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 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.

  • Subjects / Keywords
  • Graduation date
    Fall 2022
  • Type of Item
  • Degree
    Master of Science
  • DOI
  • License
    This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial 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.