Symmetry Enrichment in Three-Dimensional Topological Phases
SSymmetry Enrichment in Three-Dimensional Topological Phases
Shang-Qiang Ning, Zheng-Xin Liu, and Peng Ye
3, 4, ∗ Institute for Advanced Study, Tsinghua University, Beijing, China, 100084 Department of Physics, Renmin University of China, Beijing, China, 100872 Department of Physics, University of Illinois at Urbana-Champaign, IL 61801, USA Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, IL 61801, USA
While two-dimensional symmetry-enriched topological phases (
SET s) have been studied inten-sively and systematically, three-dimensional ones are still open issues. We propose an algorithmicapproach of imposing global symmetry G s on gauge theories (denoted by GT ) with gauge group G g .The resulting symmetric gauge theories are dubbed “symmetry-enriched gauge theories” ( SEG ),which may be served as low-energy effective theories of three-dimensional symmetric topologicalquantum spin liquids. We focus on
SEG s with gauge group G g = Z N × Z N × · · · and on-siteunitary symmetry group G s = Z K × Z K × · · · or G s = U(1) × Z K × · · · . Each SEG ( G g , G s ) isdescribed in the path integral formalism associated with certain symmetry assignment. From thepath-integral expression, we propose how to physically diagnose the ground state properties (i.e., SET orders) of
SEG s in experiments of charge-loop braidings (patterns of symmetry fractionaliza-tion) and the mixed multi-loop braidings among deconfined loop excitations and confined symmetryfluxes. From these symmetry-enriched properties, one can obtain the map from
SEG s to
SET s. Bygiving full dynamics to background gauge fields,
SEG s may be eventually promoted to a set of newgauge theories (denoted by GT ∗ ). Based on their gauge groups, GT ∗ s may be further regroupedinto different classes each of which is labeled by a gauge group G ∗ g . Finally, a web of gauge theoriesinvolving GT , SEG , SET and GT ∗ is achieved. We demonstrate the above symmetry-enrichmentphysics and the web of gauge theories through many concrete examples. I. INTRODUCTION
Recently, the field of gapped phases with symmetryhas been drawing a lot of attentions in condensed mat-ter physics. There are two kinds of symmetric gappedphases: symmetry-protected topological phases (
SPT )and symmetry-enriched topological phases (
SET ). SPT phases are short-range entangled [1] with a global sym-metry and have been studied intensively in strongly-correlated bosonic systems [1–35]. Much progress hasalso been made in two-dimensional (2D)
SET s [36–47],which are partially driven by tremendous efforts in quan-tum spin liquids (QSL) [36, 48] that respect a certainglobal symmetry (e.g., spatial reflection, time-reversal,Ising Z , U(1) and SU(2) spin rotations, etc.). In con-trast to SPT s, SET s are long-range entangled [1] and sup-port emergent excitations, such as anyons in 2D systems.Furthermore, quantum numbers carried by emergent ex-citations may be fractionalized. Experimentally, it is ofinterest to detect patterns of such symmetry fractional-ization, which may help us characterize QSLs [48]. In ad-dition to the usual global symmetry, there are also
SET senriched by a new kind of symmetry dubbed “topologi-cal (anyonic)” symmetry [43, 49–60]. This symmetry de-notes an automorphism of the topological data (braidingstatistics, quantum dimensions, etc.). A typical exampleis that Z topological order in two dimensions is invari-ant under e - m exchange operation, namely, an electric-magnetic duality in discrete gauge theories [49, 50]. ∗ Corresponding author: [email protected]
Despite much success in 2D
SET s, three-dimensional(3D)
SET physics, especially the underlying generalframework, is still poorly understood so far, partiallydue to the presence of spatially extended loop excita-tions [61]. In physical literatures, some attempts havebeen made, including 3D U(1) QSLs and Z QSLs withsymmetry, e.g., in Ref. [62–65]. Field theories of 3D
SET swith either time-reversal or 180 ◦ spin rotation about y -axis were studied where the dynamical axion electromag-netic action term is considered [18, 66]. The boundaryanomaly of some 3D SET s was viewed as 2D anomalous
SET s with anyonic symmetry [67]. In Ref. [68, 69], a di-mension reduction point of view was proposed to demon-strate how symmetry is fractionalized on loop excita-tions. In Ref. [70], the notion of “2D anyonic symmetry”was generalized to 3D “charge-loop excitation symme-try” (
Charles ) which is a permutation operation amongparticle excitations and among loop excitations. As typ-ical examples of 3D
SET s with U(1) and time-reversal,fractional topological insulators were constructed via aparton construction with gauge confinement [70].In this paper, we study 3D
SET s with Abelian topo-logical orders [71] that are encoded by deconfined dis-crete Abelian gauge theories [72]. We focus on discreteAbelian gauge group G g = Z N × Z N × · · · and on-siteunitary Abelian symmetry group G s = Z K × Z K × · · · or G s = U(1) × Z K × · · · . Physically, these 3D SET scan be viewed as 3D gapped QSLs that are enriched byunbroken on-site symmetry G s . Given a gauge group G g , there are usually many topologically distinct gaugetheories (denoted by GT ) including one untwisted andseveral twisted ones [23, 73], as shown in Fig. 1. Af-ter imposing global symmetry group, the resulting gauge a r X i v : . [ c ond - m a t . s t r- e l ] S e p field theory is called “symmetry-enriched gauge theory”( SEG ). Quantitatively, an
SEG is defined through twokey ingredients:1. an action that consists of topological terms (of one-form or two-form Abelian gauge fields) only;2. symmetry assignment via a specific minimal cou-pling to background gauge fields (denoted by { A i } with i = 1 , , · · · , where A i externally imposes sym-metry fluxes in Z K i symmetry subgroup).We also stress that an anomaly-free SEG must simultane-ously satisfy the following two stringent conditions [74]:1. global symmetry is preserved;2. gauge invariance is guaranteed on a closed space-time manifold.We use the notation
SEG ( G g , G s ) to denote such an SEG .Then we try to provide answers to the following ques-tions:1. What is the path-integral formalism of an
SEG ?And what is the “parent” GT of each SEG ?2. What is the relation between
SEG and
SET ? Howcan we probe symmetry-enriched properties in ex-periments?3. What is the resulting new gauge theory (denotedby GT ∗ ) after giving full dynamics [75] to { A i } ?To answer the first question is nothing but to lookfor anomaly-free SEG s that meet the above definitionand conditions. Following the 5-step general procedure(Sec. II C), the path-integral formalism of each
SEG canbe constructed, which is efficient for the practical pur-pose. Each
SEG can be identified as a descendant of some GT (i.e., “parent”). Many concrete examples, includingthe simplest case SEG ( Z , Z ), are calculated explicitlyin this paper. The method we will provide is doable formore general cases, some of which are collected in Ap-pendix.In the second question, a complete description of an SET order requires the information of both topologicalorders and symmetry enrichment. In this sense, the totalnumber of
SEG s is generically larger than that of distinct
SET orders. For example, two anomaly-free
SEG s, maypossibly give rise to the same
SET order. If two
SEG shave the same topological order, a practical way to probesymmetry enrichment is to insert symmetry fluxes intothe 3D bulk and perform Aharonov-Bohm experimentsbetween symmetry fluxes (flux loop formed by A i ) andbosons that are charged in the symmetry group. In addi-tion, one should also perform the mixed version of three-loop braiding experiment [26, 76] among symmetry fluxesand gauge fluxes (i.e., loop excitations). Through thesethought experiments, one may find the relations betweendifferent SEG s. If two
SEG s share the same bulk topo-logical order data as well the same symmetry-enriched G g …. GT GT GT …. SEG SEG SEG SEG SEG N/A
SEG …. GT ⇤ GT ⇤ GT ⇤ ….…. G ⇤ g G ⇤ g SET SET SET SET …. SET GT ⇤ G ⇤ g GT ⇤ FIG. 1. (Color online) Schematic representation of a webof gauge theories with global symmetry. We start with adiscrete gauge group G g that generates several topologicallydistinct gauge theories ( GT s) one of which is untwisted. Wethen assign symmetry charge of symmetry group G s to topo-logical currents of gauge theories through coupling to back-ground gauge fields. There are usually many different waysof symmetry assignment, each of which is represented by acolored arrow. Within each specific symmetry assignment,we obtain many SEG s. For example,
SEG , SEG , and SEG belong to the same symmetry assignment (marked by ma-genta arrows) in GT . It is generically possible that someof symmetry assignment do not provide SEG descendants for GT , which we mark as “N/A”. To identify SET s, we needto further study ground state properties of
SEG s. We mayexternally insert symmetry fluxes into the system and per-form Aharonov-Bohm experiments and the mixed version ofthree-loop braiding experiments. Two
SEG s (e.g.,
SEG and SEG ) may possibly describe the same SET phase. Further,by giving full dynamics to the background gauge fields, theresulting new gauge theories (denoted by GT ∗ ) are generated.Since basis transformations are allowed, there should be ingeneral many-to-one correspondence between SEG s and GT ∗ s.Finally, all GT ∗ s may be regrouped via identifying their gaugegroups (denoted by G ∗ g ). properties, they belong to the same SET ordered phase.Otherwise, they belong to two different
SET phases (seeFig. 1).For the third question, we note that in the action of an
SEG , { A I } is a set of non-dynamical background gaugefields. Symmetry fluxes formed by them are confinedloop objects that are externally imposed into the bulk.These loop objects are fundamentally different from thegauge fluxes that are deconfined bulk loop excitations.Therefore, the usual basis transformations (mathemati-cally represented by unimodular matrices of a general lin-ear group) on gauge field variables are strictly prohibited[8] if the transformations mix gauge fluxes and symme-try fluxes. However, if we give full dynamics to { A I } [75],then, the action actually represents a new gauge theory(denoted by GT ∗ ) and does not describe a SEG any more.In GT ∗ s, symmetry fluxes are legitimate deconfined bulkloop excitations and arbitrary basis transformations areallowed. As a result, it is possible that the actions oftwo SEG s may be rigorously mapped to each other viabasis transformations, both of which lead to the same GT ∗ . This set of gauge theories “ GT ∗ , GT ∗ , · · · ” may befurther regrouped by identifying their gauge groups (de-noted by G ∗ g , G ∗ g , · · · ). Finally, a web of gauge theoriesis obtained, as schematically shown in Fig. 1.The remainder of the paper is organized as follows.Sec. II is devoted to general discussions on GT s, topo-logical interactions and global symmetry. Especially, inSec. II C, the 5-step general procedure is introduced indetail. Some calculation details in Sec. II D,II E,II F willbe useful for quantitatively understanding the remain-ing sections, especially, Sec. III. For readers who areonly interested in the final results, these details maybe either skipped or gone through quickly. In Sec. III,many simple examples are studied in details, including SEG ( Z , Z K ), SEG ( Z × Z , Z ), and SEG ( Z × Z , U(1)).In Sec. IV, physical characterization of
SEG s is studied,including symmetry fractionalization and mixed three-loop braiding statistics among gauge fluxes and symme-try fluxes. In this way, we may achieve the map from
SEG to SET as schematically shown in Fig. 1. Simple exam-ples are given, including
SEG ( Z , Z K ) with K ∈ Z even and K ∈ Z odd . In Sec. V, full dynamics is given tothe background gauge field, which promotes SEG s to GT ∗ s. Again, the discussions are followed by some sim-ple examples including SEG ( Z , Z ), SEG ( Z , Z ) and SEG ( Z × Z , Z ). Summary and outlook are made inSec. VI. More technical details and concrete examplesare collected in Appendix. II. GAUGE THEORIES, TOPOLOGICALINTERACTIONS, AND GLOBAL SYMMETRYA. Inter-“layer” topological interactions andaddition of “trivial” layers
In the continuum limit, gauge theories with discretegauge groups can be written in terms of the followingmulti-component topological BF term [77]: S = i (cid:88) I,J Λ IJ π ˆ M b I ∧ da J , (1)where { b I } and { a I } are two sets of 2-form and 1-formU(1) gauge fields respectively. I = 1 , , · · · , n . Λ IJ issome n × n integer matrix, which may not be symmetricbut the determinant of Λ must be nonzero: DetΛ (cid:54) = 0[78]. In comparison to Horowitz’s action term [77], herewe do not consider b I ∧ b J . M is the 4D closed space-time (with imaginary time) manifold where our topologi-cal phases are defined. In the following, the notation M will be neglected from the action for the sake of simplic-ity. ………… type-I layers (Gauge theories) type-II layers (Trivial layers) FIG. 2. (Color online) A schematic representation of “layers”(Sec. II A) and the general procedure (Sec. II C). Each “layer”denotes a 3D system. It should be noted that all layers arestacked together in the same
3D spatial region although theyare not so in this figure. GT before imposing symmetry residesin type-I layers. Type-II layers are described by level-1 BFterms before imposing symmetry. By “trivial”, we mean thatthese layers do not carry gauge groups. The dashed curvesrepresent topological interactions between layers. Actually,three-layer and four-layer topological interactions should alsobe considered. There are two independent general linear transforma-tions represented by two unimodular matrices W, Ω ∈ GL ( n, Z ) that “rotate” loop lattice and charge latticerespectively. Therefore, Λ can always be sent into its canonical form via: W ΛΩ T = diag( N , N , · · · , N I , · · · , N n ) , (2)where { N I } are a set of positive integers. The superscript“ T ” denotes “transpose”. It is in sharp contrast to themulti-component Chern-Simons theory [71] where W =Ω and the above diagonalized basis usually doesn’t exist.In the remaining parts of this paper, we work in this newbasis unless otherwise specified. In this new basis, each I labels a “layer system” as schematically shown in the“type-I layers” in Fig. 2 (N.B., the word “layer” actuallydenotes a 3D spatial region). N I is the level of the BFterm in the I -th layer. { b I } and { a I } , as two sets of gauge fields, are subjectto the following Dirac quantization conditions:12 π ˆ M db I ∈ Z , (3)12 π ˆ M da I ∈ Z , (4)where M and M denote 3D and 2D closed manifoldsembedded in M respectively. These two equations willplay important roles in the following discussions.The BF term in the canonical form is a field theory of untwisted G g = Z N × Z N ×· · ·× Z N n gauge theory wherelayers are decoupled to each other. However, there aretopological interactions that can couple them together: S = i (cid:88) I N I π ˆ b I ∧ da I + i (cid:88) IJK q IJK π ˆ a I ∧ a J ∧ da K + i (cid:88) IJKL t IJKL π ˆ a I ∧ a J ∧ a K ∧ a L , (5)where { q IJK } and { t IJKL } are two sets of coefficients.These newly introduced action terms are topologicalsince their expressions are wedge products of differen-tial forms. Recently a lot of progress has been madebased on these topological terms in gauge theories aswell as SPT phases [13, 23, 79–81]. The presence ofinterlayer topological interactions leads to twisted G g gauge theories. Since these new topological terms areexplicitly not gauge invariant (even in a closed mani-fold) alone, the definitions of usual gauge transforma-tions on { b I } must be properly modified [to appear inEq. (10)]. To be a legitimate GT action, { q IJK } and { t IJKL } are expected to be quantized and compact (i.e.,periodic), which eventually leads to finite number of dis-tinct GT s before global symmetry is imposed. All of themare classified by the fourth group cohomology with U(1)coefficient: H ( Z N × Z N · · · , U(1)) = (cid:81)
I Now, let us consider how to impose global symmetrygroup G s = Z K × Z K × · · · × Z K m . In topologicalquantum field theory, there is a 1-form topological cur-rent J I for each I : ∗ J I = π db I , where ∗ denotes theHodge dual operation. It is conserved automatically since d = 0. The fact that the total particle number is inte-gral is nicely guaranteed by Dirac quantization condition(3). Therefore, a natural definition of global symmetryis to enforce that the symmetry charge is carried by thistopological current. This is the so-called hydrodynamicalapproach that was applied successfully in the fractionalquantum Hall effect with the multi-component Chern-Simons theory description [71]. This is also a key step ofthe topological quantum field theory description of SPTs[13]. In order to identify global symmetry, a backgroundgauge field A i is turned on. Mathematically, a mini-mal coupling term between background gauge fields andtopological currents is introduced into the action (6): S sym. = i (cid:80) nI (cid:80) mi L Ii ´ J I ∧∗ A i , where L Ii is an n × m in-teger matrix. By noting that the total symmetry group G s = Z K × Z K ×· · · , the background 1-form U(1) gaugefield A i is subject to the following constraints: K i π ˆ M A i ∈ Z for Z K i symmetry subgroup , (7)where M denotes a closed spacetime loop. As men-tioned in Sec. II A, trivial layers in Eq. (6) may be takeninto consideration once symmetry is imposed. Therefore,the index I in S sym. is allowed to be larger than n . Oncethe topological current carries symmetry charge, a newset of stringent constraints on the coefficients { q IJK } and { t IJKL } will be imposed such that global symmetry iscompatible with gauge invariance principle, the quanti-zation and periodicity of { q IJK } and { t IJKL } may bechanged dramatically after global symmetry is imposed.It means that, one GT may generate many distinct SEG descendants after symmetry is imposed, which manifestlyshows patterns of symmetry enrichment (see Fig. 1). Ifsymmetry is not imposed, those distinct SEG s become in-distinguishable and reduce back to the same parent GT . C. Summary of the 5-step general procedure Based on the preparation done in Sec. II A and II B,in this part, we summarize the general procedure for ob-taining SEG s and connecting them to their parent GT s.There are five main steps. Step-1 . Add trivial layers (i.e., type-II in Fig. 2).Mathematically, trivial layers are described by Eq. (6). Step-2 . Assign symmetry via the minimal couplingterms ( ∼ J ∧ ∗ A ). Symmetry assignment can be eithermade purely inside type-I or purely inside type-II or both[82]. Step-3 . Add all possible topological interactionsamong layers via the topological terms with coefficients { q IJK } and { t IJKL } in Eq. (5) and the indices I, J, K, · · · are extended to all layers including trivial layers. InFig. 2, only two-layer interactions (denoted by dashedlines) are drawn for simplicity. However, generic three-layer and four-layer interactions should also be taken intoconsiderations. Step-4 . Consider all consistent conditions and de-termine the quantization and periodicity of coefficientsof topological interactions. These consistent conditionsare (i) Dirac quantization conditions; (ii) “small” gaugetransformations; (iii) “large” gauge transformations; (iv) shift operation of coefficients that leads to coefficient pe-riodicity; (v) . total symmetry charge for Z K i subgroupis conserved mod K i . Once the above four steps aredone, the path-integral expressions and symmetry assign-ment for SEG s are obtained. Definitions and quantitativestudies of these consistent conditions will be provided inSec. II D,II E,II F, and Appendix A. Step-5 . Regroup all SEG s obtained above into distinct GT s in Fig. 1. For example, in Fig. 1, SEG , ··· , are SEG descendants of GT , while, SEG is a SEG descendant of GT . If gauge group is G g = Z N that will be calculatedin Sec. III A, this step can be skipped for the reason thatthere is only one Z N GT , i.e., the untwisted GT . If gaugegroup contains more than one Z N s, e.g., G g = Z N × Z N ,usually gauge theories have twisted versions. Under thecircumstances, the role of Step-5 becomes critical. Wewill discuss pertinent details in Sec. III B. D. General calculation on G g = Z N × Z N with nosymmetry In the following, we present some useful calculation de-tails on gauge theories with G g = Z N × Z N and demon-strate, especially, what the consistent conditions listedin Step-4 of Sec. II C are, at quantitative level. Severalmathematical notations are introduced and will be fre-quently used in the remaining parts of this paper. Allother calculation details are present in Appendix A.Consider the following two-layer BF theories withinter-layer topological couplings in the form of “ aada ”: S = (cid:88) I =1 iN I π ˆ b I ∧ da I + i q π ˆ a ∧ a ∧ da + i ¯ q π ˆ a ∧ a ∧ da , (8)where q ≡ q and ¯ q ≡ q . Since a a da and a a da are linearly independent, we may study them separately.First consider ¯ q = 0. The action is invariant under thefollowing gauge transformations parametrized by scalars { χ I } and vectors { V I } : a I −→ a I + dχ I , (9) b I −→ b I + dV I − q πN I (cid:15) IJ χ J ∧ da , (10)where (cid:15) = − (cid:15) = 1. It is clear that the usual gaugetransformations of b I [77] are modified through addinga q -dependent term in Eq. (10). As usual, the gaugeparameters χ I and V I satisfy the following conditions:12 π ˆ M dχ I ∈ Z , π ˆ M dV I ∈ Z . (11)Once the integers on the r.h.s. are nonzero, the associ-ated gauge transformations are said to be “large”. Letus investigate the integral π ´ M db I .Under the above modified gauge transformations (10),the integral will be changed by the amount below (for I = 1, M = M × M is considered):12 π ˆ M db −→ π ˆ M db − q π N ˆ S dχ ˆ M da = 12 π ˆ M db − q π N × π(cid:96) × π(cid:96) (cid:48) , (12)where (cid:96) , (cid:96) (cid:48) ∈ Z , and, the Dirac quantization condition(4) and gauge parameter condition (11) are applied. Inorder to be consistent with the Dirac quantization con-dition (3), the change amount must be integral, namely, q must be divisible by N . Similarly, q is also divisibleby N due to:12 π ˆ M db −→ π ˆ M db + q π N ˆ S dχ ˆ M da = 12 π ˆ M db + q π N × π(cid:96) (cid:48)(cid:48) × π(cid:96) (cid:48)(cid:48)(cid:48) , (13)where (cid:96) (cid:48)(cid:48) , (cid:96) (cid:48)(cid:48)(cid:48) ∈ Z . Hence, q = kN N N , k ∈ Z , where thesymbol “ N ” denotes the greatest common divisor of N and N .Below, we will show that k has a periodicity N andthereby q is compactified: q ∼ q + N N . Let us considerthe following redundancy due to shift operations:12 π ˆ db −→ π ˆ db − N ˜ K π N ˆ a ∧ da , (14)12 π ˆ db −→ π ˆ db + N ˜ K π N ˆ a ∧ da , (15) k −→ k + ˜ K + ˜ K . (16)Under the above shift operation, the total action (8) is in-variant. Again, in order to be consistent with Dirac quan-tization (3), the change amount of the integral π ´ M db I should be integral, namely: N ˜ K π N ˆ M a ∧ da ∈ Z , (17) N ˜ K π N ˆ M a ∧ da ∈ Z . (18)We may apply the Dirac quantization condition (4) andthe quantized Wilson loop N I π ´ M a I ∈ Z that is ob-tained via equations of motion of b I . As a result, twoconstraints are achieved: ˜ K /N ∈ Z , ˜ K /N ∈ Z . Byusing Bezout’s lemma, the minimal periodicity of k isgiven by the greatest common divisor (GCD) of N and N , which is still N . As a result, we obtain the condi-tions on q if symmetry is not taken into consideration. q = k N N N mod N N , k ∈ Z N . (19)Similarly, for ¯ q π a ∧ a ∧ da term, we also have the samequantization and the same periodicity:¯ q = k N N N mod N N , k ∈ Z N . (20)In conclusion, we have ( Z N ) different kinds of gaugetheories with G g = Z N × Z N . E. General calculation on G g = Z N × Z N with G s = Z K × Z K -(I) To impose the symmetry, we add the following couplingterm into S in Eq. (8) (again, we consider ¯ q = 0 only): (cid:88) i i π ˆ A i ∧ db i , (21)where A i is subject to the constraints in Eq (7). Thiscoupling term simply means that the first layer carries Z K symmetry while the second layer carries Z K sym-metry. The total symmetry group G s = Z K × Z K .Our goal is to determine all legitimate values of q inthe presence of global symmetry. And we expect that theperiod of q is in general larger than the original gaugetheory with no symmetry, which leads to a set of SEG s.The key observation is that the change amounts of theintegral π ´ M db I in both gauge transformations andshift operations should not only be integral [in order tobe consistent with the Dirac quantization condition (3)]but also be multiple of K i such that the coupling term(21) is gauge invariant modular 2 π . Physically, it canbe understood via the definition of the integral. Thisintegral is nothing but the total symmetry charge of theassociated symmetry group. Since the total symmetrycharge of Z K i is allowed to be changed by K i while stillrespecting symmetry. This is a peculiar feature of cyclicsymmetry group, compared to U(1) symmetry.More quantitatively, with symmetry taken into ac-count, from Eqs. (12, 13), we may obtain the quantiza-tion of q : q = kN N K K GCD( N K ,N K ) with k ∈ Z such that thechange amounts are multiple of K i . Then, with thesenew quantized values, the shift operations (14,15) arechanged to:12 π ˆ db −→ π ˆ db − ˜ K N K K ´ a ∧ da π GCD( N K , N K ) , (22)12 π ˆ db −→ π ˆ db + ˜ K N K K ´ a ∧ da π GCD( N K , N K ) . (23)The change amounts should be quantized at K inEq. (22) and K in Eq. (23), respectively, such thatsymmetry is kept. We may apply the Dirac quan-tization condition (4) and the quantized Wilson loop N I K I π ´ M a I ∈ Z that is obtained via equations of mo-tion of b I in the presence of A I background. As a result,two necessary and sufficient constraints are achieved: ˜ K GCD( N K ,N K ) ∈ Z , ˜ K GCD( N K ,N K ) ∈ Z . By usingBezout’s lemma, the minimal periodicity of k is givenby GCD of GCD( N K , N K ) and GCD( N K , N K ), which is still GCD( N K , N K ). Therefore, once sym-metry is imposed, q is changed from Eq. (19) to: q = k N N K K GCD( N K , N K ) mod N N K K , with k ∈ Z GCD( N K ,N K ) (24)which gives GCD( N K , N K ) SEG s. In other words,the allowed values of q are enriched by symmetry. For ¯ q term, the conditions are completely the same as q , whichleads to another GCD( N K , N K ) SEG s.¯ q = k N N K K GCD( N K , N K ) mod N N K K , with k ∈ Z GCD( N K ,N K ) . (25)In short, before imposing symmetry, according toEqs. (19,20), there are ( N ) distinct GT s with gaugegroup G g = Z N × Z N . After imposing symmetry group G s = Z K × Z K , according to Eqs. (24,25), there are[GCD( N K , N K )] distinct SEG s if the symmetry as-signment is given by Eq. (21). Likewise, one can considerthat Z K and Z K symmetry charges are carried by thesecond layer and the first layer respectively, i.e., Eq. (21)is changed to: i π ˆ ( A ∧ db + A ∧ db ) . (26)Then, there will be [GCD( N K , N K )] new SEG s. F. General calculation on G g = Z N × Z N with G s = Z K × Z K -(II) In the following, we alter the definition of symmetryassignment and still consider a a da first. The couplingterm in Eq. (21) is now changed to: i π ˆ ( A + A ) ∧ db (27)which means that both Z N and Z N symmetry chargesare carried by the first layer. We will show that (LCMstands for “least common multiple”): q = k LCM( N K , N K , N ) mod N N LCM( K , K ) , with k ∈ Z N N K ,K N K ,N K ,N (28)meaning that the total number of SEG s are N N LCM( K ,K )LCM( N K ,N K ,N ) if (i) both symmetry chargesare carried by the first layer shown in Eq. (27) and (ii) a a da is considered (i.e., ¯ q = 0). As a side note, byexchanging 1 ↔ 2, the above result directly impliesthat the total number of SEG s are N N LCM( K ,K )LCM( N K ,N K ,N ) if (i) both symmetry charges are carried by the secondlayer [replacing b in Eq. (27) by b ] and (ii) a a da isconsidered (i.e., q = 0):¯ q = k LCM( N K , N K , N ) mod N N LCM( K , K ) , with k ∈ Z N N K ,K N K ,N K ,N . (29)Let us present several key steps towards Eq. (28) be-low. The change amount in Eq. (12) should be divisi-ble simultaneously by K and K such that symmetry iskept. Meanwhile, the change amount in Eq. (13) shouldbe integral in order to be consistent with Dirac quantiza-tion condition (3). Therefore, q should be quantized as: q = k LCM( N K , N K , N ) with k ∈ Z . Then, withthese new quantized values, the shift operations (14,15)are changed to:12 π ˆ db −→ π db + 14 π N ˜ K LCM( N K , N K , N ) ˆ a ∧ da , (30)12 π ˆ db −→ π db − π N ˜ K LCM( N K , N K , N ) ˆ a ∧ da . (31)Again, the change amount in Eq. (30) should be divisi-ble simultaneously by K and K such that symmetry iskept. The change amount in Eq. (31) should be integralsuch that Dirac quantization condition (3) is satisfied.Before evaluating the integral, the Wilson loop of a maybe obtained via equation of motion of b : N K K π GCD( K , K ) ˆ M a ∈ Z , (32)where Eq. (7) and Bezout’s lemma are applied. The Wil-son loop of a may be obtained via equation of motionof b : N π ˆ M a ∈ Z . (33)With this preparation, we may calculate the changeamounts in Eqs. (30,31) and obtain the conditions on˜ K and ˜ K : LCM( N K , N K , N ) N N LCM( K , K ) ˜ K ∈ Z , (34)LCM( N K , N K , N ) N N LCM( K , K ) ˜ K ∈ Z . (35)Therefore, by using Bezout’s lemma, the minimal pe-riodicity of k can be fixed and k is thus compactified: k ∈ Z N N K ,K N K ,N K ,N .Following the similar procedure, we may obtain theresults for the remaining two cases: (i). a a da (labeledby ¯ q ) and both symmetry charges are in the first layer; TABLE I. SEG ( Z , Z K ). Both gauge group and symmetrygroup are in the same layer (the first layer). There is no non-trivial symmetry enrichment but a trivial stacking of symme-try and gauge theory.Symmetryassignment Z Z K Gauge Symmetry GT q/ π a a da ¯ q/ π a a da SEG K K (ii). a a da (labeled by q ) and both symmetry chargesare in the second layer. For (a), ¯ q is given by:¯ q = k LCM( N K , N K , N ) mod N N LCM( K , K ) , with k ∈ Z N N K ,K N K ,N K ,N . (36)For (b), q is given by: q = k LCM( N K , N K , N ) mod N N LCM( K , K ) , with k ∈ Z N N K ,K N K ,N K ,N . (37) III. TYPICAL EXAMPLES OFSYMMETRY-ENRICHED GAUGE THEORIES In this section, through a few concrete examples, weapply the general procedure shown in Sec. II C and con-struct SEG s that satisfy the definition and conditionslisted in Sec. I. Useful technical details are present inSec. II D,II E,II F and Appendix A. More examples arecollected in Appendix B. A. SEG ( Z , Z K ) We begin with G g = Z N and G s = Z K . The com-mon features of this class are that: (i) there is only onegauge theory before imposing global symmetry; (ii) thereare two complementary choices of symmetry assignment[82], namely, Z K is either in the first layer or in the sec-ond layer (trivial layer). More concretely, before impos-ing global symmetry, there is only one Z N gauge theorysince all additional topological terms like aada, aaaa van-ish identically. Despite that, we still formally explicitlyadd a a da and a a da in all tables in order to seewhether or not these topological terms will eventuallyhave chance to be nonvanishing after symmetry is takeninto consideration. Since we only have one cyclic sym-metry subgroup, i.e., G s = Z K , inclusion of two layers(the second one is a trivial layer in a sense that the levelof b da term is 1) is enough in the current simple cases.We choose N = K = 2 which was studied thoroughlyin Ref. [69] via a completely different approach. Theresults are collected in Tables I and II ( K = 2). In Ta-ble I, the symmetry charge is carried by the first layer. TABLE II. SEG ( Z , Z K ). Gauge group and symmetry groupare in different layers. K ∈ Z odd ( Z even ) for first (second)sub-table.Symmetryassignment ( K odd ) Gauge Symmetry Z Z K GT q/ π a a da ¯ q/ π a a da SEG K K ( K eve n ) Gauge Symmetry Z Z K GT q/ π a a da ¯ q/ π a a da SEG K KK mod 2 K K mod 2 K Before imposing symmetry, we find that both q and ¯ q are 0 mod 2, indicating that topological interactions be-tween layers are irrelevant. Mathematically, this conclu-sion can be achieved from Eqs. (19,20) by simply setting N = 2 , N = 1. Physically, it means that there is onlyone Z GT which is described by the BF term with level-2: i π ´ b ∧ da . After symmetry is imposed, both q and¯ q are 0 mod 4. This conclusion can be easily obtainedby setting N = 2 , K = 2 , N = K = 1 in Eq. (24).Physically, after imposing symmetry, for each topologi-cal interaction, there is still only one choice of the coeffi-cient but which is always connected to zero via a periodicshift. As a result, the total number of SEG s from this ta-ble is just one although the periodicity of both q and ¯ q is enhanced by symmetry.In Table II ( K = 2), the symmetry charge is carriedby the second layer that is a trivial layer. In this case,we find that there are 2 distinct choices for both q and¯ q : either 0 mod 4 or 2 mod 4. Quantitatively, this resultcan be obtained by simply setting N = 2 , N = 1 , K =1 , K = 2 in Eqs. (24,25). As a result, there are in to-tal 2 SEG s from this table. Among them, the SEG with q = ¯ q = 2 mod 4 can be simply regarded as stacking ofsymmetry enrichments from ( q, ¯ q )=(2 mod 4 , q, ¯ q )=(0 mod 4 , a a da and a a da topological interactions are presentin this SEG .In summary, there are 1 + 2 = 5 SEG s with G g = Z and G s = Z . One of them, labeled by (2 , 2) in TableII can be regarded as stacking of symmetry enrichmentpatterns of (0 , 2) and (2 , K in TablesI and II, there are in total five SEG s, just like K = 2 case.For odd K , there are two SEG s only. One is from TableI where symmetry group is in the same layer as gaugegroup. The other one is from Table II where gauge groupand symmetry group are in different layers. B. SEG ( Z × Z , Z ) The calculation in Sec. III A only involves one gaugegroup. Therefore, before imposing symmetry group,there is only one gauge theory, i.e., the untwistedone. In the following, we calculate SEGs with G g = Z × Z and G s = Z . Before imposing symmetry,there are already four topologically distinct GT s labeledby ( q, ¯ q ) = (0 mod 4 , , , , N = N = 2.Under this circumstances, Step-5 in Sec. II C cannot beskipped. All SEGs are listed in Table III, where threedifferent ways of symmetry assignment are considered.Taking the first symmetry assignment ( Z symmetryis assigned to the first layer, see the first subtable of Ta-ble III) as an example, there are two choices of q aftersymmetry is imposed: either 0 mod 8 or 4 mod 8. Thisresult can be easily obtained by setting N = 2 , K =2 , N = 2 , K = 1 in Eq. (24). Similarly, there arealso two choices of ¯ q . Therefore, totally there are 2 SEG s from the first subtable of Table III. However, onemay wonder what is the parent gauge theory ( GT ) foreach choice. This line of thinking is the goal of Step-5in Sec. II C. Interestingly, both choices of q mathemati-cally belong to the sequence “0 mod 4”. In other words,0 mod 8 and 4 mod 8, both of which belong to the se-quence 0 mod 4 and thus are indistinguishable beforeimposing symmetry, become distinguishable after sym-metry is imposed. This is nothing but a consequence ofsymmetry enrichment.Meanwhile, both choices do not match the sequence“2 mod 4” at all, which is indicated by the mark “N/A”in the table. Similar analysis can be applied to a a da .This phenomenon tells us that, Z × Z GT labeled by( q, ¯ q ) = (2 mod 4 , SEG descen-dants if symmetry is assigned to either the first layer (thefirst subtable of Table III) or the second layer (the sec-ond subtable of Table III) . Both layers are of type-I inFig. 2. One may wonder what will happen if we still en-force G s on this twisted GT in such kinds of symmetry as-signment. Can the gauge group and symmetry group becompatible with each other simultaneously? To answerthese questions, recalling the general procedure shown inSec. II C, there are several conditions (symmetry require-ment and gauge invariance) listed in Step-4 that deter-mine SEG ( G g , G s ). Therefore, if there is a SEG replacingthe mark “N/A”, it either breaks symmetry or preservessymmetry but violates gauge invariance principle. Thelatter case is an anomalous SEG and possibly realizableon the boundary of some (4+1)D system.In the third subtable of Table III, symmetry is assignedto the third layer, i.e., the type-II layer in Fig. 2. It isclear that there are 8 linearly independent topological in-teraction terms that can be applied [83]. In this symme-try assignment, each topological interaction term has twochoices of its coefficient: either 0 mod 4 or 2 mod 4 (for a a da and a a da , the result can be obtained from the TABLE III. SEG ( Z × Z , Z ). Three different ways of symmetry assignment are considered. Interestingly, all SEGs in firstand second ways of symmetry assignment come from the untwisted Z × Z GT only. “N/A” means that SEGs do not exist.Those states necessarily either break symmetry or violate gauge invariance principle. For the former, the ground states shouldbe discrete-symmetry-breaking phases. The latter may exist on the boundary of some (4+1)D systems.Symmetryassignment Gauge Symmetry Z Z Z GT q π a a da q π a a da SEG Symmetryassignment Gauge Symmetry Z Z Z GT q π a a da q π a a da SEG Symmetryassignment Gauge Symmetry Z Z Z GT a a da a a da a a da a a da a a da a a da a a da a a da SEG general calculation in Appendix A 1 and A 2). Therefore,totally, there are 2 SEG s. Interestingly, for those four SEG s with topological interactions a a da and a a da only, they can be simply regarded as stacking of a twisted Z × Z gauge theory and a direct product state with Z symmetry. C. SEG ( Z × Z , U(1)) In this part, we discuss the gauge theory Z × Z enriched by the continuous symmetry U(1). The re-sult can be obtained by following the general calculationin Appendix A 3, A 4, and A 5. Similar to the case of SEG ( Z × Z , Z ), we consider 3 ways to assign the sym-metry, as shown in Table IV. Considering the first sym-metry assignment (U(1) is assigned at the first layer), wefind that there is only one SEG ( Z × Z , U(1)) for bothinteraction terms with q = ¯ q = 0. In other words, this SEG is a descendant of the untwisted Z × Z GT with q = ¯ q = 0, while all other three twisted GT s do not have SEG descendants in this symmetry assignment . Similarly,for the second symmetry assignment, there is also onlyone SEG and it is also a descendant of the untwisted GT .However, there is one subtle feature that is absent fordiscrete symmetry group. q = ¯ q = 0 means that q and ¯ q are absolutly zero with no periodicity (or periodicity=0formally) after symmetry is imposed. We note that pe-riodicity is always nonzero in all previous examples withdiscrete symmetry group. It means that if we start withan untwisted GT but with q = 4, the resulting gauge the-ory after imposing U(1) symmetry either breaks symme-try or violates gauge invariance principle. For the lattercase, the theory can be regarded as an anomalous SEG which is possibly realizable on the boundary of certain(4+1)D systems.Now we consider the third symmetry assignment (thelast row in Table IV) which is much more complex.There are 8 linearly independent topological interactionterms of aada type [83]. We find that there are 2 SEG ( Z × Z , U(1)). Each coefficient of a a da , a a da and a a da has two choices while the coefficient of other aada interaction terms vanish identically after period-icity shift, which leads to 2 SEG s. For SEG s whereonly a a da a a da are considered (other topologicalinteraction terms vanish), they can be simply regardedas the stacking of a twisted Z × Z gauge theory anda direct product state with U(1) symmetry. For SEG swith at least a a da topological interaction term, theyare more interesting ones since they induce the nontriv-ial couplings between type-I layers and type-II layers asshown in Fig. 2. IV. PROBING SET ORDERS In Sec. III, we have constructed anomaly-free SEG s ina few concrete examples. In this section, we probe SET orders possessed by the ground states of SEG s. Then, themap from SEG s to SET s in Fig. 1 is achieved. In order toidentify SET order in a given SEG , one should know thetopological orders and symmetry-enriched properties.Given a gauge group G g , the total number of topologi-cal orders is generically smaller than that of GT s that areclassified by H ( G g , U(1)). Intuitively, the labelings ofgauge fluxes / gauge charges probably have redundancyfrom the aspect of topological orders. For example, if G g = Z × Z , there are four GT s. However, at least0 TABLE IV. SEG ( Z × Z , U(1)) When the symmetry is assigned at the first and second layer, there is only one SEG :( q, ¯ q )=(0,0).The notation N/A denotes that there is no SEG descendant for the specific symmetry assignment. 