Heavenly metrics, BPS indices and twistors

Recently, T. Bridgeland defined a complex hyperkähler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann–Hilbert problem determined by the Donaldson–Thomas invariants. This metric is encoded in a function \(W(z,\theta )\) satisfying a heavenly equation, or a potential \(F(z,\theta )\) satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both W and F in terms of solutions of that system. These expressions are recognized as conformal limits of the ‘instanton generating potential’ and ‘contact potential’ appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce’s original construction of F as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called \(\tau \) function for arbitrary values of the fiber coordinates \(\theta \), in terms of a suitable two-variable generalization of Barnes’ G function.

1. Since hyperkähler (HK) and quaternion-Kähler (QK) spaces are parametrized by solutions of nonlinear differential equations often known as heavenly equations, we refer to such quaternionic metrics as ‘heavenly,’ irrespective of their precise nature. See [1, 2] for the original descriptions in the four-dimensional HK and QK cases, and [3] for a recent discussion of the higher dimensional case.

2. In the context of class S field theories, the full space of stability conditions \(\mathrm{Stab}({\mathcal {D}})\) [16] can be interpreted as the total space of the Coulomb branch fibration over the conformal manifold, but the significance of the complex HK metric on \({\mathcal {M}}\) is yet to be understood.

3. The factor \(1/(2\pi )^2\) appearing in this and a few other equations is due to our normalization of the coordinates \(z^a\) and \(\theta ^a\) which coincides with normalization used in [18], but differs by the factor \(2\pi \mathrm {i}\) from the one in [12, 13].

4. In [12, 13], W has been called the Joyce function, but we prefer to reserve this attribution for F.

5. Here, \(\sigma _{\gamma }\) is a quadratic refinement of the skew-symmetric pairing on the charge lattice \(\Gamma \), a technical device which allows to trivialize the twisted group law \({\mathcal {X}}_\gamma \,{\mathcal {X}}_{\gamma '} =(-1)^{\langle \gamma ,\gamma '\rangle } {\mathcal {X}}_{\gamma +\gamma '}\).

6. The index ‘sf’ stands for ‘semi-flat,’ since the HK metric encoded by the twistor lines (2.6) is flat along the fibers parametrized by \(\theta ^a\). In fact, in contrast to the HK metrics appearing in the physical context before taking the conformal limit, it is also flat along the base \({\mathcal {S}}\) so that the full complex HK space encoded by (2.6) is simply \(\mathbb {C}^{2m}\) endowed with a flat metric of signature (2m, 2m).

7. Our convention is opposite to the one in [12, 18, 21], which can be reached by flipping the sign of the skew-symmetric pairing \(\langle \gamma ,\gamma '\rangle \).

8. The zero mode of F does not contribute to the vector field \({\hat{f}}\) and may be set to 0.

9. The quadratic refinement \(\sigma _{\gamma _i}\) was not included in [19], which relied on a slightly different version of rational DT invariants, which lacks the property of being deformation invariant, see, e.g., [35] for details.

10. Note that this iteration procedure is distinct from the usual iteration of the TBA equations (2.10) which leads to a formal series in powers of \({\mathcal {X}}^{\mathrm{sf}}_{\gamma }\). This second expansion is the subject of the next subsection.

11. Alternatively, one can sum over unrooted labeled tress, in which case the coefficient \(1/|\mathrm{Aut}({\mathcal {T}})|\) in \(K_{\mathcal {T}}\) is replaced by 1/n!.

12. We remark that this change of variables can be done already in (3.17), i.e., it has a generalization to any tree \({\mathcal {T}}\). In that case one defines \(w_{ij}=z_{j} u_i-z_i u_{j}\) for any edge \(e:i\rightarrow j\). Then, the Jacobian is given by \(\left( \sum _i z_i\right) \frac{\prod _{e:i\rightarrow j}z_i z_j}{\prod _i z_i}\) and leads to the integral \(\int _{R_{\mathcal {T}}} \prod _{e:i\rightarrow j} \frac{\mathrm {d}w_{ij}}{w_{ij}}\). The results presented in (3.21) and (3.22) correspond to the particular case of the tree of the simplest topology \(\bullet \!\text{--- }\!\bullet \!\text{-- }\cdots \text{-- }\!\bullet \!\text{--- }\!\bullet \,\).

13. The first formula corrects a sign misprint in [44, Eq.(25)]. The second is obtained from [44, Eq.(20)] by twice integrating by parts. Note that the second integration produces the boundary contribution \(-\frac{1}{12}\log (z)\).

We are indebted to Tom Bridgeland for his nice set of lectures at the work of the Simons collaboration on Special Holonomy in Geometry, Analysis and Physics (Jan 11–13, 2021), which stimulated our interest in this topic, and for subsequent correspondence. We are also grateful to Andy Neitzke for discussions.

Authors and Affiliations

1. Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, 34095, Montpellier, France

   Sergei Alexandrov

2. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589 CNRS-Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, 75005, Paris, France

   Boris Pioline

Corresponding author

Correspondence to Boris Pioline.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Proofs

In this appendix, we present several proofs which have been omitted in the main text.

1.1 Heavenly equation

Let us show that the function W (3.3) satisfies the heavenly equation (2.20). As in the proof of the section condition, we do this with help of an infinite series generated by iterating the relation (3.5). Namely, using (3.8), we get

(A.1)

where in the last line we inverted the labeling of first \(n'\) variables and set \(n=n'+n''\). \(\square \)

1.2 Isomonodromy equation

To prove that the isomonodromy equation (2.15), we first rewrite it in terms of the potentials F and W which can be achieved by multiplying by \(e^{2\pi \mathrm {i}\theta _\gamma }\) and summing over the lattice. This leads to the following equation

where the differential is supposed to act only on the variables \(z^a\). Substituting the expression (3.4) for the Joyce potential and using (3.8) and equation (3.2), one arrives at the condition

(A.3)

Then, note that the first term on the l.h.s. can be rewritten as

(A.4)

Substituting this result into (A.3) and bringing all terms to the l.h.s., the resulting expression becomes

(A.5)

where in the last line we inverted the labeling of the variables labeled from \(n'+1\) to \(n'+n''\) on the previous line. \(\square \)

1.3 Conjecture 1 at low rank

Here, we prove Conjecture 1 presented in Sect. 3.2 in the simplest cases \(n=3\) and \(n=4\). Let us denote \(\gamma _{ij}=\langle \gamma _i,\gamma _j\rangle \) and \(k_{ij}=\frac{t_i t_j}{t_j-t_i}\) so that \(K_{ij}=\gamma _{ij} k_{ij}/(2\pi \mathrm {i})\). The functions \(k_{ij}\) satisfy

The proof can be reduced essentially to a repetitive use of this identity.

1.3.1 \(n=3\)

In this case, we need to show that

This is a direct consequence of (A.6) because it ensures that the expression in the round brackets on the r.h.s. is equal to \(2k_{12}k_{23}\).

1.3.2 \(n=4\)

In this case, we need to prove two relations

Proceeding as in (A.7), in each relation we rewrite all terms so that they have the same product of \(\gamma _{ij}\) factors.

The first relation is then equivalent to

The last 8 terms on the l.h.s. combined pairwise using the identity (A.6) give

The last two terms can in turn be combined with the third and fourth terms in (A.10) so that the full l.h.s. becomes

Due to (A.6), the sum of the last two terms here is equal to the first one so that one indeed obtains \(4 k_{12}k_{23}k_{34}\) as required.

The second relation (A.9) follows from

To prove it, we note that combining the following pairs of terms on the l.h.s., (2,3), (4,5), (8,9), (1,11), (6,10), (5,12), one finds

This further can be rewritten as

Since the sum of the last two terms is combined into the first one, one indeed obtains \(4k_{12}k_{13}k_{14}\) as required.

1.4 Iterated integrals

Here, we prove that the functions \(J(z_1,\dots , z_n)\) defined in (3.19) satisfies the axioms (a)–(d) spelled out in (2.17).

(a) The homogeneity and holomorphicity are manifest from the representation on the second line of (3.20).

(b) Let us denote \(z_{k,l}=\sum _{i=k}^l z_i\) and \(z=z_{1,n}\). Then, bringing all terms in the recursion relation (2.17b) to one side, one finds

Since the last line sums up to \(-2\pi \mathrm {i}\sum _{k=1}^{n}\frac{1}{t_k}\left( z_k\mathrm {d}z-z\mathrm {d}z_k\right) \), the total expression reduces to

Changing variables to \(t_i=\mathrm {i}z_i/(s u_i)\) with \(s, u_i\in \mathbb {R}^+\) and \(\sum _{i=1}^n u_i=1\), the integral factorizes exactly as in (3.20) into

Since the integral over s vanishes, this proves the recursion relation (2.17b).

(c) The vanishing property (2.17c) is a consequence of the identity

To show that this holds, let \(u_i=1/t_i\), and consider

We can show that \(S_n=0\) inductively on n. For \(n=1\) and \(n=2\), this is clear. For \(n>2\), the residue of \(S_n\) at \(u_{n-1}=u_n\) is equal to \(S_{n-2}(u_1,\dots , u_{n-2})\), which vanishes by the induction hypothesis. By symmetry, the same holds for the residue at \(u_i=u_j\). Since \(S_n\) is a rational function with no poles, it must be a constant. And since it vanishes at large \(u_i\), it vanishes identically.

(d) The growth condition is evident from the representation (3.22) since singularities can only arise when the integration region \(R_n(z_1,\dots , z_n)\) touches one of the divisors \(w_i=0\), leading at most to logarithmic singularities. The behavior as \(z_k\rightarrow 0\) is easily read off from (3.20):

where we used the explicit form of \(J_2\) (2.18), whereas \(J_n(z_1,\dots , z_n)\) stays finite as \(z_k\rightarrow 0\) for \(1<k<n\), as the logarithmic divergences at \(u_k=z_k u_{k\pm 1}/z_{k\pm 1}\) cancel. \(\square \)

1.5 Refined potentials

Here, we prove that the refined versions of the Plebański and Joyce potentials, (5.8) and (5.9), satisfy the deformed heavenly equation (5.4) and isomonodromy equation (5.6), respectively. As in the unrefined case, the proof relies on an asymptotic expansion. But the difference is that now this will be the expansion in powers of \({\mathcal {X}}^{\mathrm{sf}}_\gamma \).

Thus, the starting point to establish (5.4) is the formal series (5.10) which implies

(A.22)

(A.23)

where in the last line we, as usual, set \(n=n'+n''\). Substituting these results into the deformed heavenly equation, one obtains

(A.24)

where

It is straightforward to verify that \(S_{ab}\) actually vanishes. Indeed, denoting \(Q_{m,a}=\sum _{k=1}^m q_{k,a}\), we have

Thus, the deformed heavenly equation (5.4) holds.

In a similar way, to establish (5.6), we compute

(A.27)

where \(Z_k=Z_{\gamma _k}\), \(\zeta _m=\sum _{k=1}^m Z_k\) and the function \(\kappa (x)\) was defined at the beginning of Sect. 5. Further noting that

(A.28)

we see that (A.27) can be rewritten as

(A.29)

where

This proves the deformed isomonodromy equation (5.6).

Special functions and useful identities

In this appendix, we collect the definitions and some properties of several special functions as well as various useful identities used in Sect. 4.

1.1 Miscellaneous

1.1.1 Bernoulli polynomials

The Bernoulli polynomials have the following generating function

The first few polynomials are given by

At \(x=0\) they reduce to Bernoulli numbers \(B_n\) and have the following symmetry property

Importantly, the Bernoulli polynomials arise in the inversion formula for polylogarithms, namely,

where

Note that for \(\,\mathrm{Im}\,x\ne 0\) or \(x\notin \mathbb {Z}\), the bracket satisfies \([-x]=1-[x]\).

1.1.2 Useful identities

For \(\,\mathrm{Re}\,a,\,\mathrm{Re}\,b>0\), one has

1.2 Integral representations of Gamma and Barnes functions

1.2.1 Gamma function

The logarithm of the Gamma function has two integral representations due to Binet which are both valid for \(\,\mathrm{Re}\,z>0\) [41, p248-250]:

• first Binet formula

  $$\begin{aligned} \log \Gamma (z) = \left( z-\frac{1}{2}\right) \log z - z + \frac{1}{2}\,\log (2\pi ) + \int _0^\infty \frac{\mathrm {d}s}{s} \left( {1\over 2}-\frac{1}{s}+\frac{1}{e^s-1} \right) e^{-z s}, \end{aligned}$$

  (B.7)

• second Binet formula

  $$\begin{aligned} \log \Gamma (z) = \left( z-\frac{1}{2}\right) \log z - z + \frac{1}{2}\,\log (2\pi ) - \frac{z}{\pi } \int _0^{\infty } \frac{\mathrm {d}s}{s^2+z^2}\, \log \left( 1- e^{-2\pi s} \right) . \end{aligned}$$

  (B.8)

The first formula has the following useful generalization due to Hermite, valid for \(\,\mathrm{Re}\,z>0\) and \(\,\mathrm{Re}\,(z+\eta )>0\) [42]

which follows from (B.7) by using the identities (B.6).

1.2.2 Barnes function

The Barnes function (see, e.g., [43]) is the unique meromorphic function G(z) which satisfies the functional equation

the convexity condition \(\frac{d^3}{dz^3}\log G(z)\ge 0\) for all \(z>1\), and the normalization condition \(G(1)=1\). It has the following useful properties:

• its logarithmic derivative is

  $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}z} \log G(z+1) = \frac{1}{2}\log (2\pi ) -z + \frac{1}{2} + z \frac{\mathrm {d}}{\mathrm {d}z} \log \Gamma (z), \end{aligned}$$

  (B.11)

• Kinkelin’s reflection formula

  $$\begin{aligned} \log \frac{G(1-z)}{G(1+z)} = z \log \left( \frac{\sin (\pi z)}{2\pi z}\right) - \int _0^z \pi x\, \log \sin (\pi x)\, \mathrm {d}x, \end{aligned}$$

  (B.12)

• two integral representations of Binet type, both valid for valid for \(\,\mathrm{Re}\,z>0\) [44],^{Footnote 13}

  $$\begin{aligned} \begin{aligned} \log G(z+1)=&\,z\log \Gamma (z)+ \frac{z^2}{4} -\frac{B_2(z)}{2}\, \log z +\zeta '(-1) -\frac{1}{12} \\&\,- \int _0^\infty \frac{\mathrm {d}s}{s^2} \left( \frac{1}{2}+ \frac{1}{s} +\frac{s}{12} - \frac{1}{1-e^{-s}} \right) e^{-zs}, \end{aligned} \end{aligned}$$

  (B.13)

  and

  $$\begin{aligned} \begin{aligned} \log G(z+1) =&\, \frac{1}{2}\left( z^2-\frac{1}{6}\right) \log z-\frac{3z^2}{4} + \zeta '(-1) + \frac{z}{2}\, \log (2\pi )\\&\, +\frac{1}{2\pi ^2}\int _0^{\infty } \frac{s \mathrm {d}s }{s^2+z^2}\, \Bigl [ 2\pi s \log \left( 1-e^{-2\pi s} \right) - \mathrm{Li}_2\left( e^{-2\pi s} \right) \Bigr ]. \end{aligned}\nonumber \\ \end{aligned}$$

  (B.14)

  where \(B_2(z)\) is the Bernoulli polynomial and \(\zeta '(-1)\) is related to Glaisher’s constant \(\gamma _E\) via \(\zeta '(-1)=\frac{1}{12}-\log \gamma _E\).

We will need a generalization of the first integral representation similar to Hermite’s generalization (B.9) of the first Binet formula. It is given by the following

Proposition 1

For \(\,\mathrm{Re}\,z>0\) and \(\,\mathrm{Re}\,(z+\eta )>0\), one has

Proof

First, let us note the following identity

which can be established by integration by parts and the use of the first Binet formula (B.7). Therefore, the integral representation (B.13) can be rewritten as

Let us now substitute \(z\rightarrow z+\eta \). Then, the last integral term differs from the one in (B.15) by the following contribution

where we used (B.6). Combining this result with terms in the first line of (B.17) after the replacement \(z\rightarrow z+\eta \), one reproduces the first line in (B.15), which completes the proof. \(\square \)

1.3 Generalized Gamma function

The generalized Gamma function \(\Lambda (z,\eta )\) is a function on \(\mathbb {C}^\times \times \mathbb {C}\) defined by [28]

This function has the following properties:

• twisted periodicity

  $$\begin{aligned} \Lambda (z,\eta +1) = \frac{z+\eta }{z}\, \Lambda (z,\eta ), \end{aligned}$$

  (B.20)

• reflection property

  $$\begin{aligned} \Lambda (-z,\eta ) \, \Lambda (z,1-\eta ) = {\left\{ \begin{array}{ll} \left( 1-e^{2\pi \mathrm {i}( z-\eta )} \right) ^{-1}, \quad &{} \,\mathrm{Im}\,z>0,\\ \left( 1-e^{2\pi \mathrm {i}(\eta -z)} \right) ^{-1}, \quad &{} \,\mathrm{Im}\,z<0, \end{array}\right. } \end{aligned}$$

  (B.21)

• two integral representations directly following from the generalized version of the first Binet formula (B.9) and the second Binet formula (B.8), respectively,

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

  (B.22)
  $$\begin{aligned}= & {} \, \left( z+\eta -{1\over 2}\right) \log \frac{z+\eta }{z}\nonumber \\&-\eta -\frac{z+\eta }{\pi }\int _0^\infty \mathrm {d}s\,\frac{ \log \left( 1-e^{-2\pi s}\right) }{s^2+(z+\eta )^2}, \end{aligned}$$

  (B.23)

  where the first representation is valid for \(\,\mathrm{Re}\,z>0\), \(\,\mathrm{Re}\,(z+\eta )>0\), whereas the second requires only the second condition.

• asymptotic series at large z [12, §8.3]

  $$\begin{aligned} \log \Lambda (z,\eta ) = \sum _{k= 2}^\infty \frac{(-1)^k B_k(\eta )}{k(k-1)}\, z^{1-k}. \end{aligned}$$

  (B.24)

  It follows from (B.22) by using \(\frac{s e^{x s}}{e^s-1}=\sum _{k=0}^{\infty } \frac{s^k}{k!} B_k(x)\) and integrating term by term using \(\int _0^\infty s^n e^{-zs}\mathrm {d}s= n!/ z^{n+1}\). Note that this asymptotic expansion is consistent with the periodicity relation (B.20), since \(B_k(\eta +1)-B_k(\eta )=k \eta ^{k-1}\).

1.4 Generalized Barnes function

We define the generalized Barnes function \(\Upsilon (z,\eta )\) on \(\mathbb {C}^\times \times \mathbb {C}\) by

where G(z) is the Barnes function. This definition generalizes the function \(\Upsilon (z)\) introduced in [21, Eq.(17)] which is obtained from (B.25) setting \(\eta =0\). It has the following properties:

• relation to the generalized Gamma function (B.19)

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

  (B.26)

  which follows from the property of the Barnes function (B.11),

• twisted periodicity

  $$\begin{aligned} \Upsilon (z,\eta +1) = (z+\eta )^{-\eta } \Upsilon (z,\eta ), \end{aligned}$$

  (B.27)

• reflection property

  $$\begin{aligned}&\log \frac{\Upsilon (-z,\eta )}{\Upsilon (z,1-\eta )} \nonumber \\&\quad = -\frac{\pi \mathrm {i}}{12}\,B_2(\eta )+ {\left\{ \begin{array}{ll} z\log \left( 1-e^{2\pi \mathrm {i}( z-\eta )} \right) +\frac{1}{2\pi \mathrm {i}}\, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}( z-\eta )} \right) , &{} \,\mathrm{Im}\,z>0\\ z\log \left( 1-e^{2\pi \mathrm {i}(\eta -z)} \right) -\frac{1}{2\pi \mathrm {i}}\, \mathrm{Li}_2\left( e^{2\pi \mathrm {i}(\eta -z)}\right) , &{} \,\mathrm{Im}\,z<0, \end{array}\right. }\nonumber \\ \end{aligned}$$

  (B.28)

  which is a consequence of (B.12),

• two integral representations of Binet type

  $$\begin{aligned} \log \Upsilon (z,\eta )= & {} \int _0^\infty \frac{\mathrm {d}s}{s} \left( \frac{1}{s^2} - \frac{1}{2} \,B_2(\eta ) - \frac{\eta (e^s-1)+1}{(e^s-1)^2}\,e^{(1-\eta )s} \right) e^{-z s}\qquad \quad \quad \end{aligned}$$

  (B.29)
  $$\begin{aligned}&-\frac{1}{2} \,B_2(\eta ) \log z \nonumber \\= & {} \frac{z^2}{2}\, \log \frac{z+\eta }{z}-{1\over 2}\,B_2(\eta )\, \log (z+\eta )-\frac{z\eta }{2}+\frac{\eta ^2}{4} \nonumber \\&+\int _0^{\infty } \frac{\mathrm {d}s }{s^2+(z+\eta )^2} \left[ \frac{1}{\pi }\left( s^2+\eta (z+\eta )\right) \log \left( 1-e^{-2\pi s} \right) \right. \nonumber \\&\left. - \frac{s}{2\pi ^2}\, \mathrm{Li}_2\left( e^{-2\pi s} \right) \right] , \end{aligned}$$

  (B.30)

  directly following from (B.15), (B.9) and (B.14), (B.8), respectively,

• asymptotic series at large z

  $$\begin{aligned} \log \Upsilon (z,\eta ) = -\frac{1}{2}\, B_2(\eta ) \log z + \sum _{k= 3}^\infty \frac{ (-1)^k B_k(\eta )}{k(k-2)}\, z^{2-k}, \end{aligned}$$

  (B.31)

  which follows from (B.29) taking into account

  $$\begin{aligned} \frac{1}{s^2} - \frac{1}{2}\, B_2(\eta ) - \frac{\eta (e^s-1)+1}{(e^s-1)^2}\,e^{(1-\eta )s} = \sum _{k=3}^{\infty } \frac{(k-1)}{k!}\, B_k(1-\eta )\, s^{k-2}\nonumber \\ \end{aligned}$$

  (B.32)

  and integrating term by term. For \(\eta =0\), this reduces to the result given in [21, §5.4].

Solution in the uncoupled finite case

Here, we provide some intermediate steps in deriving (4.6) and (4.7).

1.1 Darboux coordinates

In the uncoupled case, \({\mathcal {X}}_{\gamma '}\) appearing in the r.h.s. of the integral equation (1.4) can be replaced by \({\mathcal {X}}^{\mathrm{sf}}_{\gamma '}\). Then, after combining contributions of opposite charges, this equation takes the form

where

and we changed the integration variable \(t'=\mathrm {i}Z_{\gamma }/(s+\mathrm {i}\vartheta _\gamma )\). Splitting each integral into two parts, one obtains

where in the last term we flipped the sign of s. The last two terms can be combined using (B.4) at \(n=1\), leading to the contribution

If \(\,\mathrm{Re}\,\vartheta _\gamma \in (0,1)\), the integral is easily evaluated to

Combining this with the first term in (C.3), one recognizes the integral representation (B.23) of the generalized Gamma function shifted by a logarithmic term

where in the last step we used the twisted periodicity property (B.20). If however \(\,\mathrm{Re}\,\vartheta _\gamma \notin (0,1)\), there is an additional contribution given by

where \(\epsilon _\gamma =\text{ sgn }(\,\mathrm{Re}\,\vartheta _\gamma )\) and we introduced two finite sets \(S_+=\{1,\dots , \lfloor \,\mathrm{Re}\,\vartheta _\gamma \rfloor \}\) and \(S_-=\{0,\dots , 1+\lfloor \,\mathrm{Re}\,\vartheta _\gamma \rfloor \}\). Adding this contribution to (C.6) and again using repeatedly the twisted periodicity (B.20), one finally obtains

Substituting this result into (C.1), one gets formula (4.6) given in the main text.

1.2 \(\tau \) function

The starting point for deriving the \(\tau \) function is formula (3.29). In terms of the integer BPS indices (and after discarding an irrelevant function of \(\theta ^a\) coming from \(\log (\pm n)\) in the last term where n counts the reducibility of the charge), it takes the form

The next steps are similar to the one in the previous appendix. Setting \(t'=\mathrm {i}Z_{\gamma }/(s+\mathrm {i}\vartheta _\gamma )\) and combining the contributions of opposite charges, one finds

For \(\,\mathrm{Re}\,\vartheta _\gamma \in (0,1)\), the second integral is evaluated to

Substituting this back into (C.10), one recognizes the square bracket as the integral representation (B.30) of the generalized Barnes function \(\Upsilon (z,\eta )\) shifted by a logarithmic term

where in the last step we used the twisted periodicity property (B.27). For \(\,\mathrm{Re}\,\vartheta _\gamma \notin (0,1)\), there are additional contributions which can be computed as in (C.7) and amount to replacing \(\vartheta _\gamma \) by \([\vartheta _\gamma ]\), consistently with the periodicity of the original integral representation. As a result, one arrives at formula (4.7).

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