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Gravitational anomalies can be realized on the boundary of topologicallyordered states in one higher dimension and are described by topological ordersin one higher dimension. In this paper, we try to develop a general theory forboth topological order and gravitational anomaly in any dimensions. (1) Weintroduce the notion of BF category to describe the braiding and fusionproperties of topological excitations that can be point-like, string-like, etc.A subset of BF categories -- closed BF categories -- classify topologicalorders in any dimensions, while generic BF categories classify (potentially)anomalous topological orders that can appear at a boundary of a gapped quantumliquid in one higher dimension. (2) We introduce topological path integralbased on tensor network to realize those topological orders. (3) Bosonictopological orders have an important topological invariant: the vector bundlesof the degenerate ground states over the moduli spaces of closed spaces withdifferent metrics. They may fully characterize topological orders. (4) Weconjecture that a topological order has a gappable boundary iff the abovementioned vector bundles are flat. (5) We find a holographic phenomenon thatevery topological order with a gappable boundary can be uniquely determined bythe knowledge of the boundary. As a consequence, BF categories in differentdimensions form a (monoid) cochain complex, that reveals the structure andrelation of topological orders and gravitational anomalies in differentdimensions. We also studied the simplest kind of bosonic topological ordersthat have no non-trivial topological excitations. We find that this kind oftopological 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).

Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions