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On Algebraic Structures Implicit in Topological Quantum Field Theories

L. Crane, D. Yetter

TL;DR

Crane and Yetter show that physically meaningful $3$D and $4$D TQFTs inherently encode algebraic structures: a Hopf algebra object in the circle category for factorizable $3$D theories and a Hopf category in a tensor bicategory for factorizable $4$D theories. They formalize factorizability via cobordism-based monoidal structures and demonstrate how higher-dimensional structures arise from categorical analogues of algebraic notions, establishing a dimensional ladder between Hopf algebras and Hopf categories. The work presents concrete constructions (e.g., left-right crossed bimodules) and proves a central 4D theorem linking Hopf categories to 4D TQFTs, while outlining avenues to reconstruct 4D theories from these algebraic objects and proposing trialgebras as a route to quantum groups and Donaldson–Floer theory. Overall, the paper deepens the connection between path-integral gauge ideas and categorical algebra, supporting conjectures about the algebraic underpinnings of 4D TQFTs and their relation to geometric topology.

Abstract

We show that reasonably well behaved 3d and 4D TQFts must contain certain algebraic structures. In 4D, we find both Hopf categories and trialgebras.

On Algebraic Structures Implicit in Topological Quantum Field Theories

TL;DR

Crane and Yetter show that physically meaningful D and D TQFTs inherently encode algebraic structures: a Hopf algebra object in the circle category for factorizable D theories and a Hopf category in a tensor bicategory for factorizable D theories. They formalize factorizability via cobordism-based monoidal structures and demonstrate how higher-dimensional structures arise from categorical analogues of algebraic notions, establishing a dimensional ladder between Hopf algebras and Hopf categories. The work presents concrete constructions (e.g., left-right crossed bimodules) and proves a central 4D theorem linking Hopf categories to 4D TQFTs, while outlining avenues to reconstruct 4D theories from these algebraic objects and proposing trialgebras as a route to quantum groups and Donaldson–Floer theory. Overall, the paper deepens the connection between path-integral gauge ideas and categorical algebra, supporting conjectures about the algebraic underpinnings of 4D TQFTs and their relation to geometric topology.

Abstract

We show that reasonably well behaved 3d and 4D TQFts must contain certain algebraic structures. In 4D, we find both Hopf categories and trialgebras.

Paper Structure

This paper contains 8 sections, 3 theorems, 1 equation.

Table of Contents

  1. INTRODUCTION
  2. FACTORIZABLE 3D-TQFT
  3. THE HOPF ALGEBRA OBJECT
  4. HOPF CATEGORIES
  5. Categories and Algebras
  6. Hopf Categories
  7. FACTORIZABLE 4D-TQFT AND THE HOPF CATEGORY
  8. ADDITIONAL STRUCTURES AND CONCLUSIONS

Key Result

Theorem 3.1

For any factorizable 3D-TQFT ${\cal T}$, the object $O_ {\cal T}$ admits a natural structure as a Hopf algebra object in the category on the circle. There is a natural isomorphism with the dual object, which is also a Hopf algebra object.

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • ...and 5 more