quantum gauge anomalies
and generalized Chern-Simons terms
in hiral gauge theory
Torsten S hmidt
quantum gauge anomalies
and generalized Chern-Simons terms
in hiral gauge theory
Torsten S hmidt
Dissertation
an der Fakulat fur Physik
der Ludwig-Maximilians-Universitat
Mun hen
vorgelegt von
Torsten S hmidt
aus Mun hen
Zweitguta hter: Prof. Dr. Dieter Lust
Vorsitzender: Prof. Dr. Ivo Sa hs
weiteres Mitglied: Prof. Dr. Otmar Biebel
Ersatzmitglied: Prof. Dr. Dorothee S haile
The purpose of this thesis is to investigate the interplay of anomaly an ellation and
generalized Chern-Simons terms in four-dimensional hiral gauge theory. The in lusion
of generalized Chern-Simons terms and additional axioni ouplings allows to relax the
onstraintswhi hareotherwiseimposedbyanomaly-freedom. Therehasbeenalotofre ent
interest in the phenomenology of these additional ouplings. Possible models that make
use of thisare provided byinterse ting brane modelsin orientifold ompa ti ations of the
type II stringtheories. If themassof theanomalousU(1)-gauge bosonis low enough, these
modelspredi t smallsignalsthat might be dete tableinnear-future ollider experiments.
We startwitha detaileddis ussionof generalized Chern-Simonsterms and establishthe
onne tion of generalized Chern-Simons terms with the an ellation of anomalies via the
Green-S hwarzme hanism. Withthisat hand,we investigate thesituationingeneralN =1
supersymmetri eldtheorieswithgeneralized Chern-Simonsterms. Twosimple onsisten y
onditions are shown to en ode strong onstraints on the allowed anomalies for di erent
types of gauge groups. The results even apply to N = 1 matter- oupled supergravity
generalizingpreviouslyknowna tions.
In N = 1 supersymmetry or in theories without supersymmtry, the rigid symmtries of
theve torands alarse torarenotdire tlyrelated. Therigidsymmetrygroupisasubsetof
the produ t of the symple ti dualitytransformations that a t on theve tor elds and the
isometrygroup ofthe s alarmanifoldof the hiralmultiplets. Ifnontrivialele ti /magneti
dualitytransformationsare involved, the eldsbefore and after su h a symmetry operation
are not related by a lo al eld transformation. In order to use the standard pro edure for
gauging a rigid symmetry, one therefore rst has to swit h to a symple ti duality frame
in whi h the relevant symmetries a t by lo al eld transformations only. This obviously
breaks the original duality ovarian e. Re ently an alternative method has been proposed
that allows one to formally maintain the fullduality ovarian e at ea h step of the gauging
pro edure. This method requires the extension of the usual gauge degrees of freedom and
the existing formalism in order to allow for the an ellation of quantum gauge anomalies
via the Green-S hwarz me hanism. The results might be relevant for ertain N = 1 ux
ompa ti ationswith anomalousfermioni spe trum.
Attheendofthisthesiswe ommentonapuzzleintheliteratureonsupersymmetri eld
theorieswithmassivetensor elds. Theseo urnaturallyinthelow-energye e tivea tionof
ertainIIBorientifold ompa ti ationswith uxes,wheretheygive riseto s alarpotentials
that arenot of thestandard supersymmetry form. Thepotential ontainsa term that does
notarisefromeliminatinganauxiliary eld. Wewill larifytheoriginofthistermanddisplay
therelationto astandardD-term potential. Inan appendixitis expli itlyshownhowthese
lowenergye e tivea tionsmightbe onne tedtotheformulationoffour-didmensionalgauge
In dieser Dissertation untersu hen wir die Rolle verallgemeinerte Chern-Simons Terme
in vierdimensionalen hiralen Ei htheorien, genauer, wie Anomlien weggehoben werden
konnen. Unter Einbeziehung von verallgemeinertenChern-SimonsTermenundzusatzli hen
axionis hen Kopplungen ist man in der Lage die Bedingungen, die Abwesenheit von
Anomalien garantieren, zu ents harfen. Phanomenologis he Modelle, die gerade diese
Art von Kopplungen beinhalten, sind seit einiger Zeit Mittelpunkt reger Untersu hungen.
Mogli heRealisierungenfurentspre hendeModellesindzumBeispieldur hsi hs hneidende
Branen-Modelle in Orientifoldkompakti zierungen von Typ II Stringtheorien gegeben. Die
Vorhersagenderphanomenologis henUntersu hungendieserModellekonntensogarinnaher
Zukunft in Kollisionsexperimenten na hgepruft werden, falls nur die Masse des anomalen
U(1)-Ei hbosons kleingenug ist.
Na h einerkurzenEinfuhrunginQuantenanomaliendiskutierenwirim Detaildie
verall-gemeinertenChern-SimonsTermeunderlauternunterwel henUmstandensiemitHilfeeines
Me hanismus na h Green und S hwarz zum Wegfall von Anomalien fuhren konnen. Diese
ersten Ergebnisse erlauben eine umfassende Untersu hung der entspre henden Situation in
allgemeinen N = 1 supersymmetris hen Feldtheorien mit verallgemeinerten Chern-Simons
Termen. Wiegezeigt wird,konnendie starken Anforderungen,die si hausderAbwesenheit
von Anomalien unters hiedli her Ei hgruppen ergeben, dur h zwei einfa he Bedingungen
zum Ausdru kgebra ht werden. Dies giltebenfalls inN =1Supergravitationstheorien mit
Kopplungenanmassive Felder, bekannte Wirkungen verallgemeinernd.
Globale Symmetrien jener Sektoren, die Vektorfelder undSkalarfelder enthalten, stehen
in N = 1 Supersymmetrie oder in ni ht supersymmetris hen Theorien in keiner direkten
Verbindung. Die globale Symmetriegruppeist eineUntergruppe desProdukts der
symplek-tis henDualitatstransformationen,die auf die Vektorfelderwirken undderIsometriegruppe
der skalaren Mannigfaltigkeit der hiralen Multipletts dar. Ni htriviale Transformationen
der elektis h/magnetis hen Dualitat wirken derart auf Felder, dass diese ni ht mehr in
Dualitatsrahmen we hseln, in dem die Felder uber lokale Transformationen untereinander
in Beziehung stehen. Dies bri ht o ensi htli h die ursprungli he Dualitatskovarianz. Vor
ni ht all zu langer Zeit wurde eine alternative Methode vorges hlagen, die es erlaubt,
bei jedem S hritt des Ei hprozesses die volle formale Dualitatskovarianz zu bewahren.
Diese Methode verlangt eine Erweiterung der gewohnli hen Ei hfreiheitsgrade und die
Einfuhrung neuer Felder. Auf diese Art wird eine neue Formulierung der Ei htheorien in
vier Dimensionen errei ht. In einem der Hauptteile der Dissertation werden wir sehen, wie
genau nun dieser Formalismusmodi ziert werden muss, damit au h Quantenanomalien mit
Hilfe des Me hanismus na h Green und S hwarz entfernt werden konnen. Diese Resultate
sindrelevantfurgewisse N =1Flusskompakti zierungenmitanomalemFermionspektrum.
