Quantization of classical spectral curves via topological recursion

Bertrand Eynard (1, 2) , Elba Garcia-Failde (3) , Olivier Marchal (4, 5, 6, 7) , Nicolas Orantin (8)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 IHES - Institut des Hautes Études Scientifiques
3 UFR Mathématiques et informatique [Sciences] - Université Paris Cité
4 UJM - Université Jean Monnet - Saint-Étienne
5 ICJ - Institut Camille Jordan
6 PSPM - Probabilités, statistique, physique mathématique
7 IUF - Institut universitaire de France
8 UNIGE - Université de Genève = University of Geneva

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

Keywords: Topological Recursion; Quantum curves; Spectral curves; Painlevé equations

Domains: Mathematical Physics [math-ph]; Algebraic Geometry [math.AG]; Exactly Solvable and Integrable Systems [nlin.SI]; High Energy Physics - Theory [hep-th]

Bertrand Eynard, Elba Garcia-Failde, Olivier Marchal, Nicolas Orantin. Quantization of classical spectral curves via topological recursion. Communications in Mathematical Physics, 2024. ⟨hal-03254701v2⟩