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On the Absence of Symmetric Simple Conformal Boundary Conditions

Pengcheng Wei, Yunqin Zheng

TL;DR

We study when an anomaly-free internal symmetry in a 2d CFT can be realized by a simple conformal boundary. Using the Symmetry TFT framework, we derive a three-part criterion (Simplicity, Symmetry, Conformal) and a concrete three-step procedure to diagnose obstructions in a given theory. Applying this to the compact boson, diagonal minimal models, and selected WZW models reveals both violations and satisfactions of the lore: generic finite subgroups like $\\mathbb{Z}_p^m\\times\\mathbb{Z}_q^w$ are not preserved by simple conformal boundaries except at special radii, while in Ising$^3$/S$_3$ and certain WZW cases, symmetry-preserving simple boundaries do exist. The results demonstrate that the lore is not universally valid, but the SymTFT framework provides a precise diagnostic and constructive path for symmetry-preserving boundaries across a range of 1+1d CFTs.

Abstract

Non-trivial 't Hooft anomaly obstructs the existence of a simple symmetric conformal boundary condition in a CFT. Conversely, there is a common piece of lore that trivial 't Hooft anomaly promises the existence of a simple symmetry conformal boundary condition in a given CFT. Recently, counter examples to this lore was realized in tetracritical Ising CFT [1] and compact boson [2] -- the simple conformal boundary conditions preserving certain anomaly-free subsymmetry are absent in these CFTs. In this work, we uncover the underlying reason for the absence of these boundary conditions in counter examples, and propose a criterion diagnosing when the lore fails for any given 2d CFT. The Symmetry TFT description for boundary conditions plays a crucial role.

On the Absence of Symmetric Simple Conformal Boundary Conditions

TL;DR

We study when an anomaly-free internal symmetry in a 2d CFT can be realized by a simple conformal boundary. Using the Symmetry TFT framework, we derive a three-part criterion (Simplicity, Symmetry, Conformal) and a concrete three-step procedure to diagnose obstructions in a given theory. Applying this to the compact boson, diagonal minimal models, and selected WZW models reveals both violations and satisfactions of the lore: generic finite subgroups like are not preserved by simple conformal boundaries except at special radii, while in Ising/S and certain WZW cases, symmetry-preserving simple boundaries do exist. The results demonstrate that the lore is not universally valid, but the SymTFT framework provides a precise diagnostic and constructive path for symmetry-preserving boundaries across a range of 1+1d CFTs.

Abstract

Non-trivial 't Hooft anomaly obstructs the existence of a simple symmetric conformal boundary condition in a CFT. Conversely, there is a common piece of lore that trivial 't Hooft anomaly promises the existence of a simple symmetry conformal boundary condition in a given CFT. Recently, counter examples to this lore was realized in tetracritical Ising CFT [1] and compact boson [2] -- the simple conformal boundary conditions preserving certain anomaly-free subsymmetry are absent in these CFTs. In this work, we uncover the underlying reason for the absence of these boundary conditions in counter examples, and propose a criterion diagnosing when the lore fails for any given 2d CFT. The Symmetry TFT description for boundary conditions plays a crucial role.
Paper Structure (41 sections, 141 equations, 6 figures, 6 tables)

This paper contains 41 sections, 141 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Lore:
  3. Symmetry TFT for Boundary States
  4. SymTFT for a CFT
  5. SymTFT for a Boundary CFT
  6. SymTFT for a Boundary State
  7. Criteria for Simple Symmetric Conformal Boundaries
  8. Simpleness Condition
  9. Symmetric Condition
  10. Symmetric condition:
  11. Conformal Condition
  12. Conformal condition:
  13. Criteria
  14. Improved Lore:
  15. The Lore Violated: $\mathbb{Z}_p^m\times\mathbb{Z}_q^w$ in Compact Boson
  16. ...and 26 more sections

Figures (6)

  • Figure 1: Symmetry TFT for 2d CFT without boundaries.
  • Figure 2: SymTFT for 2d boundary CFT.
  • Figure 3: SymTFT for a boundary state
  • Figure 4: Left: rough boundary and Dirichlet boundary condition; Right: smooth boundary and Neumann boundary condition
  • Figure 5: Left: The cubic lattice where the 3d TQFT is defined on. Right: One term $\widetilde{\omega}\Delta\omega$ in the action \ref{['S3 lattice']}
  • ...and 1 more figures