Resurgence of Faddeev's quantum dilogarithm
aa r X i v : . [ m a t h - ph ] O c t RESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM
STAVROS GAROUFALIDIS AND RINAT KASHAEV
Abstract.
The quantum dilogarithm function of Faddeev is a special function that playsa key role as the building block of quantum invariants of knots and 3-manifolds, of quan-tum Teichm¨uller theory and of complex Chern–Simons theory. Motivated by conjectures onresurgence and the recent interest in wall-crossing phenomena, we prove that the Borel sum-mation of a formal power series solution of a linear difference equation produces Faddeev’squantum dilogarithm. Along the way, we give an explicit formula for the Borel transform,a meromorphic function in the Borel plane, locate its poles and residues and describe theStokes phenomenon of its Laplace transforms along the Stokes rays.
Contents
1. Introduction 11.1. Our results 32. Proofs 62.1. The formal power series solution of the difference equation 62.2. The Borel transform 72.3. Bounds 92.4. The Laplace transform 93. Useful properties of the dilogarithm function 12Acknowledgments 13References 13 Introduction
A well-known problem in quantum topology is the Volume Conjecture which asserts thatthe Kashaev invariant of a hyperbolic knot grows exponentially at a rate proportional tothe volume of the knot [Kas94, Kas95, Kas97]. There are several strengthenings of thisconjecture that involve the analytic properties of the asymptotics of the Kashaev invariantto all orders (see e.g., [DGLZ09, GZ] and references therein). Such factorially divergentformal power series have been conjectured to lead to resurgent functions [Gar08], and thisin turn leads to astonishing numerically testable conjectures [GMnP, GZ, GGMn]. TheKashaev invariant of a knot is a finite state-sum whose building block is the quantum n factorial ( q ; q ) n = Q nj =1 (1 − q j ), evaluated at complex roots of unity. The latter is intimatelyrelated to another special function, the Faddeev quantum dilogarithm [Fad95], evaluated at Date : 4 October, 2020.
Key words and phrases:
Quantum dilogarithm, Faddeev, asymptotics, difference equations, resurgence,quasi-periodic functions, Borel transform, Laplace transform, Stokes phenomenon, wall crossing, quantumTeichm¨uller theory, quantum hyperbolic geometry, complex Chern-Simons theory, monogenic functions. rational points. Although the conjectured resurgence properties of quantum knot invariantsare largely unproven, in an unfinished manuscript from 2006 we studied the resurgenceproperties of their building block, namely the Faddeev quantum dilogarithm. This specialfunction plays a key role in quantum Teichm¨uller theory [Kas98, AK14] and complex Chern–Simons theory [BDP14, DGG14, Dim16]. Since there is renewed interest in this subject withapplications to resurgence and wall-crossing phenomena (see for instance [Kon]), we decidedto update our manuscript and make it widely available.To begin the story, in the quantization of Teichm¨uller theory one considers the differenceequation f τ ( z − iπτ ) = (1 + e z ) f τ ( z + iπτ ) (1)whose motivation is explained in detail in [Kas98, Prop.8] and also in [AK14, Prop.1, Eqn.(9)](after some minor change in notation). The above difference equation appears, among otherplaces, in quantum integrable systems (see Ruijsenaars [Rui97, Eqn.(1.17)]) and in holomor-phic dynamics (see Marmi–Sauzin [MS03]).It it easy to see (see Lemma 2.1 below) that if f τ ( z ) satisfies Equation (1) and the limitingvalue lim z →−∞ f τ ( z ) = 1, then for a fixed z , f τ ( z ) admits an asymptotic expansion of theform log f τ ( z ) ∼ πiτ Li ( − e z ) + ˆ φ τ ( z ) , ( τ →
