Reconstructing WKB from topological recursion

Bouchard, Vincent; Eynard, Bertrand

doi:10.5802/jep.58

Reconstructing WKB from topological recursion
[De la récurrence topologique à WKB]

Bouchard, Vincent ¹ ; Eynard, Bertrand ²

¹ Department of Mathematical & Statistical Sciences, University of Alberta, 632 CAB Edmonton, Alberta, Canada T6G 2G1
² Institut de Physique Théorique, CEA Saclay 91191 Gif-sur-Yvette cedex, France

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 845-908.

Résumé
Abstract

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 : 2016-10-04
Accepté le : 2017-07-25
Publié le : 2017-08-17

DOI : 10.5802/jep.58

Classification : 14H70, 81Q20, 81S10, 30F30
Keywords: Topological recursion, WKB, quantum curves, quantization
Mot clés : Récurrence topologique, WKB, courbes quantiques, quantification

Affiliations des auteurs :

Bouchard, Vincent ¹ ; Eynard, Bertrand ²

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     pages = {845--908},
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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/

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