Quantum curves from refined topological recursion: the genus 0 case. (English) Zbl 1532.81049
Summary: We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing new degrees of freedom in the process. For such curves, we prove the fundamental properties of the recursion analogous to the unrefined case. We show the quantization of spectral curves due to Iwaki-Koike-Takei can be generalized to this setting and give the explicit formula, which turns out to be related to the unrefined case by a simple transformation. For an important collection of examples, we write down the quantum curves and find that in the Nekrasov-Shatashvili limit, they take an especially simple form.
MSC:
| 81S10 | Geometry and quantization, symplectic methods |
| 03D30 | Other degrees and reducibilities in computability and recursion theory |
| 81T32 | Matrix models and tensor models for quantum field theory |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 68W32 | Algorithms on strings |
| 14H45 | Special algebraic curves and curves of low genus |
| 15A04 | Linear transformations, semilinear transformations |
References:
| [1] | Aganagic, M.; Dijkgraaf, R.; Klemm, A.; Mariño, M.; Vafa, C., Topological strings and integrable hierarchies, Commun. Math. Phys., 261, 451-516 (2006) · Zbl 1095.81049 |
| [2] | Aganagic, M.; Cheng, M. C.N.; Dijkgraaf, R.; Krefl, D.; Vafa, C., Quantum geometry of refined topological strings, J. High Energy Phys., 11, Article 019 pp. (2012) · Zbl 1397.83117 |
| [3] | Alday, L. F.; Gaiotto, D.; Tachikawa, Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., 91, 167-197 (2010) · Zbl 1185.81111 |
| [4] | Alim, M.; Hollands, L.; Tulli, I., Quantum curves, resurgence and exact WKB · Zbl 1515.81183 |
| [5] | Andersen, J. E.; Borot, G.; Chekhov, L.; Orantin, N., The ABCD of topological recursion · Zbl 1541.81089 |
| [6] | Barbieri, A.; Bridgeland, T.; Stoppa, J., A quantized Riemann-Hilbert problem in Donaldson-Thomas theory, Int. Math. Res. Not., 2022, 3417-3456 (12 2020) · Zbl 1494.14055 |
| [7] | A. Beilinson, V. Drinfeld, “Opers,” 0501398. |
| [8] | Bonelli, G.; Maruyoshi, K.; Tanzini, A., Quantum Hitchin systems via β -deformed matrix models, Commun. Math. Phys., 358, 3, 1041-1064 (2018) · Zbl 1386.81140 |
| [9] | Borot, G.; Shadrin, S., Blobbed topological recursion: properties and applications, Math. Proc. Camb. Philos. Soc., 162, 39-87 (May 2016) · Zbl 1396.14031 |
| [10] | Borot, G.; Eynard, B.; Mulase, M.; Safnuk, B., A matrix model for simple Hurwitz numbers, and topological recursion, J. Geom. Phys., 61, 522-540 (2011) · Zbl 1225.14043 |
| [11] | Borot, G.; Bouchard, V.; Chidambaram, N. K.; Creutzig, T., Whittaker vectors for \(\mathcal{W} \)-algebras from topological recursion · Zbl 07815192 |
| [12] | Borot, G.; Bouchard, V.; Chidambaram, N. K.; Creutzig, T.; Noshchenko, D., Higher Airy structures, W-algebras and topological recursion |
| [13] | G. Borot, B. Eynard, N. Orantin, Abstract loop equations, topological recursion, and applications, 0601.194. · Zbl 1329.14074 |
| [14] | Bouchard, V.; Eynard, B., Think globally, compute locally, J. High Energy Phys., 2013 (Feb 2013) · Zbl 1342.81513 |
| [15] | Bouchard, V.; Eynard, B., Reconstructing wkb from topological recursion, J. Éc. Polytech. Math., 4, 845-908 (2017) · Zbl 1426.14009 |
| [16] | Bouchard, V.; Mariño, M., Hurwitz numbers, matrix models and enumerative geometry, Proc. Symp. Pure Math., 78, 263-283 (2008) · Zbl 1151.14335 |
| [17] | Bouchard, V.; Osuga, K., Supereigenvalue models and topological recursion, J. High Energy Phys., 04, Article 138 pp. (2018) · Zbl 1390.83378 |
| [18] | Bouchard, V.; Osuga, K., \( \mathcal{N} = 1\) super topological recursion, Lett. Math. Phys., 111 (Nov 2021) · Zbl 1481.81011 |
| [19] | Bouchard, V.; Klemm, A.; Mariño, M.; Pasquetti, S., Remodeling the B-model, Commun. Math. Phys., 287, 117-178 (2009) · Zbl 1178.81214 |
| [20] | Bouchard, V.; Klemm, A.; Mariño, M.; Pasquetti, S., Topological open strings on orbifolds, Commun. Math. Phys., 296, 589-623 (2010) · Zbl 1203.14042 |
| [21] | Bouchard, V.; Ciosmak, P.; Hadasz, L.; Osuga, K.; Ruba, B.; Sułkowski, P., Super quantum Airy structures, Commun. Math. Phys., 380, 449-522 (Oct 2020) · Zbl 1480.81106 |
| [22] | Bridgeland, T., Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math., 216, 69-124 (Dec 2018) · Zbl 1423.14309 |
| [23] | Bridgeland, T., Geometry from Donaldson-Thomas invariants · Zbl 1468.14078 |
| [24] | Bridgeland, T.; Smith, I., Quadratic differentials as stability conditions, Publ. Math. IHÉS, 121, 1, 155-278 (2015) · Zbl 1328.14025 |
| [25] | Brini, A.; Mariño, M.; Stevan, S., The uses of the refined matrix model recursion, J. Math. Phys., 52, Article 052305 pp. (May 2011) · Zbl 1317.81216 |
| [26] | Brini, A.; Eynard, B.; Mariño, M., Torus knots and mirror symmetry, Ann. Henri Poincaré, 13, 1873-1910 (2012) · Zbl 1256.81086 |
| [27] | Chekhov, L., Logarithmic potential β-ensembles and Feynman graphs, Proc. Steklov Inst. Math., 272, 1, 58-74 (2011) · Zbl 1227.81233 |
| [28] | Chekhov, L.; Eynard, B., Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys., 2006, Article 026 pp. (Dec 2006) · Zbl 1226.81138 |
| [29] | Chekhov, L.; Eynard, B.; Orantin, N., Free energy topological expansion for the 2-matrix model, J. High Energy Phys., 2006, Article 053 pp. (Dec 2006) · Zbl 1226.81250 |
| [30] | Chekhov, L.; Eynard, B.; Marchal, O., Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach, Theor. Math. Phys., 166, 141-185 (Feb 2011) · Zbl 1354.81014 |
| [31] | Chekhov, L.; Eynard, B.; Marchal, O., Topological expansion of the Bethe ansatz, and quantum algebraic geometry · Zbl 1354.81014 |
| [32] | Ciosmak, P.; Hadasz, L.; Manabe, M.; Sułkowski, P., Singular vector structure of quantum curves, (2016 AMS von Neumann Symposium Topological Recursion and Its Influence in Analysis, Geometry, and Topology (11 2017)) |
| [33] | Dijkgraaf, R.; Vafa, C., Toda theories, matrix models, topological strings, and N=2 gauge systems |
| [34] | Do, N.; Norbury, P., Topological recursion for irregular spectral curves, J. Lond. Math. Soc., 97, 398-426 (Mar 2018) · Zbl 1459.14019 |
| [35] | Dumitrescu, O.; Mulase, M., Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, Lett. Math. Phys., 104, 635-671 (2014) · Zbl 1296.14026 |
| [36] | Dumitrescu, O.; Mulase, M., Lectures on the topological recursion for Higgs bundles and quantum curves, (The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles (2018), World Scientific), 103-198 · Zbl 1401.14001 |
| [37] | Dumitrescu, O.; Mulase, M., Interplay between opers, quantum curves, WKB analysis, and Higgs bundles, SIGMA, 17, Article 036 pp. (2021) · Zbl 1469.14067 |
| [38] | Dumitriu, I.; Edelman, A., Matrix models for beta ensembles, J. Math. Phys., 43, 11, 5830-5847 (2002) · Zbl 1060.82020 |
| [39] | Dunin-Barkowski, P.; Orantin, N.; Shadrin, S.; Spitz, L., Identification of the Givental formula with the spectral curve topological recursion procedure, Commun. Math. Phys., 328, 669-700 (2014) · Zbl 1293.53090 |
| [40] | Eynard, B., Counting Surfaces, Progress in Mathematical Physics., vol. 70 (2016), Springer · Zbl 1338.81005 |
| [41] | Eynard, B., A short overview of the “topological recursion” |
| [42] | Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys., 1, 2 (2007) · Zbl 1161.14026 |
| [43] | Eynard, B.; Orantin, N., Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture, Commun. Math. Phys., 337, 2, 483-567 (2015) · Zbl 1365.14072 |
| [44] | Eynard, B.; Orantin, N., Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models · Zbl 1365.14072 |
| [45] | Eynard, B.; Orantin, N., Algebraic methods in random matrices and enumerative geometry · Zbl 1134.81040 |
| [46] | Eynard, B.; Mulase, M.; Safnuk, B., The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, Publ. Res. Inst. Math. Sci., 47, 2, 629-670 (2011) · Zbl 1225.14022 |
| [47] | Eynard, B.; Garcia-Failde, E.; Marchal, O.; Orantin, N., Quantization of classical spectral curves via topological recursion · Zbl 07845299 |
| [48] | Fang, B.; Liu, C.-C. M.; Zong, Z., On the remodeling conjecture for toric Calabi-Yau 3-orbifolds, J. Am. Math. Soc., 33, 1, 135-222 (2020) · Zbl 1444.14093 |
| [49] | Gaiotto, D.; Moore, G. W.; Neitzke, A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., 234, 239-403 (2013) · Zbl 1358.81150 |
| [50] | Grassi, A.; Gu, J.; Mariño, M., Non-perturbative approaches to the quantum Seiberg-Witten curve, J. High Energy Phys., 07, Article 106 pp. (2020) · Zbl 1451.81352 |
| [51] | Grassi, A.; Hao, Q.; Neitzke, A., Exact WKB methods in SU(2) N_f = 1, J. High Energy Phys., 01, Article 046 pp. (2022) · Zbl 1521.81085 |
| [52] | Gu, J.; Jockers, H.; Klemm, A.; Soroush, M., Knot invariants from topological recursion on augmentation varieties, Commun. Math. Phys., 336, 2, 987-1051 (2015) · Zbl 1323.57008 |
| [53] | Gukov, S.; Sulkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys., 02, Article 070 pp. (2012) · Zbl 1309.81220 |
| [54] | Hollands, L.; Kidwai, O., Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials, Adv. Theor. Math. Phys., 22, 1713-1822 (June 2019) · Zbl 07430956 |
| [55] | Hollands, L.; Neitzke, A., Exact WKB and abelianization for the \(T_3\) equation, Commun. Math. Phys., 380, 1, 131-186 (2020) · Zbl 1455.81025 |
| [56] | Hollands, L.; Rüter, P.; Szabo, R. J., A geometric recipe for twisted superpotentials, J. High Energy Phys., 12, Article 164 pp. (2021) · Zbl 1521.81390 |
| [57] | Iwaki, K., 2-Parameter τ-function for the first Painlevé equation: topological recursion and direct monodromy problem via exact WKB analysis, Commun. Math. Phys., 377, 2, 1047-1098 (2020) · Zbl 1448.81324 |
| [58] | Iwaki, K.; Kidwai, O., Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies, Adv. Math., 398, Article 108191 pp. (2022) · Zbl 1486.81157 |
| [59] | K. Iwaki, O. Kidwai, “Topological recursion and uncoupled BPS structures II: Voros symbols and the τ-function,” 2108.06995. · Zbl 1521.81394 |
| [60] | Iwaki, K.; Marchal, O., Painlevé 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas, Ann. Henri Poincaré, 18, 8, 2581-2620 (2017) · Zbl 1385.34067 |
| [61] | Iwaki, K.; Nakanishi, T., Exact WKB analysis and cluster algebras, J. Phys. A, Math. Theor., 47, 47, Article 474009 pp. (2014) · Zbl 1311.81113 |
| [62] | Iwaki, K.; Saenz, A., Quantum curve and the first Painleve equation, SIGMA, 12, Article 011 pp. (2016) · Zbl 1339.34096 |
| [63] | 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, 1 (2019), xyz004 · Zbl 1490.81078 |
| [64] | Iwaki, K.; Koike, T.; Takei, Y., Voros coefficients for the hypergeometric differential equations and Eynard-Orantin’s topological recursion - part I: for the Weber equation · Zbl 1525.34132 |
| [65] | Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1, 1-23 (1992) · Zbl 0756.35081 |
| [66] | Kontsevich, M.; Soibelman, Y., Airy structures and symplectic geometry of topological recursion · Zbl 1448.53078 |
| [67] | Manabe, M.; Sułkowski, P., Quantum curves and conformal field theory, Phys. Rev. D, 95 (Jun 2017) |
| [68] | Marchal, O., One-cut solution of the β ensembles in the zhukovsky variable, J. Stat. Mech., 2012, Article P01011 pp. (Jan 2012) |
| [69] | Mirzakhani, M., Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167, 1, 179-222 (2006) · Zbl 1125.30039 |
| [70] | Mirzakhani, M., Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Am. Math. Soc., 20, 01, 1-24 (2007) · Zbl 1120.32008 |
| [71] | M. Mulase, B. Safnuk, Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy, 0601.194. · Zbl 1144.14030 |
| [72] | Nekrasov, N., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys., 7, 5, 831-864 (2003) · Zbl 1056.81068 |
| [73] | Nekrasov, N.; Rosly, A.; Shatashvili, S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nucl. Phys. B, Proc. Suppl., 216, 69-93 (2011) |
| [74] | Nekrasov, N. A.; Shatashvili, S. L., Quantization of integrable systems and four dimensional gauge theories, (16th International Congress on Mathematical Physics (8 2009)), 265-289 · Zbl 1214.83049 |
| [75] | Norbury, P.; Scott, N., Gromov-Witten invariants of \(\mathbb{P}^1\) and Eynard-Orantin invariants, Geom. Topol., 18, 1865-1910 (Oct 2014) · Zbl 1308.05011 |
| [76] | Osuga, K., Super topological recursion and Gaiotto vectors for superconformal blocks, Lett. Math. Phys., 112, 3, 48 (2022) · Zbl 1490.17037 |
| [77] | Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surv. Differ. Geom., 1, 243-310 (1991) · Zbl 0757.53049 |
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