Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
Liang Kong, Xiao-Gang Wen
TL;DR
This work develops a comprehensive framework—BF_n categories—to unify the braiding and fusion of topological excitations across all spatial dimensions and to classify gravitational anomalies. By treating topological orders and anomalies as higher-category data, the authors connect boundary phenomena to bulk theories through a boundary-bulk (center) correspondence, yielding a cochain complex across dimensions. They introduce closed and exact BF categories, relate them to gravitational inflow and gapped boundaries, and propose that tensor-network path integrals realize large classes of exact and closed BF categories in any dimension. A holographic, TN-based program is outlined for constructing and probing topological orders, including invertible and E8-like phases, and for diagnosing gravitational anomalies via partition-function invariants. The framework provides a principled, multi-dimensional language to classify, realize, and measure topological orders and their gravitational anomalies, with concrete low-dimensional examples and extensive discussion of defects, domain walls, and higher-categorical structures.
Abstract
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 both 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 $\mathbb{Z}$ class in 2+1D (with gapless edge), a $\mathbb{Z}_2$ class in 4+1D (with gappable boundary), and a $\mathbb{Z}\oplus \mathbb{Z}$ class in 6+1D (with gapless boundary).