Higher dimensional quantum Hall effect as A-class topological insulator

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ScienceDirect

Nuclear Physics B 886 (2014) 952–1002

www.elsevier.com/locate/nuclphysb

Higher

dimensional

quantum

Hall

effect

as

A-class

topological

insulator

Kazuki Hasebe

1

DepartmentofPhysics,StanfordUniversity,Stanford,CA 94305,USA Received 23April2014;accepted 10July2014

Availableonline 21July2014 Editor: StephanStieberger

Abstract

WeperformadetailstudyofhigherdimensionalquantumHalleffectsandA-classtopological insula-torswithemphasisontheirrelationstonon-commutativegeometry.Therearetwodifferentformulations ofnon-commutativegeometryforhigherdimensionalfuzzyspheres: theordinarycommutatorformulation andquantumNambubracketformulation.Correspondingtotheseformulations,weintroducetwokindsof monopolegaugefields: non-abeliangaugefieldandantisymmetrictensorgaugefield,whichrespectively realizethenon-commutativegeometryoffuzzysphereinthelowestLandaulevel.Weestablish connec-tionbetweenthetwotypesofmonopolegaugefieldsthroughChern–Simonsterm,andderiveexplicitform oftensormonopolegaugefieldswithhigherstring-likesingularity.Theconnectionbetweentwotypesof monopoleisappliedtogeneralizetheconceptoffluxattachmentinquantumHalleffecttoA-class topo-logicalinsulator.WeproposetensortypeChern–Simonstheoryastheeffectivefieldtheoryformembranes inA-classtopologicalinsulators.Membranesturnouttobefractionallychargedobjectsandthephase en-tanglementmediatedbytensorgaugefieldtransformsthemembranestatisticstobeanyonic.Theindex theoremsupportsthedimensionalhierarchyofA-classtopologicalinsulator.AnalogiestoD-branephysics ofstringtheoryarediscussedtoo.

E-mailaddress:[email protected].

1 _On_leave_from_Kagawa_National_College_of_Technology,_Takuma-cho,_Mitoyo,_Kagawa_769-1192,_Japan._After 31 March 2014,emailto [email protected].

http://dx.doi.org/10.1016/j.nuclphysb.2014.07.011

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1. Introduction

Aboutadecadeago,thetimereversalsymmetriccounterpartofquantumHalleffect, quan-tum spin Hall effect, was theoretically proposedand experimentally discovered [1–4]. Since then,topologicalstatesofmatterhavebeenvigorouslyinvestigated[seeRefs.[5–7]asreviews]. Now,weunderstandthereexistavarietyoftopologicalcousinsofquantumHalleffect,suchas topologicalinsulatorswithtimereversalsymmetryandtopologicalsuperconductorswith parti-cleholesymmetry.BasedonageneralizedAltlandandZirnbauerrandommatrix,asystematic classificationofthebandtopologicalinsulatorswasexhaustedinthetopologicalperiodictableof ten-foldway[8–11],wherewereadilyfindtopologicalinsulatorsinanydimensionwithor with-outthreediscretesymmetries,timereversal,particle–hole,andchiral.Forinstance,thequantum Halleffectisassignedtothelowestdimensional(2D)entityoftheA-classtopologicalinsulators thatdonotrespectany ofthethreediscretesymmetriesandliveinarbitraryevendimensional space.TheA-classtopologicalinsulatorsareregardedasahigherdimensionalcounterpartofthe quantumHalleffect.

Recently,severaltheoreticalrealizationsof fractionalversionoftopological insulatorshave beenproposed[12,13],andtwogroupsindependentlyappliedthe non-commutativegeometry techniquestofractional topologicalinsulators [14,15]generalizingthe techniquesused in2D quantum Halleffect[16–19].In theworks, theyproposedquantum Nambugeometry [20,21]

asunderlingmathematicsoftopologicalinsulators.Inparticular,closerelationsbetween quan-tumNambubracketinevendimensionsandA-classtopologicalinsulatorwerepointedout in Ref. [14]where monopolein the momentumspace generates the non-commutativity of den-sityoperators.SinceA-classtopologicalinsulatorsareanaturalhigherdimensionalcounterpart of quantum Hall effect, A-class topological insulators give a good startingpoint to seehow non-commutativegeometryworksintopologicalinsulatorsbeforediscussingmore “complicat-ed”topological insulators,such as AIIclass.2 Beforethe discoveryof topological insulators, 4D generalizationof quantum Halleffectwas theoreticallyproposed inthe SU(2)monopole backgroundby Zhang andHu [28]as a generalizationof the Haldane’s quantum Halleffect ontwo-sphere[29].Ingeneral,higherdimensionalquantumHalleffectsarerealizedin(color) monopolebackgroundcompatiblewiththeholonomygroupofthebase-manifold onwhichthe systemisdefined[30–32].Sincethereexistsmagneticfieldof monopole,higherdimensional quantumHalleffectsnecessarilybreaktime-reversalsymmetryasA-classtopologicalinsulators areoughttodo.ThehigherdimensionalquantumHalleffectcanbeconsideredasarealization ofA-classtopologicalinsulatorwithLandaulevels.3Fromthisperspective,werevisitthehigher dimensionalquantumHalleffectthatisrealizedonarbitraryeven-dimensionalsphere[32,33]. Intheset-upofquantumHalleffectonS2k,theSO(2k)non-abelianmonopoleisadopted,and thesystemrealizesinterestingmathematicalstructures.Forinstance,thenon-abelianmonopole mathematicallycorrespondstothesphere-bundleoversphere[34]wheretheS2k−1-bundleover thebase-manifoldS2k givestheSO(2k)structuregroup.Innon-commutativegeometrypointof view,thesystemcanberegardedasaphysicalset-upofhigherdimensionalfuzzysphereinthe lowestLandaulevel.4Interestingly,higherdimensionalquantumHalleffectsareevenrelatedto supersymmetry[37,38]andtwistortheory[30,39,40].

2 _Recently,_AII_topological_insulators_with_Landau_level_were_constructed_in_Refs._[25–27]_.

3 _In_this_sense,_the_4D_quantum_Hall_effect_was_the_firstly_{“discovered”}_higher_dimensional_topological_insulator. 4 _Such_physical_description_of_fuzzy_sphere_in_monopole_background_is_{“consistent”}_with _the_dielectric_effect_of D-brane[35,36].

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Thoughintheformerarticles,thenon-abelianmonopolesareadoptedintheconstructionof the higherdimensionalquantum Halleffect, theremaybeanothermonopolerealization.That istouseantisymmetrictensor U (1)monopole.TensorU (1)monopoleisamonopole[41,42]

whosegaugegroupisU (1)butgaugefieldisnotavectorbutanantisymmetrictensor.5While the non-abelianmonopole corresponds toan extension of the Dirac monopole byincreasing theinternalgaugedegreesoffreedom,thetensormonopolemanifestsanotherextensionofthe Diracmonopolebyincreasingtheexternalindices.Therefore,theremaybetworeasonable gen-eralizationsofquantumHalleffect,oneisbasedonthenon-abelianmonopoleandtheotheris basedonthetensormonopole.Onemaybeimmediatelyinclinedtoaskthefollowingquestions. WhatdoesquantumHalleffectintensormonopolebackgroundlooklikeandwhatkindof non-commutativegeometrywillemergeinthelowestLandaulevel?Ifhigherdimensionalquantum Halleffecthastworeasonablegeneralizations,isthereanyconnectionbetweenthem?Forsuch questions,theprecedentresearchesofnon-commutativegeometrygiveasuggestivehint: there aretwo(superficially)differentformulationsforhigherdimensionalfuzzysphere[22–24],one ofwhichistheordinarycommutatorformulationandtheotheristhequantumNambubracket formulation.Inspiredbytheobservation,weestablishconnectionbetweenthenon-abelianand tensormonopoleandanswertothequestionsinthiswork.

