《The General Theory of Employment,Interest and Money》读后感
《The General Theory of Employment,Interest and Money》译为《就业、利息和货币通论》,是凯恩斯的经典著作。此书我看的是纸质英文原版,由上海外语教育出版社出版。由于此书部分内容较难懂,我阅读了很长时间,部分内容看的是中文译本,并且因为学识能力有限,以下是我就自己所能理解的内容谈谈对此书的感想。
凯恩斯在19xx年完成了试图挽救资本主义制度的经典著作——《就业、利息和货币通论》,标志着凯恩斯主义这一独立的理论体系的形成。在此书中,凯恩斯修正了传统西方就业理论的核心——萨伊定理,同时试图推翻传统的有关就业理论赖以组成的劳动市场论,利息论,货币论的三部分,并且提出了治理危机的对策。
一讲到资本主义的市场经济,很多人便认为是纯粹的自由经济,不需要政府的管理,而实际上任何体制下的经济,都不可能完全脱离政府的宏观管理。只有将市场的自我调节作用与政府的宏观调控作用相结合,才能使经济真正健康地发展起来。因此《通论》它在当时的重要性是不可估量的:首先,它所提出的建议和政策在一定的程度上使西方发达资本主义国家避免了19xx年经济危机所造成的灭顶之灾,并使它们在二战后约为20年处于相对繁荣的状态。其次,它促成了布雷顿森林体系的建立,对稳定战后西方国际经济秩序作出巨大贡献。
此外,它还奠定了西方宏观经济学的基础。最后,它为西方经济学的许多研究主题提供了学术源泉。
在此书中,凯恩斯认为,失业包括传统经济学认为的摩擦失业、自愿失业,还有非自愿失业,而失业主要源于有效需求不足。有效需求包括投资需求和消费需求,而有效需求不足则是它们共同作用的结果,后者又是因为消费倾向,对资本未来收益预期,对货币流动偏好这三个心理因素的作用。书中凯恩斯主张政府要想办法促进有效需求。失业的直接原因:当社会对企业生产的产品需求不断减少时,资本家就不会再增加投资了,这时就业量就会减少,从而导致工人失业。而失业的终极原因就是资产所有者的货币愿望过强。如果中央银行的货币供应量减少,而资产所有者的货币愿望不变,资本家为了自身的最大利润,就不会进行充分的投资;如果投资减少,有效需求就不能够充分实现,国民收入和就业量也就会相应地呈现较低水平,这时便产生失业。因此,他提出节俭有害论:在非充分就业存在情形下,消费的增加会引起储蓄同步增加,从而引起资本积累的增加。反之,节俭则可能导致贫困。凯恩斯在非充分就业的前提下,提出了对消费和节俭经济功能的全新认识。
本书共分六篇24章。第一篇引论中,凯恩斯批评了李嘉图及其以后的资产阶级经济学家如约翰、穆勒、马歇尔、庇古等人的两个“前提”:第一,工资等于劳动力的边际产物;第二,当就业量不变时,工资的效用正好等于该就业量的边际负效用,这仅适用于一种特例,而不能适用于普遍情况。进而他分析了“有效需求原则”:总供给函数和总需求函数相交时的数值,就业量就是这个交点值。第二篇中,他则主要阐明了预期、所得、储蓄、投资的定义,以及使用者成本等问题。第三篇研究了消费者倾向,分析消费倾向的主客观因素、边际
消费倾向与乘数的关系。第四篇关于投资引诱,分析了资本边际效率、长期预期状态、偏好与利率、资本性质以及利息与货币的特征。第五篇货币工资与物价,阐述了货币工资的改变、就业函数与物价的问题。第六篇引用了几篇短论,分析了商业循环,论述了重商主义、禁止高利贷法、加印货币以及消费不足论。最后是结束语,《通论》所引起的社会哲学。而起中则是“有效需求理论”是本书最核心精彩的部分,凯恩斯运用总量分析的方法,对总收入、总需求、总供给、投资、消费、就业水平、物价水平等一系列总量相互关系进行研究的同时,独辟蹊径地创造了三个基本心理因素定理,精辟的分析出导致现实失业与萧条的原因,在此基础上又提出了国家干预经济的主张。
凯恩斯认为:除非刺激需求,否则经济就不可能增长,不可能达到充分就业,因此他建议政府要经常干预市场,特别当经济衰退时更要采取赤字预算和灵活的财政政策。而他的消费函数和预期的不确定性为国家干预提供了强有力的理论,经济政策应以财政政策为主。
正如凯恩斯自己所说的那样,《通论》的基本内容其实是很明晰的。它主要说明:当资本主义市场经济由于存在着多余的生产能力和资源而引起经济危机和大量失业时,国家应该如何使用财政政策和货币政策来进行有效的宏观调控,以便真正有效地消除这些问题。不难发现,《通论》它只是为了解决资本主义市场经济的问题而撰写的;同时,凯恩斯本人的立场和思想的局限性,也使他提出的解决问题的方案局限在这种市场经济所容许的范围以内。然而,尽管如此,由于资本主义市场经济和我国的社会主义市场经济在市场经济这一点上具有共同之处,所以《通论》的内容还是存在着可以为我国所借鉴之处的。具体说来,至少有两点:
第一,《通论》有力地证实:市场不是万能的。由于市场中各个利益主体的经济行为不可能完全协调一致,所以经济萧条(乃至经济危机)和大量失业的“市场失灵”的现象会时常存在。因此,国家有必要使用宏观调控的手段,将市场调节与国家宏观调空有机结合起来。
第二,《通论》也说明了作为宏观调控手段的货币政策和财政政策,应该通过何种途径来消除这些“市场失灵”的现象。凯恩斯认为,“市场失灵”现象的主要原因是投资不足,而投资不足所引发的收入下降又导致了消费的萎缩。货币政策的推行便是通过利息率的降低来刺激投资的增长,进而拉动消费的增长。当货币政策刺激投资增长的力度不足以完全消除“失灵”现象时,国家就必须同时使用财政政策的手段,把资金直接投资于“公共工程”等,以便扩大内需,从而达到消除“失灵”现象的目的。
由于体制转轨,产业结构失调,东南亚金融风暴等多种原因,我国的社会主义市场经济目前也面临着一定程度的失业、产品积压,资源短缺等“市场失灵”现象。就如何通过宏观经济政策来解决这些现象而言,《通论》中的上述两点值得我们加以参考、利用。
以上就是我在阅读此书中的感想和心得,在此感谢高宏的周老师、熊老师在本学期的辛勤教导,使我对高级宏观经济学有了一些基础的认识,也学习了很多高宏的模型,对于经济问题的数理模型分析有了大体的印象。在今后的学习中,我会继续运用高宏的知识,为我的研究生学习打下坚实的基础。
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aDIAS-STP-91-29UdeM-LPN-TH-71/91ONTHEGENERALSTRUCTUREOFHAMILTONIANREDUCTIONSOFTHEWZNWTHEORYL.Feh?er*,L.O’Raifeartaigh,P.RuelleandI.TsutsuiDublinInstituteforAdvancedStudies10BurlingtonRoad,Dublin4,IrelandA.WipfInstitutf¨urTheoretischePhysikEidgen¨ossischeTechnischeHochschuleH¨onggerberg,Z¨urichCH-8093,SwitzerlandAbstractThestructureofHamiltonianreductionsoftheWess-Zumino-Novikov-Witten(WZNW)theoryby?rstclassKac-Moodyconstraintsisanalyzedindetail.Liealge-braicconditionsaregivenforensuringthepresenceofexactintegrability,conformalinvarianceandW-symmetryinthereducedtheories.ALagrangean,gaugedWZNWimplementationofthereductionisestablishedinthegeneralcaseandtherebythepath
integralaswellastheBRSTformalismaresetupforstudyingthequantumversionofthereduction.Thegeneralresultsareappliedtoanumberofexamples.Inparticular,aWpurely-algebra?rstisclassassociatedconstraints.toeachTheembeddingimportanceofslof(2)theseintoslthe(2)simplesystemsLieisalgebrasdemonstratedbyusingbyshowingthattheyunderlietheWnl-algebrasaswell.NewgeneralizedTodatheoriesarefoundwhosechiralalgebrasaretheW-algebrasbelongingtothehalf-integralsl(2)em-beddings,andtheW-symmetryofthee?ectiveactionofthosegeneralizedTodatheoriesassociatedwiththeintegralgradingsisexhibitedexplicitly.
Contents
1.Introduction...................................................................3
2.GeneralstructureofKMandWZNWreductions..............................11
2.1.FirstclassandconformallyinvariantKMconstraints......................11
2.2.LagrangeanrealizationoftheHamiltonianreduction......................15
2.3.E?ective?eldtheoriesfromleft-rightdualreductions......................18
G3.PolynomialityinKMreductionsandtheWS-algebras.........................26
3.1.Asu?cientconditionforpolynomiality...................................26
3.2.ThepolynomialityoftheDiracbracket...................................31
Gl3.4.TheWSinterpretationoftheWn-algebras................................41G3.3.FirstclassconstraintsfortheWS-algebras................................34
4.GeneralizedTodatheories.....................................................46
4.1.GeneralizedTodatheoriesassociatedwithintegralgradings...............46
4.2.GeneralizedTodatheoriesforhalf-integralsl(2)embeddings...............48
4.3.TwoexamplesofgeneralizedTodatheories................................53
5.QuantumframeworkforWZNWreductions...................................57
5.1.Path-integralforconstrainedWZNWtheory..............................57
5.2.E?ectivetheoryinthephysicalgauge.....................................60
H5.3.TheW-symmetryofthegeneralizedTodaactionIe?(b)...................62
5.4.BRSTformalismforWZNWreductions...................................64
5.5.TheVirasorocentreintwoexamples......................................67
6.Discussion....................................................................70Appendices
A:Asolvablebutnotnilpotentgaugealgebra.................................73
B:H-compatiblesl(2)andthenon-degeneracycondition......................77
C:H-compatiblesl(2)embeddingsandhalvings...............................80
References.......................................................................87
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1.Introduction
DuetotheirintimaterelationshipwithLiealgebras,thevariousone-andtwo-dimensionalTodasystemsareamongthemostimportantmodelsofthetheoryofin-tegrablenon-linearequations[1-19].Inparticular,thestandardconformalToda?eldtheories,whicharegivenbytheLagrangean
LToda(?)=κi?
2|αKi|2ij???????l
jm2iexp
i=1?1
reducedWZNWtheories.ThiswayoflookingatTodatheorieshasalsonumerousotheradvantages,describedindetailin[13].
TheconstrainedWZNW(KM)settingofthestandardTodatheories(W-algebras)allowsforgeneralizations,someofwhichhavealreadybeeninvestigated[14-18,26-29].AnimportantrecentdevelopmentistherealizationthatitispossibletoassociateageneralizedW-algebratoeveryembeddingoftheLiealgebrasl(2)intothesimpleLiealgebras[16-18].ThestandardW-algebra,occurringinTodatheory,correspondstothesocalledprincipalsl(2).Infact,thesegeneralizedW-algebrascanbeobtainedfromtheKMalgebrabyconstrainingthecurrenttothehighestweightgauge,whichhasbeenoriginallyintroducedin[13]fordescribingthestandardcase.Anotherinteresting
ldevelopmentistheWn-algebrasintroducedbyBershadsky[26]andfurtherstudiedin
2[28].Itisknownthatthesimplestnon-trivialcaseW3,whichwasoriginallyproposedby
Polyakov[27],fallsintoaspecialcaseoftheW-algebrasobtainedbythesl(2)embeddingsmentionedabove.Ithasnotbeenclear,however,astowhetherthetwoclassesofW-algebrasarerelatedingeneral,ortowhatextentonecanfurthergeneralizetheKMreductiontoachievenewW-algebras.
Inthepresentpaper,weundertakethe?rstsystematicstudyoftheHamiltonianreductionsoftheWZNWtheory,aimingatuncoveringthegeneralstructureofthereduc-tionand,atthesametime,trytoanswertheabovequestion.Variousdi?erentquestionsarisingfromthismainproblemarealsoaddressed(seeContents),andsomeofthemcanbeexaminedonitsownright.Asthisprovidesourmotivationandinfactmostofthelaterdevelopmentsoriginatefromit,wewishtorecallherethemainpointsoftheWZNW→Todareductionbeforegivingamoredetailedoutlineofthecontent.
TomakecontactwiththeTodatheories,weconsidertheWZNWtheory*
?κTr(g?1dg)3,SWZ(g)=3B3(1.2)
forasimple,maximallynon-compact,connectedrealLiegroupG.Inotherwords,weassumethatthesimpleLiealgebra,G,correspondingtoGallowsforaCartandecom-positionoverthe?eldofrealnumbers.The?eldequationoftheWZNWtheorycanbewrittenintheequivalentforms
??J=0
2(x0or?=0,?+J(1.3)±x1).TheWZNW?eldgisperiodicinx1withperiod2πr.
4
where
J=κ?+g·g?1,?=?κg?1??g.andJ(1.4)
?,Theseequationsexpresstheconservationoftheleft-andrightKMcurrents,JandJ
respectively.ThegeneralsolutionoftheWZNW?eldequationisgivenbythesimpleformula
g(x+,x?)=gL(x+)·gR(x?),(1.5)
wheregLandgRarearbitraryG-valuedfunctions,i.e.,constrainedonlybytheboundaryconditionimposedong.
LetnowM?,M0andM+bethestandardgeneratorsoftheprincipalsl(2)subalge-braofG[32].ByconsideringtheeigenspacesGmofM0intheadjointofG,adM0=[M0,],onecande?neagradingofGbytheeigenvaluesm.Undertheprincipalsl(2)thisgradingisanintegralgrading,infactthespinsoccurringinthedecompositionoftheadjointofGaretheexponentsofG,whicharerelatedtotheordersoftheindependentCasimirsbyashiftby1.Itisalsoworthnotingthatthegrade0partof
G=G++G0+G?,G±=N?m=1G±m,(1.6)
isaCartansubalgebra,and(byusingsomeautomorphismoftheLiealgebra)onecanassumethatthegeneratorM0isgivenbytheformulaM0=1
whereg0,±arefromthesubgroupsG0,±ofGcorrespondingtotheLiesubalgebrasG0,±,respectively.InthisframeworktheToda?elds?iaregivenbythemiddle-pieceoftheGaussdecomposition,g0=exp[1
to
J(x)→ea(x+)J(x)e?a(x+)+κ(ea(x))′e?a(x),++(1.10)
wherea(x+)∈G+isanarbitrarychiralparameterfunction.*Theconstraints(1.7)arechoseninsuchawaythatthefollowingVirasorogenerator
LM0(x)≡LKM(x)?Tr(M0J′(x)),whereLKM(x)=1
*Throughoutthepaper,thenotationf′=2?1fisusedforeveryfunctionf,includingthespatialδ-functions.Forachiralfunctionf(x+)onehasthenf′=?+f.
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clearnowthatitwouldhavebeenpossibletode?netheW-algebraastheDiracbracketalgebraofthecomponentsofjredin(1.12)inthe?rstplace.Oncethispointisrealized,anaturalgeneralizationarisesimmediately[16-18].Namely,onecanassociateaclassicalW-algebratoanysl(2)subalgebraS={M?,M0,M+}ofanysimpleLiealgebraG,byde?ningittobetheDiracbracketalgebraofthecomponentsofjredin(1.12),whereonesimplysubstitutesthegeneratorsM±ofthearbitrarysl(2)subalgebraSforthoseoftheprincipalsl(2).Asweshallseeinthispaper,thisDiracbracketalgebraisapolynomialextensionoftheVirasoroalgebrabyprimary?elds,whoseconformalweightsarerelatedtothespinsoccurringinthedecompositionoftheadjointofGunderSbyashiftby1,incompleteanalogywiththecaseoftheprincipalsl(2).Weshalldesignatethegeneralized
GW-algebraassociatedtothesl(2)embeddingSasWS.
WiththemainfeaturesoftheWZNW→Todareductionandtheabovede?nition
GoftheWS-algebrasatourdisposal,nowwesketchthephilosophyandtheoutlineofthepresentpaper.Westartbygivingthemostimportantassumptionunderlyingourinvestigations,whichisthatweconsiderthosereductionswhichcanbeobtainedbyimposing?rstclassKMconstraintsgeneralizingtheonesin(1.7).Tobemoreprecise,ourmostgeneralconstraintsrestrictthecurrenttotakethefollowingform:
J(x)=κM+j(x),withj(x)∈Γ⊥,(1.13)
anysl(2)structurehere.Thewholeanalysisisbasedonrequiringthe?rst-classnessofthesystemoflinearKMconstraintscorrespondingthepair(Γ,M)accordingto(1.13).However,this?rst-classnessassumptionisnotasrestrictiveasoneperhapsmightthinkat?rstsight.Infact,asfarasweknow,our?rstclassmethodiscapableofcoveringallHamiltonianreductionsoftheWZNWtheoryconsideredtodate.Themanytechnicaladvantagesofusingpurely?rstclassKMconstraintswillbeapparent.
Theinvestigationsinthispaperareorganizedaccordingtothreedistinctlevelsofgenerality.Atthemostgenerallevelweonlymakethe?rst-classnessassumptionanddeducethefollowingresults.First,wegiveacompleteLiealgebraicanalysisoftheconditionsonthepair(Γ,M)imposedbythe?rst-classnessoftheconstraints.WeshallseethatΓin(1.13)hastobeasubalgebraofGonwhichtheCartan-Killingformvanishes,andthateverysuchsubalgebraissolvable.TheLiesubalgebraΓwillbereferredtoasthe‘gaugealgebra’ofthereduction.ForagivenΓ,the?rst-classnessimposesafurther
8whereMissomeconstantelementoftheunderlyingsimpleLiealgebraG,andΓ⊥isthesubspaceconsistingoftheLiealgebraelementstraceorthogonaltosomesubspaceΓofG.Wenotethatearlierin(1.7a)wehavechosenΓ=G+andM=M?,butwedonotneed
conditionontheelementM,andweshalldescribethespaceoftheallowedM’s.Second,weestablishagaugedWZNWimplementationofthereduction,generalizingtheonefoundpreviouslyinthestandardcase[13].ThisgaugedWZNWsettingofthereductionwillbe?rstseenclassically,butitwillbealsoestablishedinthequantumtheorybyconsideringthephasespacepathintegraloftheconstrainedWZNWtheory.Third,thegaugedWZNWframeworkwillbeusedtosetuptheBRSTformalismforthequantumHamiltonianreductioninthegeneralcase.Fourth,bymakingtheadditionalassumptionthattheleftandrightgaugealgebrasaredualtoeachotherwithrespecttotheCartan-Killingform,wewillbeabletogiveadetailedlocalanalysisofthee?ectivetheoriesresultingfromthereduction.Thisdualityassumptionwillalsoberelatedtotheparityinvarianceofthee?ectivetheories,whichissatis?edinthestandardTodacasewheretheleftandrightgaugealgebrasareG+andG?in(1.6),respectively.Ingeneral,theWZNWreductionnotonlyallowsustomakecontactwithknowntheories,liketheTodatheoryin(1.1),wherethesimplicityandthelargesymmetryofthe‘parent’WZNWtheoryarefullyexploitedforanalyzingthem,butalsoleadstonewtheorieswhichare‘integrablebyconstruction’.
Atthenextlevelofgenerality,westudytheconformallyinvariantreductions.ThebasicideahereisthatonecanguaranteetheconformalinvarianceofthereducedtheorybyexhibitingaVirasorodensitysuchthatthecorrespondingconformalactionpreservestheconstraintsin(1.13).Generalizing(1.11),weassumethatthisVirasorodensityisoftheform
LH(x)=LKM(x)?Tr(HJ′(x)),(1.14)
whereHissomeLiealgebraelement,tobedeterminedfromtheconditionthatLHweaklycommuteswiththe?rstclassconstraints.Weshalldescribetherelationswhichareimposedonthetripleofquantities(Γ,M,H)bythisrequirement,andtherebyobtainaLiealgebraicsu?cientconditionforconformalinvariance.
Atthethirdlevelofgenerality,wedealwithpolynomialreductionsandW-algebras.Theabovementionedsu?cientconditionforconformalinvarianceisaguaranteeforLHbeingagaugeinvariantdi?erentialpolynomial.Weshallprovideanadditionalconditiononthetripleofquantities(Γ,M,H)whichallowsonetoconstructoutofthecurrentin(1.13)acompletesetofgaugeinvariantdi?erentialpolynomialsbymeansofapoly-nomialgauge?xingalgorithm.TheKMPoissonbracketalgebraofthegaugeinvariantdi?erentialpolynomialsyieldsapolynomialextensionoftheVirasoroalgebrageneratedbyLH.Themostimportantapplicationofoursu?cientconditionforpolynomiality
GconcernstheWS-algebrasmentionedpreviously.
