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Resurgence of Refined Topological Strings and Dual Partition Functions
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We study the resurgent structure of the refined topological string partition function on a non-compact Calabi–Yau threefold, at large orders in the string coupling constant $g_s$ and fixed refinement parameter $\mathsf{b}$. For $\mathsf{b}\neq 1$, the Borel transform admits two families of simple poles, corresponding to integral periods rescaled by $\mathsf{b}$ and $1/\mathsf{b}$. We show that the corresponding Stokes automorphism is expressed in terms of a generalization of the non-compact quantum dilogarithm, and we conjecture that the Stokes constants are determined by the refined Donaldson–Thomas invariants counting spin-$j$ BPS states. This jump in the refined topological string partition function is a special case (unit five-brane charge) of a more general transformation property of wave functions on quantum twisted tori introduced in earlier work by two of the authors. We show that this property follows from the transformation of a suitable refined dual partition function across BPS rays, defined by extending the Moyal star product to the realm of contact geometry.
Keywords: resurgence, topological string theory, Borel resummation, Stokes automorphism. 
@article{SIGMA_2024_20_a72,
     author = {Sergey Alexandrov and Marcos Mari\~no and Boris Pioline},
     title = {Resurgence of {Refined} {Topological} {Strings} and {Dual} {Partition} {Functions}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a72/}
}
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Sergey Alexandrov; Marcos Mariño; Boris Pioline. Resurgence of Refined Topological Strings and Dual Partition Functions. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a72/

[1] Aganagic M., Bouchard V., Klemm A., “Topological strings and (almost) modular forms”, Comm. Math. Phys., 277 (2008), 771–819, arXiv: hep-th/0607100 | DOI | MR | Zbl

[2] Aganagic M., Dijkgraaf R., Klemm A., Mariño M., Vafa C., “Topological strings and integrable hierarchies”, Comm. Math. Phys., 261 (2006), 451–516, arXiv: hep-th/0312085 | DOI | MR | Zbl

[3] Aganagic M., Klemm A., Mariño M., Vafa C., “The topological vertex”, Comm. Math. Phys., 254 (2005), 425–478, arXiv: hep-th/0305132 | DOI | MR | Zbl

[4] Alexandrov S., “D-instantons and twistors: some exact results”, J. Phys. A, 42 (2009), 335402, 26 pp., arXiv: 0902.2761 | DOI | MR | Zbl

[5] Alexandrov S., Banerjee S., “Dualities and fivebrane instantons”, J. High Energy Phys., 2014:11 (2014), 040, 39 pp., arXiv: 1405.0291 | DOI | MR | Zbl

[6] Alexandrov S., Banerjee S., “Fivebrane instantons in Calabi–Yau compactifications”, Phys. Rev. D, 90 (2014), 041902, 5 pp., arXiv: 1403.1265 | DOI | MR

[7] Alexandrov S., Bendriss K., “Hypermultiplet metric and NS5-instantons”, J. High Energy Phys., 2024:1 (2024), 140, 45 pp., arXiv: 2309.14440 | DOI | MR

[8] Alexandrov S., Manschot J., Pioline B., “S-duality and refined BPS indices”, Comm. Math. Phys., 380 (2020), 755–810, arXiv: 1910.03098 | DOI | MR | Zbl

[9] Alexandrov S., Persson D., Pioline B., “Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces”, J. High Energy Phys., 2011:3 (2011), 111, 73 pp., arXiv: 1010.5792 | DOI | MR

[10] Alexandrov S., Persson D., Pioline B., “On the topology of the hypermultiplet moduli space in type II/CY string vacua”, Phys. Rev. D, 83 (2011), 026001, 5 pp., arXiv: 1009.3026 | DOI

[11] Alexandrov S., Persson D., Pioline B., “Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence”, J. High Energy Phys., 2011:12 (2011), 027, 64 pp., arXiv: 1110.0466 | DOI | MR | Zbl

[12] Alexandrov S., Pioline B., “Theta series, wall-crossing and quantum dilogarithm identities”, Lett. Math. Phys., 106 (2016), 1037–1066, arXiv: 1511.02892 | DOI | MR | Zbl

[13] Alexandrov S., Pioline B., “Heavenly metrics, BPS indices and twistors”, Lett. Math. Phys., 111 (2021), 116, 41 pp., arXiv: 2104.10540 | DOI | MR | Zbl

