Twisted Higgs Bundles, Topological Recursion, and Quantum Curves
Date
2023-08-17
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
ORCID
0009-0009-3092-6385
Type
Thesis
Degree Level
Doctoral
Abstract
The focus of this thesis is on the interplay between Higgs bundles and topological recursion. Our interests lie in the relationship between quantum curves and the quantization of Hitchin spectral curves, and also the relationship between Eynard-Orantin differentials and the geometry of the Hitchin moduli space.
We give an overview of existing results in the literature on quantum curves, covering the necessary material to construct a quantum curve from a meromorphic SL(2, C)-Hitchin spectral curve. Starting from the quantum curve, we offer a new perspective on the quantization that includes the spectral correspondence and C∗-action. We view the quantization as a procedure that happens on the spectral curve, rather than the base. This idea frames quantization around the tautological section, rather than the Higgs field.
Previous works relating meromorphic Higgs bundles to topological recursion have considered non-singular models to allow the recursion to be done on a smooth Riemann surface. In this thesis, we start from an L-twisted Higgs bundle. By studying the deformation theory of the L-twisted moduli space, we interpret L as meromorphic data on a subbundle of an ordinary Higgs bundle. We encode this meromorphic data as a b-structure on the base Riemann surface and spectral curve. We then propose a so-called twisted recursion on the spectral curve, where the Eynard-Orantin differentials live in the twisted cotangent bundle. We show that the g = 0 twisted Eynard-Orantin differentials compute the Taylor expansion of the period matrix of a Hitchin spectral curve, mirroring a result for ordinary Higgs bundles and topological recursion. In particular, this shows that the geometry of the spectral curve is independent of the ambient space in which it resides.
Description
Keywords
Algebraic geometry, Quantum theory, Moduli space, Higgs bundle, Hitchin system, Topological recursion, Quantization, Quantum curve, Spectral curve, b-geometry
Citation
Degree
Doctor of Philosophy (Ph.D.)
Department
Mathematics and Statistics
Program
Mathematics