Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime

Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach towards a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic \particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multi-loop braidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loop braidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601 (2018)]. A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are $\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}$, and $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}$. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.

The order parameter, which is designed for characterizing orders, is one of fundamental concepts of many-body physics. Symmetry-breaking orders are characterized by local order parameters-local functions of spacetime. However, topological orders (e.g., fractional quantum Hall states) [1][2][3][4] in gapped systems are characterized by intrinsically nonlocal order parameters, such as adiabatic quantum phases accumulated by braiding topological excitations (e.g., anyons) [5].
In topological orders, topological excitations are usually geometrically compact manifold-like after taking continuum limit, such as point-like particle excitations, string-like loop excitations, etc. 1 Braiding phases of topological excitations are proportional to integer-valued invariants of knots or links formed by world-lines of particles and world-sheets of loops, thereby being quantized and robust against local perturbations. In addition to topological orders, braiding phases have also been applied to characterization of symmetry protected topological (SPT) phases [7][8][9][10][11] despite that SPT bulk excitations are topologically trivial. The core reason is that SPTs can be properly dualized to specific topological orders [7].
One quantitatively efficient and powerful approach to braiding phases is topological quantum field theory (TQFT) [12]. For example, braiding data of two-dimensional topological orders are encoded in (2 + 1)D Chern-Simons theory [2,5,13]. In this paper, we focus on deconfined phases of (3 + 1)D discrete gauge theories 2 [14][15][16][17][18] with Abelian gauge group G = n i=1 Z Ni where n denotes the total number of cyclic subgroups. Such phases of matter are usually called " n i Z Ni topological order". For example, the ground state of the three-dimensional toric code model [19,20] admits Z 2 topological order. By using group representation and conjugacy class [21], we may label topological excitations via gauge charges and gauge fluxes. More specifically, there are totally n i=1 N i distinct bosonic particles carrying gauge charges and n i=1 N i distinct loops carrying gauge fluxes. Without loss of generality, it is enough to consider braiding phases among n distinct elementary particles (denoted as e 1 , e 2 , · · · , e i , · · · , e n ) carrying the unit gauge charge of a specific gauge subgroup and n distinct elementary loops (denoted as m 1 , m 2 , · · · , m i , · · · , m n ) carrying the unit gauge flux of a specific gauge subgroup. When more than one particle (loop) are simultaneously involved in the same braiding process, superscripts will be added properly to e i (m i ).
In the first class, within each gauge subgroup, e.g., Z Ni , there is a well-defined particle-loop braiding phase Θ H i = 1 For non-manifold-like excitations, Ref. [6] provides some examples in a class of exotic stabilizer codes that support spatially extended excitations with restricted mobility and deformability. 2 In order to reconcile different conventions in condensed matter physics and high energy physics, we take the following convention: gauge theories/field theories are always associated with spacetime dimensions but quantum states, "Hamiltonian-type" lattice model, or topological phases of matter (topological order, SPTs, etc.) are associated with spatial dimensions only. 2π Ni mod 2π when the Hopf linking invariant is one 3 . Here, the Hopf link is formed by an elementary particle's trajectory γ ei and an elementary loop m i . This braiding phase always exists since it physically encodes the cyclic group structure of Z Ni . For the whole gauge group, the root braiding phases of the first class form a set {Θ H i } G with i = 1, · · · , n. A subscript G is added for specifying the gauge group G. Apparently, all braiding phases in the set belong to distinct gauge subgroups, thereby being mutually compatible and linearly independent. In the language of TQFT, one may compute braiding phases from gauge-invariant correlation functions of Wilson operators of the topological BF theories with action S BF =´ i Ni 2π B i ∧ dA i (abbreviated as BdA) [16,[46][47][48][49].
Here, the 1-form A i and 2-form B i are compact U(1) gauge fields describing the loop current ( 1 2π * dA i ) and particle current ( 1 2π * dB i ) degrees of freedom, respectively. As a natural higher-dimensional generalization of the Chern-Simons theory, the BF theory has been broadly applied to condensed matter systems, such as superconductors [16], bosonic and fractional topological insulators and more general 3D SPTs [49][50][51][52][53].
In the second class, i.e., multi-loop (three or four) braidings, all objects involved in the braidings are loops. More specifically, a three-loop braiding [26] consists of three elementary loops, and lead to a set of braiding phases Z Ni with n ≥ 2. As mentioned above, when G is given, Θ H i always exists. Here j, k, l indicate that three elementary loops (denoted by m 1 j , m 2 k , m b l ) respectively carry the elementary gauge flux of Z Nj , Z N k , Z N l gauge subgroups. Geometrically, the loop m b l right after the symbol "|", which carries elementary gauge flux of Z N l gauge subgroup, is called "base loop" [26]. The latter is simultaneously hopfly linked to the other two loops, i.e., m 1 j , m 2 k . Under this geometric setting, the three-loop braiding can be regarded as an anyonic braiding process on the Seifert surface bounded by the base loop. Braiding phases in the set {Θ H i ; Θ 3L j,k|l } G satisfy a series of remarkably elegant constraints such that mutually compatible braiding datasets can be unambiguously determined. The result was obtained in Ref. [26] by means of general properties of discrete gauge group and adiabaticity of braiding processes. The same result can also be obtained from TQFTs with topological terms of BdA + AAdA form [27][28][29][30][31][32][33][34]44]. Moreover, in the second class, if we consider four loops from four distinct gauge subgroups, the four-loop braiding phases form a set {Θ H i ; Θ 4L j,k,l,m } G , where G = n i=1 Z Ni with n ≥ 4, and the four loops carry elementary gauge fluxes of four different gauge subgroups Z Nj , Z N k , Z N l , Z Nm respectively. The fourloop braiding is associated with the quadruple linking number of surfaces, thereby being quantized [28]. The corresponding TQFTs can be symbolically expressed as BdA + AAAA. [27][28][29][30][31][32][33][34]44].
In the third class, i.e., the particle-loop-loop braiding or Borromean Rings (BR) braiding [45], an elementary particle carrying unit gauge charge of Z N k gauge subgroup moves around two loops (denoted by m 1 i , m 2 j ) that respectively carry unit gauge fluxes of Z Ni and Z Nj , such that the particle's trajectory γ e k and the two loops together form a Borromean Rings link, or general Brunnian link. The corresponding braiding phase is denoted as Θ BR i,j|k , which is proportional to the Milnor's triple linking numberμ [45,54,55]. Likewise, one may define a set of braiding phases: {Θ H i ; Θ BR j,k|l } G , where G = n i=1 Z Ni with n ≥ 3. The corresponding TQFTs can be symbolically expressed as BdA + AAB [45].
In this paper, we put all root braiding processes, which are denoted by {Θ H i ; Θ 3L j,k|l ; Θ 4L m,n,o,p ; Θ BR q,r|s } G , in arbitrary combinations and try to exhaust all topological orders. We find that, not all possible gauge group assignments (i.e., the subscripts i, j, k, l, · · · ) are realizable. Not all braiding phases in a given set are linearly independent. As we reviewed above, within each class, compatible braiding phases have been studied via various approaches. By "compatible", we mean that these braiding processes can be supported in the same system. In other words, the compatible braiding phases together as a set of braidings characterize a legitimate topological order. If there are two mutually incompatible braiding processes in the set, then both braiding processes must always lead to two trivial braiding phases, i.e., 0 mod 2π regardless the values of linking numbers of the braidings. As we will show, gauge invariance is broken if any one of two braidings has a nontrivial braiding phase. Therefore, in order to exhaust all legitimate topological orders, it is sufficient to find all sets of braidings formed by mutually compatible braiding processes. For this purpose, in this paper, through TQFT approach, we compute all braiding processes in a unified framework, and figure out all cases of incompatibility that are tightly related to gauge noninvariance. Especially, we focus on cases of incompatibility that occur when two or three distinct classes of root braiding processes have nontrivial braiding phases. Compatible braiding phases for different gauge groups are summarized in Table I Table II (G = Z N1 ×Z N2 ×Z N3 ), and Table III ( More general cases with more than four Z N gauge subgroups can be straightforwardly analyzed by applying the results in these four tables, as shown in Sec. III D. In addition to braiding phases, in these tables, we also provide the corresponding TQFTs and definitions of gauge transformations therein. All other sets of braiding phases are incompatible and not realizable. Some typical examples of incompatibility will be analyzed in details in this paper. The remainder of this paper is structured as follows. In Sec. II, we concretely analyze root braiding processes one by one, in order to lay the foundation for the forthcoming discussions on compatibility. In Sec. III, by combining all root braiding processes together, we study the corresponding TQFTs and extract all sets of compatible braiding processes. Then, in Sec. IV, some sets of incompatible braiding processes are illustrated in some concrete examples. In Sec. V, along the same line, we concisely discuss compatible braidings and TQFTs in (4 + 1)-dimensional topological orders.
Conclusions are made in Sec. VI. Several technical details are collected in Appendices.

