The Hermitian axiom on two-dimensional topological quantum field theories

TL;DR

This paper explores the algebraic structures of Hermitian and unitary two-dimensional topological quantum field theories, providing new structure theorems and clarifying previous results in the field.

Contribution

It introduces a detailed algebraic characterization of Hermitian and unitary 2D TQFTs, extending the correspondence with Frobenius algebras and clarifying existing results.

Findings

Algebraic objects for Hermitian and unitary TQFTs identified

Structure theorems for these TQFTs established

Clarification of older results on unitary TQFTs achieved

Abstract

In this paper, we examine Atiyah's Hermitian axiom for two-dimensional complex topological quantum field theories. Building on the correspondence between 2D TQFTs and Frobenius algebras, we find the algebraic objects corresponding to Hermitian and unitary TQFTs respectively and prove structure theorems about them. We then clarify a few older results on unitary TQFTs using our structure theorems.