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SIGMA 20 (2024), 073, 34 pages arXiv:2311.17638
https://doi.org/10.3842/SIGMA.2024.073
Resurgence of Refined Topological Strings and Dual Partition Functions
Sergey Alexandrov a, Marcos Mariño b and Boris Pioline c
a) Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, F-34095, Montpellier, France
b) Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211 Switzerland
c) Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589, CNRS-Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France
Received December 13, 2023, in final form August 02, 2024; Published online August 06, 2024
Abstract
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.
Key words: resurgence; topological string theory; Borel resummation; Stokes automorphism.
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References
- Aganagic M., Bouchard V., Klemm A., Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008), 771-819, arXiv:hep-th/0607100.
- 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.
- Aganagic M., Klemm A., Mariño M., Vafa C., The topological vertex, Comm. Math. Phys. 254 (2005), 425-478, arXiv:hep-th/0305132.
- Alexandrov S., D-instantons and twistors: some exact results, J. Phys. A 42 (2009), 335402, 26 pages, arXiv:0902.2761.
- Alexandrov S., Banerjee S., Dualities and fivebrane instantons, J. High Energy Phys. 2014 (2014), no. 11, 040, 39 pages, arXiv:1405.0291.
- Alexandrov S., Banerjee S., Fivebrane instantons in Calabi-Yau compactifications, Phys. Rev. D 90 (2014), 041902, 5 pages, arXiv:1403.1265.
- Alexandrov S., Bendriss K., Hypermultiplet metric and NS5-instantons, J. High Energy Phys. 2024 (2024), no. 1, 140, 45 pages, arXiv:2309.14440.
- Alexandrov S., Manschot J., Pioline B., S-duality and refined BPS indices, Comm. Math. Phys. 380 (2020), 755-810, arXiv:1910.03098.
- Alexandrov S., Persson D., Pioline B., Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces, J. High Energy Phys. 2011 (2011), no. 3, 111, 73 pages, arXiv:1010.5792.
- 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 pages, arXiv:1009.3026.
- Alexandrov S., Persson D., Pioline B., Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence, J. High Energy Phys. 2011 (2011), no. 12, 027, 64 pages, arXiv:1110.0466.
- Alexandrov S., Pioline B., Theta series, wall-crossing and quantum dilogarithm identities, Lett. Math. Phys. 106 (2016), 1037-1066, arXiv:1511.02892.
- Alexandrov S., Pioline B., Heavenly metrics, BPS indices and twistors, Lett. Math. Phys. 111 (2021), 116, 41 pages, arXiv:2104.10540.
- Alexandrov S., Pioline B., Saueressig F., Vandoren S., D-instantons and twistors, J. High Energy Phys. 2009 (2009), no. 3, 044, 40 pages, arXiv:0812.4219.
- Alexandrov S., Pioline B., Saueressig F., Vandoren S., Linear perturbations of quaternionic metrics, Comm. Math. Phys. 296 (2010), 353-403, arXiv:0810.1675.
- Alim M., Hollands L., Tulli I., Quantum curves, resurgence and exact WKB, SIGMA 19 (2023), 009, 82 pages, arXiv:2203.08249.
- Alim M., Länge J.D., Polynomial structure of the (open) topological string partition function, J. High Energy Phys. 2007 (2007), no. 10, 045, 13 pages, arXiv:0708.2886.
- 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.
- Andersen J.E., Kashaev R., A TQFT from quantum Teichmüller theory, Comm. Math. Phys. 330 (2014), 887-934, arXiv:1109.6295.
- 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.
- 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.
- 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.
- Bouchard V., Klemm A., Mariño M., Pasquetti S., Remodeling the B-model, Comm. Math. Phys. 287 (2009), 117-178, arXiv:0709.1453.
- Bridgeland T., Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math. 216 (2019), 69-124, arXiv:1611.03697.
- Bridgeland T., Tau functions from Joyce structures, arXiv:2303.07061.
- Bridgeland T., Strachan I.A.B., Complex hyperkähler structures defined by Donaldson-Thomas invariants, Lett. Math. Phys. 111 (2021), 54, 24 pages, arXiv:2006.13059.
- 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.
- Choi J., Katz S., Klemm A., The refined BPS index from stable pair invariants, Comm. Math. Phys. 328 (2014), 903-954, arXiv:1210.4403.
- Chuang W., Quantum Riemann-Hilbert problems for the resolved conifold, J. Geom. Phys. 190 (2023), 104860, 16 pages, arXiv:2203.00294.
- Codesido S., Grassi A., Mariño M., Spectral theory and mirror curves of higher genus, Ann. Henri Poincaré 18 (2017), 559-622, arXiv:1507.02096.
- Coman I., Longhi P., Teschner J., From quantum curves to topological string partition functions II, arXiv:2004.04585.
- Coman I., Pomoni E., Teschner J., From quantum curves to topological string partition functions, Comm. Math. Phys. 399 (2023), 1501-1548, arXiv:1811.01978.
- 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.
- 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.
- Couso-Santamaría R., Mariño M., Schiappa R., Resurgence matches quantization, J. Phys. A 50 (2017), 145402, 34 pages, arXiv:1610.06782.
- Delabaere E., Dillinger H., Pham F., Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier (Grenoble) 43 (1993), 163-199.
- Dijkgraaf R., Hollands L., Sułkowski P., Vafa C., Supersymmetric gauge theories, intersecting branes and free fermions, J. High Energy Phys. 2008 (2008), no. 2, 106, 57 pages, arXiv:0709.4446.
- Dimofte T., Gukov S., Refined, motivic, and quantum, Lett. Math. Phys. 91 (2010), 1-27, arXiv:0904.1420.
- Drukker N., Mariño M., Putrov P., Nonperturbative aspects of ABJM theory, J. High Energy Phys. 2011 (2011), no. 11, 141, 29 pages, arXiv:1103.4844.
- Eynard B., Garcia-Failde E., Marchal O., Orantin N., Quantization of classical spectral curves via topological recursion, Comm. Math. Phys. 405 (2024), 116, 118 pages, arXiv:2106.04339.
- 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.
- Eynard B., Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452, arXiv:math-ph/0702045.
- Faddeev L.D., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254, arXiv:hep-th/9504111.
- 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.
- Ferrara S., Sabharwal S., Quaternionic manifolds for type ${\rm II}$ superstring vacua of Calabi-Yau spaces, Nuclear Phys. B 332 (1990), 317-332.
- 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.
- Gaiotto D., Moore G.W., Neitzke A., Framed BPS states, Adv. Theor. Math. Phys. 17 (2013), 241-397, arXiv:1006.0146.
- Gamayun O., Iorgov N., Lisovyy O., Conformal field theory of Painlevé VI, J. High Energy Phys. 2012 (2012), no. 10, 038, 24 pages, arXiv:1207.0787.
- Gamayun O., Iorgov N., Lisovyy O., How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A 46 (2013), 335203, 29 pages, arXiv:1302.1832.
- Garoufalidis S., Kashaev R., Evaluation of state integrals at rational points, Commun. Number Theory Phys. 9 (2015), 549-582, arXiv:1411.6062.
- 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.
- Gopakumar R., Vafa C., M-theory and topological strings. II, arXiv:hep-th/9812127.
- Grassi A., Hao Q., Neitzke A., Exponential networks, WKB and topological string, SIGMA 19 (2023), 064, 44 pages, arXiv:2201.11594.
- Grassi A., Hatsuda Y., Mariño M., Topological strings from quantum mechanics, Ann. Henri Poincaré 17 (2016), 3177-3235, arXiv:1410.3382.
- Grimm T.W., Klemm A., Mariño M., Weiss M., Direct integration of the topological string, J. High Energy Phys. 2007 (2007), no. 8, 058, 78 pages, arXiv:hep-th/0702187.
- Gu J., Relations between Stokes constants of unrefined and Nekrasov-Shatashvili topological strings, J. High Energy Phys. 2024 (2024), no. 5, 199, 29 pages, arXiv:2307.02079.
- 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 pages, arXiv:2305.19916.
- Gu J., Mariño M., Peacock patterns and new integer invariants in topological string theory, SciPost Phys. 12 (2022), 058, 51 pages, arXiv:2104.07437.
- Gu J., Mariño M., Exact multi-instantons in topological string theory, SciPost Phys. 15 (2023), 179, 36 pages, arXiv:2211.01403.
- Gu J., Mariño M., On the resurgent structure of quantum periods, SciPost Phys. 15 (2023), 035, 40 pages, arXiv:2211.03871.
- Haghighat B., Klemm A., Rauch M., Integrability of the holomorphic anomaly equations, J. High Energy Phys. 2008 (2008), no. 10, 097, 37 pages, arXiv:0809.1674.
- Hatsuda Y., Okuyama K., Resummations and non-perturbative corrections, J. High Energy Phys. 2015 (2015), no. 9, 051, 28 pages, arXiv:1505.07460.
- Hollowood T., Iqbal A., Vafa C., Matrix models, geometric engineering and elliptic genera, J. High Energy Phys. 2008 (2008), no. 3, 069, 81 pages, arXiv:hep-th/0310272.
- 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.
- Huang M., Katz S., Klemm A., Towards refining the topological strings on compact Calabi-Yau 3-folds, J. High Energy Phys. 2021 (2021), no. 3, 266, 87 pages, arXiv:2010.02910.
- Huang M., Klemm A., Holomorphic anomaly in gauge theories and matrix models, J. High Energy Phys. 2007 (2007), no. 9, 054, 33 pages, arXiv:hep-th/0605195.
- Huang M., Klemm A., Direct integration for general $\Omega$ backgrounds, Adv. Theor. Math. Phys. 16 (2012), 805-849, arXiv:1009.1126.
- Iqbal A., Kozçaz C., Vafa C., The refined topological vertex, J. High Energy Phys. 2009 (2009), no. 10, 069, 58 pages, arXiv:hep-th/0701156.
- Iwaki K., Mariño M., Resurgent structure of the topological string and the first Painlevé equation, SIGMA 20 (2024), 028, 21 pages, arXiv:2307.02080.
- Kidwai O., Osuga K., Quantum curves from refined topological recursion: the genus 0 case, Adv. Math. 432 (2023), 109253, 52 pages, arXiv:2204.12431.
- Klemm A., The B-model approach to topological string theory on Calabi-Yau n-folds, in B-model Gromov-Witten theory, Trends Math., Birkhäuser, Cham, 2018, 79-397.
- Klemm A., Zaslow E., Local mirror symmetry at higher genus, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, AMS/IP Stud. Adv. Math., Vol. 23, American Mathematical Society, Providence, RI, 2001, 183-207, arXiv:hep-th/9906046.
- Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
- Krefl D., Walcher J., Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. 95 (2011), 67-88, arXiv:1007.0263.
- LeBrun C., Fano manifolds, contact structures, and quaternionic geometry, Internat. J. Math. 6 (1995), 419-437, arXiv:dg-ga/9409001.
- Mariño M., Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, J. High Energy Phys. 2008 (2008), no. 12, 114, 56 pages, arXiv:0805.3033.
- Mariño M., Open string amplitudes and large order behavior in topological string theory, J. High Energy Phys. 2008 (2008), no. 3, 060, 34 pages, arXiv:hep-th/0612127.
- Mariño M., Instantons and large $N$. An introduction to non-perturbative methods in quantum field theory, Cambridge University Press, Cambridge, 2015.
- 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.
- 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.
- Mariño M., Schwick M., Non-perturbative real topological strings, arXiv:2309.12046.
- 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.
- 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.
- Mozgovoy S., Pioline B., Attractor invariants, brane tilings and crystals, Ann. Inst. Fourier (Grenoble), to appear, arXiv:2012.14358.
- Neitzke A., On a hyperholomorphic line bundle over the Coulomb branch, arXiv:1110.1619.
- Nekrasov N., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831-864, arXiv:hep-th/0206161.
- Nekrasov N., Okounkov A., Seiberg-Witten theory and random partitions, in The Unity of Mathematics, Progr. Math., Vol. 244, Birkhäuser, Boston, MA, 2006, 525-596, arXiv:hep-th/0306238.
- Nekrasov N., Okounkov A., Membranes and sheaves, Algebr. Geom. 3 (2016), 320-369, arXiv:1404.2323.
- Nekrasov N., Shatashvili S., Quantization of integrable systems and four dimensional gauge theories, in XVIth International Congress on Mathematical Physics, World Scientific Publishing, Hackensack, NJ, 2010, 265-289, arXiv:0908.4052.
- Nekrasov N., Witten E., The omega deformation, branes, integrability and Liouville theory, J. High Energy Phys. 2010 (2010), no. 9, 092, 82 pages, arXiv:1002.0888.
- 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.
- Shenker S.H., The strength of nonperturbative effects in string theory, in Random Surfaces and Quantum Gravity, NATO Adv. Sci. Inst. Ser. B: Phys., Vol. 262, Plenum, New York, 1991, 191-200.
- Swann A., HyperKähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421-450.
- Yamaguchi S., Yau S.-T., Topological string partition functions as polynomials, J. High Energy Phys. 2004 (2004), no. 7, 047, 20 pages, arXiv:hep-th/0406078.
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