II. REVIEW ON ROOT BRAIDING PROCESSES AND GAUGE TRANSFORMATIONS
In this section, we review TQFTs of root braiding processes. We emphasize the correspondence between root braiding processes, topological terms and braiding phases, which is illustrated as the following triangle:

Topological Term Braiding Phase
A braiding process can be identified from a topological term or a braiding phase, and vice versa. In this manner, we can study the braiding processes within the framework of TQFT. More concretely, in order to extract braiding phases from a TQFT, one can either add gauge-invariant source terms [30,45] or study algebra of Wilson operators [34,40], such that the braiding phases of a given braiding process are connected to a linking / knot invariant formed by the spacetime trajectories of particles and loops.
A. Microscopic origins of discrete gauge groups, topological excitations, and TQFT actions As our basic goal is to utilize TQFTs and braiding processes to characterize and classify topological orders of underlying quantum many-body systems, it is very important to identify microscopic origins of input data of TQFTs and braiding processes.
Hamiltonian realization, especially exactly solvable model proposal, is always the most powerful stimulus of the progress of topological orders. For example, toric code model proposed by Kitaev [20] elegantly unveils all key properties of nonchiral Abelian topological orders in (2 + 1)D. String-net models constructed by Levin and Wen [56] are applied to exhaust non-Abelian topological orders in (2+1)D. In (3+1)D, topological orders that are within Dijkgraaf-Witten cohomology classification [57] have Hamiltonian realization in Ref. [38], which uses 4-cocycles to cover all particle-loop and multiloop braidings. However, exactly solvable models for Borromean Rings braiding [45] are not known, which is an interesting future direction. Ref [58] and [59] present strategies for constructing higher-dimensional Abelian and non-Abelian topological phases via coupling quantum wires, which may shed light on constructing exactly solvable lattice model for BR braiding.
While exactly solvable models have Hamiltonian form, the Hamiltonians often look very intricate and unrealistic (four-spin, six-spin interactions, and more). Furthermore, it is unclear for us to rigorously connect the Hamiltonians to TQFTs. The latter have been proved to be a very powerful machine to study topological orders since the discovery of the fractional quantum Hall effect. But, to the best of our knowledge, it is highly impossible to perform a standard perturbation theory to renormalize the intricate interacting electron system of the ν = 1/3 Laughlin state to a beautiful Chern-Simons gauge theory´3 4π AdA where A is an emergent gauge field. Although there are many effective ways, e.g., parton construction and hydrodynamical approach, to handle stronglycorrelated physics, it is still kind of mysterious to derive emergent dynamical gauge fields from the very beginning.
In order to identify microscopic origins, below we will provide an effective way of thinking, following the spirit of the previous works [16,29,34,50,52,60]. From this effective derivation, we can find how discrete gauge groups, topological terms, quantized Wilson integral arise from a quantum many-body system. We can also find particle excitations and loop excitations are gauge charges and gauge fluxes of gauge groups. Below, we take´BdA + AAB as an example by means of exotic boson condensate and vortexline condensate. The detailed derivation is given in Appendix A. Here we just briefly sketch the key idea.
We start from a multi-layer condensate in 3D space in which one layer is in vortexline condensation phase (from disordering a 3D superfluid) while the others are in charge condensation phases ("bosonic superconductors"): The former can be regarded as the Higgs phase of two-form gauge fields while the latter can be regarded as the usual Higgs phase of 1-form gauge fields. where The vector-like phase angle Θ µ describes the phase field of vortexline condensation [50], while θ 1 and θ 2 are the usual phase angles of boson condensation. The coefficients ρ 1 , ρ 2 , ρ 3 represent phase rigidity of condensates. The last term of the above Lagrangian couple three condensates together in a gauge-invariant fashion. By introducing Hubbard-Stratonovich fields Σ 3 µν , j 1 , j 2 and Lagrange multiplier fields ξ I and η I , we obtain: (2) Integrating out Θ µ , θ 1 and θ 2 yields constraints in the path-integral measure. These constraints can be solved by introducing 1-form gauge field A 3 , 2-form gauge fields B 1 and B 2 respectively: The physical meaning of Σ 3 µν is the current of loop while those of j 1 λ and j 2 ρ are the currents of particles of layer 1 and 2 respectively. In this manner, one can figure out the microscopic origins of particle and loop excitations. Substituting Σ 3 µν , j 1 λ and j 2 ρ into Lagrangian (2) and integrating out the Lagrange multiplier fields, we end up with (4) Finally, we effectively obtain the TQFT action for a BR braiding: In this theory, the particle excitations are e 1 , e 2 , e 3 carrying gauge charges of Z N1 , Z N2 , Z N3 and minimally couple to A 1 , A 2 , A 3 . The loop excitations are m 1 , m 2 , m 3 carrying gauge fluxes of Z N1 , Z N2 , Z N3 and minimally couple to B 1 , B 2 , B 3 .

B. Particle-loop braiding and BdA term
The particle-loop braiding is essentially a quantized Aharonov-Bohm effect in a discrete gauge theory where local interactions are completely screened and long-range statistical interactions lead to nontrivial braiding phases. Given a Z N1 gauge subgroup, an elementary particle e 1 is braided around a static elementary loop m 1 . The trajectory γ e1 and m 1 as a whole form a Hopf link, as shown in Fig. 1. The corresponding braiding phase is given by Ni for unit Hopf linking number. The classification of equivalent trajectories of e 1 is essentially related to the mathematics of fundamental group and link homotopy, which was preliminarily introduced in the Supplemental Materials of Ref. [45]. The corresponding TQFT is the following multi-component BF action 4 where N i are positive integers that specify the discrete gauge group G.
µν B i µν dx µ dx ν } are 1-form and 2-form compact U(1) gauge fields respectively. Although one may expect a general matrix formalism ∼ K ij B i dA j , the matrix K can always be sent to a diagonal matrix with positive elements via two independent GL(n, Z) transformations. The two transformations respectively act on B fields and A fields (A relevant discussion on basis transformations can be found in Refs. [30,53].). The action (5) keeps invariant up to boundary terms under gauge transformations where {χ i } and {V i } are respectively 0-form and 1-form compact U(1) gauge parameters with´dχ ∈ 2πZ and dV ∈ 2πZ. Once these integrals are nonzero, the corresponding gauge transformations are said to be "large". The coefficient quantization, i.e., N 1 is integral, is guaranteed by the invariance of the partition function Z = DBDAe i´N 1 2π BdA under large gauge transformations on any compact oriented manifold. subgroups is two: For this gauge group, we consider a three-loop braiding process ( Fig. 2) with braiding phase denoted as Θ 3L 2,2|1 . The TQFT action for this three-loop braiding process is S is invariant up to boundary terms under gauge transformations The coefficient q 122 is quantized and periodic: q 122 = kN1N2 N12 , where k ∈ Z N12 and N 12 is the greatest common divisor (GCD) of N 1 and N 2 .
The coefficient is q 123 = kN1N2 N123 where k ∈ Z N123 and N 123 is the GCD of N 1 , N 2 and N 3 .
, four-loop braiding processes are realizable, in which four loops carry the unit gauge fluxes of Z N1 , Z N2 , Z N3 , Z N4 gauge subgroups respectively. The corresponding TQFT action is The action (11) is invariant up to boundary terms under gauge transformations where is the Levi-Civita symbol with 1234 = − 1324 = 1.
where k ∈ Z N1234 and N 1234 is the GCD of N 1 , N 2 , N 3 and N 4 . Till now, we have only reviewed gauge transformations in this review section. It is hard to visualize four-loop braiding in three-dimensional real space. We recommend Fig. 6 of Ref. [28].

D. Borromean Rings braiding and AAB term
Last, we consider the particle-loop-loop braiding or Borromean Rings braiding with its braiding phase denoted as Θ BR i,j|k [45]. Likewise, the input data of BR braiding are Abelian gauge group G = n i=1 Z Ni but with n ≥ 3, such that all particles and loops can be labeled by gauge charges and gauge fluxes in a specific gauge subgroup Z Ni . This braiding is beyond Dijkgraaf-Witten gauge theory classification H 4 (G, U (1)). The latter only includes braiding phases of particle-loop braidings and multi-loop braidings. By further taking BR braiding into account, we need to study proper combinations of all braidings together to exhaust all topological orders.
The corresponding topological term of BR braidings is A i A j B k . For example, consider a BR braiding process shown in Fig. 4, the corresponding TQFT action is where p 12,3 = l12,3N1N2N3 N123 , l 12,3 ∈ Z N123 , N 123 is the GCD of N 1 , N 2 and N 3 . Ref. [45] points out that l 12,3 = −l 21,3 and l ij,k = 0 if any of the two indices are same. The quantization of p 12,3 is due to the large gauge invariance. The gauge transformations for (13) are where X 3 , Y 1 and Y 2 are so-called shift terms. The Dirac quantization of A 3 , B 1 and B 2 requires that 1 2π´d X 3 ∈ Z, 1 2π´d Y 1 ∈ Z and 1 2π´d Y 2 ∈ Z. The Z Ni cyclic group structures are encoded in the cyclic Wilson integrals of A 1 , A 2 and B 3 respectively:

III. COMPATIBLE BRAIDING PROCESSES
Since we have reviewed the correspondence between topological terms, braiding processes and braiding phases, we may naively think that we can design a system which can support arbitrary combinations of braiding processes. However, this is not true. In other words, there are illegitimate combinations in which the braiding processes are mutually incompatible. Such incompatibility reveals that some topological terms are forbidden to form a legitimate TQFT action.
In this section, we will compute the sets of compatible braiding phases for different gauge groups, from which the compatible braiding processes can be read out. For this purpose, we seek for a legitimate TQFT action consisting of as many topological terms as possible for a given gauge group.
In order to verify a TQFT action is legitimate or not, we need to take care of the following aspects. First of all, a legitimate TQFT action should be invariant up to boundary terms under proper gauge transformations. Second, the gauge transformations are required to preserve the Z Ni cyclic group structure. This requirement means that, if the Z Ni cyclic group structure is encoded in¸A are 0-form and 1-form compact U(1) gauge parameters with´dχ i ∈ 2πZ and´dV i ∈ 2πZ respectively. From the perspective of microscopic origins (Sec. II A), if A i has the above standard gauge transformation, the i-th layer condensate must be a charge-N i boson condensate that higgses the Wilson integrals of A i down to Z Ni , and B i comes from the duality transformation shown in Sec. II A and Appendix A; likewise, if B i has the above standard gauge transformation, the i-th layer condensate must be a charge-N i vortexline condensate that higgses the Wilson integrals of B i down to Z Ni , and A i comes from the duality transformation shown in Sec. II A and Appendix A. In summary, in a legitimate action, at least one of gauge fields (A i and B i ), for a given i, should have the above standard gauge transformations and thus have the Z Ni quantized Wilson integrals. For example, see Eq. (6), Eq. (8), Eq. (10), Eq. (12), Eq. (14).
Next, the gauge transformation of each gauge field should respect the Dirac quantization. Last but not least, a legitimate TQFT action should consist of topological terms with nontrivial coefficients. If the coefficient of a topological term is identical to 0, otherwise the action cannot be gauge invariant under gauge transformations, this topological term is actually incompatible with others in the action.
When G = Z N1 , the elementary particle (loop) is e 1 (m 1 ) carrying unit gauge charge (flux) of Z N1 gauge group. The only nontrivial braiding process in this case is the particle-loop braiding described by The gauge transformations for (16) are The braiding phase of this particle-loop braiding is When G = Z N1 ×Z N2 , the elementary particles (loops) are denoted as e 1 and e 2 (m 1 and m 2 ) carrying unit gauge charge (flux) of Z N1 and Z N2 gauge subgroups respectively. Beside particle-loop braidings, three-loop braidings can be supported. There are 4 kinds of three-loop braiding phases in this case, described by two linearly independent AAdA terms: 2,1|1 : m b l , m 1 j and m 2 k respectively carry unit gauge flux of Z N1 , Z N2 and Z N1 gauge subgroup.
The first two braiding processes are described by A 1 A 2 dA 2 and the remainder by A 2 A 1 dA 1 . All these three-loop braidings and particle-loop braidings are compatible, described by a TQFT action where q 122 = kN1N2 N12 , k ∈ Z N12 and q 211 = k N2N1 N12 , k ∈ Z N12 . N 12 is the GCD of N 1 and N 2 . The gauge transformations for (19) are The phases of particle-loop braidings and three-loop braidings are Legitimate TQFT actions for G = Z N1 and G = Z N1 × Z N2 are summarized in Table I. The sets of compatible braiding phases for G = Z N1 and G = 2 i=1 Z Ni are respectively and where the underlines denote the linear dependence of two braiding phases, for example, , the elementary particles (loops) are e 1 , e 2 and e 3 (m 1 , m 2 and m 3 ) carrying unit gauge charge (flux) of Z N1 , Z N2 and Z N3 gauge subgroups respectively. Besides particle-loop braidings and three-loop braidings discussed in Sec. III A, BR braidings described by AAB terms and three-loop braidings described by A m A n dA o terms (m, n, o are mutually different) are realizable. For the reason why BR braiding is absent when G = 2 i=1 Z Ni , one can refer to Sec. IV A.
The compatibility issues of G = 3 i=1 Z Ni are discussed as follows. First of all, if we neglect BR braidings, all particleloop braidings and three-loop braidings are compatible, we can write down a legitimate TQFT action for all root braiding processes except BR braidings: where q ijk = lNiNj Nij , l ∈ Z N ijk , N ij (N ijk ) is the GCD of N i , N j (and N k ). Next, we consider a BR braiding, say, Θ BR 1,2|3 described by an A 1 A 2 B 3 term. All particle-loop braidings are compatible with this BR braiding. However, among three-loop braiding phases, only Θ 3L The reasons for this incompatibility between three-loop braidings and BR braidings are discussed in Sec. IV C, IV D and IV E. The total TQFT action for Θ BR 1,2|3 and its compatible braiding processes isˆ3 where p 12,3 = lN1N2N3 N123 , l ∈ Z N123 and q ijj = l NiNj Nij , l ∈ Z Nij . Furthermore, if we consider two different BR braidings, we find that it is impossible to write down a legitimate TQFT action which contains two different AAB terms. The reason is explained in Sec. IV B. Nevertheless, if the gauge group is G = n i=1 Z Ni with n ≥ 4, legitimate TQFT actions for two different BR braidings are possible, which is detailed in Sec. III C.
By checking all combinations of braiding processes with the criteria of compatibility, we summarized all legitimate TQFT actions for G = Z N1 × Z N2 × Z N3 and corresponding gauge transformations in Table II. The sets of compatible braiding phases for G = 3 i=1 Z Ni can be summarized as and where the underlines denote the linear dependence between two compatible braiding phases: Similar to the case in Sec. III B, particle-loop braidings and multi-loop braidings are compatible with each other when TABLE I. Compatible braiding phases and TQFTs when G = ZN 1 and G = ZN 1 × ZN 2 . Each row represents a class of topological order(s). The 1 st row stands for a topological order whose braiding data is given by Θ H i Z N 1 with i = 1, 2, 3. The 2 nd row represents topological orders which are characterized by Θ H i , any combinations of compatible Θ 3L 's . The coefficient of AAdA term is given by qijj = nN i N j N ij , n ∈ ZN ij , which is determined by the large gauge invariance. χ and V are 0-form and 1-form compact U(1) gauge parameters with´dχ ∈ 2πZ and´dV ∈ 2πZ.
we take BR braidings into account, we need to treat them carefully. If these BR braidings involve only one kind of elementary particles, say, e 1 , correspond- corresponding to the set of compatible braiding phases with r = 1 and s = 1. In other words, any multi-loop braiding which involves elementary loop m 1 is incompatible with BR braidings which involve elementary particle e 1 .
If we consider BR braidings which involve two kinds of elementary particles, e.g., e 1 and e 2 , the legitimate TQFT action isˆ4 which excludes multi-loop braidings which involve elementary loops m 1 and m 2 . The compatible braiding phases form a set: , etc., respectively characterize a topological order. The coefficients are given by q ijk = , l ij,k ∈ ZN ijk , as a result of large gauge invariance. l ij,k = −l ji,k and l ij,k = 0 if any of the two indices are the same. The Levi-Civita symbol is defined by Compatible braiding phases TQFT actions Gauge transformations

