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arXiv:2501.07787

SymSETs and self-dualities under gauging non-invertible symmetries

Da-Chuan Lu, Zhengdi Sun, Zhang, Zipei

Published Jan 14, 2025• 6 citations
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Abstract

The self-duality defects under discrete gauging in a categorical symmetry $\mathcal{C}$ can be classified by inequivalent ways of enriching the bulk SymTFT of $\mathcal{C}$ with $\mathbb{Z}_2$ 0-form symmetry. The resulting Symmetry Enriched Topological (SET) orders will be referred to as $\textit{SymSETs}$ and are parameterized by choices of $\mathbb{Z}_2$ symmetries, as well as symmetry fractionalization classes and discrete torsions. In this work, we consider self-dualities under gauging $\textit{non-invertible}$ $0$-form symmetries in $2$-dim QFTs and explore their SymSETs. Unlike the simpler case of self-dualities under gauging finite Abelian groups, the SymSETs here generally admit multiple choices of fractionalization classes. We provide a direct construction of the SymSET from a given duality defect using its $\textit{relative center}$. Using the SymSET, we show explicitly that changing fractionalization classes can change fusion rules of the duality defect besides its $F$-symbols. We consider three concrete examples: the maximal gauging of $\operatorname{Rep} H_8$, the non-maximal gauging of the duality defect $\mathcal{N}$ in $\operatorname{Rep} H_8$ and $\operatorname{Rep} D_8$ respectively. The latter two cases each result in 6 fusion categories with two types of fusion rules related by changing fractionalization class. In particular, two self-dualities of $\operatorname{Rep} D_8$ related by changing the fractionalization class lead to $\operatorname{Rep} D_{16}$ and $\operatorname{Rep} SD_{16}$ respectively. Finally, we study the physical implications such as the spin selection rules and the SPT phases for the aforementioned categories.

