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

 Author / Creator
 Weller, Quinten

The topological recursion is a construction in algebraic geometry that takes in the data of a socalled 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 noncompact 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 GromovWitten invariants of $\mathbb{CP}^1$, and mirror curves.

 Graduation date
 Fall 2022

 Type of Item
 Thesis

 Degree
 Master of Science

 License
 This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for noncommercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.