9
GLetusrememberthat,foranarbitrarysl(2)subalgebraSofG,theWS-algebracan
bede?nedastheDiracbracketalgebraofthehighestweightcurrentin(1.12)realizedbypurelysecondclassconstraints.However,weshallseeinthispaperthatthesesecondclassconstraintscanbereplacedbypurely?rstclassconstraintseveninthecaseofarbitrary,integralorhalf-integral,sl(2)embeddings.Sincethe?rstclassconstraints
Gsatisfyoursu?cientconditionforpolynomiality,wecanrealizetheWS-algebraasthe
KMPoissonbracketalgebraofthecorrespondinggaugeinvariantdi?erentialpolynomials.
GAfterhavingourhandson?rstclassKMconstraintsleadingtotheWS-algebras,we
shallimmediatelyapplyourgeneralconstructiontoexhibitingreducedWZNWtheoriesrealizingtheseW-algebrasastheirchiralalgebrasforarbitrarysl(2)-embeddings.Inthenon-trivialcaseofhalf-integralsl(2)-embeddings,thesegeneralizedTodatheoriesrepresentanewclassofintegrablemodels,whichwillbestudiedinsomedetail.Itisalso
GworthnotingthatrealizingtheWS-algebraasaKMPoissonbracketalgebraofgauge
invariantdi?erentialpolynomialsshouldinprincipleallowforquantizingitthroughtheKMrepresentationtheory,forexamplebyusingthegeneralBRSTformalismwhichwillbesetupinthispaper.Asa?rststep,weshallgiveaconciseformulafortheVirasorocentreofthisalgebraintermsoftheleveloftheunderlyingKMalgebra.
GTheexistenceofpurely?rstclassKMconstraintsleadingtotheWS-algebramight
beperhapssurprizingtothereader,sinceearlierin[16]itwasclaimedtobeinevitablytotheirclaim,wewilldemonstratethatitispossibleandinfacteasytoobtainthe
Gappropriate?rstclassconstraintswhichleadtoWS.Roughlyspeaking,thiswillbenecessarytouseatleastsomesecondclassconstraintsfromtheverybeginning,whenGreducingtheKMalgebratoWSinthecaseofahalf-integralsl(2)embedding.Contrary
achievedbydiscarding‘half’ofthoseconstraintswhichformthesecondclasspartinthemixedsystemoftheconstraintsimposedin[16].Themixedsystemofconstraintscanberecoveredbyapartialgauge?xingofourpurely?rstclassKMconstraints.Similarly,
l-algebra,arealsoamixedsysteminBershadsky’sconstraints[26],usedtode?netheWn
theabovesense,i.e.,itcontainsboth?rstandsecondclassparts.Wecanalsoreplacetheseconstraintsbypurely?rstclassoneswithoutchangingthe?nalreducedphase
lspace.Inthisprocedureweshalluncoverthehiddensl(2)structureoftheWn-algebras,
G-algebras.namely,weshallidentifythemingeneralasfurtherreductionsofparticularWS
ThestudyofWZNWreductionsembracesvarioussubjects,suchasintegrablemod-els,W-algebrasandtheir?eldtheoreticrealizations.Wehopethatthereaderswithdi?erentinterestswill?ndrelevantresultsthroughoutthispaper,and?ndaninterplayofgeneralconsiderationsandinvestigationsofnumerousexamples.
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2.GeneralstructureofKMandWZNWreductions
ThepurposeofthischapteristoinvestigatethegeneralstructureofthosereductionsoftheKMphasespaceandcorrespondingreductionsofthefullWZNWtheorywhichcanbede?nedbyimposing?rstclassconstraintssettingcertaincurrentcomponentstoconstantvalues.Intherestofthepaper,weassumethattheWZNWgroup,G,isaconnectedrealLiegroupwhoseLiealgebra,G,isanon-compactrealformofacomplexsimpleLiealgebra,Gc.Weshall?rstuncovertheLiealgebraicimplicationsoftheconstraintsbeing?rstclass,andalsodiscussasu?cientconditionwhichmaybeusedtoensuretheirconformalinvariance.Inparticular,weshallseewhythecompactrealformisoutsideourframework.WethensetupagaugedWZNWtheorywhichprovidesaLagrangeanrealizationoftheWZNWreduction,forthecaseofgeneral?rstclassconstraints.Finally,weshalldescribethee?ective?eldtheoriesresultingfromthereductioninsomedetailinanimportantspecialcase,namelywhentheleftandrightKMcurrentsareconstrainedforsuchsubalgebrasofGwhicharedualtoeachotherwithrespecttotheCartan-Killingform.
2.1.FirstclassandconformallyinvariantKMconstraints
HereweanalyzethegeneralformoftheKMconstraintswhichwillbeusedsub-
?sequentlytoreducetheWZNWtheory.TheanalysisappliestoeachcurrentJandJ
separatelysowechooseoneofthem,Jsay,forde?niteness.To?xtheconventions,we?rstnotethattheKMPoissonbracketreads
{?u,J(x)?,?v,J(y)?}|x0=y0=?[u,v],J(x)?δ(x1?y1)+κ?u,v?δ′(x1?y1),(2.1)whereuandvarearbitrarygeneratorsofGandtheinnerproduct?u,v?=Tr(u·v)isnormalizedsothatthelongrootsofGchavelengthsquared2.Thisnormalizationmeansthatintermsoftheadjointrepresentationonehas?u,v?=1
2
pointoutthattheKMPoissonbrackettogetherwithallthesubsequentrelationswhichfollowfromitholdinthesameformbothontheusualcanonicalphasespaceandonthe
11×traceinthede?ningrepresentationfortheBlandDlseries.Wealsowishto
spaceoftheclassicalsolutionsofthetheory.ThisistheadvantageofusingequaltimePoissonbracketsandspatialδ-functionsevenonthelatterspace,whereJ(x)dependsonx=(x0,x1)onlythroughx+(seethefootnoteonpage7).
TheKMreductionweconsiderisde?nedbyrequiringtheconstrainedcurrenttobeofthefollowingspecialform:
J(x)=κM+j(x),withj(x)∈Γ⊥,(2.2)
whereΓissomelinearsubspaceandMissomeelementofG.Equivalently,theconstraintscanbegivenas
φγ(x)=?γ,J(x)??κ?γ,M?=0,?γ∈Γ.(2.3)
Inwords,ourconstraintssetthecurrentcomponentscorrespondingtoΓtoconstantvalues.Itisclearbothfrom(2.2)and(2.3)thatMcanbeshiftedbyanarbitraryelementfromthespaceΓ⊥withoutchangingtheactualcontentoftheconstraints.Thisambiguityisunessential,sinceonecan?xM,forexample,byrequiringthatitisfromsomegivenlinearcomplementofΓ⊥inG,whichcanbechosenbyconvention.
Inourmethodweassumethattheabovesystemofconstraintsis?rstclass,andnowweanalyzethecontentofthiscondition.Immediatelyfrom(2.1),wehave*{φα(x),φβ(y)}=φ[α,β](x)δ(x1?y1)+ωM(α,β)δ(x1?y1)+?α,β?δ′(x1?y1),ofG:
ωM(u,v)≡?M,[u,v]?,?u,v∈G.(2.5)(2.4)wherethesecondtermcontainstherestrictiontoΓofthefollowinganti-symmetric2-form
Itisevidentfrom(2.4)thattheconstraintsare?rstclassif,andonlyif,wehave
[α,β]∈Γ,?α,β?=0andωM(α,β)=0,for?α,β∈Γ.(2.6)ThismeansthatthelinearsubspaceΓhastobeasubalgebraonwhichtheCartan-KillingformandωMvanish.Itiseasytoseethatthethreeconditionsin(2.6)canbeequivalentlywrittenas
[Γ,Γ⊥]?Γ⊥,Γ?Γ⊥and[M,Γ]?Γ⊥,(2.7)
respectively.SubalgebrasΓsatisfyingΓ?Γ⊥existineveryrealformofthecomplexsimpleLiealgebrasexceptthecompactone,sinceforthecompactrealformtheCartan-Killinginnerproductis(negative)de?nite.
WenotethatforagivenΓthethirdconditionandtheambiguityinchoosingMcanbeconciselysummarizedbythe(equivalent)statementthat
M∈[Γ,Γ]⊥/Γ⊥.(2.8)
Theconstraintsde?nedbythezeroelementofthisfactor-spaceareinasensetrivial.Itisclearthat,forasubalgebraΓsuchthatΓ?Γ⊥,theabovefactor-spacecontainsnon-zeroelementsifandonlyif[Γ,Γ]=Γ.ActuallythisisalwayssobecauseΓ?Γ⊥impliesthatΓisasolvablesubalgebraofG.Toprovethis,we?rstnotethatifΓisnotsolvablethen,byLevi’stheorem[33],itcontainsasemi-simplesubalgebra,inwhichonecan?ndeitheranso(3,R)oransl(2,R)subalgebra.FromthisoneseesthatthereexistsatleastonegeneratorλofΓforwhichtheoperatoradλisdiagonalizablewithrealeigenvalues.Itcannotbethatalleigenvaluesofadλare0sinceGisasimpleLiealgebra,andfromthisonegetsthat?λ,λ?=0,whichcontradictsΓ?Γ⊥.ThereforeonecanconcludethatΓisnecessarilyasolvablesubalgebraofG.
Thesecondconditionin(2.6)canbesatis?edforexamplebyassumingthateveryγ∈ΓisanilpotentelementofG.ThisistrueintheconcreteinstancesofthereductionstudiedinChapters3and4.WenotethatinthiscaseΓisactuallyanilpotentLiealgebra,byEngel’stheorem[33].However,thenilpotencyofΓisnotnecessaryforsatisfyingΓ?Γ⊥.Infact,asolvablebutnotnilpotentΓcanbefoundinAppendixA.
Thecurrentcomponentsconstrainedin(2.3)arethein?nitesimalgeneratorsoftheKMtransformationscorrespondingtothesubalgebraΓ,whichactontheKMphasespaceas
J(x)?→eai(x+)γiJ(x)e?ai(x+)γi+(eai(x+)γi′?ai(x+)γi)e,(2.9)
wheretheai(x+)areparameterfunctionsandthereisasummationoversomebasisγiofΓ.Ofcourse,the?rstclassconditionsareequivalenttothestatementthattheconstraintsurface,consistingofcurrentsoftheform(2.2),isleftinvariantbytheabovetransformations.Fromthepointofviewofthereducedtheory,thesetransformationsaretoberegardedasgaugetransformations,whichmeansthatthereducedphasespacecanbeidenti?edasthespaceofgaugeorbitsintheconstraintsurface.Takingthisintoaccount,weshalloftenrefertoΓasthegaugealgebraofthereduction.
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Wenextdiscussasu?cientconditionfortheconformalinvarianceoftheconstraints.WeassumethatM∈/Γ⊥fromnowon.ThestandardconformalsymmetrygeneratedbytheSugawaraVirasorodensityLKM(x)isthenbrokenbytheconstraints(2.3),sincetheysetsomecomponentofthecurrent,whichhasspin1,toanon-zeroconstant.TheideaistocircumventthisapparentviolationofconformalinvariancebychangingthestandardactionoftheconformalgroupontheKMphasespacetoonewhichdoesleavetheconstraintsurfaceinvariant.OnecantrytogeneratethenewconformalactionbychangingtheusualKMVirasorodensitytothenewVirasorodensity
LH(x)=LKM(x)??H,J′(x)?,(2.10)
whereHissomeelementofG.TheconformalactiongeneratedbyLH(x)operatesontheKMphasespaceas
?δf,HJ(x)≡?dy1f(y+){LH(y),J(x)}(2.11)??=f(x+)J′(x)+f′(x+)J(x)+[H,J(x)]+f′′(x+)H,
foranyparameterfunctionf(x+),correspondingtotheconformalcoordinatetransfor-mationδfx+=?f(x+).Inparticular,j(x)in(2.2)transformsunderthisnewconformalactionaccordingto
??δf,Hj(x)=f(x+)j′(x)+f′′(x+)H+f′(x+)j(x)+[H,j(x)]+([H,M]+M),(2.12)
andourconditionisthatthisvariationshouldbeinΓ⊥,whichmeansthatthisconformalactionpreservestheconstraintsurface.From(2.12),oneseesthatthisisequivalenttohavingthefollowingrelations:
H∈Γ⊥,[H,Γ⊥]?Γ⊥and([H,M]+M)∈Γ⊥.(2.13)
Inconclusion,theexistenceofanoperatorHsatisfyingtheserelationsisasu?cientconditionfortheconformalinvarianceoftheKMreductionobtainedbyimposing(2.3).Theconditionsin(2.13)areequivalenttoLH(x)beingagaugeinvariantquantity,induc-ingacorrespondingconformalactiononthereducedphasespace.Obviously,thesecondrelationin(2.13)isequivalentto
[H,Γ]?Γ.(2.14)
AnelementH∈GiscalleddiagonalizableifthelinearoperatoradHpossessesacompletesetofeigenvectorsinG.BytheeigenspacesofadH,suchanelementde?nesa
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gradingofG,andbelowweshallrefertoadiagonalizableelementasagradingoperatorofG.Intheexampleswestudylater,conformalinvariancewillbeensuredbytheexistenceofagradingoperatorsubjectto(2.13).
IfHisagradingoperatorsatisfying(2.13)thenitisalwayspossibletoshiftMbysomeelementofΓ⊥(i.e.,withoutchangingthephysics)sothatthenewMsatis?es
[H,M]=?M,(2.15)
insteadofthelastconditionin(2.13).ItisalsoclearthatifHisagradingoperatorthenonecantakegradedbasesinΓandΓ⊥,sincetheseareinvariantsubspacesunderadH.Onre-inserting(2.15)into(2.12)itthenfollowsthatallcomponentsofj(x)areprimary?eldswithrespecttotheconformalactiongeneratedbyLH(x),withtheexceptionoftheH-component,whichalsosurvivestheconstraintsaccordingtothe?rstconditionin(2.13).
Asanexample,letusnowconsidersomearbitrarygradingoperatorHanddenotebyGmtheeigensubspacecorrespondingtotheeigenvaluemofadH.ThenthegradedsubalgebraG≥n,whichisde?nedtobethedirectsumofthesubspacesGmforallm≥n,willqualifyasagaugealgebraΓforanyn>0fromthespectrumofadH.InthiscaseΓ⊥=G>?nandthefactorspace[Γ,Γ]⊥/Γ⊥,whichisthespaceoftheallowedM’s,canberepresentedasthedirectsumofG?nandthatgradedsubspaceofG<?nwhichisorthogonalto[Γ,Γ].Itiseasytoseethatoneobtainsconformallyinvariant?rstclassconstraintsbychoosingMtobeanygradedelementfromthisfactorspace.Indeed,ifthegradeofMis?mthenLH/myieldsaVirasorodensityweaklycommutingwiththecorrespondingconstraints.
Insummary,inthissectionwehaveseenthatonecanassociatea?rstclasssystemofKMconstraintstoanypair(Γ,M)subjectto(2.6)byrequiringtheconstrainedcurrenttotaketheform(2.2),andthattheconformalinvarianceofthissystemofconstraintsisguaranteedifonecan?ndanoperatorHsuchthatthetriple(Γ,M,H)satis?estheconditionsin(2.13).
2.2.LagrangeanrealizationoftheHamiltonianreduction
WeshallexhibithereagaugedWZNWtheoryprovidingtheLagrangeanrealizationofthoseHamiltonianreductionsoftheWZNWtheorywhichcanbede?nedbyimposing
15
?ofthetheory.It?rstclassconstraintsofthetype(2.3)ontheKMcurrentsJandJ
shouldbenotedthat,intherestofthischapter,wedonotassumethattheconstraintsareconformallyinvariant.
Tode?netheWZNWreduction,wecanchooseleftandrightconstraintscompletelyindependently.Weshalldenotethepairsconsistingofanappropriatesubalgebraanda
?,?M?),constantmatrixcorrespondingtotheleftandrightconstraintsas(Γ,M)and(Γ
respectively.Thereducedtheoryisobtainedby?rstconstrainingtheWZNWphasespacebysetting
φi=?γi,J???γi,M?=0,and?i=??γ????γ??=0,φ?i,J?i,M(2.16)
?respectively,andthenfactorizingthewheretheγiandtheγ?iformbasesofΓandΓ,
constraintsurfacebythecanonicaltransformationsgeneratedbytheseconstraints.Onecanapplythisreductioneithertotheusualcanonicalphasespaceortothespaceofsolutionsoftheclassical?eldequation.Theseareequivalentproceduressincethetwospacesinquestionareisomorphic.Forlaterpurposewenotethattheconstraintsgeneratethefollowingchiralgaugetransformationsonthespaceofsolutions:
g(x,x)?→e+?γ(x+)+??γ?(x?)·g(x,x)·e,(2.17)
?valuedfunctions.whereγ(x+)andγ?(x?)arearbitraryΓandΓ
Forcompleteness,wewishtomentionherehowtheabovewayofreducingtheWZNWtheory?tsintothegeneraltheoryofHamiltonian(symplectic)reductionofsymmetries[34].Ingeneral,theHamiltonianreductionisobtainedbysettingthephasespacefunctionsgeneratingthesymmetrytransformationsthroughPoissonbracket(inotherwords,thecomponentsofthemomentummap)tosomeconstantvalues.Thereducedphasespaceresultsbyfactorizingthisconstraintsurfacebythesubgroupofthesymmetrygrouprespectingtheconstraints.Thesymmetrygroupweconsideristhe
?andourHamiltonianreductionisspecialinleft×rightKMgroupgeneratedbyΓ×Γ
thesensethatthefullsymmetrygrouppreservestheconstraints.Ofcourse,thelatterfactisjustareformulationofthe?rst-classnessofourconstraints.
Wenowcometothemainpointofthesection,whichisthatthereducedWZNWtheory,de?nedintheabovebyusingtheHamiltonianpicture,canbeidenti?edasthegaugeinvariantcontentofacorrespondinggaugedWZNWtheory.ThisgaugedWZNWinterpretationofthereductionwaspointedoutintheconcretecaseoftheWZNW→standardTodareductionin[13],andwebelowgeneralizethatconstructiontothepresentsituation.
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ThegaugedWZNWtheoryweareinterestedinisgivenbythefollowingactionfunctional:
??2I(g,A?,A+)≡SWZ(g)+dx?A?,?+gg?1?M?(2.18)???+?A?,gA+g?1?,+?A+,g?1??g?M
ofthisactionisthatitisinvariantunderthefollowingnon-chiralgaugetransformations:
g→αgα??1;A?→αA?α?1+α??α?1;A+→αA?+α??1+(?+α?)?α?1,
where
α=eγ(x+?respectively.Themainpropertywherethegauge?eldsA?(x)andA+(x)varyinΓandΓ,(2.19a),x?)and?(xα?=eγ+,x?),(2.19b)
?Theproofoftheinvarianceof(2.18)underforanyγ(x+,x?)∈Γandγ?(x+,x?)∈Γ.
(2.19)canproceedalongthesamelinesasforthespecialcasein[13].IntheproofonerewritesSWZ(αgα??1)byusingthewell-knownPolyakov-Wiegmannidentity[35],andinthissteponeusesthefactthattheWZNWactionvanishesfor?eldsinthesubgroups
?ThisisanobviousconsequenceoftherelationsΓ?Γ⊥ofGwithLiealgebrasΓorΓ.
??Γ?⊥.Theothercrucialpointisthatthetermsin(2.18)containingtheconstantandΓ
?areseparatelyinvariantunder(2.19).ItiseasytoseethatthismatricesMandM
followsfromthethirdconditionin(2.6).Forexample,underanin?nitesimalgaugetransformationbelongingtoα?1+γ,theterm?A?,M?changesby
δ?A?,M?=????γ,M?+ωM(γ,A?),(2.20)
whichisatotaldivergencesincethesecondtermvanishes,asbothA?andγarefromΓ.
TheEuler-Lagrangeequationderivedfrom(2.18)byvaryinggcanbewrittenequiv-alentlyas
??(?+gg?1+gA+g?1)+[A?,?+gg?1+gA+g?1]+?+A?=0,
or
?+(g?1??g+g?1A?g)?[A+,g?1??g+g?1A?g]+??A+=0,
andthe?eldequationsobtainedbyvaryingA?andA+aregivenby
?γ,?+gg?1+gA+g?1?M?=0,
17?γ∈Γ,(2.21c)(2.21b)(2.21a)
and
??=0,?γ?,g?1??g+g?1A?g?M?,?γ?∈Γ(2.21d)
WZNWtheoryandtheconstraints(2.16).Furthermore,oneseesthatsettingA±tozeroisnotacompletegauge?xing,theresidualgaugetransformationsareexactlythechiralgaugetransformationsofequation(2.17).
TheaboveargumentstellusthatthespaceofgaugeorbitsinthespaceofclassicalsolutionsofthegaugedWZNWtheory(2.18)canbenaturallyidenti?edwiththereducedrespectively.Wenownotethatbymakinguseofthegaugeinvariance,A+andA?canbesetequaltozerosimultaneously.Theimportantpointforusisthat,asiseasytosee,intheA±=0gaugeonerecoversfrom(2.21)boththe?eldequations(1.3)ofthephasespacebelongingtotheHamiltonianreductionoftheWZNWtheorydeterminedbythe?rstclassconstraints(2.16).ItcanbealsoshownthatthePoissonbracketinducedonthereducedphasespacebytheHamiltonianreductionisthesameastheonedeterminedbythegaugedWZNWaction(2.18).Insummary,weseethatthegaugedWZNWtheory(2.18)providesanaturalLagrangeanimplementationoftheWZNWreduction.
2.3.E?ective?eldtheoriesfromleft-rightdualreductions
Theaimofthissectionistodescribethee?ective?eldequationsandactionfunc-tionalsforanimportantclassofthereducedWZNWtheories.Thisclassoftheoriesis
?areobtainedbymakingtheassumptionthattheleftandrightgaugealgebrasΓandΓ
dualtoeachotherwithrespecttotheCartan-Killingform,whichmeansthatonecan
?sothatchoosebasesγi∈Γandγ?j∈Γ
?γi,γ?j?=δij.(2.22)
Thistechnicalassumptionallowsforhavingasimplegeneralalgorithmfordisentanglingtheconstraints:
φi=?γi,?+gg?1?M?=0,?i=?γ??=0,andφ?i,g?1??g?M(2.23)whichde?nethereduction.Weshallcommentonthephysicalmeaningoftheassumptionattheendofthesection,hereweonlypointoutthatitholds,e.g.,ifonechoosesΓand
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?tobetheimagesofeachotherunderaCartaninvolution*oftheunderlyingsimpleLieΓ
algebra.
Forconcreteness,letusconsiderthemaximallynon-compactrealformwhichcanbede?nedastherealspanofaChevalleybasisHi,E±αofthecorrespondingcomplexLiealgebraGc,andinthecaseoftheclassicalseriesAn,Bn,CnandDnisgivenbysl(n+1,R),so(n,n+1,R),sp(2n,R)andso(n,n,R),respectively.InthiscasetheCartaninvolutionis(?1)×transpose,operatingontheChevalleybasisaccordingto
Hi?→?HiE±α?→?E?α.(2.24)
Itisobviousthat?v,vt?>0foranynon-zerov∈GandfromthisoneseesthatΓtis
?=Γt.ItshoulddualtoΓwithrespecttotheCartan-Killingform,i.e.,(2.22)holdsforΓ
alsobementionedthatthereisaCartaninvolutionforeverynon-compactrealformofthecomplexsimpleLiealgebras,asexplainedindetailin[36].
Equation(2.22)impliesthattheleftandrightgaugealgebrasdonotintersect,andthuswecanconsideradirectsumdecompositionofGoftheform
?,G=Γ+B+Γ(2.25a)
whereBissomelinearsubspaceofG.HereBisinprincipleanarbitrarycomplementary
?inG,butonecanalwaysmakethechoicespaceto(Γ+Γ)
?⊥,B=(Γ+Γ)(2.25b)
whichisnaturalinthesensethattheCartan-Killingformisnon-degenerateonthis
B.ChoosingBaccordingto(2.25b)isespeciallywell-suitedinthecaseoftheparityinvariante?ectivetheoriesdiscussedattheendofthesection.WenotethatitmightalsobeconvenientifonecantakethespaceBtobeasubalgebraofG,butthisisnotnecessaryforourargumentsandisnotalwayspossibleeither.
Wecanassociatea‘generalizedGaussdecomposition’ofthegroupGtothedirectsumdecomposition(2.25),whichisthemaintoolofouranalysis.By‘Gaussdecomposing’anelementg∈Gaccordingto(2.25),wemeanwritingitintheform
g=a·b·c,witha=eγ,b=eβ?andc=eγ,(2.26)
whereγ,βandγ?arefromtherespectivesubspacesin(2.25).
ThereisaneighbourhoodoftheidentityinGconsistingofelementswhichallowauniquedecompositionofthissort,andinthisneighbourhoodthepiecesa,bandccanbeextractedfromgbyalgebraicoperations.(Actuallyitisalsopossibletode?nebasaproductofexponentialscorrespondingtosubspacesofB,andweshallmakeuseofthisfreedomlater,inChapter4.)WemaketheassumptionthateveryG-valued?eldweencounterisdecomposableasgin(2.26).Itiseasilyseenthatinthis‘Gaussdecompos-ablesector’thecomponentsofb(x+,x?)provideacompletesetofgaugeinvariantlocal?elds,whicharethelocal?eldsofthereducedtheoryweareafter.Belowweexplainhowtosolvetheconstraints(2.23)intheGaussdecomposablesectoroftheWZNWtheory.Moreexactly,forourmethodtowork,werestrictourselvestoconsideringthose?eldswhichvaryinsuchaGaussdecomposableneighbourhoodoftheidentitywherethematrix
Vij(b)=?γi,bγ?jb?1?(2.27)
isinvertible.Duetotheassumptions,theanalysisgiveninthefollowingyieldsalocaldescriptionofthereducedtheories.ItisclearthatforaglobaldescriptiononeshouldusepatchesonGobtainedbymultiplyingouttheGaussdecomposableneighbourhoodoftheidentity,butwedonotdealwiththisissuehere.
Firstwederivethe?eldequationofthereducedtheorybyimplementingthecon-straintsdirectlyintheWZNW?eldequation??(?+gg?1)=0.(ThisisallowedsincetheWZNWdynamicsleavestheconstraintsurfaceinvariant,i.e.,theWZNWHamiltonianweaklycommuteswiththeconstraints.)ByinsertingtheGaussdecompositionofginto(2.23)andmakinguseoftheconstraintsbeing?rstclass,theconstraintequationscanberewrittenas?γi,?+bb?1+b(?+cc?1)b?1?M?=0,(2.28)?1?1?1??γ?i,b??b+b(a??a)b?M?=0.
WiththehelpoftheinverseofVij(b)in(2.27),onecansolvetheseequationsfor?+cc?1anda?1??aintermsofb,
?+cc?1=b?1T(b)b,
where
T(b)=
?(b)=T?ij
ijand?(b)b?1,a?1??a=bT(2.29a)?1Vij(b)?γj,M??+bb?1?bγ?ib?1,?1(b)?γ?i,Vij???b?1??b?b?1γjb.M(2.29b)
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Itiseasytoobtainthee?ective?eldequationforthe?eldb(x+,x?)byusingthisexplicitformoftheconstraints.Thiscanbeachievedforexamplebynotingthat,byapplyingtheoperatorAda?1toequation(1.9)(i.e.,byconjugatingitbya?1)theWZNW?eldequationcanbewrittenintheform
[?+?A+,???A?]=0
with
A+=?+bb?1+b(?+cc?1)b?1andA?=?a?1??a.(2.31)
Thus,byinsertingtheconstraints(2.29)intotheaboveformoftheWZNWequation,weseethatthe?eldequationofthereducedtheoryisthezerocurvatureconditionofthefollowingLaxpotential:
?(b)b?1.A+(b)=?+bb?1+T(b)andA?(b)=?bT
Moreexplicitly,thee?ective?eldequationreads
?(b)b?1,T(b)]+??T(b)+b(?+T?(b))b?1=0.??(?+bb?1)+[bT(2.33)(2.32)(2.30)
Theexpressionontheleft-hand-sideof(2.33)ingeneralvariesinthefullspaceG,butnotallthecomponentsrepresentindependentequations.ThenumberoftheindependentequationsisthenumberoftheindependentcomponentsoftheWZNW?eldequationminusthenumberoftheconstraintsin(2.23),sincetheconstraintsautomaticallyimplythecorrespondingcomponentsoftheWZNWequation.Thusthereareexactlyasmanyindependentequationsin(2.33)asthenumberofthereduceddegreesoffreedom.Infact,theindependent?eldequationscanbeobtainedbytakingtheCartan-Killinginnerproductof(2.33)withabasisofthelinearspaceBin(2.25),andtheinnerproductof(2.33)withtheγiandtheγ?ivanishesasaconsequenceoftheconstraintsin(2.23)togetherwiththeindependent?eldequations.Toseethisone?rstrecallsthattheleft-hand-sideof(2.33)is,uponimposingtheconstaints,equivalenttoa?1(??J)a.Thustheinnerproduct
?)c?1withΓ,?vanishesasaconsequenceoftheofthiswithΓ,andsimilarlythatofc(?+J
?)c?1b?1,onecanconstraints.Fromthis,byusingtheidentitya?1(??J)a=?bc(?+J
?alsovanishesasaconsequenceofconcludethattheinnerproductofa?1(??J)awithΓ
theconstraintsandtheindependent?eldequations.
Atthispointwewouldliketomentioncertainspecialcaseswhentheaboveequationssimplify.Firstwenotethatifonehas
[B,Γ]?Γand
21??Γ?,[B,Γ](2.34)
then
?1?(b)=M??πΓ(b?1??b),T(b)=M?πΓ)andT?(?+bb(2.35)
whereweintroducedtheoperators
?πΓ=|γi??γ?i|iandπΓ?=?i|γ?i??γi|,(2.36)
?andassumedthatM∈Γ?andM?∈Γ.(ThelatterwhichprojectontothespacesΓandΓ,
assumptioncanbedonewithoutlossofgeneralityduetothedualitycondition(2.22)).Oneobtains(2.35)from(2.29)bytakingintoaccountthatinthiscaseVij(b)in(2.27)is
?andthustheinverseisgivenbyAdb?1.ThethematrixoftheoperatorAdbactingonΓ,
?⊥isasubalgebraofGandalsosatis?esnicestpossiblesituationoccurswhenB=(Γ+Γ)
?=M?andthus(2.33)simpli?esto(2.34).InthiscaseonesimplyhasT=MandT
??1,M]=0.??(?+bb?1)+[bMb(2.37)
ThederivativetermisnowanelementofBandbycombiningtheaboveassumptionswith
?Γ]??Γ?⊥oneseesthatthecommutatorthe?rstclassconditions[M,Γ]?Γ⊥and[M,
termin(2.37)alsovariesinB,whichensurestheconsistencyofthisequation.
Thee?ective?eldequation(2.33)isingeneralanon-linearequationforthe?eldb(x+,x?),andwecangiveaprocedurewhichcaninprinciplebeusedforproducingitsgeneralsolution.WearegoingtodothisbymakinguseofthefactthatthespaceofsolutionsofthereducedtheoryisthespaceoftheconstrainedWZNWsolutionsfactorizedbythechiralgaugetransformations,accordingtoequation(2.17).Thustheideaisto?ndthegeneralsolutionofthee?ective?eldequationby?rstparametrizing,intermsofarbitrarychiralfunctions,thoseWZNWsolutionswhichsatisfytheconstraints(2.23),andthenextractingtheb-partofthoseWZNWsolutionsbyalgebraicoperations.Inotherwords,weproposetoderivethegeneralsolutionof(2.33)bylookingattheoriginofthisequation,insteadofitsexplicitform.
Tobemoreconcrete,onecanstarttheconstructionofthegeneralsolutionby?rstGauss-decomposingthechiralfactorsofthegeneralWZNWsolutiong(x+,x?)=gL(x+)·gR(x?)as
gL(x+)=aL(x+)·bL(x+)·cL(x+),gR(x?)=aR(x?)·bR(x?)·cR(x?).(2.38)Thentheconstraintequations(2.23)become
?11?+cLc?L=bLT(bL)bL?11?anda?R??aR=bRT(bR)bR.(2.39)
22
equationsareallonehastosolvetoproducethegeneralsolutionofthee?ective?eldequation.Ifthiscanbedonebyquadraturethenthee?ective?eldequationisalsointegrablebyquadrature.Ingeneral,onecanproceedbytryingtosolve(2.39)forthefunctionscL(x+)andaR(x?)intermsofthearbitrary‘inputfunctions’bL(x+)andbR(x?).Clearly,thisinvolvesonlya?nitenumberofintegrationswheneverthegauge
?consistofnilpotentelementsofG.Thusinthiscase(2.33)isexactlyalgebrasΓandΓ
integrable,i.e.,itsgeneralsolutioncanbeobtainedbyquadrature.
Wenotethatinconcretecasessomeotherchoiceofinputfunctions,insteadofthechiralb’s,mightprovemoreconvenientfor?ndingthegeneralsolutionsofthesystemsof?rstorderequationsongLandgRgivenin(2.39)(seeforinstancethederivationofthegeneralsolutionoftheLiouvilleequationgivenin[12]).
Itisnaturaltoaskfortheactionfunctionalunderlyingthee?ective?eldtheoryobtainedbyimposingtheconstraints(2.23)ontheWZNWtheory.Infact,thee?ectiveactionisgivenbythefollowingformula:
Ie?(b)=SWZ(b)???(b)b?1,T(b)?.d2x?bT(2.40)?andInadditiontothethepurelyalgebraicproblemsofcomputingthequantitiesTandTextractingbfromg=gL·gR=a·b·c,these?rstordersystemsofordinarydi?erential
Onecanderivethefollowingconditionfortheextremumofthisaction:
?(b)b?1,T(b)]+??T(b)+b(?+T?(b))b?1?=0.?δbb?1,??(?+bb?1)+[bT(2.41)
Itisstraightforwardtocomputethis,theonlythingtorememberisthattheobjects?b?1andb?1Tbintroducedin(2.29)varyinthegaugealgebrasΓandΓ.?ThearbitrarybT
variationofb(x)isdeterminedbythearbitraryvariationofβ(x)∈B,accordingtob(x)=eβ(x),andthusweseefrom(2.41)thattheEuler-Lagrangeequationoftheaction(2.40)yieldsexactlytheindependentcomponentsofthee?ective?eldequation(2.33),whichweobtainedpreviouslybyimposingtheconstraintsdirectlyintheWZNW?eldequation.
Thee?ectiveactiongivenabovecanbederivedfromthegaugedWZNWactionI(g,A?,A+)givenin(2.18),byeliminatingthegauge?eldsA±bymeansoftheirEuler-Lagrangeequations(2.21c-d).ByusingtheGaussdecomposition,theseEuler-Lagrangeequationsbecomeequivalenttotherelations
?(b)b?1,a?1D?a=bTand
23cD+c?1=?b?1T(b)b,(2.42)
indeedbeobtainedbysubstitutingthesolutionof(2.42)forA±backintoI(g,A?,A+)withg=abc.Tothis?rstwerewriteI(abc,A?,A+)byusingthePolyakov-Wiegmannidentity[35]as
??2I(abc,A?,A+)=SWZ(b)?dx?a?1D?a,b(cD+c?1)b?1?(2.43)???.+?b?1??b,cD+c?1????+bb?1,a?1D?a?+?A?,M?+?A+,M
ThisequationcanberegardedasthegaugecovariantformofthePolyakov-Wiegmannidentity,andallbutthelasttwotermsaremanifestlygaugeinvariant.Thee?ectiveaction(2.40)isderivedfrom(2.43)togetherwith(2.42)bynoting,forexample,that???aa?1,M?isatotalderivative,whichfollowsfromthefactsthata(x)∈eΓandM∈[Γ,Γ]⊥,by(2.8).
Abovewehaveusedthe?eldequationstoeliminatethegauge?eldsfromthegaugedWZNWaction(2.18)onthegroundthatA?andA+arenotdynamical?elds,but‘Lagrangemultiplier?elds’implementingtheconstraints.However,itshouldbenotedthatwithoutfurtherassumptionstheEuler-Lagrangeequationoftheactionresultingfrom(2.18)bymeansofthiseliminationproceduredoesnotalwaysgivethee?ective?eldequation,whichcanalwaysbeobtaineddirectlyfromtheWZNW?eldequation.OnecanseethisonanexampleinwhichoneimposesconstraintsonlyononeofthechiralsectorsoftheWZNWtheory.Fromthispointofview,theroleofourassumptiononthedualityoftheleftandrightgaugealgebrasisthatitguaranteesthatthee?ectiveactionunderlyingthee?ective?eldequationcanbederivedfromI(g,A?,A+)intheabove?(b)aregivenbytheexpressionsin(2.29b)andD±denoteswherethequantitiesT(b)andTthegaugecovariantderivatives,D±=?±?A±.NowweshowthatIe?(b)in(2.40)canmanner.Toendthisdiscussion,wenotethatforg=abcthenon-degeneracyofVij(b)in(2.27)isequivalenttothenon-degeneracyofthequadraticexpression?A?,gA+g?1?iinthecomponentsofA?=Ai?i.Thisquadratictermentersintothe?γiandA+=A+γ
gaugedWZNWactiongivenby(2.18),anditsnon-degeneracyisclearlyimportantinthequantumtheory,whichweconsiderinChapter5.
Wementionedatthebeginningofthesectionthat,consideringamaximallynon-compactG,onecanmakesurethatthedualityassumptionexpressedby(2.22)holds
?tobethetransposesofeachother.HerewepointoutthatthisbychoosingΓandΓ
particularleft-rightrelatedchoiceofthegaugealgebrascanalsobeusedtoensuretheparityinvarianceofthee?ective?eldtheory.Tothis?rstwenoticethat,inthecaseofamaximallynon-compactconnectedLiegroupG,theWZNWactionSWZ(g)isinvariant
24
underanyofthefollowingtwo‘paritytransformations’g?→Pg:
(P1g)(x0,x1)≡gt(x0,?x1),and(P2g)(x0,x1)≡g?1(x0,?x1).(2.44)
?=ΓtandM?=Mttode?netheWZNWreductionthentheparityIfonechoosesΓ
?in(2.23),transformationP1simplyinterchangestheleftandrightconstraints,φandφ
andthusthecorrespondinge?ective?eldtheoryisinvariantundertheparityP1.The
?⊥,i.e.,thechoicein(2.25b),isinvariantunderthetransposeinthisspaceB=(Γ+Γ)
case,andthusthegaugeinvariant?eldbtransformsinthesamewayunderP1asgdoesin(2.44).Ofcourse,theparityinvariancecanalsobeseenonthelevelofthegaugedactionI(g,A?,A+).Namely,I(g,A?,A+)isinvariantunderP1ifoneextendsthede?nitionin(2.44)toincludethefollowingparitytransformationofthegauge?elds:
01(P1A±)(x0,x1)≡At?(x,?x).(2.45)
TheP1-invariantreductionproceduredoesnotpreservetheparitysymmetryP2,butitispossibletoconsiderreductionspreservingjustP2insteadofP1.Infact,suchreductions
?=ΓandM?=M.canbeobtainedbytakingΓ
Finally,itisobviousthattoconstructparityinvariantWZNWreductionsingeneral,forsomearbitrarybutnon-compactrealformGofthecomplexsimpleLiealgebras,onecanuse?σinsteadofthetranspose,whereσisaCartaninvolutionofG.
25
G3.PolynomialityinKMreductionsandtheWS-algebras
Inthepreviouschapterwedescribedtheconditionsfor(2.2)de?ning?rstclassconstraintsandforLH(J)in(2.10)beingagaugeinvariantquantityonthisconstraintsurface.ItisclearthattheKMPoissonbracketsofthegaugeinvariantdi?erentialpoly-nomialsofthecurrentalwayscloseonsuchpolynomialsandδ-distributions.Thealgebraofthegaugeinvariantdi?erentialpolynomialsisofspecialinterestintheconformallyin-variantcasewhenitisapolynomialextensionoftheVirasoroalgebra.InSection3.1weshallgiveanadditionalconditiononthetriple(Γ,M,H)whichallowsonetoconstructoutofthecurrentin(2.2)acompletesetofgaugeinvariantdi?erentialpolynomialsbymeansofadi?erentialpolynomialgauge?xingalgorithm.WecalltheKMreductionpolynomialifsuchapolynomialgauge?xingalgorithmisavailable,andalsocallthecorrespondinggaugesDrinfeld-Sokolov(DS)gauges,sinceourconstructionisageneral-izationoftheonegivenin[5].TheKMPoissonbracketalgebraofthegaugeinvariantdi?erentialpolynomialsbecomestheDiracbracketalgebraofthecurrentcomponentsin
GtheDSgauges,whichweconsiderinSection3.2.TheextendedconformalalgebraWSmentionedintheIntroductionisespeciallyinterestinginthatitsprimary?eldbasisis
manifestandgivenbythesl(2)structure,asweshallseeinSection3.3.Oneofourmainresultsisthatweshall?ndhere?rstclassKMconstraintsunderlyingthisalgebra,such
Gthattheysatisfyoursu?cientconditionforpolynomiality.ThuswecanrepresentWSasaKMPoissonbracketalgebraofgaugeinvariantdi?erentialpolynomials,whichin
principleallowsforitsquantizationthroughtheKMrepresentationtheory.Theimpor-
GtanceoftheWS-algebrasisclearlydemonstratedbytheresultofSection3.4,wherewe
lshowthattheWn-algebrasof[26]canbeinterpretedasfurtherreductionsofparticular
GlWS-algebras.Thismakesitpossibletoexhibitprimary?eldsfortheWn-algebrasand
GtodescribetheirstructureindetailintermsofthecorrespondingWS-algebras,whichis
thesubjectof[37].
