b) Automorphism part. It follows from (3.2) that, for Y′k ∈ A(B′,Y′), the adjoint action Ad(Ψq(Y ′ k)) is defined on Frac(A(B′,Y′)) by Ad(Ψq(Y ′ k))(Y ′ i) := Ψq(Y ′ k)Y ′ iΨq(Y ′ k) −1 = Y′iΨq ( q−2b ′ kiY′k ) Ψq(Y ′ k) −1 = Y′i \|b′ki\|∏ m=1 ( 1 + q−sgn(b ′ ki)(2m−1)Y′k )−sgn(b′ki), (3.7) and similarly, Ad ( Ψq(Y ′−1 k )−1 ) (Y′i) := Ψq ( Y′−1k )−1 Y′iΨq ( Y′−1k ) = Y′iΨq ( q2b ′ kiY′−1k )−1 Ψq ( Y′−1k ) = Y′i \|b′ki\|∏ m=1 ( 1 + qsgn(b ′ ki)(2m−1)Y′−1k )−sgn(b′ki). (3.8) By combining (3.5)–(3.8), we have the following intrinsic description of the exchange rela- tion (3.4). Proposition 3.1 ([18, 39]). We have the equality (Ad(Ψq(Y ′ k))τk,+)(Y′′i ) = ( Ad ( Ψq ( Y′−1k )−1) τk,− ) (Y′′i ), (3.9) and either side of (3.9) coincides with the right hand side of the exchange relation (3.4) with Y ′i replaced with Y′i. Remark 3.2. In [18] the case ε = 1 was employed as the definition of the exchange relation. The importance of the use of both the descriptions by ε = ±1 for quantum dilogarithm identities was clarified by [39]. We use this refinement throughout the paper. Example 3.3. Consider the sequence of mutations of quantum y-seeds of (B, Y ), (B(1), Y (1)) := (B, Y ) µk1←→ (B(2), Y (2)) µk2←→ (B(3), Y (3)). Then, for any sign-sequence (ε1, ε2), we have Yi(2) = ( Ad(Ψq(Yk1(1)ε1)ε1)τk1,ε1 ) (Yi(2)), Yi(3) = ( Ad(Ψq(Yk1(1)ε1)ε1)τk1,ε1Ad(Ψq(Yk2(2)ε2)ε2)τk2,ε2 ) (Yi(3)). (3.10) 3.4 Quantum dilogarithm identities in tropical form For any I-sequence (k1, k2, . . . , kL), set (B(1), Y (1)) := (B, Y ), and consider the sequence of mutations of quantum y-seeds of (B, Y ), (B(1), Y (1)) µk1←→ (B(2), Y (2)) µk2←→ · · · µkL←→ (B(L+ 1), Y (L+ 1)). (3.11) We say that an I-sequence (k1, k2, . . . , kL) is a ν-period of (B, Y ) if the following condition holds for the sequence (3.11) bν(i)ν(j)(L+ 1) = bij(1), Yν(i)(L+ 1) = Yi(1), i, j ∈ I. (3.12) The following theorem, essentially due to [20], tells that the periodicities of quantum y-seeds and (classical) y-seeds coincide. Classical and Quantum Dilogarithm Identities 11 Proposition 3.4 ([20]). The condition (2.10) holds for the sequence (2.9) if and only if the condition (3.12) holds for the sequence (3.11). Proof. The ‘if’ part immediately follows by formally setting q = 1 in the exchange relation (3.4) for quantum y-seeds. The ’only if’ part is proved by [20, Lemma 2.22] using [7, Theorem 6.1], when the matrix B is nondegenerate. When B is degenerate, thanks to Example 2.5 and Proposition 2.6, it is reduced to the nondegenerate case. � Now suppose that (k1, k2, . . . , kL) is a ν-period of (B, Y ). Due to the periodicity of Bν(i)ν(j)(L+ 1) = Bij(1), we have the isomorphism Frac(A(B(1),Y(1)))→ Frac(A(B(L+ 1),Y(L+ 1))), Yi(1) 7→ Yν(i)(L+ 1). Let ν also denote this isomorphism by abusing the notation. For any sign-sequence (ε1, . . . , εk), the periodicity for (3.11) is expressed as follows [39]. Ad(Ψq(Yk1(1)ε1)ε1)τk1,ε1 · · ·Ad(Ψq(YkL(L)εL)εL)τkL,εLν = idFrac(A(B(1),Y(1))). (3.13) To extract the identity involving only the quantum dilogarithm Ψq(y), we have to set (ε1, . . . , εL) to be the tropical sign-sequence of (2.9). (We also call it the tropical sign-sequence of (3.11).) The following theorem is due to [39, Theorem 5.6]. The case of simply laced finite type for certain periods was obtained by [56] with a different method. See also [50, Comments (a), p. 5], [49] for the connection to the Donaldson–Thomas invariants. We include a proof because the argument therein will be used also later. Theorem 3.5 (Quantum dilogarithm identities in tropical form [56, 39]). Suppose that (k1, . . . , kL) is a ν-period of (B, Y ), and let (ε1, . . . , εL) be the tropical sign-sequence of (3.11). Let yi(t) be the corresponding (classical) y-variables in (2.9), and let αt ∈ ZI (t = 1, . . . , L) be the vectors such that [ykt(t)] = yαt. (The vector αt is called the c-vector of ykt(t) in [24].) Then, the following identity holds Ψq(Y ε1α1)ε1 · · ·Ψq(Y εLαL)εL = 1, (3.14) where Yε1α1 , . . . ,YεLαL ∈ A(B,Y). Proof. For the choice of εt above, the periodicity of tropical y-variables implies τk1,ε1 · · · τkL,εLν = id. (3.15) Also note that Yk1(1)ε1 = Yε1α1 with α1 = ek1 and ε1 = 1. Then, push out all τkt,εt ’s to the right in (3.13) as follows Ad(Ψq(Y ε1α1)ε1)τk1,ε1Ad(Ψq(Yk2(2)ε2)ε2)τk2,ε2Ad(Ψq(Yk3(3)ε3)ε3) · · · ν = id, Ad(Ψq(Y ε1α1)ε1)Ad(Ψq(Y ε2α2)ε2)τk1,ε1τk2,ε2Ad(Ψq(Yk3(3)ε3)ε3) · · · ν = id, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Ad(Ψq(Y ε1α1)ε1) · · ·Ad(Ψq(Y εLαL)εL)τk1,ε1 · · · τkL,εLν = id. Thus, thanks to (3.15), we have for any i ∈ I Ad(Ψq(Y ε1α1)ε1 · · ·Ψq(Y εLαL)εL))(Yi(1)) = Yi(1). (3.16) If B is nondegenerate, by considering the canonical form of B, one can easily show that the only Yα which commutes with all Yi’s is 1. Therefore, (3.16) implies the identity (3.14). If B is degenerate, again thanks to Example 2.5 and Proposition 2.6, it is reduced to the nondegenerate case. � 12 R.M. Kashaev and T. Nakanishi 3.5 Quantum dilogarithm identities in universal form Let us rewrite the identity (3.14) with genuine ‘nontropical’, or universal, quantum y-variab- les Ykt(t). This generalizes the new variation of the pentagon relation (3.21) recently found by [63] and its generalization to any simply laced finite type [62]. To be more precise, the pentagon relation of [63] is expressed by the initial variables of the Y-system, while our version is expressed by the initial y-variables, so that they have different expressions. However, they coincide under a suitable identification of variables as shown in Section 3.6. A.Yu. Volkov explained us how to derive his pentagon relation and its generalization to any simply laced finite type from the tropical one using the ‘shuffle method’ in the Y-system setting [62]. Below we apply his shuffle method adapted in our cluster algebraic setting. Lemma 3.6. Under the same assumption of Theorem 3.5, the following formulas hold for t = 2, . . . , L (we call (3.18) the shuffle formula) Ψq(Ykt(t) εt)εt = Ad(Ψq(Y ε1α1)ε1 · · ·Ψq(Y εt−1αt−1)εt−1)(Ψq(Y εtαt)εt), (3.17) Ψq(Y ε1α1)ε1 · · ·Ψq(Y εtαt)εt = Ψq(Ykt(t) εt)εt · · ·Ψq(Yk1(1)ε1)ε1 . (3.18) Proof. Let us prove (3.17) for t = 3, for example. By setting i = k3 in (3.10) and repeating the argument in the proof of Theorem 3.5, we have Yk3(3) = (Ad(Ψq(Yk1(1)ε1)ε1)τk1,ε1Ad(Ψq(Yk2(2)ε2)ε2)τk2,ε2)(Yk3(3)) = Ad(Ψq(Y ε1α1)ε1Ψq(Y ε2α2)ε2)(Yα3). Then, by extending the map Ad(Ψq(Y εtαt)εt) to T̂(B,Y), we obtain (3.17) for t = 3. The general case is similar. Then, (3.18) follows from (3.17) by induction. � Applying (3.18) with t = L to the identity (3.14), we immediately obtain the universal counterpart of (3.14). Corollary 3.7 (Quantum dilogarithm identities in universal form ([63, 62])). Under the same assumption of Theorem 3.5, the following identity holds Ψq(YkL(L)εL)εL · · ·Ψq(Yk1(1)ε1)ε1 = 1. (3.19) Since (3.18) actually holds irrespective with the periodicity of the sequence (3.11), one can say that the two identities (3.14) and (3.19) are equivalent. 3.6 Example of type A2 We continue to use the data in Section 2.7. For the initial quantum y-seed (B, Y ), we have Y1Y2 = q2Y2Y1. Consider the sequence of mutations of quantum y-seeds of (B, Y ): (B(1), Y (1)) µ1←→ (B(2), Y (2)) µ2←→ · · · µ1←→ (B(6), Y (6)). Then,{ Y1(1) = Y1, Y2(1) = Y2, { Y1(2) = Y −11 , Y2(2) = Y2(1 + qY1), { Y1(3) = Y −11 (1 + qY2 + Y1Y2), Y2(3) = Y −12 (1 + q−1Y1) −1,{ Y1(4) = Y1(1 + qY2 + Y1Y2) −1, Y2(4) = q−1Y −11 Y −12 (1 + qY2), { Y1(5) = Y −12 , Y2(5) = q−1Y1Y2(1 + q−1Y2), { Y1(6) = Y2, Y2(6) = Y1. Classical and Quantum Dilogarithm Identities 13 The quantum dilogarithm identity in tropical form (3.14) is Ψq (Y1) Ψq (Y2) Ψq (Y1) −1 Ψq ( q−1Y1Y2 )−1 Ψq (Y2) −1 = 1, where we used Y e1+e2 = q−1Y1Y2. It coincides with the pentagon relation (1.3). The quantum dilogarithm identity in universal form (3.19) is Ψq (Y2) −1 Ψq ( q(1 + qY2) −1Y2Y1 )−1 Ψq ( (1 + qY2 + Y1Y2) −1Y1 )−1 ×Ψq (Y2(1 + qY1)) Ψq (Y1) = 1. (3.20) Meanwhile, the pentagon relation in [63] reads, in our convention of Ψq, Ψq ( X(1 + qY )−1 )−1 Ψq ( qX(1 + qX + qY )−1Y )−1 Ψq ( (1 + qX)−1Y )−1 ×Ψq (X) Ψq (Y ) = 1, (3.21) with Y X = q2XY . Two relations (3.20) and (3.21) coincide by identifying X = Y2(1 + qY1), Y = Y1. Remark 3.8. The relation (3.21) should be compared with the quantum pentagon relation at N th roots of unity [15], where Nth powers of the operators are central and they enter the relation as parameters. As was remarked by Bazhanov and Reshetikhin in [6], these parameters are related in exactly the same way as the arguments in the classical pentagon relation; see [6, equation (3.18)]. The quantum pentagon relation at roots of unity plays a central role in the construction of invariants of links in arbitrary 3-manifolds by using the combinatorics of triangulations [33] and in solvable 3-dimensional lattice models of Bazhanov and Baxter [2] (it is called the restricted star-triangle identity there). 4 Quantum dilogarithm identities for Φb(z) In this section we present the quantum dilogarithm identities for Φb(x). The content heavily relies on [20, 19]. 4.1 Faddeev’s quantum dilogarithm Let b be a complex number with nonzero real part. Set cb = (b+ b−1) √ −1/2, q = eπb 2 √ −1, q∨ = eπb −2 √ −1, q = (q∨)−1 = e−πb −2 √ −1. (4.1) Following [13, 14], define the Faddeev’s quantum dilogarithm Φb(z) for z ∈ C in the strip \|Im z\| < \|Im cb\| by Φb(z) = exp ( −1 4 ∫ ∞ −∞ e−2zx √ −1 sinh(xb) sinh(x/b) dx x ) , (4.2) where the singularity at x = 0 is circled from above. It is analytically continued to a meromorphic function on the entire complex plane. We have the properties (1.4) and (1.5), and also the following ones (see, e.g., [57, 64, 16, 61] for more information). (i) Symmetries: Φb(z) = Φb−1(z) = Φ−b(z). (4.3) 14 R.M. Kashaev and T. Nakanishi (ii) Recurrence relation: Φb(z ± b √ −1) = ( 1 + e2πbzq±1 )±1 Φb(z), Φb(z ± b−1 √ −1) = ( 1 + e2πb −1z(q∨)±1 )±1 Φb(z). (4.4) (iii) Unitarity: If b is real or \|b\| = 1, then \|Φb(z)\| = 1, z ∈ R. (4.5) (iv) Relation to Ψq(x): If Im b2 > 0, then Φb(z) = Ψq(e 2πbz) Ψq(e2πb −1z) . (4.6) Note that, if Im b2 > 0, then \|q\|, \|q\| < 1. 