Cluster ensembles, quantization and the dilogarithm II: The intertwiner.

V.V.Fock^{1} and A.B.Goncharov^{2}

1 ITEP, B.Cheremushkinskaya 25, 117259, Moscow, Russia, fock@math.brown.edu 2 Department of Math, Brown University, Providence RI 02906, USA,

  sasha@math.brown.edu

Abstract. This paper is the second part of our preprint “Cluster ensembles, quantization and the dilogarithm” [FG2]. Its main result is a construction, by means of the quantum dilogarithm, of certain intertwiner operators, which play the crucial role in the quantization of the cluster X -varieties and construction of the corresponding canonical representation.

Cluster ensemble (loc. cit.) is a pair (A, X) of schemes over ℤ, called cluster A- and X -varieties, related by a map p : A ⟶ X. The ring of regular functions on the cluster A-variety is the upper cluster algebra [BFZ].

Cluster A- and X -varieties are glued from families of coordinatized split algebraic tori by means of certain subtraction free rational transformations. In particular it makes sense to consider the spaces of their positive real points, denoted by A^{+} and X^{+}.

The cluster X -variety has a Poisson structure, given in any cluster coordinate system {X_{i}} by

{X_{i}, X_{j}} = ε̂_{ij}X_{i}X_{j}, ε̂_{ij} ∈ ℤ.

The Poisson tensor ε̂_{ij} depends on the choice of coordinate system. There is a canonical non-commutative deformation X_{q} of the cluster X -variety in the direction of this Poisson structure ([FG2]).

The gluing procedure underlying the definition of a cluster variety can be understood as a functor from a certain groupoid, called the cluster modular groupid G, to a category of commutative algebras.

In Section 3 we suggest a ∗-quantization of the cluster X -variety, understood as a functor from the gropoid G to the category of non-commutative topological ∗-algebras. More precisely, the coordinate systems on cluster varieties are parametrized by the objects i of the groupoid G, called seeds. To each seed i we assign two coordinatized tori, A_{i} and X_{i}. The algebra of the smooth functions on the latter admits a canonical ℏ-deformation, given by a topological Heisenberg ∗-algebra H_{i}^{ℏ}. So to define a functor we need to relate these algebras for different seeds. We write the formulas relating the generators of the algebras H_{i}^{ℏ}, but do not specify the category of topological ∗-algebras. As a result, the ∗-quantization of the cluster X -variety serves only as a motivation, and we state in Sections 3 Claims instead of Theorems when those unspecified topological algebras enter the formulations. Hopefully there will be a precise version of Section 3. However the rest of the paper does not depend on that, while motivations given in Section 3 clarify what we do next.

In Section 4 we proceed to a construction of the canonical unitary projective representation of the modular gropoid. It is realized in the Hilbert space L^{2}(A^{+}) assigned to the set of positive real points of the cluster A-variety.

For each seed i there is a Hilbert space L^{2}(A_{i}^{+}) (which is canonically identified with L^{2}(A^{+})). In Section 4.1 the Heisenberg ∗-algebra H_{i}^{ℏ} is represented in L^{2}(A_{i}^{+}). In fact a bigger algebra – the chiral double H_{i}^{ℏ} ⊗ H_{i^{o}}^{ℏ} of the Heisenberg ∗-algebra – acts on the same Hilbert space.

The morphisms in the groupoid G are defined as compositions of certain elementary ones, called mutations and symmetries. Given a mutation i → i′ we construct a unitary operator

K_{i→i′} : L^{2}(A_{i}^{+}) ⟶ L^{2}(A_{i′}^{+}).

It intertwines the actions of the Heisenberg ∗-algebras related to the seeds i and i′^{3}. (Similar itertwiners for symmetries are rather tautological). This construction is the main result of the paper. The operator K_{i→i′} is characterized by its intertwining property uniquely up to a constant.

Certain compositions of mutations and symmetries are identity morphisms in the groupoid G. So to get a representation of the modular groupoid we have to show that the corresponding compositions of the intertiners K_{i→i′} are multiples of the identity operators. This is a rather difficult problem. It is solved in [FG3], where we give another construction of the intertwiner and introduce the geometric object reflecting its properties, the cluster double. Alltogether the intertwiners give rise to a unitary projective reprersentation of the cluster modular groupoid.

The paper is organized as follows: the essential for us properties of the quantum logarithm and dilogarithm are collected, without proofs in Section 5. The proofs and more of the properties of these functions can be found in Section 4 of [FG3]. In Section 2.1 we recall, for the convinience of the reader, basic definitions/facts about cluster ensembles. Claim 2.2 delivers a quantization of the space of real positive points of the cluster X -variety. The main results of this paper is Theorem 10 providing an explicit formula for the intertwiner K_{i→i′}.

3The precise meaning we put into “intertwining” is clear from the computation carried out in the proof: we deal with the genrators of the Heisenbrg algebra only, and thus are not concerned with the nature of the topological completion, which was left unspecified in Section 3. For a different approach see [FG3].

Acknowledgments. V.F. was supported by the grants CRDF 2622; 2660. A.G. was supported by the NSF grants DMS-0099390 and DMS-0400449.

We would like to thank the referee for illuminating remarks regarding the content of the paper.

1 Cluster ensembles

1.1 Basic definitions

A seed i is a triple (I, ε, d), where I is a finite set, ε is a matrix ε_{ij}, i, j ∈ I, with ε_{ij} ∈ ℤ, and d = {d_{i}}, i ∈ I, are positive integers, such that the matrix ε̂_{ij} := ε_{ij}d_{j}^{−1} is skew-symmetric.

For a seed i we assign a torus X_{i} = (𝔾_{m})^{I} with the coordinates {X_{i}|i ∈ I} on the factors. and a Poisson structure given by

{X_{i}, X_{j}} = ε̂_{ij}X_{i}X_{j}.  (1)

Let i = (I, ε, d) and i′ = (I′, ε′, d′) be two seeds, and k ∈ I. A mutation in the direction k ∈ I is an isomorphism μ_{k} : I → I′ such that d′_{μ_{k}(i)} = d_{i}, and

ε′_{μ_{k}(i)μ_{k}(j)} = \{ \begin{matrix}−ε_{ij} & if i = k or j = k, \\ ε_{ij} & if ε_{ik}ε_{kj} ≤ 0, \\ ε_{ij} + |ε_{ik}|ε_{kj} & if ε_{ik}ε_{kj} > 0. \end{matrix}  (2)

A symmetry of a seed i = (I, ε, d) is an automorphism σ of the set I preserving the matrix ε and the numbers d_{i}. Symmetries and mutations induce rational maps between the corresponding seed X -tori, denoted by the same symbols μ_{k} and σ and given by the formulae σ^{∗}X_{σ(i)} = X_{i} and

μ_{k(i)}^{∗}X_{μ_{k}(i)} = \{ \begin{matrix}X_{k}^{−1} & if & i = k, \\ X_{i}(1 + X_{k}^{−sgn(ε_{ik})})^{−ε_{ik}} & if & i ≠ k. \end{matrix}  (3)

A seed cluster transformation is a composition of symmetries and mutations. Two seeds are equivalent if related by a cluster transformation. The equivalence class of a seed i is denoted by |i|. A seed cluster transformation induces a rational map between the two seed X -tori, called a cluster transformation map.

A cluster X -variety X_{|i|} is a scheme over ℤ obtained by gluing the seed X -tori for the seeds equivalent to a given seed i via the cluster transformation maps, and then taking the affine closure. Every seed provides a cluster X - variety with a rational coordinate system. Its coordinates are called cluster coordinates. Cluster transformation maps preserve the Poisson structure. Thus a cluster X -variety has a canonical Poisson structure.

The cluster A-varieties. Given a seed i, define a seed A-torus A_{i} := (𝔾_{m})^{I} with the standard coordinates {A_{i}|i ∈ I} on the factors. Symmetries and mutations give rise to birational maps between the seed A-tori, given by σ^{∗}A_{σ(i)} = A_{i} and

μ_{k(i)}^{∗}A_{μ_{k}(i)} = \{ \begin{matrix}A_{i} & if & i ≠ k, \\ A_{k}^{−1} (∏_{i|ε_{ki}>0} A_{i}^{ε_{ki}} + ∏_{i|ε_{ki}<0} A_{i}^{−ε_{ki}}) & if & i = k. \end{matrix} (4)

The cluster A-variety A_{|i|} is a scheme over ℤ obtained by gluing all seed A-tori for the seeds equivalent to a given seed i using the above birational isomorphisms, and taking the affine closure.

There is a map p : A ⟶ X , given in every cluster coordinate system by p^{∗}X_{k} = ∏_{I∈I} A_{i}^{ε_{ki}}.

Cluster A- and X -varieties have canonical positive atlases, so it makes sense to consider the sets of their real positive points, denoted A^{+} and X^{+}.

The cluster modular groupoid. Seed cluster transformations inducing the same map of the seed A-tori are called trivial seed cluster transformations. The cluster modular groupoid G_{|i|} is a groupoid whose objects are seeds equivalent to a given seed i, and Hom(i, i′) is the set of all seed cluster transformations from i to i′ modulo the trivial ones. Given a seed i, the cluster mapping class group Γ_{i} is the automorphism group of the object i of G_{|i|}. The group Γ_{i} acts by automorphisms of the cluster A-variety.

The quantum space X_{q}. It is a canonical non-commutative q-deformation of the cluster X -variety defined in Section 3 of [FG2].

We start from the seed quantum torus algebra T_{i}^{q}, defined as an associative ∗-algebra with generators X_{i}^{±1}, i ∈ I and q^{±1} and relations

q^{−ε̂_{ij}}X_{i}X_{j} = q^{−ε̂_{ji}}X_{j}X_{i}, ∗X_{i} = X_{i}, ∗q = q^{−1}.

Let QTor^{∗} be a category whose objects are quantum torus algebras and morphisms are ∗-homomorphisms of their fraction fields. The quantum space X_{q} is understood as a contravariant functor

η^{q} : The modular groupoid G_{|i|} ⟶ QTor^{∗}.

It assigns to a seed i the quantum torus ∗-algebra T_{i}^{q}, and to a mutation i ⟶ i′ a map of the fraction fields Frac(T_{i′}^{q}) ⟶ Frac(T_{i}^{q}), given by a qdeformation of formulas (3)^{4}: Set q_{k} := q^{1/h_{k}},

(μ_{k}^{q})^{∗}(X_{i}) := \{ \begin{matrix}X_{k}^{−1} & if & i = k, \\ X_{i} (∏_{a=1}^{|ε_{ik}|}(1 + q_{k}^{2a−1}X_{k}^{−sgn(ε_{ik})}))^{−ε_{ik}} & if & i ≠ k. \end{matrix}

4See also a more transparent and less computational definition given later on in Section 3 of [FG3].

One uses Theorem 7.2 in loc. cit. to prove that η^{q} sends trivial seed cluster transformations to the identity maps.

The chiral dual to a seed i = (I, ε, d) is a seed i^{o} := (I, −ε, d). Mutations commute with the chiral duality on seeds. Therefore a cluster X -variety X (respectively A-variety A), gives rise to the chiral dual cluster X -variety (respectively A-variety) denoted by X^{o} (respectively A^{0}). They are related, see Lemmas 1 - 2 .

The Langlands dual to a seed i = (I, ε, d) is the seed i^{∨} = (I, ε^{∨}, d^{∨}), where d_{i}^{∨} := d_{i}^{−1} and

ε_{ij}^{∨} = −ε_{ji}^{∨} := d̂_{i}^{−1}ε_{ij}d̂_{j},  d̂_{i} := d_{i}^{−1}.  (5)

The Langlands duality on seeds commutes with mutations. Therefore it gives rise to the Langlands dual cluster A-, and X -varieties, denoted A^{∨} and X^{∨}.

Below we skip the subscript |i| encoding the corresponding cluster ensemble whenever possible.

1.2 Connections between quantum X -varieties

There are three ways to alter the space X_{|i|,q}:

(i) change q to q^{−1},

(ii) change i to i^{o},

(iii) change the quantum space X_{|i|,q} to the opposite quantum space X_{|i|,q}^{opp}. (In (iii) we change every quantum torus from which we glue the space to the opposite one).

The following lemma tells that the resulting three quantum spaces are canonically isomorphic:

Lemma 1. a) There is a canonical isomorphism of quantum spaces

α_{X}^{q} : X_{|i|,q} ⟶ X_{|i|,q^{−1}}^{opp}, (α_{X}^{q})^{∗} : X_{i} ⟼ X_{i}.

(Given in on the generators of any cluster coordinate system by X_{i} ⟼ X_{i}). b) There is a canonical isomorphism of quantum spaces

i_{X}^{q} : X_{|i|,q} ⟶ X_{|i^{o}|,q^{−1}}, (i_{X}^{q})^{∗} : X_{i}^{o} ⟼ X_{i}^{−1}.

(Given in any cluster coordinate system by X_{i}^{o} ⟼ X_{i}^{−1}, where X_{i}^{o} are the generators of X_{i^{o},q^{−1}}).

c) There is a canonical isomorphism of quantum spaces

β_{X}^{q} := α_{X}^{q} ∘ i_{X}^{q} : X_{|i^{o}|,q} ⟶ X_{|i|,q}^{opp}, X_{i} ⟼ X_{i}^{o}^{−1}.

