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, JHEP04 (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 5D. 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.


I. INTRODUCTION
Originally discovered in the context of strongly correlated electron systems, such as the fractional quantum Hall effect and chiral spin liquids, topological order has been widely explored in the literature of quantum many-body physics, high energy physics, mathematical physics, and quantum information [1].While it is well-known that Ginzburg-Landau order parameters cannot characterize topological orders and their phase transitions, for the past decades, topological data, such as the number of topological excitations (i.e., anyons), ground state degeneracy, edge chiral central charge, braiding statistics, and fusion rules of anyons, have been extensively studied and utilized to characterize 3D (spacetime) topological orders.As a matter of fact, the practical goal of characterizing and classifying topological orders can be regarded as the establishment of a set of such topological data or physical observables for every type of topological order, assuming that the ground states are realized by Hamiltonians with short-range interactions and a bulk gap.
As one of the most prominent TQFTs, the Chern-Simons theory governs the low-energy physics of 3D topological orders and provides a quantitative tool for computing topological properties of anyons which are a key ingredient of topological quantum computation [13].Meanwhile, by placing the Chern-Simons theory on an open disc, the 2D boundary theory of 3D topological orders is shown to be dominated by gapless modes and mathematically described by conformal field theory (CFT) [14] with gravitational anomaly [15,16].Furthermore, the Gutzwiller projective construction of both bulk ground state wavefunctions and the associated edge CFT algebraic construction can be performed in a systematic manner [17][18][19].It should be noticed that the Chern-Simons the-in 4D and higher.In addition, twisted BF theories of 4D topological orders with various types of symmetry can be applied to 4D SET phases, leading to a field-theoretical classification and characterization of a large class of 4D SET phases, see, e.g., refs.[35,[52][53][54] and ref. [55] on the general theory.Again, twisted BF theories at the trivial level have been applied to describe the bulk effective field theory of 4D SPT phases [39,56], which, by weakly gauging, leads to the topological response theory of 4D SPT phases [57][58][59].In addition, BB and dAdA type terms were also discussed respectively for fermionic topological phases and topological response theory of bosonic topological insulators [34,38,56,[60][61][62].
By lifting spatial dimensions further, topological orders in 5D admit particle-like, loop-like, and membrane-like topological excitations, which exhibit highly unexplored features of both the formalism of TQFTs and braiding statistics [63].If we still consider the gauge group G = n i=1 Z N i , TQFTs in 5D may contain two types of (generalized) BF terms, i.e., CdA and BdB, where C is 3form, B and B are two different 2-form, A is 1-form.Therefore, for each Z N i subgroup, there are two choices for the assignment of gauge charges and corresponding BF terms, which significantly enriches topological orders.On top of two types of BF terms, there are many different twisted topological terms, e.g., AAAAA, AAAdA, AdAdA, AAC, AAAB, BBA, AdAB, AAdB, which lead to many different twisted BF theories after proper combinations in 5D, as long as gauge invariance is guaranteed [50].In ref. [63], braiding statistics encoded in all topological excitations has been investigated via computing correlation functions of Wilson operators whose expectation values rely on counting intersections of sub-manifolds in 5D.The physical theory presented in ref. [63] provides an alternative route to understanding the link theory of higher-dimensional compact manifolds.
While braiding statistics has shown how diverse the topological data of 5D topological orders are [63], the fusion rules and shrinking rules of 5D topological orders are still yet to be uncovered.
Since there are membrane topological excitations that are spatially extended excitations of two dimensions, it is natural to expect more exotic fusion and shrinking rules than that of 4D studied in ref. [51], which motivates the present paper.In this paper, we adopt the path-integral formalism of 5D topological orders constructed in ref. [63].We focus on two prototypical twisted topological terms, namely, BBA and AAAB both of which exhibit not only non-Abelian fusion rules but also hierarchical structure in shrinking rules.By hierarchical, we mean that a membrane is shrunk into particles and loops, and the loops are subsequently shrunk into a direct sum of particles.Such phenomena can only exist in 5D and higher.We realize that membrane excitations potentially possess very different topological properties, e.g., quantum dimensions if the membranes have different topologies, e.g., sphere and torus.We concretely compute all fusion rules and verify the consistency between fusion rules and shrinking rules, which leads to the algebraic structure hidden in the fusion coefficients and shrinking coefficients.Any violation of the algebraic structure may be associated with a quantum anomaly of some kind in the topological order.This paper is organized as follows.In section II, we first review the twisted BF theory with the BBA twisted term introduced in ref. [63].Then we construct Wilson operators for all topological excitations (see table I) and classify them into different equivalence classes.In section III, we calculate the fusion rules (see table II) for the BBA twisted term explicitly from the path integral.We also compute quantum dimensions (see table III) for all topological excitations.In section IV, we calculate the shrinking rules (see table IV) for the BBA twisted term.We concretely discuss the hierarchical structure in the shrinking rules.In section V, we study the general algebraic structure that determines anomaly-free topological orders.In section VI, we present several general properties (see table V) of all twisted terms in 5D.A brief summary is given in section VII.Several appendices are provided.