0 means that q or ¯ q exactlytakes zero.Symmetryassignment U(1) Gauge Symmetry Z Z GT q/ π a a da ¯ q/ π a a da SEG U(1) Gauge Symmetry Z Z GT q/ π a a da ¯ q/ π a a da SEG U(1) Gauge Symmetry Z Z GT a a da a a da a a da a a da a a da a a da a a da a a da SEG GT with q = 2 mod 4 and ¯ q = 0 mod 4 and GT with¯ q = 2 mod 4 and q = 0 mod 4 share the same topologi-cal order since both are just connected to each other viaexchanging superscripts 1 and 2.For the sake of simplicity, in this section, we will onlyconsider G g = Z N such that both GT and topologicalorder are unique. In these cases, we find that: (i) quasi-particles that carry unit gauge charge of the gauge group G g may carry fractionalized symmetry charge of the sym-metry group G s , which is classified by the second groupcohomology with G g coefficient: H ( G s , G g ); (ii) there isan interesting mixed version of three-loop braiding statis-tics among symmetry fluxes and gauge fluxes. Both fea-tures are gauge-invariant and topological, which can bedetected in experiments. A. SET orders in SEG ( Z , Z K ) with K ∈ Z even In this part, we probe SET orders with Z gauge groupand Z symmetry group in the five SEG s listed in Table Iand Table II. General even K is straightforward. Whenthe gauge group G g only includes one Z N subgroup, e.g., G g = Z , there is only one GT , i.e., the untwisted one.The topological order of the GT is dubbed “ Z N topo-logical order”, characterized by the charge-loop braidingstatistics data, i.e., the e i π/N phase accumulated by aunit gauge charge moving around a unit gauge flux. For N = 2, the phase is just e iπ = − 1. Due to this sim-plification, in order to characterize SET orders in thesefive SEG s, the only remaining task is to diagnose thesymmetry-enriched properties. From the following anal-ysis, we obtain five distinct SET orders with Z topolog-ical order and Z global symmetry. SEG SEG SEG SEG SEG G g = Z GT : Z gauge theory GT ⇤ : Z untwisted Z ⇥ Z twisted Z ⇥ Z twisted Z ⇥ Z twisted Z ⇥ Z Z ⇥ Z G g = Z GT : Z gauge theory SEG SEG GT ⇤ : Z (a) (b) G ⇤ g : Z G ⇤ g : Z SET SET SET SET SET SET FIG. 3. (Color online) Two concrete examples of webs ofgauge theories shown in Fig. 1. SEG ( Z , Z ) and SEG ( Z , Z )are shown in (a) and (b), respectively. SEG in (a) can befound in Table I. SEG , ··· , in (a) can be found in Table II. SEG in (b) can be found in Table I by setting K = 3. SEG in (b) can be found in the first subtable of Table II by setting K = 3. SEG ( Z , Z K ) with K ∈ Z even in Table I For the SEG in Table I, we may consider the followingaction in the presence of excitation terms ( K = 2 as anexample): S = i π ˆ b ∧ da + i π ˆ b ∧ dA + i ˆ b ∧ ∗ Σ + i ˆ a ∧ ∗ j , (38)where the 2-form tensor Σ represents the unit loop exci-tation current (world-sheet) of the Z gauge theory. The1-form vector j represents the unit gauge particle current1(world-line) of the Z gauge theory. Since only one layeris considered in this case, the superscripts of b , a areremoved. The background gauge field A is constrainedby Eq. (7) with K = 2. Next, integrating out b fieldleads to: π da = − ∗ Σ − π dA . Then, a can be formallysolved by adding ∗ d ∗ in both sides: a = − π ∗ d ˆ∆ Σ − A ,where the Laplacian operator ˆ∆ ≡ ∗ d ∗ d . Plugging thisexpression into the last term of Eq. (38), we obtain thefollowing effective action about excitations in the pres-ence of symmetry twist: − i ´ A ∧ ∗ j + iπ ´ j ∧ d − Σ .In this effective action, the second term characterizesthe Z topological order with charge-loop braiding phase e iπ = − 1. Mathematically, this is a Hopf term and rep-resents the long-range Aharonov-Bohm statistical inter-action between gauge fluxes (i.e., the loop excitations)and particles. The operator d − = d ˆ∆ is a formal no-tation defined as the operator inverse of d , whose exactform can be understood in momentum space by Fouriertransformations. The first term of this effective actionencodes the symmetry-enriched properties of the SEG .It indicates that the unit gauge charge carries 1 / G s = Z , which cor-responds to the second group cohomology classification H ( G s , G g ) = Z (see Appendix C for details). In summary , for the SEG given by Table I, the Z gauge charged bosons carry half quantized Z symmetrycharge. This is the first SET order we identify. SEG ( Z , Z K ) with K ∈ Z even in Table II For Table II, we first consider the q -topological inter-action term. The action in the presence of A is given by( K = 2 as an example): S = i π ˆ b ∧ da + i π ˆ b ∧ da + i π ˆ b ∧ dA + i q π ˆ a ∧ a ∧ da + i ˆ b ∧ ∗ Σ + i (cid:88) I ˆ a I ∧∗ j I , (39)where Σ and { j I } are loop excitation currents and par-ticle excitation currents of the I th layer respectively. Σ is not considered for the reason that the second layer istrivial and Σ carries 0 mod 2 π fluxes which are not de-tectable. One may first integrate out { b I } , which enforcesthat the path-integral configurations of { a I } are com-pletely fixed by excitations and the background gaugefield: a = − π ∗ d − Σ , a = − π ∗ d − σ . Here,the symbol d − has been defined in Sec. IV A 1. Thenew 2-form variable σ is defined through: σ = ∗ π dA which represents the number density / current of the π -symmetry twist induced by the background gauge field.Plugging the expressions of { a I } into { j I } -dependentterms in Eq. (39), we obtain the following effective actionterms: iπ ´ j ∧ d − Σ + i ´ j ∧ ∗ A , where the first Hopfterm indicates that the first layer has a Z topological or-der. The second term indicates that the quasiparticles in (a) (c) (d) (b) FIG. 4. (Color online) A mixed version of three-loop braid-ing process among gauge fluxes and symmetry fluxes in SEG ( Z , Z ) of Table II. Loops in red and black denote sym-metry flux loop ( σ ) and gauge flux loop (Σ ), respectively.The dashed curves shows the trajectory of one loop that movesaround another loop, both of which are linked to the baseloop. the second layer carry integer symmetry charge. In otherwords, there doesn’t exist symmetry fractionalization.Despite that, we will show that there is interest-ing mixed three-loop statistics among symmetry fluxes( σ ) and gauge fluxes (Σ ). For this purpose, pluggingthe expressions of { a I } into the q -dependent term inEq. (39), we obtain: − i qπ ´ ( ∗ d − Σ ) ∧ ( ∗ d − σ ) ∧ ( ∗ σ ) = − i ´ ( ∗ d − Σ ) ∧ πq ( ∗ d − σ ) ∧ ( ∗ σ ) which is the topologicalinvariant that characterizes the mixed three-loop statis-tics among symmetry fluxes and gauge fluxes and pro-vides important symmetry-enriched properties of SEG s.This mixed version of three-loop statistics enriches ourprevious understandings on three-loop statistics amonggauge fluxes [26–30]. Pictorially, the topological invariantcorresponds to the three-loop process shown in Fig. 4(a)where the gauge flux Σ is a base loop (a term coinedby Wang and Levin [26]). The entire process leads toBerry phase (denoted by θ σ,σ ;Σ ): θ σ,σ ;Σ = 2 × qπ = π , where q = 2 is used and the factor of 2 is due to the factthat the full braiding process accumulates two times ofhalf-braiding (exchange between σ and σ in the pres-ence of the base loop Σ ). If the base loop is pro-vided by σ instead, the topological invariant gives riseto the full braiding of another σ around a Σ as shownin Fig. 4(b), and the associated Berry phase is given by: θ σ, Σ ; σ = qπ = π mod π , where π phase ambiguity arisesfrom the possibility that Z gauge charge may be at-tached to σ such that there is π phase contribution fromthe Aharonov-Bohm phase from the topological invariant iπ ´ j ∧ d − Σ .Likewise, the ¯ q term can also be written in terms ofthe topological invariant: − i ¯ qπ ´ ( ∗ d − σ ) ∧ ( ∗ d − Σ ) ∧ ( ∗ Σ ). Pictorially, the topological invariant correspondsto the three-loop process shown in Fig. 4(c) where thesymmetry flux σ is a base loop. The entire process leadsto Berry phase (denoted by θ Σ , Σ ; σ ): θ Σ , Σ ; σ = 2 × ¯ qπ =2 π , where ¯ q = 2 is used for the SEG labeled by (0 , as the base loop, we mayobtain the Berry phase accumulated by fully braidingΣ around σ with the base loop provided by anotherΣ [see Fig. 4(d)]: θ σ, Σ ;Σ = ¯ qπ = π mod π , where π phase ambiguity arises from the possibility that Z gaugecharge may be attached to σ such that there is π phasecontribution from the Aharonov-Bohm phase from thetopological invariant iπ ´ j ∧ d − Σ . In summary , for the four SEG s given by Table II, theysupport four different SET orders. All point-particles areeither symmetry-neutral or carry integer Z symmetrycharge. In other words, symmetry is not fractionalizedand charge-loop braiding data is always trivial. However,they can be experimentally distinguished by the mixedthree-loop braiding process. In total, we obtain five dis-tinct SET orders with Z topological order and Z globalsymmetry. Likewise, for generic even K , there are alsofive SET orders. B. SET orders in SEG ( Z , Z K ) with K ∈ Z odd We consider K = 3 as an example. General odd K is straightforward. In this case, there are two distinct SEG s that are collected in Table I ( K = 3) and the firstsubtable of Table II ( K = 3) respectively. For the first SEG , the discussion is similar to that of K = 2 in Ta-ble I. We start with the action (38) and the backgroundgauge field A is now constrained by Eq. (7) with K = 3.Integrating out b, a leads to − i ´ A ∧ ∗ j + iπ ´ j ∧ d − Σwhere the first term indicates that the bosons (denotedby “ e ”) that carry unit Z gauge charge also carry 1 / Z group. However, there is no pro-jective representation (with Z coefficient) for Z symme-try group indicated by the trivial second group cohomol-ogy: H ( Z , Z ) = Z (see Appendix C), which meansthat this half-quantized symmetry charge cannot be de-tected by symmetry fluxes. The physical effect of thishalf-quantized symmetry charge is completely identicalto that of − A = 0 , π , π . The boson e that moves arounda symmetry flux with Φ A will pick up a Berry phase e i Φ A where 1 / e .However, during this process, it is possible that a gaugeflux (Φ g = 0 , π ) is dynamically excited and eventuallyattached to the symmetry flux. As a result, an addi-tional Berry phase is accumulated: e i Φ g , leading to theBerry phase e i Φ A +Φ g . After repeating the experimentsfor each Φ A sufficient times, the observer will eventuallycollect two data for each symmetry flux. If Φ A = 0, theBerry phase is either 0 or e iπ ; If Φ A = π , the Berryphase is either e i π or e i π ; If Φ A = π , the Berry phaseis either e i π