AmEndederDissertationwendenwirunseinemPunktzu,derinderLiteraturzu
super-symmetris hen Feldtheorien mit massiven Tensorfeldern angemerkt wurde. Diese Theorien
ers heinen fur gewohnli h in den e ektiven Niederenergie-Wirkungen gewisser IIB
Orien-tifold usskompati zierungenunderzeugenPotentialefurSkalarfeldervonaussergewohnli her
Form. Diese Potentiale enthalteneinenTerm,derni ht ausderEliminationeinesHilfsfeldes
resultiert. Wir werden diesen Punkt klaren und au h die Beziehung dieser Potentiale zu
gewohnli henD-TermPotentialenaufzeigen. ImAnhangzudieserArbeitistdargestellt,wie
genaudiesee ektiven Niederenergie-Wirkungenmiteinigen derzuvorerwahnten
Contents
1 Introdu tion 1
2 Quantum anomalies 5
2.1 Triangleanomaly . . . 5
2.2 Pathintegraland anomaly . . . 8
2.3 Consistent anomaly. . . 11
2.4 Can ellation ofanomalies . . . 12
3 Lie algebra ohomology and generalized Chern-Simons terms 14 3.1 GeneralizedChern-Simonsforms . . . 17
3.2 GeneralizedChern-Simonsterms andsemisimplegroups . . . 21
3.3 Appli ation: Abeliansemisimple . . . 22
3.4 Nonhomogeneous formsand anomalies . . . 23
4 N =1 Supersymmetry 27 4.1 GlobalSupersymmetry. . . 28
4.2 The gaugese tor ofN =1 supergravity . . . 33
5 Generalized Chern-Simons termsand hiral anomalies inN =1 Supersym-metry 36 5.1 GaugedisometriesandgeneralizedChern-Simonstermsinglobalsupersymmetry 37 5.2 Gaugedisometries and anomaliesinglobal N =1 supersymmetry . . . 44
5.3 GeneralizedChern-Simonsterms inSupergravity . . . 47
5.4 Redu ingto Abeliansemisimple . . . 49
5.5 Summary . . . 52
6 Symple ti ally ovariant formalismand anomalies in hiralgauge theories 54 6.1 Ele tri /magneti dualitywithoutanomalies. . . 55
6.1.1 Ele tri /magneti dualityand the onventional gauging . . . 55
6.1.2 The symple ti ally ovariant gauging . . . 57
6.2.1 Symple ti ally ovariant anomalies . . . 69
6.2.2 The new onstraint . . . 70
6.2.3 New antisymmetri tensors . . . 71
6.3 Purely ele tri gaugings . . . 74
6.4 A simpleexampleof magneti gauging . . . 75
6.5 Summary . . . 78
7 Abelian gauging and D-term potential in N =1 supersymmetry 81 8 Con lusion 88 A Appendix 92 A.1 The Lapla eequation of Liealgebra ohomology . . . 92
A.2 Appli ation: Abeliansemisimple . . . 94
B Appendix 96 C Appendix 100 D Appendix 104 D.1 The Bian hiidentity . . . 104
D.2 Gauge variationof F M . . . 105
D.3 Gauge variationof thegeneralizedChern-Simonsterm . . . 109
E Appendix 112
In quantum physi s an anomaly is thefailure of a symmetry of the lassi al theory to be a
symmetryofthefullquantum theory. In hiralgaugetheoriesananomalyofthegauge
sym-metrymayo urbe ausethe hiralityofthegaugeintera tionsmay auseloop ontributions
(e.g. to n-point fun tions)that violatethesymmetries of the lassi ala tion. Forquantum
gauge theories this is fatal, as su h a gauge anomaly leads to a loss of renormalizeability.
To avoid this, one usually hasto imposea number of nontrivial onstraints on the possible
hargesofthe hiralfermionsinsu hawaythattheanomalyisabsent. Withoutintrodu ing
anynewparti leorintera tion,thisamountstodemandingthattheanomalousFeynman
dia-grams an el. Thevanishingofallanomalousone-loopdiagramsalreadyprovidesasuÆ ient
onditionforanomaly-freedomto all loop orders[1℄.
It is possibleto relax these onstraintsifgaugevariations ofthe lassi ala tionareable
to an elsomeoftheanomalousone-loop ontributions. Inthis asethe lassi ala tionitself
annotbegaugeinvariant,of ourse. Inthesimplestexample,thea tion ontainsan axioni
ouplingof as alara(x)to the eld strengthof someve tor eld oftheform aF^F,where
a(x)transformswithashiftundersomeAbeliangaugesymmetrywithgaugeparameter(x),
i.e. Æa(x)/(x). An Abeliananomaly may be exa tly an elledbythe gauge variation of
this axioni oupling, whi h is proportional to F ^F. This is a simple four-dimensional
exampleof theGreen-S hwarzme hanism [2℄.
Thes alara(x)isusually alled\axion"anditskineti termhastobeofStu kelberg-type
inordertobegaugeinvariant,i.e. proportionalto(
a A
)
2
. TheStu kelberg oupling
im-plementstheshiftsymmetryviaanAbeliangaugebosonthatgainsamassduetoits oupling
to the axion. If the mass of su h a gauge boson is low enough and if it has suitable
inter-a tions with the Standard Model parti les,it may lead to observable signals innear-future
ollider experiments. There has re ently been quite some interest in the phenomenologi al
studies of su h anomalous Z 0
-type bosons [3{ 16℄. A natural framework for su h models is
provided byinterse ting brane modelsin type II orientifolds 1
be ause the four-dimensional
Green-S hwarz me hanism israthergeneri inthese kindofmodels[23℄.
Interestingly,the Green-S hwarzme hanism alone is oftennot enoughto an elall
on-1
tributions from gauge anomalies in these orientifold ompa ti ations [12,13℄ 2
. Espe ially
the an ellationofmixedAbeliananomaliesbetweenanomalousand non-anomalousAbelian
fa torsisingeneralnota hievedbytheGreen-S hwarzme hanismalone. Instead,oneneeds
thehelpoftopologi alterms,so- alledgeneralizedChern-Simonsterms,whi harenotgauge
invariant. In general, it is the ombination of the Green-S hwarz me hanism and the
gen-eralized Chern-Simons terms whi h possibly an els the omplete gauge anomaly. In [12℄ 3
the question was raised, how to generate the generalized Chern-Simons terms from ertain
string ompa ti ations. ItwasshownthatthegeneralizedChern-Simonstermsareageneri
featureoftheorientifoldmodelswereferredto aboveandmayleadto newobservablesignals
of Z 0
-bosons. Another possibilitywas mentionedin [26℄ where ertain ux and generalized
S herk-S hwarz ompa ti ations[27,28℄ wereusedto explainpossibleorigins. Thereisalso
the possibilityto obtainN =2 supergravitytheories with generalized Chern-Simonsterms
fromordinarydimensionalredu tionof ertain ve dimensionalN =2supergravitytheories
withtensor multiplets 4
[29℄.
It should be emphasized that the generalized Chern-Simons terms need not ne essarily
appearin ombination withtheGreen-S hwarz me hanismand anomalies. Originally,these
termswere rstdis overedinextendedgaugedsupergravitytheories[32℄whi haremanifestly
freeof anomaliesdueto theusualin ompatibilityof hiralgaugeintera tionswithextended
supersymmetry in four dimensions. This motivated the dis ussions in [26{29,33{39℄ whi h
demonstrated howgeneralized Chern-Simonsterms an elaxioni shiftsindi erent lassi al
setups. Inall these ases the absen eof gauge anomalies imposes strongrestri tionson the
form ofpossiblegaugedaxioni shiftsymmetries.
In light of the above mentioned possiblephenomenologi al appli ations and given their
generi o uren einvariousstring theory ompa ti ations, itissurprisingthatthegeneral
interplaybetweentheGreen-S hwarzme hanism,generalizedChern-SimonstermsandN =1
supersymmetry wasnot very wellunderstooduntilrather re ently. It isthe purpose of this
thesis to give a systemati a ount of these issuesas they were developed in[88℄ duringthe
pastyears.
2
Forrelatedphenomenologi alwork,seealso[14{16,24,25℄
3
Thebasi ideasarepresentedbymeansofasimpletoymodelin[13℄.
4
aboutquantumanomaliesin hiralgaugetheories. Wewillillustratehowthetrianglediagram
auses a violation of the onservation law of axial urrents. Then we will review how the
anomaly an also be understood by the Ja obian of the path integral measure under axial
transformations. Withthis at hand we will present the Wess-Zumino onsisten y ondition
and, at theend of se tion 2, we will shortly omment on some general aspe ts of anomaly
an ellation.
