  Topological orders are a prominent paradigm for describing quantum many-body
systems without symmetry-breaking orders. We present a topological quantum
field theoretical (TQFT) study on topological orders in five-dimensional
spacetime ($5$D) in which \textit{topological excitations} include not only
point-like \textit{particles}, but also two types of spatially extended
objects: closed string-like \textit{loops} and two-dimensional closed
\textit{membranes}. Especially, membranes have been rarely explored in the
literature of topological orders. By introducing higher-form gauge fields, we
construct exotic TQFT actions that include mixture of two distinct types of
$BF$ topological terms and many twisted topological terms. The gauge
transformations are properly defined and utilized to compute level quantization
and classification of TQFTs. Among all TQFTs, some are not in Dijkgraaf-Witten
cohomological classification. To characterize topological orders, we concretely
construct all braiding processes among topological excitations, which leads to
very exotic links formed by closed spacetime trajectories of particles, loops,
and membranes. For each braiding process, we construct gauge-invariant Wilson
operators and calculate the associated braiding statistical phases. As a
result, we obtain expressions of link invariants all of which have manifest
geometric interpretation. Following Wen's definition, the boundary theory of a
topological order exhibits gravitational anomaly. We expect that the
characterization and classification of 5D topological orders in this paper
encode information of 4D gravitational anomaly. Further consideration, e.g.,
putting TQFTs on 5D manifolds with boundaries, is left to future work.
