We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category C as in the Levin-Wen model, whereas the boundary is associated with a module category over C. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

Topological order describes a new kind of order in gapped quantum liquid states of matter that correspond to patterns of long-range entanglement, while gravitational anomaly describes the obstruction that a seemingly consistent low energy effective theory cannot be realized by any well defined quantum model in the same dimension. Amazingly, topological order and gravitational anomaly have a very direct relation: gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -closed BF categories -classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. ( 4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. ( 5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a Z class in 2+1D (with gapless edge), a Z2 class in 4+1D (with gappable boundary), and a Z ⊕ Z class in 6+1D (with gapless boundary).

Bose condensation is central to our understanding of quantum phases of matter. Here we review Bose condensation in topologically ordered phases (also called topological symmetry breaking), where the condensing bosons have non-trivial mutual statistics with other quasiparticles in the system. We give a non-technical overview of the relationship between the phases before and after condensation, drawing parallels with more familiar symmetry-breaking transitions. We then review two important applications of this phenomenon. First, we describe the equivalence between such condensation transitions and pairs of phases with gappable boundaries, as well as examples where multiple types of gapped boundary between the same two phases exist. Second, we discuss how such transitions can lead to global symmetries which exchange or permute anyon types. Finally we discuss the nature of the critical point, which can be mapped to a conventional phase transition in some-but not all-cases.

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group G and its group 4-cocycle ω4 ∈ H 4 [G; U (1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories. CONTENTS

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V L and V R , V L ⊗ V R is naturally a full field algebra and we introduce a notion of full field algebra over V L ⊗ V R . We study the structure of full field algebras over V L ⊗ V R using modules and intertwining operators for V L and V R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductive conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over V ⊗ V and an invariant bilinear form on this algebra.

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of n+1D local topological orders by unitary multi-fusion n-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal n-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space. A higher symmetry acts on closed submanifolds (i.e. loops and membranes, etc). A collection of such higher symmetries form a higher group. In this paper, we introduce the notion of an algebraic higher symmetry, which is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in n-dimensional space is characterized and classified by a local fusion n-category. We also show that a bosonic system with an algebraic higher symmetry can be viewed as a boundary of a bosonic topological order in one-higher dimension. This implies that the system actually has a dual-equivalent symmetry, called the categorical symmetry, which is defined by the categorical description of the one-higher dimensional bulk. This provides a holographic and entanglement view of symmetries. A categorical symmetry is fully characterized by an anomaly-free bosonic topological order in one-higher dimension. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. The holographic point of view leads to (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the duality relations between systems with different (potentially anomalous) algebraic higher symmetries, or between systems with different sets of low energy excitations; (4) the classification of gapped liquid phases for bosonic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension that characterizes the categorical symmetry. This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry. CONTENTSI. Introduction 2 II. Summary of main results 3 A. Category of topological orders 3 B. Excitations in a topological order 4 C. Holographic principle for topological order 4 D. Algebraic higher symmetry 5 E. Dual symmetry 6 F. Categorical symmetry -a holographic of view of symmetry 7 G. Emergence of algebraic higher symmetry and categorical symmetry 8 H. Categorical symmetry and duality 9 I. Gauging the algebraic higher symmetry and the corresponding R-gauge theory 10 J. Anomalous algebraic higher symmetry 11 K. Classification of gapped liquid phases for systems with a categorical symmetry 12 L. Classification of SET orders and SPT orders with an algebraic higher symmetry 13 III. An example of algebraic higher symmetries: G-gauge theory 14 A. The string operators 14 B. The point operators 15 C. A commuting-projector Hamiltonian 15 D. The point-like and string-like excitations 16 E. Exact algebraic higher symmetry 17 F. Emergent algebraic higher symmetry IV. Description of algebraic higher symmetry in symmetric product states A. Spontaneous broken and unbroken algebraic higher symmetry B. Anomaly-free algebraic higher symmetry C. The charge objects and charge creation operators for the e...

This is part one of a two-part work that relates two different approaches to two-dimensional open-closed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the properties of these algebras and prove uniqueness and existence theorems. One implication is that under certain natural assumptions, every rational closed CFT is extendable to an open-closed CFT. The relation of Cardy algebras to the solutions of the sewing constraints is the topic of part two.