or e i π . It is clear that these observed datacan be exactly obtained by considering the boson thatcarry unit gauge charge and − e − i Φ A + i Φ g .In other words, the half-quantized symmetry charge cannot be distinguished from − SEG in Table I ( K = 3), there is no symmetryfractionalization.For the second SEG (the first subtable of Table II with K = 3), since there doesn’t exist nontrivial topologicalinteractions between the two layers, this SEG is nothingbut a simple stacking of a Z gauge theory and a directproduct state with Z symmetry. By definition, it is stilla SEG but it doesn’t have interesting symmetry-enrichedproperties. In summary , both SEG s support the same SET or-der as shown schematically in Fig. 3(b). In this SET order, the topological order is Z -type. However, the Z symmetry always trivially acts on the topological orderdue to the absence of both symmetry fractionalizationand mixed three-loop braiding statistics. In other words,there is no interesting interplay beween Z topologicalorder and Z symmetry. Likewise, for generic odd K ,there is also only one SET order. V. PROMOTING SEG TO GT ∗ , BASISTRANSFORMATIONS, AND THE WEB OFGAUGE THEORIES In the above discussions, we obtained many SEG s,where the background gauge fields { A I } are treated asnon-dynamical fields. A caveat is that basis transforma-tions that mix { A I } and dynamical variables { a I } arestrictly prohibited. However, one may further give full dynamics to the background gauge fields { A I } , whichleads to the mapping from SEG s to GT ∗ as shown inFig. 1. In other words, the symmetry twist now becomesdynamical [75]. As a result, arbitrary basis transforma-tions now can be applied. It is legitimate to mix gaugefluxes and symmetry fluxes together to form a flux of anew gauge variable. A. SEG ( Z , Z ) Let us consider SEG ( Z , Z ) in Table I with K =2. The associated dynamical gauge theory of b, a, A, B (here, b = b , a = a for this single layer case) can bewritten as: S = 12 π ˆ (cid:0) B b (cid:1) (cid:18) (cid:19) ∧ d (cid:18) Aa (cid:19) , (40)where the two-form gauge field B is introduced to relaxthe holonomy of A to U(1)-valued in the path integralmeasure. According to Eq. (2), one can apply the follow-ing two unimodular matrices to send the above theory to3its canonical form: W = (cid:18) − − (cid:19) , Ω = (cid:18) (cid:19) , (41) W (cid:18) (cid:19) Ω T = (cid:18) (cid:19) (42)which directly indicates that the resulting new gauge the-ory GT ∗ after giving full dynamics to the backgroundgauge field is Z gauge theory (Fig. 3).Likewise, for Table II, the level matrix of the BF termis given by: (43)in the basis of ( b , b , B ) and ( a , a , A ). It can be diago-nalized by using the following two unimodular matrices: W = , Ω = − , (44) W Ω T = . (45)As a result, the new 1-form gauge variables are given bythe vector (˜ a , ˜ a , ˜ A ) T where, a = ˜ a , a = ˜ a − ˜ A , A = ˜ A . (46)From the canonical form (45), it is clear that the re-sulting theory after giving full dynamics to the back-ground gauge field is Z × Z gauge theory. But weshould also examine how topological interaction termstransform. Since the second layer in the new basis is atrivial layer (level-1), we may neglect all topological in-teraction terms that include ˜ a . Keeping this in mind,After the basis transformations, the topological interac-tion terms ´ iq π a ∧ a ∧ da + ´ i ¯ q π a ∧ a ∧ da aretransformed to: ˆ iq π ˜ a ∧ ˜ A ∧ d ˜ A − ˆ i ¯ q π ˜ A ∧ ˜ a ∧ d ˜ a . (47)Therefore, we reach the following conclusions. The re-sulting theory starting from SEG labeled by (0 , 0) in Ta-ble II is “untwisted” Z × Z gauge theory. The remaining SEG s lead to twisted Z × Z gauge theory after givingdynamics to the background gauge field (Fig. 3), whichis also derived in [69] from a different point of view. B. SEG ( Z , Z ) For SEG s in Table I, GT ∗ is always Z K gauge theorywhich are “untwisted”. For SEG s in Table II, for even K , GT ∗ s are Z × Z K gauge theories which have one SEG SEG SEG SEG G g = Z ⇥ Z GT : untwisted Z ⇥ Z twisted Z ⇥ Z twisted Z ⇥ Z twisted Z ⇥ Z Z ⇥ Z Z ⇥ Z ⇥ Z GT ⇤ : …… …… ………… …… …… GT ⇤ GT ⇤ GT ⇤ ……………… …… …… SEG : G ⇤ g : SET SET SET …… SET : …… …… …… FIG. 5. A skeleton of the web of gauge theories for SEG ( Z × Z , Z ). untwisted version and three twisted versions, in a similarmanner to K = 2 discussed above. But for odd K , theresulting theory GT ∗ is still Z K gauge theory since thetwo groups are isomorphic: Z × Z K ∼ = Z K when K ∈ Z odd . For example, for K = 3: (cid:18) − − (cid:19) (cid:18) (cid:19) (cid:18) − − (cid:19) = (cid:18) (cid:19) . (48)Therefore, for odd K , the resulting gauge theory is thesame as that in Table I. In other words, after giving fulldynamics to the background gauge field A , there is onlyone output: a Z K gauge theory (Fig. 3). From this sim-ple case, we see there is an interesting pattern of many-to-one correspondence between SEG s and GT ∗ s. C. SEG ( Z × Z , Z ) For SEG ( Z × Z , Z ), all SEG s are collected in Ta-ble III. Before imposing symmetry, there are already fourdistinct gauge theories. Therefore, the resulting web ofgauge theories is much more complex. A rough skeletonis shown in Fig. 5 where the resulting GT ∗ theories canbe regrouped into two gauge groups G ∗ g = Z × Z and G ∗ g = Z × Z × Z . The first gauge group arises from thefirst and second subtables of Table III while the secondgauge group arises from the third subtable of Table III.More concretely, let us consider the BF term of the firstsubtable after the background gauge field becomes fullydynamical:12 π ˆ (cid:0) B b b (cid:1) ∧ d Aa a , (49)where the two-form gauge field B is introduced to relaxthe holonomy of A to U(1)-valued in the path integral4measure. According to Eq. (2), one can apply the follow-ing two unimodular matrices to send the above theory toits canonical form: W = − − , Ω = , (50) W Ω T = (51)which indicates that G ∗ g = Z × Z . Likewise, we havethe following matrix calculation for the second subtable:12 π ˆ (cid:0) B b b (cid:1) ∧ d Aa a , (52)and W = , Ω = − − , (53) W Ω T = (54)which still leads to G ∗ g = Z × Z .For the third subtable, the BF term is given by:12 π ˆ (cid:0) B b b b (cid:1) ∧ d Aa a a , (55)where the 4 × W = − 10 1 0 00 0 1 01 0 0 0 , Ω = − 10 1 0 00 0 1 01 0 0 − , (56) W Ω T = . (57)As a result, G ∗ g = Z × Z × Z . VI. SUMMARY AND OUTLOOK In this paper, we have studied the symmetry enrich-ment through topological quantum field theory descrip-tion of three-dimensional topological phases. All phasesconstructed in this paper can be viewed as 3D gappedquantum spin liquid candidates enriched by unbrokenspin symmetry G s . Using the 5-step general procedurein Sec. II C, we have efficiently constructed symmetry-enriched gauge theories ( SEG ) with gauge group G g = Z N × Z N × · · · and symmetry group G s = Z K × Z K × · · · as well as G s = U(1) × Z K × · · · . The re-lation between SEG and its parent gauge theory GT hasbeen shown. We have also shown how to physically di-agnose the ground state properties of SEG s by investi-gating charge-loop braidings (patterns of symmetry frac-tionalization) and mixed multi-loop braiding statistics.By means of these physical detections, one can obtain aset of SET orders which represent the phase structures ofground states of SEG s. It is generally possible that two SEG s may give rise to the same SET order. Finally, byproviding full dynamics to the background gauge fields[75], the resulting new gauge theories GT ∗ s can be ob-tained and have been studied, all of which are summa-rized in a web of gauge theories (Fig. 1). Throughoutthe paper, many concrete examples have been studiedin details. From those examples, we have seen that thegeneral procedure provided in this paper is doable andefficient for the practical purpose of understanding 3D SET physics.We highlight some questions for future studies. (i) Lattice models of SEG s. Dijkgraaf-Witten models [73]and string-net models [84] have been well studied. Itis interesting to impose global symmetry (e.g., on-sitefinite unitary group) on these models in 3D. Then, lat-tice models can be regarded as an ultra-violet definitionof SEG s. Some progress on 2D SET s has been made inRef. [44, 45]. (ii) Material search and the experimen-tal fingerprint of the mixed three-loop braiding statis-tics. There are several possible experimental realizationsof Z spin liquids, such as the so-called Kitaev spin liq-uid state in the lattices in β - and γ -Li IrO [85–91]. Byfurther considering the unbroken Z Ising symmetry, theresulting ground state should exhibit SET orders. As westudied in the paper, the features of these SET s are pat-terns of symmetry fractionalization and mixed three-loopbraiding statistics. It is thus of interest to theoreticallypropose an experimental fingerprint, especially, for thethree-loop braiding statistics. (iii) Anomalous SEG s. Inour construction, by anomaly, we mean that global sym-metry and gauge invariance cannot be compatible witheach other. If both are preserved, the resulting SEG isanomaly-free as what we have calculated. As mentionedin Sec. III B, the entries with “N/A” in Table III meansthat there do not exist SEG descendants for the twistedgauge theory (with both nonzero q and ¯ q ) in the symme-try assignment (the first and second subtables) such thatboth global symmetry and gauge invariance are preservedsimultaneously. In other words, either symmetry is bro-ken or gauge invariance is violated. For the case in whichsymmetry is preserved but gauge invariance is violated,we conjecture it can be realized on the boundary of cer-tain (4+1)D systems. More careful studies in the futurealong anomaly will be meaningful. (iv) GT ∗ s originatedfrom SEG s with U(1) symmetry. In Sec. III C and Ap-pendix, some examples of SEG s with U(1) symmetry arestudied. After U(1) symmetry group becomes a dynam-ical gauge group, the resulting theory GT ∗ should admit5a mixed phenomenon generated by mixture of discretegauge group and U(1) gauge group. It will be interestingto study the properties of such a type of gauge theoryand eventually build the web (i.e., Fig. 1) of gauge the-ories for these cases. (v) SEG s with Charles symmetry[70]. Charles symmetry, which was introduced in [70], isa 3D analog of 2D anyonic (topological) symmetry. Asimple example is Z gauge theory where quasiparticle ispermuted to its antiparticle while quasi-loop is permutedto its antiloop. And there is one species of defect-charge-loop composites. This is just one gauge theory by givinga gauge group and a Charles symmetry group. It willbe interesting to investigate the possibility that there aremore than one gauge theories enriched by Charles . 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Most of derivations are similar to the previouscases except some subtle differences in the shift operations. The gauge theory before imposing the global symmetryis given by: S = (cid:88) I =1 iN I π ˆ b I ∧ da I + i ¯¯ q π ˆ a ∧ a ∧ da . (A1)The action is invariant under the following gauge transformations parametrized by scalars { χ I } and vectors { V I } : a I −→ a I + dχ I , (A2) b I −→ b I + dV I − ¯¯ q πN I (cid:15) IJ χ J ∧ da . (A3)Let us investigate the integral π ´ M db I . Under the above modified gauge transformations (A3), the integral will bechanged by the amount below (for I = 1, M = M × M is considered):12 π ˆ M db −→ π ˆ M db − ¯¯ q π N ˆ S dχ ˆ M da = 12 π ˆ M db − ¯¯ q π N × π(cid:96) × π(cid:96) (cid:48) , (A4)where (cid:96) , (cid:96) (cid:48) ∈ Z , and, the Dirac quantization condition (4) and homotopy mapping condition (11) are applied. Inorder to be consistent with the Dirac quantization condition (3), the change amount must be integral, namely, ¯¯ q mustbe divisible by N . Similarly, ¯¯ q is also divisible by N due to:12 π ˆ M db −→ π ˆ M db + ¯¯ q π N ˆ S dχ ˆ M da = 12 π ˆ M db + ¯¯ q π N × π(cid:96) (cid:48)(cid:48) × π(cid:96) (cid:48)(cid:48)(cid:48) , (A5)where (cid:96) (cid:48)(cid:48) , (cid:96) (cid:48)(cid:48)(cid:48) ∈ Z . Hence, ¯¯ q = kN N N , k ∈ Z . Below, we want to show that k has a periodicity N (i.e., GCD of N , N , N ) and thereby ¯¯ q is compactified: ¯¯ q ∼ ¯¯ q + N N N N . Let us consider the following redundancy due to shiftoperations: 12 π ˆ db −→ π ˆ db + N ˜ K π N ˆ a ∧ da , (A6)12 π ˆ db −→ π ˆ db − N ˜ K π N ˆ a ∧ da , (A7)12 π ˆ db −→ π ˆ db + N N ˜ K π N N ˆ ( da ∧ a + a ∧ da ) , (A8) k −→ k + ˜ K + ˜ K + ˜ K . (A9)Again, in order to be consistent with Dirac quantization (3), the change amount of the integral π ´ M db I should beintegral, namely: N ˜ K π N ˆ M a ∧ da ∈ Z , (A10) N ˜ K π N ˆ M a ∧ da ∈ Z , (A11) N N ˜ K π N N ˆ M ( da ∧ a + a ∧ da ) ∈ Z . (A12)We may apply the Dirac quantization condition (4) and the quantized Wilson loop N I π ´ M a I ∈ Z that is obtainedvia equations of motion of b I . As a result, three constraints are achieved: ˜ K /N ∈ Z , ˜ K /N ∈ Z , ˜ K /N ∈ Z . In8deriving the result for ˜ K , Bezout’s lemma is applied. By using Bezout’s lemma again, the minimal periodicity of k is given by GCD of N and N , which is N . As a result, we obtain the conditions on ¯¯ q if symmetry is not takeninto consideration. ¯¯ q = k N N N mod N N N N , k ∈ Z N . (A13) G g = Z N × Z N × Z N with G s = Z K × Z K × Z K To impose the symmetry, we add the following coupling term in the action (A1): (cid:88) i i π ˆ A i ∧ db i . (A14)The change amounts of the integral π ´ M db I in Eqs. (A4, A5, A6, A7, A8) should not only be integral [in orderto be consistent with the Dirac quantization condition (3)] but also be multiple of K i such that the coupling term(A14) is gauge invariant modular 2 π . More quantitatively, with symmetry taken into account, from Eqs. (A4, A5),we may obtain the quantization of ¯¯ q : ¯¯ q = kN N K K GCD( N K ,N K ) with k ∈ Z such that the change amounts are multiple of K i . Then, with these new quantized values, the shift operations (A6, A7, A8) are changed to:12 π ˆ db −→ π ˆ db + ˜ K N K K π GCD( N K , N K ) ˆ a ∧ da , (A15)12 π ˆ db −→ π ˆ db − ˜ K N K K π GCD( N K , N K ) ˆ a ∧ da , (A16)12 π ˆ db −→ π ˆ db + ˜ K N N K K π N GCD( N K , N K ) · ˆ ( da ∧ a + a ∧ da ) . (A17)After the integration over M , the change amounts should be quantized at K in Eq. (A15), K in Eq. (A16), and K in Eq. (A17). We may apply the Dirac quantization condition (4) and the quantized Wilson loop N I K I π ´ M a I ∈ Z that is obtained via equations of motion of b I in the presence of A I background. As a result, three necessaryand sufficient constraints are achieved: ˜ K GCD( N K ,N K ) ∈ Z , ˜ K GCD( N K ,N K ) ∈ Z , ˜ K N K ∈ Z . By using Bezout’slemma, the minimal periodicity of k is given by GCD of GCD( N K , N K ), GCD( N K , N K ), and N K , whichis GCD( N K , N K , N K ). As a result, once symmetry is imposed, ¯¯ q is changed from Eq. (A13) to:¯¯ q = k N N K K GCD( N K , N K ) mod N N K K GCD( N K , N K , N K )GCD( N K , N K ) , with k ∈ Z GCD( N K ,N K ,N K ) (A18)which gives GCD( N K , N K , N K ) SEG s. Since GCD( N K , N K , N K ) ≥ GCD( N , N , N ), the allowedvalues of ¯¯ q are enriched by symmetry. G g = Z N × Z N × Z N with G s = Z K × Z K × U(1) In this part, we consider U (1) symmetry. We consider the following symmetry assignment and add it in the action(A1): i π (cid:88) i ˆ A i ∧ db i + A U (1) ∧ db . (A19)where the U ( ) Wilson loop ˆ S A U (1) ∈ R (A20)meaning that the U ( ) Wilson loop can be any real value. Under the gauge transformation (A3), the change amountsof the integral π ´ M db I in Eqs. (A4, A5) should be multiple of K or K such that the coupling terms (A19) is9gauge invariant modular 2 π . More quantitatively, with symmetry taken into account, from Eqs. (A4, A5), we mayobtain the quantization of ¯¯ q : ¯¯ q = kN N K K GCD( N K ,N K ) with k ∈ Z such that the change amounts are multiple of K i . Toremove the redundancy in the possible value of ¯¯ q , we do the shift operations as that from (A15) to (A17). Similarlyto the case above, after the integration over M , the change amounts should be quantized at K in Eq. (A15), K in Eq. (A16), and zero in Eq. (A17) due to the fact that the U(1) Wilson loop can be any real value. As a result,three necessary and sufficient constraints are achieved: ˜ K GCD( N K ,N K ) ∈ Z , ˜ K GCD( N K ,N K ) ∈ Z , ˜ K = 0. By usingBezout’s lemma, the minimal period of k is GCD ( N K , N K ), i.e.¯¯ q = k N N K K GCD( N K , N K ) mod N N K K , with k ∈ Z GCD( N K ,N K ) (A21)which gives GCD( N K , N K ) SEG s. G g = Z N × Z N with G s = Z K × U(1) -(I) Here we consider the symmetry assignment and add it in the action (8): i π ˆ A K ∧ db + A U (1) ∧ db (A22)which indicates that the first layer carries the discrete symmetry Z K while the second layer carries U(1). To determinethe possible values of q in the presence of this global symmetry, We observe that the change amounts of the integral π ´ M db in Eq. (12) should be multiple of K such that the first coupling term in Eq. (A22) is gauge invariantmodular 2 π . But the key observation is that the U(1) Wilson loop (A20) is any real value, therefore, to keep thesecond coupling term in Eq. (A22) gauge invariant, the change amount π ´ M db in Eq. (13) should be strictly zero,which would be only the case that q = 0. Similarly, ¯ q = 0. Therefore, SEG only happens when q = ¯ q = 0. G g = Z N × Z N with G s = Z K × U(1) -(II) In this part, we consider the whole symmetry group G s at the same layer and add the following part in the action(8) where we first set ¯ q = 0: i π ˆ A K ∧ db + A U (1) ∧ db (A23)Similar to the case that the symmetry subgroup are assigned at different layers, in order to keep to the second termin (A23) gauge invariant, the change amount of the integral π db should be strictly zero. Therefore, q = 0. For thesimilar reason, ¯ q = 0. This symmetry assignment also only happens when q = ¯ q = 0. Appendix B: Several examples1. SEG ( Z × Z , Z ) In the main text, we illustrate the example of Z × Z gauge with Z symmetry. Here, we calculate anotherexample: G g = Z × Z with G s = Z . Before imposing symmetry, there are 4 gauge theories in total, denotedby ( q, ¯ q ):(0,0),(0,4),(4,0) and (4,4). In the first subtable of Table S5, the symmetry Z is assigned at the first layerwhere the Z gauge subgroup lives. From this table, it is clear that both q and ¯ q have four choices, resulting in 4 SEG s. Among these four choices of, say, q , we may further regroup them into two groups: { , } and { , 12 mod 16 } . The two choices in the former group are SEG descendants of GT with q = 0 mod 8 beforeimposing symmetry. The two choices in the latter group are SEG descendants of GT with q = 4 mod 8 before imposingsymmetry. In this sense, this table is sharply different from the first subtable of Table III where some entries aremarked by “N/A”.In the second subtable of Table S5, the symmetry is assigned at the second layer where the Z gauge subgrouplives. The results are similar to the second table of Table III, where some entries are marked by “N/A”. Totally, thereare 2 SEG s.0 TABLE S5. SEG ( Z × Z , Z ).Symmetryassignment Gauge Symmetry Z Z Z q/ π a a da ¯ q/ π a a da GT 0 mod 8 4 mod 8 0 mod 8 4 mod 8 SEG Gauge Symmetry Z Z Z q/ π a a da ¯ q/ π a a da GT 0 mod 8 4 mod 8 0 mod 8 4 mod 8 SEG Gauge Symmetry Z Z Z GT a a da a a da a a da a a da a a da a a da a a da a a da SEG In the third subtable of Table S5, the symmetry is assigned at the third layer where there is no gauge group.This symmetry assignment induces some new nonvanishing topological interactions involving the third layer. Thereare in total 8 kinds of topological interactions [83]. Each topological interaction contains two choices of coefficients,rendering 2 SEG s. SEG ( Z , Z × Z ) In this part, we consider SEG s whose symmetry group contains more than one cyclic subgroup. In this case, a lotof new ways of symmetry assignment exist. Specifically, we consider a relatively simple example: Z gauge theorywith Z × Z symmetry. In order to differentiate the two subgroups from each other, we introduce superscripts: G s = Z a × Z b .In Table S6, the two symmetry subgroups are assigned to the first and second layer, respectively. Before imposingsymmetry, the coefficients q, ¯ q can only take value 0 mod 2, so all topological interaction terms identically vanish.This is exactly the fact that there is only one Z gauge theory. After imposing symmetry, however, the periods ofboth q, ¯ q are enlarged from 2 to 8. Within one period, they can take either 0 or 4, resulting in 2 different SEG s.Another 2 SEG s can be obtained by simply exchanging the subscripts a, b . TABLE S6. SEG ( Z , Z × Z ). The superscripts a and b are added to distinguish the two Z subgroups. The two symmetrysubgroups are carried by the two layers respectively. There are two independent ways of symmetry assignment obtained byexchanging the auxiliary superscripts a ←→ b .Symmetryassignment Z b Z a Gauge Symmetry Z GT q/ π a a da ¯ q/ π a a da SEG In Table S7, we assign the two symmetry subgroups at the second and third layer, both of which are trivial layers.In this case, as there are three layers, we need to consider 8 different topological interactions as collected in the table.As explained also in the main text, there are only two linearly independent three-layer topological interaction termssince a a da is a a da + a a da up to a total derivative. Again, before imposing symmetry, coefficients of anykinds of topological terms identically vanish. After symmetry is considered, it turns out that these 8 topologicalinteractions generate 2 different SEG s. In addition, in Table S8, two ways to assign the two symmetry subgroups in1 TABLE S7. SEG ( Z , Z × Z ) The superscripts a and b are added to distinguish the two Z subgroups. The gauge group iscarried by the first layer, while the two symmetry subgroups by the second and third layers respectively.Symmetryassignment Z b Z a Gauge Symmetry Z GT a a da a a da a a da a a da a a da a a da a a da a a da SEG TABLE S8. SEG ( Z , Z × Z ) G s = Z × Z is carried entirely by either the first layer (the first subtable) or the second layer(the second subtable).Symmetryassignment Gauge Symmetry Z Z Z GT q/ π a a da ¯ q/ π a a da SEG Gauge Symmetry Z Z Z GT q/ π a a da ¯ q/ π a a da SEG the same layer are considered. In the first subtable, there is only one SEG . But in the second subtable, the calculationshows that there are 2 SEG s. SEG ( Z N , U(1)) To impose the U(1) symmetry to the Z N gauge theory, there are two ways, i.e. two symmetry assignments. Thefirst one is to assign the symmetry at the same layer as that where Z N gauge lives. For this symmetry assignment, itis equivalent to SET N = N , N = 1, K = 1 in the Appendix A 4, so there is only one SEG ( Z N , U(1)). The other wayis to assign it at another layer whose BF term is level-one, which is equivalent to SET N = N , N = 1, K = 1 in theAppendix A 5, so there is also only one SEG ( Z N , U(1)). SEG ( Z N , Z K × U(1)) For the Z N gauge enriched by Z K × U(1) symmetry, there are five symmetry assignments in Table S9. Four of themonly involve two layers which all have only one SEG . The fifth symmetry assignment gives rise to [GCD( N, K )] . Aswe would see below, two roots of [GCD( N, K )] come from the stacking of SEG ( Z N , Z K ) and a direct product statewith U(1) symmetry (n.b., U(1) SPT in 3D is always trivial). The third root comes from the nontrivial interaction a a da which correlates all layers together. Note that since the layers where the symmetry are assigned are level-one,exchanging the Z K and U(1) symmetry does not lead to anything new. TABLE S9. The five symmetry assignments of Z N Gauge with Z K × U(1) symmetry and the number of corresponding SEG .I II III IV V Z K Z N U(1) Gauge Symmetry Z K Z N U(1) Gauge Symmetry Z K Z N U(1) Gauge Symmetry Z K Z N U(1) Gauge Symmetry Z K Z N U(1) Gauge Symmetry SymmetryAssignment SEG N, K )] aada type topological interaction terms. To count the total number of SEG s in thissymmetry assignment, we have to determine the period of of the coefficients of these eight topological interactionterms. Below we consider each of them separately because each alone can determine a set of root SEG .1. For topological interaction a a da or a a da , the theory reduces to that of stacking SEG ( Z N , Z K ) and U(1) SPT in three dimensions. From the calculation in Appendix II E by setting N = N, N = 1 , K = 1 , K = K ,there are GCD( N, K ) different root SEG ( Z N , Z K )s from a a da and another GCD( N, K ) root SEG ( Z N , Z K )sfrom a a da . From Ref. [13] there is only one U(1) SPT in three dimensions. Therefore, there areGCD( N, K ) different SEG ( Z N , Z K × U(1))s from the topological interaction a a da and another GCD( N, K )root SEG ( Z N , Z K × U(1))s from a a da .2. For topological interaction a a da or a a da , the SEG ( Z N , Z K × U(1)) reduces to the stacking of SEG ( Z N , U(1))and Z K SPT in three dimensions. From the result in Appendix B 3 (when Z N and U(1) are not in the samelayer), we know that there is only one SEG ( Z N , U(1)) and from Ref. [13], there is only one Z K SPT. Therefore,there is only one root SEG ( Z N , Z K × U(1)) from a a da and also only one from a a da .3. For topological interaction a a da or a a da , the SEG ( Z N , Z K × U(1)) reduces to the stacking of Z N gaugetheory and Z K × U(1) SPT in three dimension. It is known that there is only one Z N gauge theory and fromRef. [13], there is only one Z K × U(1) SPT. Therefore, there is only one root SEG ( Z N , Z K × U(1)) from a a da and also only one from a a da .4. For topological interaction a a da , the symmetry assignment V is equivalent to SET N = N , N = N = 1 and K = 1 , K = K in Appendix A 3. Therefore, there are GCD( N, K ) SEG ( Z N , Z K × U(1))s in total. For anotherthree-layer topological interaction a a da , it is equivalent to exchange the layer index as 1 ←→ 3, 2 ←→ ←→ a a da , we find that the q = 0, andso there is only one SEG ( Z N , Z K × U(1)).In summary, for symmetry assignment V, each of a a d , a a da , a a da , a a da and a a da contributes only oneroot SEG ( Z N , Z K × U(1)) and each of a a da , a a da and a a da contributes GCD( N, K ) root SEG ( Z N , Z K × U(1)),so in total there are [GCD( N, K )] SEG ( Z N , Z K × U(1))s for the symmetry assignment in Table S9. SEG ( Z N × Z N , U(1)) Without U(1) symmetry, there are in total ( N ) Z N × Z N gauge theories, where N is the greatest commondivisor of N and N . With the U(1) symmetry, there are three symmetry assignments, as shown in Table S10. Forthe assignment I and II, it is equivalent to SET K = 1 in Appendix A 4. so there is only one SEG whose parent gaugetheory is untwisted Z N × Z N gauge theory. TABLE S10. The symmetry assignments of Z N × Z N gauge with U(1) symmetry and the numbers of corresponding gaugetheory and symmetry enriched gauge theory. I II III Z N Z N U(1) Gauge Symmetry Z N Z N U(1) Gauge Symmetry U(1) Z N Z N Gauge Symmetry Symmetryassignment SEG N ) For the assignment III, the number of SEG ( Z N × Z N , U(1)) is ( N ) compared to the ( N ) gauge theories. Fortopological interaction a a da or a a da , the root SEG ( Z N × Z N , U(1)) are just stacking the Z N × Z N root gaugetheories and U(1) SPT in three dimension. We know that there are ( N ) Z N × Z N gauge theories and only oneU(1) SPT in three dimensions. Therefore there are N root SEG ( Z N × Z N , U(1)) from a a da and another N root SEG ( Z N × Z N , U(1)) from a a da .For the choice of interaction a a da , a a da , a a da or a a da , there is only one SEG ( Z N × Z N , U(1)) for allcases.For topological interaction a a da , it is equivalent to SET N = K = K = 1 in Appendix A 3, so there are N SEG ( Z N × Z N , U(1))s. But for a a da , it is equivalent to exchange the layer index as 1 ←→ 3, 2 ←→ 1, 3 ←→ a a da , we find that q = 0 and so there is only one SEG ( Z N , Z K × U(1)).In summary, for symmetry assignment III, each of a a d , a a da , a a da , a a da and a a da contributes onlyone root SEG ( Z N × Z N , U(1)) and each of a a da , a a da and a a da contributes N root SEG ( Z N × Z N , U(1)),so in total there are ( N ) SEG ( Z N × Z N , U(1)) for the symmetry assignment III in Table S10. Appendix C: Calculation of H ( Z , Z ) = Z and H ( Z , Z ) = Z In this Appendix, we calculate the second group cohomology H ( G s , G g ) which describes topologically distinct pat-terns of G s symmetry fractionalization in the charge of G g (which is abelian) gauge field. Mathematically, H ( G s , G g )is a set of equivalent classes of 2-cocycles ω ( g , g ), where g , g ∈ G s and ω ( g , g ) are G g valued. The 2-cocyclesare solutions of the 2-cocycle equations: dω ( g , g , g ) = ω ( g , g ) ω − ( g g , g ) ω ( g , g g ) ω − ( g , g )= 1 . (C1)If G g = Z , then ω ( g , g ) takes value ± 1. Two 2-cocycles ω (cid:48) ( g , g ) and ω ( g , g ) are equivalent if they differ bya 2-coboundary ω (cid:48) ( g , g ) = ω ( g , g )Ω ( g , g ), withΩ ( g , g ) = Ω ( g )Ω ( g )Ω ( g g ) , (C2)where Ω ( g ) are G g variables. A 2-cocycle is said to be trivial if it is equivalent to ω ( g , g ) = 1 for all g , g ∈ G s .In the following we adopting the canonical gauge choice[1] such that ω ( E, g ) = ω ( g, E ) ≡ 1. To ensure that this isstill the case after a gauge transformation, namely, to ensure ω (cid:48) ( g, E ) = ω ( g, E )Ω ( g, E ) = 1 still holds, Ω ( E ) ≡ H ( Z , Z ) and H ( Z , Z ) using above definition. Cohomology H ( Z , Z ). If G s = Z = { E, Q } , then there is only one 2-cocycle equation, dω ( Q, Q, Q ) = ω ( Q, Q ) ω − ( E, Q ) ω ( Q, E ) ω − ( Q, Q ) = 1 . Since ω ( E, Q ) = ω ( Q, E ) = 1, above equation gives no constraint for the variable ω ( Q, Q ). Since G g = Z , ω ( Q, Q )is a free Z variable and can freely take values ± 1. On the other hand, the 2-coboundaryΩ ( Q, Q ) = Ω ( Q )Ω ( Q )Ω ( E ) = 1is trivial, so there is no gauge degrees of freedom under the canonical gauge condition. This means that ω ( Q, Q ) = 1and ω ( Q, Q ) = − H ( Z , Z ) = Z . Cohomology H ( Z , Z ). If G s = Z = { E, P, P } , substituting g , g , g by P, P , we obtain eight equations,two of which are independent. The first two equations are ω ( P, P ) ω − ( P , P ) ω ( P, P ) ω − ( P, P ) = 1 ,ω ( P, P ) ω − ( P , P ) ω ( P, E ) ω − ( P, P ) = 1 . We obtain, ω ( P, P ) = ω ( P , P ) ,ω ( P, P ) ω ( P , P ) = ω ( P, P ) . If we let ω ( P, P ) = σ, ω ( P , P ) = η , where σ, η are G g = Z variables, then ω ( P, P ) = ση .On the other hand, from equation (C2), we obtain,Ω ( P, P ) = Ω ( P )Ω ( P )Ω ( P ) = Ω ( P ) , Ω ( P, P ) = Ω ( P , P ) = Ω ( P )Ω ( P ) , Ω ( P , P ) = Ω ( P )Ω ( P )Ω ( P ) = Ω ( P ) . ( P ) = σ , Ω ( P ) = η , then we obtain a new 2-cocyle ω (cid:48) ( g , g ) = ω ( g , g )Ω ( g , g ) = 1for all g , g ∈ Z . Thus we have shown that these 2-cocyles are trivial, namely, H ( Z , Z ) = Z1