Topological Recursion and Genus One Quantum Curves: An Accessible Exploration

• Author / Creator

  Dauphinee, Tyler LJ

• We explore the connection between Eynard-Orantin Topological Recursion (EOTR) and the asymptotic solutions to differential equations constructed with the WKB method (named for its creators Wentzel, Kramers and Brillouin). Using the Airy spectral curve as an initial example, we propose a general connection between topological recursion and WKB solutions to the quantum curve generated via quantization of the defining algebraic curve. We proceed further by examining the proposed connection in the context of the genus one family of Weierstrass spectral curves. We construct the perturbative wave-function and show that it is annihilated by a differential operator which is not a quantization of the spectral curve. Furthermore, as a consequence of equivalent approaches we also obtain an infinite collection of identities relating cycle integrals of elliptic functions to quasi-modular forms.

• Subjects / Keywords

  • Elliptic
  • Modular
  • Topological recursion

• Graduation date

  Spring 2017

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R3RN30K9B

• License

  This thesis is made available by the University of Alberta Libraries 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.