Reconstructing WKB from topological recursion

TL;DR

This paper demonstrates that topological recursion can reconstruct the WKB expansion of quantum curves for a broad class of spectral curves, including higher-order cases, and explores the influence of quantization ordering.

Contribution

It proves the reconstruction of WKB expansions from topological recursion for spectral curves with no interior points in their Newton polygons, extending previous results to higher-order quantum curves.

Findings

Topological recursion reconstructs WKB expansions for a wide class of spectral curves.

Includes many higher-order quantum curves beyond the classical second order.

Explores the relationship between quantization ordering and integration divisor choices.

Abstract

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.