Reconstructing WKB from topological recursion
[De la récurrence topologique à WKB]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 845-908.

Nous montrons que la récurrence topologique permet de reconstruire le développement WKB d’une courbe quantique pour toutes les courbes spectrales dont les polygones de Newton n’ont pas de point intérieur (et qui sont lisses en tant que courbes affines). Cette classe de courbes contient presque toutes les courbes quantiques déjà étudiées dans la littérature, ainsi que beaucoup d’autres ; en particulier, beaucoup de courbes d’ordre plus élevé que 2 sont incluses dans cette classe. Nous étudions aussi la relation entre le choix d’un ordre pour la quantification de la courbe spectrale et le choix d’un diviseur pour l’intégration nécessaire à la reconstruction du développement WKB.

We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves). This includes nearly all previously known cases in the literature, and many more; in particular, it includes many quantum curves of order greater than two. We also explore the connection between the choice of ordering in the quantization of the spectral curve and the choice of integration divisor to reconstruct the WKB expansion.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.58
Classification : 14H70,  81Q20,  81S10,  30F30
Mots clés : Récurrence topologique, WKB, courbes quantiques, quantification
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     title = {Reconstructing {WKB} from topological~recursion},
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Bouchard, Vincent; Eynard, Bertrand. Reconstructing WKB from topological recursion. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 845-908. doi : 10.5802/jep.58. http://www.numdam.org/articles/10.5802/jep.58/

[1] Alexandrov, A. Open intersection numbers, Kontsevich-Penner model and cut-and-join operators, J. High Energy Phys. (2015) no. 8 (arXiv:1412.3772) | Article | MR 3402137 | Zbl 1388.81165

[2] Andersen, J. E.; Chekhov, L. O.; Norbury, P.; Penner, R. C. Models of discretized moduli spaces, cohomological field theories, and Gaussian means (2015) (arXiv:1501.05867) | Article

[3] Andersen, J. E.; Chekhov, L. O.; Norbury, P.; Penner, R. C. Topological recursion for Gaussian means and cohomological field theories (2015) (arXiv:1512.09309) | Article | Zbl 1335.81105

[4] Baker, H. F. Examples of applications of Newton’s polygon to the theory of singular points of algebraic functions, Trans. Cambridge Philos. Soc., Volume 15 (1893), pp. 403-450

[5] Beelen, P. A generalization of Baker’s theorem, Finite Fields Appl., Volume 15 (2009) no. 5, pp. 558-568 | Article | MR 2554039 | Zbl 1219.11174

[6] Beelen, P.; Pellikaan, R. The Newton polygon of plane curves with many rational points, Des. Codes Cryptogr., Volume 21 (2000) no. 1-3, pp. 41-67 | Article | MR 1801161 | Zbl 1005.14019

[7] Belliard, R.; Eynard, B. (To be published)

[8] Bergère, M.; Eynard, B. Determinantal formulae and loop equations (2009) (arXiv:0901.3273)

[9] Bergère, M.; Eynard, B. Universal scaling limits of matrix models, and (p,q) Liouville gravity (2009) (arXiv:0909.0854)

[10] Borot, G.; Eynard, B. Geometry of spectral curves and all order dispersive integrable system, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 8 (2012) (arXiv:1110.4936) | MR 3007259 | Zbl 1270.14017

[11] Borot, G.; Eynard, B. All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, Quantum Topol., Volume 6 (2015) no. 1, pp. 39-138 (arXiv:1205.2261) | Article | MR 3335006 | Zbl 1335.57019

[12] Bouchard, V.; Eynard, B. Think globally, compute locally, J. High Energy Phys. (2013) no. 2 (arXiv:1211.2302v2) | Article | MR 3046532 | Zbl 1342.81513

[13] Bouchard, V.; Eynard, B. (2014) (talk given at the workshop “Quantum curves, Hitchin systems, and the Eynard-Orantin theory” at the American Institute of Mathematics)

[14] Bouchard, V.; Hernández Serrano, D.; Liu, X.; Mulase, M. Mirror symmetry for orbifold Hurwitz numbers, J. Differential Geom., Volume 98 (2014) no. 3, pp. 375-423 (arXiv:1301.4871) | Article | MR 3263522 | Zbl 1315.53100

[15] Bouchard, V.; Hutchinson, J.; Loliencar, P.; Meiers, M.; Rupert, M. A generalized topological recursion for arbitrary ramification, Ann. Henri Poincaré, Volume 15 (2014) no. 1, pp. 143-169 | Article | MR 3147410 | Zbl 1291.81212

[16] Bouchard, V.; Klemm, M.; Mariño, M.; Pasquetti, S. Remodeling the B-model, Comm. Math. Phys., Volume 287 (2009) no. 1, pp. 117-178 | Article | MR 2480744 | Zbl 1178.81214

[17] Brezin, E.; Hikami, S. Intersection numbers of Riemann surfaces from Gaussian matrix models, J. High Energy Phys. (2007) no. 10 (arXiv:0709.3378) | Article | MR 2357882

[18] Chekhov, L.; Eynard, B.; Orantin, N. Free energy topological expansion for the 2-matrix model, J. High Energy Phys. (2006) no. 12 (arXiv:math-ph/0603003) | Article | MR 2276699 | Zbl 1226.81250

[19] Codesido, S.; Grassi, A.; Mariño, M. Spectral theory and mirror curves of higher genus, Ann. Henri Poincaré, Volume 18 (2017) no. 2, pp. 559-622 (arXiv:1507.02096) | Article | MR 3596771 | Zbl 1364.81202

[20] Dijkgraaf, R.; Fuji, H.; Manabe, M. The volume conjecture, perturbative knot invariants, and recursion relations for topological strings, Nuclear Phys. B, Volume 849 (2011) no. 1, pp. 166-211 (arXiv:1010.4542) | Article | MR 2795276 | Zbl 1215.81082

[21] Do, N.; Dyer, A.; Mathews, D. Topological recursion and a quantum curve for monotone Hurwitz numbers, J. Geom. Phys., Volume 120 (2017), pp. 19-36 (arXiv:1408.3992) | Article | MR 3712146 | Zbl 1373.14051

[22] Do, N.; Manescu, D. Quantum curves for the enumeration of ribbon graphs and hypermaps, Commun. Number Theory Phys., Volume 8 (2014) no. 4, pp. 677-701 (arXiv:1312.6869) | Article | MR 3318387 | Zbl 1366.14034

[23] Do, N.; Norbury, P. Topological recursion for irregular spectral curves (2014) (arXiv:1412.8334) | Zbl 06898538

[24] Dumitrescu, O.; Mulase, M. Quantization of spectral curves for meromorphic Higgs bundles through topological recursion (2014) (arXiv:1411.1023)

[25] Dumitrescu, O.; Mulase, M. Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, Lett. Math. Phys., Volume 104 (2014) no. 6, pp. 635-671 (arXiv:1310.6022) | Article | MR 3200933 | Zbl 1296.14026

[26] Dunin-Barkowski, P.; Mulase, M.; Norbury, P.; Popolitov, A.; Shadrin, S. Quantum spectral curve for the Gromov-Witten theory of the complex projective line, J. reine angew. Math., Volume 726 (2017), pp. 267-289 (arXiv:1312.5336) | MR 3641659 | Zbl 1364.14045

[27] Dunin-Barkowski, P.; Norbury, P.; Orantin, N.; Popolitov, A.; Shadrin, S. Dubrovin’s superpotential as a global spectral curve (2015) (arXiv:1509.06954)

[28] Dunin-Barkowski, P.; Orantin, N.; Shadrin, S.; Spitz, L. Identification of the Givental formula with the spectral curve topological recursion procedure, Comm. Math. Phys., Volume 328 (2014) no. 2, pp. 669-700 (arXiv:1211.4021) | Article | MR 3199996 | Zbl 1293.53090

[29] Eynard, B. Topological expansion for the 1-Hermitian matrix model correlation functions, J. High Energy Phys. (2004) no. 11 (arXiv:hep-th/0407261) | Article | MR 2118807

[30] Eynard, B.; Mariño, M. A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys., Volume 61 (2011) no. 7, pp. 1181-1202 (arXiv:0810.4273) | Article | MR 2788322 | Zbl 1215.81084

[31] Eynard, B.; Orantin, N. Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys., Volume 1 (2007) no. 2, pp. 347-452 (arXiv:math-ph/0702045v4) | Article | MR 2346575 | Zbl 1161.14026

[32] Eynard, B.; Orantin, N. Algebraic methods in random matrices and enumerative geometry (2008) (arXiv:0811.3531)

[33] 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, Comm. Math. Phys., Volume 337 (2015) no. 2, pp. 483-567 (arXiv:1205.1103) | Article | MR 3339157 | Zbl 1365.14072

[34] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, Ann. Sci. École Norm. Sup. (4), Volume 43 (2010) no. 4, pp. 621-658 (arXiv:math/0612510) | Article | Numdam | MR 2722511 | Zbl 1203.53090

[35] Fang, B.; Liu, C.-C. M.; Zong, Z. All genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds (2013) (arXiv:1310.4818)

[36] Fuji, H.; Gukov, S.; Sułkowski, P. Volume conjecture: refined and categorified, Adv. Theo. Math. Phys., Volume 16 (2012) no. 6, pp. 1669-1777 (arXiv:1203.2182) | Article | MR 3065082 | Zbl 1282.57016

