<table><thead><tr><th>(3, 3, 0)</th><th>2j<sub>+</sub> = 0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td></td><td>64</td><td></td><td>121</td><td></td><td>163</td><td></td><td>75</td><td></td><td>11</td><td></td><td></td><td></td></tr><tr><th>1</th><td>39</td><td></td><td>106</td><td></td><td>192</td><td></td><td>184</td><td></td><td>64</td><td></td><td>5</td><td></td><td></td></tr><tr><th>2</th><td></td><td>39</td><td></td><td>93</td><td></td><td>149</td><td></td><td>124</td><td></td><td>32</td><td></td><td>1</td><td></td></tr><tr><th>3</th><td>5</td><td></td><td>21</td><td></td><td>50</td><td></td><td>84</td><td></td><td>63</td><td></td><td>13</td><td></td><td></td></tr><tr><th>4</th><td></td><td>1</td><td></td><td>6</td><td></td><td>20</td><td></td><td>36</td><td></td><td>27</td><td></td><td>4</td><td></td></tr><tr><th>5</th><td></td><td></td><td></td><td></td><td>1</td><td></td><td>6</td><td></td><td>14</td><td></td><td>10</td><td></td><td>1</td></tr><tr><th>6</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>3</td><td></td></tr><tr><th>7</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td></tr></tbody></table>

<table><thead><tr><th>(3, 3, 1)</th><th>2j<sub>+</sub> = 0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td>29</td><td></td><td>74</td><td></td><td>130</td><td></td><td>109</td><td></td><td>26</td><td></td><td>2</td><td></td><td></td></tr><tr><th>1</th><td></td><td>59</td><td></td><td>121</td><td></td><td>175</td><td></td><td>102</td><td></td><td>18</td><td></td><td></td><td></td></tr><tr><th>2</th><td>16</td><td></td><td>50</td><td></td><td>102</td><td></td><td>124</td><td></td><td>62</td><td></td><td>6</td><td></td><td></td></tr><tr><th>3</th><td></td><td>8</td><td></td><td>26</td><td></td><td>54</td><td></td><td>67</td><td></td><td>27</td><td></td><td>2</td><td></td></tr><tr><th>4</th><td></td><td></td><td>2</td><td></td><td>8</td><td></td><td>23</td><td></td><td>28</td><td></td><td>11</td><td></td><td></td></tr><tr><th>5</th><td></td><td></td><td></td><td></td><td></td><td>2</td><td></td><td>7</td><td></td><td>11</td><td></td><td>3</td><td></td></tr><tr><th>6</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>2</td><td></td><td>3</td><td>1</td><td></td></tr><tr><th>7</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td></tr></tbody></table>

<table><thead><tr><th>(3, 3, 2)</th><th>2j<sub>+</sub> = 0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td></td><td>30</td><td></td><td>57</td><td></td><td>82</td><td></td><td>34</td><td></td><td>4</td><td></td><td></td><td></td></tr><tr><th>1</th><td>17</td><td></td><td>45</td><td></td><td>89</td><td></td><td>86</td><td></td><td>26</td><td></td><td>1</td><td></td><td></td></tr><tr><th>2</th><td></td><td>15</td><td></td><td>38</td><td></td><td>65</td><td></td><td>55</td><td></td><td>12</td><td></td><td></td><td></td></tr><tr><th>3</th><td>1</td><td></td><td>6</td><td></td><td>17</td><td></td><td>33</td><td></td><td>25</td><td></td><td>4</td><td></td><td></td></tr><tr><th>4</th><td></td><td></td><td></td><td>1</td><td></td><td>6</td><td></td><td>13</td><td></td><td>10</td><td></td><td>1</td><td></td></tr><tr><th>5</th><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>4</td><td></td><td>3</td><td></td><td></td></tr><tr><th>6</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td>1</td><td></td></tr></tbody></table>

<table><thead><tr><th>(3, 3, 3)</th><th>2j<sub>+</sub> = 0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td>6</td><td></td><td>14</td><td></td><td>30</td><td></td><td>25</td><td></td><td>3</td><td></td><td></td></tr><tr><th>1</th><td></td><td>10</td><td></td><td>22</td><td></td><td>38</td><td></td><td>19</td><td></td><td>2</td><td></td></tr><tr><th>2</th><td>2</td><td></td><td>6</td><td></td><td>16</td><td></td><td>22</td><td></td><td>10</td><td></td><td></td></tr><tr><th>3</th><td></td><td></td><td></td><td>2</td><td></td><td>6</td><td></td><td>10</td><td></td><td>3</td><td></td></tr><tr><th>4</th><td></td><td></td><td></td><td></td><td></td><td></td><td>2</td><td></td><td>3</td><td></td><td>1</td></tr><tr><th>5</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td></tr></tbody></table>

<table><thead><tr><th>(3, 3, 4)</th><th>2j<sub>+</sub> = 0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td></td><td>2</td><td>4</td><td></td><td>9</td><td></td><td>2</td><td></td><td></td></tr><tr><th>1</th><td>1</td><td></td><td>2</td><td>7</td><td>7</td><td></td><td>1</td><td></td><td></td></tr><tr><th>2</th><td></td><td></td><td></td><td>1</td><td>3</td><td></td><td>3</td><td></td><td></td></tr><tr><th>3</th><td></td><td></td><td></td><td></td><td></td><td>1</td><td>1</td><td></td><td></td></tr></tbody></table>

<table><caption>Table 18. BPS numbers of massive M-string at base degree 3 and fiber degree 3.</caption><thead><tr><th>(3, 3, 5)</th><th>2j<sub>+</sub> = 4</th><th>5</th><th>6</th></tr></thead><tbody><tr><th>2j<sub>-</sub> = 0</th><td>1</td><td></td><td>1</td></tr><tr><th>1</th><td></td><td></td><td>1</td></tr></tbody></table>

### C.4 $N_{j-j_+}^\kappa$ for a mass deformation of $T^2 \times$ the resolved $D_4$ singularity

In table 19, we report on numbers $N_{j-j_+}^\kappa$ for the massive $D_4$ case as extracted from our computations based on the methods explained in section 4. They satisfy basic consistency requirements: they are positive integers, and they follow a checkerboard pattern.

### D Sakai's mirror curve for the massive E-string

The Sakai mirror curve for the massive E-string is given by (8.16), with coefficient functions $g_2(u, \tau, \mathbf{m})$, $g_3(u, \tau, \mathbf{m})$ [72]

$$g_2(u, \tau, \mathbf{m}) = \sum_{j=0}^{4} a_j u^j, \quad g_3(u, \tau, \mathbf{m}) = \sum_{j=0}^{6} b_j u^j. \qquad (\text{D.1})$$
<figure><img src="image_5.png" /><figcaption>**Table 19.** Refined BPS invariants of the massive $D_4$ chain at base degrees $\beta = (1, 1, 1, 1)$ and fiber degree $e \le 2$.</figcaption></figure>

The coefficients are

$$a_0 = \frac{1}{12} E_4,$$

$$b_0 = \frac{1}{216} E_6,$$

$$a_1 = 0,$$

$$b_1 = -\frac{4}{E_4} A_1,$$

$$a_2 = \frac{6 \cdot 1728}{E_4(E_4^3 - E_6^2)} (-E_4 A_2 + A_1^2),$$

$$b_2 = \frac{5 \cdot 1728}{6E_4^2(E_4^3 - E_6^2)} (E_4^2 B_2 - E_6 A_1^2), \qquad (D.2)$$

........

where we have used the nine generators of the ring of $E_8$ Weyl invariant Jacobi forms introduced in subsection A.4. The other coefficients $a_3, a_4, b_3, \dots, b_6$ of the Sakai curve are not needed in our computations and can be found in [72]. Note that compared to Sakai's original notation, we have inverted $u$ so that $u \to 0$ is the large volume limit.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

[1] A. Klemm and E. Zaslow, *Local mirror symmetry at higher genus*, hep-th/9906046 [INSPIRE].

[2] J. Choi, S. Katz and A. Klemm, *The refined BPS index from stable pair invariants*, Commun. Math. Phys. 328 (2014) 903 [arXiv:1210.4403] [INSPIRE].
[3] V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, *Remodeling the B-model*, Commun. Math. Phys. **287** (2009) 117 [arXiv:0709.1453] [INSPIRE].

[4] M. Aganagic, A. Klemm, M. Mariño and C. Vafa, *The topological vertex*, Commun. Math. Phys. **254** (2005) 425 [hep-th/0305132] [INSPIRE].

[5] A. Iqbal, C. Kozcaz and C. Vafa, *The refined topological vertex*, JHEP **10** (2009) 069 [hep-th/0701156] [INSPIRE].

[6] B. Haghighat, A. Klemm and M. Rauch, *Integrability of the holomorphic anomaly equations*, JHEP **10** (2008) 097 [arXiv:0809.1674] [INSPIRE].

[7] M.-X. Huang and A. Klemm, *Direct integration for general $\Omega$ backgrounds*, Adv. Theor. Math. Phys. **16** (2012) 805 [arXiv:1009.1126] [INSPIRE].

[8] M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, *The $\Omega$ deformed B-model for rigid $N=2$ theories*, Annales Henri Poincaré **14** (2013) 425 [arXiv:1109.5728] [INSPIRE].

[9] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, *Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes*, Commun. Math. Phys. **165** (1994) 311 [hep-th/9309140] [INSPIRE].

[10] M.-X. Huang, S. Katz and A. Klemm, *Topological string on elliptic CY 3-folds and the ring of Jacobi forms*, JHEP **10** (2015) 125 [arXiv:1501.04891] [INSPIRE].

[11] R. Gopakumar and C. Vafa, *M theory and topological strings. 1*, hep-th/9809187 [INSPIRE].

[12] R. Gopakumar and C. Vafa, *M theory and topological strings. 2*, hep-th/9812127 [INSPIRE].

[13] S.H. Katz, A. Klemm and C. Vafa, *M theory, topological strings and spinning black holes*, Adv. Theor. Math. Phys. **3** (1999) 1445 [hep-th/9910181] [INSPIRE].

[14] N.A. Nekrasov and A. Okounkov, *Membranes and sheaves*, Phys. Rev. A **91** (2015) 012106 [arXiv:1401.2323].

[15] A. Iqbal and A.-K. Kashani-Poor, *Instanton counting and Chern-Simons theory*, Adv. Theor. Math. Phys. **7** (2003) 457 [hep-th/0212279] [INSPIRE].

[16] A. Iqbal and A.-K. Kashani-Poor, *SU($N$) geometries and topological string amplitudes*, Adv. Theor. Math. Phys. **10** (2006) 1 [hep-th/0306032] [INSPIRE].

[17] N.A. Nekrasov, *Seiberg-Witten prepotential from instanton counting*, Adv. Theor. Math. Phys. **7** (2003) 831 [hep-th/0206161] [INSPIRE].

[18] L.F. Alday, D. Gaiotto and Y. Tachikawa, *Liouville correlation functions from four-dimensional gauge theories*, Lett. Math. Phys. **91** (2010) 167 [arXiv:0906.3219] [INSPIRE].

[19] N. Wyllard, $A_{N-1}$ conformal Toda field theory correlation functions from conformal $N=2$ SU($N$) quiver gauge theories, JHEP **11** (2009) 002 [arXiv:0907.2189] [INSPIRE].

[20] A.-K. Kashani-Poor and J. Troost, *The toroidal block and the genus expansion*, JHEP **03** (2013) 133 [arXiv:1212.0722] [INSPIRE].

[21] F. Nieri, S. Pasquetti and F. Passerini, 3d and 5d gauge theory partition functions as $q$-deformed CFT correlators, Lett. Math. Phys. **105** (2015) 109 [arXiv:1303.2626] [INSPIRE].

[22] M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, *Gauge/Liouville triality*, arXiv:1309.1687 [INSPIRE].
[23] N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in *Proceedings, 16th International Congress on Mathematical Physics (ICMP09)*, Prague Czech Republic, 3–8 August 2009, pg. 265 [arXiv:0908.4052] [INSPIRE].

[24] A. Grassi, Y. Hatsuda and M. Mariño, *Topological strings from quantum mechanics*, *Annales Henri Poincaré* **17** (2016) 3177 [arXiv:1410.3382] [INSPIRE].

[25] Y. Hatsuda, S. Moriyama and K. Okuyama, *Exact results on the ABJM Fermi gas*, *JHEP* **10** (2012) 020 [arXiv:1207.4283] [INSPIRE].

[26] Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, *Non-perturbative effects and the refined topological string*, *JHEP* **09** (2014) 168 [arXiv:1306.1734] [INSPIRE].

[27] A. Iqbal and A.-K. Kashani-Poor, *The vertex on a strip*, *Adv. Theor. Math. Phys.* **10** (2006) 317 [hep-th/0410174] [INSPIRE].

[28] M. Huang, S. Katz and A. Klemm, *Elliptifying topological string theory*, BONN-TH-2016-27, (2016).

[29] E. Witten, *Quantum background independence in string theory*, in *Salamfest*, Trieste Italy, (1993), pg. 0257 [hep-th/9306122] [INSPIRE].

[30] T.J. Hollowood, A. Iqbal and C. Vafa, *Matrix models, geometric engineering and elliptic genera*, *JHEP* **03** (2008) 069 [hep-th/0310272] [INSPIRE].

[31] A. Klemm, P. Mayr and C. Vafa, *BPS states of exceptional noncritical strings*, *Nucl. Phys. Proc. Suppl.* **B 58** (1997) 177 [hep-th/9607139] [INSPIRE].

[32] J.J. Heckman, D.R. Morrison and C. Vafa, *On the classification of 6D SCFTs and generalized ADE orbifolds*, *JHEP* **05** (2014) 028 [Erratum ibid. **06** (2015) 017] [arXiv:1312.5746] [INSPIRE].

[33] J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, *Atomic classification of 6D SCFTs*, *Fortsch. Phys.* **63** (2015) 468 [arXiv:1502.05405] [INSPIRE].

[34] D.R. Morrison and W. Taylor, *Classifying bases for 6D F-theory models*, *Central Eur. J. Phys.* **10** (2012) 1072 [arXiv:1201.1943] [INSPIRE].

[35] M. Del Zotto, J. Gu, M. Huang, A.-K. Kashani-Poor, A. Klemm and G. Lockhart, work in progress.

[36] B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, *M-strings*, *Commun. Math. Phys.* **334** (2015) 779 [arXiv:1305.6322] [INSPIRE].

[37] D.R. Morrison and C. Vafa, *Compactifications of F-theory on Calabi-Yau threefolds*. 1, *Nucl. Phys. B* **473** (1996) 74 [hep-th/9602114] [INSPIRE].

[38] D.R. Morrison and C. Vafa, *Compactifications of F-theory on Calabi-Yau threefolds*. 2, *Nucl. Phys. B* **476** (1996) 437 [hep-th/9603161] [INSPIRE].

[39] X. Wang, G. Zhang and M.-X. Huang, *New exact quantization condition for toric Calabi-Yau geometries*, *Phys. Rev. Lett.* **115** (2015) 121601 [arXiv:1505.05360] [INSPIRE].

[40] A. Grassi and J. Gu, *BPS relations from spectral problems and blowup equations*, [arXiv:1609.05914] [INSPIRE].

[41] H. Nakajima and K. Yoshioka, *Instanton counting on blowup. II. K-theoretic partition function*, *math/0505553* [INSPIRE].
[42] L. Gottsche, H. Nakajima and K. Yoshioka, *K-theoretic Donaldson invariants via instanton counting*, *Pure Appl. Math. Quart.* **5** (2009) 1029 [math/0611945] [INSPIRE].

[43] H. Nakajima and K. Yoshioka, *Perverse coherent sheaves on blowup, III: blow-up formula from wall-crossing*, *Kyoto J. Math.* **51** (2011) 263.

[44] B. Haghighat, G. Lockhart and C. Vafa, *Fusing E-strings to heterotic strings: E + E → H*, *Phys. Rev. D* **90** (2014) 126012 [arXiv:1406.0850] [INSPIRE].

[45] M. Del Zotto and G. Lockhart, *On exceptional instanton strings*, arXiv:1609.00310 [INSPIRE].

[46] J. Harris, *Curves in projective space*, in *Séminaire de Mathématiques Supérieures* **85**, Presses de l'Université de Montréal, Montreal Canada, (1982).

[47] R. Pandharipande and R.P. Thomas, *Curve counting via stable pairs in the derived category*, *Invent. Math.* **178** (2009) 407 [arXiv:0707.2348] [INSPIRE].

[48] R. Pandharipande and R.P. Thomas, *Stable pairs and BPS invariants*, *J. Amer. Math. Soc.* **13** (2010) 267 [arXiv:0711.3899] [INSPIRE].

[49] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, *Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces*, *Commun. Math. Phys.* **167** (1995) 301 [hep-th/9308122] [INSPIRE].

[50] E. Witten, *Solutions of four-dimensional field theories via M-theory*, *Nucl. Phys. B* **500** (1997) 3 [hep-th/9703166] [INSPIRE].

[51] A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, *6D string chains*, arXiv:1504.04614 [INSPIRE].

[52] M. Eichler and D. Zagier, *The theory of Jacobi forms*, *Progr. Math.* **55**, Birkhäuser Inc., Boston U.S.A., (1985).

[53] J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, *E strings and N = 4 topological Yang-Mills theories*, *Nucl. Phys. B* **527** (1998) 581 [hep-th/9802168] [INSPIRE].

[54] M.-X. Huang, A. Klemm and M. Poretschkin, *Refined stable pair invariants for E-, M- and [p, q]-strings*, *JHEP* **11** (2013) 112 [arXiv:1308.0619] [INSPIRE].

[55] F. Benini, R. Eager, K. Hori and Y. Tachikawa, *Elliptic genera of 2d N = 2 gauge theories*, *Commun. Math. Phys.* **333** (2015) 1241 [arXiv:1308.4896] [INSPIRE].

[56] N. Bobev, M. Bullimore and H.-C. Kim, *Supersymmetric Casimir energy and the anomaly polynomial*, *JHEP* **09** (2015) 142 [arXiv:1507.08553] [INSPIRE].

[57] H. Shimizu and Y. Tachikawa, *Anomaly of strings of 6D N = (1,0) theories*, *JHEP* **11** (2016) 165 [arXiv:1608.05894] [INSPIRE].

[58] M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, *6D conformal matter*, *JHEP* **02** (2015) 054 [arXiv:1407.6359] [INSPIRE].

[59] B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, *Strings of minimal 6D SCFTs*, *Fortsch. Phys.* **63** (2015) 294 [arXiv:1412.3152] [INSPIRE].

[60] H.-C. Kim and S. Kim, *Supersymmetric vacua of mass-deformed M2-brane theory*, *Nucl. Phys. B* **839** (2010) 96 [arXiv:1001.3153] [INSPIRE].

[61] W. Cai, M.-X. Huang and K. Sun, *On the elliptic genus of three E-strings and heterotic strings*, *JHEP* **01** (2015) 079 [arXiv:1411.2801] [INSPIRE].
[62] A.B. Goncharov and R. Kenyon, *Dimers and cluster integrable systems*, arXiv:1107.5588 [inSPIRE].

[63] S. Franco, Y. Hatsuda and M. Mariño, *Exact quantization conditions for cluster integrable systems*, *J. Stat. Mech.* **06** (2016) 063107 [arXiv:1512.03061] [inSPIRE].

[64] Y. Hatsuda and M. Mariño, *Exact quantization conditions for the relativistic Toda lattice*, *JHEP* **05** (2016) 133 [arXiv:1511.02860] [inSPIRE].

[65] S. Codesido, J. Gu and M. Mariño, *Operators and higher genus mirror curves*, *JHEP* **02** (2017) 092 [arXiv:1609.00708] [inSPIRE].

[66] S. Codesido, A. Grassi and M. Mariño, *Spectral theory and mirror curves of higher genus*, *Annales Henri Poincaré* **18** (2017) 559 [arXiv:1507.02096] [inSPIRE].

[67] J. Gu, A. Klemm, M. Mariño and J. Reuter, *Exact solutions to quantum spectral curves by topological string theory*, *JHEP* **10** (2015) 025 [arXiv:1506.09176] [inSPIRE].

[68] M. Mariño and S. Zakany, *Matrix models from operators and topological strings*, *Annales Henri Poincaré* **17** (2016) 1075 [arXiv:1502.02958] [inSPIRE].

[69] R. Kashaev, M. Mariño and S. Zakany, *Matrix models from operators and topological strings*, **2**, *Annales Henri Poincaré* **17** (2016) 2741 [arXiv:1505.02243] [inSPIRE].

[70] K. Sun, X. Wang and M.-X. Huang, *Exact quantization conditions, toric Calabi-Yau and nonperturbative topological string*, *JHEP* **01** (2017) 061 [arXiv:1606.07330] [inSPIRE].

[71] J.A. Minahan and D. Nemeschansky, *Superconformal fixed points with $E_n$ global symmetry*, *Nucl. Phys. B* **489** (1997) 24 [hep-th/9610076] [inSPIRE].

[72] K. Sakai, *Topological string amplitudes for the local half K3 surface*, *Prog. Theor. Exp. Phys.* **2017** (2017) 033B09 [arXiv:1111.3967] [inSPIRE].

[73] D. Krefl and J. Walcher, *Extended holomorphic anomaly in gauge theory*, *Lett. Math. Phys.* **95** (2011) 67 [arXiv:1007.0263] [inSPIRE].

[74] J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, *The 1-2-3 of modular forms*, Universitext, Springer-Verlag, Berlin Germany, (2008).

[75] A. Dabholkar, S. Murthy and D. Zagier, *Quantum black holes, wall crossing and mock modular forms*, arXiv:1208.4074 [inSPIRE].

[76] E.T. Whittaker and G.N. Watson, *A course in modern analysis*, 4th ed., Cambridge University Press, Cambridge U.K., (1990).