Topological Recursion for Transalgebraic Spectral Curves and the TR/QC Connection

Abstract

The topological recursion is a construction in algebraic geometry that takes in the data of a so-called spectral curve, $\mathcal{S}=\left(\Sigma,x,y\right)$ where $\Sigma$ is a Riemann surface and $x,y:\Sigma\to\mathbb{C}_\infty$ are meromorphic, and recursively constructs correlators which, in applications, are then interpreted as generating functions. In many of these applications, for example the $r$-spin Hurwitz case $\mathcal{S}=\left(\mathbb{C},x(z)=z{\rm e}^{-z^r},y(z)={\rm e}^{z^r}\right)$, $x$ has essential singularities when the underlying Riemann surface is compactified. Previously, these essential singularities have been ignored and the topological recursion considered on the non-compact surface. Here we argue that it is more natural to include the essential singularities as ramification points and give the corresponding definition for topological recursion; that is, a topological recursion for \emph{transalgebraic} spectral curves rather than \emph{algebraic} spectral curves. We use this definition to shed light on the TR/QC connection, Hurwitz theory, the Gromov-Witten invariants of $\mathbb{CP}^1$, and mirror curves.