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...

Abstract. The colored HOMFLY polynomial is the quantum invariant of oriented links in S 3 associated with irreducible representations of the quantum group U q (sl N ). In this paper, using an approach to calculate quantum invariants of links via the cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. In this paper, we show that the $n+$1D "bulk" phase is given by the "center" of the $n$D boundary phase. In other words, the geometric notion of the "bulk" corresponds precisely to the algebraic notion of the "center". We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the "bulk" satisfies the same universal property as that of the "center" of an algebra in mathematics, i.e. "bulk = center". The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.Comment: 14 pages, 12 figures, This paper gives a concise explanation of one of the main results in arXiv:1502.01690. We have tried to make it easier for physicists to rea