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.
A bstract We use the intrinsic one-form and two-form global symmetries of (3+1) d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1) d . We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1) d for SU(2) gauge theory with two adjoint Weyl fermions.
We develop a constructive framework for characterizing modulated symmetries and their gauging in
d-dimensional lattice systems. The modulation profiles are organized as a finite-dimensional representation of the space group acting on a finite Abelian internal symmetry group. Since the corresponding automorphism group is finite, the nontrivial spatial action factors through a finite quotient of the space group, leading to periodic modulation patterns and providing a classification framework for discrete Abelian modulated symmetries. The symmetry-allowed local interactions are systematically obtained from a kernel construction, which also yields an algebra of open operators. Upon gauging, its elements are transmuted into extended Wilson and boundary operators while retaining their algebraic relations. In the deconfined regime, the resulting algebra realizes a mixed 't Hooft anomaly, providing a microscopic route to Abelian crystalline symmetry-enriched topological phases (SETs). These SETs do not mix electric and magnetic sectors, which transform in mutually contragredient representations, thereby preserving their braiding pairing. We illustrate the framework through explicit examples and show that several models present in the literature fit naturally within our construction.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fusion 2-category with respect to the corresponding module 2-category. We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.
We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group G.We find that, in general, a symmetry-enrichment pattern is specified by four pieces of data: ρ, a map from G to the duality symmetry group of this U(1) gauge theory which physically encodes how the symmetry permutes the fractional excitations, ν ∈ H 2 ρ [G, U T (1)], the symmetry actions on the electric charge, p ∈ H 1 [G, Z T ], indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor n over H 3 ρ [G, Z], the symmetry actions on the magnetic monopole.However, certain choices of (ρ, ν, p, n) are not physically realizable, i.e., they are anomalous.We find that there are two levels of anomalies.The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly.States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state.If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion.States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state.We apply these results to some interesting physical examples.