0) (2)where ˆ φ τ ( z ) = ∞ X n =1 (2 πi ) n − B n (1 / n )! ∂ nz Li ( − e z ) τ n − , (3)Li ( z ) = P k ≥ z k /k is Euler’s dilogarithm function and the differentiation operator ∂ nz (defined by ∂ z g ( z ) = g ′ ( z )) acts on Li ( − e z ).The goal of this paper is to identify the Borel summation of the factorially divergent seriesˆ φ τ ( z ) with Faddeev’s quantum dilogarithm function f τ ( z ) = Φ b ( z/ (2 π b )) (4)(see Corollary 1.5 below) when τ = b >
0. Along the way, we give an explicit formula forthe Borel transform G ( ξ, z ) of the power series ˆ φ τ ( z ) (see Theorem 1.1 below).It turns out that G ( ξ, z ) is a meromorphic function of ξ with poles that lie discretelyin a countable union L ( z ) of lines through the origin given in Equation (11) below. Thearrangement L ( z ) depends on z and accumulates to the imaginary axis. Such an arrangementis reminiscent of the parametric resurgence of non-linear equations (see e.g. [MS16]), theexact and perturbative invariants of Chern–Simons theory (predicted for instance in [Gar08]and Figure below Defn. 2.3 of ibid), and the wall-crossing formulas of Kontsevich–Soibelman(see [KS11] and also [Kon]).Theorem 1.3 identifies the Laplace transform of G ( · , z ) with the logarithm of Faddeev’squantum dilogarithm function f τ ( z ) given in Equation (4). We also consider the Laplacetransform of the function G ( · , z ) along any ray in the complement of L ( z ) and describe theStokes phenomenon, i.e., the change of the Laplace transform as one crosses a Stokes line R ξ m ( z ). One may think of this as an instance of a wall-crossing formula, in the spirit ofKontsevich–Soibelman. ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 3
Another noteworthy phenomenon is the Laplace transform of G ( · , z ) along the verticalrays ± i R + , which no longer lie in an open cone in the complement of L ( z ). This is the caseconsidered by Marmi–Sauzin [MS03] who prove (with a careful analysis) that the Laplacetransform f − τ (resp. f + τ ) is defined in the upper half-plane Im( τ ) > τ ) <
0) thus leading to two distinguished solutions f ± τ ( z ) of Equation (1). Figure 1.
The poles of G ( ξ, z ) in the ξ -plane are points nξ m ( z ) lie in anarrangement of lines passing through the origin.1.1. Our results.
Recall the quantum dilogarithm function of Faddeev [Fad95]Φ b ( z ) = exp (cid:18)Z R + i ǫ e − i xz x b ) sinh( x b − ) d xx (cid:19) , (5)a function with remarkable analytic properties that satisfies a pentagon identity and aninversion relation summarized in Section 3 below. Φ b ( z ) is a meromorphic, quasi-periodicfunction of z that satisfiesΦ b ( z − i b /
2) = (1 + e π b z ) Φ b ( z + i b / . (6)The function Φ b ( z ) is used as the building block of topological invariants of 3-manifolds viaquantum Teichm¨uller theory [AK14, AKa, AKb, KLV16].Consider the Borel transform B : τ C [[ τ ]] → C [[ ξ ]] , B ( τ n +1 ) = ξ n n ! . (7)Let G ( ξ, z ) = B ( ˆ φ τ ( z )) (8)denote the Borel transform of ˆ φ τ ( z ). Our first result describes a global formula for G ( ξ, z )in the complex Borel ξ -plane. STAVROS GAROUFALIDIS AND RINAT KASHAEV
Theorem 1.1.
When z ∈ C with | Im( z ) | < π and ξ ∈ C with | ξ | < π − | Im( z ) | , we have: G ( ξ, z ) = 12 πi ∞ X n =1 ( − n n (cid:18)
11 + e ξn − z + 11 + e − ξn − z (cid:19) , (9) where the right hand side is expanded as a formal power series in ξ around zero with radiusof convergence π − | Im( z ) | .It follows that G ( ξ, z ) is a meromorphic function of ξ with simple poles at ξ = nξ m ( z ) (shownin Figure 1) with residue C n , where ξ m ( z ) = z + (2 m + 1) πi, n ∈ Z ∗ = Z \ { } , m ∈ Z , C n = ( − n πin . (10)Note that the singularities of G ( · , z ) form a lattice in a countable union L ( z ) of lines throughthe origin L ( z ) = ∪ m ∈ ZR ξ m ( z ) . (11)The formula of the above theorem is similar to the one of Marmi–Sauzin [MS03, Thm. 4.3]. Itis also similar to the Euler–MacLaurin summation formula given in Costin–Garoufalidis [CG08,Thm. 2]. This is not an accident. When Im( τ ) >
0, the quantum dilogarithm has an infiniteproduct expansion (see Equation (53) below) whose logarithm can be written as an infinitesum which one can analyze with the Euler–MacLaurin summation method. However, sucha manipulation is meaningless since the product formula (53), although convergent whenIm( τ ) >
0, diverges when τ > L f )( τ ) = Z ∞ e − ξ/τ f ( ξ ) d ξ (12)of a function f ∈ L ( R ). Our next theorem concerns the analytic properties of the mero-morphic function G ( ξ, z ) and its Laplace transform ( L G )( τ, z ). For a positive real number δ , define S δ = { ξ ∈ C | dist( ξ, [0 , ∞ )) < δ } , δ S δ (13) Theorem 1.2. (a) When z ∈ C with 0 < δ < π − | Im( z ) | and ξ ∈ S δ we have: | G ( ξ, z ) | ≤ max (cid:26) , π | Im( z ) | + δ ) (cid:27) . (14)(b) Fix 0 < δ < π . Then we have: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( L G )( τ, z ) − N X n =1 (2 πi ) n − B n (1 / n )! ∂ nz Li ( − e z ) τ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ LM N (2 N )! Re( τ ) | τ | N (15)for all τ ∈ C with Re( τ ) > z ∈ C with | Im( z ) | < π − δ and all natural numbers N ,where M = M δ = 2 /δ and L = L z,δ = max n , π | Im( z ) | + δ ) o .Our next result identifies the Borel transform of G ( ξ, z ) with the quantum dilogarithmfunction. ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 5
Theorem 1.3.
When τ > and z ∈ C with | Im( z ) | < π , we have: log Φ b (cid:16) z π b (cid:17) = 12 πiτ Li ( − e z ) + ( L G )( τ, z ) (16) where b = τ and G ( ξ, z ) is as in (9) . This theorem follows from the explicit formula for G ( ξ, z ) in Theorem 1.1 which agreeswith the integral formula of Woronowicz for Φ b ( z ). Theorems 1.2 and 1.3 give the following. Corollary 1.4.
With the assumptions of part (b) of Theorem 1.2, we have: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log Φ b (cid:16) z π b (cid:17) − N X n =0 (2 πi ) n − B n (1 / n )! ∂ nz Li ( − e z ) τ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ LM N (2 N )! Re( τ ) | τ | N . (17)The special case of (17) with N = 0 is equivalent to Lemma 7.13 of [BAGPN], which itselfis an improvement of an earlier Lemma 3 of Andersen–Hansen [AH06]. An alternative proofof the above inequality (17) was given by Andersen [And].The process of replacing a factorially divergent series with its Borel transform, followedby the Laplace transform is known as Borel summation. Theorems 1.1 and 1.3 imply thefollowing. Corollary 1.5.
When τ > z ∈ C with | Im( z ) | < π , the Borel summation of theseries (3) reproduces the logarithm of Faddeev’s quantum dilogarithm function, namely,log Φ b ( z/ (2 π b )).The above corollary has some surprising consequences. A priori, a solution of (1) is well-defined up to multiplication with 2 πiτ -periodic functions, and Borel summation choosesexactly the one that agrees with the quantum dilogarithm function. What’s more, thequantum dilogarithm function satisfies the symmetry of Equation (55), and hence satisfiesa second difference equation (obtained by replacing b by b − in (6)). This, together withCorollary 1.5, implies the following. Corollary 1.6.
The Borel summation of ˆ φ τ ( z ) satisfies the additional functional equation f τ ( z − iπ ) = (1 + e zτ ) f τ ( z + iπ ) . (18)The two functional equations (1) and (18) determine f τ up to multiplication by a doublyperiodic function, and when τ > G ( · , z ). Let ρ θ = [0 , ∞ ) e iθ denote the ray in the complex plane and let( L θ f )( τ ) = Z ρ θ e − ξ/τ f ( ξ ) d ξ (19)denote the Laplace transform of a function f , integrable along ρ θ . Recall that the singularitiesof the meromorphic function G ( · , z ) are in an arrangement of lines L ( z ) through the originwhose complement C \ L ( z ) = ∪ m ∈ Z C m ( z ) is a union of open cones C m ( z ) = { ξ ∈ C ∗ | arg( ξ m − ( z )) < arg( ξ ) < arg( ξ m ( z )) } . (20) STAVROS GAROUFALIDIS AND RINAT KASHAEV
It follows that when θ ∈ C m ( z ), the Laplace transform ( L θ G )( τ, z ) is independent of θ anddefines a holomorphic function of τ for arg( ξ m − ( z )) − π/ < arg( τ ) < arg( ξ m ( z )) + π/ θ = 0 ∈ C ( z ), Theorem 1.3 implies that ( L θ G )( τ, z ) is, up to a dilogarithm term,equal to f τ ( z ) and the latter is equal tolog f τ ( z ) = log( − q e z ; q ) ∞ − log( − ˜ q e z/τ ; ˜ q ) ∞ (21)when Im( τ ) > z ) <
0, as follows from Equation (53). On the other hand, bycrossing the walls of L ( z ), we get( L π/ G )( τ, z ) − ( L G )( τ, z ) = ∞ X m =0 f m ( τ, z ) − f m +1 ( τ, z ) (22)where f m ( τ, z ) = ( L θ m ( z ) G )( τ, z ) is the Laplace transform of G ( · , z ) along a ray θ m ( z ) ∈ C m ( z ). When Re( z ) < | Im( z ) | < π and Im( τ ) >
0, the difference f m ( τ, z ) − f m +1 ( τ, z )is obtained by adding the poles − nξ − m − ( z ) with n > G ( · , z ) at the corresponding ray.Using the residue of G ( · , z ) at these points given in Theorem 1.1 and adding up, it followsthat f m ( τ, z ) − f m +1 ( τ, z ) = ∞ X n =1 πiC − n e ξ − m − ( z ) n/τ = − ∞ X n =1 ( − n n e ξ − m − ( z ) n/τ = log(1 + e ξ − m − ( z ) /τ ) = log(1 + e z/τ ˜ q m + ) . (23)This, combined with Equations (21) and (22), implies that( L π/ G )( τ, z ) = log( − q e z ; q ) ∞ (24)in agreement with the result of Marmi–Sauzin proven in Sec. 1.1 and Thm. 4.3 of [MS03].2. Proofs
The formal power series solution of the difference equation.
The followinglemma is well-known and standard (see eg [AK14, Sec. 13.4, Prop. 6]), but we include itsproof for completeness.
Lemma 2.1. If f τ ( z ) satisfies Equation (1) and lim z →−∞ f τ ( z ) = 1, then for fixed z , f τ ( z )admits an asymptotic expansion of the form (2) with ˆ φ τ ( z ) given in (3). Proof.
Letting φ τ ( z ) = log f τ ( z ), it follows that φ τ ( z + πiτ ) − φ τ ( z − πiτ ) = − log(1 + e z ) . Taylor’s theorem combined with − log(1 + e z ) = ∂ z Li ( − e z ) implies that2 sinh( πiτ ∂ z ) φ τ ( z ) = ∂ z Li ( − e z )hence, that 2 πiφ τ ( z ) = πi∂ z sinh( πiτ ∂ z ) Li ( − e z ) . ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 7
The expansion z sinh( z ) = ∞ X n =0 B n (1 /
2) (2 z ) n (2 n )!concludes the proof of the lemma. (cid:3) The Borel transform.
Consider the formal power series φ f ( τ, z ) = ∞ X n =1 B n (1 / n )! f (2 n ) ( z )(2 πi ) n − τ n − (25)for a function f analytic on z with | Im( z ) | < π , and let G f ( ξ, z ) = B ( φ f ( · , z )) denote thecorresponding Borel transform. Proposition 2.2.
We have: G f ( ξ, z ) = i π ∞ X n =1 ( − n n (cid:18) f ′′ (cid:18) z + ξn (cid:19) + f ′′ (cid:18) z − ξn (cid:19)(cid:19) . (26) Proof.
The proof is rather standard. It uses the Hadamard product ⊛ of power series (whichwas also used in [CG08]) whose definition we recall: ∞ X n =0 b n ξ n ! ⊛ ∞ X n =0 b n ξ n ! = ∞ X n =0 b n c n ξ n . (27)We have: G f ( ξ, z ) = ∞ X n =1 B n (1 / n )! f (2 n ) ( z )(2 n − πi ) n − ξ n − = ∞ X n =1 B n (1 / n )! ξ n − ! ⊛ ∞ X n =1 f (2 n ) ( z )(2 n − πi ) n − ξ n − ! = f ( ξ ) ⊛ f ( ξ, z )where f ( ξ ) = ∞ X n =1 B n (1 / n )! ξ n − f ( ξ, z ) = 2 πi ∞ X n =1 f (2 n ) ( z )(2 n − πiξ ) n − . (28)Now, since B m (1 /
2) = 0 for every odd m , we have: f ( ξ ) = ∞ X n =1 B n (1 / n )! ξ n − = 1 ξ ∞ X n =1 B n (1 / n )! ξ n = 1 ξ ∞ X n =1 B n (1 / n ! ξ n = 1 ξ (cid:18) e ξ/ ξe ξ − − (cid:19) = 1 ξ ( e ξ/ − e − ξ/ ) − ξ STAVROS GAROUFALIDIS AND RINAT KASHAEV and Taylor’s theorem gives: f ( ξ, z ) = 2 πi ∞ X n =1 f (2 n ) ( z )(2 n − πiξ ) n − = (2 πi ) − ∂ ξ ∞ X n =1 f (2 n ) ( z )(2 n )! (2 πiξ ) n = (2 πi ) − ∂ ξ (cid:18)
12 ( f ( z + 2 πiξ ) + f ( z − πiξ )) − f ( z ) (cid:19) = πi ( f ′′ ( z + 2 πiξ ) + f ′′ ( z − πiξ )) . Now, using Cauchy’s theorem, it follows that G f ( ξ, z ) = 12 πi Z γ f ( s ) f (cid:18) ξs , z (cid:19) dss where γ is a small circle around 0. The function f ( s ) has simple poles at 2 πim for m ∈ Z \{ } with residue ( − m / (2 πim ). Now, deform the integration contour to circles of increasing radiiand collect the residues. Since f ( s ) = O (1 /s ) and f ( ξ/s ) = O (1 /s ), it follows that theintegrand is O (1 /s ), thus the contribution from infinity is zero. The residue of the integrandfor m ∈ Z \ { } is given by:Res (cid:18) f ( s ) f (cid:18) ξs , z (cid:19) s , s = 2 πim (cid:19) = 12 πim f (cid:18) ξ πim , z (cid:19) Res( f ( s ) , s = 2 πim )= ( − m πim (cid:18) f ′′ (cid:18) z + ξm (cid:19) + f ′′ (cid:18) z − ξm (cid:19)(cid:19) . Thus, collecting the residues, it follows that G f ( ξ, z ) = − X m ∈ Z \ ( − m πim (cid:18) f ′′ (cid:18) z + ξm (cid:19) + f ′′ (cid:18) z − ξm (cid:19)(cid:19) = i π ∞ X m =1 ( − m m (cid:18) f ′′ (cid:18) z + ξm (cid:19) + f ′′ (cid:18) z − ξm (cid:19)(cid:19) . For fixed z and ξ , the above sum is dominated by P ∞ m =1 /m and thus the convergence isuniform on compact sets. This concludes the proof of Proposition 2.2. (cid:3) Proof of Theorem 1.1.
Apply Proposition 2.2 to the function f ( z ) = Li ( − e z ) which satisfies f ′′ ( z ) = −
11 + e − z . (cid:3) ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 9
Bounds.
In this section we give a proof of Theorem 1.2.We begin with the following lemma.
Lemma 2.3.
When z ∈ C with | Im( z ) | < π we have:1 | e z | ≤ ( z )) ≥ | sin(Im z ) | if cos(Im( z )) ≤ Proof.
With z = t + ia , we have: | e z | = e t + 2 e t cos a + 1and the right hand side, as a function of t ∈ R , has critical points in t ∈ R such that e t + cos a = 0. When cos a >
0, it follows that inf t ∈ R | e z | = 1, and when cos a ≤
0, itfollows that t is a global minimum and e t + 2 e t cos a + 1 = sin a . The result follows. (cid:3) Proof. (of Theorem 1.2) The first part follows from Equation (9), Lemma 2.3 (applied to ξ/n − z for n ∈ Z ∗ ) and the fact that P ∞ n =1 /n = π /
6. In particular, it implies that theLaplace transform ( L G )( τ, z ) is well-defined and even extends to τ ∈ C with Re( τ ) > c = 0 and c = ǫ in the notation of Theorem 5.20 of [MS16]). (cid:3) The Laplace transform.
Woronowicz [Wor00], while studying the quantum exponen-tial function via functional analysis, introduced the following function W θ ( z ) = Z R log(1 + e θξ )1 + e ξ − z d ξ (30)defined for θ > | Im( z ) | < π , and proved that (after some elementary change ofvariables) it satisfies the functional equation (1); see [Wor00, Eqn. (B.3)]. Thus, one canrelate Woronowicz’s function with the quantum dilogarithm as is done in [Kas16, Eqns. (1),(2)] without proof. The next proposition provides a formal proof of this fact. Proposition 2.4.
When τ > z ∈ C with | Im( z ) | < π , we have: W τ ( z ) = − πi log Φ b (cid:16) z π b (cid:17) , b = τ. (31) Proof.
The proof uses a mixture of real and complex analysis. Let ǫ be a positive real numbersuch that 0 < ǫ < min(1 , θ ). We remark that W θ ( z ) can be interpreted as a value of thescalar product in the complex Hilbert space L ( R ) of square integrable functions on the realline with respect to the Lebesgue measure W θ ( z ) = h f | g i = Z R f ( x ) g ( x ) d x (32)where f, g ∈ L ( R ) are defined by f ( x ) = e − ǫx log(1 + e θx ) , g ( x ) = e ǫx e x − ¯ z . (33) As the Fourier transformation (
F ψ )( x ) = Z R ψ ( y ) e πixy d y (34)is a unitary operator in L ( R ), we have the equality h f | g i = h F f | F g i . (35)By using Lemma 2.5 below, we can calculate explicitly the elements F f, F g ∈ L ( R ). Indeed,denoting ζ = 2 πx + iǫ , we have( F f )( x ) = Z R e iζy log(1 + e θy ) d y = iθζ Z R e iζy e − θy d y (36)= 2 πiζ Z R e πiζy/θ e − πy d y = πiζ Z R e ( ζθ − i )2 πiy cosh( πy ) d y (37)= πiζ cosh( πζθ − πi ) = πiζ cos( π − πζiθ ) = πiζ sin( πζiθ ) = − πζ sinh( πζθ ) (38)where in the second equality we integrated by parts, and( F g )( x ) = Z R e − iζy e y − z d y = Z R − z e − iζ ( y + z ) e y d y (39)= 2 πe − iζz Z R − z π e − πiζy e πy d y = πe − iζz Z R − z π e ( i − ζ )2 πiy cosh( πy ) d y (40)= πe − iζz cosh( πi − πζ ) = πe − iζz cos( π + πiζ ) = πe − iζz sin( − πiζ ) = πie − iζz sinh( πζ ) (41)where in the fifth equality we used the condition | Im( z ) | < π . Thus, we obtain W θ ( z ) = h F f | F g i = Z R ( F f )( x )( F g )( x ) d x = Z R + iǫ − πie − iζz πζθ ) sinh( πζ ) d ζζ (42)which implies that i π W τ ( z ) = Z R + iπ b ǫ e − i ζzπ b b ζ ) sinh( ζ / b ) d ζζ = log Φ b (cid:16) z π b (cid:17) . (43) (cid:3) The next lemma is well-known, see e.g., Godement [God15, VII, &3.15]. We will also givea proof using [GK15, Lem. 2.1].
Lemma 2.5.
When w, σ ∈ C | Im | < = { u ∈ C | | Im( u ) | < } , we have: Z R + σ e πiwz cosh( πz ) d z = 1cosh( πw ) . (44) ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 11
Proof.
By using [GK15, Lem. 2.1] with f ( z ) = e πiwz cosh( πz ) and a = i , we have f ( z + a ) f ( z ) = cosh( πz ) e πiw ( z + i ) cosh( πz + πi ) e πiwz = − e − πw (45)so that Z R + σ e πiwz cosh( πz ) d z = (cid:18)Z R + σ − Z R + σ + i (cid:19) e πiwz (1 + e − πw ) cosh( πz ) d z (46)= 2 πi Res z = i (cid:18) e πiwz (1 + e − πw ) cosh( πz ) (cid:19) (47)= πi cosh( πw ) Res z =0 (cid:18) πiz ) (cid:19) = 1cosh( πw ) (48)where, in the second equality, the application of the residue theorem is justified by the limitslim x →±∞ | f ( x + σ + it ) | ≤ (cid:12)(cid:12) e − πwt (cid:12)(cid:12) lim x →±∞ (cid:18) e π | Im( w ) || x +Re( σ ) | sinh | π ( x + Re( σ )) | (cid:19) = 0 . (49) (cid:3) The next proposition identifies Woronowicz’s formula for the quantum dilogarithm withthe Laplace transform of the function G ( ξ, z ). Proposition 2.6.
When τ > z ∈ C with | Im( z ) | < π , we have: W τ ( z ) = − τ Li ( − e z ) − πi ( L G )( τ, z ) . (50) Proof.
We have: W τ ( z ) = Z ∞−∞ log(1 + e ξ/τ ) e ξ − z + 1 d ξ = Z −∞ log(1 + e ξ/τ )1 + e ξ − z d ξ + Z ∞ log(1 + e ξ/τ )1 + e ξ − z d ξ = Z ∞ (cid:18) log(1 + e ξ/τ )1 + e ξ − z + log(1 + e − ξ/τ )1 + e − ξ − z (cid:19) d ξ = 1 τ Z ∞ ξ e ξ − z d ξ + Z ∞ log(1 + e − ξ/τ ) (cid:18)
11 + e ξ − z + 11 + e − ξ − z (cid:19) d ξ = 1 τ Z ∞ ξ e ξ − z d ξ − Z ∞ ∞ X n =1 ( − n n e − nξ/τ (cid:18)
11 + e ξ − z + 11 + e − ξ − z (cid:19) d ξ where the last equality follows from expanding the logarithm. Rescaling ξ → ξ/n and usingthe identity Z ∞ ξ e ξ − z d ξ = − Li ( − e z ) (51) (which can be verified for instance by integrating by parts) it follows that W τ ( z ) + 1 τ Li ( − e z ) = − Z ∞ e − ξ/τ ∞ X n =1 ( − n n (cid:18)
11 + e ξn − z + 11 + e − ξn − z (cid:19) d ξ = − πi Z ∞ e − ξ/τ G ( ξ, z ) d ξ = − πi ( L G )( τ, z )where we used Equation (9). (cid:3) We are now ready to give a proof of Theorem 1.3.
Proof. (of Theorem 1.3) Propositions 2.4 and Equation (50) imply that − πi log Φ b (cid:16) z π b (cid:17) = W τ ( z ) = − τ Li ( − e z ) − πi ( L G )( τ, z ) (52)which is equivalent to (16). (cid:3) Useful properties of the dilogarithm function
In this section we collect some useful properties of the quantum dilogarithm functionΦ b ( z ) = ( e πb ( z + c b ) ; q ) ∞ ( e πb − ( z − c b ) ; ˜ q ) ∞ (53)where q = e πi b , ˜ q = e − πi b − , c b = i b + b − ) , Im( b ) > . (54)The above function can also be defined in the lower half-plane Im( b ) < b ( z ) = Φ b − ( z ) , (55)and remarkably the function of b ∈ C \ R admits an extension to b ∈ C ′ = C \ ( −∞ , b ( z ) = Φ − b ( z ) . (56)Φ b ( z ) is a meromorphic function of z withpoles: c b + i N b + i N b − , zeros: − c b − i N b − i N b − . It satisfies the inversion relationΦ b ( z ) Φ b ( − z ) = e πiz Φ b (0) , Φ b (0) = q ˜ q − . It is a quasi-periodic function satisfyingΦ b ( z − i b /
2) = (1 + e π b z ) Φ b ( z + i b /
2) (57)which, due to the symmetry (55), implies a second functional equationΦ b ( z + i b /
2) = (1 + e − π b z ) Φ b ( z − i b /
2) (58)obtained from (57) by replacing b by b − . Note that Equation (57) (resp. (58)) implies thatthe function Φ b ( z/ (2 π b )) satisfies Equation (1) (resp. (18)) with τ = b . ESURGENCE OF FADDEEV’S QUANTUM DILOGARITHM 13
Acknowledgments
Theorems 1.1 and 1.3 were part of an unpublished manuscript from 2006, written during avisit of the first author to Geneva. The authors wish to thank the University of Geneva andthe International Mathematics Center at SUSTech University, Shenzhen for their hospitality.The results of the paper were presented in a Resurgence Conference in Miami in 2020. Theauthors wish to thank the organizers for their hospitality.S.G. wishes to thank David Sauzin for enlightening conversations and for a careful readingof the manuscript.This work is partially supported by the Swiss National Science Foundation research pro-gram NCCR The Mathematics of Physics (SwissMAP) and the ERC Synergy grant Recursiveand Exact New Quantum Theory (ReNew Quantum).
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