[14] Alexandrov S., Pioline B., Saueressig F., Vandoren S., “D-instantons and twistors”, J. High Energy Phys., 2009:3 (2009), 044, 40 pp., arXiv: 0812.4219 | DOI | MR

[15] Alexandrov S., Pioline B., Saueressig F., Vandoren S., “Linear perturbations of quaternionic metrics”, Comm. Math. Phys., 296 (2010), 353–403, arXiv: 0810.1675 | DOI | MR | Zbl

[16] Alim M., Hollands L., Tulli I., SIGMA, 19 (2023), 009, 82 pp., arXiv: 2203.08249 | DOI | MR | Zbl

[17] Alim M., Länge J.D., “Polynomial structure of the (open) topological string partition function”, J. High Energy Phys., 2007:10 (2007), 045, 13 pp., arXiv: 0708.2886 | DOI | MR

[18] Alim M., Saha A., Teschner J., Tulli I., “Mathematical structures of non-perturbative topological string theory: from GW to DT invariants”, Comm. Math. Phys., 399 (2023), 1039–1101, arXiv: 2109.06878 | DOI | MR | Zbl

[19] Andersen J.E., Kashaev R., “A TQFT from quantum Teichmüller theory”, Comm. Math. Phys., 330 (2014), 887–934, arXiv: 1109.6295 | DOI | MR | Zbl

[20] Antoniadis I., Florakis I., Hohenegger S., Narain K.S., Zein Assi A., “Worldsheet realization of the refined topological string”, Nuclear Phys. B, 875 (2013), 101–133, arXiv: 1302.6993 | DOI | MR | Zbl

[21] Barbieri A., Bridgeland T., Stoppa J., “A quantized Riemann–Hilbert problem in Donaldson–Thomas theory”, Int. Math. Res. Not., 2022 (2022), 3417–3456, arXiv: 1905.00748 | DOI | MR | Zbl

[22] Bershadsky M., Cecotti S., Ooguri H., Vafa C., “Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes”, Comm. Math. Phys., 165 (1994), 311–427, arXiv: hep-th/9309140 | DOI | MR | Zbl

[23] Bouchard V., Klemm A., Mariño M., Pasquetti S., “Remodeling the B-model”, Comm. Math. Phys., 287 (2009), 117–178, arXiv: 0709.1453 | DOI | MR | Zbl

[24] Bridgeland T., “Riemann–Hilbert problems from Donaldson–Thomas theory”, Invent. Math., 216 (2019), 69–124, arXiv: 1611.03697 | DOI | MR | Zbl

[25] Bridgeland T., Tau functions from Joyce structures, arXiv: 2303.07061

[26] Bridgeland T., Strachan I.A.B., “Complex hyperkähler structures defined by Donaldson–Thomas invariants”, Lett. Math. Phys., 111 (2021), 54, 24 pp., arXiv: 2006.13059 | DOI | MR | Zbl

[27] Cecotti S., Neitzke A., Vafa C., “Twistorial topological strings and a $tt^*$ geometry for $\mathcal{N}=2$ theories in $4d$”, Adv. Theor. Math. Phys., 20 (2016), 193–312, arXiv: 1412.4793 | DOI | MR | Zbl

[28] Choi J., Katz S., Klemm A., “The refined BPS index from stable pair invariants”, Comm. Math. Phys., 328 (2014), 903–954, arXiv: 1210.4403 | DOI | MR | Zbl

[29] Chuang W., “Quantum Riemann–Hilbert problems for the resolved conifold”, J. Geom. Phys., 190 (2023), 104860, 16 pp., arXiv: 2203.00294 | DOI | MR | Zbl

[30] Codesido S., Grassi A., Mariño M., “Spectral theory and mirror curves of higher genus”, Ann. Henri Poincaré, 18 (2017), 559–622, arXiv: 1507.02096 | DOI | MR | Zbl

[31] Coman I., Longhi P., Teschner J., From quantum curves to topological string partition functions II, arXiv: 2004.04585

[32] Coman I., Pomoni E., Teschner J., “From quantum curves to topological string partition functions”, Comm. Math. Phys., 399 (2023), 1501–1548, arXiv: 1811.01978 | DOI | MR | Zbl

[33] Couso-Santamaría R., Edelstein J.D., Schiappa R., Vonk M., “Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local $\mathbb{CP}^2$”, Comm. Math. Phys., 338 (2015), 285–346, arXiv: 1407.4821 | DOI | MR | Zbl

[34] Couso-Santamaría R., Edelstein J.D., Schiappa R., Vonk M., “Resurgent transseries and the holomorphic anomaly”, Ann. Henri Poincaré, 17 (2016), 331–399, arXiv: 1308.1695 | DOI | MR | Zbl

[35] Couso-Santamaría R., Mariño M., Schiappa R., “Resurgence matches quantization”, J. Phys. A, 50 (2017), 145402, 34 pp., arXiv: 1610.06782 | DOI | MR | Zbl

[36] Delabaere E., Dillinger H., Pham F., “Résurgence de Voros et périodes des courbes hyperelliptiques”, Ann. Inst. Fourier (Grenoble), 43 (1993), 163–199 | DOI | MR | Zbl

[37] Dijkgraaf R., Hollands L., Sułkowski P., Vafa C., “Supersymmetric gauge theories, intersecting branes and free fermions”, J. High Energy Phys., 2008:2 (2008), 106, 57 pp., arXiv: 0709.4446 | DOI | MR

[38] Dimofte T., Gukov S., “Refined, motivic, and quantum”, Lett. Math. Phys., 91 (2010), 1–27, arXiv: 0904.1420 | DOI | MR | Zbl

[39] Drukker N., Mariño M., Putrov P., “Nonperturbative aspects of ABJM theory”, J. High Energy Phys., 2011:11 (2011), 141, 29 pp., arXiv: 1103.4844 | DOI | MR | Zbl

[40] Eynard B., Garcia-Failde E., Marchal O., Orantin N., “Quantization of classical spectral curves via topological recursion”, Comm. Math. Phys., 405 (2024), 116, 118 pp., arXiv: 2106.04339 | DOI | MR

[41] Eynard B., Mariño M., “A holomorphic and background independent partition function for matrix models and topological strings”, J. Geom. Phys., 61 (2011), 1181–1202, arXiv: 0810.4273 | DOI | MR | Zbl

[42] Eynard B., Orantin N., “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys., 1 (2007), 347–452, arXiv: math-ph/0702045 | DOI | MR | Zbl

[43] Faddeev L.D., “Discrete Heisenberg–Weyl group and modular group”, Lett. Math. Phys., 34 (1995), 249–254, arXiv: hep-th/9504111 | DOI | MR | Zbl

[44] Faddeev L.D., Kashaev R.M., Volkov A.Yu., “Strongly coupled quantum discrete Liouville theory. I Algebraic approach and duality”, Comm. Math. Phys., 219 (2001), 199–219, arXiv: hep-th/0006156 | DOI | MR | Zbl

[45] Ferrara S., Sabharwal S., “Quaternionic manifolds for type ${\rm II}$ superstring vacua of Calabi–Yau spaces”, Nuclear Phys. B, 332 (1990), 317–332 | DOI | MR

[46] Gaiotto D., Moore G.W., Neitzke A., “Four-dimensional wall-crossing via three-dimensional field theory”, Comm. Math. Phys., 299 (2010), 163–224, arXiv: 0807.4723 | DOI | MR | Zbl

[47] Gaiotto D., Moore G.W., Neitzke A., “Framed BPS states”, Adv. Theor. Math. Phys., 17 (2013), 241–397, arXiv: 1006.0146 | DOI | MR | Zbl

[48] Gamayun O., Iorgov N., Lisovyy O., “Conformal field theory of Painlevé VI”, J. High Energy Phys., 2012:10 (2012), 038, 24 pp., arXiv: 1207.0787 | DOI | MR

[49] Gamayun O., Iorgov N., Lisovyy O., “How instanton combinatorics solves Painlevé VI, V and IIIs”, J. Phys. A, 46 (2013), 335203, 29 pp., arXiv: 1302.1832 | DOI | MR | Zbl

[50] Garoufalidis S., Kashaev R., “Evaluation of state integrals at rational points”, Commun. Number Theory Phys., 9 (2015), 549–582, arXiv: 1411.6062 | DOI | MR | Zbl

[51] Ghoshal D., Vafa C., “$c=1$ string as the topological theory of the conifold”, Nuclear Phys. B, 453 (1995), 121–128, arXiv: hep-th/9506122 | DOI | MR | Zbl

[52] Gopakumar R., Vafa C., M-theory and topological strings. II, arXiv: hep-th/9812127

[53] Grassi A., Hao Q., Neitzke A., “Exponential networks, {WKB} and topological string”, SIGMA, 19 (2023), 064, 44 pp., arXiv: 2201.11594 | DOI | MR

[54] Grassi A., Hatsuda Y., Mariño M., “Topological strings from quantum mechanics”, Ann. Henri Poincaré, 17 (2016), 3177–3235, arXiv: 1410.3382 | DOI | MR | Zbl

[55] Grimm T.W., Klemm A., Mariño M., Weiss M., “Direct integration of the topological string”, J. High Energy Phys., 2007:8 (2007), 058, 78 pp., arXiv: hep-th/0702187 | DOI | MR | Zbl

[56] Gu J., “Relations between Stokes constants of unrefined and Nekrasov–Shatashvili topological strings”, J. High Energy Phys., 2024:5 (2024), 199, 29 pp., arXiv: 2307.02079 | DOI | MR

[57] Gu J., Kashani-Poor A.K., Klemm A., Mariño M., “Non-perturbative topological string theory on compact Calabi–Yau 3-folds”, SciPost Phys., 16 (2024), 079, 84 pp., arXiv: 2305.19916 | DOI | MR

[58] Gu J., Mariño M., “Peacock patterns and new integer invariants in topological string theory”, SciPost Phys., 12 (2022), 058, 51 pp., arXiv: 2104.07437 | DOI | MR

[59] Gu J., Mariño M., “Exact multi-instantons in topological string theory”, SciPost Phys., 15 (2023), 179, 36 pp., arXiv: 2211.01403 | DOI | MR

[60] Gu J., Mariño M., “On the resurgent structure of quantum periods”, SciPost Phys., 15 (2023), 035, 40 pp., arXiv: 2211.03871 | DOI | MR

[61] Haghighat B., Klemm A., Rauch M., “Integrability of the holomorphic anomaly equations”, J. High Energy Phys., 2008:10 (2008), 097, 37 pp., arXiv: 0809.1674 | DOI | MR | Zbl

[62] Hatsuda Y., Okuyama K., “Resummations and non-perturbative corrections”, J. High Energy Phys., 2015:9 (2015), 051, 28 pp., arXiv: 1505.07460 | DOI | MR

[63] Hollowood T., Iqbal A., Vafa C., “Matrix models, geometric engineering and elliptic genera”, J. High Energy Phys., 2008:3 (2008), 069, 81 pp., arXiv: hep-th/0310272 | DOI | MR

[64] Huang M., Kashani-Poor A.K., Klemm A., “The $\Omega$ deformed B-model for rigid $\mathcal{N}=2$ theories”, Ann. Henri Poincaré, 14 (2013), 425–497, arXiv: 1109.5728 | DOI | MR | Zbl

[65] Huang M., Katz S., Klemm A., “Towards refining the topological strings on compact Calabi–Yau 3-folds”, J. High Energy Phys., 2021:3 (2021), 266, 87 pp., arXiv: 2010.02910 | DOI | MR

[66] Huang M., Klemm A., “Holomorphic anomaly in gauge theories and matrix models”, J. High Energy Phys., 2007:9 (2007), 054, 33 pp., arXiv: hep-th/0605195 | DOI | MR

[67] Huang M., Klemm A., “Direct integration for general $\Omega$ backgrounds”, Adv. Theor. Math. Phys., 16 (2012), 805–849, arXiv: 1009.1126 | DOI | MR | Zbl

[68] Iqbal A., Koz{ç}az C., Vafa C., “The refined topological vertex”, J. High Energy Phys., 2009:10 (2009), 069, 58 pp., arXiv: hep-th/0701156 | DOI | MR

[69] Iwaki K., Mariño M., “Resurgent structure of the topological string and the first Painlevé equation”, SIGMA, 20 (2024), 028, 21 pp., arXiv: 2307.02080 | DOI | MR

[70] Kidwai O., Osuga K., “Quantum curves from refined topological recursion: the genus 0 case”, Adv. Math., 432 (2023), 109253, 52 pp., arXiv: 2204.12431 | DOI | MR | Zbl

[71] Klemm A., “The B-model approach to topological string theory on Calabi–Yau n-folds”, B-model Gromov–Witten theory, Trends Math., Birkhäuser, Cham, 2018, 79–397 | DOI | MR | Zbl

[72] Klemm A., Zaslow E., “Local mirror symmetry at higher genus”, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, AMS/IP Stud. Adv. Math., 23, American Mathematical Society, Providence, RI, 2001, 183–207, arXiv: hep-th/9906046 | DOI | MR | Zbl

[73] Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv: 0811.2435

[74] Krefl D., Walcher J., “Extended holomorphic anomaly in gauge theory”, Lett. Math. Phys., 95 (2011), 67–88, arXiv: 1007.0263 | DOI | MR | Zbl

[75] LeBrun C., “Fano manifolds, contact structures, and quaternionic geometry”, Internat. J. Math., 6 (1995), 419–437, arXiv: dg-ga/9409001 | DOI | MR | Zbl

[76] Mariño M., “Nonperturbative effects and nonperturbative definitions in matrix models and topological strings”, J. High Energy Phys., 2008:12 (2008), 114, 56 pp., arXiv: 0805.3033 | DOI | MR | Zbl

[77] Mariño M., “Open string amplitudes and large order behavior in topological string theory”, J. High Energy Phys., 2008:3 (2008), 060, 34 pp., arXiv: hep-th/0612127 | DOI | MR

[78] Mariño M., Instantons and large $N$. An introduction to non-perturbative methods in quantum field theory, Cambridge University Press, Cambridge, 2015 | DOI | MR | Zbl

[79] Mariño M., From resurgence to BPS states, Talk given at the conference Strings 2019 (Brussels) https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2019/2_M_Marino.pdf

[80] Mariño M., Schiappa R., Weiss M., “Nonperturbative effects and the large-order behavior of matrix models and topological strings”, Commun. Number Theory Phys., 2 (2008), 349–419, arXiv: 0711.1954 | DOI | MR | Zbl

[81] Mariño M., Schwick M., Non-perturbative real topological strings, arXiv: 2309.12046

[82] Maulik D., Nekrasov N., Okounkov A., Pandharipande R., “Gromov–Witten theory and Donaldson–Thomas theory. I”, Compos. Math., 142 (2006), 1263–1285, arXiv: math.AG/0312059 | DOI | MR | Zbl

[83] Maulik D., Nekrasov N., Okounkov A., Pandharipande R., “Gromov–Witten theory and Donaldson–Thomas theory. II”, Compos. Math., 142 (2006), 1286–1304, arXiv: math.AG/0406092 | DOI | MR | Zbl

[84] Mozgovoy S., Pioline B., “Attractor invariants, brane tilings and crystals”, Ann. Inst. Fourier (Grenoble) (to appear) , arXiv: 2012.14358

[85] Neitzke A., On a hyperholomorphic line bundle over the Coulomb branch, arXiv: 1110.1619

[86] Nekrasov N., “Seiberg–Witten prepotential from instanton counting”, Adv. Theor. Math. Phys., 7 (2003), 831–864, arXiv: hep-th/0206161 | DOI | MR | Zbl

[87] Nekrasov N., Okounkov A., “Seiberg–Witten theory and random partitions”, The Unity of Mathematics, Progr. Math., 244, Birkhäuser, Boston, MA, 2006, 525–596, arXiv: hep-th/0306238 | DOI | MR | Zbl

[88] Nekrasov N., Okounkov A., “Membranes and sheaves”, Algebr. Geom., 3 (2016), 320–369, arXiv: 1404.2323 | DOI | MR | Zbl

[89] Nekrasov N., Shatashvili S., “Quantization of integrable systems and four dimensional gauge theories”, XVIth International Congress on Mathematical Physics, World Scientific Publishing, Hackensack, NJ, 2010, 265–289, arXiv: 0908.4052 | DOI | MR | Zbl

[90] Nekrasov N., Witten E., “The omega deformation, branes, integrability and Liouville theory”, J. High Energy Phys., 2010:9 (2010), 092, 82 pp., arXiv: 1002.0888 | DOI | MR | Zbl

[91] Pasquetti S., Schiappa R., “Borel and Stokes nonperturbative phenomena in topological string theory and $c=1$ matrix models”, Ann. Henri Poincaré, 11 (2010), 351–431, arXiv: 0907.4082 | DOI | MR | Zbl

[92] Shenker S.H., “The strength of nonperturbative effects in string theory”, Random Surfaces and Quantum Gravity, NATO Adv. Sci. Inst. Ser. B: Phys., 262, Plenum, New York, 1991, 191–200 | DOI | MR

[93] Swann A., “HyperKähler and quaternionic Kähler geometry”, Math. Ann., 289 (1991), 421–450 | DOI | MR | Zbl

[94] Yamaguchi S., Yau S.-T., “Topological string partition functions as polynomials”, J. High Energy Phys., 2004:7 (2004), 047, 20 pp., arXiv: hep-th/0406078 | DOI | MR