Extension of WKB-Topological Recursion Connection

Abstract

It has been proven in other sources that spectral curves, $(\Sigma,x,y)$, where $\Sigma$ is a compact Riemann surface, and meromophic functions $x$ and $y$ satisfy a polynomial equation (and subject to certain admissibility conditions), can be used with the topological recursion to construct the WKB expansion for the quantization of said curve. In this paper we prove an extension of that connection for spectral curves, $(\Sigma,u,y)$, where $u$ is meromorphic only on an open region of $\Sigma$, and $x=e^u$ may or may not be meromorphic on $\Sigma$, so long as $y du$ is meromorphic on $\Sigma$; we will see that the admissibility condition still holds, and that there are added constraints. We provide a rigorous proof for dealing with spectral curves where $u$ is meromorphic on $\Sigma$, but provide only a conceptual argument and affirmative examples for dealing with spectral curves where $u$ is not meromorphic on $\Sigma$.