A gapped topological phase may have gapped or gapless boundaries. This paper introduces a systematic way to determine the allowed gapped and gapless boundaries from the data that characterizes the bulk topological phase. This is achieved by introducing a non-invertible gravitational anomaly.
The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.
In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.
We study boundary conditions of topological sigma models with the goal of generalizing the concepts of anomalous symmetry and symmetry protected topological order. We find a version of 't Hooft's anomaly matching conditions on the renormalization group flow of boundaries of invertible topological sigma models and discuss several examples of anomalous boundary theories. We also comment on bulk topological transitions in dynamical sigma models and argue that one can, with care, use topological data to draw sigma model phase diagrams.
Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in (1+1)D and (2+1)D that realize these new phases and explore their anomalous interfaces.
We review a many-body formulation of gravitational anomalies as obstructions to realizing the complete low-energy data by a local {\it bosonic} lattice quantum system of the same dimension. In this formulation, an anomalous theory can be realized at the boundary of a gapped {\it bosonic} lattice bulk, whose topological order gives a one-to-one classification of the anomaly. An invertible bulk reproduces conventional anomaly inflow; a bulk with non-trivial topological excitations yields a non-invertible generalization. To describe the holographic theory of anomalies, we explain the distinction between uniquely defined topological order and many versions of topological quantum field theory and discuss the boundary-to-bulk map. As an application of the holographic theory of anomalies, we describe morphisms between local quantum theories (such as quantum field theories). These morphisms are built from a bulk domain wall and an invertible boundary junction, which give rise to a sandwich picture. Chiral boson/fermion theory, Laughlin edges, and boundaries of Z2 higher gauge theory with bosonic or fermionic point-like excitations illustrate the different kinds of information about non-invertible anomalies contained in transport coefficients, superselection sectors, and condensation data. In particular, a fermionic theory can be viewed as a theory with a particular non-invertible gravitational anomaly. Those examples also reveal that {\it symmetry ∼ gravitational anomaly ∼ topological order in one higher dimension}.
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.