Cooking up Quantum Curves from Airy Structures

• Author / Creator

  Ooms, Adrien

• In the framework of topological recursion (TR), the quantum curve conjecture
  relates the initial spectral curve to a differential equation – the quantum curve
  – satisfied by the TR wave-function. Several admissibility conditions have been
  put forward to explain which input of spectral curve effectively produces a quantum
  curve.
  In this thesis we propose a new way to prove the existence of quantum
  curves for genus zero spectral curves. This new approach is based on some early
  calculations relating the quantum curve and Virasoro constraints, and on the
  more recent Airy structure reformulation of TR in which Virasoro constraints
  play a central role.
  The Airy structure approach gives a relation between the quantum curve
  wave function and the Airy structure partition function via the specialization
  map. We explain how the specialization map extended to differential operators
  could be the key to relate some generic combination of Virasoro constraints to
  the quantum curve differential equation. In this context, admissibility conditions
  arise when looking at which spectral curve produces a generic operator that
  can be specialized. Interestingly, when specialization works we always recover
  the expected quantum curve.
  Although our method suggests a way to check the existence of any quantum
  curve, in practice it remains limited because of the pedestrian approach
  of specializing differential operators. For curves slightly more complicated than
  the easier examples, our expressions quickly get too complicated to manipulate.
  However we think that the connection which we outline here is interesting and
  could probably be extended further with a more technical treatment.

• Subjects / Keywords

  • Quantum Curves
  • Topological Recursion
  • Mathematical Physics

• Graduation date

  Fall 2019

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/r3-4vh0-0069

• License

  Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.