All Symmetries and Boundaries of the Two-Dimensional Free Scalar from Continuous Flat-Gauging
Abstract
We propose a remote detectability criterion for two-dimensional conformal field theories, which asserts that any local operator outside of the product of the left- and right-moving Virasoro algebras can be detected by some topological line defect. We discuss its relationship to existing results in the conformal net literature. We then use this criterion to argue that the known SU(2) moduli space of unitary conformal boundary conditions and the SO(4) moduli space of unitary topological line defects of the SU(2)1 Wess-Zumino-Witten model completely capture all unitary conformal boundaries and unitary topological lines of the theory. Invoking the fact that the non-compact boson conformal field theory can be obtained from SU(2)1 by "flat-gauging" a U(1) symmetry, we are also able to completely classify the unitary conformal boundaries and unitary topological line defects of the two-dimensional free scalar field. In particular, we find a 4-dimensional moduli space of topological line defects, most of which were not known previously. We support our proposed classifications by writing down concrete Lagrangian descriptions of all of the boundaries and lines of the non-compact boson. For every boundary condition, we calculate the corresponding boundary state, and for each pair of boundary conditions, we calculate the corresponding interval Hilbert space. We verify that the annulus partition functions of the theory respect open/closed duality. We also calculate the fusion rules of the topological line defects, their action on local operators, their action on the boundary conditions by parallel fusion, and their corresponding twisted-sector Hilbert spaces.
Related
- Flat Gauging of Continuous (Non-invertible) Symmetries and Non-compact BF SymTFT for Compact Boson
- Gauging Modulated Symmetries: Bond Algebras, Higher-Form Symmetries, and Symmetry-Enriched Topological Order
- Non-invertible Selection Rules from Generalized Discrete Gauging of Finite Non-Abelian Symmetries
- Symmetry defects and gauging for quantum states with matrix product unitary symmetries
- Lattice 2-group symmetries: operators, defects, and gauging
BibTeX
@misc{choi2026symmetries,
title = {{All Symmetries and Boundaries of the Two-Dimensional Free Scalar from Continuous Flat-Gauging}},
author = {Yichul Choi and Ho Tat Lam and Brandon C. Rayhaun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40818},
eprint = {2610.40818},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40818},
}
RIS
TY - UNPB
TI - All Symmetries and Boundaries of the Two-Dimensional Free Scalar from Continuous Flat-Gauging
AU - Yichul Choi
AU - Ho Tat Lam
AU - Brandon C. Rayhaun
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40818
UR - https://arxiv.org/abs/2610.40818
AB - We propose a remote detectability criterion for two-dimensional conformal field theories, which asserts that any local operator outside of the product of the left- and right-moving Virasoro algebras can be detected by some topological line defect. We discuss its relationship to existing results in the conformal net literature. We then use this criterion to argue that the known SU(2) moduli space of unitary conformal boundary conditions and the SO(4) moduli space of unitary topological line defects of the SU(2)_1 Wess-Zumino-Witten model completely capture all unitary conformal boundaries and unitary topological lines of the theory. Invoking the fact that the non-compact boson conformal field theory can be obtained from SU(2)_1 by "flat-gauging" a U(1) symmetry, we are also able to completely classify the unitary conformal boundaries and unitary topological line defects of the two-dimensional free scalar field. In particular, we find a 4-dimensional moduli space of topological line defects, most of which were not known previously. We support our proposed classifications by writing down concrete Lagrangian descriptions of all of the boundaries and lines of the non-compact boson. For every boundary condition, we calculate the corresponding boundary state, and for each pair of boundary conditions, we calculate the corresponding interval Hilbert space. We verify that the annulus partition functions of the theory respect open/closed duality. We also calculate the fusion rules of the topological line defects, their action on local operators, their action on the boundary conditions by parallel fusion, and their corresponding twisted-sector Hilbert spaces.
ER -