  Topological orders, as an emergent phenomenon, can be realized in strongly
correlated condensed matter systems or strongly-coupled quantum field theories.
In this condensed-matter oriented field-theoretical paper, by considering a
cyclic gauge group $G=\prod_{i=1}^n\mathbb{Z}_{N_i}$, we study topological
orders in five (spacetime) dimensions in which topological excitations include
not only point-like \textit{particles}, but also spatially extended objects:
closed-string-like loops and two-dimensional closed membranes. Their low-energy
effective theories are topological quantum field theories (TQFT) with various
exotic topological terms consisting of Abelian gauge fields of $1$-form $A$,
$2$-form $B$, and $3$-form $C$. In all these TQFTs, there are two types of $BF$
terms with twists, some of which are beyond Dijkgraaf-Witten gauge theories,
i.e., not classified by $H^{5}(G, \mathbb{R}/\mathbb{Z})$. Braiding statistics
among topological excitations is exhausted by following the gauge invariance
principle and combining topological terms that are mutually compatible. In this
way, we obtain the complete list of TQFTs within the present theoretical
framework. Since the boundary theory of a topological order exhibits
gravitational anomaly following Wen's definition, the boundary theory cannot be
realized alone unless a higher-dimensional bulk is added for canceling out the
anomaly. Therefore, our studies are also an attempt towards a more complete
list of gravitational anomalies and impossible quantum spin liquids in four
dimensions with gauge group $G$. The physical theory of this paper also
indicates that there are highly unexplored connections between mathematical
link theory of spatially extended objects and their realization in topological
orders.