II. TOPOLOGICAL EXCITATIONS AND WILSON OPERATORS FOR THE BBA TWISTED TERM
A. Twisted BF theory with the BBA twisted term Any gauge group G = n i=1 Z N i with n ⩾ 3 can support BBA term in TQFT action.For simplicity, we consider G = (Z 2 ) 3 and take the TQFT action to be [63] where q is a periodic and quantized coefficient.A 3 is a 1-form gauge field.B 1 , B 2 , B1 , and B2 are 2-form gauge fields.C 3 is a 3-form gauge field.We note that there are two kinds of BF terms in the action (1), i.e., BdB and CdA, so the action represents a mixed BF theory following the definition in ref. [63].The coefficient q is quantized and periodically identified: , where N 123 is the greatest common divisor of N 1 , N 2 and N 3 , and conditions: dB 1 = 0, dB 2 = 0 and dA 3 = 0.The gauge transformations are given by: where χ 3 , V i , Ṽ i and T 3 are 0-form, 1-form, 1-form and 2-form gauge parameters respectively.
They satisfy the following compactness conditions: and ´dT 3 ∈ 2πZ, where the integrals are performed over compact submanifolds.

B. Wilson operators for topological excitations
Topological excitations in 5D spacetime consist of point-like particle excitations, one-dimensional closed loop excitations, and two-dimensional closed membrane excitations.We will simply call them particles, loops, and membranes unless otherwise specified.Since loops and membranes are spatially extended excitations, they can have different shapes in space such as self-linking loops, knotted loops, or membranes in the shape of a torus with multiple handles.We consider some simplest shapes, which are S 1 loop, S 2 sphere and T 2 torus (with genus= 1), as shown in figure 1.
In TQFTs, topological excitations are represented by Wilson operators.In other words, by respectively using gauge fields of Wilson operators for particles.A particle carrying unit gauge charge of 1-form gauge field A 3 is called a Z N 3 particle, which can be represented by the following gauge invariant Wilson operator: Here we use Z N 3 gauge charge.γ is the closed world-line of the particle.Since the action (1) contains only one 1-form gauge field, i.e., A 3 , there is only one kind of particle with unit gauge charge, which is For instance, the antiparticle of P 001 is represented by P001 .In section III, we will show that for G = (Z 2 ) 3 , the antiparticle of any excitation is itself because the fusion of two identical excitations can be written as c where c is just a constant.Also, n i only takes 0 or 1 mod 2 due to the Z 2 cyclic group structure, which largely simplifies our analysis.
Wilson operators for loops.Similarly, a Z N 1 loop carries unit gauge charge of the 2-form gauge field B 1 , and can be represented by where σ is the closed world sheet of the loop excitation.Such a loop is a pure loop that only carries gauge charges of either one or more 2-form gauge fields.In contrast, we can further consider a decorated loop that simultaneously carries gauge charges of both 1-form gauge fields and 2-form gauge fields.For instance, a (Z N 1 , Z N 2 ) loop decorated by a Z N 3 particle is represented by Here, the superscript "001" denotes a Z N 3 particle decoration with γ ∈ σ, and the subscript "110" means that the loop simultaneously carries unit gauge charges of B 1 gauge field and B 2 gauge excitation operator for excitation equivalent excitations M S 001, 001 = M S 001,ij0 001 i, j = 0, 1 i, j = 0, 1 i, j = 0, 1 field.If there is no decoration at all, L 000 110 is a pure loop and can be rewritten as L 110 for the notational simplicity.
Wilson operators for membranes.As mentioned above, we will consider membrane excitations of two different shapes: the membrane denoted by M S is in the shape of S 2 and the membrane denoted by M T is in the shape of T 2 .For instance, one may write a Wilson (volume) operator as M S 001 = exp i ´ω C 3 , where the world volume ω is an S 2 × S 1 manifold in 5D spacetime.However, the Wilson operator we construct above is not gauge invariant because C 3 is transformed with nontrivial shifts as shown in eq. ( 4).The correct form should be: where N is a normalization factor and in section III we will show that N = 2.We define d −1 B 1 and A is an open area on ω) whose legitimacy relies on extra constraints: ´σ B 1 = 0 mod 2π, ´σ B 1 = 0 mod 2π, where σ is an arbitrary closed surface on ω.To enforce these constraints, we introduce two Dirac's delta-functional in eq. ( 8): Similarly, the operator for a Z N 3 membrane of T 2 shape can be written as: where ω is a T 2 × S 1 manifold in 5D spacetime, and in section IV we will show that Ñ = 4.
Symbolically, once we write down the Wilson operator for an S 2 membrane, we can formally express the Wilson operator for a T 2 membrane simply through replacing M S , N and ω by M T , Ñ and ω.
Just like loop excitations that can be decorated by particles, membranes can be decorated by particles and loops simultaneously.For instance, a Z N 3 membrane decorated by a Z N 3 particle and a Z N 1 loop is The number 001 and 100 in the superscript respectively denote a Z N 3 particle decoration and a Z N 1 loop decoration respectively.The subscript 001 means that the membrane carries a unit gauge charge of 3-form gauge field C 3 .δ ´σ B 1 in eq. ( 11) plays an important role because it induces an equivalence relation between M S 001,100 001 and M S 001,000 001 , which will be discussed in section II C. If there is no particle decoration at all, M S 000,100 001 can be rewritten as M S ,100 001 for the notational simplicity.If there is no loop decoration at all, M S 001,000 001 can be rewritten as M S 001, 001 for the notational simplicity.If all decorations are absent, the pure membrane M S 000,000 001 is directly replaced by M S 001 .Wilson operators for M ST -excitations.We have already introduced two kinds of membranes: S 2 membranes and T 2 membranes.For the completeness of fusion rules to be discussed shortly, it is vital to incorporate another type of membrane formed by fusing an S 2 membrane and a T 2 membrane.The resulting object is no longer a manifold.The corresponding Wilson operator is denoted as M ST :

C. Equivalence classes of Wilson operators
In the above discussion, we mentioned that a Z N 3 membrane that is decorated by a Z N 3 particle and a Z N 1 loop can be represented by eq. ( 11), and it is equivalent to a Z N 3 membrane that is only decorated by a Z N 3 particle, because of the constraint δ ´σ B 1 : For simplicity, we omit 000 and use M S 001, 001 to represent M S 001,000

001
. We say M S 001,100 001 and M S 001, 001 are equivalent to each other because they always give the same answers when we calculate any gaugeinvariant correlation functions in the path integral.More concretely, the correlation function of M S 001,100 001 and any operator O can be written as where where a, b and c denote topological excitations.N ab c ∈ Z is fusion coefficient.In the condensed matter literature, a fusion process is said to be Abelian if there is only one c such that N ab c is nonzero, e.g., a ⊗ b = c.Otherwise, the fusion process is said to be non-Abelian, e.g., a ⊗ b = c 1 ⊕ c 2 .We follow this nomenclature in this paper.For an excitation a, if a ⊗ b is always Abelian for any b, we call a an Abelian excitation.Otherwise, we call a a non-Abelian excitation.In this paper, we use Wilson operators to represent topological excitations, the fusion process (15) can be interpreted as the equivalence of expectation values: In the path-integral representation, the above identity is represented by: If the decomposition ( 17) is successfully performed, all fusion coefficients can be obtained.From the above formula, we can also verify that:

B. Examples
In the following, through several examples, we will concretely compute the fusion rules of the twisted BF theory with the BBA twisted term whose action is given in eq. ( 1).All fusion rules for the BBA twisted term are summarized in table II.
Fusing a Z N 3 particle and a Z N 3 particle.Using eq. ( 17), we can calculate that Integrating out Lagrange's multipliers, we obtain constraints for A 3 , B 1 , and B 2 respectively: we have exp i2 ´γ A 3 = 1, thus ⟨P 001 ⊗P 001 ⟩ = 1 = ⟨1⟩, i.e., P 001 ⊗P 001 = 1 .This result indicates that fusing two P 001 's leads to the trivial excitation.It also means the antiparticle of P 001 is itself, which is consistent with the group structure of Z 2 cyclic group.
Fusing a Z N 3 particle and a Z N 1 loop.In the path-integral representation, fusion of a Z N 3 particle and a Z N 1 loop is represented by: This result tells us that a Z N 3 decorated Z N 1 loop can be constructed by fusing a Z N 3 particle and a Z N 1 loop.
Fusing a Z N 1 loop and a Z N 3 membrane.Fusion of a Z N 1 loop and a Z N 3 membrane is where ).This is reasonable because M S 001 is equivalent to M S ,100 001 due to the delta functional δ ´σ B 1 .Thus, L 100 behaves trivially on M S 001 .
Fusing a Z N 3 membrane and a Z N 3 membrane.Fusion of a Z N 3 membrane and a Z N 3 membrane is Integrating out Lagrange's multipliers, i.e., B1 , B2 , and C 3 , we obtain ´σ where we add a hat for each gauge field in order to specify the configurations after the Lagrange's multipliers are integrated.As a result, where two redundant delta functionals have been dropped as one delta functional is sufficient for each constraint.The summation B1 , B2 , Â3 denotes the functional integration over the configurations after integrating Lagrange's multipliers.The effective partition function Z eff = B1 , B2 , Â3 exp(iS eff ) with the effective action: In deriving eq. ( 23), the exponential term "exp i2 × ´C3 " has been integrated.Since ´σ B1 = with k 1 being an integer and there exists another integer Likewise, for ´ω d −1 B2 B1 , we have similar discussion and we conclude that: can also be dropped.By further noting that the delta functionals in eq. ( 23) can be expanded via Fourier-like transformations (i = 1, 2): we finally obtain the following relation between fusion input and outcome: It is apparent that this is a non-Abelian fusion with multiple fusion outcomes.Fusing two identical Z N 3 membranes renders one vacuum (i.e., trivial excitation), one Z N 1 loop, one Z N 2 loop, and one Fusing an M ST -excitation and a Z N 3 membrane.Fusion of an M ST and a Z N 3 membrane is Similar to the derivation of eq. ( 26), we can obtain the fusion rule below: As we mentioned above, M ST is constructed by fusing an M S 001 and an M T 001 .So we can also calculate M ST ⊗ M S 001 in an alternative way: where we have applied the fusion rule in eq. ( 26) and the properties of fusion rules listed in eq. ( 18).

C. Fusion table and quantum dimensions
For the BBA twisted term, we can exhaust all combinations of two topological excitations and calculate their fusion rules.We list all possible fusion processes and the corresponding fusion rules in table II.There are 8 Abelian excitations and 6 non-Abelian excitations.We can see that this fusion table is symmetric, which is consistent with the condition a ⊗ b = b ⊗ a.
Thus the quantum dimension of M S 001, 001 is 2. We can obtain all matrices N a from table II and then calculate the quantum dimensions of all excitations, as shown in table III.We conclude that for a non-Abelian excitation, the quantum dimension is larger than 1.For an Abelian excitation, the quantum dimension is always 1.
FIG. 3. Typical shrinking rules for loops and S 2 membranes in the 5D twisted BF theory with the BBA twisted term.Shrinking an L 100 and shrinking an L 001 100 respectively lead to a vacuum and a P 001 .Shrinking an M S 001 and shrinking an M S 001, 001 respectively lead to a direct sum of 2 vacua and a direct sum of 2 P 001 's.

IV. SHRINKING RULES FOR THE BBA TWISTED TERM
A. General discussion on shrinking in 5D and its hierarchical structure In 3D topological orders, shrinking rules are absent because all topological excitations are point-like particles, i.e., anyons.However, in 4D and higher, spatially extended excitations exist, such that the inclusion of shrinking processes makes sense.While loop excitations in 4D brings us exotic braiding statistics, as reviewed in section I, one may wonder what are the resulting outcomes by adiabatically shrinking loop excitations.Indeed, a recent study in 4D indicates that the topological data given by shrinking rules significantly enrich the physics of higher-dimensional topological orders [51].More concretely, the world-sheet of a loop excitation is an spacetime manifold.If we shrink the loop to a point, then the world-sheet is simultaneously shrunk to a world-line, i.e., T 2 → S 1 , which can be either the world-line of a particle or many distinct particles.The former case is an Abelian shrinking process while the latter is a non-Abelian shrinking process.
In 5D topological orders, spatially extended excitations include both loops and membranes, and the latter can have different shapes, e.g., S 2 and T 2 .Therefore, we expect the underlying shrinking rules will be significantly diversified, as shown in figures 3, 4, and 5. Intuitively, an S 2 membrane can be directly shrunk to particles while a T 2 membrane can be shrunk to loops and particles first, and then we can further shrink these loops to particles.As membranes exist in topological orders of 5D and higher, this hierarchical structure of shrinking rules can only appear in 5D and higher.
A typical example of hierarchical shrinking is shown in figure 4.
It seems that two successive steps are required to shrink T 2 membranes completely into particles.But it is not always the case.As shown in figure 5, some T 2 membranes in the twisted BF theory with the AAAdA twisted term, upon being shrunk in the first step, only lead to trivial loops and some particles.As trivial loops and trivial particles are equivalent to each other and all of them are identical to a vacuum, these T 2 membranes are completely shrunk to particles including vacua in the first step.If we need two steps to shrink a T 2 membrane to particles, we say this T 2 membrane has hierarchical shrinking rules.
To quantitatively calculate shrinking rules, we define a shrinking operator S and the associated shrinking coefficient S a b as below: where the shrinking coefficient S a b ∈ Z. X 1 and X 2 are respectively spacetime trajectories of excitation a before and after shrinking, which respects X 2 ⊂ X 1 .The summation ⊕ b exhausts all 14 topologically distinct excitations.We leave details of X 1 and X 2 to section IV B and IV C where concrete calculations are given.
In the path-integral representation, we may compute the following expectation value in the ground state: If the integer S a b = 1 for some topological excitation b and vanishes for any other excitations, the shrinking is said to be Abelian.If S a b is nonzero for more than one topological excitations or if there exists b such that S a b > 1, the shrinking is said to be non-Abelian.We collect all shrinking rules in the shrinking table IV.

B. Shrinking rules without hierarchical structure
Next, we present several typical examples in table IV.In this subsection, we study shrinking rules without hierarchical structure.X 1 and X 2 meet the following conditions: (i) if a is a particle (1 and P 001 ), then the world-line of the particle is unchanged, i.e., X 1 = X 2 = γ, which means that particles are unshrinkable; (ii) if a is a loop, then the world-sheet of the loop is shrunk into a world-line, i.e., X 1 = σ, X 2 = γ; (iii) if a is an S 2 membrane (M S 001 and M S 001, 001 ), then the worldvolume of the membrane is shrunk into a world-line and the manifold σ in the delta functionals is also shrunk into a world-line, i.e., X 1 = (ω, σ), X 2 = (γ, γ).
Shrinking a Z N 1 loop.Shrinking a Z N 1 loop results a collapse of its world sheet to a world-line, which is σ → γ.From the path integral, we obtain So we conclude that S (L 100 ) = 1, which means that a Z N 1 loop denoted as L 100 is shrunk into a vacuum.
Shrinking a Z N 1 loop decorated by Z N 3 particle.Similarly, the shrinking rule for an L 001 100 loop can be obtained Therefore, particle decoration is unaffected by the shrinking process.Also notice that S L 001 100 = S (L 100 ⊗ P 001 ) = S (L 100 ) ⊗ S (P 001 ) = P 001 (35) implying that shrinking rules respect the fusion rules.
Shrinking a Z N 3 membrane in the shape of S 2 .From path integral we obtain In other words, the shrinking process renders a superposition of 2 vacua when shrinking a Z N 3 membrane in the shape of S 2 .This is reasonable because this membrane is a non-Abelian excitation and has a quantum dimension of 2. If we consider 2 • 1 as an excitation and derive its fusion coefficient matrix N 2•1 , we will find that N 2•1 is 2 times identity matrix since fusion coefficient matrix of 1 is an identity matrix.Thus the greatest eigenvalue of N 2•1 is 2 and we conclude that 2 • 1 has quantum dimension 2. It may be seen as preserving the quantum dimension.As a matter of fact, by exhausting all shrinking processes, we can verify that all shrinking processes preserve quantum dimension.
C. Hierarchical shrinking process (i): shrinking a T 2 membrane Spatially, when shrinking a T 2 torus, we will first get an S 1 closed line and then get a point.
Thus for a T 2 membrane, we can first shrink it into loops and particles and then shrink them to particles.If a T 2 membrane gives nontrivial loops after the first shrinking process, we will further shrink these loops into particles.This is aforementioned hierarchical shrinking.We define that S will shrink ω = T 2 × S 1 to S 1 × S 1 first and then S shrinks σ = S 1 × S 1 to S 1 .Also notice that Wilson operator carries δ ´σ B i , where σ is a submanifold of the world volume ω.Since when we shrink the world volume ω to world sheet S 1 × S 1 , σ is still a submanifold of this world sheet, we do not need to shrink σ in δ ´σ B i .Thus we have By further expanding delta functionals, we have If we regard 1 as a trivial loop, shrinking M T 001 once gives us a superposition of a vacuum and all three pure loops in table I.We also see that the normalization factor 4 ensures that the shrinking coefficients are integers.
Since the above shrinking process leads to nontrivial loops, we can continue to shrink the loops to particles (the symbol S 2 = SS): In summary, after two steps of shrinking operation, we finally shrink a Z N 3 membrane in the shape of T 2 to 4 vacua, which is different from shrinking an S 2 membrane.

D. Hierarchical shrinking process (ii): shrinking an M ST -excitation
For an M ST -excitation, we define that shrinking operator S shrinks ω = S 2 × S 1 and ω = T 2 × S 1 to S 1 and S 1 × S 1 respectively first.Then we can continue to shrink all S 1 × S 1 world sheets to S 1 .Thus we obtain If we regard 1 as a trivial loop, then shrinking M ST once will give a 2 times superposition of vacua and all pure loops.Next, we continue to shrink all loops to particles: Finally, we get 8 vacua.
If we consider shrinking an M S 001 and an M T 001 before fusing them, we have Similarly, we have This shows that our definition for shrinking an M ST -excitation respect the fusion rules.extended topological excitations eventually shrunk to a vacuum or several vacua.We can also verify that quantum dimensions are preserved in the shrinking process.

V. ANOMALY-FREE ALGEBRAIC STRUCTURE IN FUSION RULES AND SHRINKING RULES
It is fundamentally important to study the algebraic structure encoded in topological data that characterize topological orders.Especially, all data, e.g., fusion coefficients and shrinking coefficients, should be consistent to each other such that they can together define an anomaly-free topological order.Otherwise, we expect that there should be a quantum anomaly of some kind that obstructs the existence of such topological order.In this section, we study the consistency relations between fusion and shrinking coefficients with the help of the results of fusion and shrinking rules obtained through the path-integral approach in the previous sections.

Using table II and table IV, we can verify that shrinking rules and hierarchical shrinking rules
all respect the fusion rules: Eq. ( 44) also implies that there are some algebraic relationships between the fusion coefficient and shrinking coefficient.Using eq. ( 15) and eq.( 32), we first calculate S (a ⊗ b): Keep shrinking, we will get S 2 (a ⊗ b): Now we calculate S (a) ⊗ S (b): Similarly, we obtain S 2 (a) ⊗ S 2 (a): Comparing eq. ( 45) and eq.( 47), eq. ( 46) and eq.( 48), we obtain Eq. ( 49) describes the first shrinking process and c can be either loops or particles.Eq. ( 50) describes the second shrinking process and c must be particles.
Notice that eq. ( 49) and eq.( 50) are not independent, eq. ( 50) can also be derived from eq. ( 49) directly by acting a shrinking operator on both sides of eq. ( 49).Only eq. ( 49) is the fundamental relationship between fusion and shrinking coefficients and eq.( 50) can be seen as the descendant of eq. ( 49).In a topological order higher than 5D, we may derive other descendants describing hierarchical shrinking processes from eq. ( 49) by similarly acting more shrinking operators.

VI. GENERAL DISCUSSIONS ON ALL TWISTED TERMS IN 5D
In the above analysis, we have only considered the twisted BF theory with the BBA twisted term, but there are many other legitimate twisted terms in 5D topological orders, e.g., AAAB, AAAAA, AAAdA, AAC, AdAdA, AdAB, and AAdB.If gauge-invariant combinations among them are taken into account, there are even more twisted BF theories as shown in ref. [63].In principle, we can construct equivalence classes of Wilson operators, and compute fusion and shrinking rules explicitly.For example, we find AAAB also simultaneously supports non-Abelian fusion rules, non-Abelian shrinking rules and hierarchical shrinking rules (see appendix B).As shown in appendix C, we find that a simple method allows us to determine the general properties of fusion and shrinking rules from gauge transformations and some simple Wilson operators.We summarize these properties in table V and discuss this table below.
• Both BBA and AAAB twisted terms simultaneously support non-Abelian fusion rules, non-Abelian shrinking rules, and hierarchical shrinking rules.According to the discussion in appendix C, delta functionals carried by Wilson operators determine the type of fusion and shrinking rules.We find that in BBA and AAAB, Wilson operators for membranes carry δ ´σ B i , which will lead to hierarchical shrinking rules.The presence of hierarchical shrinking rules indicates that some fusion and shrinking rules are non-Abelian.
• AAAAA, AAAdA, and AAC support non-Abelian fusion rules and non-Abelian shrinking rules but they do not support hierarchical shrinking rules.As studied in ref. [63], the twisted BF theories with these twisted terms contain only type-I BF term, i.e., CdA.Since there are no B i fields, none of membranes carry δ ´σ B i and thus they do not have hierarchical shrinking.A pure T 2 membrane can only be shrunk to a trivial loop (vacuum).Although δ ´σ B i is absent, some Wilson operators for membranes carry δ ´γ A i , which can lead to non-Abelian fusion and shrinking rules.
• AdAdA, AdAB, and AAdB twisted terms only support Abelian fusion and Abelian shrinking rules.Some Wilson operators may contain nontrivial terms besides A i or B i or C i to compensate for the nontrivial shift in gauge transformation, but they do not carry delta functionals that lead to non-Abelian fusion rules and non-Abelian shrinking rules.
The presence of hierarchical shrinking leads to an interesting phenomenon: S 2 and T 2 membranes have different quantum dimensions.In a theory without hierarchical shrinking, they only differ by shape.Remember we request the fusion coefficients and the shrinking coefficients are all integers, this affects the normalization factors of Wilson operators.In the above analysis, we have seen that an S 2 membrane and a T 2 membrane carrying the same gauge charges have different normalization factors as long as the T 2 membrane can shrink to nontrivial loops.Different normalization factors will lead to different fusion coefficients and thus lead to different fusion coefficient matrices.Since quantum dimension is defined as the greatest eigenvalue of the fusion coefficient matrix, S 2 and T 2 membranes have different quantum dimensions.This is absent in the theory without hierarchical shrinking, since the S 2 and T 2 membranes carrying the same gauge charges have identical normalization factors.To be more specific, each gauge charge of C i field contributes an extra 2 to the normalization factor of a T 2 membrane.This can be clearly viewed from AAAB twisted term shown in appendix B. From eq. (B8), we can see when the membrane carries n units of gauge charges of C i field, then we have Ñabc = 2 n N abc .This result also applies to the BBA twisted term since there are only Z N 3 membranes and we do find that M T 001 has normalization factor 4 while M S 001 has 2.

VII. SUMMARY AND OUTLOOK
In this paper, as a series of work on 5D topological orders, we concentrated on the fusion and shrinking rules of 5D topological orders by means of the path-integral formalism of TQFTs proposed in ref. [63].More concretely, we constructed Wilson operators for all topologically distinct excitations by carefully investigating equivalence classes among all possible gauge-invariant Wilson operators.Based on Wilson operators, we may derive fusion rules, quantum dimensions, and shrinking rules.We found exotic non-Abelian fusion and non-Abelian shrinking, and especially identified exotic hierarchical structure in shrinking processes.The underlying algebraic structure of the fusion coefficients and shrinking coefficients was also found in terms of several exact algebraic equations about the coefficients, which sets an anomaly-free condition on 5D topological orders.Below we present several future directions motivated by this paper.(i) By adopting the strategy in ref. [55], we may explore symmetry-enrichment in 5D topological orders.While the idea of "mixed 3-loop statistics" plays an important role in charactering and classifying SET phases in ref. [55], it will be interesting to generalize such mixed type of braiding statistics to probe SET order in higher dimensions, especially when higher-form symmetry [64] is considered and higher-form response gauge fields are involved.(ii) We have only considered topological excitations in a simple shape such as particles, simple unknotted loops, S 2 membranes, T 2 membranes, and composite of the two membranes.It is important to explore more complicated spatially extended excitations, e.g., T 2 membranes with higher genus in order to see if there are more exotic topological data that should be involved for completeness.(iii) Finally, it will be interesting to explore fermionic topological orders by putting TQFTs on a spacetime manifold with spin structure.
We obtain that B after dropping some boundary terms.When we consider fusing two M S 100 , we will encounter For other excitations, we have similar calculations.
In short, when calculating fusion rules, we have ⟨exp i2 , where X is the world-line(sheet or volume) and f i is the gauge field.

TQFT action and gauge transformations for AAAB twisted term
For the AAAB twisted term, the index of all A and B fields must be different.Thus the simplest case is take the gauge group G = (Z 2 ) 4 .Then we can write down the TQFT action: where The gauge transformations are: where ϵ is the antisymmetric tensor and i, j, k = 1, 2, 3.

Nonequivalent Wilson operators for excitations
Now we can construct Wilson operators for different excitations.We will illustrate it in 3 different cases: Abelian excitations(particles and loops), membranes, and M ST -excitations.
particles and loops.All particles and loops are Abelian excitations because we can see that in eq.(B2), A i and B 4 transform normally (without nontrivial shifts).Thus we can easily write down their Wilson operators in a general way: where a, b, c, d = 0, 1 and they take value independently.Notice that when d = 0, L abc0 000d = P abc0 , when a = b = c = d = 0, L abc0 000d = 1.For simplicity, we use P abc to denote P abc0 in the following discussion.From eq. (B5), we can find that there are 2 4 = 16 nonequivalent excitations. .Any addition that appears in subscript and superscript is addition mod 2, which means 1+1 = 0 and 0+1 = 1.Notice that Possible values of (a ′′ b ′′ c ′′ ) still follows eq.(B9), so two of a ′′ , b ′′ and c ′′ will be forced to zero once (abc) is given, the last remaining one is determined by When (abc) is not equal to ãb c , then we obtain superscript is omitted since it is (000, ).This can be seen from the delta functionals that ensure all decorations are trivial.
We conclude that M abc,ã bc is equivalent to M ST abc,ã bc when (abc) ̸ = ãb c .Now we obtain all equivalence classes.There are 16 Abelian excitations and 14 + 14 + 56 = 84 non-Abelian excitations.

Fusion rules for the AAAB twisted term
The discussion above shows that any possible excitation can be represented by one of the following expressions: Thus if we want to derive all fusion rules for all nonequivalent excitations, we can try to obtain the fusion rules for these 5 expressions.We can formally express the fusion table as table VI.Once we can explicitly write down the formulas of every element in table VI, we obtain the complete fusion rules.
There are 15 elements in table VI, we will show how to derive the fusion rules between them in 3 different cases: fusion concerning Abelian excitations(particles and loops), fusion concerning two membranes and fusion concerning M ST -excitations.As for fusing an S 2 membrane and a T 2 membrane, we have: Now we obtain three more elements in table VI.
Fusion rules for M ST -excitations.At the beginning of section III, we illustrate that fusion rules in our theory should satisfy commutative and associative law, i.e., fusion rules satisfy eq. ( 18).We can use this powerful condition to calculate fusion rules for M ST -excitations.
For example M ST a ′′ b ′′ c ′′ , abc,abc and M ST abc,ã bc are equivalent to: Thus we have Now for any two given excitations, we can obtain their fusion rules.

Shrinking rules for the AAAB twisted term
We have already defined shrinking operator S for different excitations in section IV, we continue to use this definition.Through our calculation, we find that shrinking results of all S 2 membranes contain 4 different kinds of particles.This is different from the BBA twisted term where all pure membranes are shrunk to vacua.Shrinking an M ST -excitation.When (abc) = ãb c , shrinking rules for M ST -excitations can Basic nontrivial Wilson operators are Other Wilson operators that have nontrivial terms and delta functionals can always be constructed from the above three basic nontrivial Wilson operators.We can see that there exist δ ´γ A 1 , δ ´γ A 2 and δ ´ω C 3 , but δ ´σ B i is absents.Thus we can conclude that AAC twisted terms have non-Abelian fusion rules and non-Abelian shrinking rules, but do not have hierarchical shrinking rules.
Similarly, for AAAAA twisted term, TQFT action is Gauge transformations are For AAAdA twisted term, TQFT action is Gauge transformations are Basic nontrivial Wilson operators are Thus AAAdA twisted terms have non-Abelian fusion rules and non-Abelian shrinking rules, but do not have hierarchical shrinking rules.
For AdAdA twisted term, TQFT action is Gauge transformations are All fields transform without nontrivial shifts and thus all Wilson operators do not have nontrivial terms and delta functionals.Thus AdAdA twisted term only has Abelian fusion rules and Abelian shrinking rules.
For AdAB twisted term, TQFT action is Basic nontrivial Wilson operators are Although some Wilson operators have nontrivial terms, delta functionals are still absent.These nontrivial terms cannot change fusion and shrinking rules.Thus AdAB twisted term only has Abelian fusion rules and Abelian shrinking rules.
For AAdB twisted term, notice that is the partition function and D[ABC] denotes configurations of all gauge fields A 3 , B 1 , B1 , B 2 , B2 , C3 .The constraint δ ´σ B 1 means that in the functional integration, the integration over B 1 is constrained in the subspace in which B 1 is flat such that exp i ´σ B 1 = 1.Immediately, one can realize that ⟨OM S 001,100 001 ⟩ = ⟨OM S 001, 001 ⟩.Since O is an arbitrary operator, M S 001,100 001 and M S 001, 001 are indistinguishable in all gauge-invariant observables.In this sense, we conclude that these two operators as well as the associated two topological excitations are said to belong to the same equivalence class.Similarly, noticing that δ ´σ B 2 enforces exp i ´σ B 2 = 1 in the functional integration, we can obtain another equivalence class between M S 001,010 001 and M S 001,110 001 .In conclusion, we have M S 001, 001 = M S 001" simply means that they are indistinguishable in all topological data and thus belong to the same equivalence class.By exhausting all such equivalence classes, we obtain 14 topologically distinct excitations belonging to 14 topologically distinct equivalence classes, which are listed in table I. III.FUSION RULES FOR THE BBA TWISTED TERM A. Fusion rules and the path-integral representation Fusion rules are important properties in topological orders.Several prototypical fusion processes are pictorially illustrated in figure 2 and detailed calculations of these fusion rules are shown in section III B. A general fusion process can be symbolically written as:

FIG. 2 .
FIG. 2. Some typical fusion rules for the 5D twisted BF theory with the BBA twisted term.Fusing two P 001 's leads to a vacuum (trivial excitation).Fusing a L 100 and a P 001 leads to a decorated loop L 001 100 .Fusing two M S 001 's leads to a direct sum of a vacuum and three different loops denoted by L 100 , L 010 and L 110 .The first two fusion rules shown in this figure are Abelian fusion rules since each of them only contains a single fusion channel.The last fusion rule is non-Abelian fusion since it has multiple fusion channels.