4.2 Representation of quantum y-variables Let us recall a representation of quantum y-variables as differential operators in [20, 19]. We continue to use the data (4.1). In view of (4.1), we further set ~ = πb2, q = e~ √ −1. (4.7) To any quantum y-seed (B′, Y ′) of (B, Y ) we associate operators û′ = (û′i)i∈I and p̂′ = (p̂′i)i∈I satisfying the relations [û′i, û ′ j ] = [p̂′i, p̂ ′ j ] = 0, [p̂′i, û ′ j ] = ~√ −1 δij . (4.8) The algebra of operators û′ and p̂′ has a natural representation on the Hilbert space L2(RI): (û′if)(u′) = u′if(u′), (p̂′if)(u′) = ~√ −1 ∂f(u′) ∂u′i , u′ ∈ RI . (4.9) Using Dirac’s notation f(u′) = 〈u′\|f〉, we have formally 〈u′\|û′i\|f〉 = u′i〈u′\|f〉, 〈u′\|p̂′i\|f〉 = ~√ −1 ∂ ∂u′i 〈u′\|f〉, or simply 〈u′\|û′i = u′i〈u′\|, 〈u′\|p̂′i = ~√ −1 ∂ ∂u′i 〈u′\|. The set of generalized vectors {〈u′\|}u′∈RI will be called the local coordinates of (B′, Y ′). Define ŵ′i = ∑ j∈I b′jiû ′ j , D̂′i = p̂′i + ŵ′i, Ŷ′i = exp D̂′i. (4.10) The following relations hold [D̂′i, D̂ ′ j ] = 2~ √ −1b′ji, Ŷ′iŶ ′ j = q2b ′ jiŶ′jŶ ′ i. (4.11) Also recall the following general fact, which is a special case of the Baker–Campbell–Hausdorff formula: For any noncommutative variables A and B such that [A,B] = C and [C,A] = [C,B] = 0, we have eAeB = eC/2eA+B. Thus, we have a representation of T(B,Y) on L2(RI) with Y′α 7→ Ŷ′α := exp ( αD̂′ ) , αD̂′ := ∑ i∈I αiD̂ ′ i. (4.12) Classical and Quantum Dilogarithm Identities 15 4.3 Decomposition of mutations Here we present a result which is analogous to that of Section 3.3. Let (B′, Y ′) and (B′′, Y ′′) be a pair of quantum y-seeds of (B, Y ) such that (B′′, Y ′′) = µk(B ′, Y ′). (a) Monomial part. For each ε = ±1, consider the following map ρk,ε : RI → RI , (u′) 7→ (u′′), u′′i = −u ′ k + ∑ j∈I [−εb′jk]+u′j , i = k, u′i, i 6= k. (4.13) Let ρ∗k,ε be the induced map in the space of functions L2(RI), ρ∗k,ε : L2(RI)→ L2(RI), f 7→ f ◦ ρk,ε, or, formally, 〈u′\|ρ∗k,ε = 〈ρk,ε(u′)\| = 〈u′′\|, by which we relate the local coordinates of (B′, Y ′) and (B′′, Y ′′). For any linear operator Ô acting on L2(RI), let Ad(ρ∗k,ε)(Ô) := ρ∗k,εÔ(ρ∗k,ε) −1. In other words, it is defined by the commutative diagram L2(RI) ρ∗k,ε−−−−→ L2(RI) Ô y yAd(ρ∗k,ε)(Ô) L2(RI) ρ∗k,ε−−−−→ L2(RI). Then, we have Ad(ρ∗k,ε)(û ′′ i ) = −û ′ k + ∑ j∈I [−εb′jk]+û′j , i = k, û′i, i 6= k, (4.14) Ad(ρ∗k,ε)(ŵ ′′ i ) = { −ŵ′k, i = k, ŵ′i + [εb′ki]+ŵ ′ k, i 6= k, (4.15) Ad(ρ∗k,ε)(p̂ ′′ i ) = { −p̂′k, i = k, p̂′i + [εb′ki]+p̂ ′ k, i 6= k, (4.16) Ad(ρ∗k,ε)(D̂ ′′ i ) = { −D̂′k, i = k, D̂′i + [εb′ki]+D̂ ′ k, i 6= k, (4.17) where (4.17) follows from (4.16) and (4.15). It follows from (4.17) that Ad(ρ∗k,ε)(Ŷ ′′ i ) = { Ŷ′k −1, i = k, Ŷ′ei+[εb′ki]+ek , i 6= k, which coincides with (3.5). 16 R.M. Kashaev and T. Nakanishi Remark 4.1. The transformation of (4.13) is the one for the g-vectors in [24] if ε is the tropical sign of y′k. Similarly, for w′i = ∑ j∈I b ′ jiu ′ j , the induced transformation w′′i = { −w′k, i = k, w′i + [εb′ki]+w ′ k, i 6= k, is the one for the c-vectors in [24], and it is the logarithmic form of the tropical exchange relation (2.8). They are known to be dual in the following sense [18, 54]∑ i∈I u′′iw ′′ i = ∑ i∈I u′iw ′ i. (b) Automorphism part. We set D̂′i = 1 2πb D̂′i. (4.18) Then, we have Ŷ′i −1Φb(εD̂ ′ j)Ŷ ′ i = Φb ( εD̂′j − ε √ −1bb′ji ) . Thus, thanks to the recurrence relation (4.4), one obtains, for each ε = ±1, Ad(Φb(εD̂ ′ k) ε)(Ŷ′i) = Ŷ′i \|b′ki\|∏ m=1 ( 1 + q−εsgn(b ′ ki)(2m−1)Ŷ′k ε )−εsgn(b′ki) by an analogous calculation to (3.7) and (3.8). In summary, we have a parallel statement to Proposition 3.1. Proposition 4.2 ([20, 19]). We have the equality (Ad(Φb(D̂ ′ k))Ad(ρ∗k,+))(Ŷ′′i ) = ( Ad ( Φb(−D̂′k)−1 ) Ad(ρ∗k,−) ) (Ŷ′′i ), (4.19) and either side of (4.19) coincides with the right hand side of the exchange relation (3.4) with Y ′i replaced with Ŷ′i. 4.4 Dual operators Following [13] and [20], we define the operators Ẑ′i which are ‘dual’ to Ŷ′i in the sense of the first equality of (4.3). In the situation in (4.10), we define Ẑ′i = exp(b−2D̂′i). Then, the following relations hold Ẑ′iẐ ′ j = (q∨)2b ′ jiẐ′jẐ ′ i, Ŷ′iẐ ′ j = Ẑ′jŶ ′ i. (4.20) Remark 4.3. The duality between Ŷ′i and Ẑ′i is not manifest because of our preference for b over b−1 in (4.10) through (4.7). To see it manifestly, we set D̂′i = 1 γ ( γ2 4π √ −1 ∂̂′i + ŵ′i ) , Ŷ′i = exp ( 2πbD̂′i ) , Ẑ′i = exp ( 2πb−1D̂′i ) , Classical and