A bstract We investigate symmetry topological field theories (SymTFTs) of non-abelian and non-invertible symmetries and the different Lagrangian algebras associated with a given Drinfeld center. For several examples we analyze the condensable algebras of the Drinfeld center to identify the intrinsically gapless symmetry protected topological (igSPT) phases. In the previous work, the relation between igSPT phases and resolving anomalies by embedding an anomalous symmetry inside a larger fusion category was demonstrated. Here we present more examples of this mechanism that involve both group-like and categorical symmetries.
A bstract We investigate symmetry topological field theories (SymTFTs) of non-abelian and non-invertible symmetries and the different Lagrangian algebras associated with a given Drinfeld center. For several examples we analyze the condensable algebras of the Drinfeld center to identify the intrinsically gapless symmetry protected topological (igSPT) phases. In the previous work, the relation between igSPT phases and resolving anomalies by embedding an anomalous symmetry inside a larger fusion category was demonstrated. Here we present more examples of this mechanism that involve both group-like and categorical symmetries.
We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.
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 [L. Bhardwaj et al. SciPost Phys. 19, 056 (2025); L. Bhardwaj et al. arXiv:2502.20440] on gapped phases in (2+1)d and provide a systematic 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 between 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 \mathbb{Z}_2 ℤ 2 , 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 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 numerous 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, S_3 S 3 , and D_8 D 8 DW theories.
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.
A bosonic non-invertible Symmetry Protected Topological (NISPT) phase in (1+1)-dim is referred to as intrinsic if it cannot be mapped, under discrete gauging, to a gapped phase with any invertible symmetry, that is, if it is protected by a non-group-theoretical fusion category symmetry. We construct the intrinsic NISPT phases by performing discrete gauging in a partial SSB phase with a fusion category symmetry that has a certain mixed anomaly. Sometimes, the anomaly of that symmetry category can be alternatively understood as a self-anomaly of a proper categorical sub-symmetry; when this is the case, the same gauging provides an anomaly resolution of this anomalous categorical sub-symmetry. This allows us to construct intrinsic gapless SPT (igSPT) phases, where the anomalous faithfully acting symmetry is non-invertible; and we refer to such igSPT phases as igNISPT phases. We provide two concrete lattice models realizing an intrinsic NISPT phase and an igNISPT phase, respectively. We also generalize the construction of intrinsic NISPT phases to (3+1)-dim.
Anomalies of global symmetries provide important information on the quantum dynamics. We show the dynamical constraints can be organized into three classes: genuine anomalies, fractional topological responses, and integer responses that can be realized in symmetry-protected topological (SPT) phases. Coset symmetry can be present in many physical systems including quantum spin liquids, and the coset symmetry can be a non-invertible symmetry. We introduce twists in coset symmetries, which modify the fusion rules and the generalized Frobenius-Schur indicators. We call such coset symmetries twisted coset symmetries, and they are labeled by the quadruple (G,K,\omega_{D+1},\alpha_D) ( G , K , ω D + 1 , α D ) in D D spacetime dimensions where G G is a group and K\subset G K ⊂ G is a discrete subgroup, \omega_{D+1} ω D + 1 is a (D+1) ( D + 1 ) -cocycle for group G G , and \alpha_{D} α D is a D D -cochain for group K K . We present several examples with twisted coset symmetries using lattice models and field theory, including both gapped and gapless systems (such as gapless symmetry-protected topological phases). We investigate the anomalies of general twisted coset symmetry, which presents obstructions to realizing the coset symmetry in (gapped) symmetry-protected topological phases. We show that finite coset symmetry G/K G / K becomes anomalous when G G cannot be expressed as the bicrossed product G=H\Join K G = H ⋈ K , and such anomalous coset symmetry leads to symmetry-enforced gaplessness in generic spacetime dimensions. We illustrate examples of anomalous coset symmetries with A_5/\Z_2 symmetry, with realizations in lattice models.
Anomalies of (1+1)D categorical symmetries
We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category A describes a (2+1)D topological order whose gapped boundaries enumerate all (1+1)D gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the \emph{electric} Lagrangian algebra, that describes a phase with A fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in A. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where A is not spontaneously broken at all, which means that A is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to A being anomaly-free. As an application, we consider the Drinfeld centers of ZN×ZN Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.