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Topological symmetry in quantum field theory

Daniel S. Freed, Gregory W. Moore, Constantin Teleman

TL;DR

The paper develops a unified, topologically flavored framework for internal quantum field theory symmetries using quiches (σ,ρ) to separate topological symmetry data from concrete realizations as module actions on field theories. It builds a universal defect calculus grounded in fully local topological field theories, enabling explicit composition laws for defects, domain walls, and boundaries via the cobordism hypothesis and tangential structures. It then analyzes quotients (gauging) and dualities in depth, providing concrete constructions and examples from finite gauge theories, DW twists, and higher-group symmetries, and introducing dual quiches to capture finite electromagnetic duality. The work places finite homotopy theories as computable laboratories, outlines extensions to nonfinite or higher-categorical symmetries, and offers a blueprint for applying topological methods to symmetry, defects, and dualities across quantum field theories. These insights promise a coherent, scalable framework for understanding and manipulating symmetry actions in QFTs via topological and categorical data.

Abstract

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.

Topological symmetry in quantum field theory

TL;DR

The paper develops a unified, topologically flavored framework for internal quantum field theory symmetries using quiches (σ,ρ) to separate topological symmetry data from concrete realizations as module actions on field theories. It builds a universal defect calculus grounded in fully local topological field theories, enabling explicit composition laws for defects, domain walls, and boundaries via the cobordism hypothesis and tangential structures. It then analyzes quotients (gauging) and dualities in depth, providing concrete constructions and examples from finite gauge theories, DW twists, and higher-group symmetries, and introducing dual quiches to capture finite electromagnetic duality. The work places finite homotopy theories as computable laboratories, outlines extensions to nonfinite or higher-categorical symmetries, and offers a blueprint for applying topological methods to symmetry, defects, and dualities across quantum field theories. These insights promise a coherent, scalable framework for understanding and manipulating symmetry actions in QFTs via topological and categorical data.

Abstract

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.
Paper Structure (36 sections, 2 theorems, 124 equations, 35 figures)

This paper contains 36 sections, 2 theorems, 124 equations, 35 figures.

Table of Contents

  1. Groups and algebras of symmetries
  2. Fibering over $BG$
  3. Algebras of symmetries
  4. Quotients
  5. Projective symmetries
  6. Higher algebra
  7. Formal structures in field theory
  8. Axioms
  9. Domain walls
  10. Boundary theories, anomalies, and anomalous theories
  11. Defects
  12. Tangential structures and the passage from local to global defects
  13. Symmetry in field theory
  14. Abstract topological symmetry data in field theory
  15. Concrete realization of topological symmetry in field theory
  16. ...and 21 more sections

Key Result

Proposition 1

There is an isomorphism of right $\sigma$-modules

Figures (35)

  • Figure 1: The sandwich picture of $(\sigma ,\rho )$ acting on $F$: the $(\sigma ,\rho )$-module structure on $F$
  • Figure 2: The quiche $(\sigma ,\rho )$, a $\rho$-defect $D_1$ (on the boundary), and $(\sigma ,\rho )$-defects $D_2$ (in the bulk) and $D_3$ (emanating from the boundary)
  • Figure 3: A domain wall $\delta \colon\sigma _1\to \sigma _2$
  • Figure 4: Domain walls in the manifold $W$
  • Figure 5: (a) a right boundary theory (b) a left boundary theory
  • ...and 30 more figures

Theorems & Definitions (109)

  • Remark 1
  • Example 1
  • Example 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 1
  • Remark 5
  • Remark 6
  • Definition 2
  • ...and 99 more