S-duality resurgence in SL(2) Chern-Simons theory

We find that an S-duality in SL(2) Chern-Simons theory for hyperbolic 3-manifolds emerges by the Borel resummation of a semiclassical expansion around a particular flat connection associated to the hyperbolic structure. We demonstrate it numerically with two representative examples of hyperbolic 3-manifolds.


Introduction and Summary
Perturbative expansions in quantum mechanics/quantum field theories are in general asymptotic expansions with zero radius of convergence. Typically, their coefficients grow factorially. To know the information on physical observables at finite coupling, we thus need a resummation method of asymptotic expansions. A systematic approach to construct a complete trans-series expansion in general situation is a so-called resurgent analysis (For reviews, see [1][2][3] for instance). The resurgence theory implies that the perturbative sector and the non-perturbative sectors are not independent but interrelated to each other.
The Chern-Simons (CS) theory is an example of exactly solvable quantum field theories [4] and its perturbative/non-perturbative aspects have been extensively studied last three decades in various contexts of theoretical physics. Applications of the topological theory include 3 dimensional quantum gravity [5], topological strings [6], 3 dimensional superconformal field theories [7,8] and mathematical physics [9].
It is a natural idea to apply the resurgence technique to the Chern-Simons theory and see how the resurgence helps us to understand (or find) some aspects (new aspects) of Chern-Simons theory. From this motivation, a refinement of a CS invariant was addressed in [10] (see also [11,12] for a different perspective). In this paper, we study another mysterious aspect of CS theory, an S-duality [13] when the gauge group is complex SL (2). Although there are already several hints on the S-duality from state-integral models for the complex CS theory, 3d/3d correspondence and etc, our resurgent analysis gives more direct evidence and more precise statement for it. Note that the similar hidden S-duality structure also appears in the context of the so-called "Topological Strings/Spectral Theory" correspondence [14][15][16][17].
Let us briefly summarize our main statement in this paper. We find that the perturbative expansion around a saddle point corresponding to a particular flat connection, A = A conj defined in (2.4), is Borel summable, and its Borel resummation has the S-duality property, while the resummations around the other saddle points do not. State-integral models do not seem to provide its simple explanation. 1 In 3d/3d correspondence [7,8], the S-duality is related to the manifest b ↔ 1/b symmetry of a curved background called squashed 3-sphere S 3 b [18] where b denotes a squashing parameter. Our analysis also provides supporting evidence for the conjecture in [19][20][21] (see also recent discussion in [22]) saying that only the flat connection A conj on M contributes to the S 3 b partition function of the corresponding 3d theory T [M ] in (2.25).
The rest of the paper is organized as follows. In section 2, we introduce two SL(2) flat connections, A geom and A conj , on hyperbolic 3-manifolds M and perturbative expansions around them. As tools to compute the perturbative expansions, the volume conjecture and state-integral models are reviewed. In section 3, we perform the Borel-Padé resummation of the perturbative expansions for two hyperbolic 3-manifolds, the figure-eight knot complement and a closed 3-manifold called Thurston manifold, and check the S-duality of the resummation of the perturbative expansion around A conj . We also provide a heuristic understanding of the S-duality by embedding the perturbative expansion to an unitary complex CS theory where the symmetry is manifest in Lagrangian.

Perturbative invariants from Complex Chern-Simons theory
We consider an asymptotic expansion of following formal path integral In perturbation, we need to choose a flat-connection A α and let the formal perturbative expansion in 1/k around it be For a hyperbolic 3-manifold M , there are two special SL(2, C) flat connections, A geom and A conj , associated to the unique hyperbolic metric on M normalized as R µν = −2g µν .
Here ω and e are a spin-connection and a dreibein respectively constructed from the hyperbolic metric. Both of them can be considered as so(3)-valued 1-forms and they form an sl(2)-valued 1-form. The hyperbolicity condition, R µν = −2g µν , implies that both of A geom and A conj are flat connections. One basic characteristic of them is that A geom (A conj ) gives the exponentially largest (smallest) classical contribution for real large k ∈ R + : for any flat connection A α . In particular, we have Here vol(M ) is a topological invariant called hyperbolic volume defined as the volume measured in the unique hyperbolic metric. For these isolated irreducible flat connections, the perturbative expansion takes the following form [10] The two perturbative expansions for α = geom or conj are especially related by In principle, the formal perturbative expansion around a given flat connection can be computed by summing up contributions from Feynman diagrams. For the computation, we need to fix a gauge symmetry by introducing a metric on a 3-manifold. The final sum should be independent on the choice due to the topological property of the theory but each contribution might depend on the choice and the computation requires the full knowledge on the spectrum of Laplacian on the 3-manifold with respect to the metric as for usual quantum field theories. There are simpler methods fully using topological property of the Chern-Simons theory. In the subsequent sections, we review two approaches.