Proof. Apparently each of the three maps is an isomorphism of the corresponding seed quantum tori algebras. For example, in the case b) we have

i_{X}^{q} ((q^{−1})^{−ε_{ij}^{o}}X_{i}^{o}X_{j}^{o}) = q^{−ε̂_{ij}}X_{i}^{−1}X_{j}^{−1}.

So we need to check that they commute with the mutations.

a) Let us assume first that ε_{ik} = a < 0. The claim results from the fact that the following two compositions are equal (observe that α^{∗} is an antiautomorphism):

X_{i} ⟼^{μ_{k}^{∗}} X_{i} ∏_{b=1}^{a}(1 + q_{k}^{2b−1}X_{k}) ⟼^{α^{∗}} ∏_{b=1}^{a}(1 + q_{k}^{2b−1}X_{k})X_{i};

X_{i} ⟼^{α^{∗}} X_{i} ⟼^{(μ_{k}^{o})^{∗}} X_{i} ∏_{b=1}^{a}(1 + q_{k}^{−(2b−1)}X_{k}).

The computation in the case ε_{ik} > 0 is similar.

b) To check that i_{X}^{q} ∘ μ_{k}^{∗} = μ_{k}^{∗} ∘ i_{X}^{q} we calculate each of the maps on the generator X_{i}. Let us assume ε_{ik} = −a < 0. Then ε_{ik}^{o} = a, and one has

i_{X}^{q} ∘ μ_{k}^{∗}(X_{i}^{o}′) = X_{i}^{−1} ∏_{b=1}^{a}(1 + q^{2b−1}X_{k}^{−1}),

μ_{k}^{∗} ∘ i_{X}^{q} (X_{i}^{o}′) = (X_{i} ∏_{b=1}^{a}(1 + q^{−(2b−1)}X_{k}^{−1})^{−1})^{−1} =

= ∏_{b=1}^{a}(1 + q^{−(2b−1)}X_{k}^{−1})X_{i}^{−1} = X_{i}^{−1} ∏_{b=1}^{a}(1 + q^{2b−1}X_{k}^{−1}).

The case ε_{ik} > 0 is similar. One can deduce it to the case ε_{ik} < 0 since μ_{k} ∘ μ_{k} = Id, and ε′_{ik} = −ε_{ik}. The part b) is proved.

c) Follows from a) and b). The lemma is proved.

Lemma 2. The cluster ensembles related to the seeds i and i^{o} are canonically isomorphic as pairs of varieties. The isomorphism is provided by the following maps:

Id : A_{|i|} ⟶ A_{i^{o}}; i_{X} : X_{|i|} ⟶ X_{|i^{o}|}.

Proof. In a given cluster coordinate system our maps are obviously isomorphisms. The compatibility with X -cluster transformations is the part b) of Lemma 1; for the A-cluster transformations we have

A_{k}^{o}A^{o}′_{k} = ∏_{ε_{ki}^{o}>0}(A_{i}^{o})^{ε_{ki}^{o}} + ∏_{ε_{ki}^{o}<0}(A_{i}^{o})^{−ε_{ki}^{o}} = ∏_{ε_{ki}<0} A_{i}^{−ε_{ki}} + ∏_{ε_{ki}>0} A_{i}^{ε_{ki}} = A_{k}A′_{k}

Compatibility with the projection p is clear. The lemma is proved.

2 Motivation: ∗-quantization of cluster X -varieties

2.1 ∗-quantization of the space X^{+} via the quantum logarithm

Let {X_{i}} be coordinates on the cluster X -variety corresponding to a seed i. Since by definition the functions X_{i} are strictly positive at the points of X^{+}, we can introduce the logarithmic coordinates x_{i} := log X_{i} on X^{+}. For every seed i they provide an isomorphism

β_{i} : X_{i}^{+} ⟶^{∼} ℝ^{I}; t ⟼ {x_{i}(t)}.

For a mutation μ_{k} : i → i′ there is a gluing map

β_{i→i′} : X_{i}^{+} ⟶ X_{i′}^{+}, β_{i→i′}(x′_{i}) = \{ \begin{matrix}x_{i} − ε_{ik} log(1 + e^{−sgn(ε_{ik})x_{k}}) & i ≠ k, \\ −x_{k} & i = k \end{matrix}

(6)

To prepare the soil for quantization, let us look at this from a different point of view. Let Com^{∗} be the category of commutative topological ∗-algebras over ℂ. Recall the cluster modular groupoid G_{|i|}. There is a contravariant functor

β : G_{|i|} ⟶ Com^{∗}.

Namely, we assign to a seed i a commutative topological ∗-algebra S(X_{i}^{+}) of smooth complex valued functions in X_{i}^{+} with ∗f := \overline{f}, and to a mutation i → i′ a homomorphism β_{i→i′}^{∗} : S(X_{i′}^{+}) ⟶ S(X_{i}^{+}).

Let C be a category whose morphisms are ℂ-vector spaces. Projectivisation P C of the category C as a new category with the same objects as C, and morphisms given by Hom_{P C}(C_{1}, C_{2}) := Hom_{C}(C_{1}, C_{2})/U(1), where U(1) is the multiplicative group of complex numbers with absolute value 1. A projective functor F : G → C is a functor from G to P C.

Let C^{∗} be the category of topological ∗-algebras. Two functors F_{1}, F_{2} : C ⟶ C^{∗} essentially coincide if there exists a third functor F and natural transformations F_{1} → F, F_{2} → F providing for every object C dense inclusions F_{1}(C) ↪ F (C), F_{1}(C) ↪ F (C).

Definition 3. A quantization of the space X_{|i|}^{+} is a family of contravariant projective functors

κ_{|i|}^{ℏ} : G_{|i|} ⟶ C^{∗}

depending smoothly on a real parameter ℏ, related to the original Poisson manifold X_{|i|}^{+} as follows:

i) The limit κ_{|i|} := lim_{ℏ→0} κ_{|i|}^{ℏ} exists and essentially coincides with the functor β defining X_{|i|}^{+}.

ii) The Poisson bracket given by lim_{ℏ→0}[f_{1}, f_{2}]/ℏ is defined and coincides with the one on X_{|i|}^{+}.

Let us define a quantization functor κ^{ℏ} = κ_{|i|}^{ℏ}. We assign to every seed i the Heisenberg ∗–algebra H_{i}^{ℏ}. It is a topological ∗–algebra over ℂ generated by the elements x_{i} such that

[x_{j}, x_{k}] = 2πiℏε̂_{jk}; x_{j}^{∗} = x_{j}; q = e^{πiℏ}.

Further, let us assign to mutation μ_{k} : i → i′ a homomorphism of topological ∗-algebras

κ^{ℏ}(μ_{k}) : H_{i′}^{ℏ} ⟶ H_{i}^{ℏ}.  (7)

We employ the quantum logarithm ϕ^{ℏ}(z), see (22). Denote by x′_{i} the generators of H_{i′}^{ℏ}. Set

ℏ_{k} := d̂_{k}ℏ;  κ^{ℏ}(μ_{k}) : x′_{i} ⟼ \{ \begin{matrix}x_{i} − ε_{ik}ϕ^{ℏ_{k}}(−sgn(ε_{ik})x_{k}) & if k ≠ i, \\ −x_{i} & if k = i. \end{matrix} (8)

Claim. a) Formulas (8) provide a morphism of ∗–algebras κ_{|i|}^{ℏ}(μ_{k}) : H_{i′}^{ℏ} ⟶ H_{i}^{ℏ}.

b) The collection of ∗-algebras {H_{i}} and morphisms {κ_{|i|}^{ℏ}(μ_{k})} provide a quantization functor

κ_{|i|}^{ℏ} : G_{|i|} ⟶ P C^{∗}.

c) Let ℏ^{∨} := 1/ℏ. Then there are isomorphisms

H_{i}^{ℏ} ⟶ H_{i^{∨}}^{ℏ^{∨}},  x_{i} ⟼ x_{i}^{∨} := \frac{x_{i}}{d̂_{i}ℏ};  (9)

They give rise to a natural transformation of functors κ_{|i|}^{ℏ} ⟶ κ_{|i^{∨}|}^{ℏ^{∨}}.

Justification. a) Property A3 of the function ϕ^{ℏ}(x), see Section 4, guarantees that the morphism κ^{ℏ}(μ_{k}) preserves the real structure. It follows from Property A1 that when ℏ → 0 the limit of the quantum formula (8) exists and coincides with the mutation formula (6) for the logarithmic coordinates x_{i}.

b) To check that we have a functor one needs to check first that mutation formulas are compatible with the transformations κ^{ℏ}(μ_{k}). This is a straitforward calculation using Property A5. Then one has to check that the defining relations for the groupoid G_{|i|} are mapped to zero. Here we need the results of Sections 3.2-3.3 of [FG2] and the following well known lemma:

Lemma 4. Suppose that A, B are selfadjoint operators, [A, B] = −λ is a scalar, and f(z) is a continuous function with primitive F (z). Then

e^{A+f(B)} = e^{A}exp \{\frac{1}{λ} ∫_{B}^{B+λ} f(z)dz\} := e^{A}exp (\frac{F (B + λ) − F (B)}{λ}) .

c) Thanks to formula (5) the map x_{i} ⟼ x_{i}^{∨} is an ∗–algebra homomorphism:

[x_{i}^{∨}, x_{j}^{∨}] = \frac{[x_{i}, x_{j}]}{d̂_{i}d̂_{j}ℏ^{2}} = 2πiℏ^{∨}ε̂_{ij}/d̂_{i}d̂_{j} =^{(5)} 2πiℏ^{∨}ε_{ij}^{∨}.

To verify that it commutes with mutation homomorphisms we use Properties A2 and A4 of the function ϕ^{ℏ}(x), observing that

\frac{x_{i} + |ε_{ik}|ϕ^{ℏ_{k}}(x_{k})}{d̂_{i}ℏ} = \frac{x_{i}}{d̂_{i}ℏ} + \frac{|d̂_{i}^{−1}ε_{ik}d̂_{k}|ϕ^{d̂_{k}ℏ}(x_{k})}{d̂_{k}ℏ} =^{(5)+A4} x_{i}^{∨} + |ε_{ik}^{∨}|ϕ^{ℏ_{k}^{∨}}(x_{k}^{∨}).

2.2 Modular double of a cluster X -variety and ∗-quantization of the space X^{+}

Set

q := e^{πiℏ}, q^{∨} := e^{πi/ℏ}, ℏ ∈ ℝ.

Definition 5. The modular double X_{|i|,q} × X_{|i^{∨}|,q^{∨}} of a quantum cluster X - variety X_{|i|,q} is a contravariant functor

η_{|i|}^{q} ⊗ η_{|i^{∨}|}^{q^{∨}} : G_{|i|} ⟶ QTor^{∗}.

So we assign to a seed i a quantum torus algebra T_{i}^{q} ⊗ T_{i^{∨}}^{q^{∨}}, and to a mutation μ_{k} : i → i′ a positive ∗-homomorphism of the fraction fields of the quantum torus algebras

η_{|i|}^{q}(μ_{k}) ⊗ η_{|i^{∨}|}^{q^{∨}}(μ_{k}) : 𝕋_{i′}^{q} ⊗ 𝕋_{i′^{∨}}^{q^{∨}} ⟶ 𝕋_{i}^{q} ⊗ 𝕋_{i^{∨}}^{q^{∨}}, 𝕋 := Frac(T).

We want to relate the modular double X_{|i|,q}×X_{|i^{∨}|,q^{∨}} with the quantization of the space X_{|i|}^{+}. We are going to define a natural transformation of functors

η_{|i|}^{q} ⊗ η_{E^{∨}}^{q^{∨}} ⟶ κ_{|i|}^{ℏ}.

We use the following easy fact. Assume that [y_{i}, y_{j}] is a scalar. Then we have

e^{y_{i}}e^{y_{j}} = e^{[y_{i},y_{j}]}e^{y_{j}}e^{y_{i}}.  (10)

Let i be a seed. Denote by X_{i} the generators of 𝕋_{i}^{q}, and by X_{i}^{∨} the generators of 𝕋_{i^{∨}}^{q^{∨}}. It is easy to check using (10) that there are the following homomorphisms:

l_{i} : 𝕋_{i}^{q} ⟶ H_{i}^{h}, X_{i} ⟼ e^{x_{i}}, and l_{i}^{∨} : 𝕋_{i^{∨}}^{q^{∨}} ⟶ H_{i}^{ℏ}, X_{i}^{∨} ⟼ e^{x_{i}^{∨}}.

They evidently commute with the ∗-structures. Their images commute. Indeed, since ε̂_{ij} ∈ ℤ one has e^{[x_{i},x_{j}/ℏ]} = e^{2πiε̂_{ij}} = 1. So e^{x_{i}} commutes with e^{x_{j}/ℏ}. Therefore they give rise to a homomorphism of the tensor product:

L_{i} := l_{i} ⊗ l_{i}^{∨} : 𝕋_{i}^{q} ⊗ 𝕋_{i^{∨}}^{q^{∨}} ⟶ H_{i}^{ℏ}.

