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Conformal TBA for Resolved Conifolds

Abstract

We revisit the Riemann–Hilbert problem determined by Donaldson–Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2–D0-brane bound states. We explore various prescriptions to make the sum well defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the \(\tau \) function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz integral representation for Faddeev’s quantum dilogarithm.

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Notes

  1. Namely a map \(\sigma : \Gamma \rightarrow \mathbb {C}^\times \) such that \(\sigma _\gamma \, \sigma _{\gamma '} = (-1)^{\langle \gamma ,\gamma '\rangle } \sigma _{\gamma +\gamma '}\).

  2. In contrast to the RH problems in [4, 12], we relax the meromorphy condition by excluding the locus \(q=1\). This weaker condition is necessary in order that solutions exist at all, in fact the solution found in [4] fails to satisfy the stronger meromorphy condition for non-vanishing fiber coordinates.

  3. To avoid cluttering, we will also omit the brackets on \(\vartheta _\beta \) concentrating on the domain where they belong to the interval (0, 1).

  4. For the generalized conifold with \((N_0,N_1)=(2,1)\), also known as the suspended pinch point singularity, the GV invariants take values \(\{1,1,-1\}\) for the three rational curves, while \(\chi _X=3\) (see example 5.14 in [11]).

  5. Note that this assumption can easily be dropped. Indeed, if it does not hold, one shifts the summation variable in (4.2) as \(k\rightarrow k-n\) with \(n=\lfloor \,\mathrm{Re}\,(v/w)\rfloor \), so that one gets

    $$\begin{aligned} S(v,w,\varphi )=e^{-2\pi \mathrm {i}n\varphi }S(v',w,\varphi ), \end{aligned}$$

    where \(v'=v-nw\). The resulting exponential factor is combined with a similar factor in (4.7) to produce the full v, whereas in all three possible results in (4.8) \(v'\) can be replaced by v due to their invariance under integral shifts of v/w. Thus, one arrives at the same result.

  6. Recall that \(n_0\) was defined below (3.2) as \(-{1\over 2}\, \chi _X\), here we need twice this value. Note also that the functions \({\mathcal {Y}}^{(1)}_\beta \), \({\mathcal {Y}}^{(2)}_\beta \) defined here should not be confused the functions defined in (D.2c). Here they include the contribution of only one \(\beta \in B\), whereas in Appendix D they combine the contributions of \(\beta \) and \(-\beta \). We hope that this abuse of notation will not lead to any confusion.

  7. The KS transformations across \(\ell _{\pm \delta }\) are generated only by the contribution from \(\beta =0\). In this case, the discontinuities are captured by the first term in the second equation of (4.16) which is proportional to \({\mathcal {J}}_{1,1}({\hat{w}},0,\varphi ,0)\). This function differs from the function \({\tilde{{\mathcal {J}}}}_{1,1}({\hat{w}},\varphi )\) defined in (C.2) by a regular term, hence its discontinuities are described by Proposition 2.

  8. This calculation can also be done using the function S. Indeed, one has

    $$\begin{aligned} \sum _{k\ne 0} \frac{e^{2\pi n k\varphi }}{k}=\lim _{v\rightarrow 0}\left( S(v,1,\varphi )-\frac{1}{v}\right) =-2\pi \mathrm {i}\varphi +\pi \mathrm {i}\epsilon . \end{aligned}$$
  9. In [4] it was called ‘Joyce function’ and was denoted by J. Note that the normalization of the coordinates \((z,\theta )\) in loc. cit. differs from our conventions by a factor of \(2\pi \mathrm {i}\); moreover, the potential (4.32) is odd in the fiber coordinates due to (A.2) and the property of the square brackets noticed below (A.4).

  10. This also demonstrates that, contrary to the claim in [4], \({\mathcal {X}}_\gamma ^\mathrm{Br}\) are not meromorphic functions. It is possible that this fact was overlooked because for \(\varphi =0\), the case where [12, Prop. 4.6] has been derived, the singularity lies at \(w/t=0\), which is outside of the domain of consideration, whereas for non-vanishing \(\varphi \) it moves inside the complex plane.

  11. See also [33, §4.3] and [13] for a streamlined computation using Mellin transform. In comparing these references it is useful to note that \(\frac{\zeta '(2)}{2\pi ^2} -\frac{1}{12} ( \gamma _E +\log 2\pi ) = \zeta '(-1) - \frac{1}{12}\).

  12. Note that the condition \(\beta \in B^+\) is consistent with the Kähler cone condition \(\,\mathrm{Im}\,z_\beta >0\).

  13. Equation (5.15) defines the prepotential only up to quadratic terms in the moduli; we added the second term by hand so as to reach a perfect match with \(F_0\).

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Acknowledgements

We are grateful to Tom Bridgeland and Joerg Teschner for very useful correspondence. As this work was being completed, we learnt that M. Alim, A. Saha, and I. Tulli were investigating a closely related problem [42]. We thank them for agreeing to coordinate the release on arXiv.

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Appendices

A Special Functions and Useful Identities

In this appendix, we collect various useful definitions, properties, and identities (some of which were already stated in [8, §B]).

Bernoulli Polynomials

The Bernoulli polynomials have the following generating function

$$\begin{aligned} \sum _{n=0}^\infty \frac{x^n}{n!}\, B_n(\xi )= \frac{x\, e^{\xi x}}{e^x-1}. \end{aligned}$$
(A.1)

They have the following symmetry property

$$\begin{aligned} B_n(1-x)=(-1)^n B_n(x) \end{aligned}$$
(A.2)

and at \(x=0\) they reduce to Bernoulli numbers \(B_n\). Importantly, the Bernoulli polynomials arise in the inversion formula for polylogarithms: namely, for any \(n\ge 2\) and \(x\in \mathbb {C}\), or \(n\ge 0\) and \(x\in \mathbb {C}\backslash \mathbb {Z}\),

$$\begin{aligned} \mathrm{Li}_n(e^{2\pi \mathrm {i}x})+(-1)^n\mathrm{Li}_n(e^{-2\pi \mathrm {i}x})=-\frac{(2\pi \mathrm {i})^n}{n!}\, B_n([x]), \end{aligned}$$
(A.3)

where we use the principal branch definition of \(\mathrm{Li}_s(z)\), and define

$$\begin{aligned}{}[x]= {\left\{ \begin{array}{ll} x-\lfloor \,\mathrm{Re}\,x\rfloor , &{} \quad \text{ if } \,\mathrm{Im}\,x\ge 0, \\ x+\lfloor -\,\mathrm{Re}\,x\rfloor +1, &{} \quad \text{ if } \,\mathrm{Im}\,x< 0. \end{array}\right. } \end{aligned}$$
(A.4)

Note that for \(\,\mathrm{Im}\,x\ne 0\) or \(x\notin \mathbb {Z}\), the bracket satisfies \([-x]=1-[x]\), consistently with (A.3) and (A.2). For integer \(n<0\), we have instead

$$\begin{aligned} \mathrm{Li}_n(e^{2\pi \mathrm {i}x})+(-1)^n\mathrm{Li}_n(e^{-2\pi \mathrm {i}x})= 0 \end{aligned}$$
(A.5)

for all \(x\in \mathbb {C}\backslash \mathbb {Z}\).

Useful Identities

For \(d\in \mathbb {Z}\), \(0<\,\mathrm{Re}\,z<\,\mathrm{Re}\,\omega _1\), \(\,\mathrm{Im}\,(z/\omega _1)>0\) and the contour C going along the real axis but avoiding the origin from above, one has [12, Eqs.(36),(38)]

$$\begin{aligned}&\int _C \mathrm {d}s\,\frac{e^{zs}\, s^{-d}}{e^{\omega _1s}-1}=\left( \frac{\omega _1}{2\pi \mathrm {i}}\right) ^{d-1}\mathrm{Li}_d\left( e^{2\pi \mathrm {i}z/\omega _1}\right) , \end{aligned}$$
(A.6a)
$$\begin{aligned}&\int _C \mathrm {d}s\,\frac{e^{(z+\omega _1)s}\, s^{1-d}}{(e^{\omega _1s}-1)^2}= -\frac{\mathrm {d}}{\mathrm {d}\omega _1}\left[ \left( \frac{\omega _1}{2\pi \mathrm {i}}\right) ^{d-1}\mathrm{Li}_d\left( e^{2\pi \mathrm {i}z/\omega _1}\right) \right] , \end{aligned}$$
(A.6b)
$$\begin{aligned}&\int _C \mathrm {d}s\,\frac{\left( e^{\omega _1s}+1\right) e^{(z+\omega _1)s}\, s^{2-d}}{(e^{\omega _1s}-1)^3}= \frac{\mathrm {d}^2}{\mathrm {d}\omega _1^2}\left[ \left( \frac{\omega _1}{2\pi \mathrm {i}}\right) ^{d-1}\mathrm{Li}_d\left( e^{2\pi \mathrm {i}z/\omega _1}\right) \right] .\qquad \end{aligned}$$
(A.6c)

For \(\alpha \in \mathbb {C}\backslash \mathbb {Z}\), one has [36, 1.445.6]

$$\begin{aligned} \sum _{k=1}^\infty \frac{\cos (2\pi kx)}{k^2-\alpha ^2}= & {} \frac{1}{2\alpha ^2}- \frac{\pi }{2\alpha }\, \frac{\cos (2\pi \alpha ({1\over 2}-[x]))}{\sin (\pi \alpha )}\, . \end{aligned}$$
(A.7)

Generalized Gamma and Barnes Functions

The generalized Gamma function \(\Lambda (z,\eta )\) and generalized Barnes functions \(\Upsilon (z,\eta )\) are both functions on \(\mathbb {C}^\times \times \mathbb {C}\) defined by [8, 28]

$$\begin{aligned} \Lambda (z,\eta )= & {} \frac{e^z\,\Gamma (z+\eta )}{\sqrt{2\pi } z^{z+\eta -1/2}}\, . \end{aligned}$$
(A.8)
$$\begin{aligned} \Upsilon (z,\eta )= & {} \frac{e^{\frac{3}{4}z^2- \zeta '(-1)}\, G(z+\eta +1) }{(2\pi )^{z/2} \,z^{\frac{1}{2} z^2} \bigl [\Gamma (z+\eta ) \bigr ]^\eta }\, , \end{aligned}$$
(A.9)

where G(z) is the Barnes function (see, e.g., [37]). The two functions are related to each other through

$$\begin{aligned} \frac{\partial }{\partial z} \log \Upsilon (z,\eta ) = z \frac{\partial }{\partial z} \log \Lambda (z,\eta ). \end{aligned}$$
(A.10)

They have the following integral representations analogous to the first Binet formula for Gamma function and valid for \(\,\mathrm{Re}\,z>0\), \(\,\mathrm{Re}\,(z+\eta )>0\) [8, §B]

$$\begin{aligned} \log \Lambda (z,\eta )= \int _0^\infty \frac{\mathrm {d}s}{s} \left( \eta -{1\over 2}-\frac{1}{s}+\frac{e^{(1-\eta )s}}{e^s-1} \right) e^{-z s}\, , \end{aligned}$$
(A.11)
$$\begin{aligned} \log \Upsilon (z,\eta )= & {} \int _0^\infty \frac{\mathrm {d}s}{s} \left( \frac{1}{s^2} - \frac{1}{2} \,B_2(\eta ) - \frac{\eta (e^s-1)+1}{(e^s-1)^2}\,e^{(1-\eta )s} \right) e^{-z s}\nonumber \\&-\frac{1}{2} \,B_2(\eta ) \log z. \end{aligned}$$
(A.12)

B Special Functions Relevant for Bridgeland’s Solution

In this appendix, we define a class of functions which appear in the solution discussed in Sect. 3. We also provide their relations to multiple sine functions, quantum dilogarithm, and present new integral identities for some of these functions.

We define the following set of functions

$$\begin{aligned} {\mathcal {F}}_{n,m}(z|\omega _1,\omega _2)=(-1)^{n+m}\int _C \frac{\mathrm {d}s}{s}\, e^{z s}\, \mathrm{Li}_{-n}(e^{-\omega _1 s})\, \mathrm{Li}_{-m}(e^{-\omega _2 s}), \end{aligned}$$
(B.1)

where the contour C follows the real axis from \(-\infty \) to \(\infty \) avoiding the origin from above. We consider these functions for nm non-negative integers, in which case the integral is convergent provided \(\,\mathrm{Re}\,(\omega _1),\,\mathrm{Re}\,(\omega _2)>0\) and \(0<\,\mathrm{Re}\,(z)<\,\mathrm{Re}\,(\omega _1+\omega _2)\). For reference, we list

$$\begin{aligned} \mathrm{Li}_0(x)=\frac{x}{1-x}\, , \qquad \mathrm{Li}_{-1}(x)=\frac{x}{(1-x)^2}\, , \qquad \mathrm{Li}_{-2}(x)=\frac{x(1+x)}{(1-x)^3}\, . \end{aligned}$$
(B.2)

In the special cases \((n,m)=(0,0)\) and (1, 0), one reproduces the functions \(\log F(z|\omega _1,\omega _2)\) and \(\log G(z|\omega _1,\omega _2)\), respectively, introduced in [12, §4] in terms of the multiple sine functions \(S_r(z|\omega _1,\dots ,\omega _r)\),

$$\begin{aligned} e^{{\mathcal {F}}_{0,0}(z|\omega _1,\omega _2)}= & {} F(z|\omega _1,\omega _2):= e^{-\frac{\pi \mathrm {i}}{2}\, B_{2,2}(z|\omega _1,\omega _2)}S_2(z|\omega _1,\omega _2), \end{aligned}$$
(B.3)
$$\begin{aligned} e^{{\mathcal {F}}_{1,0}(z|\omega _1,\omega _2)}= & {} G(z|\omega _1,\omega _2):= e^{\frac{\pi \mathrm {i}}{6}\, B_{3,3}(z+\omega _1|\omega _1,\omega _1,\omega _2)}S_3(z+\omega _1|\omega _1,\omega _1,\omega _2). \nonumber \\ \end{aligned}$$
(B.4)

Here, the multiple Bernoulli polynomials \(B_{r,m}\) are defined through the generating function generalizing (A.1),

$$\begin{aligned} \sum _{n=0}^\infty \frac{x^n}{n!}\, B_{r,n}(\xi |\omega _1,\dots ,\omega _r)= \frac{x^r\, e^{\xi x}}{\prod _{j=1}^r (e^{\omega _j x}-1)}\, . \end{aligned}$$
(B.5)

In particular, one has

$$\begin{aligned} B_{2,2}(z|\omega _1,\omega _2)= & {} \frac{z^2}{\omega _1\omega _2}-\left( \frac{1}{\omega _1}+\frac{1}{\omega _2}\right) z +\frac{1}{6}\left( \frac{\omega _2}{\omega _1}+\frac{\omega _1}{\omega _2}\right) +{1\over 2}\, , \nonumber \\ B_{3,3}(z|\omega _1,\omega _2,\omega _3)= & {} \frac{z^3}{\omega _1\omega _2\omega _3} -\frac{3z^2(\omega _1+\omega _2+\omega _3)}{2\omega _1\omega _2\omega _3}\nonumber \\&+\frac{\omega _1^2+\omega _2^2+\omega _3^2+3(\omega _1\omega _2+\omega _1\omega _3+\omega _2\omega _3)}{2\omega _1\omega _2\omega _3} z\nonumber \\&- \frac{(\omega _1+\omega _2+\omega _3)(\omega _1\omega _2+\omega _1\omega _3+\omega _2\omega _3)}{4 \omega _1\omega _2\omega _3}\, . \end{aligned}$$
(B.6)

We refer to [24] for the definition and main properties of the multiple sine functions, and to [38] for an integral representation established for these functions, which implies the relations (B.3), (B.4). Note that all these functions are invariant under a simultaneous rescaling of the arguments z and \(\omega _i\).

Double Sine, Faddeev’s Quantum Dilogarithm and Woronowicz Integral

It will be useful to express the function \(F(z|\omega _1,\omega _2)\) in terms of Faddeev’s quantum dilogarithm [39], which will allow us to derive a new integral representation for this function. The quantum dilogarithm is defined by

$$\begin{aligned} \Phi _b(x) = \exp \left[ \frac{1}{4} \int _{C} \frac{e^{-2\mathrm {i}x t}}{\sinh (b\, t) \sinh (t/b)} \frac{\mathrm {d}t}{t} \right] . \end{aligned}$$
(B.7)

Comparing this integral with (B.1) for \(m=n=0\) where the integration variable s is changed to \(t=\frac{s}{2}\, \sqrt{\omega _1\omega _2}\), we get

$$\begin{aligned} F(z|\omega _1,\omega _2) = \Phi _{\sqrt{\omega _1/\omega _2}}\left( \frac{\mathrm {i}}{\sqrt{\omega _1\omega _2}} \left( z-\frac{\omega _1+\omega _2}{2}\right) \right) . \end{aligned}$$
(B.8)

On the other hand, \(\Phi _b\) admits a different integral representation [40] originally due to Woronowicz [41],

$$\begin{aligned} \Phi _b(x) = \exp \left( \frac{\mathrm {i}}{2\pi }\, W_{1/b^2}(2\pi b x)\right) , \qquad W_\theta (z) = \int _{\mathbb {R}} \frac{ \log (1+e^{\theta \xi })}{1+e^{\xi -z}} \mathrm {d}\xi \, . \end{aligned}$$
(B.9)

Upon folding the integral over \(\mathbb {R}\) onto the positive axis, one gets

$$\begin{aligned} W_{\theta }(z) =-\theta \, \mathrm{Li}_2(-e^{z}) + \int _0^{\infty } \log (1+e^{-\theta \xi }) \left( \frac{1}{1+e^{\xi -z}} + \frac{1}{1+e^{-\xi -z}} \right) \mathrm {d}\xi \, .\nonumber \\ \end{aligned}$$
(B.10)

Setting \(b^2=\omega _1/\omega _2\) and \(\xi =2\pi s \,\omega _1/\omega _2\), one arrives at the following integral representation,

$$\begin{aligned} \log F(z|\omega _1,\omega _2)= & {} \frac{\omega _2}{2\pi \mathrm {i}\omega _1} \mathrm{Li}_2\left( e^{\frac{2\pi \mathrm {i}}{\omega _2}(z-\frac{\omega _1}{2})}\right) \nonumber \\&+ \frac{\mathrm {i}\omega _1}{\omega _2} \int _0^{\infty } \left( \frac{ \log (1{+}e^{-2\pi s})}{1{-}e^{\frac{2\pi }{\omega _2} (\omega _1 s{-} \mathrm {i}(z-\frac{\omega _1}{2}) )}} {+} \frac{ \log (1{+}e^{-2\pi s})}{1{-}e^{-\frac{2\pi }{\omega _2} ( \omega _1 s+ \mathrm {i}(z-\frac{\omega _1}{2}))}} \right) \mathrm {d}s \, .\nonumber \\ \end{aligned}$$
(B.11)

In order to remove the half-period shift of z, we shift the integration variable \(s\mapsto s\pm \mathrm {i}/2\) in each of the two terms on the second line, which gives

$$\begin{aligned} \log F(z|\omega _1,\omega _2)= & {} \frac{\omega _2}{2\pi \mathrm {i}\omega _1} \mathrm{Li}_2\left( e^{\frac{2\pi \mathrm {i}}{\omega _2}(z-\frac{\omega _1}{2})}\right) + \sum _{\epsilon =\pm }\frac{\mathrm {i}\omega _1}{\omega _2} \int _{\epsilon \mathrm {i}/2}^{\infty } \frac{\log (1-e^{-2\pi s})}{1-e^{\frac{2\pi }{\omega _2}(\epsilon \omega _1 s - \mathrm {i}z) }} \,\mathrm {d}s.\nonumber \\ \end{aligned}$$
(B.12)

Further decomposing \( \int _{\epsilon \mathrm {i}/2}^\infty = \int _{0}^\infty - \int _0^{\epsilon \mathrm {i}/2}\) and sending \(s\mapsto -s\) in the integral from 0 to \(-\mathrm {i}/2\), this can be rewritten as

$$\begin{aligned}&\log F(z|\omega _1,\omega _2) =\frac{\omega _2}{2\pi \mathrm {i}\omega _1} \mathrm{Li}_2\left( e^{\frac{2\pi \mathrm {i}}{\omega _2}(z-\frac{\omega _1}{2})}\right) +\frac{2\pi \mathrm {i}\omega _1}{\omega _2} \int _0^{\mathrm {i}/2} \frac{\left( s-\frac{\mathrm {i}}{2}\right) \mathrm {d}s}{1-e^{\frac{2\pi }{\omega _2}(\omega _1 s- \mathrm {i}z) }} \nonumber \\&\quad +\frac{\mathrm {i}\omega _1}{\omega _2}\int _0^\infty \mathrm {d}s\, \log \left( 1-e^{-2\pi s}\right) \left[ \frac{1}{1-e^{-\frac{2\pi }{\omega _2}(\omega _1 s+\mathrm {i}z)}} +\frac{1}{1-e^{\frac{2\pi }{\omega _2}(\omega _1 s-\mathrm {i}z)}}\right] .\qquad \end{aligned}$$
(B.13)

Finally, using the identity (easily established by integrating by parts)

$$\begin{aligned} \int _0^x \frac{(s-x)\mathrm {d}s}{1-e^{as-y}}=-\frac{x}{a}\, \log \left( 1-e^y\right) +\frac{1}{a^2}\, \left( \mathrm{Li}_2\left( e^{y-ax}\right) -\mathrm{Li}_2\left( e^y\right) \right) , \end{aligned}$$
(B.14)

one obtains

$$\begin{aligned}&\log F(z|\omega _1,\omega _2) = {1\over 2}\, \log \left( 1-e^{2\pi \mathrm {i}z/\omega _2}\right) +\frac{\omega _2}{2\pi \mathrm {i}\omega _1}\, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}z/\omega _2}\right) \nonumber \\&\quad +\frac{\mathrm {i}\omega _1}{\omega _2}\int _0^\infty \mathrm {d}s\, \log \left( 1-e^{-2\pi s}\right) \left[ \frac{1}{1-e^{-\frac{2\pi }{\omega _2}(\omega _1 s+\mathrm {i}z)}} +\frac{1}{1-e^{\frac{2\pi }{\omega _2}(\omega _1 s-\mathrm {i}z)}}\right] .\qquad \quad \end{aligned}$$
(B.15)

This integral representation plays a key role in relating two types of solution of the RH problem as discussed in Sect. 4.5 because the last line can be equivalently rewritten in terms of the functions (C.1) as

$$\begin{aligned} \frac{\mathrm {i}\omega _2}{2\omega _1}\left( {\mathcal {J}}_{1,0}\left( -\frac{\omega _1}{\omega _2},-\frac{z}{\omega _2},0,0\right) - {\mathcal {J}}_{1,0}\left( -\frac{\omega _1}{\omega _2},\frac{z}{\omega _2},0,0\right) \right) -\frac{\pi \mathrm {i}\omega _2}{12\omega _1}\, . \end{aligned}$$
(B.16)

Woronowicz-Type Integral Representations for the Triple Sine Function

By comparing Bridgeland’s expressions for \({\mathcal {Y}}_0\) and \(\log \tau \) with our solution, we arrive at the following conjecture, supported by numerical checks for random values of the arguments:

Conjecture 1

One has the following integral representations for the triple sine function for two coinciding periods,

$$\begin{aligned}&\log S_3 (z|\omega _1,\omega _1,\omega _2) = -\frac{\pi \mathrm {i}}{6}\, B_{3,3}(z|\omega _1,\omega _1,\omega _2) +{1\over 2}\left( 1-\frac{z}{\omega _1}\right) \log \left( 1-e^{2\pi \mathrm {i}z/\omega _1}\right) \nonumber \\&\quad -\frac{1}{2\pi \mathrm {i}}\left( \frac{z-\omega _1}{ \omega _2}+{1\over 2}\right) \, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}z/\omega _1}\right) -\frac{\omega _1}{2\pi ^2\omega _2}\, \mathrm{Li}_3\left( e^{2\pi \mathrm {i}z/\omega _1}\right) \nonumber \\&\quad +\frac{\omega _2^2}{\omega _1^2}\int _0^\infty \mathrm {d}s\, \log \left( 1-e^{-2\pi s}\right) \left[ \frac{s+\frac{\mathrm {i}(z-\omega _1)}{\omega _2}}{e^{-\frac{2\pi }{\omega _1}(\omega _2 s+\mathrm {i}z)}-1} -\frac{s-\frac{\mathrm {i}(z-\omega _1)}{\omega _2}}{e^{\frac{2\pi }{\omega _1}(\omega _2 s-\mathrm {i}z)}-1}\right] \end{aligned}$$
(B.17)

and

$$\begin{aligned}&\log S_3 (z|\omega _1,\omega _1,\omega _2) = -\frac{\pi \mathrm {i}}{6}\, B_{3,3}(z|\omega _1,\omega _1,\omega _2) +\frac{\omega _2^2}{4\pi ^2\omega _1^2}\, \mathrm{Li}_3\left( e^{2\pi \mathrm {i}z/\omega _2}\right) \nonumber \\&\qquad -\frac{1}{12}\, \log \left( 1-e^{2\pi \mathrm {i}(z-\omega _1)/\omega _2 }\right) \nonumber \\&\qquad - \frac{\omega _1}{4\pi \omega _2}\int _0^\infty \mathrm {d}s\, \frac{\bigl (\mathrm{Li}_2 \left( e^{-2\pi s}\right) -2\pi s\log \left( 1-e^{-2\pi s}\right) \bigr )\, \sinh (2\pi s\, \omega _1/\omega _2) }{\sinh \left( \frac{\pi }{\omega _2}(\omega _1 (s-\mathrm {i})+\mathrm {i}z)\right) \sinh \left( \frac{\pi }{\omega _2}(\omega _1 (s+\mathrm {i})-\mathrm {i}z)\right) }\, ,\nonumber \\ \end{aligned}$$
(B.18)

where the triple Bernoulli polynomial \(B_{3,3}\) is given in (B.6).

Note that the integrals in (B.17) and (B.18) can be expressed as linear combinations of functions \({\mathcal {J}}_{m,n}\) and \({\mathcal {I}}_n\) introduced in Appendix C.

C Special Functions Relevant for the New Solution

In this appendix, we define a class of functions which appear in the solution obtained in Sect. 4. We also describe their analytic properties, establish various identities between them, and derive their asymptotic expansion.

Definition and Analytic Properties

For m positive integer and n non-negative integer, we define

$$\begin{aligned} {\mathcal {J}}_{m,n}(x,y,\xi ,\eta )=(-1)^{n+m+1}\int _0^\infty \mathrm {d}s\, \frac{s^n \, \mathrm{Li}_m\left( e^{-2\pi (1- \xi /x)s+2\pi \mathrm {i}(\eta -\xi y/x) }\right) }{\tanh (\pi (s-\mathrm {i}y)/x)}\, . \nonumber \\ \end{aligned}$$
(C.1)

The integral converges provided \(\,\mathrm{Re}\,(\xi /x)<1\) and \(kx+\mathrm {i}y \notin \mathbb {R}^+\) for any \(k\in \mathbb {Z}\). We also introduce a slightly modified version of these functions which removes the pole at \(y=0\) appearing for \({\mathcal {J}}_{m,0}\). Namely, we define

$$\begin{aligned} {\tilde{{\mathcal {J}}}}_{m,n}(x,\xi )=(-1)^{n+m+1}\int _0^\infty \mathrm {d}s\,s^n\left( \frac{\mathrm{Li}_m\left( e^{-2\pi (1- \xi /x)s}\right) }{\tanh (\pi s/x)} -\frac{x}{\pi s}\,\mathrm{Li}_m\left( e^{-2\pi s}\right) \right) , \nonumber \\ \end{aligned}$$
(C.2)

which converges for \(\,\mathrm{Re}\,(\xi /x)<1\) and \(\,\mathrm{Re}\,(x)\ne 0\).

The analytic continuation to the full complex \(\xi \) plane is easily obtained by rewriting the integrals as

$$\begin{aligned} {\mathcal {J}}_{m,n}(x,y,\xi ,\eta )= & {} \varepsilon (-1)^{n+m} \left[ 2 \int _0^\infty \mathrm {d}s\, \frac{s^n \,\mathrm{Li}_m\left( e^{-2\pi (1- \xi /x)s+2\pi \mathrm {i}( \eta -\xi y/x) }\right) }{1-e^{2\pi \varepsilon (s-\mathrm {i}y)/x}} \right. \nonumber \\&\left. \qquad -\frac{n!}{(2\pi )^{n+1}}\, \frac{\mathrm{Li}_{m+n+1}\left( e^{2\pi \mathrm {i}( \eta -\xi y/x)}\right) }{(1- \xi /x)^{n+1}}\right] , \end{aligned}$$
(C.3)
$$\begin{aligned} {\tilde{{\mathcal {J}}}}_{m,n}(x,\xi )= & {} \varepsilon (-1)^{n+m}\left[ 2\int _0^\infty \mathrm {d}s\,s^n\left( \frac{\mathrm{Li}_m\left( e^{-2\pi (1- \xi /x)s}\right) }{1-e^{2\pi \varepsilon s/x}} -\frac{x}{\pi s}\,\mathrm{Li}_m\left( e^{-2\pi s}\right) \right) \right. \nonumber \\&\left. \qquad -\frac{n!}{(2\pi )^{n+1}}\, \frac{\zeta (m+n+1)}{(1- \xi /x)^{n+1}}\right] , \end{aligned}$$
(C.4)

where we have set \(\varepsilon =\text{ sgn }(\,\mathrm{Re}\,x)\). The integrals are now convergent for any \(\xi \), whereas the pole at \(\xi =x\) reflects the original obstruction.

The analytic structure in the variables x and y is more complicated. As indicated above, the integrand in (C.1) has a series of poles at \(s=\mathrm {i}(y+kx)\), \(k\in \mathbb {Z}\), which can cross the integration contour as x and y are varied As a result, \({\mathcal {J}}_{m,n}\) is a multi-valued function: it jumps when the arguments (xy) cross one of the hypersurfaces in \(\mathbb {C}^2\) given by

$$\begin{aligned} \,\mathrm{Re}\,(y+k x)=0, \qquad \,\mathrm{Im}\,(y+k x) <0. \end{aligned}$$
(C.5)

The jump is given by the following proposition which follows from the residue evaluation

Proposition 1

Let \(x_k,y_k\) satisfy (C.5). Then, the difference between values of \({\mathcal {J}}_{m,n}\) evaluated from the two sides of the hypersurface with \(\pm \,\mathrm{Re}\,(y+kx)>0\), respectively, is

$$\begin{aligned} \Delta {\mathcal {J}}_{m,n}(x_k,y_k,\xi ,\eta )= 2\mathrm {i}^{n-1} (-1)^{n+m} x_k(y_k+kx_k)^n \,\mathrm{Li}_m\left( e^{2\pi \mathrm {i}(\eta -y_k++k(\xi -x_k))}\right) . \end{aligned}$$

A similar picture holds for \({\tilde{{\mathcal {J}}}}_{m,n}\) with the difference that now all poles hit the integration contour at the same hypersurface. It is described by

Proposition 2

Let \(x\in \mathrm {i}\mathbb {R}^-\). Then, the difference between values of \({\tilde{{\mathcal {J}}}}_{m,n}\) evaluated from the two sides of this line with \(\pm \,\mathrm{Re}\,x>0\), respectively, is

$$\begin{aligned} \Delta {\tilde{{\mathcal {J}}}}_{m,n}(x,\xi )= 2\mathrm {i}^{n-1} (-1)^{m+n} x^{n+1}\sum _{k=1}^\infty k^n\mathrm{Li}_m\left( e^{2\pi k(\xi -x)}\right) . \end{aligned}$$

Identities

By differentiating under the sign of the integral and integrating by parts, it is straightforward to prove

Proposition 3

$$\begin{aligned} \partial _\eta {\mathcal {J}}_{m,n}= & {} -2\pi \mathrm {i}{\mathcal {J}}_{m-1,n}, \\ \partial _\xi {\mathcal {J}}_{m,n}= & {} \frac{2\pi }{x}\, {\mathcal {J}}_{m-1,n+1}+\frac{2\pi \mathrm {i}y}{x}\, {\mathcal {J}}_{m-1,n}, \\ \partial _y{\mathcal {J}}_{m,n}= & {} -\mathrm {i}n {\mathcal {J}}_{m,n-1}+2\pi \mathrm {i}{\mathcal {J}}_{m-1,n} +\delta _{n,0}(-1)^{m+n}\,\frac{\mathrm{Li}_m\left( e^{2\pi \mathrm {i}(\eta -\xi y/x) }\right) }{\tan (\pi y/x)}\, , \\ \partial _x{\mathcal {J}}_{m,n}= & {} -\frac{2\pi }{x}\, {\mathcal {J}}_{m-1,n+1}-\frac{2\pi \mathrm {i}y}{x}\, {\mathcal {J}}_{m-1,n} +\frac{n+1}{x}\, {\mathcal {J}}_{m,n}+\frac{\mathrm {i}n y}{x}\, {\mathcal {J}}_{m,n-1} \\&+\delta _{n,0}(-1)^{m+n+1}\,\frac{y}{x}\,\frac{\mathrm{Li}_m\left( e^{2\pi \mathrm {i}(\eta -\xi y/x) }\right) }{\tan (\pi y/x)}\, , \end{aligned}$$

and

$$\begin{aligned} \partial _x{\tilde{{\mathcal {J}}}}_{m,n}= & {} \frac{n+1}{x}\,{\tilde{{\mathcal {J}}}}_{m,n}-\frac{2\pi }{x}\, {\tilde{{\mathcal {J}}}}_{m-1,n+1}, \\ (\partial _x+\partial _\xi ){\tilde{{\mathcal {J}}}}_{m,n}= & {} \frac{n+1}{x}\, {\tilde{{\mathcal {J}}}}_{m,n} +(-1)^{m+n+1}\,\frac{n!\,\zeta (m+n)}{\pi (2\pi )^n}\,. \end{aligned}$$

In particular, we have the following relations

$$\begin{aligned} (x\partial _x+y\partial _y){\mathcal {J}}_{m,n}= & {} (n+1){\mathcal {J}}_{m,n}-2\pi {\mathcal {J}}_{m-1,n+1}, \nonumber \\ (\partial _y+\partial _\eta ){\mathcal {J}}_{m,n}= & {} -\mathrm {i}n {\mathcal {J}}_{m,n-1} +\delta _{n,0}(-1)^{m+n}\,\frac{\mathrm{Li}_m\left( e^{2\pi \mathrm {i}(\eta -\xi y/x) }\right) }{\tan (\pi y/x)}\, , \nonumber \\ (\partial _x+\partial _\xi ){\mathcal {J}}_{m,n}= & {} \frac{n+1}{x}\, {\mathcal {J}}_{m,n}+\frac{\mathrm {i}n y}{x}\, {\mathcal {J}}_{m,n-1}\nonumber \\&+\delta _{n,0}(-1)^{m+n+1}\,\frac{y}{x}\,\frac{\mathrm{Li}_m\left( e^{2\pi \mathrm {i}(\eta -\xi y/x) }\right) }{\tan (\pi y/x)}\, . \end{aligned}$$
(C.6)

Asymptotic Expansion

In the discussion of the \(\tau \) function, we use the asymptotic expansions of the functions \({\mathcal {J}}_{m,n}\) and \({\tilde{{\mathcal {J}}}}_{m,n}\) at vanishing values of \(\xi \) and \(\eta \). They are given by the following

Proposition 4

At large x and y such that x/y is kept fixed, one has the following asymptotic expansions

$$\begin{aligned}&{\mathcal {J}}_{m,n}(x,y,0,0) +{\mathcal {J}}_{m,n}(x,-y,0,0)\nonumber \\&\quad =\frac{4(-1)^{n+m}}{(2\pi )^{n+1}} \sum _{k=1}^\infty \frac{(n{+}2k{-}1)!}{ (2k{-}1)!}\, \frac{\zeta (m{+}n{+}2k)}{x^{2k-1}}\mathrm{Li}_{1-2k}\left( e^{2\pi \mathrm {i}y/x}\right) , \nonumber \\&{\tilde{{\mathcal {J}}}}_{m,n}(x,0) = 2(-1)^{n+m+1}\sum _{k=1}^\infty \frac{(n+2k-1)!}{(2\pi )^{n+1}(2k)!}\zeta (m+n+2k)\,B_{2k}\, x^{1-2k}.\nonumber \\ \end{aligned}$$
(C.7)

Proof

Note that in the limit we are interested in, we have

$$\begin{aligned}&\frac{1}{\tanh (\pi (s-\mathrm {i}y)/x)}=1+2\mathrm{Li}_0\left( e^{-2\pi (s-\mathrm {i}y)/x}\right) \nonumber \\&\quad =1+2\sum _{k=0}^\infty \frac{(-1)^k}{k!}\, \mathrm{Li}_{-k}\left( e^{2\pi \mathrm {i}y/x}\right) \left( \frac{2\pi s}{x}\right) ^k. \end{aligned}$$
(C.8)

Therefore, after setting \(\xi =\eta =0\) in (C.1) and using

$$\begin{aligned} \int _0^\infty \mathrm {d}s\, s^n \,\mathrm{Li}_m\left( e^{-2\pi s}\right) =\frac{n!}{(2\pi )^{n+1}}\, \zeta (m+n+1), \end{aligned}$$
(C.9)

we obtain the following asymptotic expansion

$$\begin{aligned} \begin{aligned} {\mathcal {J}}_{m,n}(x,y,0,0)=&\, (-1)^{n+m+1}\left[ \frac{n!}{(2\pi )^{n+1}}\, \zeta (m+n+1) \right. \\&\, \left. +2\sum _{k=0}^\infty \frac{(-1)^k(n+k)!}{(2\pi )^{n+1} k!}\, \zeta (m{+}n{+}k{+}1)\,\mathrm{Li}_{-k}\left( e^{2\pi \mathrm {i}y/x}\right) x^{-k}\right] . \end{aligned} \nonumber \\ \end{aligned}$$
(C.10)

To combine two such expansions evaluated at \(\pm y\), we use (A.3) and (A.5). As a result, the first term in (C.10) is canceled by the \(k=0\) contribution, the terms with k even vanish, and terms with k odd are doubled. This gives exactly the first statement of the proposition.

To get the second statement, we follow the same steps except that, instead of (C.8), one starts with the expansion

$$\begin{aligned} \frac{1}{\tanh (\pi s/x)} =\frac{x}{\pi s}+2\sum _{k=1}^\infty \frac{B_{2k}}{(2k)!}\,\left( \frac{2\pi s}{x}\right) ^{2k-1}. \end{aligned}$$
(C.11)

Its first term cancels the last term in (C.2) and the remaining series can be easily integrated using (C.9) which gives the desired statement. \(\square \)

Finally, we also introduce functions appearing in the expression for the \(\tau \) function

$$\begin{aligned} {\mathcal {I}}_n(x,\xi ,\eta )=\int _0^{\pi \mathrm {i}x}\mathrm {d}y\left( \frac{\mathrm{Li}_n\left( e^{-2\xi y+2\pi \mathrm {i}\eta }\right) }{\tanh (y)}-\frac{\mathrm{Li}_n\left( e^{2\pi \mathrm {i}\eta }\right) }{y}\right) +\mathrm{Li}_n\left( e^{2\pi \mathrm {i}\eta }\right) \log x. \nonumber \\ \end{aligned}$$
(C.12)

Note their special value at \(\xi =0\):

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_n(x,0,\eta )=&\,\Bigl (\log \left( 1-e^{2\pi \mathrm {i}x}\right) -\pi \mathrm {i}x-\log (-2\pi \mathrm {i})\Bigr )\mathrm{Li}_n\left( e^{2\pi \mathrm {i}\eta }\right) \!. \end{aligned} \end{aligned}$$
(C.13)

D Details on Summation of Finite Case Results

In this appendix, we analyze the approach toward definition of the Darboux coordinates (2.19) and their \(\tau \) function based on the attempt to perform the sum over charges in the results emerging in the finite case, (2.13) and (2.14). Throughout we use the shorthand notations \({\hat{w}}=w/t\) and \({\hat{v}}_\beta =v_\beta /t\), and restrict to the region in the moduli space where \(\,\mathrm{Re}\,{\hat{w}}<\,\mathrm{Re}\,{\hat{v}}_\beta <0\) for all \(\beta \in B^+\) which allows us to apply the integral representations (A.11) and (A.12). Furthermore, we assume that \(\,\mathrm{Re}\,\vartheta _\beta \in (0,1)\), while \(\,\mathrm{Re}\,\varphi =0\) so that we can identify \([\varphi ]\) with \(\varphi \) (see (A.4)).

Darboux Coordinates

Using the finite case solution (2.13) and the BPS data given in (2.17), one finds that the functions (2.19) take the following form

$$\begin{aligned} {\mathcal {Y}}_i=\sum _{\beta \in B^+} n_\beta \beta _i {\mathcal {Y}}^{(1)}_\beta , \qquad {\mathcal {Y}}_0=\sum _{\beta \in B^+} n_\beta {\mathcal {Y}}^{(2)}_\beta -\frac{\chi (X)}{2} {\mathcal {Y}}^{(2)}_0, \end{aligned}$$
(D.1)

where we denoted

$$\begin{aligned} {\mathcal {Y}}^{(1)}_\beta= & {} \sum _{k=1}^{\infty } \log \Lambda ({\hat{v}}_\beta -k{\hat{w}},1-\vartheta _\beta +k\varphi )\nonumber \\&\quad -\sum _{k=0}^{\infty } \log \Lambda (-{\hat{v}}_\beta -k{\hat{w}},\vartheta _\beta +k\varphi ), \end{aligned}$$
(D.2a)
$$\begin{aligned} {\mathcal {Y}}^{(2)}_\beta= & {} -\sum _{k=1}^{\infty } k \log \Lambda ({\hat{v}}_\beta -k{\hat{w}},1-\vartheta _\beta +k\varphi )\nonumber \\&\quad -\sum _{k=0}^{\infty } k\log \Lambda (-{\hat{v}}_\beta -k{\hat{w}},\vartheta _\beta +k\varphi ), \end{aligned}$$
(D.2b)
$$\begin{aligned} {\mathcal {Y}}^{(2)}_\beta= & {} -2\sum _{k=1}^{\infty } k \log \Lambda (-k{\hat{w}},k\varphi ). \end{aligned}$$
(D.2c)

Under our assumptions, the real part of the arguments of all \(\Lambda \)-functions are positive, therefore one can apply the integral representation (A.11). Substituting it into (D.2c) and exchanging the sum over k with the integral, one obtains

$$\begin{aligned}&{\mathcal {Y}}^{(1)}_\beta = -\int _0^\infty \frac{\mathrm {d}s}{s}\left[ \left( \frac{e^{(\vartheta _\beta -\varphi )s}}{\left( e^{({\hat{w}}-\varphi )s}-1\right) \left( e^{s}-1\right) } +\frac{{1\over 2}-\frac{1}{s}-\vartheta _\beta {+}\varphi }{e^{{\hat{w}}s}-1}{-}\frac{\varphi \, e^{{\hat{w}}s}}{\left( e^{{\hat{w}}s}-1\right) ^2}\right) e^{({\hat{w}}-{\hat{v}}_\beta )s}\right. \nonumber \\&\quad \left. -\left( \frac{e^{(1-\vartheta _\beta )s}}{\left( e^{({\hat{w}}-\varphi )s}-1\right) \left( e^{s}-1\right) } -\frac{{1\over 2}+\frac{1}{s}-\vartheta _\beta }{e^{{\hat{w}}s}-1}-\frac{\varphi \, e^{{\hat{w}}s}}{\left( e^{{\hat{w}}s}-1\right) ^2}\right) e^{{\hat{v}}_\beta s}\right] , \end{aligned}$$
(D.3a)
$$\begin{aligned}&{\mathcal {Y}}^{(2)}_\beta = -\int _0^\infty \frac{\mathrm {d}s}{s}\left[ \left( \frac{e^{(\vartheta _\beta -\varphi )s}}{\left( e^{({\hat{w}}-\varphi )s}-1\right) ^2\left( e^{s}-1\right) } +\frac{{1\over 2}-\frac{1}{s}-\vartheta _\beta }{(e^{{\hat{w}}s}-1)^2}-\frac{\varphi \left( e^{{\hat{w}}s}+1\right) }{\left( e^{{\hat{w}}s}-1\right) ^3}\right) e^{({\hat{w}}-{\hat{v}}_\beta )s} \right. \nonumber \\&\quad \left. +\left( \frac{e^{(1-\vartheta _\beta -\varphi )s}}{\left( e^{({\hat{w}}-\varphi )s}-1\right) ^2\left( e^{s}-1\right) } -\frac{{1\over 2}+\frac{1}{s}-\vartheta _\beta }{(e^{{\hat{w}}s}-1)^2}-\frac{\varphi \left( e^{{\hat{w}}s}+1\right) }{\left( e^{{\hat{w}}s}-1\right) ^3}\right) e^{({\hat{w}}+{\hat{v}}_\beta )s}\right] , \end{aligned}$$
(D.3b)
$$\begin{aligned}&\quad {\mathcal {Y}}^{(2)}_0= \!-\!2\int _0^\infty \frac{\mathrm {d}s}{s}\left( \frac{e^{(1-\varphi )s}}{(e^{({\hat{w}}-\varphi )s}\!-\!1)^2\left( e^{s}-1\right) } \!-\!\frac{{1\over 2}+\frac{1}{s}}{(e^{{\hat{w}}s}-1)^2} \!-\!\frac{\varphi \left( e^{{\hat{w}}s}+1\right) }{(e^{{\hat{w}}s}\!-\!1)^3}\right) e^{{\hat{w}}s}. \end{aligned}$$
(D.3c)

Let us first analyze \({\mathcal {Y}}^{(1)}_\beta \) for vanishing \(\vartheta _\beta \) and \(\varphi \). It is easy to check that each round bracket in the integrand behaves near \(s=0\) as \(1/(12 {\hat{w}})\). While each such term gives rise to a divergent integral, the contributions of the two brackets cancel against each other so that the whole integral is convergent. Changing variable \(s\mapsto -s\) in the second term, we can recombine the two integrals over \(\mathbb {R}^+\) into a single integral over the real axis. Although the integrand has a pole at \(s=0\), the cancelation of this divergence in the original integral implies that the integral should be defined through the principle value prescription. Thus, we have

$$\begin{aligned} \begin{aligned} {\mathcal {Y}}^{(1)}_\beta =&\, \text{ P.v. }\int _{-\infty }^\infty \frac{\mathrm {d}s}{s}\left( \frac{1}{e^s-1}+{1\over 2}-\frac{1}{s}\right) \frac{e^{-{\hat{v}}_\beta s}}{e^{-{\hat{w}}s}-1} \\ =&\, \int _C \frac{\mathrm {d}s}{s}\left( \frac{1}{e^s-1}+{1\over 2}-\frac{1}{s}\right) \frac{e^{-{\hat{v}}_\beta s}}{e^{-{\hat{w}}s}-1}-\frac{\pi \mathrm {i}}{12 {\hat{w}}}\, , \end{aligned} \end{aligned}$$
(D.4)

where in the second line the contour C goes along the real axis avoiding the origin from above. Noticing that the first term in the round brackets gives rise to the function \({\mathcal {F}}_{0,0}\) (B.1), while the other two terms can be evaluated using (A.6), one finally obtains

$$\begin{aligned} {\mathcal {Y}}^{(1)}_\beta = {\mathcal {F}}_{0,0}(-{\hat{v}}_\beta |-{\hat{w}}, 1)+\frac{{\hat{w}}}{2\pi \mathrm {i}}\, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}{\hat{v}}_\beta /{\hat{w}}}\right) -{1\over 2}\log \left( 1-e^{2\pi \mathrm {i}{\hat{v}}_\beta /{\hat{w}}}\right) -\frac{\pi \mathrm {i}}{12 {\hat{w}}} . \nonumber \\ \end{aligned}$$
(D.5)

Given the invariance of \({\mathcal {F}}_{n,m}\) under simultaneous rescaling of all its arguments, this result coincides with the function B(vwt) in [12, Thm 5.2].

The situation with \({\mathcal {Y}}^{(2)}_\beta \) and \({\mathcal {Y}}^{(2)}_0\) is more complicated. In contrast to \({\mathcal {Y}}^{(1)}_\beta \), the integrands in (D.3b) and (D.3c) behave near \(s=0\) as \(1/(6{\hat{w}}^2 s^2)\). Hence, both integrals are divergent. The only exception is the case where the condition (3.1) is satisfied because it ensures that in the combination entering \({\mathcal {Y}}_0\) (D.1) the divergence is canceled. Then, one can repeat the same steps as in (D.7) and arrive at the expression

$$\begin{aligned} {\mathcal {Y}}_0= & {} \sum _{\beta \in B^+} n_\beta \left[ {\mathcal {F}}_{1,0}(-{\hat{v}}_\beta |-{\hat{w}},1)-\frac{{\hat{w}}}{2\pi ^2}\, \mathrm{Li}_3\left( e^{2\pi \mathrm {i}{\hat{v}}_\beta /{\hat{w}}}\right) \right. \nonumber \\&\left. +\frac{1-2{\hat{v}}_\beta }{4\pi \mathrm {i}}\, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}{\hat{v}}_\beta /{\hat{w}}}\right) +\frac{{\hat{v}}}{2{\hat{w}}} \log \left( 1-e^{2\pi \mathrm {i}{\hat{v}}_\beta /{\hat{w}}}\right) \right] \nonumber \\&-\frac{\chi _X}{2}\left[ {\mathcal {F}}_{1,0}(0|-{\hat{w}},1)-\frac{{\hat{w}}\zeta (3)}{2 \pi ^2}+\frac{\zeta (2)}{4\pi \mathrm {i}}\right] +\sum _{\beta \in B^+} n_\beta \,\frac{\pi \mathrm {i}{\hat{v}}_\beta }{12 {\hat{w}}^2}\, , \end{aligned}$$
(D.6)

where the two square brackets can be seen as coming from \({\mathcal {Y}}^{(2)}_\beta \) and \({\mathcal {Y}}^{(2)}_0\), respectively, while the last term results from the residue at \(s=0\). For the case of the conifold (2.18), this result perfectly agrees with the function D(vwt) in [12, Thm 5.2].

For more general CY threefolds which do not obey the condition (3.1), to define \({\mathcal {Y}}^{(2)}_\beta \) and \({\mathcal {Y}}^{(2)}_0\), we have to ignore the divergence at \(s=0\) and postulate that they are given by the integrals along the contour C shifted away from the origin. Then, \({\mathcal {Y}}^{(2)}_\beta \) and \({\mathcal {Y}}^{(2)}_0\) equal to the first and second square brackets in (D.6), respectively. Note that in this way the last term gets dropped.

After switching on the variables \(\vartheta _\beta \) and \(\varphi \), the same problem with divergence arises already for all three functions (D.3c) and even in the conifold case. For instance, the integrand in \({\mathcal {Y}}^{(1)}_\beta \) near \(s=0\) behaves as \(\frac{\varphi (2(\vartheta _\beta {\hat{w}}-\varphi {\hat{v}}_\beta )-{\hat{w}})}{{\hat{w}}^2({\hat{w}}-\varphi )s^2}\). To define all three functions, we proceed as above: we rewrite them as an integral along the real axis and declare that the integration contour is given by C. This gives

$$\begin{aligned} {\mathcal {Y}}^{(1)}_\beta= & {} \int _C\frac{\mathrm {d}s}{s}\left[ \frac{e^{-({\hat{v}}_\beta -\vartheta _\beta )s}}{\left( e^{-({\hat{w}}-\varphi )s}-1\right) \left( e^{s}-1\right) } +\frac{\left( {1\over 2}-\frac{1}{s}-\vartheta _\beta \right) e^{-{\hat{w}}s}}{e^{-{\hat{w}}s}-1}+\frac{\varphi \, e^{-({\hat{v}}_\beta +{\hat{w}})s}}{\left( e^{-{\hat{w}}s}-1\right) ^2}\right] , \nonumber \\ {\mathcal {Y}}^{(2)}_\beta= & {} -\int _C\frac{\mathrm {d}s}{s}\left[ \frac{e^{-({\hat{v}}_\beta +{\hat{w}}-\vartheta _\beta -\varphi )s}}{\left( e^{-({\hat{w}}-\varphi )s}-1\right) ^2\left( e^{s}-1\right) } +\frac{\left( {1\over 2}-\frac{1}{s}-\vartheta _\beta \right) e^{-({\hat{v}}_\beta +{\hat{w}})s}}{\left( e^{-{\hat{w}}s}-1\right) ^2}\right. \nonumber \\&\left. +\frac{\varphi \, e^{-({\hat{v}}_\beta +{\hat{w}})s}\left( e^{-{\hat{w}}s}+1\right) }{\left( e^{-{\hat{w}}s}-1\right) ^3}\right] , \nonumber \\ {\mathcal {Y}}^{(2)}_0= & {} -\int _C \frac{\mathrm {d}s}{s}\left[ \frac{e^{-({\hat{v}}_\beta -\varphi )s}}{(e^{-({\hat{w}}-\varphi )s}-1)^2\left( e^{s}-1\right) } +\frac{\left( {1\over 2}-\frac{1}{s}\right) e^{-{\hat{w}}s}}{(e^{-{\hat{w}}s}-1)^2}\right. \nonumber \\&\left. +\frac{\varphi \, e^{-{\hat{w}}s}\left( e^{-{\hat{w}}s}+1\right) }{(e^{-{\hat{w}}s}-1)^3}\right] +\Delta {\mathcal {J}}_2, \end{aligned}$$
(D.7)

where

$$\begin{aligned} \Delta {\mathcal {J}}_2=-\int _0^\infty \frac{\mathrm {d}s}{s}\left[ \frac{e^{-({\hat{w}}-\varphi )s}}{(e^{-({\hat{w}}-\varphi )s}-1)^2}-\frac{e^{-{\hat{w}}s}}{(e^{-{\hat{w}}s}-1)^2}\right] . \end{aligned}$$
(D.8)

The contribution \(\Delta {\mathcal {J}}_2\) is still divergent. It arises because the integrand in (D.3c) is not antisymmetric with respect to \(s\rightarrow -s\). Proceeding as for the divergence at \(s=0\), we simply drop this contribution so that one has \({\mathcal {Y}}^{(2)}_0={\mathcal {Y}}^{(2)}_{\beta =0}\). Finally, computing the integrals using (A.6), taking into account the homogeneity of the functions \({\mathcal {F}}_{n,m}\), and substituting the resulting expressions into (D.1), one arrives at the expressions (3.2) given in the main text.

\(\tau \) Function

The analysis of the \(\tau \) function is similar, but slightly more complicated. Its expression (2.14) implies that

$$\begin{aligned} \log \tau = \sum _{\beta \in B^+}n_\beta \, {\mathcal {T}}_\beta -\frac{\chi (X)}{2}\, {\mathcal {T}}_0, \end{aligned}$$
(D.9)

where

$$\begin{aligned} {\mathcal {T}}_\beta= & {} \sum _{k=1}^{\infty } \log \Upsilon ({\hat{v}}_\beta -k{\hat{w}},1-\vartheta _\beta +k\varphi )+\sum _{k=0}^{\infty } \log \Upsilon (-{\hat{v}}_\beta -k{\hat{w}},\vartheta _\beta +k\varphi ), \nonumber \\ \end{aligned}$$
(D.10a)
$$\begin{aligned} {\mathcal {T}}_0= & {} 2\sum _{k=1}^{\infty } \log \Upsilon (-k{\hat{w}},k\varphi ). \end{aligned}$$
(D.10b)

Using the integral representation (A.12) and performing the sum over k under the integral, the two functions in (D.10) can be rewritten as

$$\begin{aligned} {\mathcal {T}}_\beta= & {} \int _0^\infty \frac{\mathrm {d}s}{s}\left[ \left( \frac{\frac{1}{s^2}-{1\over 2}\, B_2(\vartheta _\beta )}{1-e^{{\hat{w}}s}} +\frac{\left( \vartheta _\beta (1-e^s)-1\right) e^{(1-\vartheta _\beta )s}}{(e^s-1)^2(1-e^{({\hat{w}}-\varphi )s})}\right. \right. \nonumber \\&\left. \left. -\frac{\varphi \, e^{(1-\vartheta _\beta -\varphi +{\hat{w}})s}}{(e^s-1)(1-e^{({\hat{w}}-\varphi )s})^2} \right) e^{{\hat{v}}_\beta s} \right. \nonumber \\+ & {} \left( \frac{\frac{1}{s^2}-{1\over 2}\, B_2(1-\vartheta _\beta )}{1-e^{{\hat{w}}s}} +\frac{\left( (1-\vartheta _\beta )(1-e^s)-1\right) e^{(\vartheta _\beta -\varphi )s}}{(e^s-1)^2(1-e^{({\hat{w}}-\varphi )s})}\right. \nonumber \\&\left. -\frac{\varphi \, e^{(\vartheta _\beta -\varphi )s}}{(e^s-1)(1-e^{({\hat{w}}-\varphi )s})^2} \right) e^{({\hat{w}}-{\hat{v}}_\beta ) s} \nonumber \\+ & {} \left. \!\!\frac{\varphi (1\!-\!2\vartheta _\beta )\,e^{{\hat{w}}s}}{2(1\!-\!e^{{\hat{w}}s})^2}\left( e^{{\hat{v}}_\beta s}\!-\!e^{-\!{\hat{v}}_\beta s}\right) \!-\!\frac{\varphi ^2 e^{{\hat{w}}s}(e^{{\hat{w}}s}\!+\!1)}{2(1-e^{{\hat{w}}s})^3}\left( e^{{\hat{v}}_\beta s}\!\!+\!e^{-\!{\hat{v}}_\beta s}\right) \right] \!+\!\!{\mathcal {S}}_\beta ,\, \end{aligned}$$
(D.11a)
$$\begin{aligned} {\mathcal {T}}_0= & {} 2\int _0^\infty \frac{\mathrm {d}s}{s}\left[ \left( \frac{\frac{1}{s^2}{-}\frac{1}{12}}{1{-}e^{{\hat{w}}s}} {-}\frac{e^{(1-\varphi )s}}{(e^s-1)^2(1{-}e^{({\hat{w}}{-}\varphi )s})} {-}\frac{\varphi \, e^{(1-\varphi )s}}{(e^s{-}1)(1{-}e^{({\hat{w}}{-}\varphi )s})^2}\right) e^{{\hat{w}}s} \right. \nonumber \\&\left. +\frac{\varphi \, e^{{\hat{w}}s}}{2(1-e^{{\hat{w}}s})^2}-\frac{\varphi ^2\, e^{{\hat{w}}s}(e^{{\hat{w}}s}+1)}{2(1-e^{{\hat{w}}s})^3} \right] +{\mathcal {S}}_0, \end{aligned}$$
(D.11b)

where

$$\begin{aligned} {\mathcal {S}}_\beta= & {} -{1\over 2}\, B_2(\vartheta _\beta )\log (-{\hat{v}}_\beta )- \sum _{k=1}^{\infty }\Biggl [ B_2(k\varphi )\log ((k{\hat{w}})^2-{\hat{v}}_\beta ^2) \Biggr . \nonumber \\&\left. +2k\varphi \left( \log ({\hat{v}}_\beta -k{\hat{w}})-\vartheta \log \frac{1+\frac{{\hat{v}}_\beta }{k{\hat{w}}}}{1-\frac{{\hat{v}}_\beta }{k{\hat{w}}}}\right) +(\vartheta ^2-\vartheta )\log ((k{\hat{w}})^2-{\hat{v}}_\beta ^2)\right] \nonumber \\ \end{aligned}$$
(D.12a)
$$\begin{aligned} {\mathcal {S}}_0= & {} -\sum _{k=1}^{\infty }B_2(k\varphi )\log (-k{\hat{w}}). \end{aligned}$$
(D.12b)

When the condition (3.1) is satisfied and for \(\vartheta _\beta =\varphi =0\), one has

$$\begin{aligned} \begin{aligned} \sum _{\beta \in B^+} n_\beta {\mathcal {S}}_\beta - \frac{\chi _X}{2}\,{\mathcal {S}}_0=&\, -\frac{1}{12}\sum _{\beta \in B^+} n_\beta \left[ \log (-{\hat{v}}_\beta )+\sum _{k=1}^\infty \log \left( 1-\frac{{\hat{v}}_\beta ^2}{(k{\hat{w}})^2}\right) \right] \\ =&\, -\frac{1}{12}\sum _{\beta \in B^+} n_\beta \left[ \log \frac{{\hat{w}}}{2\pi \mathrm {i}}- \frac{\pi \mathrm {i}v_\beta }{w}+\log \left( 1-e^{2\pi \mathrm {i}v_\beta /w}\right) \right] . \end{aligned} \nonumber \\ \end{aligned}$$
(D.13)

Furthermore, the integral resulting from the combination \(\sum _{\beta \in B^+} n_\beta {\mathcal {T}}_\beta -\frac{\chi _X}{2}\,{\mathcal {T}}_0\) is convergent and can be treated in the same way as for Darboux coordinates. Namely, one can rewrite it as an integral over the real axis using the principle value prescription and evaluate it in terms of the functions \({\mathcal {F}}_{n,m}\), using (A.6) and the identity

$$\begin{aligned} \int _0^\infty \frac{\mathrm {d}s}{s}\left[ \frac{1}{s^2} -\frac{e^{s}}{(e^{s}-1)^2} +\frac{1}{6{\hat{w}}s}+\frac{1}{6\left( e^{-{\hat{w}}s}-1\right) }\right] =\frac{1}{12}\, \log \frac{- w}{2\pi } -\zeta '(-1).\nonumber \\ \end{aligned}$$
(D.14)

The result is

$$\begin{aligned} \begin{aligned} \log \tau =&\, \sum _{\beta \in B^+} n_\beta \left[ {\mathcal {F}}_{1,0}(-{\hat{v}}_\beta |-{\hat{w}},1)+\frac{w^2}{(2\pi \mathrm {i}t)^2}\,\mathrm{Li}_3\left( e^{2\pi \mathrm {i}v_\beta /w}\right) +\frac{\pi \mathrm {i}}{12}\, \frac{v_\beta }{w} \right] \\&\, -\frac{\chi _X}{2}\left[ {\mathcal {F}}_{1,0}(0|-{\hat{w}},1)+\frac{w^2}{(2\pi \mathrm {i}t)^2}\,\zeta (3)+\zeta '(-1)\right] , \end{aligned}\qquad \end{aligned}$$
(D.15)

which coincides with [12, Thm 1.2] for the conifold case, up to an irrelevant constant term.

In more general cases, we again encounter divergences which now arise both in the integrals at \(s=0\) and in the series (D.12). Ignoring them, one gets (3.3).

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Alexandrov, S., Pioline, B. Conformal TBA for Resolved Conifolds. Ann. Henri Poincaré 23, 1909–1949 (2022). https://doi.org/10.1007/s00023-021-01129-x

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