bibkey: lu2024galitydefects authors: Da-Chuan Lu; Zhengdi Sun; Zipei Zhang year: 2024 title: “Exploring G-ality defects in 2-dim QFTs” doi: 10.1007/JHEP11(2025)081 url: https://arxiv.org/abs/2406.12151v3 claim: “For A = Z_N x Z_N the paper searches for beta-invertible S_3 subgroups of the anyon permutation symmetries O(A + A^) of the SymTFT, finds none for N < 20 containing a factor 2 or 3, proves this for even N, and conjectures that there is no beta-invertible S_3 symmetry when N is a multiple of 3.” strata_touched:

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Exploring G-ality defects in 2-dim QFTs

D.-C. Lu, Z. Sun and Z. Zhang, “Exploring G-ality defects in 2-dim QFTs”, arXiv:2406.12151 (v1 2024-06-17, v2 2025-03-19, v3 2025-10-25 matching the published version); JHEP 11 (2025) 081. Subjects: hep-th (primary), cond-mat.str-el, math-ph.

The paper studies G-ality defects of a 2d QFT with a finite Abelian symmetry A, built from twisted gaugings, through the 3d SymTFT Z(Vec_A). Its anyons are the pairs (a, â) ∈ A ⊕ Â, and an anyon permutation symmetry is written as a block matrix U = (α β; γ δ) with β : Â → A, preserving the self-statistics and the braiding. U is called β-invertible when β is invertible; a G-ality extension needs a subgroup G⁽⁰⁾ ≅ G all of whose non-identity elements are β-invertible. For A = ℤ_N × ℤ_N and G = S₃ the paper states in §1.1:

we numerically search for β-invertible S_3 symmetries in the SymTFT for N < 20. We find that for any N containing a factor of 2 or 3, there are no β-invertible S_3 symmetries in the SymTFT, thus there cannot be S_3-ality defects for these N. We are able to prove this analytically when N is even, and we conjecture this is true generally for N being a multiple of 3 based on our numerical evidence.

and in §5.1, after the proof for N = 2^r: “For N = 3^r, however, at the moment we do not know the numbers and the generic forms of inequivalent ℤ_3 ⊂ S_3 generator T’s for generic r, we therefore conjecture that this is true based on the explicit result with r = 1,2,3.”

Verified locator

• DOI: https://doi.org/10.1007/JHEP11(2025)081
• URL: https://arxiv.org/abs/2406.12151v3 (full text retrieved 2026-09-29).
• Location: §1.1, Example III, for the statement of the conjecture; the SymTFT review section for the block form of U and the definition of β-invertibility; §5.1 for the proof for even N and the conjecture for N = 3^r.