Topological fieldtheory description of the quantum Halleffect [44,45]has brought great progressinunderstandingnon-perturbativeaspectsofquantumHalleffect.TheChern–Simons effectivefield theorynaturally describesthe flux attachmentthat electron andChern–Simons fluxesarecombinedtoyielda“newparticle”calledcompositeboson[46,47],andthefractional quantumHalleffectisregardedasasuperfluidstateofthecompositebosons[45].The fundamen-talobjectoftheA-classtopologicalinsulatorturnsouttobemembrane-likeobjects.Basedon theconnectionbetweenthenon-abelianandtensormonopoles,weproposeatensortypeChern– SimonsfieldtheoryasaneffectivefieldtheoryoftheA-classtopologicalinsulator.Interestingly, while we startfromthe non-abelianquantum mechanicsin(2k+1)D space–time,the tensor Chern–Simons fieldtheory isdefinedin(4k−1)D space–time.Membraneshaveafractional chargeandobeyanyonicstatistics.ThegroundstateofA-classtopologicalinsulatorsisregarded as asuperfluidstate of compositemembrane atmagicvaluesofthe fillingfactor.Wediscuss dimensional condensationofmembraneswithemphasisonitsrelationtobrane-democracyof stringtheory.

Themaingoalofthispaperistointegratesofarlooselyconnectedsubjects,suchas Nambu-bracket, tensortopologicalfieldtheory andphysicsof quantum Halleffect, tohavean entire pictureof A-class topologicalinsulator [Fig. 1].Thoughwe shareseveralterminologies with stringtheory such as p-branes andC field, the presentanalysis is not directlyrelatedtothe stringtheory:we donotuseeitherstringsorD-branes.Aboutarealizationoftopological insula-torsinstringtheory,onemayconsultRefs.[48,49].ForCfieldrealizationofnon-commutative geometryonM-brane,seeRefs.[50–52].

The paper is organized as follows. In Section 2, we briefly reviewthe basic mathematics ofthefuzzysphereanditsphysicalrealizationinthelowestLandaulevel.Section3describes thetwo mathematical formulations for higher dimensional fuzzy spheres.Weintroduce non-abelianmonopolequantum Halleffectwithor withoutspindegrees offreedom inSection4. Section5discussestheconnectionbetweenthetensorandnon-abelianmonopoles,andgivesa tensormonopolerealizationofthequantumNambugeometry.InSection6,theChern–Simons

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Fig. 1.CorrespondencebetweenmathematicsandphysicsofhigherdimensionalquantumHalleffectsandA-class topo-logicalinsulators.

tensorfieldtheoryisproposedasaneffectivefieldtheoryofA-classtopologicalinsulator,where weclarifythefractionalchargeandanyonicstatisticsofmembranes.Wealsodiscussthe hierar-chicalpropertyofmembranesandA-classtopologicalinsulator.Section7isdevotedtosummary anddiscussions.

2. FuzzysphereandDiracmonopole

Here,webrieflyreviewhowthefuzzygeometryemergesinthecontextofthelowestLandau levelphysicsbyusingthefuzzytwo-sphereandDiracmonopolesystem.Theobservationwill beatemplateforhigherdimensionalfuzzysphereinthesubsequentsections.

Thefuzzytwo-sphere[53–55]isafuzzymanifoldwhosecoordinatesXi (i=1,2,3)satisfy theSU(2)algebra: [Xi, Xj] =iαij kXk, (1) and XiXi= α 2 2 I (I+2)=r2 1+2 I . (2)

Here,αistheunitofnon-commutativelengthandI (integer)specifiestheradiusofthefuzzy two-sphereras

r=α

2I. (3)

Thefuzzysphereisrealizedas thelowestLandaulevelphysics.Wewill showhowfuzzy ge-ometryemergesonatwo-sphereinDirac monopolebackgroundbothfromthe Lagrangeand Hamiltonformalisms.

2.1. HopfmapandLagrangeformalism

TheLagrangianfortheelectrononatwo-sphereinmonopolebackgroundisgivenby

L=M

(5)

wherexi (i=1,2,3)aresubjecttoaconstraint

xixi=r2, (5)

andAi denotetheDiracmonopolegaugefield

Ai= − I

2r(r+x3)

ij3xj, (6)

withDirac monopolechargeI /2 (I integer)[62].Relationtothenon-commutativegeometry willbetransparentbyintroducingtheHopfspinor.TheHopfspinoristhetwo-componentspinor thatinducesthe(1st)HopfmapS3−→S1 S2:

φ→xi=α 2φ †_σi_φ, (7) with φ†φ=I. (8)

xi (7)automaticallysatisfytheconditionoftwo-sphere:

xixi= α 2 2 φ†φ2=r2. (9)

TheHopfspinorφtakestheform

φ= I 2r(r+x3) r+x3 x1+ix2 eiχ (10)

witheiχ denotingU (1)phasefactor,andthemonopolegaugefield(6)canbederivedas

A=Aidxi= −iφ†dφ. (11)

InthelowestLandaulevel,thekineticenergyisquenchedandtheLagrangian(4)isreducedto thefollowingform:

LLLL= −Aixi˙ =iφ†

d

dtφ. (12)

WeregardtheHopfspinorasthefundamentalvariableandderivethecanonicalmomentumofφ

asiφ∗from(12)toapplythequantizationcondition:

φα, φβ∗=δαβ. (13)

Afterthequantization,theHopfspinorbecomestotheSchwingeroperatorofharmonicoscillator expressedas6

φα, φβ∗→ ∂ ∂φα

, φβ, (14)

andthecoordinatesonatwo-sphere(7)turnouttobethefollowingoperators

6 _We_can_derive_the_same_result_in_the_Hamilton_formalism._The_lowest_Landau_level_eigenstates_are_given_by_the holomorphicfunctionofφ,anditscomplexconjugateiseffectivelyrepresentedbythederivativeofφ.

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Xi=α

2φ

t_σi ∂

∂φ, (15)

whichsatisfythefuzzytwo-spherealgebra(1),andthecondition(8)isrewrittenas

φt ∂

∂φ=I. (16)

OnecanreadilyshowthatEq.(15)with(16)indeedsatisfies(2).Theemergenceoffuzzysphere isbasedontheHopf–SchwingeroperatorandthePaulimatricesintheLagrangeformalism.

2.2. Hamiltonformalismandangularmomentum

The3DHamiltonianforaparticleingaugefieldisgenerallygivenby

H= − 1 2MDi 2_{= −} 1 2M ∂2 ∂r2− 1 Mr ∂ ∂r + 1 2Mr2Λi 2_, (17) whereDi representthecovariantderivative:

Di=∂i+iAi, (18)

andΛi denotethecovariantangularmomentum:

Λi= −iij kxjDk. (19)

TheHamiltonianforaparticleontwo-sphere(rconst.)isgivenby

H= 1

2Mr2Λi

2_. ₍₂₀₎

WiththeU (1)monopoleatthecenterofthesphere,thetotalangularmomentumLiisgivenby

thesumofthecovariantangularmomentumandtheangularmomentumofthemonopolegauge field: Li=Λi+r2Fi=Λi+ 1 αxi, (21) where Fi=ij k∂jAk= I 2r3xi. (22)

SinceLi aretheconservedangularmomentum,theysatisfytheSU(2)algebra

[Li, Lj] =iij kLk. (23)

In the lowestLandau level, the kineticterm is quenched Λi=0, and thenxi (∝Fi)can be

identifiedwithLi:

Xi=αLi. (24)

ItisobviousthatXi satisfythefuzzytwo-spherealgebra(1).WithuseofLij=ij kLk,(24)is writtenas

Xi=α

2ij kLj k. (25)

NoticetheconstructionoffuzzyspherecoordinatesintheHamiltonformalismisbasedonthe angularmomentum.

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Consequently, there are two ways tosee the emergence of fuzzy sphere, oneof which is theHopf–Schwingerconstruction(15)intheLagrangeformalism,andtheotheristheangular momentumconstruction(25)intheHamiltonformalism.

3. Non-commutativegeometryinhigherdimensions

3.1. Fuzzyspherealgebra

Asdiscussedabove,thecoordinatesoffuzzytwo-spherearegivenbytheSO(3)vector oper-atorsthatsatisfy

[Xi, Xj] =iαij kXk,

and itsminimal representationis the 2×2 Pauli matrices. SincePauli matricesare equal to theSO(3)gammamatrices,itmaybenaturaltoadoptthe SO(2k+1)gammamatricesasthe coordinatesof S_F2k withminimum radius.ForS_F2k withlargerradius,the SO(2k+1)gamma matricesGa(a=1,2,· · ·,2k+1)offullysymmetricrepresentation,7

k [I 2, I 2,· · ·, I 2],isadopted asthefuzzycoordinates[56,57].IndeedXa≡αGasatisfy

2k+1 a=1 XaXa= α2 4 I (I+2k)=r 2 1+2k I , (26)

which representsthe condition of constantradius of fuzzy sphere.In the limit I → ∞ with fixedr,(26)isreducedtotheclassicalconditionof2k-sphere,_a2k₌+₁1xaxa=r2.

One shouldnoticehowever,thereisabigdifference betweenthefuzzytwo-sphere andits higher dimensionalcounterpart[58–61].ThoughtheSO(3)gammamatricesareequivalentto theSU(2)generatorsandformaclosedalgebrabythemselves,theSO(2k+1)(k≥2)gamma matricesXadonotsatisfyaclosedalgebraamongthemselvesbuttheircommutatorsyield“new”

operators,theSO(2k+1)generatorsXab:

[Xa, Xb] =iαXab. (27)

The appearanceofXabsuggeststhatthegeometry ofhigherdimensionalfuzzyspherecannot

simplybeunderstoodonlybytheoriginalcoordinates.Toconstructaclosedalgebraforhigher dimensionalfuzzysphere,weneedtoincorporateXabtohaveanenlargedalgebra

[Xa, Xbc] = −iα(δabXc−δacXb),

[Xab, Xcd] =iα(δacXbd−δadXbc+δbdXac−δbcXad), (28)

inwhichXaandXabamounttotheSO(2k+2)algebra.Aroundthenorthpole,(27)reducesto

[Xμ, Xν] =iαημνiXi, (29)

whereημνi denotestheexpansioncoefficient(fork=2,ημνi isgivenbythe’tHooftsymbol) andXi standfortheSO(2k)generatorsrelatedtoXμν bytherelation

Xμν= k(2k−1)

i=1

ημνiXi. (30)

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Theextra-degreesoffreedomisdescribedbytheoperatorsXi,andcanbeinterpretedasthefuzzy

fibre-bundleoverS2k.SincethecorrespondingalgebraofS_F2kistheSO(2k+2)algebra,thefuzzy fibredescribedbytheSO(2k)algebraisidentifiedwithS_F2k−2.Duetotheexistenceofthefuzzy bundle,theclassicalcounterpartof S2_Fk isnotsimplygivenbyS2kSO(2k+1)/SO(2k)but

SO(2k)/U (k)fibrationoverS2k[59]:

S_F2kSO(2k+1)/U (k)∼S2k⊗SO(2k)/U (k). (31)

Here,∼denotesthe localequivalence. TheSO(2k)/U (k)-fibreisthe classicalcounterpartof theextrafuzzyspaceS_F2k−2.Asweshallseelater,suchextradegreesoffreedomcorrespondto (fuzzy)membraneexcitation.

Thoughinthecommutatorformulation,theexistenceofthefuzzyfibreisexplicit,the commu-tatorformulationisrather“awkward”inthesensethealgebradoesnotclosewithintheoriginal fuzzycoordinates.TheNambubracketgivesamoresophisticatedformulation.Inthed dimen-sion,quantumNambubracket(orNambu–Heisenbergbracket)[21–24]isdefinedas

[Xa1, Xa2,· · ·, Xan] ≡X[a1Xa2· · ·Xan], (32) wherea1,a2,· · ·,an=1,2,· · ·,d (n≤d),8 andthebracketfor thelowindicesrepresentsthe

fullyanti-symmetriccombination aboutthe indices.Wehave n!terms onthe right-hand side of(32).Forinstance,

[Xa1Xa2] =Xa1Xa2−Xa2Xa1,

[Xa1Xa2Xa3] =Xa1Xa2Xa3−Xa1Xa3Xa2+Xa2Xa3Xa1−Xa2Xa1Xa3 +Xa3Xa1Xa2−Xa3Xa2Xa1

In the quantum Nambu bracket formulation,9 _the _{non-commutative} _algebra _for _S2k

F is given by[22–24] [Xa₁, Xa₂, Xa₃,· · ·, Xa_2k] =ikC(k, I )α2k−1a₁a2a3···a2k+1Xa2k+1, (33) where C(k, I )=(2k)!!(I+2k−2)!! 22k−1_I_!! . (34)

Thus,theextraoperatorsXab do notappearinthequantum Nambubracketformulation,and

theclosureofalgebraisguaranteedonlybytheoriginalfuzzycoordinates.Theextrafuzzy-fibre degreesoffreedomseemtobecompletely“hidden”inthequantumNambubracket.Aroundthe north-poleX2k+1r,(33)isreducedtothequantumNambubracketforthenon-commutative plane: [Xμ1, Xμ2, Xμ3,· · ·, Xμ2k] =i k2k μ1μ2μ3···μ2k, (35) where ≡α I 2C(k, I ) 1 2k =r (2k)!!(I+2k−2)!! I!!I2k−1 1 2k _I_∼∞ ∼ √r I. (36)

8 _For_n_{> d}_,_due_to_the_{anti-symmetric}_property,_quantum_Nambu_bracket_always_vanishes. 9 ₍₃₃₎_essentially_comes_from_the_property_of_the_SO₍₂_k₊₁₎_gamma_matrices,_γ

1γ2γ3· · ·γ2k=ikγ2k+1.Formore detailpropertiesofquantumNambubracket,see Appendix B.

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Forinstance, k=1: =r 2 I 1 2 , k=2: =r 8(I+2) I3 1 4 , k=3: =r 48(I+2)(I+4) I5 1 6 . (37)

3.2. Twomonopoleset-upsforhigherdimensionalfuzzysphere

AsdiscussedinSection2,thefuzzytwo-sphereisrealizedintheDiracmonopolebackground. The easiestwaytofind whatkindofmonopole correspondstonon-commutativegeometry is tofindtheright-handsideofthenon-commutativealgebra.Forinstance,thefuzzytwo-sphere algebraisgivenby

[Xi, Xj] =iαij kXk, (38)

andonecanreadofftheU (1)monopolefieldstrengthfromitsright-handside:

Fij

1

r3ij kxk. (39)

Forhigherdimensionalfuzzysphere,incorrespondencetothetwonon-commutative formula-tions,wewillobtaintwodifferenttypesofmonopoles.

• Non-abelianmonopole

Aroundthenorthpole,thecommutationrelationbetweenthefuzzycoordinates(27)becomes to

[Xμ, Xν] =iαXμν,

where the right-hand side is the SO(2k) generators. This suggests the SO(2k) non-abelian monopolefieldstrength:

Fμν

1

r2Σμν, (40)

whereΣμν denotestheSO(2k)matrixgenerators.Thus,we canidentifyonemonopoleset-up

forS_F2k withtheSO(2k)non-abelianmonopole. • Tensormonopoleset-up

Meanwhile,theright-handsideofthequantumNambubracketformulation [Xa1, Xa2,· · ·, Xa2k] =i

k_{C(k, I )α}2k−1

a1a2···a2ka2k+1Xa2k+1, impliesantisymmetrictensormonopolefieldstrength:

Ga1a2···a2k 1

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Heretwo comments are added. Firstly,eventhough thereare two differentnon-commutative formulations,theydescribethesamenon-commutativeobject,i.e.thefuzzysphere,andthenthe twodifferenttypesofmonopolesareexpectedtodescribesamephysicalsystemcorresponding tofuzzy sphere. In otherwords, the non-abelian andthe tensormonopoles are two different physicalset-ups for the same system. Theyare expected tobe “equal”insome sense.Their connectionwillbeclarifiedinSection5.Secondly,thoughthequantumNambualgebraveilsthe “extra”degrees offreedomof fuzzy-bundle,(2k−1)rankfield(41)impliesthe existenceof

(2k−2)-branewhose(2k−1)-fromcurrentnaturallycoupledto(2k−1)ranktensorfield.This observationwillbeimportantinconstructingtheChern–SimonstensorfieldtheoryinSection6.

4. Non-abelianmonopoleandhigherdimensionalquantumHalleffect

Here,we give non-abelianmonopole realization for higher dimensional quantum Hall ef-fect [32,33].The SO(2k) monopole gaugegroup is adopted so as tobe compatible with the holonomyofthebase-manifoldS2k.10

4.1. SO(2k)non-abelian monopole

FirstletusintroducethegeneralizedHopfmap:

xa=αΨ†ΓaΨ, (42)

wherexa(a=1,2,· · ·,2k+1)aresubjecttotheconditionofS2k:

xaxa=r2, (43)

andΓa(a=1,2,· · ·,2k+1)denotetheSO(2k+1)gammamatrices: Γi= 0 iγi −iγi 0 , Γ2k= 0 1₂k−1 1₂k−1 0 , Γ2k+1= 12k−1 0 0 −12k−1 , (44)

withSO(2k−1)gammamatricesγi (i=1,2,· · ·,2k−1).TheSO(2k)generators Σμν≡ −i

1

4[Γμ, Γν] (45)

taketheformof

Σμν= Σ_μν+ 0 0 Σ_μν− , (46)

wheretheSO(2k)Weylgeneratorsare

Σ_μν± =Σ_ij±, Σ_i,±₂_k= −i1 2γiγj,± 1 2γi (i=j ) (47)

10 _The_present_monopole_set-up_is_quite_similar_to_the_{Kaluza–Klein}_monopole_in_the_sense_that_the_geometrical informa-tiondeterminesthecorrespondingmonopolegaugegroup.Kaluza–Kleinmonopoleaccompanieswiththespontaneous compactificationoftheKaluza–Kleintheory [63,64],andtheisometryofthecompactifiedspaceistransferedtothe gaugesymmetryoftheuncompactifiedspace.Forinstance,S2k−1compactificationyieldstheSO(2k)gaugesymmetry ofnon-abelian monopole [65].

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Notice that the SO(2k) Weyl generators (47) consist of the SO(2k−1) generators and the

SO(2k−1)gammamatrices.The2kcomponentspinorΨ thatsatisfies(42)isgivenby

Ψ = 1 2r(r+x2k+1) (r+x2k+1)12k−1 x2k12k−1−ixiγi ψ, (48)

whereψisa2k−1componentnormalizedcomplexspinorψ†ψ=I.WithuseofΨ,theSO(2k)

non-abeliangaugefields[66–70]canbederivedbytheformula

A= −iΨ†dΨ, (49) whereA=Aadxawith Aμ= − 1 r(r+x2k+1) Σ_μν+xν (μ, ν=1,2,· · ·,2k), A2k+1=0. (50)

ThefieldstrengthF =dA+iA2orFab=∂aAb−∂bAa+i[Aa,Ab](F =1₂Fabdxa∧dxb)is evaluatedas11 Fμν= − 1 r2xμAν+ 1 r2xνAμ+ 1 r2Σμν+, Fμ,2k+1= 1 r2(r+x2k+1)Aμ. (54)

Aroundthe northpole, x2k+1/r1 xμ/r0,thefieldstrength(54)isreduced to(40).Itis

obviousthatundertheSO(2k)gaugetransformation

Ψ → g 0 0 g Ψ, (55) withg g= 1 1−x2k+12 (x2k12k−1+ixiγi), (56)

AandF aretransformedas

A→g†Ag−ig†dg,

F →g†F g. (57)

11 _The_component_fields_of_A

aandFabarerespectivelygivenby Aa= μ<ν AaμνΣμν+, Fab= μ<ν FabμνΣμν+, (51) where Aaμν= − 1 r(r+x2k+1) (δaμxν−δaνxμ) (52) and Fρσμν= 1 r3_(r₊_x 2k+1) (δρμxσxν−δρνxσxμ+δbμxρxν−δσ νxρxμ)+ 1 r2(δρμδσ ν−δρνδσ μ), Fρμν,2k+1= − 1 r3(δρμxν−δρνxμ). (53)

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Thehomotopytheoremguaranteesthenon-trivialbundletopologyoftheSO(2k)monopoleon

S2k12:

π2k−1

SO(2k)Z, (58)

whichismeasuredbythekthChern-number:

ck= 1 k!(2π )k S2k trFk. (59)

Inlowdimensions,(59)yields

ck=1= 1 2π S2 trF, ck=2= 1 8π2 S4 trF2, ck=3= 1 48π3 S6 trF3, ck₌4= 1 384π4 S8 trF4. (60)

FortheSO(2k)fullysymmetricrepresentation

k [I 2, I 2,· · ·, I

2],theChern-numbersarecalculated as[36] ck=1=I, ck=2= 1 6I (I+1)(I+2), ck=3= 1 360(I+1)(I+2) 2_(I₊ 3)(I+4), ck=4= 1 302 400I (I+1)(I+2) 2_(I₊₃₎2_(I₊₄₎2_(I₊₅_)(I₊₆_), ₍₆₁₎

whichcorrespondtothemonopolechargeorthenumberofmagneticfluxesonspheres.

4.2. Non-commutativegeometryinthelowestLandaulevel

FollowingtothesimilarstepinSection2.1,wecanfindhowhigherdimensionalfuzzysphere geometryemerges inthelowestLandaulevel.Itshouldbenotedsincethemonopolegaugefield isnon-abelian,andthentheparticleonS2k carriestheSO(2k)colordegreesoffreedomlikea “quark”.TheLagrangianisgivenby

L=M

2 x˙ax˙a− ˙xaAa, (62)

12 _For_k₌₂_,_{4 we}_have_two_Z_s:_π

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wherexaxa=r2.InthelowestLandaulevel,theLagrangianisreducedto L=iΨ†d

dtΨ, (63)

withΨ (48).ByimposingthecanonicalquantizationconditiononΨ andΨ∗,xa(42)are

effec-tivelyrepresentedbytheoperators

Xa=α 2Ψ t_Γa ∂ ∂Ψ, (64) whichsatisfy [Xa, Xb] =iαXab, (65) where Xab=αΨtΣab ∂ ∂Ψ, (66)

withΣab= −₄i[Γa,Γb].XaandXabamountto(2k+1)+k(2k+1)=(k+1)(2k+1)generators of theSO(2k+2)algebra,andXabbringthe“extra”degreesoffreedomoffuzzyfibreS_F2k−2

overS2k.Itshouldbenotedthatthecoordinatesoftheexternalspaceandthoseoftheinternal spacearerelatedby(65)andtheyaresamesizematricesoftheSO(2k+2)generators.Since theyaresimilarlytreatedinthefuzzyalgebra,thereisnoreasontodistinguishtheexternaland internalspacesinthelowestLandaulevel.Itmaybemorenaturaltoconsideranenlargedspace that includes bothexternalandinternalspaces.Since thefuzzy-fibrecoordinates Xab are the

SO(2k+1)generators,Xabcanberepresentedas

Xab=αLab. (67)

MeanwhileLab∼r2FabinthelowestLandaulevel(seeSection5.4).Fromtheserelations,we have

Xab∼αr2Fab, (68)

whichsuggeststhenon-abelianfieldstrengthisequivalenttothefuzzy-fibre[seeFig. 2].This identificationcoincideswiththeintuitivepicturethatthefuzzy-fibrerealizesasthenon-abelian fluxofthemonopole.Inthe2DquantumHallliquid,theU (1)magneticfluxpenetrationinduces achargedexcitationatthepointwherethefluxispierced.Similarlyinhigherdimensional quan-tumHallliquid,thenon-abelianfluxpenetrationinducesapoint-likeexcitationonS2k.Though the excitationis“point”likeonS2k,the non-abelian fluxmatrixaccommodatestheS_F2k−2 ge-ometryasitsinternalstructure.Rememberthatthereisnodistinctionbetweentheexternaland internalspacesinthelowestLandaulevel,andsothe“internal”spaceS_F2k−2canberegarded as anextended(2k−2)-dimensional object,(2k−2)-brane,intheenlarged(4k−2)-dimensional space.Inthissense,thenon-abelianfluxpenetrationinduces(2k−2)-branelikeexcitation.

4.3. TheSO(2k+1)Landaumodel

Ind-dimensionalspace,one-particleHamiltonianundertheinfluenceofgaugefieldisgiven by H= − 1 2M d a=1 Da2= − 1 2Mr 1−d ∂ ∂rr d−1 ∂ ∂r + 1 2Mr2 a<b Λab2, (69)

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Fig. 2.TheinternalgeometryoftheSO(2k)non-abelianfluxisequivalenttothefuzzy-fibreS_F2k−2,andtheS_F2k−2

correspondsto(2k−2)-braneintheenlarged(4k−2)-dimensional space.

withDa=∂a+iAaandΛab= −ixaDb+ixbDa(a,b=1,2,· · ·,d).Λabsatisfy

[Λab, Λcd] =i(δacΛbd+δbdΛac−δbcΛad−δadΛbc)

−i(xaxcFbd+xbxdFac−xbxcFad−xaxdFbc), (70)

where Fab are the components of the field strength, Fab = −i[Da,Db]=∂aAb−∂bAa+ i[Aa,Ab].SincetheSO(2k)non-abelianmonopole(50)islocatedatthecenterofd=2k+1 dimensionalspace,itsfieldstrengthisradiallydistributedandthesystemrespectstheSO(2k+1)

rotationalsymmetry.WecanconstructtheconservedSO(2k+1)angularmomentumas

Lab=Λab+r2Fab. (71)

ItisstraightforwardtoverifythatLabactastheSO(2k+1)generators:

[Lab, Mcd] =i(δacMbd+δbdMac−δbcMad−δadMbc), (72)

whereMab=Lab,Λab,Fab. Foraparticle on 2k-sphere,(69) is reduced tothe SO(2k+1)

LandauHamiltonian: H= 1 2Mr2 a<b Λab2. (73)

DuetotheexistenceoftheSO(2k+1)symmetry,onemayreadilyderivetheeigenvaluesof(73)

byagrouptheoreticalmethod.WiththeorthogonalityΛabFab=FabΛab,(73)isrewrittenas

H= 1 2Mr2 a<b Lab2− a<b Fab2 = 1 2Mr2 a<b Lab2− μ<ν Σ_μν+2 , (74)

where_a<bFab2=_μ<νΣ_μν±2wasused.Weadoptthefullysymmetricrepresentation

(I /2)≡ k I 2, I 2,· · ·, I 2 (75) fortheSO(2k)Casimirμ<νΣμν+

2

,andtheirreduciblerepresentation

(n, I /2)≡ k n+I 2, I 2, I 2,· · ·, I 2 (76)

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fortheSO(2k+1)Casimir_a<bLab2(ndenotestheLandaulevelindex),andthentheenergy

eigenvaluesarederivedas13

En= 1 2Mr2 C2k₊1(n, I /2)−C2k(I /2) = 1 2Mr2 n(n+2k−1)+I n+1 2k , (78) whereC2k+1(n,I /2)andC2k(I /2)respectivelyrepresenttheSO(2k+1)andSO(2k)Casimir

eigenvaluesfor(n,I /2)and(I /2):

C2k+1(n, I /2)=n2+n(I+2k−1)+ 1 4I k(I+2k), (79a) C2k(I /2)= μ<ν Σ_μν±2=1 4I k(I+2k−2). (79b)

ThedegeneracyinthenthLandaulevelisgivenby

Dn(k, I )= 2n+I+2k−1 (2k−1)!! (n+k−1)! n!(k−1)! (I+2k−3)!! (I−1)!! ·(n+I+2k−2)! (n+I+k−1)! k−2 l=1 (I+2l) (I+l)! k−1 l=1 l! (2l)!. (80)

InparticularforthelowestLandaulevel(n=0),therepresentationisreducedtotheSO(2k+1)

fullysymmetricspinorrepr.(I /2),andthedegeneracybecomesto

DLLL(k, I )= k l=1 l i=1 I+l+i−1 l+i−1 . (81) Inlowdimensions, k=1: DLLL(1, I )=I+1, k=2: DLLL(2, I )= 1 6(I+1)(I+2)(I+3), k=3: DLLL(3, I )= 1 360(I+1)(I+2)(I+3) 2_(I₊₄_)(I₊₅_), k=4: DLLL(4, I )= 1 302 400(I+1)(I+2)(I +3) 2 ×(I+4)2(I+5)2(I+6)(I +7). (82)

One maynoticethat thelowestLandauleveldegeneracy(82)andthe Chernnumber(61)are relatedbythefollowingsimpleformula:

13 _In_the_{thermodynamic}_limit,_r,_I_{→ ∞}_with_{I /r}2_fixed,_the_energy_eigenvalues₍₇₈₎_are_reduced_to

En→ I 2Mr2 n+1 2k . (77)

ThelowestLandaulevelenergy,ELLL=₄_MrI 2k,isequaltoktimesthelowestLandaulevelenergyofthe2D(planar) Landaumodel, ₂B_M= I

4Mr2.Thisisbecausethatinthethermodynamiclimit,the2kD fuzzysphereisreducedtok copiesof2D non-commutativeplane.

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ck(I )=DLLL(k, I−1). (83) Thisrelationisindeedguaranteedbytheindextheoremforarbitraryk[seeSection4.4].

4.4. TheSO(2k+1)spinorLandaumodelandindextheorem

Here,weconsideraspinorparticleonS2kintheSO(2k)monopolebackground.Thespinor particlecarriestheSO(2k+1)spindegreesoffreedomcoupledtotheexternalSO(2k)magnetic fieldthroughZeemanterm.WeanalyzetheSO(2k+1)spinorLandauproblemwithuseofthe formulationexploredbyDolan[71].

Inthepresencethegaugefield,theDiracoperatorond-dimensionalcurvedmanifoldis gen-erallygivenby

D=γαDα=eαμγμ(∂α+iωα+iAα), (84)

whereαstandfortheintrinsiccoordinatesofthemanifold,ωαdenotethespinconnectionofthe

manifold,μrepresentthecoordinatesofthed-dimensionalflatEuclideanspace,andγμarethe

SO(d)gammamatrices:

γμ, γν=2δμν (μ, ν=1,2,· · ·, d). (85)

Forsymmetric(≡torsionfree)manifold,thesquareoftheDiracoperatorisgivenbythe follow-ingLichnerowiczformula[72]:

(−iD)2= −+Fαβ⊗σαβ+R

4 (86)

wheretheLaplacianandthefieldstrengthFαβarerespectivelygivenby =_√1 g∇α √ ggαβ∇β =gαβ∇α∇β−Γγαβ∇γ , Fαβ=∂αAβ−∂βAα+i[Aα,Aβ], (87)

andRdenotesthescalarcurvature.Thesecondtermontheright-handsideof(86),σαβFαβ= eaαebβσabFαβ,representstheZeemanterm.AsreadilyverifiedfromtheLichnerowiczformula,

intheabsenceoftheZeemanterm,theDiracoperatordoesnothavezero-eigenvalueson mani-foldswithpositivescalarcurvature,sincetheeigenvaluesofLaplacianaresemi-positivedefinite. Meanwhileinthepresenceofthegaugefieldstrength,theZeemantermmaycancelthe contri-butionfromthecurvature termtogivezero-eigenvaluesfor (−iD)2.Thiscancellationindeed occursinthepresentcase,andthezero-modesoftheDiracoperatorareidentifiedwiththe low-estLandau level basis states whose spin direction isopposite tothe external magnetic field. WhenthegaugegroupisidenticaltotheholonomygroupofthecosetMG/H,(86)canbe expressedbythegrouptheoreticalquantities[71]:

(−iD)2=C(G)−C(H, R)+R

8, (88)

where C(G) represents (quadratic) Casimir for the isometry group G and C(H,R) denotes (quadratic) Casimir for the holonomy group H made by the gauge group representation R. With(88)we are abletoderivetheeigenvaluesof (−iD)2byusing asimplegroup theoreti-calmethod.

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ForS2kSO(2k+1)/SO(2k),weproposetheSO(2k+1)spinorLandauHamiltonianas H= 1 2M(−iD) 2₌ 1 2M(C2k+1−C2k)+ 1 8Mk(2k−1), (89)

whereweusedtheRicciscalarofS2k14

R=2k(2k−1). (90)

Fortheirreduciblerepresentations

(n, J )≡ k n+J, J,· · ·, J, forSO(2k+1) (91a) I 2 ≡ k I 2, I 2,· · ·, I 2 forSO(2k), (91b)

theCasimireigenvaluesarerespectivelygivenby

C2k+1(n, J )=n2+n(2J+2k−1)+kJ (J+k), (92a) C2k I 2 =kI 2 I 2+k−1 , (92b)

andtheeigenvaluesof(89)arederivedas

E(n, J )= 1 2M n2+n(2J+2k−1)+k J (J+k)−I 2 I 2+k−1 + 1 8Mk(2k−1), (93)

andthenthLandauleveldegeneracyisobtainedas

Dn(k,2J )=2n+2J+2k−1 (2k−1)!! (n+k−1)! n!(k−1)! · k−1 i=1 (2J+2i−1)· k i=2 n+2J+2k−i 2k−i · k−2 l=1 k i=l+2 2J+2k−i−l 2k−i−l . (94)

Forthespinorparticle,15wetake

J=I

2 ± 1

2, (97)

14 _The _SO₍₂_k) _Casimir _for _the _fundamental _{representation} _(79b) ₍_I ₌₁₎ _is _equal _to _the _Ricci _scalar _of _S2k_:

μ<νσμν2=k₄(2k−1)=1₈R.

15 _For_the_scalar_particle,_we_substitute

J=I

2 (95)

to (89)toderivetheenergyeigenvalues (78):

H− 1 8Mk(2k−1)= 1 2M C2k+1(n, J )−C2k(I /2)_J=I 2 = 1 2M n2+n(I+2k−1)+1 2I k . (96)

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wherefor+(↑spinstate),I≥0,whilefor−(↓spinstate),I≥1.Thisimpliesthatthespin polarizationduetotheZeemaneffecteffectivelychangesthestrengthofmagneticfluxby±1₂ accordingtothedirectionofspin.Inaccordancewith±sector,(93)isblockdiagonalizedas

E₊(n) 0 0 E₋(n) , (98) whereE_±(n)≡E(n,J )_J₌I 2±12: E₊(n)= 1 2M n2+n(I+2k)+k(I+k), E₋(n)= 1 2M n2+n(I+2k−2), (99)

whosedegeneraciesarerespectivelygivenbyDn(k,I +1)andDn(k,I −1)throughthe

for-mula(94).16Inlowdimensions,(99)readsas

S2: E₊(n) 0 0 E₋(n) k=1 = 1 2M (n+1)(n+I+1) 0 0 n(n+I ) , S4: E₊(n) 0 0 E₋(n) k=2 = 1 2M n2+n(I+4)+2(I+2) 0 0 n2+n(I+2) , S6: E₊(n) 0 0 E₋(n) k=3 = 1 2M n2+n(I+6)+3(I+3) 0 0 n2+n(I+4) , S8: E₊(n) 0 0 E₋(n) k₌4 = 1 2M n2+n(I+8)+4(I+4) 0 0 n2+n(I+6) . (101) TheLandaulevelenergyspectrumisboundedbyzeroforthelowestLandaulevelbasisstates (n=0)with↓spin:

E₋(n=0)=0, (102)

andthenumberofthezero-energystatesisgivenby

DLLL(k, I−1). (103)

SincetheHamiltonianisthesquareoftheDiracoperator,thezero-energyeigenstatescorrespond tothezero-modesoftheDiracoperator:

Ind(iD)=DLLL(k, I−1). (104)

Theindextheoremtellsthatthenumberofzero-modesisequaltothetopologicalchargeofthe non-trivialgaugeconfiguration:

Ind(iD)=ck. (105)

16 _It_can_be_confirmed_that_E

+(n)|I=0(99)andDn(k,2J=I+1)|I=0=Dn(k,1)(94)respectivelyreproducethe

eigenvaluesandthedegeneracyofthefreeDiracoperatorwithoutgaugefield [73–76]:

2ME₊(n)_I₌₀=n+k, Dn(k,1)=2k n+2k−1 n . (100)

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In the presentcase, ck denotesthe kthChernnumberof theSO(2k) monopole(59).Wethus

verified(83)forarbitraryk.

4.5. Laughlin-likewavefunction

ForhigherdimensionalquantumHalleffect,theparticlescarrytheSO(2k)colordegreesof freedomwiththegeometryS_F2k−2,andthetotalspacewillbegivenby

(x,y)∈S2k×S2k−2, (106)

where x=(x1,x2,· · ·,x2k+1)with 2k+1

a=1 xaxa=r2denotesthebase-manifoldS2k while y= (y1,y2,· · ·,y2k−2)with

2k−1

i=1 yiyi =r2 representsthe coordinateson(2k−2)-dimensional

internalspaceS2k−2(whichisregardedastheclassicalcounterpartoffuzzybundlecoordinates

Xi (30)).ThecoordinatesofthetotalspaceS2k⊗S2k−2isrepresentedby

Ψ (x)= 1 2r(r+x2k+1) (r+x2k₊1)ψ (x2k+iγixi)ψ , (107)

whereψdenotes2k−1_component_spinor_giving_the_internal_coordinates_by_the_relation:

ψ†γiψ=yi. (108)

ThelowestLandaulevelbasisstatescanbeconstructedbytakingafullysymmetricproductof thecomponentsofΨ (x): Ψm1,m2,···,m2k(x)= 1 √ m1!m2! · · ·m2k! Ψm1 1 (x)Ψ m2 2 (x)· · ·Ψ m2k 2k (x), (109)

withm1+m2+ · · · +m2k=I.Form=1 theparticlesoccupyallthelowestLandaulevelstates

onS2k,andsothetotalparticlenumberNisgivenby

N≡d(k, I )≡ D(k, I ) D(k−1, I )= (k−1)! (2k−1)! (I+2k−1)! (I+k−1)! ∼I k_, (110) whereD(k,I )denotesthenumberofstatesofthetotalspaceS_F2k,andD(k−1,I )standsforthe numberofstatesofthefuzzy-fibreS_F2k−2.ForI /2→mI /2,thestatenumberonS2kchangesas

d(k, mI )= D(k, mI ) D(k−1, mI )= (k−1)! (2k−1)! (mI+2k−1)! (mI+k−1)! ∼(mI ) k_. ₍₁₁₁₎

WithuseoftheSlaterdeterminant,theLaughlin-likegroundstatewavefunctionisconstructedas

ΨLin(x1,x2,· · ·,xN)=

A1A2···ANΨA1(x1)ΨA2(x2)· · ·ΨAN(xN) m

, (112)

whereA=(m1,m2,· · ·,m2k)andmistakenasanoddinteger tokeeptheFermistatisticsof

theparticles.WhenthepowerofΨAchangesfrom1 tom,themonopolechargechangesfrom I tomI,andthenΨLincorrespondstothegroundstateof2kD quantumHallliquidatthefilling factor: ν2k= N d(k, mI ) 1 mk. (113)

Noticethatsincemisanoddinter,ν2kisalsotheinverseofanoddinteger.Fromtheperspective

oftheoriginalbase-manifoldS2k,ΨLlindenotestheincompressibleliquidmadeoftheparticles. However, fromtheemergent (4k−1)D space–time pointofview, the particlecorrespondsto

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5. Tensormonopolefieldsfromnon-abelianmonopolefields

Wediscussedthenon-abelianmonopoleswhosegaugegroupiscompatiblewiththeholonomy ofsphere.Inthissection,weintroduceanothertypeofmonopole,thetensormonopole[41,42]

whosegaugegroupisU (1)andgaugefieldisanantisymmetrictensor.17

5.1. Tensormonopolefields

Tobeginwith,wereviewseveralbasicpropertiesofn-formtensorgaugefield[42]:

Cn= 1

n!Ca1a2···andxa1dxa2· · ·dxan (114)

whereCa1a2···anrepresentatotallyantisymmetrictensorgaugefield.NoticethatCa1a2···anisnot amatrix-valuedgaugefieldbutatensorextensionof the U (1)gaugefield. Like theordinary

U (1)gaugetheory,thefieldstrengthisdefinedas

Gn+1=dCn= 1 (n+1)!Ga1a2···an+1dxa1dxa2· · ·dxan+1, (115) where Ga1a2···an+1= 1 n!∂[a1Ca2···an+1]. (116) Forinstance,

n=2: Gabc=∂aCbc+∂bCca+∂cCab,

n=3: Gabcd=∂aCbcd−∂bCcda+∂cCdab−∂dCabd. (117)

TheU (1)gaugesymmetryisincorporatedinthefollowingway.TheU (1)gaugetransformation isgivenby Cn→Cn+dΛn−1, (118) with Λn−1= 1 (n−1)!Λa1a2···an−1dxa1dxa2· · ·dxan−1. (119) ItisobviousthatthefieldstrengthGisinvariantunder(118).Intermsofthetenorcomponents, thegaugetransformationisrepresentedas

Ca1a2···an→Ca1a2···an+ 1

(n−1)!∂[a1Λa2···an]. (120)

Forinstance,

n=2: Cab→Cab+∂aΛb−∂bΛa,

n=3: Cabc→Cabc+∂aΛbc+∂bΛca+∂cΛab. (121)

17 _The_{antisymmetric}_tensor_gauge_field_is_realized_as_a_solution_of_the_{Kalb–Ramond}_equation_and_also_referred_to_as theKalb–Ramondfield [43].

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Table 1

Relationsbetweenthenon-abelianmonopoleandthetensormonopole. Non-abelian monopole Tensor monopole

Sphere S2k S2k

Gauge group SO(2k) U (1)

Rank of gauge field 1 2k−1

Rank of field strength 2 2k

Itisasimpleexercisetoseethat(117)isinvariantunder(121).ThefieldstrengthoftheU (1)

tensormonopolelocatedattheoriginof(n+2)D Euclideanspaceisgivenby

Ga1a2···an+1=g 1

rn+2a1a2···an+2xan+2, (122) wheregdenotesthechargeof U (1)tensormonopole.Theintegralof thegaugefieldstrength overSn_yields Sn+1 Gn+1=gA Sn+1, (123)

whereA(Sn+1)representstheareaofSn+1.

5.2. Correspondencebetweenfieldstrengthsofmonopoles

Thenon-abelianandtensormonopolesaretwodifferentextensionsoftheDiracmonopolein termsofinternalandexternalindices.AsdiscussedinSection3,thereisnoreasonabledistinction betweentheexternalandinternalspacesinthe lowestLandaulevel,andso itisexpectedthat non-abelianandtensormonopolesshouldbe“equivalent”insomesense.Interestingly,forthe

SU(2)monopole and3-rank tensormonopole,their connectionhasalreadybeen pointedout, atleast forfundamentalrepresentation(quaternions)[77]andforthe integralform[78].Asa naturalgeneralizationoftheseresults,weestablishconnectionbetweentensorandnon-abelian monopolesforfullysymmetricrepresentationinarbitraryevendimension.Inthefollowing,we takenasanoddinteger,n=2k−1 andthemonopoleatthecenterofS2k[Table 1].Thetensor monopolegaugefield(122)takesthefollowingform:

Ga1a2···a2k=gk 1

r2k+1a1a2···a2k+1xa2k+1. (124) Wefixtheratiobetweentwomonopolecharges,ck(59)andgk,byimposingthecondition:

S2k G2k=tr S2k Fk. (125) From S2k G2k=gkA S2k (126) with AS2k= 2 k₊1_πk (2k−1)!!, (127)

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therelationbetweentwomonopolechargesisdeterminedas

gk= (2k)!

2k+1ck. (128)

Eq.(125)israther“trivial”,sincewearealwaysabletoimpose(125)byfixingtheratiobetween thetwo monopolecharges.What wereally needtoverify isthe localnon-abelianandtensor monopolerelation:

G2k=trFk. (129)

To prove (129) we take a bruteforce method: We substitute the explicit form of F (54) to the right-hand sideof (129) tosee whether we canderive G(124) on the left-handside un-derthe identification(128).Forthe component relationbetweenGa1a2···a2k (a1,a2,· · ·,a2k= 1,2,· · ·,2k+1)andFab,thelocalrelation(129)canberewrittenas18

Ga1a2···a2k= 1 2ka1a2···a2k+1ba₁ba₂···ba_2ka2k+1tr(Fba₁ba₂· · ·Fba_2k₋₁ba_2k). (132) Forinstance, G12···2k= 1 2kμ1μ2···μ2ktr(Fμ1μ2· · ·Fμ2k−1μ2k), (133) whereμ1,μ2,· · ·,μ2k =1,2,· · ·,2k.We substitute (54)to the right-hand side of (133) and

performastraightforwardcalculationwithuseoftheformulaeforthe SO(2k)matrices(240), andthenwefindtheright-handsideof(133)gives

G12_···2k= (2k)!

2k+1_r2k+1x2k+1. (134)

Inthecovariantnotation,(134)isexpressedas

Ga₁a2···a2k= (2k)! 2k₊1_r2k₊1a1a2···a2k+1xa2k+1 (135) or G2k= 1

2k₊1_r2k₊1a1a2···a2k+1xa2k+1dxa1dxa2· · ·dxa2k. (136) Forinstance, U (1): Gij= 1 2r3ij kxk (i, j, k=1,2,3), SU(2): Gabcd= 3 r5abcdexe (a, b, c, d, e=1,2,3,4,5), SO(6): Ga₁a2···a6= 45 r7a1a2···a6a7xa7 (a1, a2,· · ·, a7=1,2,· · ·,7), SO(8): Ga1a2a3···a8= 1260 r9 a1a2···a8a9xa9 (a1, a2,· · ·, a9=1,2,· · ·,9). (137) 18 _Here,_we_used G2k= 1 (2k)!Ga1a2···a2kdxa1dxa2· · ·dxa2k (130) and trFk= 1 2ktr(Fa1a2· · ·Fa2k−1a2k)dxa1dxa2· · ·dxa2k. (131)

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WethusdemonstratedthederivationofthetensormonopolegaugefieldGfromtrFk. Further-moreintermsofageneralsymmetricrepresentationoftheSO(2k)19

(I /2)≡ k I 2, I 2,· · ·, I 2 ,

wecanderiveagenericexpressionfortheU (1)tensorfieldstrengthas

Ga1a2···a2k= (2k)!I 2k+2 C(k, I )D(k−1, I ) 1 r2k+1a1a2···a2k+1xa2k+1 =I 2C(k, I )D(k−1, I )G (I=1) a1a2···a2k, (138)

where C(k,I )and G(Ia1=as1···)a2k+1 are respectivelygiven by (34) and(135). Here,we used the formulae for the symmetricrepresentation(241).One canconfirmthe symmetric representa-tion(138)forI=1 reproduces(135)bytheformula

DLLL(k, I=1)=

(2k)!! k! =2

k_.

(139) With(139)andthefollowingformulaaboutthelowestLandauleveldegeneracy

C(k, I )DLLL(k−1, I )=

(2k)!

2k_I DLLL(k, I−1), (140)

wefinallyfindthatGtakesanamazinglysimpleform20:

Ga₁a2···a2k=ck(I )·G

(I=1)

a1a2···a2k, (143)

where G(I=1) isgivenby (135)andthe relation(83)was used.From (143),we canreadoff thetensormonopolechargeasgk=(₂2k+k)1!ck(I ),whichisconsistentwiththeresult(128).Inlow dimensions,wehave Gij= 1 2r3I ij kxk, Gabcd= 1 2r5I (I+1)(I+2)abcdexe, Ga₁a2···a6= 1 8r7I (I+1)(I+2) 2_(I₊ 3)(I+4)a₁a2···a7xa7, Ga1a2a3···a8= 1 240r9I (I+1)(I+2) 2_(I₊ 3)2(I+4)2(I+5)(I+6)a1a2···a9xa9. (144) Thus,weverifiedthelocalnon-abelianandtensormonopolecorrespondence(129)forgeneric fullysymmetricrepresentationinarbitraryevendimension.

19 _I₌_{1 corresponds}_to_the_spinor_{representation.} 20 _In_differential_form,₍₁₄₃₎_is_represented_as

G2k=

1

2k+1_r2k+1ck(I )a1a2···a2k+1xa2k+1dxa1dxa2· · ·dxa2k=ck(I )G(I=1), (141)

andhencethenormalizedU (1)tensormonopolechargeqk(I )≡ 1

S2kG (I=1) 2k

S2kG2k,isidenticaltotheChernnumber:

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5.3. Correspondencebetweengaugefieldsofmonopoles

Fornon-abeliangaugefield,wehave[79]

trFk=dL_CS(2k−1)[A], (145)

whereL(_CS2k−1)representstheChern–Simonsterm

L(_CS2k−1)[A] =k

1 0

dttrAt dA+it2A2k−1. (146) Meanwhileforthetensormonopolegaugefield,wehaveseen

G2k=dC2k−1. (147)

Fromthenon-abelianandtensormonopolecorrespondence(129),it isobviousthat thetensor monopolegaugefieldisidenticaltothenon-abelianChern–Simonsterm:

C2k−1=tr L(_CS2k−1)[A]. (148) Forinstance, C1=trA, C3=tr AdA+2 3iA 3 =tr AF −1 3iA 3 , C5=tr A(dA)2+3 2iA 3_dA₋3 5A 5 =tr AF2−1 2iA 3_F₋ 1 10A 5 , C7=tr A(dA)3+8 5iA 3_(dA)2₊4 5iA(AdA) 2₋ 2A5dA−4 7iA 7 =tr AF3−2 5iA 3_F2₋1 5iAF A 2_F₋1 5A 5_F ₊ 1 35iA 7 . (149)

Noticethat tr(A3F2)=tr(AF A2F ),since Aand F are matrix-valued quantities andare not commutative.Forcomponentsof(149),wehave

Ci=trAi, Cabc=tr A_[a∂bAc]+ 2 3iA[aAbAc] =1 2tr A_[aFbc]− 2 3iA[aAbAc] , Cabcde= 1 4tr A_[aFbcFde]−iA[aAbAcFde]− 2 5A[aAbAcAdAe] , Ca₁a2···a7= 1 8tr A_[a1Fa2a3Fa4a5Fa6a7] −4 5iA[a1Aa2Aa3Fa4a5Fa6a7]− 2 5iA[a1Fa2a3Aa4Aa5Fa6a7] −4 5A[a1Aa2Aa3Aa4Aa5Fa6a7]+ 8 35iA[a1Aa2Aa3Aa4Aa5Aa6Aa7] . (150)

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TheSO(2k)gaugetransformationactsasthe U (1)gaugetransformationforC2k−1.For

in-stancek=2,thenon-abelian(SU(2))gaugetransformation(57)actstoC3as

C3→C3−id trAdgg†+1 3tr g†dg3. (151)

Thesecondtermontheright-handsideisthetotalderivative.Thethirdtermsatisfies21

dtrg†dg3= −trg†dg4=0, (152)

and is locally expressed as a totalderivative (Poincaré lemma). Consequently, (151) can be rewritteninthefollowingform

C3→C3+dΛ2. (153)

Ingeneral,theSO(2k)gaugetransformationactsasU (1)gaugetransformationtotensorgauge field(seeAppendix Cformoredetails):

C2k−1→C2k−1+dΛ2k−2. (154)

Forpracticalapplications,itisimportanttoderivetheexplicitformofthetensormonopole gaugefield.Withuseofthegeneralformula(150),wederivethetensormonopolegaugefield from the non-abelianmonopole in low dimensions.We substitute the non-abelian monopole field(50)totheright-handsideoftheformula(150).Afteralongbutstraightforward calcula-tionsusing traceformulaeofgammamatrices,weobtain thefollowingexpressionsforspinor representation: Ci= − 1 2r(r+x3) ij3xj, Cabc= −1 r3 1 r+x5+ r (r+x5)2 abcd5xd, Cabcde= − 9 r5 1 r+x7+ r (r+x7)2+ 2 3 r2 (r+x7)3 abcdef7xf, Ca1a2···a7= − 180 r7 1 r+x9 + r (r+x9)2 +4 5 r2 (r+x9)3+ 2 5 r3 (r+x9)4 a1a2···a89xa8. (155) Noticethat(2k−1)ranktensormonopolegaugefieldexhibitskthpowerstring-likesingularity. Similarlyforfullysymmetricrepresentation,weobtain

Ci= − I 2r(r+x3) ij3xj, Cabc= − 1 6r3I (I+1)(I+2) 1 r+x5 + r (r+x5)2 abcd5xd, Cabcde= − 1 40r5I (I+1)(I+2) 2_(I₊ 3)(I+4)

21 _tr_(α2n₎₌_{0 for}_any_one-form_α₌_dx aαa.

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× 1 r+x7+ r (r+x7)2+ 2 3 r2 (r+x7)3 abcdef7xf Ca1a2···a7= − 1 1680r7I (I+1)(I+2) 2_(I₊ 3)2(I+4)2(I+5)(I+6) × 1 r+x9+ r (r+x9)2+ 4 5 r2 (r+x9)3+ 2 5 r3 (r+x9)4 a1a2···a89xa8. (156) ForI =1, (156) isreduced to(155).Onemayalso confirmthat(156) indeedgives thefield strength(144)throughtheformula:

Ga₁a2···a2k= 1

(2k−1)!∂[a1Ca2···a2k−1]. (157)

5.4. QuantumNambugeometryviatensormonopole

InthelowestLandaulevel,thecovariantangularmomentumisquenched,andthenwehave theidentification:

Lab=Λab+r2Fab∼r2Fab. (158)

In 3D, two rank antisymmetrictensor isequivalent tovector, andthe angular momentum is directlyrelatedtothecoordinatesoffuzzytwo-sphere(24).Howeverinhigherdimensions,two rankantisymmetrictensorisnolongerequivalenttovectorandtheangularmomentumdoesnot seemtoapparentlyberelatedtothecoordinatesoffuzzysphere.AsmentionedinSection3.2, thequantumNambubracketimpliestheexistenceoftensormonopoleandwehaveshownthe non-abelianandtensormonopolecorrespondence(129)or

1

r2k+1xa= 2

(2k)!ck

aa1a2···a2ktr(Fa1a2· · ·Fa2k−1a2k). (159) Theidentification(158)suggeststhat(159)becomesto

Xa= I (2k)!ck

αaa1a2···a2k(La1a2La3a4· · ·La2k−1a2k) (160) inthelowestLandaulevel,andthecoordinatesofhigherdimensionalspherearenowregarded astheoperators.Eq.(160)isanaturalgeneralizationof(25).

LetusconsiderthealgebraforXa.Forthispurpose,itisusefultoadopttheanalogybetween thealgebrasofXa andthecovariantderivatives−iDa [31].ForS2_F case,thealgebraof Xi is givenby

[Xi, Xj] =iαij kXk, (161)

whilethecovariantderivativegives [−iDi,−iDj] = −iFij= −i 1

αr2ij kxk. (162)

Onemaynoticetheanalogy:

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Thisanalogycanholdinhigherdimensions[seeSection3.2],andforevaluationoftheNambu bracketforXaweutilizethefollowingidentification:

[Xa1, Xa2,· · ·, Xa2k] ↔

1

DLLL(k−1, I )

−(αr)2k[−iDa1,−iDa2,· · ·,−iDa2k]. (164) Theright-handsidegives

[−iDa1,−iDa2,· · ·,−iDa2k] = 1

2ka1a2···a2k+1ba₁ba₂···ba_2ka2k+1 × [−iDb_a

1,−iDba2][−iDba3,−iDba4] · · · [−iDba2k−1,−iDba2k] = −i1 2 k a1a2···a2k+1ba₁ba₂···ba_2ka2k+1Fba₁ba₂Fba₃ba₄· · ·Fba_2k₋₁ba_2k, (165) andthetraceisevaluatedas

tr[−iDa1,−iDa2,· · ·,−iDa2k] =(−i)k(2k)!

2k+1DLLL(k, I−1)·a1a2···a2k+1 1

r2k+1xa2k+1. (166) Duetotherelation(140),weobtain

[Xa1, Xa2,· · ·, Xa2k] =i

k_{C(k, I )α}2k−1

a1a2···a2k+1Xa2k+1, (167) whichisexactlyequaltothequantumNambualgebraforfuzzysphere(33).

6. FluxattachmentandtensorChern–Simonsfieldtheoryformembranes

HerewediscussphysicalpropertiesofA-classtopologicalinsulatorbasedonChern–Simons tensorfieldtheory.Wewillseeexoticconceptsin2DquantumHalleffectarenaturally general-izedinhigherdimensions:

• Fluxattachmentandcompositeparticles[46,47,45]

• Effectivetopologicalfieldtheory[44,45]

• Fractionalstatisticsofquasi-particleexcitations[80]

• Haldane–Halperinhierarchy[29,81] ..

.

6.1. Basicobservations

Before going to the details, we summarize basicobservations about the relevant physical conceptsandassociatedmathematicsinhigherdimensions.

• (2k−1)ranktensorgaugefieldand(2k−2)-brane

The(2k−1)rankgaugefieldisnaturallycoupledtothe(2k−1)rankcurrentof(2k−2)-brane. ThemembranedegreesoffreedomisautomaticallyincorporatedinthegeometryofS_F2k asthe fuzzyfibreS_F2k−2overS2k: