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.
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 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.
Accepted:
Published online:
DOI: 10.5802/jep.58
Keywords: Topological recursion, WKB, quantum curves, quantization
Mot clés : Récurrence topologique, WKB, courbes quantiques, quantification
Vincent Bouchard 1; Bertrand Eynard 2
@article{JEP_2017__4__845_0, author = {Vincent Bouchard and Bertrand Eynard}, title = {Reconstructing {WKB} from topological~recursion}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {845--908}, publisher = {\'Ecole polytechnique}, volume = {4}, year = {2017}, doi = {10.5802/jep.58}, mrnumber = {3694097}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.58/} }
TY - JOUR AU - Vincent Bouchard AU - Bertrand Eynard TI - Reconstructing WKB from topological recursion JO - Journal de l’École polytechnique — Mathématiques PY - 2017 SP - 845 EP - 908 VL - 4 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.58/ DO - 10.5802/jep.58 LA - en ID - JEP_2017__4__845_0 ER -
%0 Journal Article %A Vincent Bouchard %A Bertrand Eynard %T Reconstructing WKB from topological recursion %J Journal de l’École polytechnique — Mathématiques %D 2017 %P 845-908 %V 4 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.58/ %R 10.5802/jep.58 %G en %F JEP_2017__4__845_0
Vincent Bouchard; Bertrand Eynard. Reconstructing WKB from topological recursion. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 845-908. doi : 10.5802/jep.58. https://jep.centre-mersenne.org/articles/10.5802/jep.58/
[1] - “Open intersection numbers, Kontsevich-Penner model and cut-and-join operators”, J. High Energy Phys. (2015) no. 8, arXiv:1412.3772 | DOI | MR | Zbl
[2] - “Models of discretized moduli spaces, cohomological field theories, and Gaussian means” (2015), arXiv:1501.05867 | DOI | Zbl
[3] - “Topological recursion for Gaussian means and cohomological field theories” (2015), arXiv:1512.09309 | DOI | Zbl
[4] - “Examples of applications of Newton’s polygon to the theory of singular points of algebraic functions”, Trans. Cambridge Philos. Soc. 15 (1893), p. 403-450
[5] - “A generalization of Baker’s theorem”, Finite Fields Appl. 15 (2009) no. 5, p. 558-568 | DOI | MR | Zbl
[6] - “The Newton polygon of plane curves with many rational points”, Des. Codes Cryptogr. 21 (2000) no. 1-3, p. 41-67 | DOI | MR | Zbl
[7]
, To be published[8] - “Determinantal formulae and loop equations” (2009), arXiv:0901.3273 | Zbl
[9] - “Universal scaling limits of matrix models, and Liouville gravity” (2009), arXiv:0909.0854 | Zbl
[10] - “Geometry of spectral curves and all order dispersive integrable system”, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), arXiv:1110.4936 | MR | Zbl
[11] - “All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials”, Quantum Topol. 6 (2015) no. 1, p. 39-138, arXiv:1205.2261 | DOI | MR | Zbl
[12] - “Think globally, compute locally”, J. High Energy Phys. (2013) no. 2, arXiv:1211.2302v2 | DOI | MR | Zbl
[13]
(2014), talk given at the workshop “Quantum curves, Hitchin systems, and the Eynard-Orantin theory” at the American Institute of Mathematics[14] - “Mirror symmetry for orbifold Hurwitz numbers”, J. Differential Geom. 98 (2014) no. 3, p. 375-423, arXiv:1301.4871 | DOI | MR | Zbl
[15] - “A generalized topological recursion for arbitrary ramification”, Ann. Henri Poincaré 15 (2014) no. 1, p. 143-169 | DOI | MR | Zbl
[16] - “Remodeling the B-model”, Comm. Math. Phys. 287 (2009) no. 1, p. 117-178 | DOI | MR | Zbl
[17] - “Intersection numbers of Riemann surfaces from Gaussian matrix models”, J. High Energy Phys. (2007) no. 10, arXiv:0709.3378 | DOI | MR | Zbl
[18] - “Free energy topological expansion for the 2-matrix model”, J. High Energy Phys. (2006) no. 12, arXiv:math-ph/0603003 | DOI | MR | Zbl
[19] - “Spectral theory and mirror curves of higher genus”, Ann. Henri Poincaré 18 (2017) no. 2, p. 559-622, arXiv:1507.02096 | DOI | MR | Zbl
[20] - “The volume conjecture, perturbative knot invariants, and recursion relations for topological strings”, Nuclear Phys. B 849 (2011) no. 1, p. 166-211, arXiv:1010.4542 | DOI | MR | Zbl
[21] - “Topological recursion and a quantum curve for monotone Hurwitz numbers”, J. Geom. Phys. 120 (2017), p. 19-36, arXiv:1408.3992 | DOI | MR | Zbl
[22] - “Quantum curves for the enumeration of ribbon graphs and hypermaps”, Commun. Number Theory Phys. 8 (2014) no. 4, p. 677-701, arXiv:1312.6869 | DOI | MR | Zbl
[23] - “Topological recursion for irregular spectral curves” (2014), arXiv:1412.8334 | Zbl
[24] - “Quantization of spectral curves for meromorphic Higgs bundles through topological recursion” (2014), arXiv:1411.1023
[25] - “Quantum curves for Hitchin fibrations and the Eynard-Orantin theory”, Lett. Math. Phys. 104 (2014) no. 6, p. 635-671, arXiv:1310.6022 | DOI | MR | Zbl
[26] - “Quantum spectral curve for the Gromov-Witten theory of the complex projective line”, J. reine angew. Math. 726 (2017), p. 267-289, arXiv:1312.5336 | MR | Zbl
[27] - “Dubrovin’s superpotential as a global spectral curve” (2015), arXiv:1509.06954
[28] - “Identification of the Givental formula with the spectral curve topological recursion procedure”, Comm. Math. Phys. 328 (2014) no. 2, p. 669-700, arXiv:1211.4021 | DOI | MR | Zbl
[29] - “Topological expansion for the -Hermitian matrix model correlation functions”, J. High Energy Phys. (2004) no. 11, arXiv:hep-th/0407261 | DOI | MR
[30] - “A holomorphic and background independent partition function for matrix models and topological strings”, J. Geom. Phys. 61 (2011) no. 7, p. 1181-1202, arXiv:0810.4273 | DOI | MR | Zbl
[31] - “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys. 1 (2007) no. 2, p. 347-452, arXiv:math-ph/0702045v4 | DOI | MR | Zbl
[32] - “Algebraic methods in random matrices and enumerative geometry” (2008), arXiv:0811.3531 | Zbl
[33] - “Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture”, Comm. Math. Phys. 337 (2015) no. 2, p. 483-567, arXiv:1205.1103 | DOI | MR | Zbl
[34] - “Tautological relations and the -spin Witten conjecture”, Ann. Sci. École Norm. Sup. (4) 43 (2010) no. 4, p. 621-658, arXiv:math/0612510 | DOI | Numdam | MR | Zbl
[35] - “All genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds” (2013), arXiv:1310.4818
[36] - “Volume conjecture: refined and categorified”, Adv. Theo. Math. Phys. 16 (2012) no. 6, p. 1669-1777, arXiv:1203.2182 | DOI | MR | Zbl
[37] - “Super--polynomial for knots and BPS states”, Nuclear Phys. B 867 (2013) no. 2, p. 506-546, arXiv:1205.1515 | DOI | MR | Zbl
[38] - “Difference and differential equations for the colored Jones function”, J. Knot Theory Ramifications 17 (2008) no. 4, p. 495-510, arXiv:math/0306229 | DOI | MR | Zbl
[39] - “Knots, BPS states, and algebraic curves”, Comm. Math. Phys. 346 (2016) no. 1, p. 75-113, arXiv:1504.06327 | DOI | MR | Zbl
[40] - “Topological strings from quantum mechanics”, Ann. Henri Poincaré 17 (2016) no. 11, p. 3177-3235, arXiv:1410.3382 | DOI | MR | Zbl
[41] - “Exact solutions to quantum spectral curves by topological string theory”, J. High Energy Phys. (2015) no. 10, arXiv:1506.09176 | MR | Zbl
[42] - “A-polynomial, B-model, and quantization”, J. High Energy Phys. (2012) no. 02, arXiv:1108.0002 | DOI | MR | Zbl
[43] - “Operators from mirror curves and the quantum dilogarithm”, Comm. Math. Phys. 346 (2016) no. 3, p. 967-994, arXiv:1501.01014 | DOI | MR | Zbl
[44] - “Matrix models from operators and topological strings, 2”, Ann. Henri Poincaré 17 (2016) no. 10, p. 2741-2781, arXiv:1505.02243 | DOI | MR | Zbl
[45] - “Quantization condition from exact WKB for difference equations”, J. High Energy Phys. (2016) no. 06, arXiv:1604.01690 | DOI | MR | Zbl
[46] - “Virasoro constraints and topological recursion for Grothendieck’s dessin counting”, Lett. Math. Phys. 105 (2015) no. 8, p. 1057-1084, arXiv:1406.5976 | DOI | MR | Zbl
[47] - “Quantum curves for simple Hurwitz numbers of an arbitrary base curve” (2013), arXiv:1304.0015
[48] - “Open string amplitudes and large order behavior in topological string theory”, J. High Energy Phys. (2008) no. 03, arXiv:hep-th/0612127 | DOI | MR
[49] - “Spectral theory and mirror symmetry” (2015), arXiv:1506.07757
[50] - Random matrices, Pure and Applied Mathematics, vol. 142, Elsevier/Academic Press, Amsterdam, 2004
[51] - “The Eynard-Orantin recursion for the total ancestor potential”, Duke Math. J. 163 (2014) no. 9, p. 1795-1824, arXiv:1211.5847 | DOI | MR | Zbl
[52] - “The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures”, Commun. Number Theory Phys. 7 (2013) no. 1, p. 125-143, arXiv:1301.5580 | DOI | Zbl
[53] - “Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion”, Adv. Theo. Math. Phys. 19 (2015) no. 5, p. 955-1015, arXiv:1210.3006 | DOI | Zbl
[54] - “Polynomials representing Eynard-Orantin invariants”, Q. J. Math. 64 (2013) no. 2, p. 515-546, arXiv:1001.0449 | DOI | MR | Zbl
[55] - “Symplectic invariants, Virasoro constraints and Givental decomposition” (2008), arXiv:0808.0635
[56] - “Topological recursion for open intersection numbers”, Commun. Number Theory Phys. 10 (2016) no. 4, p. 833-857, arXiv:1601.04049 | DOI | MR | Zbl
[57] - “On the statistical distribution of the widths and spacings of nuclear resonance levels”, Proc. Cambridge Phil. Soc. (4) 47 (1951), p. 790-798 | DOI | Zbl
[58] - “Algebraic geometry associated with matrix models of two-dimensional gravity”, in Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, p. 235-269 | Zbl
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