Volume conjecture. In an asymptotic limit N ∈ Z → ∞, for a hyperbolic knot K in S 3 . Similarly, in an asymptotic limit N ∈ 2Z + 1 → ∞, (2.11) Here 4 1 denotes the figure-eight knot, the simplest hyperbolic knot. For a closed 3-manifold M = (S 3 \K) p obtained by taking Dehn surgery along a knot K with slope p ∈ Z, the SO(3) WRT invariant is given as following formula

From SL(2) state-integral models
Another simple approach is to use state-integral models based on ideal triangulation and Dehn filling representation of 3-manifolds. Decomposing a 3-manifold into basic building blocks, ideal tetrahedra and solid-torus, the SL(2) CS partition functin can be computed by gluing the wave-functions on them. As a topological field theory, the phase spaces associated to the boundaries of basic building blocks are finite dimensional non-compact symplectic varieties and the wave functions depend on the finite number of continuous position variables and the gluing of the wave functions is realized as an integration over the boundary variables. We refer to [19,[31][32][33][34] for state-integral models for knot complements based on its ideal triangulation and its extension [21,30] to closed 3-manifolds by incorporating Dehn filling operation. We give explicit expressions for the state-integral model for two simple hyperbolic 3-manifolds, the figure-eight knot complement and a closed hyperbolic 3-manifold called Thurston manifold.
where := 2πi k . (2.14) The 3-manifold has a torus boundary and there is a conventional canonical choice for basis of the boundary 1-cycles called meridian (µ) and longitude (λ).
This function has the following interesting S-duality We will discuss some basic properties of the quantum dilogarithm in appendix A. Using the semiclassical expansion of Φ b (z) (see (A.8)), we have where B n (x) is the n-th Bernoulli polynomial. Note that B n (1/2) is vanishing for all odd n. The state-integral is then written in the following form where the leading classical part is At u = 0, there are two saddle points, z geom and z conj Expanding the integrand in (2.20) around these saddle points, one obtains the following perturbative expansions: where V is the hyperbolic volume of the knot complement: The state-integral can be interpleted as a squashed 3-sphere partition function of a 3d N = 2 gauge theory T [S 3 \4 1 ] associated to the knot complement upon a proper choice of integral contour. In this identification, the formal SL(2) CS level k is related to the squashing parameter b by the relation k = b −2 . Following [8], we define T [M ] :=3d theory obtained from a twisted compacitification of 6d A 1 (2,0) theory on a hyperbolic 3-manifold M . (2.25) According to [8], The subscript in u(1) 0 denotes the CS level for the gauge u(1) symmetry. The theory has SU (2) Φ × U (1) J symmetry where the SU (2) rotates two chiral fields and U (1) J is the topological symmetry whose conserved charge is the monopole flux of the gauge U (1). It is argued that the symmetry is enhanced to SU (3) at the IR fixed point [35]. To find a contour of the state-integral relevant to the gauge theory, let us first briefly summarize the localization on S 3 b [18] in our notation.
• Fayet-Iliopoulos (FI) term for the u(1) with parameter ζ : exp(−2iπζσ) Here σ is a real scalar in a vector multiplet coupled to the u(1) symmetry. Applying the localization formulae, Here ∆ 1 and ∆ 2 are the R-charge choices for two chiral multiplets. ζ 1 is the real mass for a Cartan u(1) of the SU (2) flavor symmetry and the ζ 2 is the FI parameter, which can be considered as the real mass for the u(1) J symmetry. This expression is equivalent to the state-integral (2.13) when we choose with the following change of variables Since the σ and ζ i are real variables, the relation tell us that the state-integral model can be interpreted as S 3 b partition function of T [S 3 \4 1 ] when integrated over following contour Note that the saddle point z conj in (2.22) asymptotically touch the contour in the limit b → 0. This may imply that the squashed 3-sphere partition function of In this case, as a side remark, the geometric R-charge choice 2 coincides with the conformal R-charge at infrared (IR) fixed point and the state-integral at u = 0 gives the S 3 b partition function of the IR superconformal field theory.
Thurston manifold. Let (S 3 \4 1 ) p be a closed 3-manifold obtained by Dehn surgery along figure-eight knot with integral slope p.
The state-integral model for the closed 3-manifold is [21] (2.33) When p = −5, the 3-manifold (S 3 \4 1 ) p=−5 is called the Thurston manifold, which is known to be the second smallest hyperbolic 3-manifold with In this case, there are two saddle points (z conj ± , u conj ± ) corresponding to the flat connection A conj Two saddle points are related by the Weyl-reflection of SL(2) and the perturbative expansions around two saddle points are identical to all order The Z conj pert k; (S 3 \4 1 ) p=−5 is the sum of contributions from two saddle points Interestingly, there is a simper integral expression which reproduce the same perturbative expansion [36]. Let One saddle point for the integral is One can check that the perturbative expansion of Z around the saddle point gives the same perturbative expansion with Z conj pert k; (S 3 \4 1 ) p=−5 . With a proper choice of integral contour, the state-integral Z can be interpreted as the partition function of a 3d gauge theory T [Thurston] on a squashed 3-sphere. The theory T [Thurston] is field-theorectically described as [36] T [Thurston] = u(1) −7/2 coupled to a chrial Φ . (2.40) From a localization, we have Replacing the integration variable σ by z : Except the factor √ 2, the remaining factor is purely phase factor which can be removed by a local counterterm and thus negligible. The factor √ 2 may come from a topological degree of freedom coupled to the system. Modulo the contribution from topological degree of freedom, the Z is S 3 b -partition function of the T [Thurston] theory when integrated over following contour Note that 4π 7 = 1.7952... So the contour Γ Thurston is very close to the saddle point in (2.39) in the limit b → 0 and can be smoothly deformed to touch the saddle point. In this case, the geometric R-charge choice (∆ = 3 7 ) is different from the IR conformal R-charge determined by F-maximization [37].

Borel resummation method
Here we discuss the resummation for the perturbative expansions (2.23) and (2.36) (or (2.7) more generally). The important fact is that all of these perturbative expansions are divergent series. Therefore one needs a resummation method to get a finite value for given k. The standard way to do so is the Borel summation method. We briefly review it at the beginning in this section.
Let us consider a formal perturbative series of the form We assume that the perturbative coefficient f n factorially diverges in n → ∞. Therefore this perturbative expansion is a formal divergent series. The Borel transform of this series expansion is defined by Note that this infinite sum is now convergent. We can analytically continue it to the complex ζ-plane except for its singularities. We then define the Borel sum by the Laplace transform: The asymptotic expansion of this Borel sum reproduces the original divergent series (3.1). The Borel sum gives a meaning of the formal divergent series. If there are no singularities on the integration contour (i.e., on the positive real axis), the Laplace transform in the Borel sum is well-defined. In this case, f (k) is called Borel summable. However, we often encounter the situation that the integrand has singularities on ζ ∈ R + . This case is called non Borel summable. In the non Borel summable case, we deform the integration contour, and define a new deformed Borel sum by where θ is chosen to avoid the singularities. In our case, it is sufficient to consider the case where θ is very close to 0 in order to avoid singularities on the positive real axis. We denote it as where is a small constant. Unless the contour hits a singularity, the Laplace intergal does not depend on . If the Borel transform has singularities on the positive real axis, the deformed Borel sums S ± f (k) do not agree with each other: The discontinuity of the Borel sums is called the Stokes phenomenon.
In practical computations, we know only the first several values of f n . If we have f n up to n = 2n max , then the Borel transform (3.2) is truncated at n = 2n max : This finite sum still gives a good approximation of Bf (ζ) inside the convergence circle. To perform the Borel resummation, however, we have to integrate it along the whole positive real axis. This means that we need the information on Bf (ζ) outside the convergence circle.
To resolve this problem, the Padé approximant is usually used. We replace the finite sum of the Borel sum by its "diagonal" Padé approximant 3 where P nmax (ζ) and Q nmax (ζ) are degree-n max polynomials. Then, we can extrapolate the Padé approximant outside the convergence circle. The Padé approximant also tells us the (approximate) singularity structure of the Borel transform. This numerically powerful procedure is often called the Borel-Padé resummation.

Figure-eight knot complement
Let us start with the case of the figure-eight knot complement. Note that this case has been studied in [10] briefly, but we find that there are a few small mistakes in their analysis. We  Figure 1. The pole structure of Q nmax (ζ) for n max = 100 (left) and for n max = 120 (right) in the figure-eight knot complements. Some of poles depend on n max , and they should not be the true singularities of BZ conj pert (ζ). In the current case, we can conclude that BZ conj pert (ζ) does not have any singularities on the positive real axis but has on the negative real axis and on the imaginary axis.
re-analyze it here in much more detail. As a consequence, we arrive at a different conclusion from theirs.
We want to perform the Borel(-Padé) resummation for the perturbative expansion (2.23). As we will see just below, the perturbative expansion Z conj pert (k) turns out to be Borel summable, and we find that its Borel resummation recovers the S-duality for k ↔ 1/k. On the other hand, the Borel resummation of Z geom pert (k) does not.
Resumming the perturbative series. As in (2.23), the perturbative expansions in the state-integral (2.13) at u = 0 are given by (3.10) In spite of this simple relation, their resummations have quite different properties. Following the method in [10], we computed the exact values of a conj n up to n = 240. The first observation is that Z conj pert (k) is an alternating sum, while Z geom pert (k) is a nonalternating one. This implies that Z conj pert (k) is Borel summable, while Z geom pert (k) is not. To check this in detail, we analyze the singularities for the Borel-Padé transform 4 BZ conj pert (ζ) ≈ P nmax (ζ)/Q nmax (ζ). In figure 1, we show the pole structure of the denominator Q nmax (ζ) of the Padé approximant for n max = 100 and n max = 120. These figures strongly suggest 4 Here the Borel transforms BZ geom, conj pert (ζ) are defined by that the Borel transform BZ conj pert (ζ) has no singularities on the positive real axis. Using the relation (3.10), one easily finds the relation BZ geom pert (ζ) = BZ conj pert (−ζ) .

(3.11)
Since BZ conj pert (ζ) has singularities on the negative real axis, 5 we conclude that Z geom pert (k) is not Borel summable.
Let us proceed to the Borel resummation. What we actually do is the Borel-Padé resummation for 2n max = 240: For a given value of k, we can evaluate the Borel-Padé resummation by this equation. For example, the value at k = 1 reads SZ conj pert (k = 1) ≈ 0.379567579522536528565367 . . . .

(3.13)
We compare this value with the direct evaluation of the state-integral (2.13) along the contour in (2.31). For u = 0 and k = 1, we can deform the integration contour to the real axis, and the exact value of the state-integral was evaluated in [38] (3.14) We find agreement with 22-digit accuracy. 6 More interestingly, we observe that the Borel resummation SZ conj pert (k) has the S-duality relation: SZ conj pert (k) = SZ conj pert (1/k). (3.15) In fact, we show explicit values of SZ conj pert (k) and SZ conj pert (1/k) for various k's in table 1. We also confirmed that all these values are in good agreement with the direct evaluation of the state-integral (2.13) for the contour (2.31).
The integrand of the original state-integral (2.13) possesses this symmetry manifestly, but the perturbative expansion in k → ∞, of course, makes this symmetry invisible. After the Borel resummation, the symmetry is precisely restored! We emphasize that to perform the Borel resummation, we use only the perturbative data in k → ∞. Nevertheless the resummation "knows" the information in the opposite regime k → 0. This fact is surprising and unexpected. In fact, the authors in [10] did not expect this property. 5 BZ conj pert (ζ) also seems to have singularities at ζ = ±2πi. These are not important in our analysis. 6 In [10], the authors conclude that the Borel resummation of Z conj pert (k) does not reproduce the exact value of the state-integral. This conflicts our conclusion here. The discrepancy comes from the exponential factor in (3.9). In [10], the exponential factor in Z conj pert (k) is e kV 2π . It is however obvious that the exponential factor in Z conj pert (k) must be e − kV 2π because Z conj pert (k) is exponentially small in the semiclassical limit k → ∞. This factor is crucially important to reproduce the exact result for finite k as well as the S-duality restoration below.  Figure 2. The real part of S + Z geom pert (k) does not recover the S-duality, while the imaginary part does.
Next, let us discuss the Borel resummation of Z geom pert (k). As we have already seen, Z geom pert (k) is not Borel summable. Therefore we have to consider the deformed Borel resummations (3.5). In the actual computation, we use the Borel-Padé resummations: where the Padé approximant is the same function appearing in Z conj pert (k). These Borel resummations turn out to be complex-valued. For example, the values at k = 1 are given by S ± Z geom pert (1) ≈ 1.0526393020 ± 0.5693505539i . Moreover, we observe that the Borel resummations S ± Z geom pert (k) do not have the S-dual symmetry: We show the k-dependence of the real and imaginary parts of S + Z geom pert (k) in figure 2. Though S ± Z geom pert (k) do not have the S-dual relation totally, their imaginary part seems to have it. This is because the imaginary part is precisely related to the Borel sum SZ conj pert (k). In fact, the standard resurgent analysis (see [2] for instance) tells us that the difference of S ± Z geom pert (k) is given by where S is called a Stokes constant. As we will see below, in our case we have S = 3i.
Large order behavior. Finally, we discuss the large order behavior of the perturbative expansion. From the resurgent analysis, the large order behavior of Z conj pert (k) provides the information on the other saddle Z geom pert (k). More precisely, as in [10], we expect the large order behavior Since we have a conj n up to n = 240, we can extract the information on A, S and a geom n very precisely from this formula.
To know A, we look at a relation To accelerate the convergence of this sequence, we use the Richardson extrapolation. For the analysis of the large order behavior by using the Richardson extrapolation, see [39]. Let us define the m-th Richardson transform of a given sequence f n by If the sequence f n behaves as then the Richardson transform of f n behaves as Therefore the convergence speed is improved.
In the current case, we apply the 80th Richardson transform 7 to the sequence na conj n /a conj n+1 , and find the convergent value As found in [10], the exact value of A is given by the difference of the actions of the two saddles A geom and A conj , We find remarkable agreement with |A − R 80 [159a conj 159 /a conj 160 ]| ∼ O(10 −98 ). This A is also related to a singularity on the Borel transform BZ conj pert (ζ). The closest singularity of the Padé approximant of BZ conj pert (ζ) on the negative real axis from the origin 8 is which is indeed in agreement with A.
Once the exact value of A is known, we can extract S by Using the Richardson transform of b n again, we find This strongly suggest that the exact value of S is Note that our obtained value is different from the one in [10]. The value in [10] is S [GMP] ≈ 7.51989i. 9 The evidence of our result here is that the discontinuity (3.20) holds only for S = 3i.
Repeating this way, one can confirm the large order relation (3.21) with very high numerical accuracy.

Thurston manifold
Borel resummation. In this case, the perturbative expansions of the state-integral (2.33) (or (2.38)) take the forms  Recall that we have the relation (2.8). Also, we can compute the perturbative expansion of the state-integral (2.38) around the saddle (2.39). As mentioned before, the result coincides with Z conj pert (k): In the following, we mainly focus on the resummation of Z pert (k). It is very likely that Z pert (k) is Borel summable. It is observed that the closest singularity from the origin is located at ζ ≈ −0.1563 + 0.5846i. We also confirm the S-duality restoration in the Borel resummation, as shown in table 2. 10 In the computation, we keep all the numerical values sufficiently high precision.
In the case of the Thurston manifold, the perturbative expansion Z geom pert (k) is also Borel summable. We again observe that the Borel resummation of Z geom pert (k) does not reproduce the S-dual relation. For k = √ 2, we have Large order behavior. Let us proceed to the large order behavior. As in (3.21), we assume the large order behavior of the form Using the first 100 coefficients, we find the numerical values A ≈ −0.1561897001 + 0.5841922570i, (3.42) All of these values are stable in the 17th Richardson transform at least up to this digit. One can see that the value of A coincides with the closest singularity (3.36), as in the figure-eight knot complement. Our analysis implies that b 1 seems purely imaginary and that b 2 seems real. So far, it is unclear to us the relation between this large order behavior and the saddle-point approximation in the state-integral (2.38) (or (2.33)). This is not a main purpose in this paper. It would be interesting to explore it in more detail.

Physical reasoning of the S-duality
In the previous subsection, we saw that the perturbative expansion around the saddle corresponding to the flat connection A = A conj is Borel summable and that its Borel resummation has the S-duality. We also observed that the perturbation around A geom is not Borel summable for the figure eight knot complement, but Borel summable for the Thurston manifold. In general, it is not simple to say whether the perturbation around a given flat connection A α is Borel summable or not. Nevertheless, we can say that for the particular connection A conj , the perturbative expansion Z conj pert is always Borel summable. The reason is as follows. If Z conj pert is not Borel summable, then it has to receive non-perturbative corrections to cancel the ambiguity of the Borel sum. However, the inequality (2.5) shows that there are no saddles, whose exponentiated classical actions are smaller than that for A conj : This means that Z conj pert does not receive any non-perturbative corrections, and we conclude that Z conj pert must be Borel summable. Our conjecture here is that the Borel resummation SZ conj pert has the S-dual symmetric structure in k ↔ 1/k. In this sense, the flat connection A conj is very special.
The emergence of the S-duality after the Borel resummation of Z conj pert is somewhat surprising since there is no such a symmetry in the path integral (2.1). One heuristic explanation of this surprise is the following. First, notice that the integrand in (2.1) is not unitary for complex gauge field A and the path-integral makes sense only at the perturbative level. As will be explained below, there is a unitary SL(2) CS theory whose partition function has the same asymptotic expansion as (2.1) in a certain limit of coupling in the theory. So the unitary theory can be considered as a non-perturbative completion of the formal perturbative CS partition function in (2.1). Further, the unitary complex CS theory has an S-duality as a manifest symmetry. The Borel resummation somehow knows the nonperturbative completion and it gives the non-perturbative answer which has the S-duality symmetry.
Let us explain the unitary complex CS theory in more detail. The complex SL(2) CS theory depends on two CS levels, K and σ, whose action is given by For the invariance under the large gauge transformation, K should be an integer: For the unitarity of the theory, σ is either real or purely imaginary.
σ ∈ R or σ ∈ iR . (3.46) In [40], it is conjectured that if we choose then the asymptotic expansion of the partition function of the complex SL(2) CS theory in a singular limit b → 0 is equivalent to the formal perturbative expansion in (2.1) with identification k = b −2 . So, the SL(2) CS theory with K = 1 can be considered as a non-perturbative completion of the formal path integral in (2.1). After the substitution in In the limit b → 0, the second term in (3.48) vanishes and the action reduced to a CS action only with A with a quantum parameter = 2πi(1 + b 2 ). The resulting action is equivalent to the action in (2.1) except for k = b −2 is replaced by (1 + b 2 ) −1 . In a quantization of CS theory, the relevant quantum parameter is q := e instead of and the difference between two actions disappears. This is an heuristic derivation of the conjecture in [40].
For b = 1, the function reduces to the classical (di)logarithm: Φ b=1 (z) = exp iz log(1 − e 2πz ) + i 2π Li 2 (e 2πz ) . (A.5) Note that compared to the compact quantum dilogarithm, the non-compact one is welldefined even for |q| = 1. For |q| < 1, it is constructed by two copies of the compact quantum dilogarithm: where q := e 2πib 2 ,q := e 2πi/b 2 . (A.7) As in the compact case, we can expand Φ b (z) around = 2πib 2 = 0, where B n (x) is the Bernoulli polynomial. Note that the compact function φ q (−q 1/2 e 2πbz ) also has the same semiclassical expansion: This is a consequence of the relation (A.6). At the semiclassical level, we cannot distinguish the non-compact function Φ b (z) from the compact one φ q (−q 1/2 e 2πbz ). However, one should keep in mind that the equations (A.8) and (A.9) mean the equalities in the asymptotic sense in → 0. We know, of course, Φ b (z) = φ q (−q 1/2 e 2πbz ) for finite .
Resummation. As shown in [41], the semiclassical expansion on the right hand side in (A.8) or (A.9) is resummed exactly. The resummed function turns out to reproduce the non-compact quantum dilogarithm, not the compact one. Let us denote the semiclassical expansion as We stress that this symmetry restoration is far from obvious in the integral representation (A.12) or (A.13). We have checked it numerically. Finally, one can also numerically confirm that this resummation reproduces the original non-compact quantum dilogarithm: