  As a series of work about 5D (spacetime) topological orders, here we employ
the path-integral formalism of 5D topological quantum field theory (TQFT)
established in Zhang and Ye, JHEP 04 (2022) 138 to explore non-Abelian fusion
rules, hierarchical shrinking rules and quantum dimensions of particle-like,
loop-like and membrane-like topological excitations in 5D topological orders.
To illustrate, we focus on a prototypical example of twisted $BF$ theories that
comprise the twisted topological terms of the $BBA$ type. First, we classify
topological excitations by establishing equivalence classes among all
gauge-invariant Wilson operators. Then, we compute fusion rules from the
path-integral and find that fusion rules may be non-Abelian; that is, the
fusion outcome can be a direct sum of distinct excitations. We further compute
shrinking rules. Especially, we discover exotic hierarchical structures hidden
in shrinking processes of 5D or higher: a membrane is shrunk into particles and
loops, and the loops are subsequently shrunk into a direct sum of particles. We
obtain the algebraic structure of shrinking coefficients and fusion
coefficients. We compute the quantum dimensions of all excitations and find
that sphere-like membranes and torus-like membranes differ not only by their
shapes but also by their quantum dimensions. We further study the algebraic
structure that determines anomaly-free conditions on fusion coefficients and
shrinking coefficients. Besides $BBA$, we explore general properties of all
twisted terms in $5$D. Together with braiding statistics reported before, the
theoretical progress here paves the way toward characterizing and classifying
topological orders in higher dimensions where topological excitations consist
of both particles and spatially extended objects.