D. Results for general gauge groups
In fact, our previous discussions can be easily generalized to the case where the number of Z Ni subgroups are arbitrary. As we can see in the previous main text, compatibility is always guaranteed for particle-loop braidings and multi-loop braidings, or, particle-loop braidings and BR braidings. The general rule of incompatibility is that if two braiding processes (multi-loop braiding and BR braiding, or two different BR braidings ) involve particle and loop that carry gauge charge and gauge flux of the same Z Ni gauge subgroup, these two braiding processes are incompatible with each other. In the language of TQFT, a Z Ni gauge subgroup is associated with a topological term B i dA i . In a legitimate TQFT action, given the BF term B i dA i , only A i or B i , not both, can appear in the twisted terms (i.e., AAdA, AAAA and AAB). Otherwise, the TQFT action would be illegitimate, e.g., violating the gauge invariance.
Viewed from the microscopic origin of topological terms, the rule of incompatibility is natural. Each layer of condensate (corresponding to each Z Ni gauge subgroup) is either in the charge condensation or in the vortexline condensation, never both. The Z Ni charge condensation implies particles that carry gauge charge e i (corresponding to A i ), while the Z Ni string condensation results in loops that carry gauge flux m i (corresponding to B i ). If a multi-loop braiding or BR braiding involves the Z Ni particle, it is incompatible with other braiding process that involves the Z Ni loop, and vice versa.

IV. INCOMPATIBLE BRAIDING PROCESSES
In Sec. III, we discuss the compatible braiding processes in different cases. We find that, when the gauge group is given, the set of compatible braiding processes is only a subset of {all possible root braiding processes}. This means that some braiding processes are mutually incompatible.
For example, . We can see that, among all possible root braiding phases, Θ 3L i,j|k with one index equal to 3, Θ BR 2,3|1 and Θ BR 3,1|2 are excluded from this set, because they are not compatible with Θ BR 1,2|3 . In this section, we will demonstrate several examples of such incompatibility between braiding processes via the TQFT perspective. The basic idea is: if an action consists of topological terms corresponding to incompatible braiding processes, it would never be a legitimate TQFT theory.

A. Absence of BR braidings when
In Sec. II D, we review the BR braiding in which a particle carrying unit Z N3 gauge charge moves around two loops which carry unit Z N1 and Z N2 gauge flux respectively. Such a braiding process is only possible when there are three distinct Z Ni gauge subgroups. It is natural to ask: why it is impossible when there are only two Z Ni gauge subgroups, i.e., G = 2 i=1 Z Ni ? Without loss of generality, we can consider the following braiding process: a particle e 2 carrying unit Z N2 gauge charge moves around two loops m 1 and m 2 which carry unit Z N1 and Z N2 gauge flux respectively such that the trajectory of e 1 and two loops form a Borromean Rings link. One may naively think that, this braiding process, along with its braiding phase Θ BR 12,2 , is described by the topological term A 1 A 2 B 2 , analogous to the case in Sec. II D. The corresponding TQFT action should be However, the action (40) is not a legitimate TQFT action because we cannot find gauge transformations which respects gauge invariance and Z N cyclic group structure simultaneously. We are going to illustrate this in details.
In the action (40), B 1 serves as a Lagrange multiplier imposing a local constraint dA 1 = 0. This means that the gauge transformation of A 1 is A 1 → A 1 +dχ 1 . Thus, the Z N1 cyclic group structure is encoded in¸A i ∈ 2π Ni Z Ni . The Z N2 cyclic group structure requires that at least one of A 2 → A 2 + dχ 2 and B 2 → B 2 + dV 2 holds.
First, we assume that the gauge transformations are where Under the gauge transformations (41), the variation of action (40) (boundary terms are neglected) is which should be an integral of total derivative terms in order to be gauge invariant. Focus on the dχ 1 dχ 2 B 2 term which is not a total derivative term: we want to eliminate it by subtraction or by absorbing it into a total derivative term. If we want to eliminate dχ 1 dχ 2 B 2 by subtraction, the only way is to require since dχ 1 dχ 2 Y 2 is the only term containing dχ 1 dχ 2 in (42). However, Eq. (43) means that Y 2 = −B 2 + · · · , thus the gauge transformation of B 2 is which is not a proper gauge transformation for B 2 . If we want to absorb dχ 1 dχ 2 B 2 to a total derivative term, we could make use of However, this attempt fails since there is no −χ 1 dχ 2 dB 2 term in ∆S. Therefore, (41) is not the proper gauge transformations for S. Next, we assume that gauge transformations are , l ij,k ∈ ZN ijk as a result of large gauge invariance. l ij,k = −l ji,k and l ij,k = 0 if any of the two indices are the same. The Levi-Civita symbol is defined by a 1 a 2 ···an = 1≤i<j≤n sgn (aj − ai) where sgn (x) = 1, 0, or −1 if x > 0, x = 0, or x < 0.

