Jhep01(2021)173

JHEP01(2021)173 Springer October 9, 2020 January 27, 2021 : December 3, 2020 : : Received Accepted Published , a,d Published for SISSA by https://doi.org/10.1007/JHEP01(2021)173 and Ryo Yokokura d [email protected] , Muneto Nitta . 3 a,b,c 2009.14368 The Authors. Eﬀective Field Theories, Global Symmetries, Topological Field Theories, c

We investigate a higher-group structure of massless axion electrodynamics in , [email protected] dimensions. By using the background gauging method, we show that the higher- Keio University, Hiyoshi 4-1-1, Yokohama,E-mail: Kanagawa 223-8521, Japan [email protected] Tsukuba 305-0801, Japan Graduate University for Advanced StudiesTsukuba 305-0801, (Sokendai), Japan RIKEN iTHEMS, RIKEN, Wako 351-0198, Japan Department of Physics & Research and Education Center for Natural Sciences, KEK Theory Center, b c d a Open Access Article funded by SCOAP Keywords: Topological States of Matter ArXiv ePrint: (3 + 1) form symmetries necessarily have aand global semistrict exhibit 3-group (2-crossed ’t module) HooftHooft structure, anomalies anomaly of between 0-form thegroup structure. and 3-group. 1-form In symmetries, particular, which we is find specific a to cubic the mixed higher- ’t Abstract: Yoshimasa Hidaka, Global 3-group symmetry andaxion ’t electrodynamics Hooft anomalies in JHEP01(2021)173 16 27 28 10 10 27 4 17 2-form symmetry 22 11 26 16 8 9 U(1) 7 12 6 – i – 8 8 5 1-form symmetry 23 19 25 26 N Z 5 13 21 1-form symmetry 32 2-form symmetry 33 1-form symmetry and 30 1-form symmetry 29 25 N 29 U(1) N Z U(1) Z 0-form symmetry 22 2-form symmetry 0-form symmetry N 1 Z N Z U(1) 4 0-form transformation 1-form transformation N N B.3.1 Correlation functionB.3.2 of 0-form and Correlation 1-form functionB.3.3 symmetry of generators 1-form symmetry Correlation generators function of 0-form and 2-form symmetry generators Z Z 3.2.1 Gauging3.2.2 electric Gauging 3.1.1 Constraint3.1.2 on background gauge field Background gauging as insertion of symmetry generators 2.2.1 2.2.2 Electric 2.2.3 Magnetic 2.2.4 C.1 Axiom ofC.2 3-group Example ofC.3 3-group Lie algebraC.4 of 3-group 3-group gauge theory B.3 Correlation functions of symmetry generators A.1 ’t HooftA.2 loop Worldsheet of axionic string B.1 B.2 4.2 Global 3-group4.3 symmetry for axion Gauging electrodynamics 3-group symmetry 3.3 Gauging all symmetries 4.1 Correlation functions of symmetry generators 3.1 Gauging 3.2 Gauging electric 2.2 Higher-form symmetries 2.1 Action C 3-group gauge theory B Correlation functions 5 Summary and discussion A ’t Hooft loop and worldsheet of axionic string 4 Global 3-group symmetry and its gauging in axion electrodynamics 3 Background gauging and ’t Hooft anomalies Contents 1 Introduction 2 Higher-form symmetries in axion electrodynamics JHEP01(2021)173 – 8 37 ]. ]). By ]. This 39 36 – – 17 , 37 ]. 33 12 ] for the axion 24 , 22 ] as a review). 23 41 ] (see also refs. [ 44 15 7 – 1 ]. If a domain wall is 29 – ]. There are induced electric ] (see also refs. [ 25 40 , 30 32 , , 43 17 35 , 31 21 ] (see also ref. [ , 14 14 – 13 36 – ]. If an axionic domain wall encloses a 43 12 34 12 16 H 33 41 – 1 – ]. Furthermore, this topological coupling leads to 17 , 16 , 14 – 12 39 : enclosing elements by surfaces ]. One of the characteristic effects is the Witten effect [ G : taking interior of topological objects 2 21 ∂ as 2-group – ) 18 and 1 L, H D.6.5 Symmetry groups parameterizing symmetry generators D.6.1 Symmetry transformations D.6.2 Diagrammatic expressionsD.6.3 of symmetry transformations Diagrammatic expressionD.6.4 of actions Diagrammatic expression of Peiffer lifting ∂ ( Another characteristic effect is the so-called anomalous Hall effect for the axion due One of the characteristic features of the axion electrodynamics is a topological cou- D.6 Global 3-group symmetry and symmetry generators D.1 Elements ofD.2 groups as topological objects D.3 Action of D.4 Peiffer lifting:D.5 braiding of elements of ] as a review). In condensed matter physics, the axion electrodynamics can describe currents whose directions are perpendicularthe to axion. both Since of the the electricso-called electric current anomaly flux flows inflow and to the mechanism the axionic gradient ofthese of strings, non-trivial axionic this phenomena strings effect of [ is extendedelectrodynamics related objects to has due the to also the been topological investigated coupling, as the a axion simple model of string theory [ domain wall enclosing the magnetic monopole is called ato monopole the bag modification [ ofplaced the in Maxwell-Ampère the law electricwhose [ flux direction is background, perpendicular electric toof currents the the are axionic electric strings flux. induced in on the This electric effect the flux also domain background arises [ wall in the presence non-trivial effects on extended objectstemporally in the extended axion objects electrodynamics. suchionic There as strings are magnetic [ spatially or monopoles,due axionic to domain the walls, modification and ofmagnetic ax- the monopole, electric electric Gauss charges law are [ induced on the axionic domain wall [ magneto-electric responses in topological matter [ pling between the axionof and massive Dirac photon. fermionsthe coupled This Maxwell-Ampère with coupling law them, originates [ which from modifies the the chiral electric anomaly Gauss law and Axion electrodynamics has beento widely condensed investigated matter from physics. particlethe In physics particle strong and physics, CP cosmology the problem, axion11 and has it been is introduced to a resolve candidate of cold dark matter [ 1 Introduction D Diagrammatic expression of 3-group JHEP01(2021)173 ], ], 88 with 49 – , ) higher- 75 47 , 42 , and actions G, H, L -form symme- ( G p → -dimensional topo- H ]. Therefore, higher- 1) ]). For − is the spacetime dimen- 100 p 51 ]. Such an interpretation ], deformation of current – ]. Here, a charged object − D 98 D 43 56 53 – , a map , ( – ) 89 52 54 , , groups with maps between them. G, H 42 ( 26 ], it was shown that the axion elec- n 101 ]. Thus, it becomes possible to understand 73 – -dimensional, where – 2 – p 61 ] (see also related topics [ 42 – ] and quantum chromodynamics [ 40 99 ]. One of the particular properties of the 3-group is the presence of -groups are given by a set of n 102 ), charged objects are . The higher-groups have been recently applied to various aspects of the- H ] can be regarded as broken phases of higher-form symmetries [ ,...,D 1 60 , and – ], which are extensions of conventional groups describing ordinary (0-form) sym- G 57 74 = 0 [ p on In the previous paper by the present authors [ A more elegant description of higher-form symmetries can be given by so-called Recently, the notion of symmetries has been generalized to ones for extended objects, What are the underlying structures for the above peculiar effects for the extended G of the winding numbersymmetries can of have axionic a semistrict strings.correlation 3-group (or functions 2-crossed Furthermore, of module) we the structure3-group found symmetry by to analyzing generators. that the the the We 3-group hereaftermaps higher-form for between abbreviate simplicity. them the The [ semistrict 3-group is a set of three groups the effects in the axion electrodynamics. trodynamics possesses a 0-form symmetry,symmetry, an and electric a 1-form 2-form symmetry, a symmetry.axion. magnetic Here, 1-form the The 0-form electric symmetryelectric and is and a magnetic magnetic shift 1-form symmetry fluxes, symmetries of respectively. are the related The to 2-form conservation symmetry laws is of the conservation law of oretical physics, such as highereffective gauge theories theories where of charged gapped objectsalgebra or are for gapless extended tensorial [ topological currents [ matterform symmetries [ and higher-group structures may provide us with new understandings of phenomena of extended objects in terms of higher-form symmetries. groups metries. Here, higher For example, a 2-group is given by a set of two groups is a 1-dimensional Wilsonphase. loop Another whose application of vacuum higher-form expectationphases symmetries is value [ that is Abelian finite topologically ordered inin which the the Coulomb charged objectgauge theories is based a on worldline those of symmetries [ an anyon. One can further classify phases of logical objects, while the conventional symmetriessince can be they understood act as 0-form ontries symmetries, local give 0-dimensional us objects, newthe i.e., aspects pure of local Maxwell modern fields. theory physics. ashas Such For Nambu-Goldstone been higher-form example, bosons generalized symme- we [ to can non-relativistic understand cases photons in as well [ by symmetries, it istended objects. plausible to consider symmetries whose transformationscalled act higher-form symmetries on [ ex- tries ( sions. Symmetry generators acting on the charged objects are objects? One ofmodel-independent, the candidates and is universal thechiral understandings notion symmetry of of and physical symmetries, itsunderstand phenomena. giving these anomaly us effects. in non-perturbative, In However,rather the fact, its than axion symmetry the extended electrodynamics transformation objects. are acts on essential only If notions local we to fields try to understand the effects on extended objects JHEP01(2021)173 ] , ]. ]. ]. 42 42 83 106 105 , , – 82 62 , , 103 . We give 79 61 5 ]. The absence ], which implies -dimensional axion 107 107 , 99 , we consider the back- (3+1) 3 , we review the axion electrodynamics 2 – 3 – , we give explicit forms of the ’t Hooft loop and worldsheet A , we discuss the other gauging procedure, which is based on the . Another property is that there must be a map from two elements 4 ]. We show that a naive gauging violates the invariance under the L , H 100 , , G 99 , on 42 G , which is called the Peiffer lifting. L to This paper is organized as follows. In section As a consequence of the background gauging, we determine ’t Hooft anomalies of the Next, we show that the above background gauging with the modified gauge transforma- There are at least two methods to establish the background gauging. One is to es- In this paper, we investigate the higher-group structure in the ], it is important to determine the ’t Hooft anomalies for the axion electrodynamics. H 3-group gauge theory. We showsults that in both the of axion the electrodynamics. gaugingfour methods appendices. Finally, we give In summarize rise appendix this toof paper the the same in axionic re- section strings which2-form are symmetry, charged respectively. objects We of show the detailed magnetic derivations of 1-form correlation symmetry and functions the used the obstruction to the simultaneous gauging of the 0-formand and higher-form the symmetries 1-form in symmetries. ground this gauging of system the in higher-form symmetries detail.for that are the In consistent dynamical with section the fields. gaugesymmetries. invariance We further In determine section the ’t Hooft anomalies of the higher-form is a mixed ’t Hooft anomalythe for 0- the axion, and which 2-form prevents symmetries. usgauging from The of a second simultaneous the gauging is of two forknown 1-form the anomalies symmetries. photon, for which the These forbids axion twoThe a and third anomalies simultaneous is photon are a in cubic extensions the ’t of Hooft absence anomaly, previously of that is the so-called topological 2-group coupling anomaly [ [ 3-group rather than the gauge invariancetransformation of laws the axion and and field photon.result strengths, By in comparing we the the confirm same gauge that physics. these two independent methods higher-form symmetries. We find that there are three kinds of the ’t Hooft anomalies. One tions can be sufficiently described byTo a this 3-group end, gauge theory we formulatedwe establish in assume refs. the the [ 3-group global gauge 3-groupmation structure theory laws in for of the the the axion axion background electrodynamics. gauge electrodynamics. The fields gauge Here, are transfor- determined by the basic quantities of the transformations of the axion andof photon, apparent inconsistencies which should requires be modificationssponding avoided of to [ the the gauge 3-groupon transformation structure. the laws corre- higher-form We should symmetriesthe note and 3-group that the structure. gauge this invariance, gauging but procedure it is does based not a priori assume Since the ’t Hooft anomalies constrainand possible phase describe structure anomalous of a phase given factors theory107 [ in correlation functions of symmetry generators [ tablish gauged actionsgenerators with [ couplings between background gauge fields and symmetry in electrodynamics in more detailthe by higher-form introducing symmetries. background Theanomalies, gauge background which fields gauging are corresponding enables us obstructions to to to describe dynamical the gauging ’t of Hooft global symmetries [ actions of JHEP01(2021)173 is ? (2.5) (2.2) (2.3) (2.4) (2.1) periodicity , where . We regard π 0 2 ?da ∧ is a spin manifold a coupling constant da 4 . e M and is an integer. Although da -dimensional axion electro- ∧ φE ?dφ q , which is transformed as ∧ ]. An invariant operator under a φ da 2 (3 + 1) dφ π , N . ) 8 109 . , P ) ( D -dimensional spacetime manifold, e.g., − φ P . 2 ( | Z φE 108 denote φ , π iq dλ. da 2 e | 2 | ∼ 41 (3 + 1) 2 + , we review the notions of the 3-group, the ∈ e 1 π da a | 2 can be a multi-valued function on a closed loop C – 4 – ) := dφ ) as a + → ) is a point operator, P C 1-form gauge field ) + 2 , P 4 2 and a Z | ( P 2.2 a decay constant of the axion, 2 φ M φE ( | dφ q v φ | ( dφ 2 U(1) I | 2 v 4 . In appendix M ] Z B 12 − the photon, = an integer. -form gauge symmetry [ a gauge symmetry, and the axion as a circle valued pseudo-scalar field. S 1) N itself. Here, the invariance requires that − ( ) in the spacetime: U(1) P ] in detail. In particular, we carefully discuss the symmetry groups for the ( P φ 101 is a single-valued function, ) is the axion, P φ , The photon is described by a φE with the winding number, q ( Physically, the nonvanishing winding numbera implies topologically the quantized existence charge. of a string object with rather than I C We have assumed that the mass-dimensionthe of the periodicity scalar as field isunderstood a normalized as as redundancy a ofthe the redundant axion. transformation in eq. In ( other words, the redundancy can be the Minkowski spacetime. Throughoutsuch this that paper, the we axion assumeat photon that each coupling point term is well-defined. The axion has a Here, of the photon, and the Hodge star operator. We refer to 2.1 Action Here, we give an actiona of gauge the field massless of electrodynamics,The in action which we has regard the the form photon [ as dynamics [ higher-form symmetries. After givingshow an the action existence of of the theBianchi massless identities higher-form axion of symmetries electrodynamics, the by we axion thegenerators, and equations photon. and of symmetry We also motion groups present (EOM) for the and the charged higher-form objects, symmetries. symmetry Lie algebra of thediagrammatic 3-group, expression and of the the 3-group 3-group gauge in theories. appendix We also2 give an intuitive and Higher-form symmetries inIn axion this electrodynamics section, we review the higher-form symmetries in in this paper in appendix JHEP01(2021)173 as S (2.9) (2.6) (2.7) (2.8) (2.11) (2.10) 1-form, , we can . This is C N S S Z , because of 0 Ω is the following φE . Since the gauge ) j . , P ( f φE λ C j S R V ∼ Z 0-form, electric is topological: it is invariant aE π ]. In the following, we show ) , iq ) N e V V ) := ( Z ( 101 = V ) + 2 ( φE φE , P da Q a Q ( φE C φ C λ S R is a boundary of a surface N R , πin . 2 aE C Z aE Z e iq π iq π e lead to the following closed current 3-form 2 e 2 da, Q ∈ ) := ∈ = ∧ ) := V – 5 – = 0 a a f da , C with a four-dimensional subspace valued, the gauge parameter can have a winding 2 C , S S S π 0 da Z N ∂ /N R Z 8 R φ aE Ω ∧ q ∂ ( − aE πin 2 iq da W 2-form symmetries. They are associated with the EOM e e 2 ( π N ? dφ 8 φE 2 ) = v − U U(1) C V → V ∪ , − 0-form symmetry, which is a shift symmetry of the axion. The aE N and use the Dirac quantization condition on a closed surface as := is not globally well-defined. q Z ( d ? dφ . Such a transformation with a nonvanishing winding number is a 2 da φE should be an integer. When Z W v j π gauge parameter, which satisfies 2 aE q 1-form, and ∈ is the field strength. In general, the field strength is a globally well-defined U(1) 0-form symmetry dλ da U(1) C N = R is a Z is a 3-dimensional closed subspace. The charge f λ V Here, under a small deformation the Stokes theorem. A gaugeunitary invariant object, observable given by the current EOM of the axion, and conserved charge, that there are fourmagnetic kinds of the higher-formor symmetries: Bianchi identities of the axion and photon. 2.2.1 First, we consider the bearing in mind that 2.2 Higher-form symmetries Here, we review higher-form symmetries in this system [ It physically means thatnothing there but can the be Dirac athe quantization field magnetic condition. strength monopole as Throughout in this the paper, interior we of simply denote where closed two-form that may not be the exact form, and it is quantized on a closed surface where a charge rewrite the Wilson loop by using the Stokes theorem as parameter is circle valued rathernumber than called a large gaugetransformation transformation. is a An Wilson operator loop, that is invariant under the large gauge Here, JHEP01(2021)173 ]. ), is d ? Z 112 2 1 2.3 e . We gauge (2.14) (2.15) (2.16) (2.12) (2.13) ∈ − ) φE as P α , U(1) V V should satisfy = . dimensions [ i 0 V ) φE Ω P Link ( α ∂ , is the 4-dimensional (2 + 1) φE 0 V q and try to define it by Ω ( . . We can also define I ) ) − h V V V Ω , . . , /N ) satisfying da da /N φE P , ∧ ∧ φ 0 V iα V da da Ω e with a real parameter ( πin 0 V V 2 ) as Ω Ω , e V φE R R Link ( ( ( V U Q φE 2 2 = 1 φE φE φE q = π π φE in U 8 8 φ da V iNα iNα iα ∧ e , we show the derivation in detail. da Ω πin − − 2 ) = da e e ∧ ∂ e 0 a ]. We define this term by using an auxiliary Ω R V Ω B.1 – 6 – := := = ) = R ∂ i V 2 φE 111 da da ) , π 2 φE , ∧ ∧ 8 π 0-form symmetry. The symmetry transformation is P a a 8 V ∪ iNα φE , , iNα V V 110 − N iα R R − e φE e Z /N e ( q φ ( 2 2 φE φE π π I φE 8 8 ) πin iNα iNα U 2 . In appendix V e − − with a boundary , ( P e e V on a spin manifold. Therefore, the parameter /N φE φ Ω parameterizes the topological object. This object is topological 1-form symmetry originated from the EOM of the photon, is the 4-dimensional closed subspace, and Z U and 2 1-form symmetry ) πin , which would imply the conservation of electric fluxes modified by ) N N 2 0 V V ’ denotes a vacuum expectation value (VEV), and π e Z N Z Ω ( hi (2 Z = 0 − · ∈ ( φE is a topological unitary object. by using another 4-dimensional subspace 2 U ∪ da h since there is a conserved current. However, the symmetry group is restricted da ∈ with an opposite orientation. By the Dirac quantization condition, the integral V /N φE ∧ ∧ 1 φ . U 0 V a da N V Ω πin dφ R 2 Z ∧ U(1) 2 e by the large gauge invariance or the Dirac quantization condition of the π Ω = Ω N ∈ 2 4 φE da π N The charged object for the symmetry is the 0-dimensional point object in eq. ( One might think that the group parameterizing the symmetry is a continuous group 8 This requirement is the same as the quantization of the Chern-Simons term in Ω + φE Z iNα 1 R iα − 2.2.2 Electric Second, we show a da Here, the symbol ‘ a linking number of subspace is e and therefore this symmetrygenerated is by a the topological unitary object and is expressed by the correlation function, The difference should be invisible, so that we require the following condition, where However, the integrale has an ambiguity of the choice of using a gauge invariant4-dimensional integrand subspace [ to field. This ischarge due is to not the large facta gauge that topological invariant. the unitary Let current object us isfocus consider not on this gauge the problem invariant, gauge in and variant detail. the term conserved We consider meaning that Therefore, such as where JHEP01(2021)173 . , ) aE φda S α . In ( = 0 S as ) R (2.20) (2.21) (2.22) (2.17) (2.18) (2.19) C S aM , N 2 V dda , Q S π : ) aE 4 S iα leads to ( aM − S aE group instead e U . V Link ( Q i , ) ) a N S C 1-form symmetry. 1-form symmetry, Z ( N aM , . πin q i Q 2 ) , the parameter a aE e C = Z q iα ). The transformation , ( 2 U(1) of the charge e U(1) ) . ) = da π 2.6 W aM S h S A q S (2 ) = , ( R , which is a closed worldline /N . S T ) π ∈ ) 1 , /N h 2 C C da ) a , a , C ∧ S , da iα S e πin dφ ∧ aM ( 2 ) = . The restriction on the group can S q , e S ( V ( Link ( ( dφ R aM T Link ( φda V aE aE = 1 N q R aM 2 is parameterized by a a aM π aE q da 4 Q ,U a ∧ iα πin ,U aE iα 2 − as dφ aM U e e e j aE V j R – 7 – . Since the charged object is a 1-dimensional = S = = aM S Z N i i q 2 Z is linked with a surface ) ) π aE C C 4 φda is a real parameter that will be determined by the B.2 C , , iα ) = S transformation of the ’t Hooft loop by ) = R S − is an integer by the Dirac quantization condition. Note aE ( aE S aM e N q ( q 2 α ( ( π 1-form symmetry as the magnetic aM aE 1-form symmetry. We refer to this 1-form symmetry as aM 4 U(1) . aE T q W iα ) N ) N − S Z S e Z , , U(1) a ∈ 1-form symmetry iα /N da, Q a e aE ( π φda, Q 1 2 iα 2 πin e 2 aM π U(1) N e = 4 U ( h − 1-form symmetry, since the symmetry is related to a conservation of the aE aM j U N h Z ? da group due to the gauge variant integrand 2 1 is a 3-dimensional closed subspace. Since e V = U(1) If the worldline of the monopole The charged object for the symmetry is a Wilson loop in eq. ( aE j Since the charged object is a 1-dimensionalHereafter, object, we the refer symmetry is to a this since it is related to the conservation law of the magnetic fluxes. of a magnetic monopole. Here, that the explicit form of the ’t Hooft loopthe is charge shown detects in the appendix terms monopole of charge the correlationproperty function can be of expressed the as ’t a Hooft loop and the symmetry generator, this The corresponding closed currentgiven 2-form, by conserved charge, and symmetry generator are respectively. The charged object is an ’t Hooft loop object, the symmetrythe is electric a electric fluxes. 2.2.3 Magnetic Third, we discuss a 1-form symmetry due to the Bianchi identity of the photon, law is given by The derivation is shown in appendix The condition that the integral does not depend on the auxiliary subspace where should be chosen as be shown as follows. Wein try a to gauge define the invariantlarge integral way, gauge of where the invariance. gauge We define variant term the integral by using a 3-dimensional subspace given by respectively. The topological unitaryof object a the axion. The closed current 2-form, conserved charge, and topological unitary object are JHEP01(2021)173 , ) ) Z S π , 2 (2.24) (2.23) φM q ( V , ) . In terms of the C N N ) . ( i Z Z Q S U(1) U(1) ) ( φ Group S iα , . e φM . Note that the explicit A Q φM ) ) ) q ) S a ) = ( S C P , C , , C ( V R C , φ h φ ) aM φM aE φE S q q iα , iq ( ( iq ]. We show that the invariance of e C e e ( T V Link ( 42 Charged object φM Link ( 0-form symmetry, which is introduced φM ) N φM πq ) q da Z ,U φ ∧ a φda iα = 2 2 2 φM e π π N N j 8 4 C – 8 – = dφ − − ]. Although this constraint is already known, we da dφ Z i C C S ) R 40 ?da ?dφ R R 2 S 2 1 a φ v e π ) = π , 2 2 ( − iα iα C ( S ( e e R V φM R q a ( φM φ N parameterizes the symmetry generator. The charged object . πin V N 2 Symmetry generator πin ) 1 2 e 2-form symmetry originated from the Bianchi identity of the C e , φ U(1) dφ, Q iα ∈ e U(1) π aE φE ( 1 aM φM φ 2 U U U U 0-form symmetry iα φM . Higher-form symmetries of the massless axion electrodynamics. e = N U Form h Z . The corresponding current 1-form, conserved charge, and symmetry φM 1-form 0-form j 2-form symmetry 1-form 2-form = 0 Table 1 gauge field with a constraint [ is a 2-dimensional closed subspace. In the presence of the axionic string, the U(1) ddφ S U(1) We can regard this as a symmetry transformation of the worldsheet of the axionic 3.1 Gauging First, we couple a backgroundas gauge a field of the here show the derivation ofalso the note constraint a explicitly relation in between our the case background for gauge self-containedness. field We and the symmetry generator. In this section, wecouple consider the the action background ofsponding gauging to the of the the axion higher-form symmetries higher-form electrodynamicsthe following symmetries. gauged with ref. [ action the We under backgroundleads the gauge to gauge modifications fields transformations of corre- of the the gauge axion transformation and laws of photon the (up background to gauge fields. We summarize the higher-formduced symmetries in of this section the in massless table axion electrodynamics intro- 3 Background gauging and ’t Hooft anomalies string, since the axionic stringcorrelation is function, a the source transformation of law a is topological given object by respectively. Here, for the symmetry generatorwhere is a worldsheetwinding of number the of axionic theform string axion of becomes denoted the as worldsheet of the axionic string is shown in appendix Finally, we consider a axion, generator are given by 2.2.4 JHEP01(2021)173 . a ], φE 1 φE 0 as A 110 (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) 4 M αdA π. 2 = 5 on a closed φE 1 mod Z ∂X A π 2 ∧ φE 0 ∈ da dA φE 0 ∧ ∧ Λ becomes a total deriva- satisfying a d da 2 5 -dimensional integral [ C ) is not invariant under the π ∧ R N φE 1 X 8 A , 3.1 da is locally expressed as ), we have chosen the auxiliary + Z 5 ∧ π X (3 + 1) 2.5 2 φE 1 Z , φE . Therefore, the gauge field A ? dφ ∈ 2 . j 2 φE 0 π . 4 v 8 Nα /N Λ , the gauge invariance may be preserved φE 0 M ). Since the gauge transformation of = 0 . At the linearized level, the background φE 0 d R in eq. ( = 4 φE 1 N dA + 2.5 gauge field that is transformed as a φE 1 dA M Z = 1 A ∧ Z φE 1 ∧ vanishes: ). While the integrand is manifestly invariant = φE 1 dA A α − , in which da – 9 – A but da U(1) ∧ φE 1 φE 1 S := ∧ 2.13 ∧ in eq. ( → A da = 0 = da φE a is a 2 dλ NA ∧ U(1) 5 φE 1 4 F φE 1 φE 1 Z A da M φE 1 Z A R 5 gauge transformation of the photon, or equivalently, by the dA A 2 by using gauge invariant integrand. We define the term ∧ X π Z ” when we discuss the definitions of actions by using 5- 8 Nα π φE 1 2 φE . However, the coupling in eq. ( j 2 U(1) π A N C 4 8 ∧ M ). Here, Z , := 0-form gauge parameter that satisfies da on an auxiliary 5-dimensional manifold + 2.1 φE 1 φE 1 ∧ S A φE 1 A a U(1) = , we have the condition A 4 ∧ . In order to discuss the large gauge invariance, we would like to define . Z . M ∧ π R lin -dimensional Chern-Simons term by using da lin , , 2 2 0 is a 0 da ∧ π is a 5-dimensional manifold without boundaries. Under the normalization N S S 8 a ∧ ∈ . The ambiguity of the choice of the auxiliary space does not exist if the following 5 φE 0 4 is a parameter that will be determined below. 5 a in Z Λ M (2 + 1) 4 α X Z 0 φE M The gauge transformation of the coupling da 2 R π dA 2 N ∧ 8 π C N 8 should satisfy As a consequence, the field strength of space condition is satisfied: where R Hereafter, we omitdimensional “mod manifolds. Noteof that the this definitionwhich is we a have natural already discussed extensionunder in of the eq. gauge the ( transformation definition of the photon the term if we impose theHere, flat condition tive under the condition,transformation. but This this problem totala is derivative caused may by not the vanish presence under of a the large gauge gauge variant integrand where one-dimensional manifold gauge transformation of theleads to photon the term proportional to to the action in eq. ( First, let us derive the constraintby the on invariance the under background the gauge field.fact that The the constraint is global required symmetrygauging is could not be done1-form by gauge adding field a coupling of the conserved current with a background 3.1.1 Constraint on background gauge field JHEP01(2021)173 , φE 1 (3.9) (3.7) (3.8) A (3.10) , since , since 0-form . ) ) π. 0 V V 2 N (Ω . Here, the Z (Ω . 0 0 , as a conse- δ Z δ φE 1 5 mod φ φ π . A X 2 N πn corresponds to a πn 2 φE 0 − ∈ da = φE 0 dφ ∧ da = 2 + Λ Λ ∧ . In the viewpoint of d φ φE 0 da p φE 0 1-form gauge field. This Λ − + ∧ da A with → D N ) ∧ 2-form symmetry V φE 1 Z dφ φE 1 A 1 φE A , φ J A , → p − 5 φE 0 U(1) D − Z ) implies Λ D as Z φE 1 dφ M V ( N 2 3.5 A Z 5 π φE 1 ) by choosing N -form global symmetry is a finite group + X 8 0 A Z ) = 2 . Here, we have introduced the delta func- ). We can further gauge the φE 0 p 2.12 ) π N − A V 8 3.5 0-form symmetry at a linearized level of ( D and 1 V → + δ – 10 – N ( φ p , Z δ φE 0 2 in eq. ( N Z | πn -dimensional manifold 2 ∧ ) 0 p in eq. ( da Ω J | πN = 2 ∂ ,A 2 is a 4-dimensional subspace whose boundary is − D e 1 . In particular, we can obtain the symmetry generator φE 1 2 M ∈ V φE 1 φE 0 D A Z φE 1 ( Ω Λ A 1-form symmetry and + -dimensional spacetime d da A 2 | D N ∧ + Z electric 1-form symmetry. As we see below, we need to gauge φE 1 V → V ∪ da φE 1 and a A N . Here, A ∧ ) Z − J V → dφ . Note that the condition in eq. ( ], and the configuration of the symmetry generators is expressed by dφ (Ω ) by choosing 5 | 0 0 Z 2 42 ) φE 1 2 Z Ω v dδ V -form A ∂ 2 φ ) , ( p π 1 N 4 8 /N δ πn M 2-form symmetry simultaneously in order to preserve the gauge invariance for φ − Z -form such that, in = 2 πin D ) = − p 2 ( 0 e We have explained the gauging the = U(1) ( 1 φE , whose gauge field need to be a flat connection. (Ω 0 0 φE N S 3.2 Gauging electric Next, we gauge the the the axion. for a the symmetry generator, thetopological gauge deformation transformation dδ NA the background gauge field U tional cal fields. 3.1.2 Background gaugingWe as can insertion of interpret symmetry thethe generators background spacetime gauging [ as a network of the symmetry generator in quence of Therefore, the gauged action is invariant under the gauge transformations of dynami- gauge transformations of themanifest, axion we define and a photon. gauged action In by using order the to 5-dimensional make action the as gauge invariance The action, in particular the last term, does not depend on the choice of couple the background gauge fieldaxion to is the shifted action under by a replacing gauge transformation of We can confirm that the action with the background gauge field is invariant under the construction is consistent withZ the fact that the and derived thesymmetry condition at of a non-linear level, which can be done as in ordinary gauge theories. We can We refer to the 1-form gauge field with this condition as the JHEP01(2021)173 shift (3.17) (3.14) (3.15) (3.16) (3.11) (3.12) (3.13) . The π . This 2 aE Z 2 . . ) π B ) 2 aE 2 . In order to − aE 2 ∈ B . aE B 2 da − B − aE 1 aE 1 ∧ Λ da da ( d ( with + Λ da is gauge invariant, we S ∧ . . In fact, the deviation . a R 4 ∧ with the normalization ) π Z ) da M 2 aE 2 aE 2 π R → aE 2 N 2 B aE aE B 1 2 π B N 2 B ∧ ∧ ∈ dB − ]). We can define the gauged ) up to , a − aE da φ = j , but we have chosen an auxiliary aE da 2 aE 1 115 da ∧ ( 4 φ ( B Λ – φ M . aE 2 4 R ∧ dφ N − are . , 5 M 113 Z + as a Z = 0 X dφ da aE 1 NB 1-form symmetry, which can be done by shift of π ( Z 5 N 2 5 2 shift of aE π 1 π 2 X N aE 2 ∧ N X dB . At the linearized level, the coupling would 2 8 ∈ π Z π B Z ) N 2 4 , and 2 = dB aE 2 + aE aE π 2 1 → N B = aE – 11 – 1 8 B aE 2 := 2 B dB | aE 1 , and the deviation is aE 2 − 2 , ∧ aE 3 ) = . Since the global symmetry is parameterized by the NB B B is absent if aE 2 aE 2 da H by using a 5-dimensional space as aE ∧ ( 5 2 B aE − 1 aE 2 ,B B B X ∧ B ∧ 1-form symmetry aE 1 aE da dB 2 | − φda 4 Λ B 2 dφ such that the coupling 4 N d e 1 M . However, this is generally not invariant under the φda 5 . The gauged action suffers from the ambiguity of the choice ∧ da 2 R Z M 2 Z ( 5 + Z π aE aE Z 2 N 2 4 + X ∧ 1 2 2 πN B B aE 4 2 2 φda ) 4 π π | N N 4 4 ∧ M B 8 aE R 2 = M dφ is not invariant under the | R B aE → 2 j 2 2 aE 2 v aE 2 π − 4 N 4 aE 2 B B . Therefore, we require that the 2-form gauge field is constrained by the M 4 B R da ∧ Z ∧ ( M π φ is a 1-form gauge parameter with the normalization Z 2 aE aE 2 2 4 − B ∈ M aE 1 4 φB Z = Λ 4 2 M aE 1 R E π due to the term M N We can discuss the problem by using the 5-dimensional action whose integrand is 1 group, there is a similar constraint on R 8 π N dB φ 4 S 2 N π S N 8 This action is manifestly invariant under5-dimensional the spacetime of the spacetime: Therefore, we cannot gauge the 1-form symmetry by itself. manifestly gauge invariant (see,topological e.g., term in recent a refs. 5-dimensional [ spacetime Here, action can lead to the couplingis at the linearized level. However, the non-linear term The gauge transformation laws of At the nonlinear level,gauged the action gauging would be could be done by replacing The ambiguityR of the choice of 1-form gauge field as which means that the field strength vanishes, derive the condition for define the term Here, we consider the gaugingintroducing a of 2-form the gauge electric field Z be written as of 3.2.1 Gauging electric JHEP01(2021)173 → aE 2 (3.25) (3.21) (3.22) (3.23) (3.24) (3.18) (3.19) (3.20) , B φM 3 . Since the C Z π under 2 2-form symmetry . ∈ is associated to a . φM 3 ) Z aE 1 C . π φM 2 Λ aE 2 2 d U(1) Λ φM 3 B . d ∈ ∧ C V − aE 2 R ∧ aE 1 B aE 1 da Λ ( dφ ∧ Λ π . 4 d N 4 ∧ 1-form symmetry. We can cancel ) by modifying the field strength . M aE 2 ) ∧ : φM Z Z − 3 , N 5 B π , which couples to the closed current aE 2 is a total derivative, the large gauge π Z aE 1 π 3.14 1 2 X dC aE 2 φM 2 2 N B 4 Λ , 2-form symmetry simultaneously. This φM ∈ 3 Λ B ∧ π Z φM − + 3 d + N C 4 ∧ π C S dφ 2 + φM 3 da 2-form symmetry. − 5 ∧ in eq. ( ( φM 3 aE 1 U(1) = ∈ X dC Λ ∧ φM 3 3 Z aE 2 dC aE 2 φM π ∧ C – 12 – B φM j N π 3 2 U(1) 1 B dC dφ = 2 4 C ∧ 5 dφ Ω → ∧ − M Z X − ∧ 5 R R aE 1 Z aE 2 S 2 φM Z φM 2 3 Λ of the as φMaE 4 π φM B N Λ C π 8 = π j 1 G N d 2 ∧ 2 4 2 dφ φM S M + π j − → 1 dφ 2 Z 5 + φM = X 3 φM 2 φM 3 R C S 2-form symmetry Λ 2 d dC π φM N = 8 j → 2 d S Ω U(1) φM 3 ) as Z C 3.21 , is a 2-form gauge parameter that is normalized as 2-form symmetry aE 1 is a 4-dimensional closed subspace. The gauge transformation law of the 3-form Λ φM 2 d in eq. ( Ω Λ U(1) + We now resolve the problem of the gauging of the Before discussing the resolution, we consider the gauging of the φM 3 aE 2 Note that the additional transformation does not violate the normalization of The modification requires anB additional gauge transformation law of the problematic term dC The gauged action does not depend on the choice of where gauge transformation of the coupling invariance of the couplingdefine is the nontrivial. coupling on In a order 5-dimensional to manifold as show the large gauge invariance, we where gauge field is of the Here, the 3-form gauge field is normalized by the Dirac quantization condition This problem can be resolvedis by because gauging the the problematicclosed term current 1-form independently. We introduce a 3-form gauge field 3.2.2 Gauging JHEP01(2021)173 aE 2 B π (3.30) (3.28) (3.29) (3.26) (3.27) ∧ 2 φMaE 4 aE 2 . G Z B mod π ∧ . ∧ 2 ) Z ∈ π dφ φE 1 N 2 φMaE 4 5 is A G X φMaE 4 2-form symmetries. ∈ R Z − G φM ∧ 3 aE 1 π 2 π . π 2 aE 1 N ∧ 2 8 dφ dC dφ dB ) ( U(1) 2-form symmetries. We ∈ 5 5 dB ∧ ∧ φE 1 X X . ∧ dλ Z Z 1-form symmetry in order to mod ) A . dφ da C ) π π R − U(1) 1 1 π dλ aE 2 1 ∧ 2 2 φMaE 4 2 aE 2 ) B ∧ dφ G U(1) − − ( B − aE 1 , or equivalently, the violation of − 5 φE 1 ∧ with φE 0 5 Z − 2 2 A 1-form, and aE 1 Z | | dB X da dφ dA ( 0-form symmetry. The gauged action dλ − da π ∧ N 5 1 aE aE 2 2 1-form symmetries requires the gauging 4 ( dB 2 ∧ Z Z + N B B M Z dφ ∧ ) ∧ φE 0 N a Z ( − Z ) 1-form, and − π − Z 2 1 ) aE 2 π 2 da dA 1 N π → aE 2 2 B N 2 4 da aE da 1 2 ∧ ∧ | | − a – 13 – B π Z B − 2 2 ) 4 − e e 1 1 − dφ da − 2 2 mod aE 2 2 da = da ( N 1 π da da B + + 2 ( 1 ( 4 ∧ ∧ aE 2 2 ) after the partial integration. The deviation under the 2 π − | | ∧ ) ∧ 0-form and 4 − B 0-form, electric da ) φM 3 dφ φE 1 φE 1 da + ∧ N | dφ 3.28 ∧ ( da N by gauging the A aE 2 2 A Z 5 ) 2 dC Z v ∧ ∧ B 2 dλ X − φM 3 − ) Z ∧ φE 1 E, − ∧ 4 ) da 1 2 0-form, electric A dφ dC aE 2 dφ | S π M da in eq. ( N ( ∧ φE 1 φE 1 ( 2 8 B − Z ∧ 5 N 2 v A A X Z ∧ − − aE dφ 2 + 4 Z dφ φE 1 ) 2 − ( B 4 2 M = A π da 2 N Z π φE 1 M ( 8 ∧ 2 N π π dφ N 1 8 Z 2 A 8 2 ( a ∧ E, π 1 N 5 − π − + 4 1 ∧ S Z 2 5 5 dφ Z = Z Z dφ 5 φE 1 − ( Z Z 2 Z = 5 A Z E, Z = = 4 1 Z 2 , 0 M π 2 On the other hand, in the 5-dimensional action, the ambiguity of the choice of the Let us try to gauge the N R S 8 π N 2 8 π N 4 5-dimensional space can be expressed as the violation of the gauge invariance maylarge be gauge transformation seen of as the follows. photon The problematic term is However, this gaugingthe depends gauge on invariance in the the choice 4-dimensional of action. In terms of the 4-dimensional action, all of the higher-form symmetries. We deform the action would be 3.3 Gauging allWe symmetries now gaugeshow the that we should simultaneouslypreserve gauge the the invariance magnetic under thesimultaneous gauge gauging transformation of of the the photon. In other words, the is canceled out as This gauged action has no ambiguity since the problematic term Eventually, the gauged action can be defined on the 5-dimensions as JHEP01(2021)173 aM 2 B is φM 3 (3.36) (3.34) (3.35) (3.31) (3.32) (3.33) 1-form dC aM 2 ∧ B . The nor- U(1) ) . Z as dynamical π that is coupled φE 1 . 2 Z aE 1 A . Thus, the field aE 2 1-form symmetry φM 3 π Λ ∈ aM 2 B − . We expect that 2 aE 2 C ∧ B a aE ∧ 2 B : ∈ , we gauge it in the ) dφ aM 1 2 B to eliminate the prob- U(1) aE ( 2 Λ φE 1 5 aM and 2 φE 0 E, 2 ∧ B d 1 X A Λ , B aE 1 E, Z S 0 ∧ d π 1 da Λ R φE 1 , , S N 2 π 0 1-form symmetry, since the 1 Z + A ∧ ∧ 2 φE 1 S . − π . Including the term canceling ) A N 2 − φE 1 dφ π aE φE 0 2 aM 2 5 A N aM 2 2 ∈ U(1) , ( Λ B 2 X | d Z π dB − dB N − aE 1 2 aM 1 2 aE 2-form gauge field 2 + . Note that the modified gauge trans- ∧ Λ π N B − aM 2 da dB d 4 ∧ aE 1 φE 1 da − ∧ Λ + 5 aE − 2 A dB U(1) aE d 2 ( X B da Z φE 0 B π | da + aM 2 N ∧ 2 5 2 φE 0 π 1 ∧ B ) e – 14 – 1 X dA in the last line a mixed ’t Hooft anomaly of the 2 Λ aE 2 2 − Z 1-form symmetry in ∧ by the Dirac quantization condition. In order to aE π 2 1-form symmetry in Z da → B + π N . Here, the gauge transformation law of 2 + 1 π aE 2 B 2 N ∧ 2 Z S da 2 → | B ) + π aM aM 2 2 − − U(1) ∈ 2 = N U(1) B B φE 1 2 φE 0 φE 1 1 aE 2 da ), we gauge the magnetic 1-form symmetry by introducing aM 1 ∈ π M A Λ ∧ ( A aM 2 B 1 4 φM Λ 3 5 π B S d − 5 N − X 2 da 3.31 aM Z 2 Z ]. However, the second term, dC ∧ 4 + Z and + dφ dφ π | ∧ M 42 1 ( dB 2 R da 2 5 is shifted under the gauge transformation of ). Since we have already gauged the electric 2 aM 2 ) should be replaced with v V 5 φE 0 aM 1 π X φE 1 1 1-form gauge parameter normalized as . R a + 2 B X Z Λ Λ A 2 Z d d 4 2 da 3.30 is π + preserve the Dirac quantization condition of 3.33 1 → E, 2 π π π M 1 N 1 + 1 2 , 8 2 S Z 5 d U(1) for simplicity. We introduce a 0 aM Z 2 aM 2 aM V 2 S − = = + + R φE 1 S Z B B B A = = M is a in eq. ( 1 aM 2 manifestly invariant under the large gauge transformations, we again define the j → S , the photon da M, aM 1 2 M 1 as 1 φE 1 Λ E, E, S 1 1 A , , Now, we gauge the magnetic Before gauging the magnetic 0 0 aM j S S with formations of In order to makeshould the be gauged modified action as gauge invariant, the gauge transformation law in strength the problematic term in eq. ( the following term, coupling in the 5-dimensional space as lematic term in eq. ( to where malization of make we can eliminateproblematic the term term is proportional by tosymmetry gauging the closed the 2-form current magnetic of the magnetic original action axion, which just expressesvariables the simultaneously [ fact that we cannot regard is problematic since it depends on the dynamical field (q-number) The first term JHEP01(2021)173 , ) ) φM 3 3.38 (3.39) (3.37) (3.38) ,C π 2 . aM 2 ] 5 ,B ,X φMaE 4 mod ) is now absent, G aE 2 φM 3 1-form symmetry. ∧ ,C ,B ) 3.30 N aM 2 φE 1 Z φE 1 aM,φE 3 ,B A A ( H aE 2 − ∧ ,B ) . dφ φE 1 ( aE 2 5 aE 2 B X B φ,a,A Z [ is identified as − 2 ∧ π 1 2 M, da 1 aM φE 1 2 ( aM 2 E, − 5 A B 1 π. , Z ) 0 π -form symmetries. The third one can be 2 Z dB N 2 1 ]. This anomaly means the obstruction to iS aE 2 π e 1 ∧ ] 2 B − 107 mod aE − 2 + φ, a aM – 15 – 2 [ B da 5 D ( aE dB Z 2 ), respectively. In the next section, we derive the Z ∧ Z ). The gauge transformation laws are given by B ), respectively. The field strengths are determined ) := ]. π 0-form symmetry and the electric 1 ∧ N 2 3.23 is non-zero. Physically, it means that we have a fraction- aE 17 2 N 3.34 3.24 aE − 2 B Z ] = aE 2 B 5 aM,φE − 3 B φM 3 ∧ H ∧ -form and magnetic ,X ) and ( da 1 ), ( dC φE 1 φE 1 φM ), and ( 3 ∧ A ∧ A 3.37 ) 5 3.34 ,C Z φE 1 3.35 Z φE 1 ), ( A aM 2 2 A 5 ) ), ( Z π − N ,B 3.13 Z (2 aE π 2 dφ 1 3.15 ), ( 2 ( + 5 ,B Z = 3.6 ), ( Z φE 1 2 A 3.7 π [ N In order to see this effect, we consider the following partition function given by the Further, the existence of the 2-group anomaly implies the existence of a fractionally The existence of the ’t Hooft anomalies forbids a trivial gapped vacuum. In our case, Thus, we have successfully gauged the higher-form symmetries. Furthermore, we have In summary, we have introduced the background gauge fields 8 Z ally charged particle on thewall, domain which was wall proposed if in we add ref. the [ magnetic fieldgauged through action the in domain eq. ( trivial gapped vacuum is stillpreserving forbidden. the For symmetries, example, we if can acan have gapped be topologically vacuum degenerated ordered is on realized phases, a while whose compact ground states spatial manifold. charged particle where this requirement is satisfiedistence by of the the existence massless ofthe axion the 0-form corresponds and massless to 2-form axion thesponds symmetries. and existence to photon. of Likewise, the the the The existence mixedsymmetries. existence ex- of anomaly of If the between the we mixed massless deform anomaly photon the between corre- system the with electric preserving and these magnetic higher-form 1-form symmetries, any second term is thatof of the the pair of photon,identified the which as electric prohibits the the so-called 2-groupthe simultaneous simultaneous anomaly dynamical gauging [ of gauging the above gauge transformationgauge laws theory. and field strength from theobtained viewpoint the ’t of Hooft anomalies the foris 3-group the the higher-form symmetries. mixed The ’t first term Hooft in anomalies eq. ( of the axion, which has been discussed previously. The The remaining ambiguity in the right-hand side represents thewhose ’t Hooft action anomalies. iseqs. ( given inby eq. eqs. ( ( By adding the term, the problematic operator-valued ambiguity in eq. ( Accordingly, the gauge invariant field strength for JHEP01(2021)173 = (4.2) (4.1) aE 2 (3.40) B . Since ) , ] ) aE 2 V , φ,a [ ( i B ) 1 iS δ , e ∧ φ i ) . Note that this ) ∩ V S N φ πn ( ) φE 1 2 2 n δ N S a A ∩ S ∧ in the path integral. ], which give us the n = ) π N V 2 a V which is given by the −V − ( ( Ω 1 101 ) φE 1 , , − δ , → S i ∧ A ( ) ) a /N 2 /N δ S a a S 4 aM 2 , the right-hand side implies ( ∧ n n 2 V M ) φ δ φ ( R V dB ∧ 1 ( ) ( φ 1 δ πin ]. The correlation functions give πin n δ V V ∩S a 2 2 ( N a ∧ ∧ 1 − − a N in δ ) πn e 101 e 4 2 ∧ ( ( − a M aE 2 e . By setting ] 4 R + aM B aM M a φ R U n U − ) = 1 N φ, a a ) φ [ V i n in S N a , and D 1 , da − h ( i i in e φ 5 ) ) /N Z /N − φ – 16 – a X S S with the fractional charge e R , , → N h πin π πin ) 1 2 2 2 /N /N = = V e a a e ( ( V ∩ S 5 (Ω , we obtain πin πin 0 aE φE 2 2 δ e e ,X U U φ ( ( = 0 h h 0 , N πn aE aE 0 = = 2 , φM 3 U U ) ) ) − C S V V φ ( , , 2 δ /N /N a φ φ , and N πn πin πin 2 2 2 = 0 . Note that we only consider the correlation functions of the symmetry , e e ) ( ( V is a delta function 3-form on the closed line aM 2 ( φE φE B.3 ) 1 B U U δ . S h h , ( φ is a normalization factor such that ) 2 δ S N 3.3 πn ( N ∧ 2 2 δ ) It has been shown that the correlation functions of the symmetry generators are not a V Z ( N πn 1 2 sider fractionally charged objectssection due to the intersection of theindependent, symmetry but generators related in toelectric each 1-form other. symmetry First, generators the induce correlation a functions magnetic of 1-form the symmetry 0-form generator: and We review the correlationgroup functions structure of in the the symmetry higher-formalgebra symmetries. generators in ordinary [ This quantum is field ain theories. natural appendix generalization The of details current generators of the which derivations are are summarized not intersected to each other. Therefore, we do not have to con- on a mathematical procedure,of independent dynamical of fields the in gaugingcoincides the based with previous the on one the section. in gauge the We invariance previous confirm4.1 section. that the gauging Correlation of functions this of section symmetry generators In this section, we3-group derive gauge the theory. background First,the gauging we correlation by functions review a of the the differentus global symmetry ingredients approach 3-group generators of based symmetry [ the on byformulated 3-group. the the for Next, a structure given we of 3-group. establish the We should 3-group remark gauge that theory, this which 3-group can gauge be theory is based the existence of the Wilsonfractionally loop charged on particle doesof not the arise background as fields long i.e., as the we symmetry do generators. not4 consider the intersection Global 3-group symmetry and its gauging in axion electrodynamics where we have redefined On the right-hand side,term we related have to the theδ term 2-group anomaly Here, JHEP01(2021)173 ) . 1 i S ) ) + 2 , re- 1 V (4.4) (4.3) ( S S S 1 , V ]. The δ ( = a 2 /N δ 0 a S 101 . This sign a πn V ) n 2 πin ∂ S ( 2 − V π ), respectively. e ( ( 2 1 = and − 4.2 aE 1 dδ V U B = − ) . 2 = i 1 ) V B C and ) = ∩ S , Ω ) S . Here, we summarize the ) φ ( ∂ 1 S 2 ) and eq. ( ) is that the electric flux can iα ( S and δ C e 2 ( δ 4.1 4.2 )) a −V 2 ( φM N , πn S 2 ( U . This sign matches the choice of h 2 ]. By the Maxwell-Ampère law, the ) /N δ 1 0 a = ) and ( 0 a S n = are as follows: 2 30 a n V , i ) ( ) B 4.1 1 πin C 21 2 , −} ) + , dδ e φ , 1 ( by the same procedure. The minus sign in − 13 S iα ( ]. In the presence of the axionic string and the e ) {− 2 φM ( 1 – 17 – δ S U a ) = 101 h , φM 1 n ) means that the magnetic field can be induced by ]. ( G, ., S U = /N π ( ) a 1 N 2 i 2 4.3 ∂ 24 V → ) δ , , 2 πin ) is due to the minus sign in = 2 S H : they are not necessarily Lie groups. /N , e 23 2 , ( φ L 2 4.2 ∂ → B /N aE 0 a πin 17 L , 2 U ( e πin ( 12 , and 2 e in eq. ( H ( φE ). ]: if the domain wall encloses a magnetic monopole, the domain wall , S U aE h G 101 U 3.34 −V ) are 4- and 3-dimensional subspaces satisfying 1 S S , V /N in eq. ( ). a and )) πin V 2 2 3.34 S Ω e is due to the minus sign in ( V Three groups ( We would like to discuss a mathematical structure behind these correlation functions. Other correlation functions induce no further symmetry generators. For example, one Physically, the relation in eq. ( A physical meaning of these relations in eqs. ( Second, we consider the correlation function of the 1-form symmetry generators, which 1 1 S aE 1. δ U 0 a h 4.2 Global 3-groupHere, symmetry we for review axion electrodynamics thedetail global of 3-group the symmetry axiomsingredients for of of the the the 3-group 3-group axion is electrodynamics explained [ in appendix generators. However, thegenerate correlation a 2-form function symmetry ofno generator such the cannot a 1-form be structureextending symmetry described in the generators 2-group a by that to a 2-group. a 2-group, 3-group, Fortunately, since which we we there can explain is find below. an appropriate structure by One candidate is a 2-group,In which terms is of roughly a given by 2-group, two we groups may and describe maps the between correlation them. function of 0- and 1-form symmetry the anomalous Hall effect forelectric the field, axion the [ electric currentelectric is current induced induces [ the magnetic field. can evaluate the correlation function of the 0-form and 2-form symmetry generators, −V the background gauge fields n the electric field in the presence of the axionic string. The correlation function represents leads to a 2-form symmetry generator: Here, we have eliminated be induced by the axionicaxionic domain domain wall in wall. the Therefore, presence theeffect of correlation of the functions the magnetic can axion monopole [ be insideinduces interpreted the the as electric the Witten flux [ spectively. Here, we have eliminated theredefining 0-form the and electric integral 1-form variables symmetry of generatorsThe the by minus path sign integral in in eq.matches the ( choice of the backgroundin gauge eq. fields ( where JHEP01(2021)173 , ∈ on G ) ∈ (4.7) (4.8) (4.9) (4.5) (4.6) G iα is the 0 is triv- , e G ∈ g . g = 1 G πim/N 2 2 1 . These maps is Abelian, the e ∂ ( H ◦ , N 1 Z N → ∂ Z . = ) L , where = : iα L G on itself is defined by a e 2 . G ∈ ∂ G l , ∈ . In particular the action of 1) G . L , . We define the action of } should be nontrivial, since the 0 πinm/N . Since ∈ 2 H → = U(1) l L − πin/N . The actions are compatible with H for all = (1 2 1 H , e e is compatible with the Peiffer lifting: . − as G iβ : . on g e ,L iβ 0 1 L 2 e , and , and as the 0-, 1-, 2-form symmetry groups, ∂ G gg H πim/N H g . h, g . h = ) = 1 on L 2 ). For l { U(1) , can act on e , ∂ ( ∈ := iβ 2 G G 0 × = 4.1 h G ∂ . e } , N ◦ 0 on – 18 – ) = ( . In terms of the elements, the Peiffer lifting is , and ) = 1 Z G 1 for the axion electrodynamics. In the correlation g . g L iα . The action ∂ . Note that the requirement iα H G 2 ∈ h, h = L ∂ , e , H { 0 , e πin/N → of 2 G ∈ ∈ e . Therefore, the action of , let us specify the 3-group for the axion-photon system. . 0 g . 0 g, g H and g iβ ,H by automorphism: the actions are denoted as πim/N πim/N e 1 × 4.1 2 N 2 for h, h = ∂ L e ) implies that , we can relate the Peiffer lifting to the correlation function is given by e Z ( H ( , 1 L H 1 . . : for − 4.1 = and . H ∂ ∈ g 0 L , G G H −} G . However, the action of gg , 0 between the three groups: πin/N g ∈ } ∈ g . l 2 0 2 on e {− ), we define the action of ) ∂ = iα G 0 h, h 4.4 { , the action , e , and of and is defined by conjugation: H 1 . g . g ∂ G U(1) ∈ πim/N 2 × on e ( N G the group compositions. written as are group homomorphism, i.e.,composition they of are the compatible mapsidentity with satisfies element group in compositions. The g . h Z Peiffer lifting Action Maps Finally, we identify the Peiffer lifting. Since the Peiffer lifting generates an element of Third, we consider the action Next, we define the maps By the discussion in section following the correlation function in eq. ( 4. 3. 2. = from the two elements of trivial as in eq. ( L H Since the action of the 0-form symmetry generator on the 2-form symmetry generator is conjugation is trivial: trivial one: correlation function in eq. ( H we define these maps as follows, for all ially satisfied. respectively: function, there are no maps whichtry relate generators, the or 1-form symmetry 2-form generators symmetry to generators 0-form to symme- 1-form symmetry generators. Therefore, First, we identify the three groups JHEP01(2021)173 , , /N N 0 + (4.13) (4.10) (4.11) (4.12) 0 ). The πimm m 2 , we may e 0 L. 4.5 → 3 = m ∈ 0 } are not gauge ) } 0 m ) ↔ /N 0 0 iα aE iα m j , e e and /N πimm 0 /N 0 2 N and e . πim 2 + N = e 2 φE πinm ( / j 2 } 0 , m ) ) − , } iα ) iα 0 N → , e . In particular, it is nontrivial ) by this definition. 2 , e , e πimm iα / and the previous paper of the 0 2 /N m 0 ) and the correlation function in e , e C.1 3.23 = πim/N /N πim B.3 0 2 πimm 2 πim/N e } D.48 2 2 e ( ) e ( e 0 { πim ( , 2 iα ) , = e ) 0 ( iα , e N . e iα . 2 ]. / /N 0 0 , e 3.3 – 19 – 101 /N πim ) is 0 πin/N 2 πinm/N πimm 2 e 2 2 ( 4.5 πim − , e , 2 ) . ) . e e , e ( N iα iα 2 / }{ , e , e 0 ) 0 2 πin/N ). Therefore, we have confirmed that the compatibility in πim/N 2 iα 2 e e πimm , e 4.9 ( πim/N πim/N 2 ) can be evaluated as 2 2 { e e e /N 0 ( ( 4.5 = = { . ]. πim 2 e ( ) as a 3-group generalization of the current algebra, which imply that a 101 is Abelian and the right-hand side is symmetric under , πin/N ) 2 4.4 e iα { , e )–( ). The diagrammatic expression in eq. ( ) is unambiguous. As we discuss later, we can show that the gauge transformation = U(1) 4.1 4.3 ) is satisfied. ) suggest that we define the Peiffer lifting such that it satisfies L πim/N 2 4.5 4.3 4.10 We can understand that the correlation functions between the symmetry generators We have defined the three groups, actions, and Peiffer lifting. We should confirm that e We choose that a different definition of the Peiffer lifting from our previous paperIn to order be consistent to with define the Peiffer lifting itself in an unambiguous way, we may need to treat the spin ( 2 3 { which avoids the operator shifts in section the background gauging, althoughis the also previous consistent with definition, the axiom of thestructure 3-group explicitly. [ We leave this issue as future work. present authors [ 4.3 Gauging 3-groupHere, symmetry we consider the gaugingand the show 3-group that symmetry the gauging in based terms on of the the 3-group 3-group gauge theory gauge is theory, consistent with the gauging in eqs. ( conserved current can be abetween source symmetry of generators, anothermeaningful current. but in Note not our that case. between theinvariant. This correlation the is functions For because conserved the the currents, conserved detailed are currents discussion, physically see appendix where we have usedeq. ( eq. ( Meanwhile, the left-hand side of eq. ( obtained by the background gauging procedure in eq. ( these satisfy the axiomsis of the to 3-group, confirm summarized theright-hand in side compatibility of between eq. the ( action and Peiffer lifting in eq. ( Although this definition haseq. ambiguities ( under the shift law and field strength of the 3-form gauge field in the 3-group gauge theory match the one eq. ( Since introduce the Peiffer lifting as in eq. ( JHEP01(2021)173 , , 1 E 1 E 2 → − by M 2 de- are p B B p B ω E 2 (4.16) (4.17) (4.18) (4.19) (4.14) (4.15) − dA , . −} B 1 E 2 and , does not are con- = da (U(1) − B 0 − p p N E {− 2 , A Σ Z given in sec- 1 B da and ) NA A 1 , ! A φM 3 and E 1 and ): the coupling of ! M 1 − Λ E 2 Λ 1 2 ,C d d B A B 3.34 dφ

-form gauge field aM , 2 ∧ p − . 1 of the 3-group   ) ,B . N , independently. The field 0 A E 1 da E Z 2 π 0 Λ M aE Λ 2 2 N 2 d B B ∧ U(1) } ,B − ∧ + ) . E with 2 0 1 E 1 E 2 φE 1 M 2 and the Peiffer lifting Λ 3 and A -dimensional subspace , but the Lie algebra of Λ ,B B A d C . d ( R 1) E 1 dB E π 2 i , and + N + ( 4 ∧ Λ

B − 1 is ! E 1 + p and = A ( gauge ﬁelds, we now establish the back- Λ E 2 ( M 3 U(1) 2 ! π π } − { B M E 1 2 B N 2 N U(1) 4 × E E 1 dC 2 M 2 Λ

B for the 0- and 1-form gauge ﬁelds Λ d − B B for a N − . = U(1) , – 20 – Z

E 0 E 2 + 1 E 1 E } 2 Z ) as ) as . Λ , B Λ E π correspond to 2 E B 2 0 1 dB d 2 N ) − B ∧ Λ 3 A ,B Z -form gauge field satisfies the condition C.46 ∈ + = π ! C.57 E 1 p E 2 N does not act on 2 1 + ,C E 2 E 1 , Λ E M 1 2 )–( ) B − )–( + p ! B M { Λ 2 π Λ M 2 N d 2 d E 2 M 2 = 0 + M 1 , respectively. Here, the gauge fields NB dA ,B

) C.44 − { ,B 1 3 , − Λ 1 C.55 E 2 dB 0 2 dB d E 2 − 2 + p −} Λ Λ dA dC Λ B and ,B , + Σ d ! d ( d 1 R 0 = = = , in which a E 2 M . 2 + A M + 2 {− + 2 4 ( 0 B ! 1 B 3 B dA 3 F G Λ E A C 3 M , ., gauge ﬁelds as well as 3   C U(1) = N H H → → = → = 1 N Z to

is a discrete group. However, we can introduce a 1 3 Z ! N C A N NA → E 2 with the axion and photon are give by the combinations M 2 Z because Z B B E 2 ) denote the gauge ﬁelds of

B , since the gauge transformation laws and field strengths coincide with each other. M 2 3 U(1) C The gauge fields By using In order to establish the 3-group gauge theory, we need the Lie algebra of the 3-group in ,B 3.3 × and E 2 1 N B Therefore, we have confirmed thatics the possess higher-form the symmetries 3-group of structure the insymmetry. the axion We viewpoint electrodynam- also of have the the backgroundA gauging same of gauged the action 3-group asrespectively. the Furthermore, one in the eq. coupling ( of By the above structurefields of coincide with the the gauge gauging transformation that avoids laws operator-valued and shifts. tion field strengths, the gauge We note that it( is necessary tostrengths treat are the given two-form in gauge eqs. ( fields as a pair of gauge fields laws, they are given in eqs. ( ground gauging ofand the 3-group.Z The 1-, 2-,strained 2-, by and 3-formwith gauge proper fields normalization, respectively.termine the The gauge action transformation laws and field strengths. For the gauge transformation the axion electrodynamics. The Lieexist algebra of since embedding with the normalization JHEP01(2021)173 , 4 G ∧ ) 1 ) implies A ]. We may 4.1 − 116 , dφ ( 5 55 is deformed. Note , X R aE π 54 1 , j 2 − 26 ), which are generalizations , the gauge invariance of the and 4.4 3 1-form symmetries. H )–( 3.1.2 ] for recent discussion). Another N ∧ Z 4.1 ) 2 117 B − da ( 5 X themselves are not physical observable, and we – 21 – R π 1 2 aE j -dimensional axion electrodynamics in detail by using ] and a mathematical procedure in the 3-group gauge 0-form and electric and in the presence of another current 101 N ]. Therefore, the higher-group structure may be deformed Z (3 + 1) φE φE j j 118 There are several avenues for future work. One is to analyze physics in the background We have discussed two independent gauging procedures. One is to formulate the One comment is in order. As we have seen in section in the presence of the 3-form symmetry. becomes massive, while inthe this axion work becomes we massive by havewhich non-perturbative assumed have effects, that a there the can topological beshown axion domain that axionic is there wall domain can massless. charge. walls, be aof When In discrete the 3-form the symmetry axionic absence whose domain charged of wall object is the [ a photon, worldvolume it has been magnetic field, spatially varying axion field,backgrounds, and mass so spectra on. of It the hasunderstand axion been and shown such that photon deformations in are of non-trivial deformed thegroup, [ phase and structure their by using ’timportant higher-form direction Hooft is symmetries, to anomalies 3- discuss (see what happens e.g., for ref. higher-form symmetries [ when the axion theory. By comparing the gaugegauge transformation laws fields, and field we strengthsgive have of the the then background same shownglobal result. that 3-group the symmetry. Furthermore, above In wea two particular, simultaneous have we independent gauging determined have of gauging found the the a procedures ’t 2-group Hooft anomaly, which anomalies forbids of the This procedure does not relythe on gauge the transformation existence laws of andthe a gauge gauged 3-group invariant action field structure. preserves strengths the We bygauging have gauge the determined based invariance requirement for on that the the dynamicalglobal 3-group fields. 3-group gauge The structure theories. other in is This ref. the gauging [ can be established by using a and ’t Hooft anomalies inthe the background gauging.model We exhibiting have the found 3-group that structure the and axion ’t Hooft electrodynamics anomaliesgauged offers action of where a the the background simple 3-group. gauge fields are coupled with the symmetry generators. have discussed the correlation functions of the gauge invariant symmetry5 generators. Summary and discussion In this paper, we have studied the higher group structure of the higher-form symmetries laws corresponds to the deformationour of case, the the conservation deformation laws of ofrelation the the functions conserved gauge of currents. the transformation symmetry In lawsof generators can the in be ordinary eqs. understood ( current by algebra.that the the For cor- conservation example, law the of that correlation the function in conserved eq. currents ( respectively. Since this gauging procedurethe gives higher-form the symmetries same is action, also the the ’t same. Hooft anomalies of background fields corresponds to the invariance underconservation the laws topological of deformations, the i.e., conserved the currents. The deformation of the gauge transformation described as five dimensional actions, JHEP01(2021)173 ) A.4 (A.6) (A.2) (A.3) (A.4) (A.5) (A.1) as : S R a a 0 f ∧ 0 . ) φf , C 4 ] , ,g M : 0 S Z a ,f 2 S . a π ] ]. We can rewrite eq. ( N + 8 S a R Link ( can be expressed as a singular + 120 + and the regular part ) [ , R φ,a C [ aM S ) , a a C φ,a 1st [ ( ?dφ πq a, 3 aM iS δ , respectively. ∧ q iS e . ( φ ] e = 2 . ) ] R T aM S S dφ S a , g 4 0 πq da and da M φ, a + [ , f Z S a − R Z R D = 2 2 due to the monopole is expressed by a 2 R v – 22 – S a Z = da φ, a ) = − [ S = 0 a dda − D a i 0 ) ?f = f + Z C ( ∧ , R into the singular part 0 = ∧ a f dda a ( i aM g ) 4 d q 4 is a first order derivative action for ( C M S , M ] T Z Z Z h 2 , g aM 0 = e π 1 1 q 2 2 ( , f da S T can be understood as the 1-form that breaks the Bianchi identity, − − h a S S Z = a + R 1st a, ]. We decompose S φ, a [ , the configuration of the ’t Hooft loop 119 [ 1st φ a, a S and Now, we consider the explicit form of the ’t Hooft loop in terms of a local field. This a where can be done byby the using dual the transformation Fourier transformation, of the photon which represents the existenceformalism, we of can the express magnetic the ’t monopole Hooft current. loop as In the path integral The singular part part of Here, we require the configuration of the and a 2-form gauge fields, which are dualA.1 of ’t HooftFirst, loop we express the ’t Hoofton loop in terms of local fields. In the original formulation based A ’t Hooft loopHere, and we worldsheet summarize the of expressionsstring axionic of in string the terms ’t Hooft ofworldsheet loop local of and the fields the axionic by worldsheet string of using can the dual axionic be transformations. expressed as The line ’t and Hooft surface integrals loop of and a the 1-form R. Y. thankswork Yuji is Hirono, supported TaroAid in Kimura, for part and Scientific by Naoki18H01217 Japan Research Yamamoto (M. (KAKENHI Society N.)). for of Grants discussions. No. Promotion of This 17H06462, Science (JSPS) 18H01211 Grant-in- (Y. H.) and Acknowledgments JHEP01(2021)173 . )) S (A.7) (A.8) (A.9) da (A.15) (A.11) (A.12) (A.13) (A.14) (A.10) − 0 f ( ∧ as , where the 0 dw R f 4 ) by using the φ , which implies M ] R and π dg 1 2 A.14 [ g . δ ) − 0 . S f ] , , ∧ 0 w C C φf R ,a,ζ,h . 4 , can be expressed by a line by the dual transformation S ] , S φ M aM ,a ) R w Link ( + S da , which are 2-form fields inde- iq S 2 λ φ R ∧ e g as , π N . . Instead, we can go to the dual φ + 8 [ ) φM ) R )] dw = + can be expressed as a singular part S φ S 1st φM S [ and the regular part ) R ( πq q ) , a and φ, C 2 ( π iS , ( i S ?dφ δ 0 S 2 iS 3 e + ∧ , φ . V ] δ w f ) by integrating out e φM , = 2 − ] ∧ w q R S dφ e φM ( , a S w ). This is the expression of the desired ’t a φM φ 4 2.1 ( R dλ 4 q V πq M ( dφ d φ + M R , ζ, h [ dw, + R C V 3.10 2 R − 2 R Z D v = 2 0 w = φ ]. We can again rewrite eq. ( aM − – 23 – S f 0 Z [ g φ, a iq , we have the delta function = [ → δ ) = e ?f R 121 = S ∧ D φ ddφ a 0 w φ i = f ) Z = S 4 + S M , da = R R ∧ have non-trivial winding number, i φ 2 ddφ ) 1 ( e φM S dw 2 q d S φ into the singular part 4 ( ) and then eq. ( − , C ( M i φ V Z R h e ] φM π A.3 i 0 = q 2 ( can be written as − can be understood as the function that breaks the Bianchi identity V e i dφ : h ) S C C C . φ, v, f Z φ , [ φ D aM to a 2-form gauge field [ Z q ( along and gives us the delta function φ = T a i h w g ) C , can be locally given by a 1-form gauge field aM g Now, we consider an alternative expression of We can go back to the original action in eq. ( q . We again decompose ( φ T h of the axion Fourier transformation, of the axion, In the path integral formalism, we can express where we have assumed that The singular part A.2 Worldsheet ofSimilarly, axionic we string can express thethe original worldsheet formulation, of the the configuration of of axionic string in terms of a local field. In where we have used eq.Hooft ( loop in terms of the local field. We find that the termintegral given of by the singular part, with a gauge transformation by a 0-form gauge parameter Therefore, pendent of integral of action as follows. Bythat integrating out Here, we have introduced new dynamical valuable JHEP01(2021)173 ). . )) gives S A.20 (A.19) (A.20) (A.21) (A.22) (A.16) (A.17) (A.18) dφ h − ζ ( ∧ db 4 f M ∧ R , we can go back π a 1 2 1st ]. Note that we do can be locally given ∧ a, + f S ζ h 32 ∧ , , 4 a b , can be expressed by a : ∧ M S 31 S ζ φ Z R 4 dφ 2 ∧ M , π aM R N b , where the integral of db 8 da iq 2 λ 4 h e π N ∧ 8 M − R = − π i dλ ) ?ζ and 2 ?ζ , which are 1-form and 3-form fields implying that S π ∧ ( − ζ ∧ h N ζ ] 4 2 . e δ gauge symmetry cancels the violation. ζ 4 ) , ∧ b S dh 4 ), M − should be shifted under the gauge trans- b ). This is the expression of the worldsheet [ λda. R can be written as and 4 M δ h dλ h dφ 2 π Z 2 M i v N db, ζ 4 ) R U(1) 2 + 3.10 − A.17 → − 2 S v b = − R , aM – 24 – b ?da h − iq ∧ → dφ e φM . Instead, we can again go to the dual action. By b → q da − )] ( = 4 b ?da S dλ, h ζ V S M φ ( ∧ h R ) and eq. ( + dφ , 2 ∧ + 1 ∧ e a da 2 ) by integrating out h db R db 4 − 4 A.13 4 → φ ( M = 2.1 i ( M M is a first order derivative action for : e Z a d R ] Z h ] S 2 π i − π e 1 2 , 1 2 2 b ζ . Note that the 3-form − [ e φ b, a, ζ δ + − [ along , a, ζ, h D S = b , we have the delta function φ and R Z 1st φ a + = φ, R i S ) φ [ S , 1st φ, φM S q ( We remark that this expression of the axionic string is not invariant under eq. ( V h The inconsistency can bestring. resolved by The taking chiral intoThis mode account mechanism the charged is chiral under nothing modenot the but on need the to the anomaly consider axionic the inflowsince contribution mechanism from we [ the focus chiral on modes the in bulk the physics discussion in around this the paper, axionic string. where we have again used eq.of ( the axionic string. Therefore, the gauge invariance for the photon seems to be violated on the axionic string. We find that thesurface term integral given of by the singular part, Substituting the condition with a gauge transformation by a 1-form gauge parameter in addition to the one corresponding to eq. ( in order to make theto action the to original be action gauge inus invariant. eq. the As ( delta in function theintegrating case out of by a 2-form gauge field independent of formation of the 1-form field as We have introduced new dynamical valuable Here, JHEP01(2021)173 , P (B.7) (B.3) (B.4) (B.1) (B.2) (B.5) (B.6) and V Ω , ) . ) . V i ( P ) ( 1 . φ P i ( ?δ ) φ φE P ∧ ( iq φE . φ ) can be rewritten as + iq ) V ) e V φE ( φE can be expressed as the h P j 1 ) by adding a local counter iq ( δ e (Ω V φ P ) V h ( 0 R 4 . ) 4 V δ φE δ φ ( M P Z ) , iq 1 ). The transformation can be Z N V φE V e πin ∈ 2 2 . ?δ dj (Ω ) ) 0 + ∧ 4 2.16 δ P φ P ) ( Link ( iS 4 M , 4 V e δ Z N M φ . In order to evaluate the correlation ] ( πn V ∧ R n 1 ). Suppose 2 φ = δ φ 4 φE N n 4 = 1 φ, a [ , M 2 i M 3.10 φE πiq 2 R ) is the intersection number of N φE v 2 D 1 R P h dj e φE πiq Z V 2 B.5 iq ] + = Ω e ) = Link ( e and N – 25 – Z i . In general, it may not be taken as a boundary, as , a ) P = ) V = ) V ( P = φ = V 4 i ( ) V Ω ) δ i φ P ) ) in eq. ( P ( → (Ω (Ω φE ( P V φ φE 0 j 0 ) ( ) φ δ δ iq φ V V φE P (Ω e φ φE ( Ω 0 ) iq φE φE 4 iq δ ∂ (Ω e N V δ iq πn e 0 Z 4 dj , ) ) e 2 δ ) 4 V M V φ = /N Z V , M − φ N , (Ω πn Z 2 φ 0 φE /N [ φ δ j /N πin φ S 2 φ 4 − V e N πn Z M πin ( φ = 2 2 R πin 2 e φE ( e ( U ] + h φE φE U h U φ, a [ h 0-form transformation S is the normalization factor such that N Z N Therefore we have Here, we have regularized the trivialterm. divergence The integral which is equal to the linking number of and the redefinition We can eliminate the symmetry generator by using Here, the delta-function formboundary is of defined a in 4-dimensional eq. subspace but ( it is necessary forrewritten discussing as the transformation law. The symmetry generator can be function, we rewriteintegral the by point using delta-function operator forms. and The symmetry point operator generator in term of spacetime written by the following correlation function: Here, Here, we summarize detailed calculations of the correlation functions. B.1 We first evaluate the 0-form transformation law in eq. ( B Correlation functions JHEP01(2021)173 ) S V ( (B.8) (B.9) 1 by the (B.11) (B.12) (B.13) (B.14) (B.10) δ ) ∧ S ) ( 2 C ( , 3 ?δ ) ) δ electric 1-form S ∧ S . . 4 ( ) ( i ) is equal to zero a 2 M 2 a N S δ C R ( . C R Z ?δ 2 i R ∧ a , . δ a B.11 a ) ) iq ∧ C 4 Z iq S S + ) R e ( M V a ∈ h S 2 R ( aE ) ( iq j 1 ) 2 S 2 e δ φδ C S δ V h in eq. ( ( R , ) ) 4 a 4 ∧ 1 C S ) a S δ , M N M πn ∧ V S S Z N 2 Z ) aE ( ( πin 2 C 1 2 2 2 . ( dj δ δ ) 3 2 + 4 δ C a e 1 a Link ( a ( ∧ 2 4 iS 3 M a ) e δ N M Z N πn , which is equal to the linking number ] n πn N πn ∧ S R a 2 C 2 ( a 2 N ) = Link ( a = 2 4 n C πiq φ, a + 2 ( a [ M 2 N φδ 3 2 R e aE and D δ e π 1 4 πiq a N 2 dj 8 2 φ, a = S S iq M e Z V S R e V i − + – 26 – V Z a S as = = Z = C i i R a a a a = a ) = ) = C C C iq S S R → R R e aE V a V a a ) ) ( j ( iq iq iq 1 S 1 can be written as the boundary of a three dimensional S e S e e δ δ , V V ) ) ( S ∂ ∧ ∧ 1 V S /N Z δ , ) , a a C aE = ( /N /N N πin 3 πn a a dj can be rewritten as 2 2 δ 4 e aE a 4 ( j πin πin M + C 2 2 M S R Z e e aE a Z Z a ( ( a U iq h e aE aE N πn U U 2 h h . We can eliminate the symmetry generator by using ] + ) is the intersection number of S 1-form transformation , V C φ, a N [ B.12 Z S and S B.3 Correlation functions ofWe symmetry here generators show the derivations of the correlation functions of symmetry generators. in eq. ( of Therefore, we have Here, we have again regularized thelocal trivial counter divergence term. Furthermore, theif term we consider a closed surface without self-intersections. The integral and the redefinition where we have assumedsubspace that The Wilson loop Similarly, the symmetry generator can be rewritten as Second, we considertransformation: the correlation function, which represents the B.2 JHEP01(2021)173 V by . ) , . ) V i 1 ( ) aE 1 (B.20) (B.18) (B.19) (B.15) (B.16) (B.17) S j δ i S , 2 ∧ , )) ) S V /N S R ( aE a V 1 j 0 a ), we obtain ( δ V ∩ V 1 S ∧ N , πin R ) πin dδ ), we obtain 2 2 S a ∧ e . ). Note that we /N V ) B.11 + ( i . ( a N S ) 1 πin i B.4 n 2 V 2 aE ) aE φ ( j dδ 1 + B.22 U 1 ∧ δ ) S ∩ S πin ∩ S φE S R R 2 ) and ( j 1 V a i φ V − ( S V ) and ( ) n 1 e 2 R Ω N 1 δ 2 a ( i πin , S N φ 2 −V R B.10 V da B.3 iπn , ( ) N aM φ /N ]+ 1 πin V − n a δ 2 U 2 /N 2 a e n ∧ ) φ,a N (Ω 0 a i φ [ ]+ 0 ) n V dφ iπn δ . Here, the EOM of the photon a , iS S 2 φ,a πin S e − [ S ] ) 2 R φE /N R πin ]. We eliminate S iS j − a φ 2 a V e e n e ( ] n i i ( φ, a N φ ( 1 V ∩ V ) 0 a N [ ) πin 101 δ , 2 in 2 in ∧ S D aM e ) S φ, a φM , − − ( [ /N , V e U e U a ]. We first eliminate the 0-form symmetry ), and obtain ( Z ) ) D ) ) 1 /N n φE /N δ is equal to zero, since we have assumed that 2 2 a S S φ 0 a N ∧ , , U S S Z ) 101 – 27 – h , , B.11 V πin da πin = ( πin /N /N 2 2 N 1 2 = R a a e i /N /N δ − e ( ) 0 a 0 a i a ∧ ( e = 2 ) ) n ( πin πin i N i S φ S aE 2 2 ) S πin πin aE ) n , V e e 2 2 , U ) and ( ( S aM ( ( U S ) e e − 1 , ) , ( ( /N e U /N V 1 dδ 0 a ) ) aE aE a , /N S ∧ aE aE /N U U V V B.10 ) a , a h h , , πin U U S /N πin 2 h h V 2 φ . By the same procedures as eqs. ( πin /N e ( = = ) πin e /N /N 2 are not gauge invariant. a ( 1 2 ( φ φ = = δ e V πin e ( , 2 ( aE R πin aE aE e πin πin 2 U j φ ( aE 2 2 U /N e aE ) n e e ) 2 ( φ U 1 2 a ( ( U ) φE N V ) S , aE , U V πin iπn and φE φE V h 2 , U , − e U U h /N /N e ( h h φ a φE /N are not intersected to each other. By using eqs. ( /N j φ φ = = φE S πin πin 2 2 U πin πin e e 2 2 ( ( e e is deformed in the presence of the current and ( do not intersect. Therefore, we obtain ( φE aE V φE S U U φE aE h h U j U h h the same procedures as eqs. ( Similarly, we can evaluate the correlation function between 1-form symmetry generators, which corresponds to the anomalous Hall effect [ corresponds to the finitehave redefinition discussed which the deformation has in beenthe terms applied currents of in the gauge eq. invariant ( symmetry generators, since B.3.2 Correlation function of 1-form symmetry generators and The correlation function impliesof that the EOM of the photon, i.e., the conservation law The last term Alternatively, one can eliminate the 1-form symmetrythat generator as follows. Here, we assume First, we consider the correlation function of 0- and 1-form symmetry generators which corresponds togenerator the Witten effect [ B.3.1 Correlation function of 0-form and 1-form symmetry generators JHEP01(2021)173 )+ and 1 S φE V (B.23) (B.24) (B.25) (B.21) (B.22) ( U 1 δ a N πn 2 . , + dφ . by a topological ) a )) i C 2 ) V R ) ( 1 S 2 1 ( φ π δ 2 S 2 . δ iα , C are closed. Therefore, 4 i 0 a ∩ S R . + ) n ) 2 ). Note that the linking /N V φ C S C 0 a φE , )+ N j , πn 1 φ 2 φ S V πin ( −V iα R φ and is the transversally intersect- 2 D.48 π iα 2 , e 2 δ e e φ iα ) ( 0 a a C ( ( n V − n N ( a ( πin N e 1 aE 2 φM 2 ∧ i δ φM ) U πin U + C U 2 h aE ) again imply that the conservation C R j iS . Note again that we have discussed e , φ e is not gauge invariant, the correlation of the 4 ( φ = ] aE N M πn i iα j R 2 B.22 ) φE φM e j . ( φ C φ, a π πi U i N [ is non-zero, but the correlation between , 2 2 ) ) iα φ 2 2 e D φM − i iα e φM ) U e Z j 2 ]. h ) and ( ) shows that the correlation function of two 1- ∩ S ∩ S ( . Since ) = – 28 – S C 1 1 2 N , = S S S and φM i 123 , = B.20 /N , ) B.22 leads to U 0 a over ) −V −V i C φE ) after eliminating /N ) φ , , , ) V 0 a dj 122 φ C πin 0 a 0 a V , ( 2 , n n → iα 1 a a e φ πin N δ /N e N ( B.20 2 ) 2 ( φ iα e πin πin V e ( 2 2 aE ( πin e e φM U (Ω 2 ( ( aE ) 0 U e 1 φM U δ ) ( ) S φ U φM φM V 1 , ) , φE , which is equal to zero since both of U U N S πn V h h , 2 in the numerical factor U V /N , /N h is not deformed in the presence of a ) φ = = /N − /N V a as ( πin φ φ φE and πin 2 1 j 2 a e δ πin e C πin ( 2 ( C 2 e R e → ( aE ( φE is deformed in the presence of another ) U 2 aE U h φE S h U aE U V j h ( 1 is zero due to the integration of δ The correlation functions in eqs. ( Technically, the correlation between 0 a 4 N φM πn 2 Note that the resultingconservation correlation of function implies that the EOM of the axion,U i.e., the symmetry generators is physically meaningful. The integral ing number of we obtain The redefinition B.3.3 Correlation function ofWe also 0-form show and the 2-form correlation symmetry function generators of the 0-form and 2-form symmetry generators: of surfaces is called a surface link [ law of the deformation in terms of the gauge invariant symmetry generators. This result coincides withdeformation. eq. ( The second lineform symmetry of generators linking eq. with each ( otherwhich leads is to two consistent 2-form with symmetry generators, the diagrammatic expression in eq. ( lation, we can simply evaluate the 1-form symmetry generators by the redefinition We can consider another property of the correlation function. Remarking the following re- JHEP01(2021)173 , for 82 , ) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.1) (C.2) 2 l (C.11) (C.10) 79 2 ∂ )( 1 l , 2 is defined by L ∂ 0 , ∈ } 2 l ] (see also [ g . g by automorphisms, ) = ( . h 2 . l L 102 ) ) 1 1 l , and ∈ ( h 2 l 1 H ∂ g . l ∂ ( ( , ∈ 2 , } . ∂ 1 h 3 and − G , and , H } , h ) G 3 2 H ) = 2 l → h h 2 . ∈ , h ∈ { 1 1 ∂ 1 G L . In terms of the elements, g ( ∂ . h − , h : L, L 1 g ) { , )( 0 2 1 2 − 2 1 G h ∂ h → gg ) = 1 1 } ∈ 1 l ∈ . h 2 ( ∂ }{ , ∂ , g . 0 ( H 2 ) 3 } := ) g } 1 , h ∂ 2 G, ∂ 0 1 1 × h , h , . In particular, the action ◦ 1 − 2 1 h 1 – 29 – on , L 1 → ∂ { ) = ( h h H − g . h 1 ( ∂ 2 3 ( g . g 1 − 2 ∈ G h 1 , g . h h }{ H . l l − 1 1 1 2 2 ∂ 1 : ) satisfies the following axioms [ ∈ h h − 1 h 1 2 ( , h , h l ) g 1 1 ∂ 1 1 2 h g . l l ) = ∂ ∂ 1 is the identity element in g . h h h ( 1 −} h { l { { l h is a map . 1 , G ∂ L ======( ∈ {− −} ∈ } } } } } } , and l , 2 2 2 3 3 , respectively. They satisfy G 2 -equivalent, that is, for all l 2 g . , H h 1 L 2 1 2 , h , h , h G l {− 1 1 2 G, ., ∈ ∈ , ∂ h, ∂ h h , h h 1 1 2 l 1 1 { { , are ∂ → }{ 2 2 is an action of 1 h h and l ∂ are groups. 2 ∂ { { , . { g . . The Peiffer lifting satisfies g . h 1 H , where l, h H L ∂ 2 , L H 2 ∂ and ∈ ∂ → . G { ∈ ∈ 3 , l H L 2 2 ∈ D , , ( , and ]): 1 1 0 ∈ h h H 2 , , 1 124 , for for g . g conjugation, h for all G are group homomorphisms The Peiffer lifting The symbol The maps The maps 88 5. 3. 4. 1. 2. , in appendix C.1 Axiom ofA 3-group 3-group 83 In this section, we reviewrefer the semistrict to 3-group the or 2-crossed semistrictexample module. 3-group of Hereafter, as we the the simply 3-group.gauge 3-group. theory Next, based We first on wethat present it. show of the a ordinary Since axiom groups, Lie the and we axioms algebra give a of of a simple the diagrammatic the 3-group explanation 3-group, are of and complicated the the axiom compared of to 3-group the 3-group C 3-group gauge theory JHEP01(2021)173 . , 2 } } , ) 1 L ∂ 2 n ∂ , h and 0− 1 ∈ − l (C.20) (C.13) (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.12) l H 2 h, h ISO( ∂ { 2 { on and l ∂ 1 and transla- H − H ) := h is a set of two , n l 0 of ) ∈ 0 ) n 0 0 , O( h hh . h . ) = n H,. for ISO( 0 R and ) 2 , ∈ ∂ ) → ∈ 1 . h 1 2 − . a a ) L as the subgroup, where the ) h h . l and an embedding map ( , , which is called the Peiffer , h ( 0 h ( , ) 2 1 n ) 1 1 2 1 0 ∂ n − h ∂ ! ∂ ∂ hh ) l ( 0 2 2 O( . ll a . a ISO( H,. := ) ), ) ( , and an action ∈ ) = 2 0 2 are 2 n 2 , = l ∈ 0 1 h h ∂ ∂ ( A → ) H 0 A ( 0 C.6 2 ! . l O( | ! n 1

1 ) n L , 0 1 ∂ ( → 2 a h . . 1 ) = ) R a = ∂ 2 1 , ) A n satisﬁes these axioms. On the other hand, −→ L 2 h . ∂ l 0 1 h + a ISO( 1 ( ( ( ) ) : ) = 1 2 2 0 1 2 + 1

∈ n a ∂ 2 a ∂ ∂ ) := ( 2 1 n ) ∂ , h 2 2 ( 1 h – 30 – A H,. ∂ h ! a ( ) := ) = ISO( 1 M ( ◦ 2 2 1 1 2 2 ∂ → a ∂ 1 a A h ∈ 2 ∂ and 0 1 ( ∂ 1 1 ∂ 1 L 2 satisﬁes −→ 0 1 are deﬁned as h n ( ! A A ∂ ( 1 n a R 1 ) =

L L

2 ∂ R ∈ T n -dimensional Euclidean group (or isometry group) a = ∈ = A → 0 n 1 0 2 1 + l a h h H 1 1 ( : a h on on 0 ( , respectively. The (strict) 2-group 2 . 1 L ) = ∂ H ∂ n ∈ ∈ and l . Note that the product of two elements ) and l 0 ISO( H ( , a group homomorphism is compatible with the action, 2 2 ∂ L satisfy to 2 ∂ and → . In other words, there are a projection map ∂ 2 is generally not a 2-group. From eq. ( n , T n 1 H H 0 R ) ∂ and : ∈ 0 . H satisfies 0 G, . 1 1 ∂ The Euclidean group can be decomposed into the orthogonal group We note that the 3-group contains a 2-group ∂ → h, h . The map H and The maps and respectively. These maps aresatisfies compatible with products. In fact, the embedding map The actions of tion group and we show thatabbreviate we can decompose the Euclidean group as a 3-group. Hereafter, we is Before explaining the 3-groupthe gauge 3-group. theory, It we is give given a by an simple and non-trivial example of identity. One can easily check( that follows. In this sense, the Peiffer liftingC.2 measures the failure of Example the of Peiffer 3-group identity. actions for groups L The action of JHEP01(2021)173 )– from C.7 ) (C.26) (C.27) (C.28) (C.29) (C.30) (C.31) (C.21) (C.22) (C.23) (C.24) (C.25) n O( . , ! . , as 2 ! ! ! 2 a 1 1 1 1 ∂ a a a 1 A , but this decomposition A n ! 1 + + . 1 1 1 − 1 2 R 1 a − − 1 a a A 1 ) 1 1 2 , n 0 1 A → 2 − 1 1 − 1 2 A 0 1 A R ) A a , . 1 A A ) . h n ) , + 2 1 − 2 A 2 → + 2 ) . , 1 A a h A 1 1 1

1 1 ) ( ( 1 a h − 1 itself by using ∂ − 1

0 1 n a 2 1 1 ( ISO( A = 1 ) A 1 A − 1 = 2 − 1 ∂ − n . ∂ . ∂ ! A ( → are diﬀerent. The diﬀerence is the lack of A A 2 1 1 0 1 ! n ISO( 1 2 ! 0 1 ) 1 2 0 1 − A 2 A A 1 n A A 1 × h

ISO( a 1 − 1 1 0 1 2 ) are transformed under an action generated by A . h − 1 O( 0 1 A 2 can act on itself as h A n := = ) = ) = ) ) 0 1 – 31 – − 1 A

) 2 2 A 1 − 2 1 n h as 1 2

n h a − ( − 1 h ) 2 ) . h . O( 1 := 2 ! .A h n A 1 ∂ 1 ) = ! 1 : ISO( } 1 2 0 1 } 0 2 ! ISO( a A A . h 2 A A ( ( 0 , 1 1 1 ) 1 2 −} , h 2 0 1 ISO( 0 1 and , and 1 , h 1 ∂ , ∂ 1 A A A ) a 1 0 1 h h 1 1 A

( n

h { − 1 1 {− { A . Therefore, we can define the Peiffer lifting, ∂ h ( = 2 = = . We can measure the difference by ( 2 2 1 ∂ 2 1 1 := := ∂ h ISO( ∂ − 1 1 − 1 2 2 , h h h . h n a 2 2 ) . h R h . h 1 1 1 1 1 h h h ( A A 1 ∂ , we can construct an action of 1 ∂ by ) ). . We denote and define the action as n Finally we determine the Peiffer lifting. Since we can obtain an element of The elements of ) due to the projection n 1 C.11 which satisfies One can explicitly check( that these definitions satisfy other axioms of the 3-group ( which can be an image of as Therefore, two actions a ISO( On the other hand, the element of and and One can check that the actions are compatible with is not compatible with the product. O( Note that we would decompose JHEP01(2021)173 , g ∈ 0 (C.43) (C.35) (C.36) (C.37) (C.38) (C.39) (C.40) (C.41) (C.42) (C.32) (C.33) (C.34) . g g , , } } 2 , 3 ] } 2 , h 3 l , h . } 2 ) 1 3 , h h , ∂ 2 . l { , h , 1 h l 1 g } . { 2 ( 2 h ) 2 ∂ . { 2 ∂ 2 ) , h . h 1 ∂ by automorphisms, 2 1 h h ∂ ] = [ l , g 1 ( ) = 2 l . h 1 ∂ 2 ∈ → , l h ) . } − { ∂ 1 ] l 1 { l 0 3 . l ( } − l + ( : [ h ] 2 1 3 . + , g } 2 , h , . In terms of the elements, ] ∂ l g } } = 0 1 3 ( , h } ∈ 2 2 l 1 2 2 , and . l − , h , ∂ → is defined by the commutator, h , h , g , h , ∂ ] := [ [ , respectively. The Lie 3-group (differential ∂ 2 h ] ) , h ) l 1 0 1 2 , g h , 0 2 h ] 1 h ◦ 2 h [ h 2 ∈ 1 h , 1 { 1 × h . g , h . h . h → ∂ . g { 1 , l 2 – 32 – ∂ 1 ( h h 1 g g , ∂ g h ∂ h g h l ( 1 , − { { { , and − { : 1 g h h ∂ 1 1 = = [ = = [ = = , ∂ ∂ ∈ , g } } } } } } 0 2 3 1 2 2 ] ) = l l g 3 2 2 as h , h , h , h ] = [ is a map 1 ] 1 1 2 , h , ∂ , ∂ 2 . ∂ L on 2 h h 1 2 ( L l , h { { , respectively. They satisfy h −} h , h l g [ . 2 . The action 1 2 , 1 l { . , ∈ ∂ ∂ h g 1 ∈ h [ ∈ { g [ 2 ∈ + 1 , h {− 2 , and { 1 , ∂ g { } l 1 l 2 H . l , g , h G 1 and l and 2 are Lie algebras. . The Peiffer lifting satisfies ∂ H l h l -equivalent, that is, { , and g ∈ ∈ ∈ h 2 2 2 , , , ∈ -equivalent homomorphisms 1 1 1 are g , and h h h 2 h , is an action of . h 1 , for for ∂ . g for g are The Peiffer lifting The maps 5. 4. 3. 1. 2. We consider the (background) gauging of theand 3-group. their In gauge order transformation tothe introduce laws, the Lie we gauge algebra fields need of the2-crossed Lie module) algebra is of defined the by 3-group. the following We axioms: denote C.3 Lie algebra of 3-group JHEP01(2021)173 (C.50) (C.51) (C.52) (C.53) (C.54) (C.55) (C.56) (C.57) (C.44) (C.45) (C.46) (C.47) (C.48) (C.49) . . } , respectively, 2 2 , respectively. } Λ α i α w .B w − α 3 { 1 C = − 2 , g l = 1 3 , and h , C } ] a b v , and { 1 , v . } − { a , } , 1 Λ v , and } ) 2 i [ 1 a A , h b 1 -valued 0-, 1-, and 2-form gauge − v 0 − l , 2 u ,B h a ]. We introduce 1-, 2-, and 3-form 2 g 2 { l , B = ∧ A ) B 2 B 1 . a ∂ a 1 83 { as -valued differential forms, respectively. } , h l = gu h b l ( , + . v , + 2 1 2 , } − { -, and 0 A 1 1 , v 1 2 1 82 3 h A B Λ , a i A A = u , v e , h , ( . h -, 79 b { 2 A , ) .B ∧ .C = -, and , and a 1 l b 1 v ) 0 1 = α u , G 1 h 1 h 1 2 , h a a h ) A ) A – 33 – A 1 ∂ v , g A A A α a -, ∧ b 1 { ∧ g A , v g + w i h + + + a A 1 , 2 g . w + g . v 1 ) ) 1 = ∂ g . u 2 − 1 ( 3 ( ∧ ∂ 0 2 0 1 h 2 ( h ( a α 2 A 1 1 3 a 1 a 1 be the 1 B A α = 2 ∧ dA dB dC . l l A B C h h A ∂ α 1 α 0 1 + = = ( = = = = = ( = h w w := := := A 1 α 2 2 1 2 3 1 1 1 , } l l − i 1 − F G h h + H 2 1 , which are 1 − ∂ 2 . h 3 ∧ = , h ∂ 0 2 0 1 dh gdg dl 1 C g . C g . A g . B = 2 l h A B a + + + { v 2 1 3 , i , and 2 − , and B a g . A g . B g . C = v , a 1 1 = = = A h 0 0 A 0 1 2 3 u C A B = 1 → → → h 3 1 2 , C A B g The field strengths are defined by By the structure of the Lie algebra, the gauge transformation laws are given as follows. Here, we explain the 3-groupaxioms diagrammatically. seems The abstract, definition of but the wea 3-group show more based that intuitive on the way. the axiomstranslated In of particular, to the we the 3-group show only can thatviewpoint be one of all understood simple the of higher-form in the statement symmetries. axioms “the of group the elements 3-group are can topological” be in the Note that thesubstituting 3-group gauge theory for the axionD electrodynamics can be Diagrammatic obtained expression by of 3-group Here, we have used the following notations: Let parameters, respectively. Then the gauge transformations are given by Now, we formulate agauge 3-group fields, gauge theory [ If we write the basisthe of gauge the fields Lie can algebras be written as C.4 3-group gauge theory JHEP01(2021)173 , and (D.3) (D.4) (D.5) (D.6) (D.1) (D.2) H case by , G = 4 -dimensional 3) D − are expressed by D ( L 0 h 0 g -, and h , and 2) g H − , D G ( = g l h -, = are represented as a dotted surface, 1) − L 0 l D ∈ 0 0 ( = g = = h L l as 1 for simplicity, which is sufficient to describe L h g can be graphically expressed as follows: – 34 – = 3 L = l , and g D = , and = ∈ H l H h , ∈ G H 0 1 , and 0 = = ll = 0 , H G gg hh l g ∈ h ∈ h G , = 1 = = G 0 0 ∈ 0 gg ll hh g By using the left arrow, the group operations can be expressed as follows: The elements as symmetry groups of 0-, 1-, and 2-form symmetries. Unlike ordinary higher-form The identity elements, line, and point, respectively. Frequently, we abbreviate these identity elements to nothing. Here, the right-hand sidesblack left of arrows the represent above thegroups equations order can of are freely the move projections products. as of long We the as require they that diagrams. intersect the with elements The the of left the arrow. all of their objects.surfaces, lines, In and points, thisextending respectively. case, the objects Note the such that as groupdirection, worldvolumes, we worldsurfaces, elements e.g., can and of temporal easily worldlines along direction. have the the fourth We identify the elements of groups topological objects respectively. By theL identification, we may relate thesymmetries, groups the topological objectstopological may objects. exit as Hereafter, the we boundaries take of one-dimensional higher D.1 Elements of groups as topological objects JHEP01(2021)173 as L (D.7) (D.8) (D.9) (D.10) (D.11) (D.12) (D.13) ∈ 1 − l as intermediate , and 1 l − H l 2 h @ ∈ 1 − and h 1 , − = . Since we have regarded the G g 2 , 1 ∈ = = ∂ = = 1 1 − 1 l g h g l G 2 H 1 1 @ h g with the inverses – 35 – ======h 1 1 1 h l g , respectively. One of the properties of the inverses is and l h G g l L 1 h 1 g 1 @ and give the elements of the interior from the boundary elements: H h 1 2 h , , 1 = g = = ∂ = = = 1 1 1 = G H L ll 1 1 1 gg hh : taking interior of topological objects 2 h 1 ∂ @ h and 1 ∂ Finally, we express the inverses of the elements, D.2 Next, we consider diagrammaticelements expression of of the the groups as maps generallyother extended objects. objects, The the maps elements can be boundaries of the that we can connectstates the of the object annihilation, respectively: objects which annihilate They are explicitly described as follows: JHEP01(2021)173 . D.5 (D.18) (D.16) (D.17) (D.14) (D.15) to move L ∈ l ) l ) 1 2 2 2 can be simply described as l @ h g 1 1 0 l L h ( ( 2 gg ∈ 1 @ l = @ l , following the above diagrammatic )= 2 1 and )= 2 L 1 @ = l 2 1 l l 2 H h 2 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 l 2 @ 1 2 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 h @ l g ∈ @ )( 1 1 l 1 0 h , and h )( l g , 1 2 l as follows: = H 2 l h @ G 1 ( , 1 @ – 36 – g − @ ∈ G l ( 0 = g l = 2 l on have a 2-group structure, which we use in section ) is manifest, since an interior of an interior is nothing = @ and on 2 2 2 l G = l H h h G 2 l 1 0 C.20 2 ∈ @ 1 1 g l @ 1 and intersects with the left arrow: 1 g . h l 1 L 1 : 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 2 l g H of h l 1 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@ 1 2 − . in eq. ( ∈ @ @ g l G 2 l ∂ 2 l : enclosing elements by surfaces @ and = 1 G l g 2 ∂ ◦ 1 ∂ While the elements should intersect with the left arrow, we allow D.3 Action of Third, we express theexpressions. actions The action of the enclosing by This property implies that As an application, we can deform vertically as long as the axiom (conversely, the boundary of a boundary isis nothing). just saying The that property the of product groupof of homomorphism the the elements elements: is compatible with the product of the interior The right-hand sides of the above equations are projected diagrams. By the expression, JHEP01(2021)173 = ]. 1 0− 126 (D.22) (D.23) (D.19) (D.20) (D.21) , . h ) 125 h 1 ∂ ( intersects with 0 h 1 0 h ends on . 0 ) ) h ) ) is manifest in our dia- h l ) is satisfied: h . Since 0 1 . . h @ C.3 g C.6 ( g ( ( 1 1 2 @ @ ) can be simply understood as the h )= 0 )= C.4 h l l 1 1 h h H 2 . 1 . @ @ g g g ( g ( given in eq. ( . . h l in eq. ( h 1 g g 2 − , 1 g 0 g g – 37 – ∂ . Therefore, the line of = = 0 gg h braids with the line of = 1 = 1 = = 1 h 0 0 g g h such that the axiom in eq. ( l h l h l . h g . g 2 1 H } ) . @ . 0 @ h g g 1 -equivalence of g g @ h, h G ( acts on the line { 1 h with the action: 1 h 0 2 ∂ , 1 hh ∂ . For the relation between the 3-group and braids, see, e.g., refs. [ 1 − ) 0 . h ) While we have expressed the Peiffer lifting diagrammatically, it is non-trivial whether In particular, the axiom h , the surface of 1 h ∂ 1 ∂ (( the other axioms are satisfiedof in terms the of Peiffer the lifting diagram satisfies or all not. of We confirm the that axioms our of diagram the Peiffer lifting as follows: In the right-hand side, the line of Finally, we express the Peifferas lifting a diagrammatically. braid We of determine two the elements expression in of it D.4 Peiffer lifting: braiding of elements of gram. Furthermore, the compatibility of JHEP01(2021)173 . 1 − 3 } 3 1 . h (D.26) (D.25) (D.24) 2 ,h l ) 2 2 2 h h { 1 2 .@ . ∂ ) ( 1 ) g 1 . 2 l h g ) 2 1 1 . @ @ h G ( 1 } ∂ 1 } 0 1 @ g = 1 h ( 2 1 = ( 1 . l h 1 3 0 1 2 3 1 h h ∂ − 3 l h H 2 2 . g ◦ h, g 1 @ . h 1 ,h . )) ∂ 1 2 g )) h h 2 h 2 { l 1 { H 2 h h 1 = 2 g @ ( 1 } ∂ 1 3 = 3 @ , since )( h ) for the third expression. ( 1 1 1 ,h l . − 2 2 h 2 = 1 l 1 ) h @ 2 1 2 2 ∂ 1 1 D.10 h h h ) 3 . ∂ 1 { 2 g h @ ) h = (( ( 2 1 – 38 – = 1 l h 1 . 2 h − 3 ( . ) ∂ 1 3 1 ) ◦ 0 2 h h 1 l . h ) 1 1 h h 1 2 @ 0 ∂ 1 . )) 1 } ( h } l @ h 2 2 1 3 ( l g h h = ( 2 2 1 ( ,h l @ 1 h h 2 h , = ( − 2 h 1 1 l 1 l . l 1 1 g 2 1 1 2 ∂ 1 h 2 g ( ∂ @ 3 { 1 ) h { h 2 h 1 } ) . h 3 3 = = 1 = = 1 h )) 2 ): ): ): h 1 2 ,h } 2 ( h 0 2 h 2 3 h 3 l h 1 . h 1 h C.9 C.8 C.7 1 ) h 2 ) 1 h h, h 2 ( l 1 h { 1 { 2 h ( h l 1 @ 1 1 1 . ( h l h @ ( ( g ⇥ = Here, we have used Here, we have used Equation ( Here, we have used the property inEquation eq. ( ( Equation ( • • • JHEP01(2021)173 1 (D.28) (D.27) 1 ). Here, l l 2 , and the 0 . . and by the C.1 .@ h 1 1 3 h − @ 1 h h 0 with 2 @ h 1 2 hh ∂ h 2 . l l h 1 = 2 1 h . @ 0 H 1 ). 1 h 1 } @ 0 h 3 = 1 @ ,h D.16 h . 1 2 } h l 3 h } 2 { 1 2 2 @ ,h h h @ 1 . h h 1 is a 2-group (see appendix 1 1 }{ = h ) 2 1 0 l h , the compatibility of @ 0 h ,h , . 1 1 H,. l l h 2 { 2 h } , which may be complicated. Let us express the @ ∂ 0 → – 39 – 3 . l = L 3 2 ,h ( = 1 h @ 1 h 2 { . h l 2 2 l 2 are defined by conjugation } @ 1 0 = 3 , respectively. In order to reproduce the 2-group structure, . h L } }{ .@ ) 2 ,h 3 1 } , h l 1 h h l 1 2 1 2 ,h h . = ∂ and − 1 @ @ { l 1 h 2 } h h, H ∂ 3 = ( } 1 { }{ 2 h ): ): l 1 @ 2 }{ 2 − 1 on ,h l = h ,h 2 1 ,h l l l, h 1 1 H C.11 C.10 h 0 1 2 l l h h { as 2-group 2 @ { { of h . { @ ) 0 . = = = L, H ( In order to obtain the last equation, we have used eq. ( Equation ( Equation ( One of the advantages of the diagrammatic expression is that we can straightforwardly • • Peiffer lifting we should diagrammatically showPeiffer the identity action reproduce them, in particular the action D.5 We have shown that thecan 3-group also can describe be thethe fact diagrammatically that action expressed. the By set using them, we JHEP01(2021)173 , 1 − ) = h l 0 0 (D.31) (D.29) (D.30) hh h . } ( = ) 2 0 l ∂ ,h 2 h 1 , respectively. 0 @ : ( H l 0 H 1 h . 2 . l ∈ @ 2 h { @ , which can now be h l 2 1 1 1 l by using the following 2 0 2 l l . h h } 0 for the 3-group, we can 0 2 with = 1 . @ l . , h l 2 1 L 2 hh 1 l l ∂ 1 − 2 ∈ l @ 2 h l = h ∂ 1 } ) 1 { l l − 1 2 l l ,h = and @ 2 = = 1 with the action of = l ( 1 l 1 H h 2 l l l 0 ) ∂ 2 } ∈ 1 defined by conjugation h @ 1 l h . { h l 2 H 2 0 @ @ 2 h , ∈ l ( 1 0 = 1 h 2 – 40 – l l l h 2 2 l 2 1 1 @ @ on @ { 2 l h H by enclosing ∈ l = 2 2 0 l l l = h h . ) in order to have the fourth expression. The above defor- ) in the first line. h 0 h . As in the definition of the action as symmetry groups which parameterize the 0-, 1-, and 2-form and 0 = = D.17 0 0 . D.17 . L h 0 h . = 1 1 h . h − 1 l h , and 1 ) l ) l H 2 l 0 l ∂ 2 , 2 ( . 0 l @ G h . h ( l h 2 ) = 1 @ l l 2 ∂ The 2-group should satisfy the Peiffer identity ( 0 = D.6 Global 3-groupFinally, symmetry we and consider symmetry how generators generators to of relate the the higher-form abovewe global diagrammatic identify symmetries expressions given to by the the symmetry 3-group. In the following, Here, we have used eq. ( Here, we have usedmations eq. automatically ( show theh . compatibility of shown as follows: Second, we can simplythe reproduce deformations: the definition describe the action First, we consider the actionwhich of can be expressed as follows: definition of the action JHEP01(2021)173 , H , , and , and G ) h (D.34) (D.35) (D.36) (D.37) (D.38) (D.32) (D.33) , W g C ( , W , respectively. , ) )) 0 Φ ) = 1 , , , P P 2 V , ∂ S Φ( P , ( ) ) = 1 ) = 1 0 l, Φ ) P ( W 2 , P C V , ) , U := Ker P S C S . ( W gl C AAADKXichVI7TxtBEP58JARIwiM0SGksEIgIyZqjSURlQUMJODZI2LLujsWsuFfu1kbm5D+QP0BBA0gUET8jTVoKpJCWCqUkUpoUmdszIEDAnm7325n55rVjh66MFdF5zuh58bL3VV//wOs3bweHhkfeVeKgGTmi7ARuEK3ZVixc6YuyksoVa2EkLM92xaq9vZDqV1siimXgf1btUNQ8q+HLTelYikW1ynRSdSw3X+rUKx/qwxNUIL3yD4HZBRPF8erMAYClYCTXgyo2EMBBEx4EfCjGLizE/K3DBCFkWQ0JyyJGUusFOhhgbpOtBFtYLN3mvcG39a7U53vqM9Zsh6O4/EfMzGOSzugbXdEPOqFL+veor0T7SHNp82lnXBHWh76Olf4+y/L4VNi6ZT2Zs8ImPulcJeceaklahZPxW7t7V6W5lclkio7oN+d/SOf0nSvwW3+c42Wxsq+9+8zZ0dV6Or7P/U10l4WuPcXhTS8t7knnSZ5iy2teiqXWNfQbpLcG16ee8WFzxZ0uSl9h46afQRafJ8e8PycPQWW2YFLBXOYRmke2+vAe45jmOfmIIhaxhDLH/4I9HODQODFOjZ/Gr8zUyHU5o7izjIv/XWK3cQ==AAADKXichVI7TxtBEP58kIRHCBCaSDQWiAgUyZqjIUplJQ0lYGyQsGXdHYuz4l7crR05J/+B1JEo0oRIFIifQUNLgYRpU0WUINGkyNyeAyIosKfb/XZmvnnt2KErY0XUzRl9/U+ePhsYHBp+PvJidGz8ZSUOmpEjyk7gBtG6bcXClb4oK6lcsR5GwvJsV6zZ2x9S/VpLRLEM/FXVDkXNsxq+3JKOpVhUq8wmVcdy86VOvTJXH5umAumVvw/MHpguTlXffO0W20vBeK4PVWwigIMmPAj4UIxdWIj524AJQsiyGhKWRYyk1gt0MMTcJlsJtrBYus17g28bPanP99RnrNkOR3H5j5iZxwyd0gFd0jEd0i/6/V9fifaR5tLm0864IqyPfnlVun6U5fGp8PGW9WDOClt4q3OVnHuoJWkVTsZvfd69LL1bmUle0w+64Pz3qEtHXIHfunL2l8XKN+3dZ84nXa2n4/vc30R3WejaUxze9NLinnQe5Cm2/MtLsdS6hn6D9Nbg+tQjPmyuuNND6Sts3vQzyOLz5Jj/zsl9UJkvmFQwl3mE3iNbA5jEFGZ5ThZQxCKWUOb4O9jFd+wZh8aJcWacZ6ZGrseZwJ1l/PwDPai49w==AAADKXichVI7TxtBEP58kIRHCBCaSDQWiAgUyZqjIUplJQ0lYGyQsGXdHYuz4l7crR05J/+B1JEo0oRIFIifQUNLgYRpU0WUINGkyNyeAyIosKfb/XZmvnnt2KErY0XUzRl9/U+ePhsYHBp+PvJidGz8ZSUOmpEjyk7gBtG6bcXClb4oK6lcsR5GwvJsV6zZ2x9S/VpLRLEM/FXVDkXNsxq+3JKOpVhUq8wmVcdy86VOvTJXH5umAumVvw/MHpguTlXffO0W20vBeK4PVWwigIMmPAj4UIxdWIj524AJQsiyGhKWRYyk1gt0MMTcJlsJtrBYus17g28bPanP99RnrNkOR3H5j5iZxwyd0gFd0jEd0i/6/V9fifaR5tLm0864IqyPfnlVun6U5fGp8PGW9WDOClt4q3OVnHuoJWkVTsZvfd69LL1bmUle0w+64Pz3qEtHXIHfunL2l8XKN+3dZ84nXa2n4/vc30R3WejaUxze9NLinnQe5Cm2/MtLsdS6hn6D9Nbg+tQjPmyuuNND6Sts3vQzyOLz5Jj/zsl9UJkvmFQwl3mE3iNbA5jEFGZ5ThZQxCKWUOb4O9jFd+wZh8aJcWacZ6ZGrseZwJ1l/PwDPai49w==AAADKXichVI7TxtBEP58kPDIg1cTicaKReQ01lwaEBVKGkrA2CCBZd0di7PiXrlbG8Hp/gB/gIIGIqWI+Bk0tBRIQEuFUoJEkyJz6wtRgoL3dLvfzsw3rx07dGWsiC4LRl//s+cDg0PDL16+ej0yOjZej4N25IiaE7hBtGpbsXClL2pKKleshpGwPNsVK/bWp0y/0hFRLAN/We2EouFZLV9uSsdSLGrUy8m6Y7nFatqsv2+OlqhCehUfAzMHJeRrIRgr9GEdGwjgoA0PAj4UYxcWYv7WYIIQsqyBhGURI6n1AimGmdtmK8EWFku3eG/xbS2X+nzPfMaa7XAUl/+ImUVM0Tl9p1s6pWO6oZ//9ZVoH1kuO3zaXa4ImyN7b6r3PVkenwqf/7CezFlhEzM6V8m5h1qSVeF0+Z3d/dvq7NJU8o6+0g/O/4gu6YQr8Dt3zrdFsXSgvfvM2dbVejq+z/1NdJeFrj3D4UMvLe5J+iRPseVvXoal1rX0G2S3FteneviwueI0R9krbDz0M+jG58kx/52Tx6D+oWJSxVyk0tzHfIYGMYm3KPOcTGMO81hAjeN/wT4OcWQcG2fGhXHVNTUKOWcCfy3j+hdwaLXo . This assumption is sufficient V ⌦ ( H , and 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 W Link ( Link (( Link ( ) ,L ⊂ 2 C 1 ∂ ∂ if if if h, ( = = Im 1 / U Ker . , the corresponding symmetry generators . , ) are projected diagrams. gl Ab ) ) ) L ) S H i W ) i V i P C – 41 – g, ) ∈ ) D.38 ( ( ( S W := Φ ( 0 l V C . 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 ( P 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U W S gl 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V are a point, a closed line, and a closed surface. We ( W Φ( V h h V and h ) ) ) and ( ) S . g h l ,H = ( ( ( gl as unitary representation matrices (c-number) of 1 0 1 2 , the assumptions require that the symmetry groups = = ∂ ) H ) l D.37 R R R ( ) ) 2 W , and ∈ Im = = = V C R i i i W S ( P h D.6.5 is the Abelian part of ) ) ) ( C ( are a closed surface, a closed line and two points. The symmetry G/ . , Φ V . , 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W W ) 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V 0 S P C gl Φ Ab := ( ( , and P . P G H ) , V gl )Φ( W h P )) ) ( ∈ G S 0 ( 1 C P g g, R , ( h, , 0 ( ) P 1 ( U g , and h ( U l, h C 0 ( , 2 R as unitary representations: are reduced to S U ) h L V S ( , respectively. The right-hand sides of eqs. ( We now show the diagrammatic expressions oftransformations. the symmetry The generators and charged their objects symmetry can be diagrammatically expressed as follows. respectively. Here, denote l D.6.2 Diagrammatic expressions of symmetry transformations V Let us recall theof symmetry the transformations in groups theare higher-form expressed symmetries. by ForHere, elements topological objects generators can act on the 0-, 1-, and 2-dimensional charged objects respectively. Here, to consider the symmetry generators of theD.6.1 axion electrodynamics. Symmetry transformations form symmetries which do notassumptions have interiors, restrict and the are not symmetrygenerators. boundaries group of Further, other that we objects. assume non-trivially thatwe The parameterizes the will the restricted show symmetry in group appendix alsoand has a 3-group structure. As symmetries, respectively. In the following, we discuss symmetry generators for the higher- JHEP01(2021)173 , ) 1 S − g g, ) and ( (D.41) (D.42) (D.43) (D.44) (D.39) (D.40) 0 U and D.34 g to with the 0-, ) . 1 gl ) ) ) ) − ) L ) V g W ( P S P V W 0 C ( ( ( ( C S U 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( ( ⌦ 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W ) ) . 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V g h l 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W g ) h gl ( ( ( l ( ( and 0 1 ( 2 0 ) can be visualized by the H 1 2 ) R R R , R . R g R parameterized by ( gl 0 ) D.35 = G = = 1 U = = = − 0 g . ( )) 0 P 0 0 )) ) P 0 P U S P P , ) , g, ⌦ P ( P ( W and 0 ( ) C ) l, U ) ( l, S g ( ( ( C 2 ) 0 2 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W g, U C P ( h, U U – 42 – P 0 ( = 1 h, U ) ( U 1 1 = U = = g = ( ) 0 ) 1 = ) 1 U ) 1 ) l by surfaces 1 ) 1 ( l ) ( 2 g ) ) Φ h 2 ( U h P ( V are topological, we can deform 0 V P ( U ( 1 S 1 S U ) ( 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( U Φ( g U ) ( 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V 0 ) ) ) l . The projections of the diagram can be shown as follows: ) U P ( l ) W S ( ( 2 W g C 2 C ( ⌦ 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 U ( ( 0 U U 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 W 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 W ) has an orientation opposite to the point g ) can be described as follows: 0 ) ) ( P h 0 h ( ( U 1 1 D.33 U U For the 1-form symmetry, the diagram for the unitary representation in eq. ( As in the ordinary quantum mechanics, the symmetry generators can be unitary repre- Here, the For the 2-form symmetry, the unitaryfollowing representation diagram in and eq. its ( projection: the projection of the diagram are respectively given as follows: respectively. Since given by a surface Here, we have enclosed sentations of the symmetry groupswe which can preserve simply the replace group the1-, structures. diagrammatic and expressions This 2-form of implies symmetry the that generators.in For eq. the ( 0-form symmetry, the unitary representation JHEP01(2021)173 ). 4.1 (D.47) (D.45) (D.46) is equal ) S , 0 )) 0 ) ) ) C S P S for the elements , g . g h, g, ( , we can construct P g, ( ( . 0 ( )) ( 1 0 0 gl 0 U l, U U P ( U H , 2 ∈ U P h 0 ( , ⌦ ) } 0 0 1 h, h = ). The symmetry generators h 0 g 1 ( h, h ) h 0 1 { 0 ( : D.39 U ). 2 h ) l } = ( 1 ) 1 U . Note that this diagram can express 2 g ) , h 0 ( ) U g C ], which has been discussed in eq. ( D.44 0 h 1 ( h 0 0 U h, }{ U ( h h 0 101 1 ( h ) 1 U ) or ( ) g U 1 h ( h, h 0 { = – 43 – g U ( D.43 1 0 ) = l U ) ⌦ , the Peiffer lifting is related to the braiding of the ( h 1 2 ) 1 U g } g ( can be respectively expressed as follows: ( 0 D.4 0 ) 0 h ,h U is a set of two cylinders, and the orientation of the cylinder h 0 )) U ( 0 h 1 S ) P { U g , ), ( = 0 P 0 by using eqs. ( } ) h ( 0 ) 1 U 1 1 ∂ D.45 l = h, h h . h . In the case of the symmetry generators, the Peiffer lifting is related to g . l, Im { . from the Peiffer lifting . ( g / ( 2 gl 0 g 0 ( h U , and the diagram is reduced to eq. ( H Ab 1 U ) . For the 0-form symmetry, the symmetry generator is opposite to the one outside = . U ∈ S H ) , 0 and C ⌦ and 1 = ) ) − l ) . h h h, g h, h C ( gl . 0 . 1 g H g gg U ( ( ( 0 1 ∈ 0 As we discussed in appendix U 0 U g . h, U ( 1 elements the linking of two 1-form symmetrywe generators. consider Before the discussing linking the symmetry oflinking generators, group of elements. For the elements D.6.4 Diagrammatic expressionHere, of we Peiffer show lifting theerators. diagrammatic expression We of canh, h the describe Peiffer the lifting symmetry for the generator symmetry gen- In the diagram in eq.inside ( the Witten effect for the axionic domain walls [ Now we consider the diagrammaticby the expression of action theto symmetry generators parameterized U D.6.3 Diagrammatic expression of actions JHEP01(2021)173 , h ), H and and , i.e., ⊂ 1 2 G D.47 (D.49) (D.48) 1 ∂ ∂ ∂ . satisfying = 1 Ker Ker H h Ker G 1 , ⊂ , G 1 G ∂ g . ∂ on , and ⌦ Ker H ) = G )) )) , 4 2 L P P . Therefore, the Abelian cannot parameterize non- , , g . h . h 3 1 ( 2 0 1 G ∂ P P h should belong to ∂ = ( ( ∈ , , 1 ). The last assumption requires = )) } } g ∂ 0 0 and 0 P ,h 1 C.8 ). 0 , hh for ∂ Ker satisfy ⌦ h h, h P { { 4.3 H 2 ( ∈ ( ( ) ∂ 0 2 0 2 can be seen as follows: l, ( C = 1 U ) U can be annihilated by pair creations of 2 is an Abelian part of l C l, Ker l = h, h 2 U 2 . 2 l, ∂ @ ∂ 2 ∈ 0 ( Ab ∂ ), we have l ) 1 ( 0 – 44 – H 1 h . U C and U C.6 , 0 and h h ) = 1 1 l ( : ∂ ∂ 0 1 ) = ) must be Abelian (except for theories on manifolds with ) , where V U 0 2 C h . S ∂ ( ( . Note that these restrictions are consistent with the fact Ker 2 for the elements h, V ∂ Ab ( ∈ p > ⌦ } 1 0 Ker H h U ) ]. as h, h . Therefore, the symmetry groups are reduced to C , and { and G 42 1 themselves but subgroups of them. The subgroups are specified as H l, ∂ 2 2 ∂ L @ = 1 = 1 must be Abelian by the axiom in eq. ( ( ], which has been discussed in eq. ( 1 Ker l h Im 2 2 1 U . By the axiom in eq. ( ∂ contributes to the higher-form symmetries. In the following, we denote the / ∂ . 101 H 1 , and ∂ Ab . Note that we can consistently define the actions of g . ∂ Ker H L H and -form symmetries ( = 1 , , p Ker 1 } G ⊂ H ) = 0 ∂ , since the elements 2 2 . For example, the annihilation of l The second assumption implies that the images of The axiom of the Peiffer lifting gives us some properties of the subsets. In particular, We require three assumptions for the symmetry generators. One is that the non-trivial Let us specify the subgroups by using the assumptions. By the first assumption, ∂ ∂ = 1 h, h Im g . l { l ( 2 2 2 and that the non-trivial topology) [ trivial symmetry generators. The reasonthe can symmetry be generators simply given understood by by using our diagrams: the subset that the Peiffer lifting ∂ part of Abelian part of the symmetry generators are∂ parameterized by theKer elements of Ker ∂ symmetry generators are notsymmetry boundaries generators of do other objects. notalso have Another have boundaries. the is 3-group that The structure. the last non-trivial one is that the restricted groups D.6.5 Symmetry groupsHere, parameterizing we symmetry show generators thatgroups the groups which parameterizeG/ the symmetry generators are not the Note that the diagrammaticionic expression strings can [ describe the anomalous effect around the ax- we find thatgenerators, the which linking can of act on the 1-form symmetry generators leads to 2-form symmetry Now, we discuss the linking of the symmetry generators. By the diagram in eq. ( JHEP01(2021)173 , 1 ∂ H and Sov. Im G and (1979) G/ ∈ Phys. Rev. G (1987) 1 , 43 g ]. Phys. 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