3.1.Asu?cientconditionforpolynomiality
Letussupposethat(Γ,M,H)satisfythepreviouslygivenconditions,(2.6)and(2.13),for
J(x)=M+j(x),
26j(x)∈Γ⊥(3.1)
describingtheconstraintsurfaceofconformallyinvariant?rstclassconstraints,whereHisagradingoperatorandMissubjectto
[H,M]=?M,M∈/Γ⊥.(3.2)
Then,asweshallshow,thefollowingtwoadditionalconditions:
Γ∩KM={0},
and
Γ⊥?G>?1,(3.4a)
allowforestablishingadi?erentialpolynomialgauge?xingalgorithmwherebyonecanconstructoutofJ(x)in(3.1)acompletesetofgaugeinvariantdi?erentialpolynomials.
Beforeprovingthisresult,wediscusssomeconsequencesoftheconditions,whichweshallneedlater.InthepresentsituationΓ,Γ⊥andGaregradedbytheeigenvaluesofadH,and?rstwenotethat(3.4a)isequivalentto
G≥1?Γ.(3.4b)whereKM=Ker(adM),(3.3)
Indeed,thisfollowsfromthefactthatthespacesGhandG?haredualtoeachotherwithrespecttotheCartan-Killingform,whichisaconsequenceofitsnon-degeneracyandinvarianceunderadH.Ofcourse,hereandbelowthegradingistheonede?nedbyH,andwenotethatG±1arenon-trivialbecauseof(3.2).Theconditiongivenby(3.4a)playsatechnicalroleinourconsiderations,butperhapsitcanbearguedforalsophysically,onthebasisthatitensuresthattheconformalweightsoftheprimary?eldcomponentsofj(x)in(3.1)arenon-negativewithrespecttoLH(2.10).Second,letusobservethatinoursituationMsatisfying(3.2)isuniquelydetermined,thatis,thereisnopossibilityofshiftingitbyelementsfromΓ⊥,simplybecausetherearenograde?1elementsinΓ⊥,onaccountof(3.4a).Equation(3.3)meansthattheoperatoradMmapsΓintoΓ⊥inaninjectivemanner,andforthisreasonwereferto(3.3)asthenon-degeneracycondition.Combiningthenon-degeneracyconditionwith(3.2),(3.4a)and(2.7)weseethatourgaugealgebraΓcancontainonlypositivegrades:
Γ?G>0.(3.5)
Thisimpliesthateveryγ∈Γisrepresentedbyanilpotentoperatorinany?nitedimen-sionalrepresentationofG,andthat
G≥0?Γ⊥.
27(3.6)
Itfollowsfrom(3.2)that[H,KM]?KM,whichistellingusthatKMisalsograded,andweseefrom(3.3)and(3.4b)that
KM?G<1.(3.7)
Finally,wewishtoestablishacertainrelationshipbetweenthedimensionsofGandKM.ForthispurposeweconsideranarbitrarycomplementaryspaceTMtoKM,de?ningalineardirectsumdecomposition
G=KM+TM.
ωMtoTMisasymplecticform,inotherwords:
ωM(TM,TM)isnon?degenerate.(3.9)(3.8)Itiseasytoseethatforthe2-formωMwehaveωM(KM,G)=0,andtherestrictionof
(WenoteinpassingthatTMcanbeidenti?edwiththetangentspaceatMtothecoadjointorbitofGthroughM,andinthispictureωMbecomestheKirillov-Kostantsymplecticformoftheorbit[34].)Thenon-degeneracycondition(3.3)saysthatonecanchoosethespaceTMin(3.8)insuchawaythatΓ?TM.Onethenobtainstheinequality
dim(Γ)≤
wherethefactor112??dim(G)?dim(KM),(3.10)
jtobenotedthat,byde?nition,thesubsriptkonelementsθk∈Θdoesnotdenotethegrade,whichis(1?k).Thenormal(orreduced)formcorrespondingtoΘisgivenbythe
followingequation:
Jred(x)=M+jred(x)wherejred(x)∈Γ⊥∩Θ⊥.(3.12)
Inotherwords,thesetofreducedcurrentsisobtainedbysupplementingthe?rstclassconstraintsofequation(2.3)bythegauge?xingcondition
χθ(x)=?J(x),θ???M,θ?=0,?θ∈Θ.(3.13)
whichisdisjointfromtheimageofΓundertheoperatoradMandisinfactcomplementarytotheimage,i.e.,onehas
Γ⊥=[M,Γ]+V.WecallagaugewhichcanbeobtainedintheabovemanneraDrinfeld-Sokolov(DS)gauge.ItisnothardtoseethatthespaceV=Γ⊥∩Θ⊥isagradedsubspaceofΓ⊥(3.14)
Italsofollowsfromthenon-degeneracycondition(3.3)thatanygradedcomplementVin(3.14)canbeobtainedintheabovemanner,bymeansofusingsomeΘ.Thusitispossibletode?netheDSnormalformofthecurrentdirectlyintermsofacomplementaryspaceVaswell,ashasbeendoneinspecialcasesin[5,13,18].
Asthe?rststepinprovingthatanycurrentin(3.1)isgaugeequivalenttoonein?+ltheDSgauge,letusconsiderthegaugetransformationbygh(x+)=exp[lalh(x)γh]forsome?xedgradeh.Suppressingthesummationoverl,itcanbewrittenas*
j(x)→jgh(x)=eah·γh(j(x)+M)e?ah·γh+(eah·γh)′e?ah·γh?M.(3.15)
iTakingtheinnerproductofthisequationwiththebasisvectorsθkin(3.11)forallk≤h,
weseethatthereisnocontributionfromthederivativeterm.Wealsoseethattheonlycontributionfrom
eah·γhj(x)e?ah·γh=j(x)+[ah(x+)·γh,j(x)]+...(3.16)
listheonecomingfromthe?rstterm,sinceallcommutatorscontainingtheelementsγh
dropoutfromtheinnerproductinquestionasaconsequenceofthefollowingcrucialrelation:
li[γh,θk]∈Γ,fork≤h,(3.17)
whichfollowsfrom(3.4b)bynotingthatthegradeofthiscommutator,(1+h?k),isatleast1fork≤h.Takingtheseintoaccount,andcomputingthecontributionfromthosetwotermsinjgh(x)whichcontainMbyusing(3.11),weobtain
ii+?θk,jgh(x)?=?θk,j(x)??aih(x)δhk,forallk≤h.(3.18)
Weseefromthisequationthat
i?θk,j(x)?=0??i?θk,jgh(x)?=0,fork<h,(3.19)
and
+iaih(x)=?θh,j(x)??i?θh,jgh(x)?=0,fork=h.(3.20)
iTheselasttwoequationstellusthatifthegauge-?xingcondition?θk,j(x)?=0issatis?ed
forallk<hthenwecanensurethatthesameconditionholdsforjgh(x)fortheextended
+irangeofindicesk≤h,bychoosingaih(x)tobe?θh,j(x)?.Fromthisitiseasytosee
thattheDSgauge(3.13)canbereachedbyaniterativeprocessofgaugetransformations,
+andthegauge-parametersaih(x)areuniquepolynomialsinthecurrentateachstageof
theiteration.
Inmoredetail,letuswritethegeneralelementg(a(x+))∈eΓofthegaugegroupasaproductinorderofdescendinggrades,i.e.,as
g(a(x+))=ghn·ghn?1···gh1,
where
hn>hn?1>...>h1
isthelistofgradesoccurringinΓ.Letustheninsertthisexpressioninto
j→jg=g(j+M)g?1+g′g?1?M,
andconsiderthecondition
jg(x)=jred(x),(3.22b)
withjred(x)in(3.12),asanequationforthegauge-parametersah(x+).Oneseesfromtheaboveconsiderationsthatthisequationisuniquelysolubleforthecomponentsof
30(3.22a)(3.21b)withghi(x+)=eahi(x+)·γhi,(3.21a)
theah(x+)andthesolutionisadi?erentialpolynomialinj(x).Thisimpliesthatthecomponentsofjred(x)canalsobeuniquelycomputedfrom(3.22),andthesolutionyieldsacompletesetofgaugeinvariantdi?erentialpolynomialsofj(x),whichestablishestherequiredresult.Theaboveiterativeprocedureisinfactaconvenienttoolforcomputingthegaugeinvariantdi?erentialpolynomialsinpractice[15].Weremarkthat,ofcourse,anyuniquegauge?xingcanbeusedtode?negaugeinvariantquantities,buttheyareingeneralnotpolynomial,notevenlocalinj(x).
WealsowishtonotethatanarbitrarylinearsubspaceofGwhichisdualtoVin(3.14)withrespecttotheCartan-KillingformcanbeusedinanaturalwayasthespaceofparametersfordescribingthosecurrentdependentKMtransformationswhichpreservetheDSgauge.Infact,itispossibletogiveanalgorithmwhichcomputestheW-algebraanditsactionontheother?eldsofthecorrespondingconstrainedWZNWtheoryby?ndingthegaugepreservingKMtransformationsimplementingtheW-transformations.Thisalgorithmpresupposestheexistenceofsuchgaugeinvariantdi?erentialpolynomialswhichreducetothecurrentcomponentsintheDSgauge,whichisensuredbytheabovegauge?xingalgorithm,butitworkswithoutactuallycomputingthem.Thisissueistreatedindetailin[13,18]inspecialcases,buttheresultsgiventhereapplyalsotothegeneralsituationinvestigatedintheabove.
3.2.ThepolynomialityoftheDiracbracket
Itfollowsfromthepolynomialityofthegauge?xingthatthecomponentsofthegauge?xedcurrentjredin(3.12)generateadi?erentialpolynomialalgebraunderDiracbracket.InourproofofthepolynomialityweactuallyonlyusedthatthegradedsubspaceΘofGisdualtothegradedgaugealgebraΓwithrespecttoωMandsatis?esthecondition
([Θ,Γ])≥1?Γ,(3.23)
ilsatisfying(3.11)and(3.17).andθkwhichisequivalenttotheexistenceofthebasesγh
Wehaveseenthatthisconditionfollowsfrom(3.3)and(3.4),butitshouldbenotedthatitisamoregeneralcondition,sincetheconverseisnottrue,asisshownbyanexampleattheendofthissection.
BelowwewishtogiveadirectproofforthepolynomialityoftheDiracbracket
31
algebrabelongingtothesecondclassconstraints:
cτ(x)=?τ,J(x)?M?=0whereliτ∈{γh}∪{θk}.(3.24)
Theproofwillshedanewlightonthepolynomialitycondition.Wenotethatforcertainpurposessecondclassconstraintsmightbemorenaturaltousethan?rstclassonessinceinthesecondclassformalismonedirectlydealswiththephysical?elds.Forexample,theGWS-algebramentionedintheIntroductionisverynaturalfromthesecondclasspointofviewandcanberealizedbystartingwithanumberofdi?erent?rstclasssystemsofconstraints,asweshallseeinthenextsection.
We?rstrecallthat,byde?nition,theDiracbracketalgebraofthereducedcurrentsisuvuv{jred(x),jred(y)}?={jred(x),jred(y)}??,uv?dz1dw1{jred(x),c?(z)}??ν(z,w){cν(w),jred(y)}
?ν(3.25)
uwhere,foranyu∈G,jred(x)=?u,jred(x)?istobesubstitutedby?u,J(x)?M?under
theKMPoissonbracket,and??ν(z,w)istheinverseofthekernel
D?ν(z,w)={c?(z),cν(w)},
inthesensethat(ontheconstraintsurface)
??dx1??ν(z,x)Dνσ(x,w)=δ?σδ(z1?w1).
ν(3.26)(3.27)
ToestablishthepolynomialityoftheDiracbracket,itisusefultoconsiderthematrixdi?erentialoperatorD?ν(z)de?nedbythekernelD?ν(z,w)intheusualway,i.e.,
???D?ν(z)fν(z)=dw1D?ν(z,w)fν(w),(3.28)
νν
foravectorofsmoothfunctionsfν(z),whichareperiodicinz1.Fromthestructureoftheconstraintsin(3.24),cτ=(φγ,χθ),oneseesthatD?ν(z)isa?rstorderdi?erentialoperatorpossessingthefollowingblockstructure
???DγγD0?γθD?ν==DθDθθ?E??γ??EF?,(3.29)
whereE?istheformalHermitianconjugateofthematrixE,(E?)θγ=(Eγθ)?.ItisclearthattheDiracbracketin(3.25)isadi?erentialpolynomialinjred(x)andδ(x1?y1)
32
whenevertheinverseoperatorD?1(z),whosekernelis??ν(z,w),isadi?erentialoperatorwhosecoe?cientsaredi?erentialpolynomialsinjred(z).Ontheotherhand,weseefrom(3.29)thattheoperatorDisinvertibleifandonlyifitsblockEisinvertible,andinthatcasetheinversetakestheform
???1??1??1(E)FE?(E)(D?1)?ν=.(3.30)E?10
SinceE(z)andF(z)arepolynomial(evenlinear)injred(z)andin?zandtheinverseofF(z)doesnotoccurinD?1(z),itfollowsthatD?1(z)isapolynomialdi?erentialoperatorifandonlyifE?1(z)isapolynomialdi?erentialoperator.
ToshowthatE?1existsandisapolynomialdi?erentialoperatorwenotethatintermsofthebasisof(Γ+Θ)in(3.24)thematrixEisgivenexplicitlybythefollowingformula:
mnmnm,θn(z)=δhkδmn+?[γEγh,θ],j(z)?+?γ,θk??z.(3.31)redhkhk
Thecrucialpointisthat,bythegradingandthepropertyin(3.17),wehave
m,θn(z)=δhkδnm,Eγhkfork≤h.(3.32)
ThematrixEhasablockstructurelabelledbythe(block)rowand(block)columnindiceshandk,respectively,and(3.32)meansthattheblocksinthediagonalofEareunitmatricesandtheblocksbelowthediagonalvanish.Inotherwords,EisoftheformE=1+ε,whereεisastrictlyuppertriangularmatrix.Itisclearthatsuchamatrixdi?erentialoperatorispolynomiallyinvertible,namelybya?niteseriesoftheform
E?1=1?ε+ε2+...+(?1)NεN,(εN+1=0),(3.33)
which?nishesourproofofthepolynomialityoftheDiracbracketin(3.25).OnecanusetheargumentsintheaboveprooftosetupanalgorithmforactuallycomputingtheDiracbracket.TheproofalsoshowsthatthepolynomialityoftheDiracbracketisguaranteedwheneverEisoftheform(1+ε)withεbeingnilpotentasamatrix.Inourcasethiswasensuredbyaspecialgradingassumption,anditappearsaninterestingquestionwhetherpolynomialreductionscanbeobtainedatallwithoutusingsomegradingstructure.
ThezeroblockoccursinD?1in(3.30)becausethesecondclassconstraintsoriginatefromthegauge?xingof?rstclassones.WenotethatthepresenceofthiszeroblockimpliesthattheDiracbracketsofthegaugeinvariantquantitiescoincidewiththeiroriginalPoissonbrackets,namelyoneseesthisfromtheformulaoftheDiracbracketby
33
keepinginmindthatthegaugeinvariantquantitiesweaklycommutewiththe?rstclassconstraints.
Finally,wewanttoshowthatcondition(3.23)isweakerthan(3.3-4).Thisisbestseenbyconsideringanexample.TothisletnowGbethemaximallynon-compactrealformofacomplexsimpleLiealgebra.If{M?,M0,M+}istheprincipalsl(2)embeddingΓH≥1={M+}.Furthermore,writingH=(M0+?),we?ndfrom[H,M±]=±M±that?mustbeansl(2)singletintheadjointofG.However,inthecaseoftheprincipalsl(2)embedding,thereisnosuchsingletintheadjoint,andhenceH=M0.Butthen
M0theconditionG≥1?Γisnotful?lled.beθ=M0,andthen(3.23)holds.Toshowthatconditions(3.3-4)cannotbesatis?ed,HweprovethatagradingoperatorHforwhich[H,M?]=?M?andG≥1?Γ,doesnotexist.Firstofall,[H,M?]=?M?and?M?,M+?=0imply[H,M+]=M+,andthusinG,withcommutationrulesasin(3.34)below,wesimplychoosetheone-dimensionalgaugealgebraΓ≡{M+}andtakeM≡M?.TheωM-dualtoM+canbetakento
G3.3.FirstclassconstraintsfortheWS-algebras
LetS={M?,M0,M+}beansl(2)subalgebraofthesimpleLiealgebraG:
[M0,M±]=±M±,[M+,M?]=2M0.(3.34)
WearguedintheIntroductionthatitisnaturaltoassociateanextendedconformal
Galgebra,denotedasWS,toanysuchsl(2)embedding[16,18].Namely,wede?nedthe
GWS-algebratobetheDiracbracketalgebrageneratedbythecomponentsofthecon-strainedKMcurrentofthethefollowingspecialform:
Jred(x)=M?+jred(x),withjred(x)∈Ker(adM+),(3.35)
whichmeansthatjred(x)isalinearcombinationofthesl(2)highestweightstatesintheadjointofG.Thisde?nitionisindeednaturalinthesensethattheconformalpropertiesaremanifest,since,asweshallseebelow,withtheexceptionoftheM+-componentthespinscomponentofjred(x)turnsouttobeaprimary?eldofconformalweight(s+1)withrespecttoLM0.Beforeshowingthis,weshallconstructhere?rstclassKM
G-algebra,whichwillbeusedinChapter4toconstructconstraintsunderlyingtheWSG-algebrasastheirchiralalgebras.WegeneralizedTodatheorieswhichrealizetheWS
34
GexpecttheWS-algebrastoplayanimportantorganizingroleindescribingthe(primary
?eldcontentof)conformallyinvariantKMreductionsingeneral,andshallgiveargumentsinfavourofthisidealater.
Wewishto?ndagaugealgebraΓforwhichthetriple(Γ,H=M0,M=M?)satis?esoursu?cientconditionsforpolynomialityand(3.35)representsaDSgaugeforthecorrespondingconformallyinvariant?rstclassconstraints.WestartbynoticingthatthedimensionofsuchaΓhastosatisfytherelation
GdimKer(adM+)=dimWS=dimG?2dimΓ.(3.36)
Fromthis,sincethekernelsofadM±areofequaldimension,weobtainthat
dimΓ=1
2dimKer(adM?),(3.37)
whichmeansby(3.10)thatwearelookingforaΓofmaximaldimension.Bytherepre-sentationtheoryofsl(2),theaboveequalityisequivalentto
dimΓ=dimG≥1+1
2,(3.38)
wherethegradingisbythe,ingeneralhalf-integral,eigenvaluesofadM0.Wealsoknow,
(3.4b)and(3.5),thatforourpurposewehavetochoosethegradedLiesubalgebraΓofGinsuchawaythatG≥1?Γ?G>0.Observethatthenon-degeneracycondition(3.3)isautomaticallysatis?edforanysuchΓsinceinthepresentcaseKer(adM?)?G≤0,andM0∈Γ⊥isalsoensured,whichguaranteestheconformalinvariance,see(2.13).
Itisobviousfromtheabovethatinthespecialcaseofanintegralsl(2)subalgebra,forwhichG1
2
havethecorrectdimension.Thekeyobservationforde?ningtherequiredhalvingofG1
isnon-degenerate.This2
35toG≥1,inorderto
canbeseenasaconsequenceof(3.9),butisalsoeasytoverifydirectly.BythewellknownDarbouxnormalformofsymplecticforms[34],thereexistsa(non-unique)directsumdecomposition
G1
Q12+Q12andQ12and
2,(3.42)
byusinganysymplectichalvingoftheabovekind.Itisobviousfromtheconstructionthatthe?rstclassconstraints,
J(x)=M?+j(x)withj(x)∈Γ⊥,(3.43)
obtainedbyusingΓin(3.42)satisfythesu?cientconditionsforpolynomialitygiveninSection3.1.WiththisΓwehave
Γ⊥=G≥0+Q?1
2isthesubspaceofG?1
2=[M?,P1
Gcomponents,andwecanthusrealizetheWS-algebraasaKMPoissonbracketalgebra
ofgaugeinvariantdi?erentialpolynomials.
Thesecondclassconstraintsde?ningthehighestweightgauge(3.35)arenaturalinthesensethatinthiscaseτin(3.24)runsoverthebasisofthespaceTM?=[M+,G]whichisanaturalcomplementofKM?=Ker(adM?)inG,eq.(3.8).
algebraisgivenbythefollowingformula:
??1+?δf,Mj(x)≡?dyf(y){L(y),j(x)},redMred00GInthesecondclassformalism,theconformalactiongeneratedbyLM0ontheWS-(3.46)
wheretheparameterfunctionf(x+)referstotheconformalcoordinatetransformationδfx+=?f(x+),cf.(2.11),andjred(x)istobesubstitutedbyJ(x)?M?whenevaluatingtheKMPoissonbracketsenteringinto(3.46),likein(3.25).Toactuallyevaluate(3.46),we?rstreplaceLM0bytheobject
Lmod(x)=LM0(x)?1
2f′′′(x+)M+.(3.48)
Thisprovesthat,withtheexceptionoftheM+-component,thesl(2)highestweightcomponentsofjred(x)in(3.35)transformasconformalprimary?elds,wherebythecon-
GformalcontentofWSisdeterminedbythedecompositionoftheadjointofGunderS
intheaforementionedmanner.WeendthisdiscussionbynotingthatinthehighestweightgaugeLM0(x)becomesalinearcombinationoftheM+-componentofjred(x)andaquadraticexpressioninthecomponentscorrespondingtothesingletsofSinG.FromthisweseethatLM0(x)andtheprimary?eldscorrespondingtothesl(2)highestweight
Gstatesgiveabasisforthedi?erentialpolynomialscontainedinWS,whichisthusindeed
a(classical)W-algebrainthesenseofthegeneralideain[20].
Intheaboveweproposeda‘halvingprocedure’for?ndingpurely?rstclasscon-
GstraintsforwhichWSappearsasthealgebraofthecorrespondinggaugeinvariantdif-
ferentialpolynomials.Wenowwishtoclarifytherelationshipbetweenourmethodand
37
GtheconstructioninarecentpaperbyBaisetal[16],wheretheWS-algebrahasbeende-
scribed,inthespecialcaseofG=sl(n),byusingadi?erentmethod.Werecallthatthe
GWS-algebrahasbeenconstructedin[16]byaddingtothe?rstclassconstraintsde?nedbythepair(G≥1,M?)thesecondclassconstraints
?u,J(x)?=0,for?u∈G1
2.(3.50)
foranysl(2)subalgebra,namelybyusingourgeneralmethodofWZNWreductions.Thiswillbeelaboratedinthenextchapter.Wenotethatin[16]theauthorswereactuallyalsoledtoreplacingtheoriginalconstraintsbya?rstclasssystemofconstraints,inordertobeabletoconsidertheBRSTquantizationofthetheory.Forthispurposetheyintroducedunphysical‘auxiliary?elds’andthusconstructed?rstclassconstraintsinanextendedphasespace.However,inthatconstructiononehastocheckthattheauxiliary?elds?nallydisappearfromthephysicalquantities.Anotherimportantadvantageofourhalvingprocedureisthatitrenderstheuseofanysuchauxiliary?eldscompletelyunnecessary,sinceonecanstartbyimposingacompletesystemof?rstclassconstraintsontheKMphasespacefromtheverybeginning.WestudysomeaspectsoftheBRSTquantizationinChapter5,andweshallseethattheVirasorocentralchargegivenin[16]agreeswiththeonecomputedbytakingour?rstclassconstraintsasthestartingpoint.
GThe?rstclassconstraintsleadingtoWSarenotunique,forexampleonecanconsider
anarbitraryhalvingin(3.41)tode?neΓ.WeconjecturethattheseW-algebrasalwaysOneoftheadvantagesofourconstructionisthatbyusingonly?rstclassKMconstraintsGitiseasytoconstructgeneralizedTodatheorieswhichpossessWSastheirchiralalgebra,occurundercertainnaturalassumptionsontheconstraints.Tobemoreexact,letussupposethatwehaveconformallyinvariant?rstclassconstraintsdeterminedbythepair(Γ,M?)whereM?isanilpotentmatrixandthenon-degeneracycondition(3.3)holdstogetherwithequation(3.37).BytheJacobson-Morozovtheorem,itispossibletoextendthenilpotentgeneratorM?toansl(2)subalgebraS={M?,M0,M+}.ItisalsoworthnotingthattheconjugacyclassofSundertheautomorphismgroupofGisuniquelydeterminedbytheconjugacyclassofthenilpotentelementM?.Forthisandotherquestionsconcerningthetheoryofsl(2)embeddingsintosemi-simpleLiealgebras
38
GpolynomialsandtheiralgebraisisomorphictoWS,whereM?∈S.Wearenotyetabletoprovethisconjectureingeneral,butbelowwewishtosketchtheproofinanimportant
specialcasewhichillustratestheidea.thereadermayconsultrefs.[32,33,38,39].Weexpectthattheaboveassumptionson(Γ,M?)aresu?cientfortheexistenceofacompletesetofgaugeinvariantdi?erential
suchthatHisanintegralgradingoperatorofG.Wenotethattheseareexactlytheassumptionssatis?edbytheconstraintsinthenon-degeneratecaseofthegeneralizedTodatheoriesassociatedtointegralgradings[18].Inthiscaseequation(3.37)isactuallyautomaticallysatis?edasaconsequenceofthenon-degeneracycondition(3.3).Onecanalsoshowthatitispossibleto?ndansl(2)algebraS={M?,M0,M+}forwhichin
[H,M0]=0and[H,M+]=M+,Letusassumethatwehaveconformallyinvariant?rstclassconstraintsdescribedby(Γ,M?,H)subjecttothesu?cientconditionsforpolynomialitygiveninSection3.1,additionto[H,M?]=?M?onehas(3.51)
andthatforthissl(2)algebratherelation
HKer(adM+)?G≥0(3.52)
holds,wherethesuperscriptindicatesthatthegradingisde?nedbyH.Forthesl(2)
Hsubjectto(3.51)thelatterpropertyisinfactequivalenttoKer(adM?)?G≤0,whichis
Hjustthenon-degeneracyconditionasinourcaseΓ=G>0.Theproofofthesestatements
isgiveninAppendixB.
Weintroduceade?nitionatthispoint,whichwillbeusedintherestofthepaper.Namely,wecallansl(2)subalgebraS={M?,M0,M+}anH-compatiblesl(2)fromnowonifthereexistsanintegralgradingoperatorHsuchthat[H,M±]=±M±issatis?edtogetherwiththenon-degeneracycondition.Thenon-degeneracyconditioncanbeexpressedinvariousequivalentforms,itcanbegivenforexampleastherelationin(3.52),andits(equivalent)analogueforM?.
Turningbacktotheproblemathand,wenowpointoutthatbyusingtheH-
Hcompatiblesl(2)wehavethefollowingdirectsumdecompositionofΓ⊥=G≥0:
HHG≥0=[M?,G>0]+Ker(adM+).(3.53)
Thismeansthatthesetofcurrentsoftheform(3.35)representsaDSgaugeforthepresent?rstclassconstraints.Thisimpliestherequiredresult,thatisthattheW-
Halgebrabelongingtotheconstraintsde?nedbyΓ=G>0togetherwithanon-degenerate
39
GM?isisomorphictoWSwithM?∈S.InthisexamplebothLH(x)andLM0(x)are
gaugeinvariantdi?erentialpolynomials.AlthoughthespectrumofadHisintegralbyassumption,insomecasestheH-compatiblesl(2)isembeddedintoGinahalf-integralmanner,i.e.,thespectrumofadM0canbehalf-integralincertaincases.Weshallreturn
tothispointlater.Wefurthernotethatingeneralitisclearlyimpossibletobuildsuchansl(2)outofM?forwhichHwouldplaytheroleofM0.Itispossibletoprovethatinthosecasesthereisnofullsetofprimary?eldswithrepecttoLHwhichwouldcompletethisVirasorodensitytoageneratingsetofthecorrespondingdi?erentialpolynomialW-
Galgebra.WehaveseenthatsuchaconformalbasisismanifestforWS,whichseemsto
indicatethatinthepresentsituationtheconformalstructurede?nedbythesl(2),LM0,ispreferredincomparisontotheonede?nedbyLH.
whichwillbeusedinthenextsection.LetusconsiderthedecompositionofGunderthesl(2)subalgebraS.Ingeneral,weshall?ndsingletstatesandtheyspanaLiesubalgebraGWealsowouldliketomentionaninterestinggeneralfactabouttheWS-algebras,intheLiesubalgebraKer(adM+)ofG.LetusdenotethiszerospinsubalgebraasZ.It
iseasytoseethatwehavethesemi-directsumdecomposition
Ker(adM+)=Z+R,[Z,R]?R,[Z,Z]?Z,(3.54)
whereRisthelinearspacespannedbytherestofthehighestweightstates,whichhavenon-zerospin.ItisnothardtoprovethatthesubalgebraoftheoriginalKMalgebra
GwhichbelongstoZ,survivesthereductiontoWS.Inotherwords,theDiracbracketsof
theZ-componentsofthehighestweightgaugecurrent,jredin(3.35),coincidewiththeiroriginalKMPoissonbrackets,givenby(2.1).Furthermore,thisZKMsubalgebraacts
GontheWS-algebrabythecorrespondingoriginalKMtransformations,whichpreserve
thehighestweightgauge:
Jred(x)→eai(x+)ζiJred(x)e?ai(x+)ζi+(eai(x+)ζi′?ai(x+)ζi)e,(3.55)
GwheretheζiformabasisofZ.Inparticular,oneseesthattheWS-algebrainheritesthe
semi-directsumstructuregivenby(3.54)[16].Thepointwewishtomakeisthatitis
G-algebrabyapplyingthegeneralmethodofconformallypossibletofurtherreducetheWSinvariantKMreductionstothepresentZKMsymmetry.Inprinciple,onecangenerate
G-algebrasinthisahugenumberofnewconformallyinvariantsystemsoutoftheWSway,i.e.,byapplyingconformallyinvariantconstraintstotheirsingletKMsubalgebras.
Forexample,ifonecan?ndasubalgebraofZonwhichtheCartan-KillingformofGvanishes,thenonecanconsidertheobviouslyconformallyinvariantreductionobtained
40
importancewillbehighlightedbytheexampleofthenextsection.byconstrainingthecorrespondingcomponentsofjredin(3.35)tozero.WedonotexploreGthese‘secondary’reductionsoftheWS-algebrasinthispaper.However,theirpotential
Finally,wenotethat,forahalf-integralsl(2),onecanconsider(insteadofusingΓin(3.42))alsothoseconformallyinvariant?rstclassconstraintswhicharede?nedbythetriple(Γ,M0,M?)withanygradedΓforwhichG≥1?Γ?(G≥1+P1
theconstraintsφδ(x)=0forallδ∈?,likein(2.3).Generally,theseconstraintscomprise?rstandsecondclassparts,wherethe?rstclasspartistheonebelongingtothesubalgebraDof?de?nedbytherelationωM?(D,?)=0,(see(2.4)).Thesecond
classpartbelongstothecomplementaryspace,C,ofDin?.Infact,forl=1theconstraintsaretheusual?rstclassoneswhichyieldthestandardW-algebras,butthesecondclasspartisnon-emptyforl>1.TheaboveKMreductionissoconstructedthatitisconformallyinvariant,sincetheconstraintsweaklycommutewiththeVirasorodensityLHl(x),see(2.10),whereHl=1
l
and0≤r<l,wecantake
??mrtimes?
M0=diag?2???m,···,···,?rtimes?
2andl?rofspinm?1
2+G1+G>1.(3.59)
Fromthisandthede?nitionofωM?,thesubalgebraDcomprisingthe?rstclasspart
canalsobedecomposedinto
D=D0+D1+G>1,
where
D0=Ker(adM?)∩?0(3.61)
isthesetofthesl(2)singletsin?,andD1isasubspaceofG1whichwedonotneedtospecify.Bycombining(3.59)and(3.60),weseethatthecomplementaryspaceC,towhichthesecondclasspartbelongs,hasthestructure
C=Q0+G1(3.60)
wherethesubspaceQ0iscomplementarytoD0in?0,andP1iscomplementarytoD1inG1.The2-formωM?isnon-degenerateonCbyconstruction,andthisimpliesbythe
gradingthatthespacesQ0andP1aresymplecticallyconjugatetoeachother,whichisre?ectedbythenotation.
Weshallconstructagaugealgebra,Γ,sothatBershadsky’sconstraintswillberecov-eredbyapartialgauge?xingfromthe?rstclassonesbelongingtoΓ.Asageneralizationofthehalvingprocedureoftheprevioussection,wetakethefollowingansatz:
Γ=D+P1
2isde?nedbymeansofsomesymplectichalvingG12+Q1
2+G≥1,(3.64)
whichwouldbejustthefamiliarformula(3.42)ifD0wasnothere.Byusing(3.57)and(3.58),D0canbeidenti?edasthesetofn×nblock-diagonalmatrices,σ,ofthefollowingform:
σ=block-diag{Σ0,σ0,Σ0,.....,Σ0,σ0,Σ0},(3.65)
wheretheΣ0’sandtheσ0’sareidenticalcopiesofstrictlyuppertriangularr×rand(l?r)×(l?r)matricesrespectively.Thisimpliesthat
dimD0=1
2]?P1
2forwhichP1
toitshalf-integraleigenvalues,andthensubstractingamultipleoftheunitmatrixsoastomaketheresulttraceless.Intheadjointrepresentation,wethenhave2
43
adH=adM0onthetensors,andadH=adM0±1/2onthespinors.Wenoticefrom
thisthattheH-gradingisanintegralgrading.Infact,therelationshipbetweenthetwogradingsallowsustode?neagoodhalvingofG1
2≡G12≡G1
withrespecttoωM?.Thatthisisagoodhalving,i.e.,
itensuresthecondition(3.67),canalsobeseeneasilybyobservingthatD0hasgrade2
0intheH-grading,too.ThusweobtaintherequiredsubalgebraΓofGbyusingthisparticularP1
2+P1)ofΓ,(3.63),byimposingthepartialgauge?xing
qi∈(Q0+Q1
2conditionφqi(x)=0,)andtheφq’sarede?nedlikein(2.3).
Thisimpliesthatthereducedphasespacede?nedbytheconstraintsin(3.69)isthesameastheonedeterminedbytheoriginalconstraints(3.56).Inconclusion,ourpurely?rstclassconstraints,(3.69),havethesamephysicalcontentasBershadsky’soriginalmixedsetofconstraints,(3.56).
lFinally,wegivetherelationshipbetweenBershadsky’sWn-algebrasandthesl(2)
lsystems.HavingseenthatthereducedKMphasespacescarryingtheWn-algebrascan
berealizedbystartingfromthe?rstclassconstraintsin(3.69),itfollowsfrom(3.64)
GlthattheWn-algebrascoincidewithparticularWS-algebrasifandonlyifthespaceD0is
G2lempty,i.e.,forWnwithn=odd.InordertoestablishtheWSinterpretationofWnin
thegeneralcase,wepointoutthatthereducedphasespacecanbereachedfrom(3.69)bymeansofthefollowingtwostepprocessbasedonthesl(2)structure.Namely,onecanproceedby?rst?xingthegaugefreedomcorrespondingtothepiece(P1
phasespaceisobtainedinthesecondstepby?xingthegaugefreedomgeneratedbytheconstraintsbelongingtoD0,whichwehaveseentobethespaceoftheuppertriangular
lsingletsofS.ThuswecanconcludethatWncanberegardedasafurtherreductionof
GthecorrespondingWS,wherethe‘secondaryreduction’isofthetypementionedatthe
lendofSection3.3.Onecanexhibitprimary?eldbasesfortheWn-algebrasanddescribe
GtheirstructureindetailintermsoftheunderlyingWS-algebrasbyfurtheranalysingthe
secondaryreduction,butthisisoutsidethescopeofthepresentpaper,see[37].the?rststepisthesameastheoneobtainedbyputtingtozerothosecomponentsoftheGhighestweightgaugecurrentrepresentingWSwhichcorrespondtoD0.The?nalreduced
45
4.GeneralizedTodatheories
Letusremindourselvesthat,ashasbeendetailedintheIntroduction,thestandardconformalToda?eldtheoriescanbenaturallyregardedasreducedWZNWtheories,and
?GastheircanonicalasaconsequencethesetheoriespossessthechiralalgebrasWG×WSSsymmetries,whereSistheprincipalsl(2)subalgebraofthemaximallynon-compactreal
LiealgebraG.ItisnaturaltoseekforWZNWreductionsleadingtoe?ective?eldtheories
G?Gastheirchiralalgebrasforanysl(2)subalgebraSofanywhichwouldrealizeWS×WSsimplerealLiealgebra.Themainpurposeofthischapteristoobtain,bycombiningthe
resultsofsections2.3and3.3,generalizedTodatheoriesmeetingtheaboverequirementinthenon-trivialcaseofthehalf-integralsl(2)subalgebrasofthesimpleLiealgebras.Beforeturningtodescribingthesenewtheories,nextwebrie?yrecallthemainfeaturesofthosegeneralizedTodatheories,associatedtotheintegralgradingsofthesimpleLiealgebras,whichhavebeenstudiedbefore[3,4,14-18].Thesimplicityofthelattertheorieswillmotivatesomesubsequentdevelopments.
4.1.GeneralizedTodatheoriesassociatedwithintegralgradings
TheWZNWreductionleadingtothegeneralizedTodatheoriesinquestionissetupbyconsideringanintegralgradingoperatorHofG,andtakingthespecialcase
HΓ=G≥1and?=GH,Γ≤?1
?∈GH,M1(4.1)andanynon-zeroHM∈G?1and(4.2)
inthegeneralconstructiongiveninSection2.3.WenotethatbyanintegralgradingoperatorH∈GwemeanadiagonalizableelementwhosespectrumintheadjointofG
Hconsistsofintegersandcontains±1,andthatGndenotesthegradensubspacede?ned
HbyH.InthepresentcaseBin(2.25b)isthesubalgebraG0ofG,and,becauseofthe
gradingstructure,thepropertiesexpressedbyequation(2.34)hold.Thusthee?ective?eldequationreadsas(2.37)andthecorrespondingactionisgivenbythesimpleformula
HIe?(b)=SWZ(b)??
46??1,M?,d2x?bMb(4.3)
wherethe?eldbvariesinthelittlegroupGH0ofHinG.
Generalized,ornon-Abelian,Todatheoriesofthistypehavebeen?rstinvestigatedbyLeznovandSaveliev[1,3],whode?nedthesetheoriesbypostulatingtheirLaxpoten-tial,
?1AH+M,+=?+b·b??1,AH?=?bMb(4.4)
whichtheyobtainedbyconsideringtheproblemthatifonerequiresaG-valuedpure-gaugeLaxpotentialtotakesomespecialform,thentheconsistencyofthesystemofequationscomingfromthezerocurvatureconditionbecomesanon-trivialproblem.Incomparison,wehaveseeninSection2.3thatintheWZNWframeworktheLaxpotentialoriginatesfromthechiralzerocurvatureequation(1.9),andtheconsistencyandtheintegrabilityofthee?ectivetheoryarisingfromthereductionisautomatic.
?aretakentobeItwasshownin[3,4,16]inthespecialcasewhenH,MandM
thestandardgeneratorsofanintegralsl(2)subalgebraofG,thatthenon-AbelianTodaequationallowsforconservedchiralcurrentsunderlyingitsexactintegrability.These
GcurrentsthengeneratechiralW-algebrasofthetypeWS,forintegrallyembeddedsl(2)’s.
BymeansoftheargumentgiveninSection3.3,wecanestablishthestructureofthechiralalgebrasofawiderclassofnon-AbelianTodasystems[18].Namely,weseethatif
?in(4.2)satisfythenon-degeneracyconditionsMandM
HKer(adM)∩G≥1={0}andHKer(adM?)∩G≤?1={0},(4.5)
Gnotalwaysintegrallyembeddedones.Thusforcertainhalf-integralsl(2)algebrasWScanberealizedinageneralizedTodatheoryofthetype(4.3).Aswewouldliketohavethentheleft×rightchiralalgebraofthecorrespondinggeneralizedTodatheoryisisomor-G?G,whereS?(S+)isansl(2)subalgebraofGcontainingthenilpotentphictoWS×WS+??),respectively.TheH-compatiblesl(2)algebrasS±occurringherearegeneratorM(M
GgeneralizedTodatheorieswhichpossessWSastheirsymmetryalgebraforanarbitrary
sl(2)subalgebra,wehavetoaskwhetherthetheoriesgivenabovearealreadyenoughforthispurposeornot.Thisleadstothetechnicalquestionastowhetherforeveryhalf-integralsl(2)subalgebraS={M?,M0,M+}ofGthereexistsanintegralgradingoperatorHsuchthatSisanH-compatiblesl(2),inthesenseintroducedinSection3.3.Theanswertothisquestionisnegative,asproveninAppendixC,wheretherelationshipbetweenintegralgradingsandsl(2)subalgebrasisstudiedindetail.Thuswehaveto?ndnewintegrableconformal?eldtheoriesforourpurpose.
47
4.2.GeneralizedTodatheoriesforhalf-integralsl(2)embeddings
InthefollowingweexhibitageneralizedTodatheorypossessingtheleft×right
G?Gforanarbitrarilychosenhalf-integralsl(2)subalgebraS=chiralalgebraWS×WS{M?,M0,M+}ofthearbitrarybutnon-compactsimplerealLiealgebraG.Clearly,ifoneimposes?rstclassconstraintsofthetypedescribedinSection3.3onthecurrentsof
theWZNWtheorythentheresultinge?ective?eldtheorywillhavetherequiredchiralalgebra.WeshallchoosetheleftandrightgaugealgebrasinsuchawaytobedualtoeachotherwithrespecttotheCartan-Killingform.
Turningtothedetails,?rstwechooseadirectsumdecompositionofG1
=P?1tobegivenby22
Q?1∩G?1]andP?1∩G?1].(4.6)thesubspaces2222
Itiseasytoseethatthe2-formωM+vanishesontheabovesubspacesofG?1
2.Thuswe
cantaketheleftandrightgaugealgebrastobe
Γ=(G≥1+P1),(4.7)2
?enteringtheconstraintsgivenbyM?andM+,withtheconstantmatricesMandM
respectively.ThedualityhypothesisofSection2.3isobviouslysatis?edbythisconstruc-tion.
Inprinciple,theactionandtheLaxpotentialofthee?ectivetheorycanbeobtainedbyspecializingthegeneralformulasofSection2.3tothepresentparticularcase.Inourcase
B=Q1
?2,(4.8)andthephysicalmodes,whicharegivenbytheentriesofbinthegeneralizedGaussdecompositiong=abcwitha∈eΓandc∈eΓ,arenowconvenientlyparametrizedas
b(x)=exp[q1(x)],(4.9)2
andg0(x)∈G0,thelittlegroupofM0inG.Nextweintroducewhereq±12
somenotationwhichwillbeusefulfordescribingthee?ectivetheory.
48
designatethisoperatorasa2×2matrix:
Adg0|G
TheoperatorAdg0mapsG?1
asatwo-componentcolumnvectoru=(u1u2)twithu1∈P?1
2
2
,wecan
?1
2
2
,respectively.Analogously,weintroducethenotation
Adg?1
andQ?1
|G1
2
asacolumnvector,whoseupper
andlowercomponentsbelongtoP1
2
,respectively.
Theactionfunctionalofthee?ective?eldtheoryresultingfromtheWZNWreduc-tionathandreadsasfollows:
?
?1S2
Ie?(g0,q1)=S(g)?dx?gMg,M??WZ00+02
?(4.12a)
???1
+d2x???q1g?,?+?η1
022wheretheobjectsη±1
η1
2
2
aregivenbytheformulae
2
]+Y12·??q1
=[M?,q1
2
.(4.12b)
TheEuler-Lagrangeequationofthisactionisthezerocurvatureconditionofthefollowing
Laxpotential:
?1
AS+=M?+?+g0·g0+g0(?+q?1
?1
)g,02
2
AS?
=?
?1
g0M+g0
???q1
.
(4.13)
Theabovenew(conformallyinvariant)e?ectiveactionandLaxpotentialareamongthemainresultsofthepresentpaper.Clearly,foranintegrallyembeddedsl(2)thisactionandLaxpotentialsimplifytotheonesgivenbyequation(4.3)and(4.4).Thederivationoftheaboveformulaeisnotcompletelystraightforward,andnextwewishtosketchthemainsteps.First,letusrememberthat,by(2.29a),tospecializethegenerale?ectiveactiongivenby(2.40)andtheLaxpotentialgivenby(2.32)tooursituation,weshouldexpresstheobjects?+cc?1anda?1??aintermsofbbyusingthe
49
?,respectively.(InthepresentcaseitwouldbetedioustocomputeconstraintsonJandJ
theinversematrixofVijin(2.27),whichwouldbeneededforusingdirectly(2.29b).)ForthispurposeitturnsouttobeconvenienttoparametrizetheWZNW?eldgbyusingthegradingde?nedbythesl(2),i.e.,as
g=g+·g0·g?whereg+=a·exp[q12]·c.(4.14)
Werecallthatthe?eldsa,c,g0andq±1
+q1+q?1
222
P±1∈
2partofJ?1
=?+p?1.(4.18)2=[p12,M?]+g0·N?122
Byusingthenotationintroducedin(4.10),thevanishingoftheprojectionofJtoP?1
2,M?]+X11·?+p?12=0,(4.19)
andfromthisweobtain
?+p?12]?X12·?+q?1
2??1+X11(g0)·[M?,q1
502?.(4.21)
Asimilaranalysisappliedtotherightconstraintsyieldsthattheyareequivalenttothefollowingequation:
?1?1·??g+=?g0M+g0???q1?g+2]+Y12(g0)·??q1
],whichmadethe
?nalresultsimpler.Ofcourse,fortheaboveanalysiswehavetorestrictourselvestoa2
neighbourhoodoftheidentitywheretheoperatorsX11(g0)andY11(g0)areinvertible.
Thechoiceoftheconstraintsleadingtothee?ectivetheory(4.12)guaranteesthatthe
G?G,andthusoneshouldbeabletochiralalgebraofthistheoryistherequiredone,WS×WSexpresstheW-currentsintermsofthelocal?eldsintheaction.Tothis?rstwerecallthat
inSection3.1wehavegivenanalgorithmforconstructingthegaugeinvariantdi?erentialpolynomialsW(J).ThepointwewishtomakeisthattheexpressionofthegaugeinvariantobjectW(J)intermsofthelocal?eldsin(4.12)issimplyW(?+bb?1+T(b)),wherebisgivenby(4.9).Applyingthereasoningof[40,18]tothepresentcase,thisfollowssincethefunctionWisform-invariantunderanygaugetransformationofitsargument,andthequantity(?+bb?1+T(b))isobtainedbya(non-chiral)gaugetransformationfromJ,namelybythegaugetransformationde?nedbythe?elda?1∈eΓ,seeequations(2.31-2).(Inanalogy,whenconsideringarightmovingW-currentonegaugetransforms
??bythe?eldc∈eΓtheargumentJ.)WecaninprinciplecomputetheobjectT(b),as
explainedintheabove,andthuswehaveanalgorithmfor?ndingtheformulaeoftheW’sintermsofthelocal?eldsg0andq±1
1haveconformalweights(2),respectively.Thisassignment2
oftheconformalweightscanbeestablishedinanumberofways,onecanforexamplederiveitfromthecorrespondingconformalsymmetrytransformationoftheWZNW?eldginthegaugedWZNWtheory,seeeq.(5.30).Wealsonotethattheaction(4.12)canbe
51
madegenerallycovariantandtherebyourgeneralizedTodatheorycanbere-interpretedasatheoryoftwo-dimensionalgravitysinceφbecomesthegravitationalLiouvillemode
[14].
WewouldliketopointouttherelationshipbetweenthegeneralizedTodatheorygivenby(4.12)andcertainnon-linearintegrableequationswhichhavebeenassociatedtothehalf-integralsl(2)subalgebrasofthesimpleLiealgebrasbyLeznovandSaveliev,byusingadi?erentmethod.(See,e.g.,equation(1.24)inthereviewpaperinJ.Sov.Math.referredtoin[3].)Tothiswenotethat,inthehalf-integralcase,onecanalsoconsiderthatWZNWreductionwhichisde?nedbyimposingtheleftandrightconstraints
?in(4.7).Infact,theLaxcorrespondingtothesubalgebrasG≥1andG≤?1ofΓandΓ
potentialofthee?ective?eldtheorycorrespondingtothisWZNWreductioncoincideswiththeLaxpotentialpostulatedbyLeznovandSavelievtosetuptheirtheory.Thus,inasense,theirtheoryliesbetweentheWZNWtheoryandourgeneralizedTodatheorywhichhasbeenobtainedbyimposingalargersetof?rstclassKMconstraints.Thismeansthatthetheorygivenby(4.12)canalsoberegardedasareductionoftheirtheory.
Thereisacertainfreedominconstructinga?eldtheorypossessingtherequired
GchiralalgebraWS,forexample,onehasafreedomofchoiceinthehalvingprocedure
usedheretosetupthegaugealgebra.Thetheoriesin(4.12)obtainedbyusingdi?erenthalvingsinequation(3.41)havetheirchiralalgebrasincommon,butitisnotquiteobviousifthesetheoriesarealwayscompletelyequivalentlocalLagrangean?eldtheoriesornot.Wehavenotinvestigatedthis‘equivalenceproblem’ingeneral.
Aspecialcaseofthisproblemarisesfromthefactthatonecanexpectthatinsomecasesthetheoryin(4.12)isequivalenttooneoftheform(4.3).Thisiscertainlysointhosecaseswhenforthehalf-integralsl(2)ofM0andM±onecan?ndanintegralgrading
+G≤?1=operatorHsuchthat:(i)[H,M±]=±M±,(ii)P12HHG≤?=G,whereoneusestheM0gradingandtheH-grading01,(iv)Q?12
ontheleft-andontherighthandsidesoftheseconditions,respectively.Byde?nition,
+Q1wecallthehalvingG12
ofarenamingofthevariables,sinceinthiscasetherelevant?rstclassconstraintsareintheoverlapoftheoneswhichhavebeenconsideredfortheintegralgradingsandforthehalf-integralsl(2)’stoderivetherespectivetheories.Sincetheformoftheactionin(4.3)ismuchsimplerthantheonein(4.12),itappearsimportanttoknowthelistofthosesl(2)embeddingswhichallowforanH-compatiblehalving,i.e.,forwhichconditions(i)...(iv)canbesatis?edwithsomeintegralgradingoperatorHandhalving.Westudythisgrouptheoreticquestionforthesl(2)subalgebrasofthemaximallynon-compactrealformsoftheclassicalLiealgebrasinAppendixC.WeshowthattheexistenceofanH-compatiblehalvingisaveryrestrictiveconditiononthehalf-integralsl(2)subalgebrasofthesymplecticandorthogonalLiealgebras,wheresuchahalvingexistsonlyforthespecialsl(2)embeddingslistedattheendofAppendixC.Incontrast,itturnsoutthatforG=sl(n,R)anH-compatiblehalvingcanbefoundforeverysl(2)subalgebra,sinceinthiscaseonecanconstructsuchahalvingbyproceedingsimilarlyaswedidinSection
G3.4(see(3.68)).ThismeansthatinthecaseofG=sl(n,R)anychiralalgebraWScan
berealizedinageneralizedTodatheoryassociatedtoanintegralgrading.
Itisinterestingtoobservethatthosetheorieswhichcanbealternativelywritteninbothforms(4.3)and(4.12)allowforseveralconformalstructures.Thisissosinceinthiscaseatleasttwodi?erentVirasorodensities,namelyLHandLM0,survivetheWZNW
reduction.
4.3.TwoexamplesofgeneralizedTodatheories
Wewishtoillustrateherethegeneralconstructionoftheprevioussectionbyworkingouttwoexamples.FirstweshalldescribeageneralizedTodatheoryassociatedtothehighestrootsl(2)ofsl(n+2,R).Thisisahalf-integralsl(2)embedding,but,asweshallseeexplicitly,thetheory(4.12)caninthiscaseberecastedintheform(4.3),sincethecorrespondinghalvingisH-compatible.WenotethattheW-algebrasde?nedbythesesl(2)embeddingshavebeeninvestigatedbeforebyusingauxiliary?eldsin[29].Itisperhapsworthstressingthatourmethoddoesnotrequiretheuseofauxiliary?eldswhenreducingtheWZNWtheorytothegeneralizedTodatheorieswhichpossesstheseW-algebrasastheirsymmetryalgebras,seealsoSection5.3.AccordingtothegroupG-algebrade?nedbyatheoreticanalysisinAppendixC,thesimplestcasewhenaWShalf-integralsl(2)embeddingcannotberealizedinatheoryofthetype(4.3)isthecase
53
ofG=sp(4,R).Asoursecondexample,weshallelaborateonthegeneralizedTodatheoryin(4.12)whichrealizestheW-algebrabelongingtothehighestrootsl(2)ofsp(4,R).
i)Highestrootsl(2)ofsl(n+2,R)
IntheusualbasiswheretheCartansubalgebraconsistsofdiagonalmatrices,thesl(2)subalgebraSisgeneratedbytheelements
M0=1
istraceorthogonaltoM0andg?0isfromsl(n).WenotethatTandM0generatethecentreofthecorrespondingsubalgebra,G0.WeconsiderthehalvingofG±1
andQ±12??0pt0
=?00n0?,q120···0??0···0
=?p?0n0?,q?120···0
areinvariantunder
theadjointactionofg0,whichmeansthattheblock-matricesin(4.10)and(4.11)are2n···?0?2In2+n0···??00?n(4.24)
diagonal,andthusη±1
,q?12?].OnecanalsoverifythatX=e11212φ+ψ?1(?+q?)t·g?0·(??q)
+e1
wheredotmeansusualmatrixmultiplication.Withrespecttotheconformalstructurede?nedbyM0,eφhasweights(1,1),the?eldsqandq?havehalf-integerweights(1
respectively,ψandg?0areconformalscalars.Inparticular,weseethatφis
theLiouvillemodewithrespecttothisconformalstructure.
Infact,thehalvingconsideredin(4.25)canbewrittenliketheonein(3.68),byusingtheintegralgradingoperatorHgivenexplicitlyas
??1n+10H=M0+.0?In+1n+22),(4.27)
ItisanH-compatiblehalvingasonecanverifythatitsatis?estheconditions(i)...(iv)mentionedattheendofSection4.2,seealsoAppendixC.ItfollowsthatourreducedWZNWtheorycanalsoberegardedasageneralizedTodatheoryassociatedwiththeintegralgradingH.Inotherwords,itispossibletoidentifythee?ectiveaction(4.26)asaspecialcaseoftheonein(4.3).Toseethisinconcreteterms,itisconvenienttoparametrizethelittlegroupofHas
b=exp(q1
andS=M0?(n+2),2whereg0=eΦH·eξS1·?00??···0g?00?,···1(4.28)
2Φ?2+n
andtheCartansubalgebraisdiagonal.Thesl(2)subalgebraScorrespondingtothehighestrootofsp(4,R)isgeneratedbythematrices
1M0=
+2·2underS.Thethreesingletsgenerate
ansl(2)subalgebradi?erentfromS,sothatthelittlegroupofM0isGL(1)×SL(2).GL(1)isgeneratedbyM0itselfandthecorresponding?eldistheLiouvillemode.UsingusualGauss-parametersfortheSL(2),wecanparametrizethelittlegroupofM0as??1000ψ?ψ0αe?ψ?φM0?0e+αβe(4.32)g0=e?.?0010
0βe?ψ0e?ψ
WedecomposetheG±1
2+q12+q?1
2sothatthematricesXijandYijin(4.10)and(4.11)
possesso?-diagonalelements:
Xij=e?1
2φ?ψ(??q)·(?+q?)
?(4.35)
,+2e12φ?ψ??1β??q·q+e?
eψ+αβe?ψ
fortheLiouvillemodeφ,theconformalscalarsψ,α,βandthe?eldsq,q?withweights(12),respectively.
Itiseasytoseedirectlyfromitsformulathatitisimpossibletoobtaintheaboveactionasaspecialcaseof(4.3).Indeed,iftheexpressionin(4.35)wasobtainedfrom(4.3)thenthenon-derivativeterm?q?q(eψ+αβe?ψ)?1couldonlybegottenfromthesecondtermin(4.3),but,sinceg0andbarematricesofunitdeterminant,thistermcouldneverproducethedenominatorinthenon-derivativetermin(4.35).
56
5.QuantumframeworkforWZNWreductions
InthischapterwestudythequantumversionoftheWZNWreductionbyusingthepath-integralformalismandalsore-examinesomeoftheclassicalaspectsdiscussedinthepreviouschapters.We?rstshowthatthecon?gurationspacepath-integraloftheconstrainedWZNWtheorycanberealizedbythegaugedWZNWtheoryofSection2.2.Wethenpointoutthatthee?ectiveactionofthereducedtheory,(2.40),canbederivedbyintegratingoutthegauge?eldsinaconvenientgauge,thephysicalgauge,inwhichthegaugedegreesoffreedomarefrozen.Anontrivialfeatureofthequantumtheorymayappearinthepath-integralmeasure.Weshall?ndthatforthegeneralizedTodatheoriesassociatedwithintegralgradingsthee?ectivemeasuretakestheformdeterminedfromthesymplecticstructureofthereducedtheory.ThismeansthatinthiscasethequantumHamiltonianreductionresultsinthequantizationofthereducedclassicaltheory;inotherwords,thetwoprocedures,thereductionandthequantization,commute.WeshallalsoexhibittheW-symmetryofthee?ectiveactionforthisexample.ByusingthegaugedWZNWtheory,wecanconstructtheBRSTformalismfortheWZNWreductioninthegeneralcase.Forconformallyinvariantreductions,thisallowsforcomputingthecorrespondingVirasorocentreexplicitly.Inparticular,wederivehereaniceformula
GfortheVirasorocentreofWSforanarbitrarysl(2)embedding.Weshallverifythat
ourresultagreeswiththeoneobtainedin[16],inspiteoftheapparentdi?erenceinthestructureoftheconstraints.
5.1.Path-integralforconstrainedWZNWtheory
Inthissectionwewishtosetupthepath-integralformalismfortheconstrainedWZNWtheory.Forthis,werecallthatclassicallythereducedtheoryhasbeenobtainedbyimposingasetof?rst-classconstraintsintheHamiltonianformalism.Thuswhatweshoulddoistowritedownthepath-integraloftheWZNWtheory?rstinphasespacewiththeconstraintsimplementedandthen?ndthecorrespondingcon?gurationspaceexpression.Thephasespacepath-integralcanformallybede?nedoncethecanonicalvariablesofthetheoryarespeci?ed.Apracticalwayto?ndthecanonicalvariablesisthefollowing[41].LetusstartfromtheWZNWactionSWZ(g)in(1.2)andparametrizethegroupelementg∈Ginsomearbitraryway,g=g(ξ).Weshallregardtheparameters
57
ξa,a=1,...,dimG,asthecanonicalcoordinatesinthetheory.To?ndthecanonicalmomenta,weintroducethe2-formA=1
Tr(dgg?1)3=dA.(5.1)3
The2-formAiswell-de?nedonlylocallyonG,sincetheWess-Zumino3-formisclosedbutnotexact.FortunatelywedonotneedtospecifyAexplicitlybelow.Wenextde?neNab(ξ)by??g
??=κ0ξa
TheHamiltonianoftheWZNWtheory?Nab(ξ)(?0gg?1)b?Aab(ξ)?1ξbisthengivenbyH=?.
H=Πa?0ξa?L=1?dx1Hwith(5.3)
?)inthepath-whereweimplementthe?rstclassconstraintsbyinsertingδ(φ)andδ(φ
integral.Theδ-functionsofχandχ?refertogauge?xingconditionscorrespondingto
?,whichactasgeneratorsofgaugesymmetries.Byintroducingtheconstraints,φandφ
iLagrange-multiplier?elds,A?=Ai?i,(5.7)canbewrittenas?γiandA+=A+γ
Z=?
BychangingthemomentumvariablefromΠatoPain(5.5),themeasureacquiresadeterminantfactor,dΠ=dPdetN,andtheintegrandoftheexponentin(5.8)becomes
?)?HTr(Π?0ξ+A?φ+A+φ?1?21P+=κTr?κ
2(?1gg?12?χdΠdξdA+dA?δ(χ)δ(χ?)det|{φ,χ}|det|{φ,?}|?????2?)?H.×expidxTr(Π?0ξ+A?φ+A+φ(5.8))+A?(?1gg?1?M)?A+(g?1??).?1g+M(5.9)
SincethematrixN(ξ)isindependentofP,wecaneasilyperformtheintegrationoverPprovidedthattheremainingδ-functionsandthedeterminantfactorsarealsoP-independent.Wecanchoosethegauge?xingconditions,χandχ?,sothatthisistrue.(Forexample,thephysicalgaugewhichwewillchooseinthenextsectionful?llsthisdemand.)Thenweendupwiththefollowingformulaofthecon?gurationspacepath-integral:
Z=??χdξdetNdA+dA?δ(χ)δ(χ?)det|{φ,χ}|det|{φ,?}|eiI(g,A?,A+),(5.10)whereI(g,A?,A+)isthegaugedWZNWaction(2.18).Wenotethatthemeasureforthecoordinatesinthispath-integralistheinvariantHaarmeasure,
d?(g)=?a?dξdetN=(dgg?1)a.aa(5.11)
Thisisaconsequenceofthefactthatthephasespacemeasurein(5.7)isinvariantundercanonicaltransformationstowhichthegrouptransformationsbelong.
Theaboveformulaforthecon?gurationspacepath-integralmeansthatthegaugedWZNWtheoryprovidestheLagrangianrealizationoftheHamiltonianreduction,whichwehavealreadyseenonthebasisofaclassicalargumentinSection2.2.
59
5.2.E?ectivetheoryinthephysicalgauge
HavingseenhowtheconstrainedWZNWtheoryisrealizedasthegaugedWZNWtheory,wenextdiscussthee?ectivetheorywhichariseswhenweeliminatealltheun-physicaldegreesoffreedominaparticularlyconvenientgauge,thephysicalgauge.Weshallrederive,inthepath-integralformalism,thee?ectiveactionwhichappearedintheclassicalcontextearlierinthispaper.Forthispurpose,withinthissectionwerestrictourattentiontotheleft-rightdualreductionsconsideredinSection2.3.It,however,shouldbenotedthatthisrestrictionisnotabsolutelynecessarytogetane?ectiveac-tionbythemethodgivenbelow.Inthisrespect,itisalsoworthnotingthatPolyakov’s2-dimensionalgravityactioninthelight-conegaugecanberegardedasane?ectiveac-tioninanon-dualreduction,whichisobtainedbyimposingaconstraintonlyontheleft-currentforG=SL(2)[43,12].Wewillnotpursuethenon-dualcaseshere.
Toeliminatealltheunphysicalgaugedegreesoffreedom,wesimplygaugethemawayfromg,i.e.,wegauge?xtheGaussdecomposedgin(2.25)intotheform
g=abc→b.(5.12)
Morespeci?cally,withtheparametrizationa(x)=exp[σi(x)γi],c(x)=exp[?σi(x)?γi]wede?nethephysicalgaugeby
χi=σi=0,χ?i=σ?i=0.(5.13)
Weherenotethatforthisgaugethedeterminantfactorsin(5.8)areactuallyconstants.Nowthee?ectiveactionisobtainedbyperformingtheA±integrationsin(5.10).TheintegrationofA?givesrisetothedelta-function,
????1?1δ?γi,bA+b+?+bb?M?,(5.14)
i
withγi∈Γnormalizedbythedualitycondition(2.22).Onethennoticesthatthedelta-function(5.14)impliesexactlycondition(2.29)with?+cc?1replacedbyA+.Hence,withthehelpofthematrixVij(b)in(2.27)andT(b)in(2.29),itcanberewrittenas
???1?1(detV)δA+?bT(b)b.(5.15)
Finally,theintegrationofA+yields
Z=?d?e?(b)eIeff(b),
60(5.16)
whereIe?(b)isthee?ectiveaction(2.40)*,andd?e?(b)isthee?ectivemeasuregivenby
d?e?(b)=(detV)?1d?(g)δ(σ)δ(?σ)=(detV)?1d?(g)
*Actually,thee?ectiveactionalwaystakestheform(2.40)ifonerestrictstheWZNW
??eldtobeoftheformg=abcwitha∈eΓ,c∈eΓandbsuchthatVij(b)isinvertible.
?isnotnecessarybutcanbeusedtoensurethistechnicalThedualitybetweenΓandΓ
assumption.
61
(gauged)WZNWtheoriesevenatthequantumlevel,i.e.,includingthemeasure.ThisresulthasbeenestablishedbeforeinthespecialcaseofthestandardTodatheory(1.1)?in[44],wherethemeasured?e?(b)issimplygivenbyid?i.
Weendthissectionbynotingthatitisnotclearwhetherthemeasuredeterminedfromthesymplecticstructureofthereducedclassicaltheoryisidenticaltothee?ectivemeasure(5.17)ingeneral.Inthegeneralcasebothmeasuresinquestioncouldbecomequiteinvolvedandthusonewouldneedsomegeometricargumenttoseeiftheyareidenticalornot.
H5.3.TheW-symmetryofthegeneralizedTodaactionIe?(b)
IntheprevioussectionwehaveseenthequantumequivalenceofthegeneralizedTodatheoriesgivenby(4.3)andthecorrespondingconstrainedWZNWtheories.ItfollowsfromtheirWZNWoriginthatthegeneralizedTodatheoriespossessconserved
HSW-currents.Itisthusnaturaltoexpectthattheire?ectiveactions,Ie?in(4.3)andIe?in
(4.12),allowforsymmetrytransformationsyieldingtheW-currentsasthecorrespondingNoethercurrents.Wedemonstratebelowthatthisisindeedthecaseontheexampleofthetheoriesassociatedwithintegralgradings,whentheactiontakesasimpleform.Wehoweverbelievethattherearesymmetriesofthee?ectiveactioncorrespondingtotheconservedchiralcurrentsinheritedfromtheKMalgebraforanyreducedWZNWtheory.
Letusconsideragaugeinvariantdi?erentialpolynomialW(J)intheconstrainedWZNWtheorygivingrisetothee?ectivetheorydescribedbytheactionin(4.3).IntermsofthegeneralizedToda?eldb(x),thisconservedW-currentisgivenbythedi?erentialpolynomial
We?(β)=W(M+β),whereβ≡?+bb?1.(5.22)
Thisequality[34,15]holdsbecausetheconstrainedcurrentJand(M+β)(whichis,incidentally,justtheLaxpotentialAH+in(4.4))arerelatedbyagaugetransformation,aswehaveseen.Bychoosingsometestfunctionf(x+),wenowassociatetoWe?(β)thefollowingtransformationofthe?eldb(x):
??δWe?(x)δWb(y)=d2xf(x+)
notice,bycombiningthede?nitionin(5.23)with(5.22),that(δWb)b?1isapolynomialexpressioninf,βandtheir?+-derivativesuptosome?niteorder.
Westarttheproofbynotingthatthechangeoftheactionunderanarbitraryvariationδbisgivenbytheformula
HδIe?(b)=??dy?δbb2?1(y),b(y)HδIe?
δβ(y)
?,??β(y)?,(5.25)
andthen,fromthefactthat??We?=0on-shell,weobtainthefollowingidentity:
d2y?δWe?(x)
δβ(y)
Wethenrewritethisequationas
HδWIe?(b)=???1(y),M]?.,??β(y)+[b(y)Mb(5.27)?d2xf(x+)??We?(x),(5.28)
withtheaidoftheidentities(5.26)and(5.25).Thisthenprovesthat
HδWIe?(b)=0,(5.29)
63
sincetheintegrandin(5.28)isatotalderivative,thanksto??f=0.Onecanalsosee,fromequation(5.23),thatWe?istheNoetherchargedensitycorrespondingtothe
HsymmetrytransformationδWbofIe?(b).
5.4.BRSTformalismforWZNWreductions
SincetheconstrainedWZNWtheorycanberegardedasthegaugedWZNWtheory(2.18),oneisnaturallyledtoconstructtheBRSTformalismforthetheoryasabasisforquantization.BelowwediscusstheBRSTformalismbasedonthegaugesymmetry(2.19)andthusreturntothegeneralsituationofSection5.1wherenorelationshipbetweenthe
?issupposed.twosubalgebras,ΓandΓ,
PriortotheconstructionweherenotehowtheconformalsymmetryisrealizedinthegaugedWZNWtheorywhenthereisanoperatorHsatisfyingthecondition(2.13).(Forsimplicity,inwhatfollowswediscussthesymmetryassociatedtotheleft-movingsector.)Infact,withsuchHandachiraltestfunctionf+(x+)onecande?nethefollowingtransformation,
δg=f+?+g+?+f+Hg,
δA?=f+?+A?+?+f+[H,A?],
δA+=f+?+A++?+f+A+,(5.30)
totheconformaltransformationintheconstrainedWZNWtheorygeneratedbytheVirasorodensityLHin(2.10),ascanbecon?rmedbyobservingthat(5.30)impliestheconformalaction(2.11)forthecurrentwithf(x+)=f+(x+).WeshallderivelatertheVirasorodensityastheNoetherchargedensityintheBRSTsystem.whichleavesthegaugedWZNWactionI(g,A?,A+)invariant.Thiscorrespondsexactly
TurningtotheconstructionoftheBRSTformalism,we?rstchoosethespaceΓ??G??dualtoΓ).?whichisdualtoΓwithrespecttotheCartan-Killingform(andsimilarlyΓ
Followingthestandardprocedure[45]weintroducetwosetsofghost,anti-ghostand
?,???}.TheBRSTNakanishi-Lautrup?elds,{c∈Γ,c?+,B+∈Γ?}and{b∈Γb?,B?∈Γ
64transformationcorrespondingtothe(left-sectorofthe)localgaugetransformation(2.19)
isgivenby
δBg=?cg,
δBA?=D?c,
δBc=?c2,
withD±=?±?[A±,δBc?+=iB+,δBB+=0,δB(others)=0,(5.31)?Bfortheright-sector].Afterde?ningtheBRSTtransformationδ
inananalogousway,wewritetheBRSTactionbyaddingagauge?xingtermandaghosttermtothegaugedaction,
IBRST=I(g,A?,A+)+Igf+Ighost.(5.32)
TheadditionaltermscanbeconstructedbythemanifestlyBRSTinvariantexpression,
???2??Igf+Ighost=?iκ(δB+δB)dx?c?+,A??+?b?,A+????=κd2x?B+,A??+?B?,A+?+i?c?+,D?c?+i??b?,D+b?,(5.33)
wherewehavechosenthegauge?xingconditionsasA±=0.Thenthepath-integralfortheBRSTsystemisgivenby
?Z=d?(g)dA+dA?dcdc?+dbd?b?dB+dB?eiIBRST,(5.34)
which,uponintegrationoftheghostsandtheNakanishi-Lautrup?elds,reducesto(5.10).(Strictlyspeaking,forthiswehavetogeneralizethegauge?xingconditionsin(5.10)to
2bedependentonthegauge?elds.)Bythisconstructionthenilpotency,δB=0,andthe
BRSTinvarianceoftheaction,δBIBRST=0,areeasilychecked.
Itis,however,convenienttodealwiththesimpli?edBRSTtheoryobtainedbyperformingthetrivialintegrationsofA±andB±in(5.34),
???IBRST(g,c,?c+,b,?b?)=SWZ(g)+iκd2x?c?+,??c?+??b?,?+b?.(5.35)
Wenotethatthise?ectiveBRSTtheoryisnotmerelyasumofafreeWZNWsectorandfreeghostsectorasitappears,butratheritconsistsofthetwointerrelatedsectorsinthephysicalspacespeci?edbytheBRSTchargede?nedbelow.AtthisstagetheBRSTtransformationwhichleavesthesimpli?edBRSTaction(5.35)invariantreads
??δBg=?cg,?++c?+c),δBc?+=?πΓ?i(?+gg?1?M?)+(cc(5.36)2δBc=?c,δB(others)=0,
65
???whereπΓ?=i|γi??γi|istheprojectionoperatorontothedualspaceΓwiththe
?Bnormalizedbases,?γi,γj?=δij.FromtheassociatedconservedNoethercurrent,??j+=
0,theBRSTchargeQBisde?nedtobe
QB=?B(x)=dx+j+?dx+?c,?+gg?1?M?cc?+?.(5.37)
Thephysicalspaceisthenspeci?edbythecondition,
QB|phys?=0.(5.38)
InthesimplecaseoftheWZNWreductionwhichleadstothestandardTodatheory,theBRSTcharge(5.37)agreeswiththeonediscussedearlier[46].
InthecasewherethereisanHoperatorwhichguaranteestheconformalinvariance,theBRSTsystemalsohasthecorrespondingconformalsymmetry,
δg=f+?+g+?+f+Hg,
δc=f+?+c+?+f+[H,c],
δc?+=f+?+c?++?+f+(?c++[H,c?+]),δb=f+?+b,δ?b?=f+?+?b?,(5.39)
inheritedfromtheone(5.30)inthegaugedWZNWtheory.IftheHoperatorfurtherprovidesagrading,one?ndsfrom(5.39)thatthecurrentsofgrade?hhavethe(left-)conformalweight1?h,excepttheH-component,whichisnotaprimary?eld.Similarly,theghostsc,c?+ofgradeh,?hhavetheconformalweighth,1?h,respectively,whereastheghostsb,?bareconformalscalars.Nowwede?nethetotalVirasorodensityoperator
CLtotfromtheassociatedNoethercurrent,??j+=0,by
?Cdx+j+(x)=1
Letus?ndtheVirasorocentreofourBRSTsystem.ThetotalVirasorocentrectotisgivenbythesumofthetwocontributions,cfromtheWZNWpartandcghostfromthe
centrefromLHisgivenby
c=kdimG
g|ρ?|2,
67.44)ghostone.TheViraso(5
whereρ?istheWeylvector,givenbyhalfthesumofthepositiveroots.Fourth,wechoosethesimplepositiverootsinsuchawaythatthecorrespondingstepoperators,whichareingeneralinGcandnotinG,havenon-negativegradeswithrespecttoH.
Byusingtheaboveconventions,itisstraightforwardtoobtainthefollowingexpres-sions?11=dimΓ=
h>0
2tr(adH)=g?H,H?=g|?δ|,22(5.45)
forthecorrespondingtermsin(5.43).Substitutingtheseinto(5.43)andalso(5.44)into(5.42),
onecan?nallyestablishthefollowingniceformulaofthetotalVirasorocentre[14]:
???H√ctot=c+cghost=dimG0?12?
2=G>0?Q1
1
2,whichis2
thatinthiscase,fromtheexpressionin(5.46).Wethusobtain
k+g?δ??21?ρ??,k+g(5.47a)ctot=Nt?
where1
Nt=dimG0,andNs=dimG1
2,1},andthat?δis(1
componentscorrespondingtoG>0areconstrainedfromtheverybeginning.Intheirsystem,theconstraints(3.49)ofG1
.Accordingly,the
inthetotalVirasorocentre.auxiliary?eldsgiverisetotheextracontribution?1
2
ItisclearthataddingthistothesumoftheWZNWandghostparts(whichisofthe2form(5.46)withM0substitutedforH),rendersthetotalVirasorocentreoftheirsystemidenticaltothatofoursystem,givenby(5.47).Thisresultisnaturalifwerecallthefactthattheirreducedphasespace(aftercompletegauge?xing)isactuallyidenticaltoours.Itisobviousthatourmethod,whichisbasedonpurely?rst-classKMconstraintsanddoesnotrequireauxiliary?elds,providesasimplerwaytoreachtheidenticalreducedtheory.
69
6.Discussion
ThemainpurposeofthispaperhasbeentostudythegeneralstructureoftheHamiltonianreductionsoftheWZNWtheory.Consideringthenumberofinterestingexamplesresultingfromthereduction,thisproblemappearsimportantforthetheoryoftwo-dimensionalintegrablesystemsandinparticularforconformal?eldtheory.
OurmostimportantresultperhapsisthatweestablishedthegaugedWZNWsettingoftheHamiltonianreductionby?rstclassconstraintsinfullgenerality.ItwasthenusedheretosetuptheBRSTformalisminthegeneralcase,andforobtainingthee?ectiveactionsfortheleft-rightdualreductions.Wehopethatthegeneralframeworkwesetupwillbeusefulforfurtherstudiesofthisveryrichproblem.
TheothermajorconcernofthepaperhasbeentoinvestigatetheW-algebrasandtheir?eldtheoreticrealizationsarisingfromtheWZNWreduction.Wefound?rstclass
GKMconstraintsleadingtotheWS-algebraswhichallowedustoconstructgeneralized
Todatheoriesrealizingtheseinterestingextendedconformalalgebras.Webelievethat
Gthesl(2)-embeddingsunderlyingtheWS-algebrasaretoplayanimportantorganizing
roleingeneralforunderstandingthestructure,especiallytheprimary?eldcontent,oftheconformallyinvariantreducedKMsystems.Weillustratedthisideabyshowing
GlthattheWn-algebrasarenothingbutfurtherreductionsofWS-algebrasbelongingto
particularsl(2)-embeddings(seealso[37]).InourstudyofW-algebrasweemployedtwo(apparently)newmethods,whicharelikelytohaveawiderrangeofapplicabilitythanwhatweexploitedhere.The?rstisthemethodofsymplectichalvingwherebywe
Glconstructedpurely?rstclassKMconstraintfortheWSaswellasfortheWn-algebras.
nilpotent,thenoneshouldbuildthesl(2)containingM?andtrytoanalysethesystemintermsofthissl(2).Weusedthismethodtoinvestigate,inthenon-degeneratecase,thegeneralizedTodasytemsbelongingtointegralgradings,andalsotoprovidetheGl-interpretationoftheWn-algebras.WSThesecondiswhatwecallthesl(2)-method,whichcanbesummarizedbysayingthatifonehasconformallyinvariant?rstclassconstraintsgivenbysome(Γ,M?)withM?
Wewishtoremarkherethat,asfarasweknow,thetechnicalproblemconcerning
G-algebraswhichbelongtogrouptheoreticallyinequivalenttheinequivalenceofthoseWSsl(2)embeddingshasnotbeentackledyet.
Itiswellknown[22]thatthestandardW-algebrascanbeidenti?edasthesecond
70
PoissonbracketstructureofthegeneralizedKdVhierarchiesofDrinfeld-Sokolov[5].AsimilarrelationshipbetweenW-algebrasandKdVtypehierarchieshasbeenestablished
lveryrecentlyinmoregeneralcases[28,48,49].Inparticular,theWn-algebrashavebeen
relatedtothesocalledfractionalKdVhierarchies.ItwouldbeclearlyworthwhiletostudyingeneraltherelationshipbetweenthegeneralizedDrinfeld-Sokolovhierachiesof
G[48]andtheWS-algebrastogetherwiththeirfurtherreductions,seealso[16,17].
Wegaveagenerallocalanalysisofthee?ectivetheoriesarisingintheleft-rightdualcaseofthereduction,andinvestigatedinparticularthegeneralizedTodatheoriesobtainedbythereductioninsomedetail.InthecaseofthegeneralizedTodatheoriesassociatedwiththeintegralgradingsweexhibitedthewayinwhichtheW-symmetryoperatesasanordinarysymmetyoftheaction,anddemonstratedthatthequantumHamiltonianreductionisconsistentwiththecanonicalquantizationofthereducedclas-sicaltheory.Itwouldbenicetohavetheanalogousproblemsundercontrolalsoinmoregeneralcases.InouranalysiswerestrictedtheconsiderationstoGauss-decomposable?elds.ThefactthattheGaussdecompositionmaybreakdowncanintroduceapparentsingularitiesinthelocaldescriptionofthee?ectivetheories,buttheWZNWdescriptionisinherentlyglobalandremainsvalidfornonGauss-decomposable?eldsaswell[12,13].Itishenceaninterestingproblemtofurtheranalyzetheglobal(topological)aspectsofthephasespaceofthereducedWZNWtheories.
Weshouldalsonotethatitispossibletoremovethetechnicalassumptionofleft-rightduality.Inparticular,thestudyofpurelychiralWZNWreductionscouldbeofimportance,astheyarelikelytogivenaturalgeneralizationsofPolyakov’s2dgravityaction[43,12].
InthispaperweassumedtheexistenceofagaugeinvariantVirasorodensityLH,oftheformgivenby(2.10),forobtainingconformallyinvariantreductions.Basedonthisassumption,wecametorealizethat,whenHprovidesagradingofΓandM,thesl(2)builtoutofM=M?playsanimportantrole.However,theexampleofAppendixAindicatesthatthereisanotherclassofconformallyinvariantreductionswheretheformofthesurvivingVirasorodensityisdi?erentfromthatofanLH.ThestudyofthisnovelwayofpreservingtheconformalinvariancemayopenupanewperspectiveonconformalreductionsoftheWZNWtheoryaswellasonW-algebras.
TherearemanyfurtherinterestingquestionsrelatedtotheHamiltonianreductionsoftheWZNWtheory,whichwecouldnotmentioninthispaper.Wehopetobeabletopresentthoseinfuturepublications.
71
Acknowledgement.WewishtothankB.SpenceforasuggestionwhichhasbeencrucialforusforunderstandingtheW-symmetryoftheTodaaction.
Noteadded.After?nishingthispaper,thereappearedapreprint[50]alsoadvocatingtheimportanceofsl(2)structuresinclassifyingW-algebras.
72
AppendixA:Asolvablebutnotnilpotentgaugealgebra
InallthecasesofthereductionweconsideredinChapters3and4,thegaugealgebraΓwasagradednilpotentsubalgebraofG.Ontheotherhand,wehaveseeninSection
2.1thatthe?rst-classnessoftheconstraintsimplythatΓissolvable.WewantheretodiscussaconstrainedWZNWmodelforwhichthegaugealgebraissolvablebutnotnilpotent.Interestinglyenough,itturnsoutthatinthisexamplenoHsatisfying(2.13)existswhichwouldrendertheconstraintsconformallyinvariant.However,conformalinvariancecanstillbemaintained,showingclearlythattheexistenceofsuchanHisonlyasu?cientbutnotanecessarycondition.
WechoosetheLiealgebraGtobesl(3,R)andthegaugealgebraΓasgeneratedbythefollowingthreegenerators
??010
γ1=Eα1=?000?,
000
γ3=11(2H1+H2)+30=?002√1γ2=Eα1+α2√?0100?,002
√2?(A.1a)0?
3
2γ2,[γ2,γ3]=?13
3Eα2,2H1?√
√√
withoutlossofgenerality.
Thenextquestionistheconformalinvariance.AsdiscussedinSection2.1,asuf-?cientconditionforconformalinvarianceisprovidedbytheexistenceofa(modi?ed)VirasorodensityLH=LKM??x?H,J(x)?weaklycommutingwiththeconstraints.Forthistowork,thegeneratorHmustsatisfythethreeconditionsin(2.13).However,itisaneasymattertoshowthatthoseconditionsarecontradictoryinthepresentcase,andthereforenosuchHexists.
Theaboveanalysiscanalsobecarriedoutforthesimplergaugealgebraspannedbyγ3only.Thisgaugealgebraisobviouslynilpotent,sinceitisAbelian.Nevertheless,thepreviousconclusionsremain:ThereexistsnoHwhichwouldrenderthe?rstclassconstraintsconformallyinvariant,foranyM=0from[Γ,Γ]⊥/Γ⊥.Thisshowstheimportanceofthegaugegeneratorsbeingnilpotentoperators,ratherthanthegaugealgebrabeingnilpotent.ItwouldbeinterestingtoknowwhetherthereisalwaysanHsatisfying(2.13)forgaugealgebrasconsistingofnilpotentoperators.
AlthoughthereisnoHsuchthattheconstraintsarepreservedbyLH,wecanneverthelessconstructanotherVirasorodensityΛwhichdoespreservetheconstraints.Itisgivenby
tΛ(x)=LKM(x)???γ3,J(x)?.(A.5)
ForMgivenin(A.4),theconstraintsread
?γ1,J(x)?=?γ2,J(x)?=0,?γ3,J(x)?=?,(A.6)
andarecheckedtoweaklycommutewithΛ:{Λ(x),?γi,J(y)?}≈0ontheconstraintsurface(A.6).(Notethat,whengoingfromLKMtoΛ,wehavenotchangedtheconformalcentralcharge,whichisclassicallyzero.)ThereforeweexpectthereducedtheorytobeinvariantundertheconformaltransformationgeneratedbyΛbeingitsNoetherchargedensity.Wenowproceedtoshowthatitisindeedthecase.Beforedoingthis,wedisplaytheformofΛontheconstraintsurface:
22Λ(x)=T1(x)+T2(x),(A.7a)
T1=1
?=νYt=νY.WewritetheSL(3,R)groupelementsasandconsiderM=?YandM
?withH=span{Y,H2}theCartang=a·b·c,witha∈expΓ,b∈expHandc∈expΓ,
subalgebra.WedidnotconformtothegeneralprescriptiongiveninSection2.3,which
?inG,requiredtowriteg=abcwithb∈expBforaspaceBcomplementarytoΓ+Γ
eqs.(2.25-26).Hadwedonethat,theresultinge?ectiveactionwouldhavelookedmuchmorecomplicated.Here,wesimplytakeasetofcoordinatesinwhichtheactionlookssimple.
Thereductionyieldsane?ectivetheoryforthegroup-valued?eldb,ofwhichthee?ectiveactionisgivenby(2.40)with(2.29b).Usingtheparametrizationb=exp(αY)·exp(2βH2),theexplicitformofthee?ectiveactionis
Ie?(α,β)=??(?+α??)(??α?ν)dx?+α??α+?+β??β?2
variablescontainedina.Hencewecanjustaswellputa=1in(A.12).Doingthat,thede?nitions(A.7b)yield(A.11).Wethus?ndthefollowingexpressionforΛ:
Λ=(???+α)2tanh2β+(?+β)2.(A.13)
Itisaneasymattertoshow,byusingthe?eldequationsobtainedfromtheaction(A.9),
sinhβ?+??α+tanhβ?+β(??α?ν)+??β(?+α??)=0,
coshβ?+??β?tanhβ(??α?ν)(?+α??)=0,
thatΛisindeedchiral,satisfying
??Λ=0.
MoreoveronealsochecksthefollowingPoissonbrackets
{Λ(x),α(y)}=?(?+α??)δ(x1?y1),
1122??(A.14)(A.15){Λ(x),β(y)}=?(?+β)δ(x?y),(A.16)
whichreproducethetransformations(A.10).ThusthedensityΛfeaturesallwhatisexpectedfromtheNoetherchargedensityassociatedwiththeconformalsymmetry.
Finally,wepresenthereforcompletenessthegeneralsolutionoftheequationsofmotion(A.14).AlongthelinesofSection2.3,itcanbeobtainedasfollows:
α=(ηL+ηR)+tan?1?sinh(θ?θ)LR
AppendixB:H-compatiblesl(2)andthenon-degeneracycondition
OurpurposeinthistechnicalappendixistoanalysethenotionoftheH-compatiblesl(2)subalgebra,whichhasbeenintroducedinSection3.3.Werecallthatthesl(2)subalgebraS={M?,M0,M+}ofthesimpleLiealgebraGiscalledH-compatibleifHisanintegralgradingoperator,[H,M±]=±M±,andM±satisfythenon-degeneracy
HKer(adM±)∩G?={0}.conditions(B.1)
Notethatthesecondpropertyinthisde?nitionisequivalenttothefactthatScommuteswith(H?M0).WeproveheretheresultsstatedinSection3.3,andalsoestablishanalternativeformofthenon-degeneracycondition,whichwillbeusedinAppendixC.
Letus?rstconsideranarbitrary(notnecessarilyintegral)gradingoperatorHof
HGandsomenon-zeroelementM?fromG?1.Wewishtoshowthattoeachsuchpair
H(H,M?)thereexistsansl(2)subalgebraS={M?,M0,M+}forwhichM+∈G+1.om-
mutesToexhibittheS-tripleinquestion,weneedtheJacobson-Morozovtheorem,whichhasalreadybeenmentionedinSection3.3.Inaddition,weshallalsousethefollowinglemma,whichcanbefoundin[33](Lemma7onpage98,attributedtoMorozov).
Lemma:LetLbea?nite-dimensionalLiealgebraovera?eldofcharacteristic0andsupposeLcontainselementshandesuchthat[h,e]=?eandh∈[L,e].Thenthereexistsanelementf∈Lsuchthat
[h,f]=fand[f,e]=2h.(B.2)
Turningtotheproof,we?rstusetheJacobson-Morozovtheoremto?ndgenerators(m?,m0,m+)inGcompletingm?≡M?toansl(2)subalgebra.Wethendecomposetheelementsm0andm+intotheircomponentsofde?nitegrade,i.e.,wewrite
m0=?nmn0andm+=?nmn+,(B.3)
wherenrunsoverthespectrumofthegradingoperatorH.SinceM?isofgrade?1,itfollowsfromthesl(2)commutationrelationsthat
[m00,M?]=?M?and
770[m1+,M?]=2m0,(B.4)
andtheserelationstellusthath=m00ande=M?satisfytheconditionsoftheabove?nlemma.Thusthereexistsanelementfsatisfying(B.2),whichwecanwriteasf=nfbyusingtheH-gradingagain.Theproofis?nishedbyverifyingthatM+≡f1andM0≡m00togetherwithM?spantherequiredsl(2)subalgebraofG.
respectively,thepair(H,M±)iscallednon-degenerateifitsatis?esthecorrespondingconditionin(B.1).Fromnowon,letHbeanintegralgradingoperator.ForanelementM±ofgrade±1,
HG±1,thenthenon-degeracyofthepairs(H,M?)and(H,M+)areequivalentstatements.Thiswillfollowimmediatelyfromthesl(2)structureifweprovethatthenon-degeneracyWeclaimthatifS={M?,M0,M+}isansl(2)forwhichthegeneratorsM±arefromofthepair(H,M±)isequivalenttothefollowingequality:
HdimKer(adM±)=dimG0.(B.5)
Itisenoughtoprovethislatterstatementforapair(H,M?),sincethenforapair(H,M+)itcanbeobtainedbychangingHto?H.Toprovethisletus?rstrearrangetheidentity
dimG=dimKer(adM?)+dim[M?,G]
byusingthegradingas
dimKer(adM?)?HdimG0(B.6)
Sincebothtermsontherighthandsideofthisequationarenon-negative,weseethat
HdimKer(adM?)≥dimG0,??HH=dimG+?dim[M?,G+]??.HHH+dimG??dim[M?,G0+G?](B.7)(B.8)
andequalityisachievedhereifandonlyif
HH=dim[M?,G+]anddimG+HHH]=G?.+G?[M?,G0(B.9)
Ontheotherhand,wecanshowthatthetwoequalitiesin(B.9)areactuallyequivalenttoeachother.Toseethis,letusassumethatthesecondequalityin(B.9)isnottrue.This
HHisclearlyequivalenttotheexistenceofsomenon-zerou∈G+suchthat?u,[M?,G0+
HG?]?={0}.Bytheinvarianceandthenon-degeneracyoftheCartan-Killingform,thisisinturnequivalentto[M?,u]=0,whichmeansthatthe?rstequalityin(B.9)isnottrue.Bynoticingthatthe?rstequalityin(B.9)isjustthenon-degeneracyconditionforthe
78
pair(H,M?),wecanconcludethatthenon-degeneracyconditionisindeedequivalenttotheequalityin(B.5).
Wewishtomentionaconsequenceoftheresultsprovenintheabove.Tothisletusconsideranon-degeneratepair(H,M?).Byourmoregeneralresult,weknowthat
Hthereexistssuchansl(2)subalgebraS={M?,M0,M+}forwhichM+isfromG+1.The
pointtomentionisthatthisSisanH-compatiblesl(2)subalgebra,ashasalreadybeensatedinSection3.3.Infact,itisnoweasytoseethatthisfollowsfromtheequivalenceof
(B.1)with(B.5)bytakingintoaccountthatthekernelsofadM±areofequaldimension
bythesl(2)structure.
79
AppendixC:H-compatiblesl(2)embeddingsandhalvings
InSection3.3,weshowedthat,givenatriple(Γ,M,H)satisfyingtheconditionsfor?rst-classness,conformalinvarianceandpolynomiality(eqs.(2.6),(2.13)and(3.2-4)),
GthecorrespondingW-algebraisisomorphictoWS,providedthatHisanintegralgrading
operator.HereS={M?,M0,M+}issomesl(2)subalgebracontainingM?=M.Anaturalquestioniswhatsl(2)subalgebrasariseinthisway,orequivalently,givenan
Garbitrarysl(2)subalgebra,cantheresultingWS-algebrabeobtainedastheW-algebra
correspondingtothetriple(Γ,M,H),forsomeintegralgradingoperatorH?Whetherthisoccursornotdependsonlyonhowthesl(2)isembedded,anditisthereforeapuregroup-theoreticquestion.AccordingtoSection3.3,thesl(2)subalgebrashavingthispropertyaretheH-compatibleones.Thisappendixisdevotedtoestablishingwhenagivensl(2)embeddingisH-compatible,andifso,whatthecorrespondingHis.
Thequestionofansl(2)beingH-compatibleisverymuchrelatedtoanotherone,whichwasmentionedattheendofSection4.2.Wenotedthatinsomeinstances,ageneralizedTodatheoryassociatedtoansl(2)embeddingcouldaswellberegardedasaTodatheoryassociatedtoanintegralgradingoperatorH.Thismeansthatthee?ectiveactionofthetheoryisaspecialcaseofboth(4.12)and(4.3)atthesametime.WehaveseenthatthisisthecasewhenthecorrespondinghalvingisH-compatible,i.e.,whenthe
+G0+Q?1+G≤?1)(subscriptsLiealgebradecompositionG=(G≥1+P122HHHareM0-grades)canbenicelyrecastedintoG=G≥1+G0+G≤?1.Oursecondproblem,
addressedattheendoftheappendix,isto?ndthelistofthosesl(2)subalgebraswhichallowforanH-compatiblehalving.Clearly,ansl(2)subalgebrawhichpossessesanH-compatiblehalvingisalsoH-compatibleintheabovesense,butitwillturnoutthattheconverseisnottrue.
LetS={M?,M0,M+}beansl(2)subalgebraembeddedinamaximallynon-compactrealsimpleLiealgebraG.FortheclassicalalgebrasAl,Bl,ClandDl,theserealformsarerespectivelysl(l+1,R),so(l,l+1,R),sp(2l,R)andso(l,l,R).(WedonotconsidertheexceptionalLiealgebras.)ForStobeanH-compatiblesl(2),oneshould?ndanHinGwiththefollowingproperties:
1.adHisdiagonalizablewitheigenvaluesbeingintegers,
2.H?M0mustcommutewiththeS-triple,
80
3.dimKer(adH)=dimKer(adM±).
Weremarkthatheretheequivalenceofrelations(B.1)and(B.5),proveninthepreviousappendix,hasbeentakenintoaccount.Underconditions1-3,thedecomposition
Γ⊥=[M?,Γ]+Ker(adM+)
Hholds,whereΓ=G≥1inthe(Γ,M?,H)setting,orΓ=P1(C.1)
howeverceasestobetrueintherealcaseingeneral:inequivalentsl(2)subalgebrascanhavethesamemultipletcontentinthefundamentalofG.TheanswertotheproblemofH-compatibilitywillinfactbeprovidedbylookingmorecloselyatthedecompositionofthefundamentalofGunderthesl(2)subalgebrainquestion,aswillbeclearbelow.
Asanimmediateconsequenceofcondition2,H?M0isansl(2)invariantandcanonlydependonthevalueoftheCasimir.If,inthereductionofthefundamentalofG,a
?andHcanbespinjrepresentationoccurswithmultiplicitymj,thesl(2)generatorsM
written
?=M?j?(j)×Im,Mj
I2j+1×D(j),(C.2a)(C.2b)H=M0+?j
whereIndenotestheunitn×nmatrix,andtheD(j)’saremj×mjdiagonalmatrices.Hence,withineachirreduciblerepresentationofsl(2),HisequaltoM0shiftedbyaconstant.Obviously,thisisalsotrueintheadjointrepresentationand,inturn,thisimpliesthatadHtakesthevaluezeroatmostonceineachsl(2)multipletintheadjointofG.Fromcondition3,adHmusttakethevaluezeroexactlyonce,i.e.,eachsl(2)representationmustintersectKer(adH)exactlyonce.Inparticular,thesl(2)singletsmustbeadH-eigenvectorswithzeroeigenvalue.
ThetrivialsolutionH=M0existswheneveradM0isdiagonalizableontheintegers,
i.e.,whenthereductionofthefundamentalofGiseitherpurelytensorialorpurelyspinorial.Fromnowon,wesupposethatthereductioninvolvesbothkindsofsl(2)representations.
1)Alalgebras.
TheproblemfortheAlseriesissimpletosolvesince,inthiscase,anHalwaysexists.Asaproof,weexplicitlygiveanHwhichful?llsalltherequirements.In(C.2b),weset
D(j)=?λ·Imj(λ+1ifj∈N,2,(C.3)
whereλisaconstantthatmakesHtraceless.InordertoshowthattheHsode?nedhastherequiredproperties,werecallthatfortheAlalgebras,theadjointrepresentation
82
isobtainedbytensoringthefundamentalwithitscontragredient.Asaresult,therootsarethedi?erencesoftheweightsofthefundamental(uptoasinglet)andwehave
adH=adM0+[D(j1)?D(j2)],(C.4)
wherej1andj2arethespinsofthestatesinthefundamentalrepresentationfromwhichagivenstateintheadjointrepresentationisformed.Thattheconditions1-3aresatis?edisobviousfromthefactthatadH=adM0ontensorsandadH=adM0±
1occurringasmanytimesas?21
2on
spinors.
Letusnowlookatthemsspinorrepresentationsofspins,says1,s2,...,sms.Theproductsi×sjofanytwoofthosecontainsasinglet,andthatimpliesD(si)+D(sj)=0.Thisequalitymustholdforanypairofspinsrepresentations,whichisimpossibleunlessms≤2.
Letusconsidertherestrictiongsofthesymplecticformtothespinsrepresentations.Therestrictedformisnon-degenerate,becausetheoriginalnon-degeneratemetricisblock-diagonalwithrespecttotheeigenvaluesofthesl(2)Casimir.
83
Ifms=1,thentheHgivenbyM0±1
020
?1
2onapairofspinorss/s′.(C.7)
Conditions1-3aresatis?edsince(C.7)impliesadH=adM0onsinglets,adH=adM0±(1or0)ontensorsandadH=adM0±1
2ontensors,
H=M0onspinorsandmt≤2foranytensorrepresentationofspint.
Ifasin2),welookattherestrictiongtoftheorthogonalmetrictothespinttensors,wehavemt=2onaccountofthenon-degeneracyofgt.Fromthis,wegetat
84
oncethattherecanbenosolutionfortheBlalgebras.Indeed,thefundamentalbeingodd-dimensional,atleastonetensorrepresentationmustcomeonitsown.
Onthe2(2t+1)-dimensionalsubspacemadeupbythetwospinttensors,H?M0andgttaketheform
H?M0=±?1?,gs=?a
btbc?,(C.9)
2
whereaandcarenowsymmetric.RequiringthatH?M0beorthogonal,weagainobtaina=c=0.
Therefore,fortheorthogonalalgebras,wegetthefollowingconclusions.ThereisnosolutionfortheBlseriesifthesl(2)embeddingisnotintegral.AstotheDlseries,thesl(2)embeddingmustbesuchthat:(i)everytensorinthefundamentalofGhasamultiplicityequalto2,(ii)if(t,t′)issuchapairoftensors,theymustbethedualofeachotherwithrespecttotheorthogonalmetric.Inthiscase,Hisgiveninthefundamentalby
H=?M0+/?1
namelytheproblemofH-compatiblehalvings.Fromthede?nition,ansl(2)subalgebraallowsforanH-compatiblehalvingifinadditiontoconditions1-3onealsohas
4.P12H+G≤?1=G≤?1.
M0HInparticular,thisfourthconditionimpliesG0?G0.SowereadilyobtainthatHand
M0mustsatisfy
adH=adM0,ontensors,(C.11)
representation(condition2).Therefore,wecansimplylookatthosesolutionsofthe?rstproblemwhichsatisfy(C.11)andcheckifcondition4isfullysatis?edornot.Wegetthatthesl(2)embeddingsallowingforanH-compatiblehalvingareasfollows:Al:anysl(2)subalgebra.ThereareonlytwosolutionsforHgivenbysettingin
(C.2b):D(j)=(λ±?(j))·Imjwith?(j)=0/1sinceweknow,fromthepreviousanalysis,thatadH?adM0isaconstantinevery
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90
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