Proposition 6. For any mutation μ_{k} : i → i′ the following diagram, where the left vertical arrow is the map η^{q}(μ_{k})⊗η^{q^{∨}}(μ_{k}), and the right one is κ^{ℏ}(μ_{k}), is commutative:

𝕋_{i}^{q} ⊗ 𝕋_{i}^{q^{∨}} ⟶^{L_{i}} H_{i}^{ℏ}
η^{q,q^{∨}} ↑  ↑ κ^{ℏ}
𝕋_{i′}^{q} ⊗ 𝕋_{i′}^{q^{∨}} ⟶^{L_{i′}} H_{i′}^{ℏ}

Proof. We need Lemma 4. The case i = k is trivial, so we assume that i ≠ k. Let ε_{ik} = −a ≤ 0. Then applying the lemma we get

κ_{|i|}^{ℏ}(μ_{k})L_{i′}(X′_{i} ⊗ 1) = κ^{ℏ}(μ_{k})e^{x_{i}^{∨}} = e^{x_{i}+aϕ^{ℏ_{k}} (x_{k})} =

= \frac{e^{x_{i}}}{2πiaℏ_{k}} exp (∫_{x_{k}}^{x_{k}+2πiℏ_{k}a} aϕ^{ℏ_{k}}(z)dz) =

= \frac{e^{x_{i}}}{2πiℏ_{k}} exp (∫_{−∞}^{x_{k}}(ϕ^{ℏ_{k}}(z + 2πiℏ_{k}a) − ϕ^{ℏ_{k}}(z))dz) =^{A5}

=^{A5} \frac{e^{x_{i}}}{2πiℏ_{k}} exp (∫_{−∞}^{x_{k}} ∑_{b=1}^{a} \frac{2πiℏ_{k}}{e^{−z−iπ(2b−1)ℏ_{k}} + 1}dz) =

e^{x_{i}} ∏_{b=1}^{a}(1 + q_{k}^{2a−1}e^{x_{k}}) = L_{i} (X_{i} ∏_{b=1}^{a}(1 + q_{k}^{2a−1}X_{k})) =

= L_{i}(η_{|i|}^{q}(μ_{k}) ⊗ η_{|i^{∨}|}^{q^{∨}}(μ_{k}))(X′_{i} ⊗ 1).

The calculation in the case ε_{ik} = a ≥ 0 is similar. The proposition is proved.

Claim. The collection of homomorphisms {L_{i}} provides a morphism of functors

𝕃^{ℏ} : η_{|i|}^{q} ⊗ η_{|i^{∨}|}^{q^{∨}} ⟶ κ_{|i|}^{ℏ}.  (11)

Justification. Is given by Proposition 6.

Representations of the quantized X_{|i|}^{+}-space. The following definition serves as a motivation of the construction of intertwiners presented below.

Definition 7. A projective ∗-representation of the quantized X_{|i|}^{+}-space is the following data:

i) A projective functor

L_{|i|} : G_{|i|} ⟶ the category of Hilbert spaces.

It includes for each object i of G_{|i|} a Hilbert spaces 𝕃_{i}, and for every mutation μ_{k} : i → i′ a unitary operator, defined up to a scalar of absolute value 1:

K_{i,i′} : 𝕃_{i} ⟶ 𝕃_{i′}.

ii) A ∗-representation ρ_{i} of the Heisenberg algebra H_{i}^{ℏ} in the Hilbert space 𝕃_{i}.

iii) The operators K_{i,i′} intertwine the representations ρ_{i} and ρ_{i′}:

ρ_{i}(s) = K_{i,i′}^{−1}ρ_{i′}(κ^{ℏ}(μ_{k})(s))K_{i,i′}, s ∈ H_{i}^{ℏ}.

The morphisms of the representations of the quantum X_{|i|}^{+}-space are defined in an obvious way.

Representations of the mapping class group Γ_{|i|}. Restricting the functor ρ_{|i|} to the group of automorphisms of an object of the groupoid G_{|i|} we get a projective unitary representation of Γ_{|i|}.

The Heisenberg algebra H_{i}^{ℏ} has a family of irreducible ∗-representation by operators in a Hilbert space. These representations are characterized by the central character χ.

The collection of the Hilbert spaces {𝕃_{i}} and representations {ρ_{i}} is by no means canonical: it depends, for example, on the choice of polarization of the Heisenberg algebra. Once choosen, it determines the intertwiners K_{i,i′}. Below we introduce a canonical representation of the chiral double of the quantized space X_{|i|}^{+}, defined by using the Hilbert spaces L^{2}(A_{i}^{+}).

3 The intertwiner

3.1 A bimodule structure on functions on the A–space

Let X be an algebra. Recall that M is a bimodule over X if X acts on M from the left as well as from the right, and these two actions commute. So M is an X ⊗ X^{opp}-module, where X^{opp} is the algebra with the product x ∗ y := yx.

Let us choose a seed i. Recall the algebra ℚ[A_{i}] of regular functions on the seed torus A_{i}. We assume that q ∈ ℂ^{∗}. For each i ∈ I let us define commuting algebra homomorphisms

t_{i}^{±} : ℚ[A_{i}] ⟶ ℚ[A_{i}]; t_{i}^{±} : \{ \begin{matrix}A_{i} ⟼ q^{^{±}d̂_{i}}A_{i} \\ A_{j} ⟼ A_{j} & j ≠ i. \end{matrix}

Since ε_{ii} = 0, A_{i} does not appear in the monomial p^{∗}X_{i}, and so the operator of multiplication by p^{∗}X_{i} commutes with t_{i}^{±}. Let us define an T_{i}^{q}-bimodule structure on ℚ[A_{i}]. The left and right actions of the generator X_{i} on f ∈ ℚ[A_{i}] are given by

X_{i} ∘ f := p^{∗}X_{i} · t_{i}^{−}(f),  f ∘ X_{i} := p^{∗}X_{i} · t_{i}^{+}(f),  (12)

Lemma 8. The operators (12) provide ℚ[A_{i}] with a structure of a bimodule over the algebra T_{i}^{q}.

Proof. Observe that one has

t_{j}^{+}(p^{∗}X_{i}) · p^{∗}X_{j} = t_{i}^{−}(p^{∗}X_{j}) · p^{∗}X_{i} = qε̂_{ij}p^{∗}X_{i}p^{∗}X_{j}.

Indeed, the first term equals to q^{ε_{ij}d_{j}}p^{∗}X_{i}·p^{∗}X_{j}, and the second is q^{−ε_{ji}d_{i}}p^{∗}X_{i}· p^{∗}X_{j}. One has

q^{−ε̂_{ij}}X_{i}X_{j} ∘ f = q^{−ε̂_{ij}}p^{∗}X_{i} · t_{i}^{−}(p^{∗}X_{j}) · t_{i}^{−}t_{j}^{−}(f) = p^{∗}X_{i} · p^{∗}X_{j} · t_{i}^{−}t_{j}^{−}(f).

f ∘ q^{−ε̂_{ij}}X_{i}X_{j} = q^{−ε̂_{ij}}p^{∗}X_{j} · t_{j}^{+}(p^{∗}X_{i}) · t_{j}^{+}t_{i}^{+}(f) = p^{∗}X_{i} · p^{∗}X_{j} · t_{i}^{+}t_{j}^{+}(f). Since the right hand sides are evidently symmetric in i, j we have the desired relations, and hence the left and right actions of the quantum algebra torus. Further, the two actions commute:

X_{i} ∘ (f ∘ X_{j}) = X_{i} ∘ (t_{j}^{+}(f)p^{∗}X_{j}) = p^{∗}X_{i}t_{i}^{−}(p^{∗}X_{j})t_{i}^{−}t_{j}^{+}(f).
(X_{i} ∘ f) ∘ X_{j} = (p^{∗}X_{i}t_{i}^{−}(f)) ∘ X_{j} = p^{∗}X_{j}t_{j}^{+}(p^{∗}X_{i})t_{j}^{+}t_{i}^{−}(f).

The lemma is proved.

The logarithmic version of the bimodule structure. Since the coordinate functions A_{i} are positive on the space A^{+}, one can introduce new coordinates a_{j} := log A_{j}. They provide an isomorphism α_{i} : A_{i}^{+} ⟶^{∼} ℝ^{I}. Set da := da_{1} ∧ ⋯ ∧ da_{|I|}. There is a Hilbert space L^{2}(A_{i}^{+}) with a scalar product

(f, g) := ∫_{A_{i}^{+}} f(a)\overline{g(a)}da.

Apparently the form da changes the sign under a mutation i → i′. So the Hilbert spaces L^{2}(A_{i}^{+}) for different seeds i are naturally identified. Consider the following operators in L^{2}(A_{i}^{+}):

x_{j}^{−} := −πiℏd̂_{j} \frac{∂}{∂a_{j}} + ∑_{k} ε_{jk}a_{k}, x_{j}^{+} := πiℏd̂_{j} \frac{∂}{∂a_{j}} + ∑_{k} ε_{jk}a_{k}.

Lemma 9. The operators {x_{j}^{±}} provide the Hilbert space L^{2}(A_{i}^{+}) with a structure of a bimodule over the ∗–algebra H_{i}^{ℏ}.

Proof. These operators are selfadjoint and one has

[x_{j}^{−}, x_{k}^{−}] = 2πiℏε̂_{jk};  [x_{j}^{+}, x_{k}^{+}] = −2πiℏε̂_{jk},  [x_{j}^{−}, x_{k}^{+}] = 0

for any j, k ∈ I.

The lemma is proved.

Remark. There is an automorphism x_{j} ⟼ −x_{j} of the Heisenberg algebra H_{i}. Similarly there is an automorphism X_{j} ⟼ X_{j}^{−1} of the quantum torus algebra T_{i}^{q}.

3.2 The intertwiner via the quantum dilogarithm

Let μ_{k} : i → i′ be a mutation. By Lemma 9 for each seed i the Hilbert space L^{2}(A_{i}^{+}) has a natural H_{i}^{ℏ}-bimodule structure. According to Lemma 1, this is the same as the H_{i}^{h} ⊗ H_{i^{o}}^{ℏ}-module structure. Our goal is to define an operator

K_{i→i′} : L^{2}(A_{i}^{+}) ⟶ L^{2}(A_{i′}^{+})  (13)

intertwining the H_{i}^{ℏ} ⊗ H_{i^{o}}^{ℏ}- and H_{i′}^{ℏ} ⊗ H_{i′^{o}}^{ℏ}-module structures. By this we mean only that the operator K_{i→i′} intertwines the action of the generators of H_{i}^{ℏ}⊗H_{i^{o}}^{ℏ} withthe action of their images under the gluing map map κ_{|i|}^{ℏ}(i → i′).

The function G. Let us introduce our key function

G(a_{1}, …, a_{n}) =

∫ Φ^{ℏ_{k}}(d̂_{k}c−∑_{j} ε_{kj}a_{j})^{−1}Φ^{ℏ_{k}}(−d̂_{k}c−∑_{j} ε_{kj}a_{j}) exp (c\frac{∑_{j|ε_{kj}<0} ε_{kj}a_{j} + a_{k}}{πiℏ} ) dc.

(14)

Substituting the explicit integral expression for the function Φ^{ℏ_{k}}(z) one gets

G(a_{1}, …, a_{n}) =

= ∫ exp (∫_{Ω} \frac{exp(it ∑_{j} ε_{kj}a_{j}) sin(td̂_{k}c)}{2it sh(πt)sh(πℏ_{k}t)}dt + c\frac{∑_{j|ε_{kj}<0} ε_{kj}a_{j} + a_{k}}{πiℏ} ) dc.

We denote by (a_{1}, ..., a_{n}) the logarithmic coordinates corresponding to v, and by (a_{1}, ..., a′_{k}, ..., a_{n}) the ones corresponding to i′. Recall that only the coordinate a_{k} changes under the mutation μ_{k}. Let us define the operator K_{i→i′} by

(K_{i→i′}f)(a_{1}, …, a′_{k}, …, a_{n}) :=

∫ G(a_{1}, …, a′_{k} + a_{k}, …, a_{n})f(a_{1}, …, a_{k}, …, a_{n})da_{k}, (15)

where a′_{k} + a_{k} and a_{k} are on the k-th places.

Theorem 10. The operators K_{i→i′} intertwine the H_{i}^{ℏ}⊗H_{i^{o}}^{ℏ}-module structures on L^{2}(A_{i}^{+}) provided by Lemma 9.

Remark. We prove in [FG3] that the collection of Hilbert space L^{2}(A_{i}^{+}) and operators K_{i→i′} provide a unitary projective representation of the groupoid G_{|i|}. This implies that the operators K_{i→i′} give rise to a unitary projective representation of the cluster modular group Γ_{|i|} in L^{2}(A_{i}^{+}).

Proof. We present a computation which allows to find the function G as a unique up to a scalar function such that the corresponding integral transformation intertwines the H_{i}^{ℏ}- and H_{i′}^{ℏ}-bimodule structures on L^{2}(A_{i}^{+}) and L^{2}(A_{i′}^{+}). Recall that ε_{ij}^{o} = −ε_{ij}, so we may write ε_{ij}^{±} := ±ε_{ij} and denote by x_{i}^{±} the x_{i}-coordinates for the seeds i and i^{o}.

