Symmetry, Integrability and Geometry: Methods and Applications 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 naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.

Actually, there are at least two variants of the quantum dilogarithm. The first one Ψ q (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 − , The second variant of the quantum dilogarithm Φ b (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 Φ b (z) also satisfies properties parallel to the ones for Ψ q (x) [14,13,64,16].
(a) Asymptotic behavior: In the limit b → 0, (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 Li 2 (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 Li 2 (x) differ by logarithms (see (2.3) and (2.4)), and the noncommutativity of U , V and P , Q 'magically' turns Li 2 (x) into L(x). To clarify this phenomenon in a (much) wider situation is the main theme of this paper.

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 ε t L y kt (t) εt 1 + y kt (t) εt = 0. (1.6) A precise account will be given in Section 2.5. Here we only mention that ε 1 , . . . , ε L is a certain sequence of signs called the tropical sign-sequence. The simplest case of the cluster algebra of type A 2 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 Ψ q (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 Ψ q (x). This is the form presented by [39], and it looks as follows Ψ q (Y ε 1 α 1 ) ε 1 · · · Ψ q (Y ε L α L ) ε L = 1. (1.7) A precise account will be given in Section 3.4. The simplest case of the quantum cluster algebra of type A 2 yields the pentagon relation (1.3).
2) Identities in universal form for Ψ q (x). This is the form presented by [63,62], and it looks as follows (1.8) A precise account will be given in Section 3.5. The simplest case of type A 2 yields the new variation of the pentagon relation for Ψ q (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 Φ b (z). This is the counterpart of the form (1.7), and it looks as follows (1.9) A precise account will be given in Section 4.5. The simplest case of type A 2 yields the pentagon relation (1.5). 4) Identities in local form for Φ b (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 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 rigorousness. 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 Ψ q (x) with tropical sign-sequence. This is the counterpart of the form (1.10), and it looks as follows 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 Φ b (z) from the quantum-mechanical point of view. For the completeness, we also present it in Appendix A.
In summary, our result establishes the following scheme periods of quantum cluster algebras / / quantum dilogarithm identities semiclassical limit periods of classical cluster algebras The organization of the paper is the following. In Section 2 we present the classical dilogarithm identities obtained from periods of cluster algebras. In Section 3 we present the quantum dilogarithm identities for the quantum dilogarithm Ψ q (x). In Section 4 we present the quantum dilogarithm identities for the Faddeev's quantum dilogarithm Φ b (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 Ψ q (x). Then, we derive the classical dilogarithm identities from them in the semiclassical limit.

Classical dilogarithm identities
In this section we present the classical dilogarithm identities obtained from periods of cluster algebras following [52].

Euler and Rogers dilogarithms
Let Li 2 (x) and L(x) be the Euler and Rogers dilogarithm functions, respectively [47], Two functions are related as follows The function L(x) satisfies the property (1.1) and also the following ones

y-variables in cluster algebras
In this subsection we recall some definitions and properties of the cluster algebras with coefficients [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'.
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 = (b ij ) i,j∈I and an I-tuple of commutative variables y = (y i ) i∈I . Let P univ (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 ∈ P univ (y). For each k ∈ I, we define another pair (B ′′ , y ′′ ) = µ k (B ′ , y ′ ) of a skewsymmetric matrix B ′′ = (b ′′ ij ) i,j∈I and an I-tuple y ′′ = (y ′′ i ) i∈I with y ′′ i ∈ P univ (y), called the mutation of (B ′ , y ′ ) at k, by the following rule: (i) Mutation of matrix: otherwise.
(2.6) (ii) Exchange relation of y-variables: 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.

Tropical y-variables
Let P trop (y) be the tropical semifield of y = (y i ) i∈I , which is the Abelian multiplicative group freely generated by y endowed with the addition ⊕ i∈I There is a canonical surjective semifield homomorphism π T (the tropical evaluation) from P univ (y) to P trop (y) defined by π T (y i ) = y i and π T (α) = 1 (α ∈ Q + ). For any y-variable y ′ i of a y-seed (B ′ , y ′ ) of (B, y), let us write [y ′ i ] := π T (y ′ i ) for simplicity. We call [y ′ i ]'s the tropical 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).
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: (2.8)
Remarkably, the periodicity of y-seeds reduces to the periodicity of tropical y-variables, which is much simpler. ForĨ ⊃ I and a skew-symmetric matrixB = (b ij ) i,j∈Ĩ , we say thatB is anĨ-extension of B ifb ij = b ij 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 letĨ = I ⊔ I ′ . Define the skew-symmetric matrix Then,B is anĨ-extension of B; furthermore,B is nondegenerate. The matrixB is called the principal extension of B.
In Proposition 2.6 the periodicity of the 'external' variablesỹ i (i ∈Ĩ \ I) is nontrivial.
Two identities (2.11) and (2.12) are equivalent due to (2.5 We introduce the sign-sequence (ε 1 , . . . , ε L ) so that ε t is the tropical sign of y kt (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. (2.13)

Example of type A 1
Consider the simplest case, I = {1} and Let (k 1 , k 2 ) = (1, 1), and consider the sequence of mutations of y-seeds of (B, y), Then, Thus, (k 1 , k 2 ) is a ν-period with ν = id, which is nothing but the involution property of the mutation. Also The classical dilogarithm identity (2.13) is which is trivial.

Quantum dilogarithm identities for Ψ q (x)
In this section we present the quantum dilogarithm identities for Ψ q (x). The content heavily relies on [18,20,39].

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 commutative. 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.
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 = (b ij ) i,j∈I and an I-tuple of Accordingly, let T(B, Y) be the associated quantum torus, which is the Q(q)-algebra generated by the noncommutative variables Y α (α ∈ Z I ) with the relations Following [39], let A(B, Y) be the associated quantum affine space, which is the Q(q)-subal- is given by the following rule [18,20]: Exchange relation of quantum y-variables 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 ).

Decomposition of mutations
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 τ k,ε for each ε = ±1 by (3.5) The dependence of the map τ k,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 are given by τ k,−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 ′ ), and similarly, By combining (3.5)-(3.8), we have the following intrinsic description of the exchange relation (3.4).
Proposition 3.1 ( [18,39]). We have the equality and either side of (3.9) coincides with the right hand side of the exchange relation 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 ), Then, for any sign-sequence (ε 1 , ε 2 ), we have (3.10)

Quantum dilogarithm identities in tropical form
For any I-sequence (k 1 , k 2 , . . . , k L ), set (B(1), Y (1)) := (B, Y ), and consider the sequence of mutations of quantum y-seeds of (B, Y ), We say that an I-sequence (k 1 , k 2 , . . . , k L ) is a ν-period of (B, Y ) if the following condition holds for the sequence (3.11) The following theorem, essentially due to [20], tells that the periodicities of quantum y-seeds and (classical) y-seeds coincide. 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 [ Let ν also denote this isomorphism by abusing the notation. For any sign-sequence (ε 1 , . . . , ε k ), the periodicity for (3.11) is expressed as follows [39]. (3.13) To extract the identity involving only the quantum dilogarithm Ψ q (y), we have to set (ε 1 , . . . , ε L ) 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.

Quantum dilogarithm identities in universal form
Let us rewrite the identity (3.14) with genuine 'nontropical', or universal, quantum y-variables Y kt (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) Proof . Let us prove (3.17) for t = 3, for example. By setting i = k 3 in (3.10) and repeating the argument in the proof of Theorem 3.5, we have Then, by extending the map Ad(Ψ q (Y εtαt ) εt ) toT(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).
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.

Example of type A 2
We continue to use the data in Section 2.7. For the initial quantum y-seed (B, Y ), we have Consider the sequence of mutations of quantum y-seeds of (B, Y ): Then, The quantum dilogarithm identity in tropical form (3.14) is
Remark 3.8. The relation (3.21) should be compared with the quantum pentagon relation at N th roots of unity [15], where N th 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).

Quantum dilogarithm identities for Φ b (z)
In this section we present the quantum dilogarithm identities for Φ b (x). The content heavily relies on [20,19].

Faddeev's quantum dilogarithm
Let b be a complex number with nonzero real part. Set Following [13,14], define the Faddeev's quantum dilogarithm Φ b (z) for z ∈ C in the strip |Im z| < |Im c b | by 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).

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 To any quantum y-seed The algebra of operatorsû ′ andp ′ has a natural representation on the Hilbert space L 2 (R I ): (4.9) Using Dirac's notation f (u ′ ) = u ′ |f , we have formally The set of generalized vectors { u ′ |} u ′ ∈R I will be called the local coordinates of (B ′ , Y ′ ). Definê The following relations hold

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 ′′ ) = µ k (B ′ , Y ′ ).
(a) Monomial part. For each ε = ±1, consider the following map ρ k,ε : R I → R I , (4.13) Let ρ * k,ε be the induced map in the space of functions L 2 (R I ), or, formally, by which we relate the local coordinates of (B ′ , Y ′ ) and (B ′′ , Y ′′ ).
For any linear operatorÔ acting on L 2 (R I ), let In other words, it is defined by the commutative diagram Then, we have (4.14) where (4.17) follows from (4.16) and (4.15). It follows from (4.17) that which coincides with (3.5).
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 ′ ji u ′ j , the induced transformation 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 (b) Automorphism part. We set Then, we havê Thus, thanks to the recurrence relation (4.4), one obtains, for each ε = ±1, by an analogous calculation to (3.7) and (3.8).
In summary, we have a parallel statement to Proposition 3.1. 20,19]). We have the equality (4.19) and either side of (4.19) coincides with the right hand side of the exchange relation (3.4) with Y ′ i replaced withŶ ′ i .

Dual operators
Following [13] and [20], we define the operatorsẐ ′ i which are 'dual' toŶ ′ i in the sense of the first equality of (4.3). In the situation in (4.10), we definê Then, the following relations hold Remark 4.3. The duality betweenŶ ′ i andẐ ′ 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 where γ is an arbitrary nonzero real number and∂ ′ i satisfy [∂ ′ i ,∂ ′ j ] = 0 and [∂ ′ i ,û ′ j ] = δ ij . The following relations hold irrespective of γ Now the duality b ↔ b −1 is manifest. Further setting γ = 2πb, we haveD ′ i =D ′ i and we recover the operatorsŶ ′ i andẐ ′ 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 (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 withẐ ′ i and q ∨ , respectively.

Quantum dilogarithm identities in tropical form
Suppose that (k 1 , k 2 , . . . , k L ) is a ν-period of (B, Y ) as in Section 3.4. The identities parallel to (3.14) are available for Φ b (z) directly from (3.14).

Theorem 4.5 (Quantum dilogarithm identities in tropical form).
Under the same assumption of Theorem 3.5 (in particular, (ε 1 , . . . , ε L ) is the tropical sign-sequence of the mutation sequence), the following identity holds.
where α tD = i∈I α i (t)D i andD i 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 b 2 ≥ 0 without losing generality. By the continuity of Φ b with respect to b, it is enough to show the claim for Im b 2 > 0. Then, by (4.6), we have Then, thanks to the commutativity (4.20), the relation (4.22) factorizes into two identities where (4.23) is a specialization of (3.14), while (4.24) is equivalent to which is another specialization of (3.14).
Let us recall the result of [20,Theorem 5.4]. Suppose that b is a nonzero real number. Note that, Φ b (ε tD ′ 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 = (ε 1 , . . . , ε L ) This is equivalent to saying that the operator commutes withŶ i (1) andẐ i (1) for any i ∈ I. It was shown in [20] that, when b 2 is irrational, suchÔ 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 b 2 as well by continuity. SinceÔ 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.
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.
Proof . By the duality in Remark 4.1, the periodicity of tropical y-variables (3.15) is equivalent to 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 identities (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).

