Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hermitian axiom on two-dimensional topological quantum field theories

Honglin Zhu

TL;DR

The paper analyzes Atiyah's Hermitian axiom for 2D complex TQFTs, translating Hermitian and unitary constraints into precise algebraic structures on Frobenius algebras. It proves that a Hermitian TQFT corresponds to a complex commutative $^*$-Frobenius algebra whose involution $J$ induces a real Frobenius subalgebra $A_0$ via complexification, making Hermitian theories essentially real. In the unitary case, the real part $A_0$ decomposes as $\mathbb{R}^n$ with positive Frobenius weights, yielding a diagonalizable left-regular action and a $C^*$-Frobenius structure; the handle operator spectrum is positive, with eigenvalues $\lambda_i=1/\epsilon(e_i)$. Overall, the work unifies and clarifies prior results by showing Hermitian TQFTs are controlled by real data and unitary TQFTs are classified by positive weight data, providing a clean algebraic classification of 2D Hermitian/unitary theories.

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.

The Hermitian axiom on two-dimensional topological quantum field theories

TL;DR

The paper analyzes Atiyah's Hermitian axiom for 2D complex TQFTs, translating Hermitian and unitary constraints into precise algebraic structures on Frobenius algebras. It proves that a Hermitian TQFT corresponds to a complex commutative -Frobenius algebra whose involution induces a real Frobenius subalgebra via complexification, making Hermitian theories essentially real. In the unitary case, the real part decomposes as with positive Frobenius weights, yielding a diagonalizable left-regular action and a -Frobenius structure; the handle operator spectrum is positive, with eigenvalues . Overall, the work unifies and clarifies prior results by showing Hermitian TQFTs are controlled by real data and unitary TQFTs are classified by positive weight data, providing a clean algebraic classification of 2D Hermitian/unitary theories.

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.
Paper Structure (5 sections, 7 theorems, 26 equations)

This paper contains 5 sections, 7 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Atiyah's Axioms
  3. The Hermitian Axiom
  4. The Algebraic Structure of Hermitian TQFTs
  5. The Algebraic Structure of Unitary TQFTs

Key Result

Theorem 3.1

Suppose $M$ is a cobordism from $\Sigma_1$ to $\Sigma_2$. Let $V = Z(\Sigma_1)$, $W = Z(\Sigma_2)$ so that we can view $Z(M)$ and $Z(M^*)$ as linear transformations $Z(M): V \to W$ and $Z(M^*): W \to V$. If $h_{V}$, $h_{W}$ are the Hermitian forms on $V$ and $W$, respectively, then $Z(M^*)$ is the a for all $v \in V$, $w \in W$.

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Definition 3.2
  • Definition 4.1: Kock (kock2003)
  • Proposition 4.2: Kock (kock2003)
  • Definition 4.3
  • Theorem 4.4
  • proof
  • Theorem 4.5: Sawin (sawin1995)
  • proof
  • ...and 10 more