So we have to find G such that the integral transformation (15) induces a map of operators:

x_{i}^{′±} ⟼ \{ \begin{matrix}x_{i}^{±} − ε_{ik}^{±}ϕ^{ℏ_{k}}(−sgn(ε_{ik}^{±})x_{k}^{±}) & if i ≠ k \\ −x_{k}^{±} & if i = k. \end{matrix}  (16)

This means that we should have (changing x_{i}^{−} to −x_{i}^{−} for convenience) for i ≠ k:

πihd̂_{i} \frac{∂}{∂a_{i}} ± ∑_{j≠k} ε′_{ij}a_{j} ± ε′_{ik}a′_{k} ↦ πiℏd̂_{i} \frac{∂}{∂a_{i}} ±

∑_{j≠k} ε_{ij}a_{j} ± ε_{ik} (a_{k} − ϕ^{ℏ_{k}} (−sgn(±ε_{ik})(πiℏd̂_{k} \frac{∂}{∂a_{k}} ± ∑_{j} ε_{kj}a_{j}))) , (17)

and

πiℏd̂_{k} \frac{∂}{∂a′_{k}} ± ∑_{j} ε′_{kj}a_{j} ↦ −(πiℏd̂_{k} \frac{∂}{∂a_{k}} ± ∑_{j} ε_{kj}a_{j}).  (18)

Here we use the following conventions. The signs ± in our formulas always use either + everywhere, or − everywhere, so ∓ := −±. Thus we have one set of the equations corresponding to the upper signs and another one to the lower signs.

Observe that ε_{kk} = 0. The relation (18) is satisfied by (15) if and only if

−ε′_{kj} = ε_{kj}, and \frac{∂}{∂a′_{k}} ⟼ −\frac{∂}{∂a_{k}} .

Since these two conditions are evidently valid, we have the relation (18). Substituting (17) into (15) one gets the identities:

∫ (πiℏd̂_{i} \frac{∂G}{∂a_{i}} f + πiℏd̂_{i}G\frac{∂f}{∂a_{i}} ± ∑_{j≠k} ε′_{ij}a_{j}Gf ± ε′_{ik}a′_{k}Gf)da_{k} =

= ∫ πiℏd̂_{i}G\frac{∂f}{∂a_{i}} ± ∑_{j≠k} ε_{ij}a_{j}Gf±

± Gε_{ik} (a_{k} − ϕ^{ℏ_{k}} (sgn(ε_{ik})(∓πiℏd̂_{k} \frac{∂}{∂a_{k}} − ∑_{j} ε_{kj}a_{j})) f) da_{k}.

Since these identities should be valid for any f one gets the equations for the function G:

(πiℏd̂_{i} \frac{∂}{∂a_{i}} ± ∑_{j≠k}(ε′_{ij} − ε_{ij})a_{j} ∓ ε_{ik}a_{k}±

±ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(±πiℏd̂_{k} \frac{∂}{∂a_{k}} − ∑_{j} ε_{kj}a_{j}))) G = 0.

Let us introduce the function Ĝ related to G by the Fourier transform:

Ĝ(c) = ∫ e^{− \frac{a_{k}c}{πiℏ}} G(a_{k})da_{k}; G(a_{k}) = \frac{1}{2π^{2}ℏ} ∫ e^{\frac{a_{k}c}{πiℏ}} Ĝ(c)dc

(we omit the variables a_{1}, …, a_{k−1}, a_{k+1}, …, …, a_{n} both G and Ĝ depends on). Taking into account the relations

πiℏ ˆ\frac{∂G}{∂a_{k}} = cĜ, â_{k}G = −πiℏ\frac{∂Ĝ}{∂c}

one can get the equation for the function Ĝ:

(πiℏd̂_{i} \frac{∂}{∂a_{i}} ± ∑_{j≠k}(ε′_{ij} − ε_{ij})a_{j} ± ε_{ik}πiℏ\frac{∂}{∂c}±

±ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(±d̂_{k}c − ∑_{j} ε_{kj}a_{j}))) Ĝ = 0. (19)

Taking sum and difference of the equations corresponding to the upper and lower signs one obtains:

(2πiℏd̂_{i} \frac{∂}{∂a_{i}} + ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(d̂_{k}c − ∑_{j} ε_{kj}a_{j})) −

−ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(−d̂_{k}c − ∑_{j} ε_{kj}a_{j}))) Ĝ = 0

and

(2 ∑_{j≠k}(ε′_{ij} − ε_{ij})a_{j} + 2ε_{ik}πih\frac{∂}{∂c}+

+ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(d̂_{k}c − ∑_{j} ε_{kj}a_{j})) +

+ε_{ik}ϕ^{ℏ_{k}} (sgn(ε_{ik})(−d̂_{k}c − ∑_{j} ε_{kj}a_{j}))) Ĝ = 0.

Observe that these are the system of 2n − 2 equations on a function of n variables. So it is an overdetermined system if n > 2. Using the identities

ϕ^{ℏ_{k}}(sgn(a)b) = ϕ^{ℏ_{k}}(b) + (sgn(a) − 1)b/2; d̂_{i}ε_{ji} = −d̂_{j}ε_{ij}

they can be transformed to the form

\begin{matrix}2πi & ℏ\frac{∂ log Ĝ}{∂a_{i}} = d̂_{k}^{−1}ε_{ki}× \\ × & (ϕ^{ℏ_{k}}(d̂_{k}c − ∑_{j} ε_{kj}a_{j}) − ϕ^{ℏ_{k}}(−d̂_{k}c − ∑_{j} ε_{kj}a_{j}) + d̂_{k}c(sgn(ε_{ik}) − 1)) \end{matrix} ; (20)

and

2πiℏ\frac{∂ log Ĝ}{∂c} = 2(ε_{ik})^{−1} ∑_{j≠k}(ε_{ij} − ε′_{ij})a_{j}  (21)

−ϕ^{ℏ_{k}}(d̂_{k}c − ∑_{j} ε_{kj}a_{j}) − ϕ^{ℏ_{k}}(−d̂_{k}c − ∑_{j} ε_{kj}a_{j}) + (sgn(ε_{ik}) − 1) ∑_{j} ε_{kj}a_{j}

Taking into account that

ε′_{ij} − ε_{ij} = \frac{|ε_{ik}|ε_{kj} + ε_{ik}|ε_{kj}|}{2}, i, j ≠ k

we have the following identity:

2(ε_{ik})^{−1}(ε_{ij} − ε′_{ij}) + (sgn(ε_{ik}) − 1)ε_{kj} = (sgn(ε_{jk}) − 1)ε_{kj}

Multiplying it by a_{j} and taking the sum over j ≠ k we get

2(ε_{ik})^{−1} ∑_{j≠k}(ε_{ij} − ε′_{ij})a_{j} + (sgn(ε_{ik}) − 1) ∑_{j} ε_{kj}a_{j} = ∑_{j}(sgn(ε_{jk}) − 1)ε_{kj}a_{j},

Thus (21) is equivalent to

2πiℏ\frac{∂ log Ĝ}{∂c} = −ϕ^{ℏ_{k}}(d̂_{k}c − ∑_{j} ε_{kj}a_{j})−

− ϕ^{ℏ_{k}}(−d̂_{k}c − ∑_{j} ε_{kj}a_{j}) + (sgn(ε_{jk}) − 1) ∑_{j} ε_{kj}a_{j}

Therefore the solution of the equations (20) and (21) is given by the formula

Ĝ = CΦ^{ℏ_{k}}(d̂_{k}c−∑_{j} ε_{kj}a_{j})^{−1}Φ^{ℏ_{k}}(−d̂_{k}c−∑_{j} ε_{kj}a_{j})e^{c ∑_{j}(sgn(ε_{jk})−1))ε_{kj}a_{j}/2πiℏ},

where C is an arbitrary constant. Taking C = 2π^{2}ℏ, one obtains the desired formula (14). The statement is proved.

Lemma 11. An integral operator given by the formula (15) for certain function G intertwines the operators (16) if and only if the standard formula for the mutation of the function ε_{ij} holds.

Proof. The proof of the theorem shows that this formula, as well as the formula for mutations of the A-coordinates follow from the anzatz (15) and the mutation formulas for the quantized X-coordinates.

Representation of the modular double of the chiral double of X_{|i|,q}. Combining Claim 2.2 and Theorem 10 we see that the collection of Hilbert spaces {L_{2}(A_{i}^{+})} should provide a projective unitary ∗-representation of the modular double of the chiral double of X_{|i|,q}, defined as

X_{|i|,q} × X_{|i^{o}|,q} × X_{|i^{∨}|,q^{∨}} × X_{|i^{o}^{∨}|,q^{∨}}.

According to [FG3], the collection of Hilbert spaces {L_{2}(A_{i}^{+})} should provide a representation of the modular double D_{|i|,q} × D_{|i^{∨}|,q^{∨}} of the cluster double of the quantum cluster X -variety X_{|i|,q}. Since there is a canonical map of quantum spaces D_{|i|,q} ⟶ X_{|i|,q} × X_{|i^{o}|,q}, this implies the above claim.

4 The quantum logarithm and dilogarithm functions

The proofs of all results listed above can be found in [FG3].

Recall the dilogarithm function

Li_{2}(x) := − ∫_{0}^{x} log(1 − t)dt.

The quantum logarithm function. It is the following function:

ϕ^{ℏ}(z) := −2πℏ ∫_{Ω} \frac{e^{−ipz}}{(e^{πp} − e^{−πp})(e^{πℏp} − e^{−πℏp})}dp;  (22)

where the contour Ω goes along the real axes from −∞ to ∞ bypassing the origin from above.

Proposition 12. The function ϕ^{ℏ}(x) enjoys the following properties.

(A1)  lim_{ℏ→0} ϕ^{ℏ}(z) = log(e^{z} + 1).
(A2)  ϕ^{ℏ}(z) − ϕ^{ℏ}(−z) = z.
(A3)  \overline{ϕ^{ℏ}(z)} = ϕ^{ℏ}(\overline{z}).
(A4)  ϕ^{ℏ}(z)/ℏ = ϕ^{1/ℏ}(z/ℏ).

(A5)

ϕ^{ℏ}(z +iπℏ)−ϕ^{ℏ}(z −iπℏ) = \frac{2πiℏ}{e^{−z} + 1}, ϕ^{ℏ}(z +iπ)−ϕ^{ℏ}(z −iπ) = \frac{2πi}{e^{−z/ℏ} + 1}.

(A6) The form ϕ^{ℏ}(z)dz is meromorphic with poles at the points {πi((2m − 1) + (2n − 1)ℏ)|m, n ∈ ℕ} with residues 2πiℏ and at the points {−πi((2m − 1) + (2n − 1)ℏ)|m, n ∈ ℕ} with residues −2πiℏ.

The quantum dilogarithm. Recall the quantum dilogarithm function:

Φ^{ℏ}(z) := exp(−\frac{1}{4} ∫_{Ω} \frac{e^{−ipz}}{sh(πp)sh(πℏp)} \frac{dp}{p}).

It goes back to Barnes [Ba], and was used by Baxter [Bax], Faddeeev [Fad], and others.

Proposition 13. The function Φ^{ℏ}(x) enjoys the following properties.

(B)  2πiℏ d log Φ^{ℏ}(z) = ϕ^{ℏ}(z)dz
(B0)  lim_{ℜz→−∞} Φ^{ℏ}(z) = 1.

Here the limit is taken along a line parallel to the real axis.

(B1)  lim_{ℏ→0} Φ^{ℏ}(z)/ exp \frac{−Li_{2}(−e^{z})}{2πiℏ} = 1.

(B2)  Φ^{ℏ}(z)Φ^{ℏ}(−z) = exp (\frac{z^{2}}{4πiℏ}) e^{− \frac{πi}{12} (ℏ+ℏ^{−1})}.

(B3)  \overline{Φ^{ℏ}(z)} = (Φ^{ℏ}(\overline{z}))^{−1}. In particular |Φ^{ℏ}(z)| = 1 for z ∈ ℝ.
(B4)  Φ^{ℏ}(z) = Φ^{1/ℏ}(z/ℏ).
(B5) Φ^{ℏ}(z + 2πiℏ) = Φ^{ℏ}(z)(1 + qe^{z}),  Φ^{ℏ}(z + 2πi) = Φ^{ℏ}(z)(1 + q^{∨}e^{z/ℏ}).

(B6) The function Φ^{ℏ}(z)dz is meromorphic with poles at the points

{πi((2m − 1) + (2n − 1)ℏ)|m, n ∈ ℕ}

and zeroes at the points

{−πi((2m − 1) + (2n − 1)ℏ)|m, n ∈ ℕ}

References

[Ba] Barnes E.W., The genesis of the double gamma function, Proc. London Math. Soc. 31 (1899) 358-381.

[Bax] Baxter R., Exactly solved models in statistical mechanics, Academic Press (1982).

[BFZ] Berenstein A., Fomin S., Zelevinsky A., Cluster algebras III, Upper bounds and double Bruhat cells, math.RT/0305434. J. Amer. Math. Soc. 15 (2002), no. 2, 497–529

[Fad] Faddeev L.D., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), no. 3, 249–254.

[FCh] Fock, V. V., Chekhov, L. O., Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999), no. 3, 511–528.

[FG1] Fock V.V., Goncharov A.B., Moduli spaces of local systems and higher Teichmuller theory, Publ. Math. IHES, n. 103 (2006) 1-212.

math.AG/0311149.

[FG2] Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the

dilogarithm I, math.AG/0311245.

[FG3] Fock V.V., Goncharov A.B., The quantum dilogarithm and unitary representations of the modular groupoid, math.QA/0702397.

[FZI] Fomin S., Zelevinsky A., Cluster algebras. I, Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529

[K] Kashaev R.M., Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), no. 2, 105–115.