Quantum Dilogarithm Identities 17 where γ is an arbitrary nonzero real number and ∂̂′i satisfy [∂̂′i, ∂̂ ′ j ] = 0 and [∂̂′i, û ′ j ] = δij . The following relations hold irrespective of γ [D̂′i, D̂′j ] = √ −1 2π b′ji, Ŷ′iŶ ′ j = (q)2b ′ jiŶ′jŶ ′ i, Ẑ′iẐ ′ j = (q∨)2b ′ jiẐ′jẐ ′ i, Ŷ′iẐ ′ j = Ẑ′jŶ ′ i. Now the duality b↔ b−1 is manifest. Further setting γ = 2πb, we have D̂′i = D̂′i and we recover the operators Ŷ′i and Ẑ′i in the main text. Due to the symmetry b ↔ b−1 in (4.3) and the above remark, we immediately obtain the following from Proposition 4.2. Proposition 4.4. We have the equality (Ad(Φb(D̂ ′ k))Ad(ρ∗k,+))(Ẑ′′i ) = ( Ad ( Φb(−D̂′k)−1 ) Ad(ρ∗k,−) ) (Ẑ′′i ), (4.21) and either side of (4.21) coincides with the right hand side of the exchange relation (3.4) with Y ′i and q replaced with Ẑ′i and q∨, respectively. 4.5 Quantum dilogarithm identities in tropical form Suppose that (k1, k2, . . . , kL) is a ν-period of (B, Y ) as in Section 3.4. The identities parallel to (3.14) are available for Φb(z) directly from (3.14). Theorem 4.5 (Quantum dilogarithm identities in tropical form). Under the same assumption of Theorem 3.5 (in particular, (ε1, . . . , εL) is the tropical sign-sequence of the mutation sequence), the following identity holds. Φb(ε1α1D̂)ε1 · · ·Φb(εLαLD̂)εL = 1, (4.22) where αtD̂ = ∑ i∈I αi(t)D̂i and D̂i is the operator in (4.18) for (B, Y ). Proof. Due to the symmetry b ↔ b−1 in (4.3), one can assume that Im b2 ≥ 0 without losing generality. By the continuity of Φb with respect to b, it is enough to show the claim for Im b2 > 0. Then, by (4.6), we have Φb(εtαtD̂) = Ψq(Ŷ εtαt) Ψq(Ẑεtαt) . Then, thanks to the commutativity (4.20), the relation (4.22) factorizes into two identities Ψq(Ŷ ε1α1)ε1 · · ·Ψq(Ŷ εLαL)εL = 1, (4.23) Ψq(Ẑ εLαL)εL · · ·Ψq(Ẑ ε1α1)ε1 = 1, (4.24) where (4.23) is a specialization of (3.14), while (4.24) is equivalent to Ψq∨(Ẑε1α1)ε1 · · ·Ψq∨(ẐεLαL)εL = 1, which is another specialization of (3.14). � 18 R.M. Kashaev and T. Nakanishi 4.6 Quantum dilogarithm identities in local form Let 〈u(t)\| and D̂i(t) denote the local coordinates and the operator in (4.10) for (B(t), Y (t)), respectively. Let L2(RI)t be the Hilbert space together with the local coordinate 〈u(t)\|, so that ρ∗kt,εt : L2(RI)t+1 → L2(RI)t. For the bijection ν, we apply the same formalism as ρ. Namely, let ν : RI → RI be the coordinate transformation defined by (u(L + 1)) 7→ (u(1)) with ui(1) = uν(i)(L + 1). Define ν∗ : L2(RI)1 → L2(RI)L+1, f 7→ f ◦ ν, and Ad(ν∗)(Ô) := ν∗Ô(ν∗)−1 for any linear operator Ô acting on L2(RI). Then, Ad(ν∗)(D̂i(1)) = D̂ν(i)(L+ 1). Let us recall the result of [20, Theorem 5.4]. Suppose that b is a nonzero real number. Note that, Φb(εtD̂ ′ kt (t)) is a unitary operator by (4.5). By the periodicity assumption and Propositions 4.2 and 4.4, we have the following equalities for any sign-sequence ~ε = (ε1, . . . , εL) Ad(Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗)(Ŷi(1)) = Ŷi(1), Ad(Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗)(Ẑi(1)) = Ẑi(1). This is equivalent to saying that the operator Ô~ε,b = Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗ commutes with Ŷi(1) and Ẑi(1) for any i ∈ I. It was shown in [20] that, when b2 is irrational, such Ô~ε,b is the identity operator up to a complex scalar multiple λ~ε,b by generalizing the result of [13]; furthermore, the claim holds for rational b2 as well by continuity. Since Ô~ε,b is unitary, we have \|λ~ε,b\| = 1. Therefore, one obtains the following local form of the quantum dilogarithm identities. We call it so, since it is described by the family of local coordinates 〈u(1)\|, . . . , 〈u(L)\| associated with the mutation sequence. Theorem 4.6 (Quantum dilogarithm identities in local form [20]). Let b be a nonzero real number. For any sign-sequence ~ε = (ε1, . . . , εL), the following identity holds. Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗ = λ~ε,b, \|λ~ε,b\| = 1. (4.25) For the tropical sign-sequence, we have a stronger version of Theorem 4.6. One can obtain it as a direct corollary of Theorem 4.5, and not via Theorem 4.6. So the assumption that b is real is not necessary here. This is the identity we use to derive the corresponding classical dilogarithm identity. Theorem 4.7. For the tropical sign-sequence ~ε = (ε1, . . . , εL), the following identity holds Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗ = 1. (4.26) In particular, λ~ε,b = 1 for the tropical sign sequence. Proof. By the duality in Remark 4.1, the periodicity of tropical y-variables (3.15) is equiva- lent to ρ∗k1,ε1 · · · ρ ∗ kL,εL ν∗ = id. Multiply it from the right of (4.22). Then, repeat the argument in the proof of Theorem 3.5 in the inverse way. � In summary, for the tropical sign-sequence we have four forms of quantum dilogarithm iden- tities (3.14), (3.19), (4.25), and (4.26). The first three identities are obtained from each other without referring to the seed periodicity of (3.11). The last one is obtained from the rest by assuming the tropical periodicity (3.15). Classical and Quantum Dilogarithm Identities 19 4.7 Example of type A2 We continue to use the data in Sections 2.7 and 3.6. The quantum dilogarithm identity in tropical form (4.22) is Φb(D̂1)Φb(D̂2)Φb(D̂1) −1Φb(D̂1 + D̂2) −1Φb(D̂2) −1 = 1, [D̂1, D̂2] = √ −1 2π . By identifying Q̂ = D̂1, P̂ = D̂2, it coincides with the pentagon relation (1.5). Let us also write the relevant data for the identity (4.26) explicitly D̂kt(t) =  1 2πb (p̂1(t) + û2(t)) , t = 1, 3, 5, 1 2πb (p̂2(t) + û1(t)) , t = 2, 4. The images of û1(t + 1), û2(t + 1), ŵ1(t + 1), ŵ2(t + 1) by the map Ad(ρ∗kt,εt) are given in the order t = 1 : −û1(1), û2(1), −ŵ1(1), ŵ2(1), t = 2 : û1(2), −û2(2), ŵ1(2), −ŵ2(2), t = 3 : −û1(3) + û2(3), û2(3), −ŵ1(3), ŵ2(3) + ŵ1(3), t = 4 : û1(4), −û2(4) + û1(4), ŵ1(4) + ŵ2(4), −ŵ2(4), t = 5 : −û1(5) + û2(5), û2(5), −ŵ1(5), ŵ2(5) + ŵ1(5). 5 From quantum to classical dilogarithm identities In this section we demonstrate how the classical quantum dilogarithm identities (2.13) emerge from the quantum dilogarithm identities in local form (4.26) in the semiclassical limit. This is the main part of the paper. 5.1 Position and momentum bases We are going to evaluate the operator in the left hand side of (4.26), which is actually the identity operator, by the standard quantum physics method. Throughout Section 5 we assume that b is a nonzero real number. Recall that we set ~ = πb2 in (4.7). The asymptotic property (1.4) is written as Φb ( z 2πb ) ∼ exp (√ −1 ~ 1 2 Li2(−ez) ) , ~→ 0, (5.1) where and in the rest ∼ means the leading term for the asymptotic expansion in ~. Let (B(t), Y (t)) be the quantum Y -seed of (B(1), Y (1)) = (B, Y ) in (3.11). Let L2(RI)t be the Hilbert space together with the local coordinate 〈u(t)\| associated with (B(t), Y (t)) in the previous section. Let {\|u(t)〉 \| u(t) ∈ RI} and {\|p(t)〉 \| p(t) ∈ RI} be the standard position and the momentum bases of L2(RI)t, respectively. They satisfy the following properties, where n = \|I\|, ûi(t)\|u(t)〉 = ui(t)\|u(t)〉, p̂i\|p(t)〉 = pi(t)\|p(t)〉, 〈u(t)\|u′(t)〉 = ∏ i∈I δ(ui(t)− u′i(t)), 〈p(t)\|p′(t)〉 = (2π~)n ∏ i∈I δ(pi(t)− p′i(t)), 〈u(t)\|p(t)〉 = exp (√ −1 ~ u(t)p(t) ) , 〈p(t)\|u(t)〉 = exp ( − √ −1 ~ u(t)p(t) ) , 20 R.M. Kashaev and T. Nakanishi where u(t)p(t) := ∑ i∈I ui(t)pi(t), 1 = ∫ du(t)\|u(t)〉〈u(t)\|, 1 = ∫ dp(t) (2π~)n \|p(t)〉〈p(t)\|. (5.2) In particular, we have 〈u(t)\|D̂i(t)\|p(t)〉〈u(t)\|p(t)〉 = pi(t) + wi(t), wi(t) := ∑ j=1 bji(t)uj(t). (5.3) Let Ô be the composition of the operators in the left hand side of (4.26), namely, Ô = Φb(ε1D̂k1(1))ε1ρ∗k1,ε1 · · ·Φb(εLD̂kL(L))εLρ∗kL,εLν ∗ (= 1), where (ε1, . . . , εL) is the tropical sign-sequence. Choose any position eigenvector \|u(1)〉. Then, set the momentum eigenvector \|p̃(1)〉 such that its eigenvalues are given by p̃i(1) = wi(1) := ∑ j∈I bji(1)uj(1), (5.4) where the notation p̃(1) is used for later convenience. The condition (5.4) will be used only at the last stage when we construct the solution of the saddle point equations in Section 5.4. The main idea of our consideration is to study the semiclassical behavior of the quantum identity by using q-p symbols of operators, see for example [8]. By Dirac’s argument [12], the semiclassical limit of a q-p symbol of a unitary operator O is given by the exponential of the generating function of the canonical transformation, which quantum mechanically corresponds to the unitary inner transformation generated by O. In our case, the q-p symbol corresponds to the ‘u-p’ symbol defined by F (u(1), p̃(1)) := 〈u(1)\|Ô\|p̃(1)〉〈u(1)\|p̃(1)〉 . Below we show that the leading term of logF (u(1), p̃(1)) in the limit ~→ 0 yields the left hand side of (2.13) up to a multiplicative constant. We know a priori that its value is 0, which yields the right hand side of (2.13). 5.2 Integral expression By inserting the intermediate complete states (5.2), we obtain the following integral expression F (u(1), p̃(1)) = (2π~)−n(2L−1) ∫ dp(1)dp̃(2)du(2)dp(2)dp̃(3) · · · dp̃(L)du(L)dp(L) × 〈p̃(1)\|u(1)〉〈u(1)\|Φb(ε1D̂k1(1))ε1 \|p(1)〉〈u(1)\|p(1)〉〈u(1)\|p(1)〉〈p(1)\|ρ∗k1,ε1 \|p̃(2)〉 × 〈p̃(2)\|u(2)〉〈u(2)\|Φb(ε2D̂k2(2))ε2 \|p(2)〉〈u(2)\|p(2)〉〈u(2)\|p(2)〉〈p(2)\|ρ∗k2,ε2 \|p̃(3)〉 · · · × 〈p̃(L)\|u(L)〉〈u(L)\|Φb(εLD̂kL(L))εL \|p(L)〉〈u(L)\|p(L)〉〈u(L)\|p(L)〉〈p(L)\|ρ∗kL,εLν ∗\|p̃(1)〉. The integration over p(L) is done by (4.16), and it yields the relation p̃i(1) = { −pkL(L), ν(i) = kL, pν(i)(L) + [εLb ′ kLν(i) (L)]+pkL(L), ν(i) 6= kL. (5.5) Classical and Quantum Dilogarithm Identities 21 Similarly, the integration over p̃(t+ 1) (t = 1, . . . , L− 1) yields the relation p̃i(t+ 1) = { −pkt(t), i = kt, pi(t) + [εtb ′ kti (t)]+pkt(t), i 