Compatible braiding phases
TQFT actions Gauge transformations where X 2 and Y 1 are shift terms with¸X 2 / ∈ 2π N2 Z N2 anḑ Y 1 / ∈ 2π N1 Z N1 . The Z N2 cyclic group structure is encoded Under (46), the variation of action (40) is which should be an integral of total derivative terms. Focus on the dχ 1 A 2 dV 2 term: we want to eliminate it by subtraction or by absorbing it into a total derivative term. If we want to eliminate dχ 1 A 2 dV 2 by subtraction, the only way is to require, which means that X 2 = −A 2 + · · · hence an ill-defined gauge transformation for A 2 . If we want to absorb dχ 1 A 2 dV 2 into a total derivative term, we need a χ 1 dA 2 dV 2 term since but there is no a χ 1 dA 2 dV 2 term in ∆S, so this attempt fails. Therefore, Eq. (46) is not the proper gauge transformations for the action (40) either.
Since neither (41) nor (46) serves as proper gauge transformations, it is impossible to construct gauge transformations respecting Z N2 cyclic group structure for the action (40). This means that the action (40) is not a legitimate TQFT action. Therefore, Θ BR 1,2|2 is not even a well-defined braiding phase, not to mention its compatibility with other braiding phases. In other words, when G = 2 i=1 Z Ni , BR braidings cannot be realized.

2,3|1
In previous sections we have known that the A 1 A 2 B 3 term describes the BR braiding with a phase denoted as Θ BR 1,2|3 . One may expect that, in the same system, there could exists another BR braiding process described by an A 2 A 3 B 1 term and its braiding phase Θ BR 2,3|1 . Assumed that these two BR braidings could be supported in the same system, the TQFT action should be In the action (51), B 2 is a Lagrange multiplier imposing dA 2 = 0, i.e., A 2 → A 2 + dχ 2 . Therefore the Z N2 cyclic group structure is encoded in¸A 2 ∈ 2π N2 Z N2 . In order to preserve the Z N1 and Z N3 cyclic group structure, at least one of A i → A i + dχ i and B i → B i + dV i (i = 1, 3) is required. We find that, if the gauge transformations preserve Z N1 and Z N3 cyclic group structure, the action (51) would never be gauge invariant, i.e., the braiding processes it describes are gauge-dependent. The reason is that no matter how we modify the gauge transformations, there are "stubborn terms" in ∆S which cannot be eliminated by subtraction nor be absorbed into a total derivative term, making ∆S could never be 0 mod 2π. The details of calculation can be found in Appendix C 1. Therefore, A 1 A 2 B 3 and is incompatible with A 2 A 3 B 1 , i.e., Θ BR 2,3|1 is incompatible with Θ BR 1,2|3 . In the same manner, we can derive that any two of Θ BR 2,3|1 , Θ BR 3,1|2 , Θ BR 1,2|3 are mutually incompatible.

1,2|3
In this and following subsections, we investigate the incompatibility between three-loop braidings and BR braidings.
As the first example, we consider these two braiding processes: a three-loop braiding with phase Θ 3L 3,3|1 , corresponding to an A 1 A 3 dA 3 term, and a BR braiding with phase Θ BR 1,2|3 , corresponding to an A 1 A 2 B 3 term. If theses two braiding processes are compatible, the total TQFT action should be: In the action (52), B 1 and B 2 serve as Lagrange multipliers, imposing dA 1 = dA 2 = 0, i.e., A 1,2 → A 1,2 + dg 1,2 . The Z N1 and Z N2 cyclic group structure are encoded in¸A 1 ∈ 2π N1 Z N1 and¸A 2 ∈ 2π N2 Z N2 respectively. For the Z N3 cyclic group structure, we have two choices: we can encoded it iņ A 3 ∈ 2π N3 Z N3 , corresponding to gauge transformations or¸B 3 ∈ 2π N3 Z N3 , corresponding to gauge transformations A 1,2 →A 1,2 + dχ 1,2 , However, similar to the case in Sec. IV A and Sec. IV B, neither (53) nor (54) could make the action (52) gauge invariant up to boundary terms, due to the existence of "stubborn terms". Details of derivations can be found in Appendix C 2.
If the action is forced to be gauge invariant up to boundary terms, the Z N3 cyclic group structure would be violated. This dilemma reveals that (52) is not a legitimate TQFT action. So we conclude that D. Incompatibility: Θ 3L 2,3|1 and Θ BR

1,2|3
In this subsection, we consider these two braiding processes: a three-loop braiding corresponding to Θ 3L 2,3|1 and an A 1 A 2 dA 3 term, and a BR braiding corresponding to Θ BR

