Geodesic
Quantum Curves, Resurgence and Exact WKB
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov–Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson–Thomas invariants of the underlying Calabi–Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov–Rosly–Shatashvili proposal, in which the NS free energy is computed as a generating function of $q$-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.
Keywords: resolved conifold, topological string theory, Borel summation, difference equations, exponential spectral networks. 
@article{SIGMA_2023_19_a8,
     author = {Murad Alim and Lotte Hollands and Iv\'an Tulli},
     title = {Quantum {Curves,} {Resurgence} and {Exact} {WKB}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a8/}
}
TY  - JOUR
AU  - Murad Alim
AU  - Lotte Hollands
AU  - Iván Tulli
TI  - Quantum Curves, Resurgence and Exact WKB
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a8/
LA  - en
ID  - SIGMA_2023_19_a8
ER  - 
%0 Journal Article
%A Murad Alim
%A Lotte Hollands
%A Iván Tulli
%T Quantum Curves, Resurgence and Exact WKB
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a8/
%G en
%F SIGMA_2023_19_a8
Murad Alim; Lotte Hollands; Iván Tulli. Quantum Curves, Resurgence and Exact WKB. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a8/

[1] Aganagic M., Cheng M.C.N., Dijkgraaf R., Krefl D., Vafa C., “Quantum geometry of refined topological strings”, J. High Energy Phys., 2012:11 (2012), 019, 53 pp., arXiv: 1105.0630 | DOI | MR

[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

[3] Aganagic M., Vafa C., Mirror symmetry, D-branes and counting holomorphic discs, arXiv: hep-th/0012041

[4] Aganagic M., Yamazaki M., “Open BPS wall crossing and M-theory”, Nuclear Phys. B, 834 (2010), 258–272, arXiv: 0911.5342 | DOI | MR

[5] Alexandrov S., Persson D., Pioline B., “Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence”, J. High Energy Phys., 2011:12 (2011), 027, 65 pp., arXiv: 1110.0466 | DOI | MR

[6] Alim M., “Difference equation for the Gromov–Witten potential of the resolved conifold”, J. Geom. Phys., 183 (2023), 104688, 4 pp., arXiv: 2011.12759 | DOI | MR

[7] Alim M., Intrinsic non-perturbative topological strings, arXiv: 2102.07776

[8] Alim M., Saha A., “On the integrable hierarchy for the resolved conifold”, Bull. Lond. Math. Soc., 54 (2022), 2014–2031, arXiv: 2101.11672 | DOI | MR

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

[10] Alim M., Saha A., Tulli I., “A hyperkähler geometry associated to the BPS structure of the resolved conifold”, J. Geom. Phys., 180 (2022), 104618, 32 pp., arXiv: 2106.11976 | DOI | MR

[11] Aniceto I., Schiappa R., Vonk M., “The resurgence of instantons in string theory”, Commun. Number Theory Phys., 6 (2012), 339–496, arXiv: 1106.5922 | DOI | MR

[12] Banerjee S., Longhi P., Romo M., “Exploring 5d BPS spectra with exponential networks”, Ann. Henri Poincaré, 20 (2019), 4055–4162, arXiv: 1811.02875 | DOI | MR

[13] Banerjee S., Longhi P., Romo M., “Exponential BPS graphs and D brane counting on toric Calabi–Yau threefolds: Part I”, Comm. Math. Phys., 388 (2021), 893–945, arXiv: 1910.05296 | DOI | MR

[14] Banerjee S., Longhi P., Romo M., Exponential BPS graphs and D brane counting on toric Calabi–Yau threefolds: Part II, arXiv: 2012.09769

[15] Banerjee S., Longhi P., Romo M., “A-branes, foliations and localization”, Ann. Henri Poincaré (to appear) , arXiv: 2201.12223 | DOI

[16] Barnes E., “On the theory of multiple gamma function”, Trans. Cambridge Philos. Soc., 19, 1904, 374–425

[17] Benini F., Benvenuti S., Tachikawa Y., “Webs of five-branes and ${\mathcal N}=2$ superconformal field theories”, J. High Energy Phys., 2009:9 (2009), 052, 33 pp., arXiv: 0906.0359 | DOI | MR

[18] 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

[19] Bonelli G., Grassi A., Tanzini A., “Quantum curves and $q$-deformed Painlevé equations”, Lett. Math. Phys., 109 (2019), 1961–2001, arXiv: 1710.11603 | DOI | MR

