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.
Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d “inflow” symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the “Frobenius-Schur indicator” of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing \mathbb{Z}_{4} ℤ 4 ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} 𝖳 obeying the algebra \mathsf{T}^{2}=C 𝖳 2 = C or \mathsf{T}^{2}=(-1)^{F}C, 𝖳 2 = ( − 1 ) F C , with C C a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.
Higher-form symmetries generalize conventional global symmetries and act on lower-dimensional submanifolds of a quantum system. While Abelian topological phases can be organized by 1-form symmetries that form a group, non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras. In this work, we make this statement precise in quantum double lattice models
D(G) for finite non-Abelian groups
G. We construct the electric, magnetic, and dyonic 1-form operators directly at the lattice fixed point and show that together they form a complete nonlocal diagnostic algebra for the topological Hilbert space. Using these operators, we explicitly determine the cylinder and torus ground-state subspaces for arbitrary finite
G. Furthermore, we calculate the microscopic fusion and gluing of the 1-form symmetries and show that their topological deformation properties emerge after projection to the defect-free topological subspace. Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking and provide an operator language that may be useful for characterizing such states in quantum processors.
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.
The self-duality defects under discrete gauging in a categorical symmetry
C can be classified by inequivalent ways of enriching the bulk SymTFT of
C with
Z2 0-form symmetry. The resulting Symmetry Enriched Topological (SET) orders will be referred to as
SymSETs and are parameterized by choices of
Z2 symmetries, as well as symmetry fractionalization classes and discrete torsions. In this work, we consider self-dualities under gauging
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
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
RepH8, the non-maximal gauging of the duality defect
N in
RepH8 and
RepD8 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
RepD8 related by changing the fractionalization class lead to
RepD16 and
RepSD16 respectively. Finally, we study the physical implications such as the spin selection rules and the SPT phases for the aforementioned categories.
The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works [1,2] on gapped phases in (2+1)d and provide a system¬atic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces be¬tween topological orders, so-called “club sandwiches”, which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for Z2, and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group G with twist – so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories. This is exemplified in numer¬ous Abelian and non-Abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from Abelian, S3, and D8 DW theories.