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SymTh for non-finite symmetries

Fabio Apruzzi, Francesco Bedogna, Nicola Dondi

TL;DR

SymTh provides a bulk, non-topological framework built from a $(p+1)$-form Maxwell theory to capture non-finite symmetries of a boundary QFT, complemented by an interval sandwich construction to extract the symmetry sector. The work derives topological operators, boundary projections, and gauging mechanisms across a spectrum of examples—from abelian $(p)$-form and 2-group cases to $\mathbb{Q}/\mathbb{Z}$ non-invertible symmetries—and gives both bottom-up ($4d$ axion-Maxwell) and top-down (IIB supergravity) routes to the SymTh description. It shows how the interval limit decouples bulk dynamics while preserving symmetry data, and identifies brane configurations as UV avatars for topological defects and quantum Hall states dressing non-invertible operators. The framework unifies low- and high-dimensional models, offering a versatile holographic-compatible approach to non-invertible and higher-form symmetries with potential applications to conformal boundaries and categorical formulations of generalized symmetries.

Abstract

Symmetry topological field theory (SymTFT) is a convenient tool for studying finite generalized symmetries of a given quantum field theory (QFT). In particular, SymTFTs encode all the symmetry structures and properties, including anomalies. Recently, this tool has been applied for non-finite symmetries as well. In this paper, we take a different route, which consists of considering a free theory rather than a topological field theory in the bulk. We call it Symmetry Theory (SymTh). We study its topological operators together with the free boundary conditions. We also propose a procedure that is analogous to the sandwich construction of SymTFTs and allows us to obtain the physical QFT. We apply this to many examples, ranging from abelian $p$-form symmetries to 2-groups, and the (solvable) case of group-like symmetries in quantum mechanics. Finally, we provide a derivation of the SymTh of $\mathbb Q/ \mathbb Z$ non-invertible symmetries from the dimensional reduction of IIB supergravity on the conifold. In addition, we give an ultraviolet interpretation of the quantum Hall states dressing the non-invertible $\mathbb Q/ \mathbb Z$ topological defects, in terms of branes in the IIB supergravity background.

SymTh for non-finite symmetries

TL;DR

SymTh provides a bulk, non-topological framework built from a -form Maxwell theory to capture non-finite symmetries of a boundary QFT, complemented by an interval sandwich construction to extract the symmetry sector. The work derives topological operators, boundary projections, and gauging mechanisms across a spectrum of examples—from abelian -form and 2-group cases to non-invertible symmetries—and gives both bottom-up ( axion-Maxwell) and top-down (IIB supergravity) routes to the SymTh description. It shows how the interval limit decouples bulk dynamics while preserving symmetry data, and identifies brane configurations as UV avatars for topological defects and quantum Hall states dressing non-invertible operators. The framework unifies low- and high-dimensional models, offering a versatile holographic-compatible approach to non-invertible and higher-form symmetries with potential applications to conformal boundaries and categorical formulations of generalized symmetries.

Abstract

Symmetry topological field theory (SymTFT) is a convenient tool for studying finite generalized symmetries of a given quantum field theory (QFT). In particular, SymTFTs encode all the symmetry structures and properties, including anomalies. Recently, this tool has been applied for non-finite symmetries as well. In this paper, we take a different route, which consists of considering a free theory rather than a topological field theory in the bulk. We call it Symmetry Theory (SymTh). We study its topological operators together with the free boundary conditions. We also propose a procedure that is analogous to the sandwich construction of SymTFTs and allows us to obtain the physical QFT. We apply this to many examples, ranging from abelian -form symmetries to 2-groups, and the (solvable) case of group-like symmetries in quantum mechanics. Finally, we provide a derivation of the SymTh of non-invertible symmetries from the dimensional reduction of IIB supergravity on the conifold. In addition, we give an ultraviolet interpretation of the quantum Hall states dressing the non-invertible topological defects, in terms of branes in the IIB supergravity background.
Paper Structure (23 sections, 111 equations, 1 figure)

This paper contains 23 sections, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Note added:
  3. WarmUp: SymTh for a $U(1)$ $p$-form global symmetry
  4. Topological operators and their action
  5. Boundary conditions
  6. Sandwich Construction
  7. Truncation to Symmetry TFT
  8. Lower-dimensional examples
  9. Boundary conditions in $d = 2$
  10. $2d$ Maxwell theory
  11. $2d$ Yang-Mills theory
  12. Higher-dimensional examples
  13. Continuous 2-group in 4d
  14. Mixed case: 2-group with a discrete 1-form symmetry
  15. SymTh for $\mathbb{Q}/\mathbb{Z}$ 0- and 1-form non-invertible symmetries
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Topological operators $U$ linking with Wilson surfaces $W$ or $V$. Depending on boundary conditions the Wilson surface can end on the boundary or not. If the Wilson surface $W$ (for example) ends on the boundary it will source a non-topological operator, $\mathcal{E}$. In this case, the topological operator $U$ defines a symmetry of the boundary theory.