6= kt. (5.6) Thus, p̃(t+ 1) is now a dependent variable of p(t) by (5.6). In view of (4.10) it is natural to introduce new dependent variables ykt(t) = exp(pkt(t) + wkt(t)), t = 1, . . . , L, (5.7) where the notation yi(t) anticipates the identification with classical y-variables eventually. Then, by (5.3), we have 〈u(t)\|D̂kt(t)\|p(t)〉〈u(t)\|p(t)〉 = 1 2πb log ykt(t), (5.8) and the remaining integration has the following form F (u(1), p̃(1)) = (2π~)−n(L−1) ∫ dp(1) · · · dp(L− 1)du(2) · · · du(L) ×Φb ( log yk1(1)ε1 2πb )ε1 exp (√ −1 ~ u(1)(p(1)− p̃(1)) ) ×Φb ( log yk2(2)ε2 2πb )ε2 exp (√ −1 ~ u(2)(p(2)− p̃(2)) ) · · · ×Φb ( log ykL(L)εL 2πb )εL exp (√ −1 ~ u(L)(p(L)− p̃(L)) ) . Using (5.1), we have F (u(1), p̃(1)) ∼ (2π~)−n(L−1) ∫ dp(1) · · · dp(L− 1)du(2) · · · du(L) exp (√ −1 ~ L∑ t=1 { 1 2 εtLi2(−ykt(t)εt) + u(t)(p(t)− p̃(t)) }) . (5.9) To evaluate the integral expression (5.9) in the semiclassical limit, we apply the saddle point method. It consists of three steps. Step 1. Write the saddle point equations, that is, the extremum condition of the integrand of (5.9) for the independent variables p(1), . . . , p(L− 1) and u(2), . . . , u(L). Step 2. Find a solution of the saddle point equations. Step 3. Evaluate the integrand at the solution. 5.3 Saddle point equations Let us derive the saddle point equations for (5.9). We use the following formulas, which are obtained from (2.1), (4.10), (5.7), and (5.8), ∂ ∂pi(t) ( 1 2 εtLi2(−ykt(t)εt) ) = δikt log(1 + ykt(t) εt)−1/2, ∂ ∂ui(t) ( 1 2 εtLi2(−ykt(t)εt) ) = − log(1 + ykt(t) εt)−bkti(t)/2. (a) Extremum conditions with respect to ui(t) (t = 2, . . . , L). 22 R.M. Kashaev and T. Nakanishi By differentiating the integrand of (5.9) by ui(t), we have − log(1 + ykt(t) εt)−bkti(t)/2 + pi(t)− p̃i(t) = 0, (5.10) or, equivalently, epi(t) = ep̃i(t)(1 + ykt(t) εt)−bkti(t)/2. Combining it with (5.6), we also have ep̃i(t+1) = { (ep̃kt (t))−1, i = kt, ep̃i(t)(ep̃kt (t))[εtbkti(t)]+(1 + ykt(t) εt)−bkti(t)/2, i 6= kt. (5.11) (b) Extremum conditions with respect to pi(t) (t = 1, . . . , L− 1). By differentiating the integrand of (5.9) by pi(t), we have log(1 + ykt(t) εt)−1/2 + ukt(t)− ∑ j∈I [εtbktj(t)]+uj(t+ 1) + ukt(t+ 1) = 0, i = kt, (5.12) ui(t)− ui(t+ 1) = 0, i 6= kt, (5.13) or, equivalently, eui(t+1) = (eukt (t))−1 ∏ j∈I (euj(t))[−εtbjkt (t)]+(1 + ykt(t) εt)1/2 i = kt, eui(t), i 6= kt. (5.14) With (2.6), this also implies the following equations for wi(u) = ∑ j∈I bji(t)uj(t) ewi(t+1) = { (ewkt (t))−1, i = kt, ewi(t)(ewkt (t))[εtbkti(t)]+(1 + ykt(t) εt)−bkti(t)/2, i 6= kt, (5.15) which is identical to (5.11). 5.4 Solution Let us summarize the relevant variables and their relations schematically p̃(1) p(1) τ p̃(2) 1+y p(2) τ p̃(3) p̃(L) 1+y p(L) τ, ν p̃(1) w(1) w(2) w(3) w(L) u(1) u(2) u(3) u(L) Here, the framed variables are the initial variables and the underlined variables are the remaining integration variables which should be determined to solve the saddle point equations. This is a highly complicated systems of equations, but the relevance to the y-seed mutations of (2.9) is rather clear. To see it quickly, set yi(t) := ep̃i(t)ewi(t). Classical and Quantum Dilogarithm Identities 23 Note that p̃i(t) = pi(t) if i = kt by (5.10), therefore, it agrees with the previous definition (5.7). Then, multiply two identities (5.11) and (5.15), we have yi(t+ 1) = { ykt(t) −1, i = kt, yi(t)yk(t) [εtbkti(t)]+(1 + ykt(t) εt)−bkti(t), i 6= kt. This is nothing but (2.7). Furthermore, (5.11) and (5.15) suggest that yi(t) 1/2 = ep̃i(t) = ewi(t). Having this observation in mind, let us describe the construction of the solution more clearly. (i) (y-variables) We have ui(1) as initial data, from which wi(1) is uniquely determined. Temporarily forgetting (5.7), set yi(1) = e2wi(1), from which yi(t) (t = 2, . . . , L) are determined by the mutation sequence (2.9). (ii) (u-variables) Set ui(t) (t = 2, . . . , L) by (5.13) and (5.12). Then, (5.15) is also satisfied. (iii) (p-variables) Set p̃i(t) by ep̃i(t) = yi(t) 1/2. This forces the relation p̃i(1) = wi(1), which is guaranteed by the assumption (5.4). Then, pi(t) are determined by (5.5) and (5.6). Since p̃i(t) satisfies (5.11) by definition, (5.10) is also satisfied. (iv) (compatibility) The only thing to be checked is (5.7). Since pkt(t) = p̃kt(t) by (5.10), it is enough to show ep̃i(t) = yi(t) 1/2, ewi(t) = yi(t) 1/2. (5.16) The first equality is by definition. The second equality is true for t = 1 by definition. Then, the rest is shown by (2.7) and the square of (5.15). Thus, we obtain the desired solution of the saddle point equations. We do not argue on the uniqueness of the solution here as stated in Section 1.2. Remark 5.1. Since (5.14) is the square half of the exchange relation of the x-variables of the corresponding cluster algebras [18, Proposition 2.3], the variable eui(t) is regarded as the square half of the x-variable xi(t). 5.5 Result As the final step, we evaluate the logarithm of the integrand in (5.9) at the solution of the saddle point equations in Section 5.4. Using (5.1) and ignoring the common factor, it is given by L∑ t=1 { 1 2 εtLi2(−ykt(t)εt) + ∑ i∈I ui(t)(pi(t)− p̃i(t)) } . (5.17) Recall that pi(t)− p̃i(t) = log(1 + ykt(t) εt)−bkti(t)/2, wi(t) = 1 2 log yi(t) by (5.10) and (5.16). Then, the second term of (5.17) is rewritten as∑ i∈I ui(t)(pi(t)− p̃i(t)) = ∑ i∈I ui(t) log(1 + ykt(t) εt)−bkti(t)/2 = 1 2 (∑ i∈I bikt(t)ui(t) ) log(1 + ykt(t) εt) 24 R.M. Kashaev and T. Nakanishi = 1 2 wkt(t) log(1 + ykt(t) εt) = 1 4 εt log ykt(t) εt log(1 + ykt(t) εt). (5.18) Therefore, by (2.4), (5.17) is equal to −1 2 L∑ t=1 εtL ( ykt(t) εt 1 + ykt(t) εt ) , but we know it is 0 from the beginning. This is the classical dilogarithm identity (2.13). A Quantum dilogarithm identities in local form for Ψq(x) and their semiclassical limits In this section we present the quantum dilogarithm identities in local form for Ψq(x) with tropical sign-sequence. Then, we derive the classical dilogarithm identities from them in the semiclassical limits. The treatment is parallel to the one in Sections 4 and 5 with slight complication. A.1 Representation of quantum y-variables We consider a representation of quantum y-variables as differential operators which are quite similar to the one in Section 4.2 but slightly different. Throughout the section, let ~ be a positive real number, and λ be a complex number such that Imλ2 > 0. We reset q = eλ 2~ √ −1. (A.1) By the assumption, we have \|q\| < 1. Compare with q in (4.7), where \|q\| = 1 when b is real. This difference is due to the fact that Ψq(x) is convergent only for \|q\| < 1, while Φb(z) is well-defined also for \|q\| = 1. The phase λ is the main difference between the two cases and the source of extra complication for Ψq(x) which persists throughout the section. The asymptotic property (1.4) is written as Ψq(x) ∼ exp (√ −1 λ2~ 1 2 Li2(−x) ) , ~→ 0. (A.2) Because of λ, the argument x of the dilogarithms Li2(x) and L(x) eventually take values in C in the semiclassical limit. They are defined by analytic continuation of (2.1) and (2.2) along the integration path. To avoid the ambiguity of the branches, we assume that Imλ is sufficiently small (or, q is sufficiently close to the unit circle \|q\| = 1) so that the resulting argument x in this section is in a neighborhood of the interval (−∞, 1] for Li2(x) or [0, 1] for L(x). To any quantum y-seed (B′, Y ′) of (B, Y ) we associate operators û′ = (û′i)i∈I and p̂′ = (p̂′i)i∈I , and the local coordinates {〈u′\|}u′∈RI as in (4.8) and (4.9). We reset Ŷ′i in (4.10) as ŵ′i = ∑ j∈I b′jiû ′ j , D̂′i = p̂′i + ŵ′i, Ŷ′i = exp(λD̂′i). (A.3) The relations in (4.11) still hold with q in (A.1), and we have a representation of T(B,Y) of (4.12). Classical and Quantum Dilogarithm Identities 25 Let (B′, Y ′) and (B′′, Y ′′) be a pair of quantum y-seeds of (B, Y ) such that (B′′, Y ′′) = µk(B ′, Y ′). Let ρk,ε be the map in (4.13). Then, repeating the argument in Section 4.3, we obtain Ad(ρ∗k,ε)(Ŷ ′′ i ) = { Ŷ′k −1, i = k, Ŷ′ei+[εb′ki]+ek , i 6= k. A.2 Quantum dilogarithm identities in local form for Ψq(x) Under the same assumption and notation for Theorem 4.7, we obtain the counterpart of Theo- rem 4.7 for Ψq(x) by repeating its proof. Theorem A.1. For the tropical sign-sequence ~ε = (ε1, . . . , εL), the following identity holds Ψq(Ŷk1(1)ε1)ε1ρ∗k1,ε1 · · ·Ψq(ŶkL(L)εL)εLρ∗kL,εLν ∗ = 1. (A.4) Let Ô be the composition of the operators in the left hand side of (A.4). Again, choose any position eigenvector \|u(1)〉 and set the momentum eigenvector \|p̃(1)〉 by (5.4). Set F (u(1), p̃(1)) := 〈u(1)\|Ô\|p̃(1)〉〈u(1)\|p̃(1)〉 . Below we show that the leading term of logF (u(1), p̃(1)) in the limit ~→ 0 yields the left hand side of (2.13) up to a multiplicative constant. A.3 Integral expression Repeating the argument in Section 5.2, we obtain the following integral expression F (u(1), p̃(1)) = (2π~)−n(L−1) ∫ dp(1) · · · dp(L− 1)du(2) · · · du(L) ×Ψq (yk1(1)ε1)ε1 exp (√ −1 ~ u(1)(p(1)− p̃(1)) ) ×Ψq (yk2(2)ε2)ε2 exp (√ −1 ~ u(2)(p(2)− p̃(2)) ) · · · ×Ψq (ykL(L)εL)εL exp (√ −1 ~ u(L)(p(L)− p̃(L)) ) , where p̃(t) is the one in Section 5.2, while we reset ykt(t) = exp (λ(pkt(t) + wkt(t))) . (A.5) Using (A.2), we have F (u(1), p̃(1)) ∼ (2π~)−n(L−1) ∫ dp(1) · · · dp(L− 1)du(2) · · · du(L) × exp (√ −1 ~ L∑ t=1 { 1 2λ2 εtLi2(−ykt(t)εt) + u(t)(p(t)− p̃(t)) }) . (A.6) 26 R.M. Kashaev and T. Nakanishi A.4 Saddle point equations The saddle point equations for (A.6) are obtained in the same manner as in Section 5.3. We use the following formulas, which are obtained from (2.1), (A.3), and (A.5) ∂ ∂pi(t) ( 1 2λ2 εtLi2(−ykt(t)εt) ) = δikt 1 λ log(1 + ykt(t) εt)−1/2, ∂ ∂ui(t) ( 1 2λ2 εtLi2(−ykt(t)εt) ) = − 1 λ log(1 + ykt(t) εt)−bkti(t)/2. (a) Extremum conditions with respect to ui(t) (t = 2, . . . , L). By differentiating the integrand of (A.6) by ui(t), we have − 1 λ log(1 + ykt(t) εt)−bkti(t)/2 + pi(t)− p̃i(t) = 0. (A.7) Combining it with (5.6), we also have eλp̃i(t+1) = { (eλp̃kt (t))−1, i = kt, eλp̃i(t)(eλp̃kt (t))[εtbkti(t)]+(1 + ykt(t) εt)−bkti(t)/2, i 6= kt. (A.8) (b) Extremum conditions with respect to pi(t) (t = 1, . . . , L− 1). By differentiating the integrand of (A.6) by pi(t), we have 1 λ log(1 + ykt(t) εt)−1/2 + ukt(t) − ∑ j∈I [εtbktj(t)]+uj(t+ 1) + ukt(t+ 1) = 0, i = kt, (A.9) ui(t)− ui(t+ 1) = 0, i 6= kt. (A.10) With (2.6), this also implies the following equations for wi(u) = ∑ j∈I bji(t)uj(t). eλwi(t+1) = { (eλwkt (t))−1, i = kt, eλwi(t)(eλwkt (t))[εtbkti(t)]+(1 + ykt(t) εt)−bkti(t)/2, i 6= kt. (A.11) A.5 Solution The (complex) solution of the saddle point equations is constructed in the same manner as in Section 5.4 and given as follows. (i) (y-variables) We have ui(1) as initial data, from which wi(1) is uniquely determined. Temporarily forgetting (A.5), set yi(1) = e2λwi(1), from which yi(t) (t = 2, . . . , L) are determined by the mutation sequence (2.9). (ii) (u-variables) Set ui(t) (t = 2, . . . , L) by (A.10) and (A.9). Then, (A.11) is also satisfied. (iii) (p-variables) Set p̃i(t) by eλp̃i(t) = yi(t) 1/2. This forces the relation p̃i(1) = wi(1), which is guaranteed by the assumption (5.4). Then, pi(t) are determined by (5.5) and (5.6). Since p̃i(t) satisfies (A.8) by definition, (A.7) is also satisfied. A.6 Result The evaluation of the logarithm of the integrand in (A.6) at the solution of the saddle point equations is done in the same manner as in Section 5.5. Using (A.2) and ignoring the common factor, it is given by L∑ t=1 { 1 2 εtLi2(−ykt(t)εt) + λ2 ∑ i∈I ui(t)(pi(t)− p̃i(t)) } . Classical and Quantum Dilogarithm Identities 27 Recall that λ(pi(t)− p̃i(t)) = log(1 + ykt(t) εt)−bkti(t)/2, λwi(t) = 1 2 log yi(t). Then, repeating the calculation in (5.18), we obtain the classical dilogarithm identity (2.13) with complex argument. Taking the limit λ → 1 further, we recover the identity (2.13) with real argument. Acknowledgments We thank Vladimir V. Bazhanov, Ludwig D. Faddeev, Kentaro Nagao, Boris Pioline, and Andrei Zelevinsky for very useful discussions and comments. We especially thank Alexander Yu. Volkov for making his result in [62] available to us prior to the publication. 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[65] Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, 3–65. http://arxiv.org/abs/1006.2706 http://dx.doi.org/10.1016/0550-3213(93)90291-V http://arxiv.org/abs/1002.4884 http://arxiv.org/abs/1103.2922 http://dx.doi.org/10.1215/00277630-1260432 http://arxiv.org/abs/0909.5480 http://arxiv.org/abs/1006.0632 http://dx.doi.org/10.3842/SIGMA.2010.085 http://arxiv.org/abs/1005.4199 http://arxiv.org/abs/1101.3736 http://arxiv.org/abs/1004.0830 http://dx.doi.org/10.1017/S1474748009000176 http://arxiv.org/abs/0804.3214 http://dx.doi.org/10.1063/1.531809 http://dx.doi.org/10.1063/1.531809 http://dx.doi.org/10.1007/s10468-010-9226-6 http://arxiv.org/abs/0904.3291 http://dx.doi.org/10.1007/s00220-005-1342-5 http://arxiv.org/abs/math.QA/0312084 http://arxiv.org/abs/1104.2267 http://dx.doi.org/10.1142/S0129055X00000344 http://dx.doi.org/10.1007/978-3-540-30308-4_1 1 Introduction 1.1 Pentagon relations 1.2 Classical and quantum dilogarithm identities from cluster algebras 2 Classical dilogarithm identities 2.1 Euler and Rogers dilogarithms 2.2 y-variables in cluster algebras 2.3 Tropical y-variables 2.4 Periodicity of y-seeds 2.5 Classical dilogarithm identities 2.6 Example of type A1 2.7 Example of type A2 3 Quantum dilogarithm identities for q(x) 3.1 Quantum dilogarithm 3.2 Quantum y-variables 3.3 Decomposition of mutations 3.4 Quantum dilogarithm identities in tropical form 3.5 Quantum dilogarithm identities in universal form 3.6 Example of type A2 4 Quantum dilogarithm identities for b(z) 4.1 Faddeev's quantum dilogarithm 4.2 Representation of quantum y-variables 4.3 Decomposition of mutations 4.4 Dual operators 4.5 Quantum dilogarithm identities in tropical form 4.6 Quantum dilogarithm identities in local form 4.7 Example of type A2 5 From quantum to classical dilogarithm identities 5.1 Position and momentum bases 5.2 Integral expression 5.3 Saddle point equations 5.4 Solution 5.5 Result A Quantum dilogarithm identities in local form for q(x) and their semiclassical limits A.1 Representation of quantum y-variables A.2 Quantum dilogarithm identities in local form for q(x) A.3 Integral expression A.4 Saddle point equations A.5 Solution A.6 Result References

Classical and Quantum Dilogarithm Identities

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