1,2|3
and an A 1 A 2 B 3 term. If Θ 3L 2,3|1 is compatible with Θ BR 1,2|3 , the total TQFT action should be: where the coefficients are q 123 = k1N1N2 N12 , k 1 ∈ Z N123 and p 12,3 = k2N1N2N3 N123 , k 2 ∈ Z N123 , determined by the large gauge invariance. In (55), B 1 and B 2 serve as Lagrange multipliers, imposing dA 1 = dA 2 = 0, i.e., A 1,2 → A 1,2 +dχ 1,2 . We can find a set of gauge transformations that seems to respect both Z N3 cyclic group structure and gauge invariance (see Appendix C 3 for derivation): where Nevertheless, the action (55) is still not a legitimate TQFT theory: the coefficient of A 1 A 2 dA 3 in (55) is actually identical to 0. To see this, we first integrate out A 3 in S, i.e., sum over all possible nontrivial compactness contributions from dA 3 , leading to a constraint q 123 Since the Z N3 cyclic group structure is encoded in¸N 3 2π B 3 ∈ Z, Eq. (58) requires that i.e., (consider a 2-manifold where k 1 ∈ Z N123 , m 1 ∈ Z, m 2 ∈ Z. Since m 1 and m 2 can be arbitrary integers, the sufficient condition for Eq. (60) is On the other hand, N 12 is the integral multiple of N 123 , thus where n = N12 N123 . Notice that k 1 ∈ Z N123 , this means that Therefore the coefficient of A 1 A 2 dA 3 term is required to be identical to 0: So far, we have argued that, in order to preserve the Z N3 fusion structure with arbitrary values of N 1 , N 2 and N 3 , the coefficient of A 1 A 2 dA 3 term in (55) is required to be trivial. If the coefficient of A 1 A 2 dA 3 is nontrivial, the action (55) cannot be gauge invariant up to boundary terms while the Z N3 cyclic group structure is preserved. The restriction on the coefficient of A 1 A 2 dA 3 indicates the incompatibility between A 1 A 2 dA 3 and A 1 A 2 B 3 , i.e., the incompatibility between Θ 3L 1,2|3 and Θ BR 1,2|3 .
If the Z N3 cyclic group structure is encoded in¸A 3 ∈ 2π N3 Z N3 , i.e., A 3 → A 3 + dχ 3 , we cannot find a set of gauge transformations with A i → A i +dχ i (i = 1, 2, 3) under which the action (65) is invariant up to boundary terms. The reason is same as those in Sec. IV A and Sec. IV C. If we encode the Z N3 cyclic group structure in¸B 3 ∈ 2π N3 Z N3 , i.e., B 3 → B 3 + dV 3 , we can find a set of gauge transformations under which (65) is invariant up to boundary terms: At first glance, one may think that (66) is a proper set of gauge transformations for (65) thus the action (65) is a legitimate TQFT action. However, we argue that this is not true. We provide the following two arguments.
In the first argument, we consider the gauge transformations at limiting cases. Let us set p 12,3 = 0, i.e., turn off the A 1 A 2 B 3 term, the action (65) becomes and the corresponding gauge transformations should be According to Ref [34], gauge transformations (68) is motivated by a microscopic derivation of the action (67). However, if we set p 12,3 = 0 in (66), we cannot correctly recover the gauge transformations to (68). Therefore, we claim that (66) is not proper gauge transformations for the action (65).
In fact, if we expect that by setting q 231 = 0 or p 12,3 = 0 the gauge transformations for the action (65) would recover to (15) or (68), the gauge transformations for the action (65) should take the form of where the gauge parameters f 3 A (p 12,3 , q 231 ) and f i B (p 12,3 , q 231 ) with i = 1, 2, 3 are identical to 0 once p 12,3 = 0 or q 231 = 0. We can see that, the Z N3 cyclic group structure is no longer preserved under gauge transformation (69).
The second argument exploits the identity Using (70) we can rewrite the action (65) as (71) Similar to the case in Sec. IV D, we can integrate out A 3 and obtain a constraint Since in the gauge transformations (56) the Z N3 cyclic group structure is encoded in¸B 3 = 2π N3 Z N3 , we require that In other words, we require that (consider where k ∈ Z N123 . Since m 1 and m 2 can be arbitrary integers, we need Since N3 N23 ∈ Z, the sufficient condition for Eq. (74) to hold is Notice that N1 N123 ∈ Z and k ∈ Z N123 , therefore where n = N1 N123 ∈ Z. We see that the coefficient of A 2 A 3 dA 1 is required to be identical to 0: to ensure that the Z N3 cyclic group structure is preserved no matter what the values of N 1 , N 2 and N 3 are. This restriction on the coefficient of Since we have demonstrated the incompatibility between BR braidings and three-loop braidings, one may wonder whether BR braiding is compatible with four-loop braiding or not. The conclusion is that, when G = 4 i=1 Z Ni , no BR braiding is compatible with four-loop braiding, i.e., each Θ BR i,j|k is incompatible with Θ 4L 1,2,3,4 . The proof of this claim, which follows the same idea discussed in previous sections, 1,2,3,4 is compatible with Θ BR i,j|5 , as long as i = 5, j = 5 and i = j. Such compatibility due to an extra gauge subgroup can be checked by applying the criteria for legitimacy of TQFT actions.

V. COMPATIBLE BRAIDINGS IN (4+1)D SPACETIME
In the above discussions, we have obtained compatible braidings in (3 + 1)D spacetime. In the following, we concisely discuss compatible braidings in (4+1)D spacetime. More systematic and quantitative studies will appear in Ref. [61] separately.
A. Excitations and braiding processes in (4 + 1)D We limit the gauge group to be G = n i=1 Z Ni where n is the total number of cyclic subgroups. In (4 + 1)D spacetime, besides particles and loops, there is a kind exotic topological excitations, dubbed membranes, which are closed 2D surface objects in the continuum limit. The membrane excitations in 3-dimensional space is topologically trivial because they are impenetrable hence identified as particle excitations. However, in 4-dimensional space, the interior of a membrane excitation becomes accessible due to the extra dimension. Therefore, nontrivial braiding processes involving particles, loops and membranes are possible in (4 + 1)D spacetime.
In field theory, a Z Ni gauge theory is realized by a BF term in the continuum limit. In (3 + 1)D spacetime, the Z Ni cyclic group structure is encoded in the B i dA i term, the only possible BF term in (3 + 1)D spacetime which corresponds to the particle-loop braiding (see Sec. II B). In (4 + 1)D spacetime, there are two kinds of BF terms with distinct physical meanings: C i dA i and B i dB i , where the 3-form C i , 1-form A i , 2form B i and 2-form B i are respectively compact U(1) gauge fields. Therefore, we can encode the Z Ni cyclic group structure either in C i dA i , corresponding to a particle-membrane braiding; or in B i dB i , corresponding to a loop-loop braiding. If we consider a n i=1 Z Ni topological order in (4 + 1)D spacetime, we can even encode some Z Ni gauge subgroups in C i dA i terms while the others in B i dB i terms, which would lead to a more complicated situation in which different gauge fields can couple together resulting in many nontrivial braiding processes.
B. An overview of compatible braiding processes in (4 + 1)D Following the discussions in (3+1)D spacetime, we can investigate the compatibility of braiding processes in (4 + 1)D spacetime. In other words, we check the legitimacy of TQFT actions consisting of topological terms, then distinguish compatible/incompatible braiding processes in (4 + 1)D spacetime.
First, if all Z Ni cyclic group structures are encoded in particle-membrane braidings, i.e., C i dA i terms, the compatible braiding processes are multi-membrane braidings and particle-membrane-membrane braiding. Here are some examples. When G = 5 i=1 Z Ni , one of legitimate TQFT actions is where the quantized coefficients of topological terms are neglected. The action (79) indicates the compatibility of particle-membrane braidings and multi-membrane braidings when G = 5 i=1 Z Ni . We believe that TQFT action (79) describes braiding processes within the Dijkgraaf-Witten model. There are also braiding processes beyond the Dijkgraaf- as an analog of the BR braiding in We have known that, in (3 + 1)D, a BR braiding is incompatible with some of the multi-loop braidings (Sec. IV C, IV D and IV E). Similar incompatibility occurs in (4 + 1)D. When G = 3 i=1 Z Ni , a multi-membrane braiding described by the A 1 A 2 A 3 dA 1 term is incompatible with the loop-membrane-membrane braiding described by the C 3 A 1 A 2 term. Using the language of TQFT, we claim that a legitimate TQFT action in (4 + 1)D can only include one of Next, we consider the case in which all Z Ni cyclic group structures are encoded in B i dB i terms, i.e., loop-loop braidings. In this case, the only topological excitations are loops, thus only loop-loop braidings can be supported. We can simply write down the legitimate TQFT action: Last, we tackle the more complicated yet more interesting situation: some Z N cyclic group structures (e.g., Z Ni gauge subgroups) are encoded in Ni 2π C i dA i terms while the others (e.g., Z Ki gauge subgroups) in Ki 2π B i dB i terms. Different assignments for N i 's and K i 's correspond to different topological orders in (4 + 1)D. In this case, there are particle, loop and membrane excitations in the system. All nontrivial braiding processes in (4 + 1)D should be taken into consideration. In the followings we will demonstrate several examples of legitimate TQFT actions from which we can directly read out compatible braiding processes in (4 + 1)D.

