# An invitation to 2D TQFT and quantization of Hitchin spectral curves

## TL;DR

This paper develops a categorical formulation of 2D TQFTs using ribbon graphs and Frobenius objects, and explores the geometric quantization of Hitchin spectral curves into opers, linking quantum curves with complex geometric structures.

## Contribution

It introduces a functorial approach to 2D TQFTs via ribbon graphs and Frobenius algebras, and provides a geometric framework for quantizing Hitchin spectral curves into opers.

## Key findings

- Categorical formulation of 2D TQFTs using ribbon graphs and Frobenius objects
- Equivalence of quantum curves, opers, and projective structures for SL_2(C) on genus > 1 surfaces
- Connection between Frobenius algebra twisted recursion and topological recursion

## 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.