Fusion rules and shrinking rules of topological orders in five dimensions

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.