Example of type A 2
We continue to use the data in Sections 2.7 and 3.6.

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.

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 = πb 2 in (4.7). The asymptotic property (1.4) is written as 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 L 2 (R I ) 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) ∈ R I } and {|p(t) | p(t) ∈ R I } be the standard position and the momentum bases of L 2 (R I ) t , respectively. They satisfy the following properties, where n = |I|, where u(t)p(t) := i∈I u i (t)p i (t), In particular, we have LetÔ be the composition of the operators in the left hand side of (4.26), namely, where (ε 1 , . . . , ε L ) is the tropical sign-sequence. Choose any position eigenvector |u(1) . Then, set the momentum eigenvector |p(1) such that its eigenvalues are given bỹ where the notationp (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 Below we show that the leading term of log F (u(1),p(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).
Step 2. Find a solution of the saddle point equations.
Step 3. Evaluate the integrand at the solution.

Solution
Let us summarize the relevant variables and their relations schematicallỹ 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 Note thatp i (t) = p i (t) if i = k t by (5.10), therefore, it agrees with the previous definition (5.7). Then, multiply two identities (5.11) and (5.15), we have This is nothing but (2.7). Furthermore, (5.11) and (5.15) suggest that Having this observation in mind, let us describe the construction of the solution more clearly. (i) (y-variables) We have u i (1) as initial data, from which w i (1) is uniquely determined. Temporarily forgetting (5.7), set y i (1) = e 2w i (1) , from which y i (t) (t = 2, . . . , L) are determined by the mutation sequence (2.9).
(iv) (compatibility) The only thing to be checked is (5.7). Since p kt (t) =p kt (t) by (5.10), it is enough to show 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 e u i (t) is regarded as the square half of the x-variable x i (t).

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 Recall that by (5.10) and (5.16). Then, the second term of (5.17) is rewritten as ε t L y kt (t) εt 1 + y kt (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 Ψ q (x) and their semiclassical limits In this section we present the quantum dilogarithm identities in local form for Ψ q (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 Im λ 2 > 0.
We reset 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 Ψ q (x) is convergent only for |q| < 1, while Φ b (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 Ψ q (x) which persists throughout the section. The asymptotic property (1.4) is written as Because of λ, the argument x of the dilogarithms Li 2 (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 Im λ 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 Li 2 (x) or [0, 1] for L(x).
We resetŶ ′ i in (4.10) aŝ The relations in (4.11) still hold with q in (A.1), and we have a representation of T(B, Y) of (4.12).
Let (B ′ , Y ′ ) and (B ′′ , Y ′′ ) be a pair of quantum y-seeds of (B, Y ) such that (B ′′ , Y ′′ ) = µ k (B ′ , Y ′ ). Let ρ k,ε be the map in (4.13). Then, repeating the argument in Section 4.3, we obtain A.2 Quantum dilogarithm identities in local form for Ψ q (x) Under the same assumption and notation for Theorem 4.7, we obtain the counterpart of Theorem 4.7 for Ψ q (x) by repeating its proof.
Theorem A.1. For the tropical sign-sequence ε = (ε 1 , . . . , ε L ), the following identity holds Below we show that the leading term of log F (u(1),p(1)) in the limit → 0 yields the left hand side of (2.13) up to a multiplicative constant.
(A.8) (b) Extremum conditions with respect to p i (t) (t = 1, . . . , L − 1). By differentiating the integrand of (A.6) by p i (t), we have With (2.6), this also implies the following equations for w i (u) = j∈I b ji (t)u j (t).

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.

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 ε t Li 2 (−y kt (t) εt ) + λ 2 i∈I u i (t)(p i (t) −p i (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.