DOI
10.48550/arxiv.2501.07787

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The self-duality defects under discrete gauging in a categorical symmetry C\mathcal{C}C can be classified by inequivalent ways of enriching the bulk SymTFT of C\mathcal{C}C with Z2\mathbb{Z}_2Z2​ 0-form symmetry. The resulting Symmetry Enriched Topological (SET) orders will be referred to as SymSETs\textit{SymSETs}SymSETs and are parameterized by choices of Z2\mathbb{Z}_2Z2​ symmetries, as well as symmetry fractionalization classes and discrete torsions. In this work, we consider self-dualities under gauging non-invertible\textit{non-invertible}non-invertible 000-form symmetries in 222-dim QFTs and explore their SymSETs. Unlike the simpler case of self-dualities under gauging finite Abelian groups, the SymSETs here generally admit multiple choices of fractionalization classes. We provide a direct construction of the SymSET from a given duality defect using its relative center\textit{relative center}relative center. Using the SymSET, we show explicitly that changing fractionalization classes can change fusion rules of the duality defect besides its FFF-symbols. We consider three concrete examples: the maximal gauging of Rep⁡H8\operatorname{Rep} H_8RepH8​, the non-maximal gauging of the duality defect N\mathcal{N}N in Rep⁡H8\operatorname{Rep} H_8RepH8​ and Rep⁡D8\operatorname{Rep} D_8RepD8​ respectively. The latter two cases each result in 6 fusion categories with two types of fusion rules related by changing fractionalization class. In particular, two self-dualities of Rep⁡D8\operatorname{Rep} D_8RepD8​ related by changing the fractionalization class lead to Rep⁡D16\operatorname{Rep} D_{16}RepD16​ and Rep⁡SD16\operatorname{Rep} SD_{16}RepSD16​ respectively. Finally, we study the physical implications such as the spin selection rules and the SPT phases for the aforementioned categories.
We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions d=2 d = 2 and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed ‘t Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension d=2 d = 2 and 3. In d=2 d = 2 , these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In d=3 d = 3 , among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.
We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2 ℤ 2 symmetry. We introduce a subsystem \mathbb Z_2 ℤ 2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.
Abstract We present practical and formal methods for gauging non-invertible symmetries in (2+1)d topological quantum field theories. Along the way, we generalize various aspects of invertible 0-form gauging, including symmetry fractionalization, discrete torsion, and the fixed point theorem for symmetry action on lines. Our approach involves two complementary strands: the fusion of topological interfaces and Morita theory of fusion 2-categories. We use these methods to derive constraints on gaugeable symmetries and their duals while unifying the prescription for gauging non-invertible 0-form and 1-form symmetries and various higher structures. With a view toward recent advances in creating non-Abelian topological orders from Abelian ones, we give a simple recipe for non-invertible 0-form gauging that takes large classes of the latter to the former. We also describe conditions under which iterated gauging of invertible 0-form symmetries is equivalent to a single-step gauging of a non-invertible symmetry. We conclude with a set of concrete examples illustrating these various phenomena involving gauging symmetries of the infrared limit of the toric code.
We explicitly realize the Rep(Q8Q_8Q8​) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep(Q8Q_8Q8​) duality web. We further show that we can construct a class of lattice models with Rep(GGG) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.
A bstract We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ 2 × ℤ 2 symmetry, and has Rep( H 8 ) and Rep( D 8 ) fusion category symmetries as a result. We find that gauging the entire Rep( H 8 ) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/ R . At R = 2 \sqrt{2} 2​ 2 , which corresponds to two decoupled Ising CFTs (Ising 2 in short), the theory is self-dual under gauging the Rep( H 8 ) symmetry. This implies the existence of a topological defect line in the Ising 2 CFT obtained from half-space gauging of the Rep( H 8 ) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep( H 8 ) symmetry to form a bigger fusion category which is a ℤ 2 -extension of Rep( H 8 ). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising 2 CFT. Finally, we show that the torus partition functions of the Monster 2 CFT and Ising × Monster CFT are also invariant under gauging the Rep( H 8 ) symmetry.
A cluster of recent preprints in strongly correlated systems converges on a structural pattern worth investigating: deconfined quantum critical points (DQCPs) and beyond-Landau phase transitions in one and two spatial dimensions can be systematically organized — and in some cases *derived* — by tracking what happens to lattice anomalies and symmetry categories when internal symmetries are gauged. This synthesis draws primarily from cond-mat.str-el sources, with supporting material from cond-mat.mes-hall and cond-mat.dis-nn, and argues the following heuristic reading (not a derivation): the non-invertible symmetries that emerge after gauging anomalous onsite symmetries encode, in their fusion category structure, the full universality data of the associated critical point — including whether the transition is type-I or type-II DQCP. Concretely, we synthesize five findings: (1) the spectral-sequence classification of Lieb-Schultz-Mattis (LSM) anomalies into E∞1,2E_\infty^{1,2}E∞1,2​ versus E∞2,1E_\infty^{2,1}E∞2,1​ types, which predicts whether the dual symmetry after gauging is invertible or non-invertible [corpus:arxiv:2606.06343]; (2) the demonstration that Rep(D8D_8D8​) and Rep(H8H_8H8​) crystalline categorical symmetries, despite sharing fusion rings, differ in FFF-symbols and therefore define inequivalent DQCP universality classes [corpus:arxiv:2606.05856]; (3) the introduction of twin condensable algebras and twin phases — inequivalent phases sharing anyon content but differing in algebra structure — as a source of intrinsically beyond-Landau transitions [corpus:arxiv:2605.31602, corpus:arxiv:2605.31601]; (4) the explicit lattice construction of non-invertible symmetry-breaking phases with local order parameters obeying generalized algebraic structures, and their duality to DQCPs of invertible symmetries [corpus:arxiv:2605.27672]; and (5) the tensor-network derivation that string order parameters in 1D organize into Lagrangian algebras in the Drinfel'd center of the symmetry category [corpus:arxiv:2605.27193]. A concrete falsification path: if two DQCPs with the same fusion ring but different FFF-symbols can be shown numerically (e.g., via DMRG central-charge extraction and operator-content comparison) to lie in identical universality classes, the thesis that FFF-symbols carry physical content distinguishing transitions would be falsified. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602, 2606.04651, 2606.04748, 2606.04958, 2606.05845, 2606.05856, 2606.06343, 2606.06412, 2606.06432, 2606.06490