Title: An invitation to 2D TQFT and quantization of Hitchin spectral curves,
Abstract: This article consists of two parts. In Part 1, we present a formulation of
two-dimensional topological quantum field theories in terms of a functor from a
category of Ribbon graphs to the endofuntor category of a monoidal category.
The key point is that the category of ribbon graphs produces all Frobenius
objects. Necessary backgrounds from Frobenius algebras, topological quantum
field theories, and cohomological field theories are reviewed. A result on
Frobenius algebra twisted topological recursion is included at the end of Part
1.
In Part 2, we explain a geometric theory of quantum curves. The focus is
placed on the process of quantization as a passage from families of Hitchin
spectral curves to families of opers. To make the presentation simpler, we
unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined
on a compact Riemann surface $C$ of genus greater than $1$. In this case,
quantum curves, opers, and projective structures in $C$ all become the same
notion. Background materials on projective coordinate systems, Higgs bundles,
opers, and non-Abelian Hodge correspondence are explained.