Consider a
3 i=1 Z Ni case in which the Z N1 and Z N2 cyclic group structures are encoded in CdA while the Z K1 cyclic group structure in BdB, i.e., the gauge group is G = Z N1 × Z N2 × Z K1 , the legitimate TQFT actions is Consider again a 3 i=1 Z Ni case but this time G = Z N1 × Z K1 × Z K2 , the corresponding legitimate TQFT action is We should point out that the BC term is also a topological term in (4 + 1)D spacetime, but we do not regard that it corresponds to any braiding process in the discussion above. Naively, because the 2-form B is related to the loop current Σ via Σ = 1 2π * dB and the 3-form C is related to the particle current J via J = 1 2π * dC, one may think that BC is related to the particle-loop braiding in (4 + 1)D. But we argue that particle-loop braiding is actually trivial in (4 + 1)D spacetime thus the BC term does not describe the particle-loop braiding. To show the triviality of particle-loop braiding in (4 + 1)D spacetime, we can first consider the particle-loop braiding in (3 + 1)D spacetime in which the Hopf link formed by trajectory γ e of the particle e and the loop m (e.g., FIG. 1) cannot be unlinked in 3D space. However, in 4D space, due to an extra dimension, we can smoothly deform γ e such that it is unlinked with the loop m. This argument is analog to the fact that the nontrivial particle-particle braiding in (2 + 1)D spacetime becomes trivial in (3 + 1)D spacetime. In this manner, we claim that the particle-loop braiding in (4 + 1)D spacetime is trivial.
A question naturally arises: what is the physical effect of the BC term? We noticed that there are topological terms which can generate trivial fermions from a bosonic model: k 4π AdA with an odd integer k in (2 + 1)D and BB in (3 + 1)D. We conjecture that the effect of BC term is similar: the emergence of trivial fermions is possible in a bosonic model described by the action S ∼´CdA + BdB + BC.

VI. CONCLUSION
In this paper, when a gauge group is given, we investigate the compatibility between all three classes of root braiding processes in (3 + 1)D spacetime, i.e., particle-loop braidings, multi-loop braidings and Borromean Rings braidings. We find that not all root braiding processes are mutually compatible, which is not so obvious on a superficial level. By incompatibility we mean that two braiding processes cannot be supported in the same system, i.e., there is no legitimate topological order in (3 + 1)-dimensional spacetime characterized by incompatible braiding phases. In the language of TQFT, a TQFT action describing incompatible braiding processes is not gauge invariant while preserving the cyclic group structure of each gauge subgroup. Our conclusions are drawn as follows.
When the gauge group is G = Z N1 , only one class of root braiding processes, i.e., particle-loop braiding, is realizable. Therefore, there is no incompatibility when G = Z N1 . The set of compatible braiding phase is simply Θ H 1 Z N 1 . The corresponding TQFT action and gauge transformations are listed in Table I.
When G = 2 i=1 Z Ni , besides particle-loop braidings, three-loop braidings are realizable. In this case, all braiding processes are compatible with each other. The set of compatible braiding phase is the collection of all possible braiding phases: Table I lists the TQFT action corresponding to compatible braiding phases along with its gauge transformations. As for G = ings. We find that, particle-loop braidings are compatible with three-loop braidings or BR braidings respectively. However, incompatibility occurs between three-loop braidings and BR braidings, even between different BR braidings. In general notations, a BR braiding phase Θ BR i,j|k is incompatible with three-loop braiding phases Θ 3L n,o|m in which one of the indices {m, n, o} is equal to k. Moreover, in the case of  Table II. For the case of G = 4 i=1 Z Ni , besides the braidings mentioned above, four-loop braiding is realizable, classified as one of multi-loop braidings. Similar to the case of G = and Θ BR 1,2|3 , Θ BR 1,2|4 is respectively a subset of the sets of compatible braiding phases. Table III  and Table IV list all possible sets of compatible braiding phases, legitimate TQFT actions and corresponding gauge transformations at the case of G = 4 i=1 Z Ni . Motivated by the compatibility of braiding processes in (3 + 1)D spacetime, we initiate an attempt to the physics of braiding processes in (4 + 1)D spacetime and their compatibility. Several results are given in Sec. V. A comprehensive understanding of compatible braiding processes and TQFTs in (4 + 1)D topological orders will be discussed in Ref. [61].
There are several interesting future investigations based on the compatibility analysis of braiding phases. For example, it is interesting to study the connection between incompatibility and quantum anomaly [32]. Meanwhile, we have not considered BB term [47,50,[62][63][64][65] which can switch self-statistics (either bosonic or fermionic) of particles. In this sense, adding BB term may be useful when considering compatible braiding phases of either fermionic systems or bosonic systems with fermionic particle excitations that carry nontrivial gauge charges. Our work focuses only on topological orders where all particles are bosonic, so it is unnecessary to consider BB. In addition, BB can drastically change the gauge group. In other words, the gauge group G is simultaneously determined by the coefficients of BF term and BB term. Nevertheless, this complexity leads to superfluous difficulty in determining gauge group and does not provide new braidings among topological excitations. In addition, from experiences in twodimensional topological orders, exhausting all non-Abelian braidings with non-Abelian gauge group are very challenging if still applying the present field theory method. Algebraic tools from mathematics may be a much better way. It is definitely interesting to explore how to exhaust all legitimate topological orders with a non-Abelian gauge group.
We can derive the´BdA + AAB action from a 3-layer 3D superfluid where the layer 1 and 2 are in charge-N 1 and charge-N 2 particle condensations while the layer 3 is in chareg-N 3 vortexline condensation. To illustrate the vortexline condensation picture, we start with a normal superfluid state: Separate the phase angle into smooth part and singular part θ 3 = θ 3 s + θ 3 v and introduce a Hubbard-Stratonovich auxiliary vector field J 3 µ , the above L 3 can be expressed as Integrating out the smooth part θ 3 s leads to a constraint ∂ µ J 3 µ = 0, which can be solved by introducing a 2-form noncompact U (1) gauge filed B 3 µν : Thus L 3 is dualed to the following gauge theory The string current operator is defined by Now consider a 3-layer 3D superfluid where the layer 1 and 2 are in particle condensation while the layer 3 is in string condensation. The Lagrangian is µν , j 1 and j 2 : Further introduce Lagrange multiplier fields ξ I and η I to decouple terms like iΛε µνλρ −θ 1 ∂ λ ∂ ρ θ 2 N 3 B 3 µν : Integrating out Θ µ , θ 1 and θ 2 yield constraints These constraints can be solved by Then we obtain Integrate out ξ 3 µν : Integrate out ξ 1 λ : Since we consider the vortexline condensation for ρ 3 and particle condensation for ρ 1 and ρ 2 , i.e., ρ 3 → ∞, ρ 1 → ∞ and ρ 2 → ∞ are taken. These limit conditions enforce that η 1 λ = N 1 A 1 λ , η 2 ρ = N 2 A 2 ρ and η 3 µν = 1 2 N 3 B 3 µν . So we obtain (A21) The Maxwell kinetic term L Maxwell can be neglected since its scaling dimension is more irrelevant than the other topological terms. Finally, we can drop the overall minus sign by relabeling indices and then obtain The general expression of legitimate TQFT actions and corresponding gauge transformations are listed in where r = s, r = l, s = l, and the assignments for (i, j, k, l) are (1, 2, 3, 4), (2,3,4,1), (3,4,1,2) and (4, 1, 2, 3).
3. S with AAB terms which involve two flavors of B i . The set of compatible braiding phases is where the assignments for (i, j, k, l) are: (3, 4, 1, 2), (2, 4, 1, 3), (2, 3, 1, 4), (4, 1, 2, 3), (3,1,2,4) and (1,2,3,4). We try to find proper gauge transformations for this action: We list all 4 possible test gauge transformations respecting the Z N1 and Z N3 cyclic group structure and corresponding ∆S's in Table V and Table VI. We point out that there always exist "stubborn terms" which cannot be eliminated by subtraction not be absorbed into a total derivative term, making ∆S nonvanishing. If we impose that ∆S = 0 mod 2π, the Z N1 and Z N3 cyclic group structure are no longer respected. Such dilemma indicates that (C1) is not a legitimate TQFT action, thus A 1 A 2 B 3 and A 2 A 3 B 1 are incompatible.   Table III. By properly assigning (i, j, k, l), one can obtain all legitimate TQFT actions for G = 4 i=1 ZN i . The 1 st row: since no AAB terms in the TQFT action, no assignment for (i, j, k, l) is needed. The 2 nd row: for actions with A i A j B l + A i A k B l + A j A k B l terms, the assignments for (i, j, k, l) are (1,2,3,4), (2,3,4,1), (3,4,1,2) and (4,1,2,3). For example, by taking (i, j, k, l) = (4, 1, 2, 3), we re-obtain the 2 nd TQFT action in Table III. The 3 rd row: for actions with A i A j B k + A i A j B l terms, there are 6 assignments for (i, j, k, l): (3,4,1,2), (2,4,1,3), (2,3,1,4), (4,1,2,3), (3,1,2,4) and (1,2,3,4). By taking (i, j, k, l) = (1, 2, 3, 4), we re-obtain the 3 rd TQFT action in Table III. Compatible braiding phases TQFT actions Gauge transformations For the 2 nd column of Table VI, it may not so straightforward to see that ∆S is nonvanishing. Below we make some illustration. The gauge transformations are The variation of action (boundary terms neglected) is If we want to eliminate the A 1 A 2 dV 3 term by subtraction, we can only require that which requires However (C5) is impossible since A 1 A 2 is not exact in general. If we want to absorb A 1 A 2 dV 3 into a total derivative term, we would need a dA 1

Derivation of the incompatibility between
The TQFT action is assumed to be First, we assume that the gauge transformations are The "stubborn terms" are terms which cannot be eliminated by subtraction nor be absorbed into a total derivative term, making ∆S = 0 mod 2π. Test gauge transformations ∆S Stubborn terms The "stubborn terms" are terms which cannot be eliminated by subtraction nor be absorbed into a total derivative term, making ∆S = 0 mod 2π. Test gauge transformations ∆S Stubborn terms i.e., the Z N3 cyclic group structure is encoded in˛A Under the gauge transformations, the variation of action is (boundary terms neglected) is ∆S is expected to be boundary terms. However, we are going to prove that this is impossible: the dχ 1 dχ 2 B 3 term cannot be eliminated by subtraction nor be absorbed into a total derivative term. If dχ 1 dχ 2 B 3 can be eliminated by subtraction, it is required that i.e., but this is not allowed otherwise the gauge transformation of B 3 is ill-defined. If dχ 1 dχ 2 B 3 can be absorb into a total derivative term, ∆S should (but in fact does not) contain a −χ 1 dχ 2 dB 3 term since Therefore, the action (C6) is impossible to be gauge invariant under gauge transformation (C7), hence it is not a legitimate TQFT action.
Next, we assume that the gauge transformations are A 1,2 →A 1,2 + dχ 1,2 , The Z N3 cyclic group structure is encoded in˛B Under the gauge transformation, the variation of action is (boundary terms neglected) is Similarly, ∆S is impossible to be boundary terms since the term dχ 1 A 3 dA 3 cannot be eliminated by subtraction nor be absorbed into a total derivative term. The only way to eliminate dχ 1 A 3 dA 3 by subtraction is to require that which means that If so, the gauge transformation of A 3 is which is not well-defined. If dχ 1 A 3 dA 3 can be absorbed into a total derivative term, ∆S should (but in fact does not) contain a χ 1 dA 3 dA 3 term since Therefore, (C13) cannot be the proper gauge transformations for (52). So far, we have seen that if the Z N3 cyclic group structure is respected, we cannot find a proper set of gauge transformations under which the action (52) is invariant up to boundary terms. If the action is imposed to be gauge invariant up to boundary terms, the Z N3 cyclic group structure would be violated. This dilemma reveals that the action (52) is not a legitimate TQFT theory. In other words,

Derivation of incompatibility between
If we assume that A 1 A 2 A 3 A 4 are compatible with A 1 A 2 B 4 , the TQFT action should be In the action (C45), B 1 , B 2 and B 3 serve as Lagrange multipliers, imposing dA 1 = dA 2 = dA 3 = 0, i.e., the gauge transformations of A 1 , A 2 and A 3 are A 1,2,3 → A 1,2,3 + dχ 1,2,3 . The Z N1 , Z N2 and Z N3 cyclic group structures are encoded in¸A I ∈ 2π N I Z N I , where I = 1, 2, 3. The remaining Z N4 cyclic group structure can be encoded in¸A 4 ∈ 2π N4 Z N4 oŗ B 4 ∈ 2π N4 Z N4 , corresponding to A 4 → A 4 + dχ 4 or B 4 → B 4 + dV 4 respectively. In the following text, we are going to examine both cases.However, we will find that neither of them would result in gauge transformations under which the action (C45) is gauge invariant up to boundary terms.
First, we assume that the gauge transformations are Under (C46), the variation of action is (boundary terms neglected) We recognize that dχ 1 dχ 2 B 4 is a stubborn term, i.e., it cannot be eliminated by substraction or be absorbed into a total derivative term. Due to the existence of stubborn term, ∆S cannot be written as an integral of total derivative terms. Therefore, the action (C45) is not gauge invariant up to boundary terms under (C46).
However, this χ 1 dχ 2 dχ 3 dA 4 term does not exist in ∆S. Therefore, due to the "stubborn term" dχ 1 dχ 2 dχ 3 A 4 , the action (C45) is not gauge invariant up to boundary terms under (C46). Finally, we can conclude that the action (C45) could never be gauge invariant up to boundary terms under gauge transformations which respect Z Ni cyclic group structures, hence it is not a legitimate TQFT action. Therefore A 1 A 2 A 3 A 4 Θ 4L 1,2,3,4 is incompatible with A 1 A 2 B 4 Θ BR 1,2|4 .