[37] Fuji, H.; Gukov, S.; Sułkowski, P. Super-A-polynomial for knots and BPS states, Nuclear Phys. B, Volume 867 (2013) no. 2, pp. 506-546 (arXiv:1205.1515) | Article | MR 2992793 | Zbl 1262.81170

[38] Garoufalidis, S. Difference and differential equations for the colored Jones function, J. Knot Theory Ramifications, Volume 17 (2008) no. 4, pp. 495-510 (arXiv:math/0306229) | Article | MR 2414452 | Zbl 1155.57012

[39] Garoufalidis, S.; Kucharski, P.; Sułkowski, P. Knots, BPS states, and algebraic curves, Comm. Math. Phys., Volume 346 (2016) no. 1, pp. 75-113 (arXiv:1504.06327) | Article | MR 3528417 | Zbl 1365.57015

[40] Grassi, A.; Hatsuda, Y.; Mariño, M. Topological strings from quantum mechanics, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3177-3235 (arXiv:1410.3382) | Article | MR 3556519 | Zbl 1365.81094

[41] Gu, J.; Klemm, A.; Mariño, M.; Reuter, J. Exact solutions to quantum spectral curves by topological string theory, J. High Energy Phys. (2015) no. 10 (arXiv:1506.09176) | MR 3435614 | Zbl 1388.81411

[42] Gukov, S.; Sułkowski, P. A-polynomial, B-model, and quantization, J. High Energy Phys. (2012) no. 02 (arXiv:1108.0002) | Article | MR 2996110 | Zbl 1309.81220

[43] Kashaev, R.; Mariño, M. Operators from mirror curves and the quantum dilogarithm, Comm. Math. Phys., Volume 346 (2016) no. 3, pp. 967-994 (arXiv:1501.01014) | Article | MR 3537342 | Zbl 1348.81436

[44] Kashaev, R.; Mariño, M.; Zakany, S. Matrix models from operators and topological strings, 2, Ann. Henri Poincaré, Volume 17 (2016) no. 10, pp. 2741-2781 (arXiv:1505.02243) | Article | MR 3546986 | Zbl 1353.81104

[45] Kashani-Poor, A. Quantization condition from exact WKB for difference equations, J. High Energy Phys. (2016) no. 06 (arXiv:1604.01690) | Article | MR 3538818 | Zbl 1388.81557

[46] Kazarian, M.; Zograf, P. Virasoro constraints and topological recursion for Grothendieck’s dessin counting, Lett. Math. Phys., Volume 105 (2015) no. 8, pp. 1057-1084 (arXiv:1406.5976) | Article | MR 3366120 | Zbl 1332.37051

[47] Liu, X.; Mulase, M.; Sorkin, A. Quantum curves for simple Hurwitz numbers of an arbitrary base curve (2013) (arXiv:1304.0015)

[48] Mariño, M. Open string amplitudes and large order behavior in topological string theory, J. High Energy Phys. (2008) no. 03 (arXiv:hep-th/0612127) | Article | MR 2391060

[49] Mariño, M. Spectral theory and mirror symmetry (2015) (arXiv:1506.07757)

[50] Mehta, M. L. Random matrices, Pure and Applied Mathematics, 142, Elsevier/Academic Press, Amsterdam, 2004

[51] Milanov, T. The Eynard-Orantin recursion for the total ancestor potential, Duke Math. J., Volume 163 (2014) no. 9, pp. 1795-1824 (arXiv:1211.5847) | Article | MR 3217767 | Zbl 1327.14051

[52] Mulase, M.; Shadrin, S.; Spitz, L. The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures, Commun. Number Theory Phys., Volume 7 (2013) no. 1, pp. 125-143 (arXiv:1301.5580) | Article | Zbl 1283.14012

[53] Mulase, M.; Sułkowski, P. Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion, Adv. Theo. Math. Phys., Volume 19 (2015) no. 5, pp. 955-1015 (arXiv:1210.3006) | Article | Zbl 1342.81458

[54] Norbury, P.; Scott, N. Polynomials representing Eynard-Orantin invariants, Q. J. Math., Volume 64 (2013) no. 2, pp. 515-546 (arXiv:1001.0449) | Article | MR 3063520 | Zbl 1269.30047

[55] Orantin, N. Symplectic invariants, Virasoro constraints and Givental decomposition (2008) (arXiv:0808.0635)

[56] Safnuk, B. Topological recursion for open intersection numbers, Commun. Number Theory Phys., Volume 10 (2016) no. 4, pp. 833-857 (arXiv:1601.04049) | Article | MR 3636676 | Zbl 06723023

[57] Wigner, E. P. On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Phil. Soc. (4), Volume 47 (1951), pp. 790-798 | Article | Zbl 0044.44203

[58] Witten, E. Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235-269 | Zbl 0812.14017

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