FIG. 4 .
FIG. 4.A typical example of hierarchical shrinking rules for T 2 membranes in the 5D twisted BF theory with the BBA twisted term.We can shrink a T 2 membrane M T 001 to a superposition of a vacuum, a L 100 , a L 010 and a L 110 first.Then we can continue to shrink them to 4 vacua.

S 2
and T 2 membranes.All membranes are non-Abelian excitations.We give a general formula M ST -excitations.Similar to BBA twisted term, M ST -excitations in AAAB twisted term are also created by fusing S 2 membranes and T 2 membranes.When we fuse an M S a ′ b ′ c ′ , abc and an M T ã′b′ c′ , ãb c , we label the result as M ST (a ′ +ã ′ )(b ′ + b′ )(c ′ +c ′ ), abc,ã bc

abc,ã
bc with different abca ′ b ′ c ′ ãb cã ′ b′ c′ can be equivalent in our calculation.These M ST -excitations can be classified into 56 equivalence classes and we are going to give two formulas to describe all nonequivalent M ST -excitations.When (abc) is equal to ãb c , i.e., a = ã, b = b, c = c, an M ST -excitation can be represented by:

Shrinking a T 2 S 2 M
membrane.Shrinking rules for T 2 membranes can be written asS M T a ′ b ′ c ′ , abc = Ñabc 2 {a ′ P a ′ 00 ⊗ b ′ P 0b ′ 0 ⊗ c ′ P 00c ′ ⊗ (1 ⊕ L 0001 ) T a ′ b ′ c ′ , abc = Ñabc N abc S M S a ′ b ′ c ′ , abc(B28)If a T 2 and an S 2 membrane carry the same gauge charges and decorations, they can eventually be shrunk to the same superposition of particles up to a global coefficient Ñabc N abc .When they only carry one unit of gauge charge, this global coefficient is 2 1 = 2, carry two units is 2 2 = 4, and carry three units is 2 3 = 8.Every unit of gauge charge contributes an extra 2 to the normalization factor Ñabc in the front of Wilson operators for T 2 membranes.

5 m=1ϵ 4 2πq N i k<l<m 5 j=1ϵ 5 n=1δ
ijklm χ j dχ k dχ l A m − 2πq N i j<k<l<m ϵ ijklm χ j dχ k dχ l dχ m (C8)Basic nontrivial Wilson operators areW i = N i exp i ˆω C 1 + 1 ijklm d −1 A j A k A l A m ˆγ A n (C9)Thus AAAAA twisted terms have non-Abelian fusion rules and non-Abelian shrinking rules, but do not have hierarchical shrinking rules.
. 1. Illustration of particle excitations and spatially extended topological excitations with different shapes in the 5D twisted BF theory with the BBA twisted term.
3considered here.The gauge fields B1 , B2 and C 3 serve as the Lagrange multipliers in action (1).B 1 , B 2 and A 3 satisfy the flat connection FIG topological excitations, which eventually leads to a finite set of topological excitations despite the infinite number of gauge-invariant Wilson operators that can be written.Below, we will demonstrate the details of some examples of Wilson operators, and leave all nonequivalent Wilson operators in table I.In this table, there are 14 topologically distinct excitations and one of them is the topologically trivial excitation denoted as 1.It is important to point out that trivial particles, trivial loops, and trivial membranes are all identical to each other and are eventually denoted as 1.
1-, 2-, and 3-form, we can construct Wilson loop operators for particle excitations, Wilson surface operators for loop excitations, and Wilson volume operators for membrane excitations.Thus, the equivalence classes of Wilson operators directly render the equivalence classes of

TABLE I .
Operators for nonequivalent excitations in 5D topological order with action S = ´N1

TABLE III .
Quantum dimension of excitations in table II.For a non-Abelian excitation, the quantum dimension is larger than 1 and equal to the normalization factor of the Wilson operator.For an Abelian excitation, the quantum dimension is always 1.

TABLE IV .
Shrinking table for the 5D twisted BF theory with the BBA twisted term.All shrinking rules for the BBA twisted term are shown in table IV.We can see that all pure loops are shrunk to vacuum, and pure S 2 membrane is shrunk to 2 vacua.Pure T 2 membrane can be ultimately shrunk to 4 vacua, M ST ultimately shrunk to 8 vacua.Thus all pure spatially

TABLE V .
Comparision of fusion and shrinking rules in different twisted terms.

TABLE VI .
Fusion rules for AAAB twisted term.The fusion table is symmetric, so we only have to write down the upper triangular part of this table.To get all information about fusion rules, we have to know what exactly these elements are.
B24) Now we can use table VII to derive the fusion rules instead of directly calculate M ST a ′′ b ′′ c ′′ , abc,abc ⊗ M S e ′ f ′ g ′ , ef g .For other fusion rules concerning M ST -excitations, such as M ST abc,ã bc ⊗ M S e ′ f ′ g ′ , ef g , we have similar discussions.So in principle, we can obtain the rest of 7 elements in table VI.
Shrinking an S 2 membrane.Shrinking rules for S 2 membranes can be written asS M S a ′ b ′ c ′ , abc =N abc {a ′ P a ′ 00 ⊗ b ′ P 0b ′ 0 ⊗ c ′ P 00c ′In the above equation, if the coefficient of the excitation is zero, we just omit it.For example terms like ⊗ 0 2 (1 ⊕ P abc ) are omitted.If we consider shrinking an M S 100, 101 , we have (P 100 ⊕ P 001 ⊕ P 110 ⊕ P 011 ) (B27)