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As a series of work about 5D (spacetime) topological orders, here we employthe path-integral formalism of 5D topological quantum field theory (TQFT)established in Zhang and Ye, JHEP 04 (2022) 138 to explore non-Abelian fusionrules, 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 thatcomprise the twisted topological terms of the $BBA$ type. First, we classifytopological excitations by establishing equivalence classes among allgauge-invariant Wilson operators. Then, we compute fusion rules from thepath-integral and find that fusion rules may be non-Abelian; that is, thefusion outcome can be a direct sum of distinct excitations. We further computeshrinking rules. Especially, we discover exotic hierarchical structures hiddenin shrinking processes of 5D or higher: a membrane is shrunk into particles andloops, and the loops are subsequently shrunk into a direct sum of particles. Weobtain the algebraic structure of shrinking coefficients and fusioncoefficients. We compute the quantum dimensions of all excitations and findthat sphere-like membranes and torus-like membranes differ not only by theirshapes but also by their quantum dimensions. We further study the algebraicstructure that determines anomaly-free conditions on fusion coefficients andshrinking coefficients. Besides $BBA$, we explore general properties of alltwisted terms in $5$D. Together with braiding statistics reported before, thetheoretical progress here paves the way toward characterizing and classifyingtopological orders in higher dimensions where topological excitations consistof both particles and spatially extended objects.

Fusion rules and shrinking rules of topological orders in five dimensions