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

[21] Bridgeland T., “Riemann–Hilbert problems for the resolved conifold”, J. Differential Geom., 115 (2020), 395–435, arXiv: 1703.02776 | DOI | MR

[22] Bullimore M., Kim H.-C., Koroteev P., “Defects and quantum Seiberg–Witten geometry”, J. High Energy Phys., 2015:5 (2015), 095, 78 pp., arXiv: 1412.6081 | DOI | MR

[23] Ceresole A., D'Auria R., Ferrara S., Lerche W., Louis J., “Picard–Fuchs equations and special geometry”, Internat. J. Modern Phys. A, 8 (1993), 79–113, arXiv: hep-th/9204035 | DOI | MR

[24] Chiang T.-M., Klemm A., Yau S.-T., Zaslow E., “Local mirror symmetry: calculations and interpretations”, Adv. Theor. Math. Phys., 3 (1999), 495–565, arXiv: hep-th/9903053 | DOI | MR

[25] Chuang W.-Y., Quantum Riemann–Hilbert problems for the resolved conifold, arXiv: 2203.00294

[26] Coman I., Longhi P., Teschner J., “From quantum curves to topological string partition functions”, Comm. Math. Phys. (to appear) , arXiv: 1811.01978 | DOI

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

[28] Couso-Santamaría R., Resurgence in topological string theory, Ph.D. Thesis, Universidade de Santiago de Compostela, 2014 http://hdl.handle.net/10347/11868

[29] 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

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

[31] Couso-Santamaría R., Schiappa R., Vaz R., “On asymptotics and resurgent structures of enumerative Gromov–Witten invariants”, Commun. Number Theory Phys., 11 (2017), 707–790, arXiv: 1605.07473 | DOI | MR

[32] Cox D.A., Katz S., Mirror symmetry and algebraic geometry, Math. Surveys Monogr., 68, Amer. Math. Soc., Providence, RI, 1999 | DOI | MR

[33] Dimofte T., Gaiotto D., Gukov S., “Gauge theories labelled by three-manifolds”, Comm. Math. Phys., 325 (2014), 367–419, arXiv: 1108.4389 | DOI | MR

[34] Dimofte T., Gukov S., Hollands L., “Vortex counting and Lagrangian 3-manifolds”, Lett. Math. Phys., 98 (2011), 225–287, arXiv: 1006.0977 | DOI | MR

[35] Dingle R.B., Morgan G.J., “${\rm WKB}$ methods for difference equations. I”, Appl. Sci. Res., 18 (1968), 221–237 | DOI | MR

[36] Dingle R.B., Morgan G.J., “${\rm WKB}$ methods for difference equations. II”, Appl. Sci. Res., 18 (1968), 238–245 | DOI | MR

[37] Eager R., Selmani S.A., Walcher J., “Exponential networks and representations of quivers”, J. High Energy Phys., 2017:8 (2017), 063, 68 pp., arXiv: 1611.06177 | DOI | MR

[38] Elliott C., Pestun V., “Multiplicative Hitchin systems and supersymmetric gauge theory”, Selecta Math. (N.S.), 25 (2019), 64, 82 pp., arXiv: 1812.05516 | DOI | MR

[39] Eynard B., Garcia-Failde E., Marchal O., Orantin N., Quantization of classical spectral curves via topological recursion, arXiv: 2106.04339

[40] Faber C., Pandharipande R., “Hodge integrals and Gromov–Witten theory”, Invent. Math., 139 (2000), 173–199, arXiv: math.AG/9810173 | DOI | MR

[41] Forbes B., Jinzenji M., “Extending the Picard–Fuchs system of local mirror symmetry”, J. Math. Phys., 46 (2005), 082302, 39 pp., arXiv: hep-th/0503098 | DOI | MR

[42] Gaiotto D., Kim H.-C., “Surface defects and instanton partition functions”, J. High Energy Phys., 2016:10 (2016), 012, 49 pp., arXiv: 1412.2781 | DOI | MR

[43] Gaiotto D., Moore G.W., Neitzke A., “Wall-crossing, Hitchin systems, and the WKB approximation”, Adv. Math., 234 (2013), 239–403, arXiv: 0907.3987 | DOI | MR

[44] Garoufalidis S., Gu J., Mariño M., “The resurgent structure of quantum knot invariants”, Comm. Math. Phys., 386 (2021), 469–493, arXiv: 2007.10190 | DOI | MR

[45] Garoufalidis S., Gu J., Mariño M., Peacock patterns and resurgence in complex Chern–Simons theory, arXiv: 2012.00062

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

[47] Garoufalidis S., Kashaev R., “Resurgence of Faddeev's quantum dilogarithm”, Topology and Geometry – a Collection of Essays Dedicated to Vladimir G Turaev, IRMA Lect. Math. Theor. Phys., 33, Eur. Math. Soc., Zürich, 2021, 257–271, arXiv: 2008.12465 | DOI | MR

[48] Gel'fand I.M., Zelevinskii A.V., Kapranov M.M., “Hypergeometric functions and toric varieties”, Funct. Anal. Appl., 23 (1989), 94–106 | DOI | MR

[49] Gopakumar R., Vafa C., M-theory and topological strings. I, arXiv: hep-th/9809187

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

[51] Gopakumar R., Vafa C., “On the gauge theory/geometry correspondence”, Adv. Theor. Math. Phys., 3 (1999), 1415–1443, arXiv: hep-th/9811131 | DOI | MR

[52] Grassi A., Hao Q., Neitzke A., Exponential NETWOrks, WKB and the topological string, arXiv: 2201.11594

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

[54] Grassi A., Mariño M., Zakany S., “Resumming the string perturbation series”, J. High Energy Phys., 2015:5 (2015), 038, 35 pp., arXiv: 1405.4214 | DOI | MR

[55] 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

[56] Hatsuda Y., Comments on exact quantization conditions and non-perturbative topological strings, arXiv: 1507.04799

[57] Hatsuda Y., “Spectral zeta function and non-perturbative effects in ABJM Fermi-gas”, J. High Energy Phys., 2015:11 (2015), 086, 33 pp., arXiv: 1503.07883 | DOI | MR

[58] Hatsuda Y., Mariño M., Moriyama S., Okuyama K., “Non-perturbative effects and the refined topological string”, J. High Energy Phys., 2014:9 (2014), 168, 42 pp., arXiv: 1306.1734 | DOI | MR

[59] Hatsuda Y., Okuyama K., “Resummations and non-perturbative corrections”, J. High Energy Phys., 2015:9 (2015), 051, 29 pp., arXiv: 1505.07460 | DOI | MR

[60] Hollands L., Kidwai O., “Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials”, Adv. Theor. Math. Phys., 22 (2018), 1713–1822, arXiv: 1710.04438 | DOI | MR

[61] Hollands L., Neitzke A., “Spectral networks and Fenchel–Nielsen coordinates”, Lett. Math. Phys., 106 (2016), 811–877, arXiv: 1312.2979 | DOI | MR

[62] Hollands L., Neitzke A., “Exact WKB and abelianization for the $T_3$ equation”, Comm. Math. Phys., 380 (2020), 131–186, arXiv: 1906.04271 | DOI | MR

[63] Hollands L., Rüter P., Szabo R.J., “A geometric recipe for twisted superpotentials”, J. High Energy Phys., 2021:12 (2021), 164, 90 pp., arXiv: 2109.14699 | DOI | MR

[64] Hori K., Vafa C., Mirror symmetry, arXiv: hep-th/0002222

[65] Hosono S., “Central charges, symplectic forms, and hypergeometric series in local mirror symmetry”, Mirror Symmetry. V, AMS/IP Stud. Adv. Math., 38, Amer. Math. Soc., Providence, RI, 2006, 405–439, arXiv: hep-th/0404043 | DOI | MR

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

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

[68] Iwaki K., Koike T., Takei Y., “Voros coefficients for the hypergeometric differential equations and Eynard–Orantin's topological recursion: Part II: For confluent family of hypergeometric equations”, J. Integrable Syst., 4 (2019), xyz004, 46 pp., arXiv: 1810.02946 | DOI | MR

[69] Iwaki K., Nakanishi T., “Exact WKB analysis and cluster algebras”, J. Phys. A, 47 (2014), 474009, 98 pp., arXiv: 1401.7094 | DOI | MR

[70] Jockers H., Mayr P., “Quantum K-theory of Calabi–Yau manifolds”, J. High Energy Phys., 2019:11 (2019), 011, 20 pp., arXiv: 1905.03548 | DOI | MR

[71] Jockers H., Mayr P., “A 3d gauge theory/quantum K-theory correspondence”, Adv. Theor. Math. Phys., 24 (2020), 327–457, arXiv: 1808.02040 | DOI | MR

[72] Kashani-Poor A.-K., “Quantization condition from exact WKB for difference equations”, J. High Energy Phys., 2016:6 (2016), 180, 34 pp., arXiv: 1604.01690 | DOI | MR

[73] Katz S., Klemm A., Vafa C., “Geometric engineering of quantum field theories”, Nuclear Phys. B, 497 (1997), 173–195, arXiv: hep-th/9609239 | DOI | MR

[74] Katz S., Mayr P., Vafa C., “Mirror symmetry and exact solution of $4$D $N=2$ gauge theories. I”, Adv. Theor. Math. Phys., 1 (1997), 53–114, arXiv: hep-th/9706110 | DOI | MR

[75] Krefl D., Mkrtchyan R.L., “Exact Chern–Simons/topological string duality”, J. High Energy Phys., 2015:10 (2015), 045, 27 pp., arXiv: 1506.03907 | DOI | MR

[76] Lerche W., “Introduction to Seiberg–Witten theory and its stringy origin”, Nuclear Phys. B Proc. Suppl., 55 (1997), 83–117, arXiv: hep-th/9611190 | DOI | MR

[77] Lockhart G., Vafa C., “Superconformal partition functions and non-perturbative topological strings”, J. High Energy Phys., 2018:10 (2018), 051, 43 pp., arXiv: 1210.5909 | DOI | MR

[78] Mariño M., “Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings”, Fortschr. Phys., 62 (2014), 455–540, arXiv: 1206.6272 | DOI | MR

[79] Mariño M., “Spectral theory and mirror symmetry”, String-Math 2016, Proc. Sympos. Pure Math., 98, Amer. Math. Soc., Providence, RI, 2018, 259–294, arXiv: 1506.07757 | DOI | MR

[80] Mariño M., Moore G., “Counting higher genus curves in a Calabi–Yau manifold”, Nuclear Phys. B, 543 (1999), 592–614, arXiv: hep-th/9808131 | DOI | MR

[81] Mariño M., Zakany S., “Exact eigenfunctions and the open topological string”, J. Phys. A, 50 (2017), 325401, 50 pp., arXiv: 1606.05297 | DOI | MR

[82] Mironov A., Morosov A., “Nekrasov functions and exact Bohr–Sommerfeld integrals”, J. High Energy Phys., 2010:4 (2010), 040, 15 pp., arXiv: 0910.5670 | DOI | MR

[83] Narukawa A., “The modular properties and the integral representations of the multiple elliptic gamma functions”, Adv. Math., 189 (2004), 247–267, arXiv: math.QA/0306164 | DOI | MR

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

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

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

[87] Nekrasov N.A, Rosly A., Shatashvili S.L., “Darboux coordinates, Yang–Yang functional, and gauge theory”, Nuclear Phys. B Proc. Suppl., 216 (2011), 69–93, arXiv: 1103.3919 | DOI | MR

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

[89] Ooguri H., Vafa C., “Knot invariants and topological strings”, Nuclear Phys. B, 577 (2000), 419–438, arXiv: hep-th/9912123 | DOI | MR

[90] Pandharipande R., “The Toda equations and the Gromov–Witten theory of the Riemann sphere”, Lett. Math. Phys., 53 (2000), 59–74, arXiv: math.AG/9912166 | DOI | MR

[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

[92] Ruijsenaars S.N.M., “On Barnes' multiple zeta and gamma functions”, Adv. Math., 156 (2000), 107–132 | DOI | MR

[93] Seiberg N., Witten E., “Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang–Mills theory”, Nuclear Phys. B, 426 (1994), 19–52 | DOI | MR

[94] Takei Y., “WKB analysis and Stokes geometry of differential equations”, Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Math., Birkhäuser/Springer, Cham, 2017, 263–304 | DOI | MR

[95] Wang X., Zhang G., Huang M.-X., “New exact quantization condition for toric Calabi–Yau geometries”, Phys. Rev. Lett., 115 (2015), 121601, 5 pp., arXiv: 1505.05360 | DOI

[96] Witten E., Quantum background independence in string theory, arXiv: hep-th/9306122