In se tion 3, we onstru t generalized Chern-Simons terms along the lines of [34℄. We
willfurthershowthatthere arenonontrivialgeneralizedChern-Simonstermsforsemisimple
gauge groups. This motivates a short dis ussion of the example of a gauge group with
the stru ture Abeliansemisimple. The se tion ends with a generalization of the method
developed in[34℄ soasto beable to in orporate anomaliesintothe formalism.
Se tion4 summarizesthemostimportantformulae on erningthegaugese torofglobal
and lo alN =1supersymmetrywhi hwillbe ofmajor on erninthesubsequent se tion5.
Aftertheintrodu toryse tions2to4,wewillapply,inse tion5,theresultsofse tion3to
gaugedisometriesonthetarget manifoldofs alar eldsinglobaland lo alN =1
supersym-metry andgeneralizepreviouswork. Therefore,webeginbygaugingan Abelianisometryin
globalN =1supersymmetryandshowwhenitisne essarytoaddgeneralizedChern-Simons
terms to the gauge se tor presented in se tion 4 su h that the resulting a tion is invariant
under the gauged isometries. After having generalized the results to gauged nonabelian
isometries, we will display under whi h onditions gauge anomalies are possibly an elled.
Furthermore,weinvestigatethe onservation ofsupersymmetryinpresen eofgauged
isome-tries. After thisis a omplished, we willextendthe resultsto N =1 supergravity. We will
illustratethe an ellationpro edurefor agaugegroup ofthe formAbeliansemisimple.
In se tion 6, we will show that four-dimensional gauge theories with Green-S hwarz
anomaly an ellationandpossiblegeneralizedChern-Simonstermsadmitaformulationthat
is manifestly ovariant withrespe t to ele tri /magneti dualitytransformations. This
gen-eralizes previous work on the symple ti ally ovariant formulation of anomaly-free gauge
theoriesand mayhave interestingappli ations, e.g.,for ux ompa ti ation with
interse t-ingbranes.
ve tor multipletsinthe N =1 super eldformalism. We ompute theD-term potentialand
showthatitisequivalent toapotentialinstandardformexplainingan earlierresult by[90℄.
The a tion an be regarded as the supersymmetrization of a spe ial Abelian gauge of the
theorypresentedinse tion 6. Thepre ise onne tion isillustratedinappendixE.
The on lusionis foundinse tion 8,and notations and onventions, aswellaste hni al
A quantum theory is alled anomalousif there is an exa t symmetry of the lassi ala tion
whi hisnotpreserved asasymmetryafterquantization. Whenforgaugetheoriesthe
quan-tuma tionisnotgaugeinvariant,thenthequantumtheoryisnotrenormalizable. Thereason
isthatso- alled Ward-identities,whi hareabsolutelyne essaryfortherenormalization
pro- edure to bewell-de ned,do nothold.
Anomalies are not only a possible feature of gauge symmetries, but may also arise for
global symmetries of the lassi al a tion. Contrary to quantum gauge theories, in the ase
of theglobalsymmetrythisis notne essarilya problembutmayinsteadlead to interesting
measurable physi al e e ts as, for example, the de ay of the pion into gamma rays shows.
Histori ally, the observed de ay rates in experiments did not mat h the theoreti al
predi -tions. Onlyon ethe ontributionoftheglobalanomalywas onsidered,verygoodagreement
between experiment and theory ould be obtained. The anomaly does not spoil
renormal-izationhere be ause noWard-identityisviolated. Thisexamplealsoshows thatan anomaly
is not simplya mathemati al problem aused bythe formalism buthas a lear physi al
in-terpretation. In fa t, an anomaly is a onsequen e of the non-invarian e of the quantum
measure in the path integral formulation as demonstrated by Fujikawa [41℄. Nevertheless,
already triangle diagrams show whether a given theory is anomalous or free of anomalies,
whi h will be reviewed in the next se tion. In se tion 2.2, we illustrate how the anomaly
appears in the path integral formalism. The onsistent anomaly is explained in se tion 2.3
andtheWess-Zumino onsisten y onditionispresented. Finally,inse tion2.4,we omment
brie y onthe an ellationof anomalies.
2.1 Triangle anomaly
Gauge symmetry and renormalizationare loselyrelated topi s. In gaugetheory,the
renor-malization pro edure makes use of identitiesthat relate di erent Green's fun tions. These
identitieswereprovenbyWard[42℄andTakahashi[43℄andarehen e alled\Ward-Takahashi
identities". The validity of the Ward-Takahashi identities is not automati when hiral
fermions are in the theory. More expli itly, one has to he k whether there are diagrams
reprodu -ing themselves re ursively at higher orders in perturbation theory. In a theory with hiral
fermions 5
thethree-point fun tions
T (q;k 1 ;k 2 ) h0jT[J 5 (q)J (k 1 )J (k 2 )℄j0i; (2.1) T (q;k 1 ;k 2 ) h0jT[P(q)J (k 1 )J (k 2 )℄j0i (2.2)
ausesu hanomaloustermsthatviolatetheWard-Takahashiidentities. HereP(q)represents
the pseudos alar urrent whi h is expli itlygiven by P =
5
. The Feynman graphsthat
illustrate(2.1) and (2.2) are, to lowest order,trianglegraphswithtwo externalphotonsand
one axialve tor inthe rst aseand a pseudos alar(if present)forthese ond ase.
ApplyingthestandardFeynmanrulestotheFeynmandiagramsdisplayedin gure1allows
Figure 1: These diagrams ause ontributions that violate expli itly the Ward-Takahashi
identities. The graphi istaken from[1℄.
5
ConsideraLagrangianwherethefermionisdenotedby and ouplestoave tor eldAandtoanaxial
ve tor eldA 5
. TheLagrangianisgivenbyL(A 5 ;A)= ( +A +A 5
5) ,forexample. Notethat
thegivenLagrangiandes ribesalsothe ouplingofave tor eld totheele tromagneti urrentrepresented
byJ =
andofanaxialve tor eld ouplingtotheaxialve tor urrentJ 5
=
one to writedowntheexpli itexpressionsfor(2.1) and (2.2), whi h aregiven by T (q;k 1 ;k 2 ) = i Z d 4 p (2) 4 tr i p m 5 i (p q) m i (p k 1 ) m + +tr i p m 5 i (p q) m i (p k 2 ) m (2.3) T (q;k 1 ;k 2 ) = i Z d 4 p (2) 4 tr i p m 5 i (p q) m i (p k 1 ) m + +tr i p m 5 i (p q) m i (p k 2 ) m (2.4) where q := k 1 +k 2
. In order to ndthe Ward-Takahashi identityfor the axial ve tor, one
hasto ompute q T . A usefulidentityis 1 p m q 5 1 p q m = 1 p m 5 + 5 1 p q m +2m 1 p m 5 1 p q m ; (2.5)
whi h an be easily proven by multiplying (2.5) from the left sideby (p
m) and from
theright sideby(p q
m). Withthehelp oftheidentity(2.5) one an repla ethe
rst two fra tions in(2.3) by theright hand side of (2.5), and it is not diÆ ultto see that
we have q T = R 1 +R 2 +2mT ; (2.6) whereR 1 andR 2
denoteintegrals thatare aused bythe rsttwoterms ontheright hand
sideof(2.5). The axialWard-Takahashiidentityis
q T = 2mT ; (2.7)
and we see that (2.6) violates (2.7) by the remainingterms R 1 and R 2 . These remaining
terms do not vanishbe ause, when written outwith thehelp of Feynmanrules, they result
inlinearlydivergentintegralsthatleadtoambiguitiesinthemomentumrouteofthetriangle
graph.
The amplitudeT
(2.2)is onvergent be ause theapparentlinearandlogarithmi
diver-gen ies disappearin the a tual omputation. The al ulation is not repeated here but an
be foundinthe lassi alle tureson anomaliesbyJa kiw ( [44,45℄) and inanytextbook on
TheresultinganomalousWard-Takahashiidentityisequivalentto themodi ed onservation
law fortheaxial urrent
J 5 = 2mP(x)+A; (2.8)
wheretheanomaly, A,isgiven by
A = e 2 (4) 2 " F F : (2.9)
ThisisthefamousAdler-Bell-Ja kiwanomaly[48,49℄,whereF
istheAbelian eldstrength
de nedbyF =2 [ A ℄ . 6
The anomaly (2.9) isindependentof thefermionmassand therefore violatesthe urrent
onservation of themasslesstheory.
The Ward-Takahashi identity of the ve tor urrents is ful lledwhi h is a onsequen e of a
hosenmomentumroute.
Observethatatta hingnewphotonlinestooneloopdiagrams,whi hisequivalenttoturning
the triangle diagram into a quadrangle or in general n-angle diagram, generates an
inte-gral that is at least logarithmi ally divergent: T
:::
for fermioni loops with more than
four external photons atta hed to it. This an be understood heuristi ally by noting that
the super ialdegrees of divergen eof the higher order diagramsare less than one and the
momentum-routingambiguitydoesnotexist forthosediagrams. Thissummarizes the
theo-rembyAdlerand Bardeen[50℄,thatstatesthatradiative orre tionsinhigherordersdonot
alter(2.8) and, thus,theanomaly is already totallydeterminedbythetrianglediagram.
2.2 Path integral and anomaly
AdlerandBardeenproposedintheirtheoremthat thefullstru tureofthe hiralanomaly is
given by thetriangleanomaly [50℄ and doesnotre eive ontributionsfrom furtherradiative
orre tions. This suggests that the anomaly shouldeven exist beyond perturbation theory.
Fujikawa was the rst to re ognize that in the path integral formalism the anomaly
orre-sponds to the Ja obian of a
5
-transformation of the quantum measure [41℄. One an see
6
Hereand inthe following, [℄and () denote,respe tively, antisymmetrizationand symmetrizationwith
\strengthone",i.e.,[ab℄= 1
2
thisasfollows: Lettherebemasslessfermioni eldsinthetheorytransformingnontrivially
under hiralgauge transformationsas
! e i 5 ; ! e i 5 : (2.10)
TheimportantstepsinFujikawa's methodare rstto de nethepathintegralmeasure more
a uratelybyde omposingthespinors and
intoeigenfun tionsoftheDira operatorand
se ond to determine the Ja obian of the path integral measure under hirality
transforma-tions. The Ja obianof in nitesimaltransformationswillbe exa tlythe anomaly.
The eigenve torsjni ofthe operator Daregiven by:
D jni = n jni; (2.11)
and thespinorsde omposea ording to
(x) = X n a n hxjni; (2.12) (x) = X n hnjxi b n ; (2.13)
where the de omposition oeÆ ients a
n and
b
n
are independent Grassmann obje ts. These
oeÆ ients at hand, we are able to re-express thepath integralmeasures D D a ording to D D = Y n Da n D b n ; (2.14)
be ause theset of eigenve tors jniis ompleteand orthonormal, i.e. hnjmi=Æ
nm
. In order
todeterminethebehaviouroftheobje tsa
n and
b
n
under hiraltransformations,we onsider
therotatedspinor
0
(x) = e i
5
(x): (2.15)
Afterde omposingbothsidesof(2.15)intotheeigenve torsjni,andusingtheorthonormality
of theeigenve tors, one ndsthat
a 0 n = X m C nm a m ; C nm := Z dxhnjxie i(x) 5 hxjmi: (2.16)
The Grassmann measure transforms with the inverse determinant and, therefore, the path
integral measure transforms with det(C nm ) 1 2
, whi h has to be determined. Making use
of detC = e trlog (C)
and onsidering in nitesimally small transformations, one an
de om-posethe logarithmaround theunitymatrix. Then,the Ja obianJ ofin nitesimal hirality
transformationsisgiven by J = e 2i R dx ~ tr( 5 ) : (2.17)
Observe, that the fun tional tra e ~
tr(
5
) is de ned through the eigenve tors ~ tr( 5 ) := P n hnjxi 5 hxjni. 7
This tra e is a tually divergent, and we have to regulate the sum. As
theregulatorwe usethe onvergent fa tor exp ( n M ) 2
and take thelimitM !1. Then,
we an manipulatetheregulatedexponent of(2.17) and afterintrodu ingunityoperators of
theform R
d 4
kjkihkjand byusing ompletenessof thesetfjnig, we nd
lim M!1 ~ tr( 5 e ( n M ) 2 ) = lim M!1 Z d 4 k (2) 4 e ikx ( 5 e ( D M ) 2 )e ikx :
We de omposethe operator D 2
into an oddpie e proportionalto [ ; ℄and an even pie e proportional to f ; g = 2g so that we have D 2 = D D + 1 4 [ ; ℄F .
Afterres alingthemomentumandde omposingtheexponential,thereisonlyone termthat
survivesin thelimitM !1 (thetermquadrati inthe eld strength),and we obtain:
~ tr( 5 ) = 1 32 2 " F F : (2.18)
Insertingthisba kinto (2.17) we indeed ndthe anomaly(2.9), orinother words,thepath
integral measuretransforms withthe Ja obian
J = e i 16 2 R dx(x)" F F : (2.19)
However, aswedidnotexpandthepathintegral,thisresultisvalidbeyondanyperturbative
expansion. In the path integral pi ture the anomaly is explained by the non-invarian e of
the path integral measure under hirality transformations. The formal reason for the
non-invarian e an be tra ed ba kto thefun tionaltra e ~ tr 5 ,whi hissingular. 7 Note,that ~
2.3 Consistent anomaly
So far we have only onsidered Abelian symmetries. If we want to generalize the above
on eptsto thenonabelian ase, thentheexpression(2.18) willof oursenolongerrepresent
thefullanomaly. The naive extension of (2.9),in whi h the eldstrength is repla ed by its
ovariant ounterpart, is not orre t be ause the ontribution of quadrangle diagrams and
pentagon diagrams, though nite, violates the nonabelian stru ture. The a ess through
diagrams be omes now more ompli atedand solet us hoose the more onvenient way by
meansof the pathintegral. As a rst step,we de ne Green'sfun tions with thehelp ofthe
generatingfun tional,whi his givenby
Z[A ℄ = Z D D e R d 4 x( +A ) ; (2.20)
where the gauge elds are treated as external elds and sour es for the fermions are
ig-nored. For the proof of renormalizeability it is suitable to use onne ted Green fun tions,
butthegeneratingfun tionalZ[A℄ ontainsboth onne tedanddis onne teddiagrams. The
onne tedGreen fun tionsaregenerated byW[A℄ de nedby
Z[A ℄ = e W[A ℄ : (2.21)
Fortheanomalyweonlyneedto onsiderthefermioni partofthetheory,so(2.21),givenby
(2.20), isreallyall we needfrom thefullquantuma tion. Letthegaugegroup be generated
by T
A
satisfying the algebra [T
A ;T B ℄ = f AB C T C , where f AB C
are the stru ture onstants.
In nitesimalgaugetransformationsthata tonthea tion(2.21)arede nedbytheoperators
X A (x) = D C A Æ ÆA C (x) := Æ C A +f AB C A B (x) Æ ÆA C (x) : (2.22)
It an be shown thatthese operatorsful lthealgebra givenby
[X A (x);X B (y)℄ = f AB C X C (x)Æ(x y); (2.23)
and thatthegauge variationof W[A℄ isgiven by
ÆW[℄ := Z d 4 x A (x)X A (x)W[A ℄ = Z d 4 x A (x)hD C A j C (x)i on. (2.24)
where hj C i on. = 1 Z[A℄ R D D ( T C )e R d 4 xL( ; ;A)
is the expe tation value of the
on-ne ted urrent. We an easily see that for an invariant quantum a tion, ÆW[℄ = 0, the
urrentis ovariantly onserved,hD C A j C i on.
=0. However,ifthetheoryisanomalous,then
thegeneratingfun tionalof onne tedGreenfun tionssatis esÆW[℄= A
A
A
,andinorder
to be onsistentwith thegaugealgebra(2.23), the anomalyhas to obey the ondition
X A (x)A B X B (y)A A = f AB C A C Æ(x y); (2.25)
whi h is the so- alled \Wess-Zumino onsisten y ondition" [51℄. We an also see that the
naive nonabelian extension of (2.9), where the Abelian eld strengths are repla ed bytheir
nonabelian ounterparts, isnot orre t be ause itviolates (2.25).
An expli itsolutionof(2.25) isgiven by
A C = 1 24 2 " tr[T C (A A + 1 2 A A A )℄; (2.26)
whi h represents exa tly Bardeen's result [52℄ found from fermion loop omputations. This
solutionisnotuniquebe auseone anaddlo alpolynomialsoftheexternalgauge eldsf[A℄
to(2.26)andobtainanothersolution. Theselo alpolynomials anbeindu ed,e.g.,whenthe
renormalizationpro edureis hanged. The2-pointGreenfun tionsoftwove tor urrents,for
example,havearenormalizationambiguitybe ausetheirLorentzinvariantextensionsto test
fun tionsarenotunique[53℄. Forthequantuma tion,thismeansthat ~
W[A℄=W[A℄+f[A℄
and the generating fun tional re eives a phase fa tor Z[A℄e if[A℄
. A phase fa tor, however,
does not a e t the transition probability and is not observable. Consequently, we an also
all a theory anomalous, if there does not exist a lo al polynomial of the external gauge
elds,su hthat(2.26) ise e tively an elled. Possiblelo alpolynomialsaregivenby
Chern-Simons terms or generalized Chern-Simons terms (depending on the dimension). In the
following se tion we will dis uss these topologi al terms, espe ially the generalized
Chern-Simons termsbe ause theseare ofspe ialinterest infourdimensions.
2.4 Can ellation of anomalies
Although there are attemptsto live with anomalous theories, seefor example[54℄ and [55℄,
in renormalizable theories, anomalies must not o ur. This implies severe restri tions on
both hiral se tors and any potential gauge anomaly in the left-handed se tor is an elled
bytheanomalyof theright-handedfermions. In hiralgaugetheories,by ontrast, anomaly
an ellationisnotautomati andthe an ellationrequiresa arefulbalan eofthefermioni
gaugequantumnumbers,as, e.g.,inthestandard model.
Anotherpossibilityto an elanomaliesistointrodu ea ountertermintothea tion,with
parti les that transform appropriately under gauge transformations su h that the anomaly
is ompensated. Asmentionedintheintrodu tion, asimpleAbelianexampleisgivenbythe
intera tion " ia(x)F F ; (2.27)
wherethes alar, a(x)varies underthegauge symmetrya ording to
Æa(x) = i(x): (2.28)
Then the variation of the intera tion (2.27) is able to an el the Abelian anomaly (2.9).
When the gauge theory is nonabelian then thefull onsistent anomaly annot be an elled
bythisme hanism. The Green-S hwarzanomaly an ellationme hanism in 10-dimensional
supergravity and super Yang-Mills theory is a sophisti ated generalization of this simple
3 Lie algebra ohomology and generalized Chern-Simons
terms
In generi e e tive eld theoriesone hass alar eld dependent fun tions appearinginfront
ofthegaugekineti terms,i.e. infrontofF
F
andF^F. Hereingeneral,thenonabelian
eldstrength two form isde nedas
F C := dA C 1 2 f AB C A A ^A B : (3.1)
Supersymmetri theories, for example, often generalize the gauge se tor to in orporate a
nontrivial gauge kineti fun tion f
AB
that depends on a set of s alar elds, as is further
explainedinse tion4.1. Compatibilitywithsupersymmetry onstrains thisfun tionand so,
for instan e, in N = 1 supersymmetry it is required to be a holomorphi fun tion of the
omplexs alarsof the hiral multiplets.
The Lagrangian will ontain a nontrivial F ^F term when the imaginary part of the
gaugekineti fun tionis nontrivial. Inthe literaturethistermis sometimesreferred to asa
\Pe ei-Quinnterm"and reads
L PQ = iImf AB F A ^F B : (3.2)
The intera tiongiven in equation (2.27) a tually represents a spe ial ase of (3.2) wherewe
just have a U(1) gauge symmetry (and hen e only one index, whi h may be dropped), and
thegauge kineti fun tionisgiven bytheaxioni s alara(x), i.e. f=4a(x).
In theremainder,theexterior produ t^isunderstood andwillnolongerbewrittenout
expli itly.
Under thegauge transformationof the onne tion one-formsA C =A C dx ,whi h read ÆA C = D C :=d C +f AB C A A B ; (3.3)
the eld strengthtwo forms(3.1) transform ovariantly, i.e. if
ÆF C = f AB C A F B : (3.4)
Clearly,theLagrangian(3.2) isinvariantunder(3.3)ifthegaugekineti fun tiontransforms
inthesymmetri produ tof two adjointrepresentations,i.e. if
Æf AB = 2 C f C(A D f B)D : (3.5)
Æf AB = 2 C f C(A D f B)D +iC AB;D D ; (3.6)
soasto allowfor onstantshiftsinthegaugekineti fun tion. HereC
AB;D
isareal onstant
tensorsatisfyingthe onstraints
C (AB;D) = 0; (3.7) 1 2 C AB;D f EF D C DB;[E f F℄A D C DA;[E f F℄B D = 0: (3.8)
Thismore generaltransformation(3.6) an be indu edifthes alar eldstransform
nontriv-iallyunderthegaugegroupandappearina ertainwayinf
AB
,butwewilladdressthislater
inmore detail.
Obviously,on eweallowfortheseshifts,theLagrangian(3.2)isnolongerinvariantunder
(3.3) and (3.6). Its variationis insteadgiven by
ÆL PQ = iC AB;D D F A F B : (3.9)
Ifweonly onsiderthe lassi ala tion,thevariation(3.9) anonlybe an elledbynewterms
addedto L
PQ
,the so alledgeneralized Chern-Simonsterms [32,34℄. Inthis se tionwe will
show how a lassi ally gaugeinvariant a tiongeneralizing(3.2) an be onstru ted byusing
the te hniques of [34℄. In the following subse tion we introdu e Lie algebra valued forms
C(A;F) and analyze them bymeans of ohomologi al te hniques. This method allows one
tounderstandtheoriginofthe onstraints(3.7) and(3.8). The onstraint(3.7) demandsthe
formsC(A;F) tobehomogeneousinthe eldstrengthand thegauge onne tionseparately.
Then, forformsC(A;F) whose oeÆ ientssatisfy (3.7) we an identifythe onstraint (3.8)
as the onstraint demanding C(A;F) to be losed with respe t to the exterior derivative.
After spe ifying the transformation properties of the gauge kineti fun tion, we are able
to onstru t the gauge invariant extension of the Pe ei-Quinn term, whi h is obtained by
in ludinggeneralized Chern-Simonsterms.
In subse tion 3.2 we ndthat there are no non-trivial generalized Chern-Simons terms
for semisimple gauge groups and present the example of a gauge group that has the form
Abeliansemisimplein se tion 3.3. The results of these subse tions are dis ussed in more
Finally,insubse tion3.4, we generalizetheformalismdevelopedin[34℄ inorderto allow
forformsthatdo notneedto satisfythe onstraint (3.7). We willseethat thePe ei-Quinn
termand the generalized Chern-Simonstermareno longer gaugeinvariant on e we give up
the onstraint (3.7). The onlypossibilityto an el the gauge non-invarian e insu h a ase
isto onsideranomalies.
Beforewe onstru t thegeneralizedChern-Simonsterms,Iwould liketo give afew
om-ments on `ordinary' Chern-Simons terms [57℄ that should illustrate the di eren e between
ordinaryandgeneralized Chern-Simonsterms. The onstru tion of`ordinary'Chern-Simons
forms isusually donebymeansof so alled hara teristi orinvariant polynomialsP
n . The
hara teristi polynomials P
n
(F) are symmetri fun tions of degree n in the eld strength
formF and invariantunderthea tionofthegaugesymmetry group. Therefore, the
hara -teristi polynomials satisfy P
n (F
g
) =P
n
(F) where we denoted the gauge transformed eld
strength byF g
. Withthehelp ofthe Bian iidentity
DF :=dF +[A;F℄ = 0; (3.10)
it an be proven that the invariant polynomialsare losed, i.e. dP
n
(F)= 0. A theorem by
Chern and Weil states that the ohomology lasses of P
n
(F) do not depend on the hoi e
of the onne tion form A and hara terize the de Rham ohomology group [58℄. Then, the
ohomology lassesof invariant polynomials P
n
(F) of degree n arefurther hara terized by
theChern-SimonstermsQ
n 1
(A;F) whi h areformsof degree (n 1), i.e.
P
n
(F) = dQ
n 1
(A;F) (3.11)
Integrals of hara teristi polynomialsare topologi alinvariants. Let us onsider, for
exam-ple, in four dimensions a hara teristi polynomial of the form P
4 (F) = tr(FF) whi h is invariantbe auseofP 4 (F g )=tr(gFg 1 gFg 1 )=tr(FF) =P 4
(F). Thenthis hara teristi
polynomialleads to the three-dimensionalChern-Simonsform Q
3 (A;F) =tr[AdA+ 3 2 A 3 ℄. 8
Observe, that Chern-Simonsformsare ingeneral odd dimensionalwhilegeneralized
Chern-Simons formslive ineven dimensionsaswe willsee.
8
Sin e the determinant is invariant underthe adjoint ofthe gauge symmetry,i.e. det() =det
g 1
()g
if g represents anelement of the gauge group, one an also obtain invariant polynomials withthe help of
thedeterminant. However,the orrespondingChern-Simonsforms are notrelatedto theoneobtainedfrom
3.1 Generalized Chern-Simons forms
GeneralizedChern-Simonsterms annot be onstru ted from hara teristi polynomials
be- ause there are no odd dimensional invariant polynomials in the eld strength. To set the
stage we onsidera ve-formC(A;F) de nedas
C(A;F) := C AB;D A D F A F B ; (3.12)
and do notlimitourselvesto fourspa etime dimensions.
Note the pe uliarstru ture of the indi esof the onstant tensor C
AB;D
: the index
or-responding to that arried by the gauge onne tion is separated from the indi es that are
arried by the eld strengths by a omma. Therefore, the onstant tensor is symmetri in
its rst two indi eswhi h is also onsistent with(3.6). Furthermore, observe that the form
C(A;F) does not represent an invariant or hara teristi polynomial as mentioned in the
ontext of ordinary Chern-Simons terms be ause C(A;F) depends expli itly on the gauge
onne tion. Thereisno problemingeneralizing(3.12) to formsofarbitrary degreeinA and
F byintrodu ing onstanttensorsoftheformC
A 1 :::A n ;D 1 :::D m
. Nevertheless,herewefo uson
theform(3.12),whi hleadsto thegaugeinvariantgeneralizationof(3.2)infourdimensions.
Using(3.1) and (3.10) we an ompute theexteriorderivativeof (3.12), whi h leadsto
dC(A;F) = C AB;D F D F A F B + 1 2 C AB;D f D EF +f D AE C DB;F +f D BE C AD;F A E A F F A F B :
Comparing this result with the onstraints (3.7) and (3.8) shows that these orrespond to
demanding thatC(A;F) ishomogenous 9
and losed, i.e. dC(A;F)=0. Ontheother hand,
we an de nean algebrai operator
(DC) AB;EF := 1 2 C AB;D f D EF C DB;[F f D E℄A C AD;[F f D E℄B ; (3.13) satisfying D 2 = 0 be ause of d 2
= 0 (this an also be dire tly proven from (3.13) by using
the Ja obi identity on the stru ture onstants). Hen e, we an say that as d 2
leads to the
de Rham ohomology, D 2
=0 leads to Lie algebra ohomology of forms C(A;F) satisfying
the onstraints (3.7) and (3.8). For a losed form C, i.e. if C
AB;D
ful ls the equations
9
Observethat we allC(A;F)ahomogeneousform,following[34℄, ifdC(A;F)ishomogeneousinAand
F separately. The onstraint (3.7) is satis ed by homogeneous forms. Homogeneity enables one to de ne
(3.7) and (3.8), the equivalen e lasses of all C 0
in the ohomology are, for some four-form
Z =Z AB F A F B ,given by C 0
=C+dZ. So if the ohomology lass is trivial, thenwe have
C=dZ and C is d-exa t. 10
We willseelaterwhen thisisthe ase.
At this point it is suitable to dis uss the transformation properties of the s alars that
appear in the gauge kineti fun tion f
AB
. We assume that the s alar elds z i
transform
undergauge transformationas
Æz i = A k i A (z); (3.14)
where the ve tor elds 11 k A = k i A i
de ne a (possibly nonlinear) realization of the gauge
groupand satisfy
k j A j k i B k j B j k i A = f AB C k i C : (3.15)
As transformations of the s alars in general indu e transformations of the gauge kineti
fun tion, letusassume that(3.14) indu esthetransformation (3.6),i.e.,
Æ(Imf AB ) := k j D j (Imf AB ) D = 2f D(A E Imf B)E D +C AB;D D : (3.16)
Then, in order to make use of the form C(A;F) as de ned in (3.12), let us onsider the
followingLie algebra-valuedform
k j D j (Imf AB )A D F A F B : (3.17)
With thehelp of the Bian hiidentity(3.10) and the variation of thegauge kineti fun tion
(3.16), this an be writtenas
k j D j (Imf AB )A D F A F B = Imf AB d(F A F B )+C AB;D A D F A F B : (3.18)
Dueto the hain-rule,we furthermore have
d(Imf AB )(z)F A F B = j (Imf AB )dz j F A F B ; (3.19) 10
NotethatfromdZwe ande nethea tionofanalgebrai operatoronZABintotalanalogywithequation
(3.13)forC AB;D ,su hthatC AB;D =(DZ) AB;D
. Thealgebrai operator(DZ)
AB;D
isde nedasinequation
(A.5),whi hforthe aseathandreads(DZ)
AB;D =2f D(A E Z B)E . 11
from whi h we subtra t(3.18) to nallyobtain j (Imf AB )(dz j k j D A D )F A F B = d (Imf AB )F A F B C AB;D A D F A F B : (3.20)
Letushave a loserlookat thisresultand ndoutaboutits impli ations.
Firstly,the left handside of (3.20) is gauge invariant be ause dz j k j D A D is the gauge
ovariantderivativeforthes alar eldsz i
,andfrom(3.16)weseethat
j (Imf AB )transforms ovariantlyasC AB;D
is a onstant. Consequently,theleft hand sideof (3.20) represents an
invariant Lagrangianin 5dimensions.
Se ondly,letus onsidertherighthandsideof(3.20). We anseethatanyshiftofC
AB;D
byanexa t (in theLiealgebra ohomology)pie e (DZ)
AB;D =2f D(A E Z B)E leadsto a shift
of the ve form C(A;F) by an exa t form dZ, as was explained in footnote 10. A ording
to (3.20), this exa t form dZ an then be absorbed by a shift Imf
AB ! Imf AB +Z AB , as
is also suggested by(3.16). Therefore, we an say that anyexa t ontributionsof C an be
absorbed bya rede nitionof thegauge kineti fun tionby a onstant imaginaryshift.
Now, that we have an invariant a tion in ve dimensions, we want to pave the way to
obtain invarian e in four dimensions. If we demand that C
AB;D
satis es the onstraints
(3.7) and (3.8),weknowthatC(A;F) is losed. It thenfollows fromPoin are'slemma that
lo ally there exists a form !, su h that C = d!. In order to ndan expli itexpression for
!,we singleout one oordinate tand require A D
(t)=tA D
with A D
dependingonlyon the
remaining oordinates. After introdu ingd t =d+ t dt and de ning H A (t) := tdA 1 2 t 2 f BC A A B A C ; (3.21)
we an verify thefollowingformulae
F C (t) = H C (t)+dtA C ; (3.22) F A (t)F B (t) = H A (t)H B (t)+2dtA (B H A) (t): (3.23)
As by assumption C(A;F) is a losedform, the parti ular t-dependent form C(A(t);F(t)),
onstru tedfrom thede nitionsmadeabove,is losed,too thereasonis thatthe onstants
C
AB;D
satisfythe onstraints (3.7) and(3.8)
. Then itis notdiÆ ultto prove that
0 = d t C(A(t);F(t))=dt t C(A(t);F(t))+dC(A(t);F(t)) = dt t C(A(t);F(t))+dC(A(t);H(t))+2tdtd(C AB;C A C A B H A ): (3.24)
These ond terminthelastlinevanishes,whi hone seesvery easilyon ethetermiswritten
inits omponentform
dC(A(t);H(t)) = d[C AB;D A D (t)H A (t)H B (t)℄: (3.25)
Ifwenowabsorbthefa tortbyres aling,At!A,thenitfollowsfromthede nition(3.21),
that H(t) ! F, and (3.25) be omes dC(A;F) whi h vanishes be ause C(A;F) is losed.
Finally,integrating (3.24) overt leaves uswith
C(A;F) = d 2C AB;D Z 1 0 dttA D A B H A (t) : (3.26)
Inserting(3.21), theintegral an be omputed,andwe nd
! = 2 3 C BC ;D A D A B (dA C 3 8 f EF C A E A F ): (3.27)
Fromtheargumentsbelow(3.20)weknowthatd[Imf
AB F
A
F B
℄ C(A;F)isagaugeinvariant
expression in ve dimensions and, onsequently, Imf
AB F A F B ! represents a gauge
invariant Lagrangian in four dimensions. Con retely, the gauge invariant extension of the
Pe ei-QuinnLagrangian reads
L PQ +L GCS = iImf AB F A F B + 2i 3 C BC ;D A D A B (dA C 3 8 f EF C A E A F ); (3.28)
wherethese ond termis theso alledgeneralized Chern-Simonsterm.
These onsiderationsarequitegeneraland allowtheextensionof thetransformationlaw
forthe gauge kineti fun tion bya onstant imaginary shiftiC
AB;D
when at the same time
thePe ei-Quinntermisa ompaniedbythegeneralizedChern-Simonsterm. Thepro edure
isnotlimitedto fourdimensionsand an beeasilygeneralizedto arbitraryeven dimensions.
The generalized Pe ei-Quinnterm then be omes the2n form f
A1A2:::An F A 1 F A 2 :::F A n and
startingfrom the (2n+1) form C(A;F) =C
A 1 :::A n ;D A D F A1 :::F An
thesame pro edureas
outlinedabove determinesthe orrespondinggeneralized Chern-Simonsform to be
! = Z 1 0 dtntC A 1 A 2 :::A n ;D A D A A 1 H A 2 (t):::H A n (t): (3.29)
The Abelian ase is simply obtained by setting all stru ture onstants to zero, and the
generalized Chern-Simonstermforan Abeliangaugetheory isgiven by
L GCS = 2i 3 C BC ;D A D A B dA C : (3.30)
3.2 Generalized Chern-Simons terms and semisimple groups
As we presented in the previous subse tion, when C
AB;D
is D-exa t it an be absorbed by
rede ning the gauge kineti fun tion and, as a onsequen e, the new Pe ei-Quinn term
be omes gaugeinvariant. Now, we will showthis is the ase for semisimplealgebras, whi h
means that the main appli ation of generalized Chern-Simons terms is for non-semisimple
gaugealgebras.
We startwiththeresult thatif
C AB;C =2f C(A D Z B)D ; (3.31)
for a onstant real symmetri matrix Z
AB
, theChern-Simons term an be reabsorbed into
thePe ei-Quinntermusing
f 0 AB =f AB +iZ AB : (3.32)
In fa t, one easily he ks that with the substitution(3.31) in thetransformation law of the
gauge kineti fun tion (3.6), the C-termsare absorbed bythe rede nition (3.32). Equation
(3.31) an be writtenas C AB;C =T C ;AB DE Z DE ; T C ;AB DE 2f C(A (D Æ E) B) : (3.33)
In the ase that the algebra is semisimple, one an always onstru t a Z
AB
su h that this
equation isvalidforanyC
AB;C : Z AB =C 2 (T) 1 AB CD T E;CD GH g EF C GH ;F ; (3.34) whereg AB andC 2 (T) 1
arethe inversesof theCartan-Killingmetri
g AB = f AC D f BD C ; (3.35)
and,respe tively,theCasimiroperatorde nedby
C 2 (T) CD EF := g AB T A;CD GH T B;GH EF : (3.36)
Theseinversesexistforsemisimplegroups. To showthat(3.34) leadsto(3.33) oneneedsthe
onstraint (3.8), whi h an be brought to thefollowingform
g HD T H 1 2 C C f DE C +T [D C E℄ =0: (3.37)
We have dropped doublet symmetri indi es here, using the notation for ontra tions of
su h doubleindi es. Furthermore,thisimplies
g AB T E T B C A =C 2 (T)C E ; (3.38)
withwhi hthementioned on lusions an easily be obtained.
Thisresult anbealsoobtainedfroma ohomologi alanalysisandwerefertheinterested
readerto appendixA.1.
3.3 Appli ation: Abeliansemisimple
The simplest nontrivial appli ation are gauge groups of the form Abeliansemisimple for
whi h one obtains an interesting result. Abelian generalized Chern-Simons terms are not
trivial,butaswe ouldshow, thepurelysemisimpleterms are. However, the dire tprodu t
ofanAbeliangaugegroupwithasemisimplegaugegroupisnottrivialagain,espe iallyithas
anontrivialmixed se tor, whi hisgoingto beinvestigatedingreaterdetailinthefollowing.
To re e ttheprodu tstru ture, wesplittheadjointindi esA;B;:::into indi esa;b; ;::: for
theAbelian part and adjoint indi es x;y;z;w;::: forthesemisimple part. Dueto thegroup
stru ture, onlythe stru ture onstants ofthe type f
xy z
are nonzero. As before, we de ne a
homogeneous ve-form C(A;F),whi his given by
C(A;F) = 2C (xb);a A a F x F b +C xy;a A a F x F y +2C (ax);y A y F a F x ; (3.39) with onstants C xb;a , C bx;a , C xy;a , C ax;y and C ya;x
. The losure relations an be dire tly
obtainedfrom (3.8) bysimplyinsertingAbelianand semisimpleindi es 12
and we are ledto
f v xu C vb;a = 0 (3.40) f v xy C bv;a = 0 (3.41) f u(y v C x)v;a = 0 (3.42) f uy v C ax;v +f xy v C av;u f xu v C av;y = 0 (3.43) f uy v C xa;v +f xy v C va;u f xu v C va;y = 0: (3.44)
These relations already lead to various interesting results. By de nition, a semisimple Lie
algebrahasnoAbelianideals. Thisimplies,inparti ular,thatthere annotbeanynon-trivial
12
nulleigenve tor of thestru ture onstants, sothat (3.40) and (3.41) imply C xb;a 0; (3.45) C bx;a 0: (3.46)
Equation (3.42) means that C
xy;a
is for ea h a, a symmetri invariant tensor inthe adjoint
representation of the semisimple part of the gauge group. C
xy;a
therefore has to be
pro-portional to the Cartan-Killing metri g
xy
of the semisimple Lie algebra. Thus, we have
C xy;a =B a g xy wheretheB a
's arearbitrary but onstant. The onlynontrivialpart of(3.39)
is C(A;F) = 2C (xy);a A a F x F y +(C ya;x +C ay;x )A x F a F y : (3.47)
What we have doneisto simply applytheformalismdeveloped earlierinthisse tionto the
mixed part of a gauge group with the stru ture Abeliansemisimple. The purely Abelian
part is not trivial and leads to the Chern-Simons term (3.30). After the ohomologi al
analysis we found that the only nontrivial generalized Chern-Simons terms in the mixed
se tor of Abeliansemisimpleare determinedbythe ve form (3.47) and, onsequently,the
generalized Chern-Simonsterms ofthe mixedse tor read
L GCS = 4i 3 C (xy);a A a A x (dA y 3 8 f rs y A r A s )+ 2i 3 C ya;x A x A y dA a + + 2i 3 C ay;x A x A a (dA y 3 8 f rs y A r A s ): (3.48)
be ause all the other omponents of the onstant tensor C vanish due to ohomologi al
reasons. Observe, that if we do not allow for o -diagonal elements of the gauge kineti
fun tion, i.e. f
ax = f
xa
=0, then thegeneralized Chern-Simonstermin the mixed se tor is
given by L GCS = 4i 3 C (xy);a A a A x (dA y 3 8 f rs y A r A s ): (3.49)
The purely semisimplepart of C an be absorbed into the gaugekineti fun tionby
rede -nition. Thismat hesthe situationen ountered in[12℄ withoutanomalies.
3.4 Nonhomogeneous forms and anomalies
IntermsofLiealgebra ohomology,the onstraintsonC(A;F),theequations(3.7)and(3.8),
is losed under thealgebrai operator D de ned in (3.13) if it satis es the onstraint (3.8).
However, is the formalism stillvalid fornonhomogeneous forms or, in other words, an the
onstraint (3.7) berelaxed?
In order to understand this, let C(A;F) be nonhomogeneous, i.e. C
(AB;D)
6= 0.
Conse-quently,C(A;F) annotbe losedeither,butisinstead
dC(A;F) = C (AB;D) F D F A F B : (3.50)
Clearly, the omputation that led to the generalized Chern-Simons term (3.27) annot be
validanymore. More pre isely,insteadof (3.24) one nowhas
C (AB;D) F D (t)F A (t)F B (t) = dt t C(A(t);F(t))+dC(A(t);H(t))+ +2tdtd C AB;D A D A B H A : (3.51)
By using(3.22) one an prove easilythatthe lefthandsidede omposes a ordingto
C (AB;D) F D (t)F A (t)F B (t) = C (AB;D) H D (t)H A (t)H B (t)+ +3dtC (AB;D) A D (t)H A (t)H B (t): (3.52)
Of oursethese ondtermontherighthandsideof(3.51)nolongervanisheseitherbut auses
the ontributionC (AB;D) H D (t)H A (t)H B
(t)that an els the orresponding terminequation
(3.52). Therefore,(3.26) re eivesanextra ontributionand isrepla ed by
C(A;F) = 3C (D;AB) Z 1 0 dtA D (t)H A (t)H B (t) 2C AB;D d Z 1 0 dtA D (t)A B (t)H A (t) : (3.53)
We see, that the nonvanishing totally symmetri part of C
AB;D
introdu es the
ve-dimensionalform Q 5 (A;F) = 3C (D;AB) Z 1 0 dtA D (t)H A (t)H B (t) (3.54)
This form is nothingelsebut the ve-dimensionalChern-Simonsterm orresponding to the
invariant polynomialP 6 (F)= C (D;AB) F D F A F B
. As thenonhomogeneous form C(A;F) is
nolonger losed,theredoesnotexistaform!,su hthatC=d!or,equivalently,the
Chern-SimonsformQ
5
isnotrepresentablebya oboundary,i.e. there isno! 0 su hthatQ 5 =d! 0 .
Consequently, the ve-dimensional form d(Imf AB F A F B
) C(A;F) annot be represented
bythe oboundary(3.28)ofhomogeneousforms. Furthermore,itisnolongergaugeinvariant
be ause Q
5
isnot gaugeinvariant. However, thisis onlya problemin theoriesthat arefree
of quantum anomalies. The solutionisgiven bythedes ent equations [59{ 62℄. By meansof
this set of equations, Stora and Zumino ould relate theChern-Simons forms Q
2n 1
to the
onsistent anomalyA
2n 2
(;A)in2n 2dimensions. The des ent equationrelevant forour
ase is Æ Q 5 (A;F) = dA(;A); (3.55)
representing the gauge variation of the Chern-Simons form as the oboundary of the
four-dimensional onsistent anomaly. Applyinga gauge variation to d(Imf
AB F A F B ) C(A;F), we have d Æ (f AB F A F B ) d 2C AB;D Æ Z 1 0 dtA D (t)A B (t)H A (t) +d[ A(;A)℄; (3.56)
whi hisequaltozerobe auseof(3.20) asthestepsleadingto(3.20)arequitegeneralanddo
notdependonC(A;F)beinghomogeneousornot. ThetensorC
AB;D
in(3.6),however,isno
longerrestri tedtoitsmixedsymmetri partalonebutnowalso ontainsatotallysymmetri
part. Therefore, it an be de omposed into its totally symmetri part C (s) AB;D and a part of mixed symmetryC (m) AB;D ,i.e. C AB;D = C (s) AB;D +C (m) AB;D : (3.57)
The generalized Chern-Simons termis still onlyproportional to the mixed symmetri part.
The totallysymmetri partis to beexa tly an elledby theanomaly as(3.56) shows. Note
that(3.54) anonlybe onsistentwith(3.55)ifthetotallysymmetri partofC
AB;D ,C (s) AB;D = C (AB;D)
is related to thequantum anomaly (we willdis uss thisin greater detailin se tion
5.2).
We see that the onstraint (3.7) an be relaxed to allow for nonhomogeneous forms
C(A;F). As a onsequen e, the four-dimensional a tion(3.28) is no longer gauge invariant
be ausethegeneralizedChern-Simonstermisstillonlyproportionaltothemixedsymmetri
part of thetensor C
AB;D
. The left overvariationproportionalto C
(AB;D)
may be an elled
aretheappropriateformsne essaryinappli ationsto anomaloustheoriesinorder to absorb
the anomaly. The ohomologi al reason is that the nonhomogeneous forms introdu e the
ve-dimensional Chern-Simonsform Q
5
into the ohomologi al dis ussion, whi h in turn is
relatedtotheanomalyinfourdimensionsbytheStora-Zuminodes entequation(3.55).
Con-sequently,thegaugevariationof(3.28) doesno longervanish,butisgiven bythenegative of
thegauge anomaly,i.e.
Æ L PQ +L GCS = A(;A): (3.58)
Thisresult goesbeyond thework of [34℄ andallows fornonhomogeneous forms.
At the end of this se tion, let us dis uss again the example of a gauge group with the
stru tureAbeliansemisimple. We setallo -diagonalelementsofthegaugekineti fun tion
to zero, i.e. f
ax
= 0. The onstraints (3.40) to (3.44) do not hange for nonhomogeneous
forms(althoughtheydonotimply losureanymore),butarenowvalidforthefull oeÆ ient
C AB;D =C (s) AB;D +C (m) AB;D
. Nevertheless,theimpli ationsdrawnfrom(3.40)to(3.44) arestill
validand, onsequently,theonlynontrivialpartofa ve-dimensionalnonhomogeneousform
C(A;F) isdetermined byC xy;a ,i.e. C xy;a 6=0. De omposingC xy;a ,we obtain C (s) xy;a =C (s) ax;y = 1 3 C xy;a ; (3.59) C (m) xy;a = 2 3 C xy;a ; (3.60) C (m) ax;y = 1 3 C xy;a : (3.61)
Thus, we see that the generalized Chern-Simons term in the mixed se tor is still given by
(3.49). However, there are new ontributions due to the totally symmetri tensors C (s)
xy;a
and C (s)
ax;y
whi h ause nontrivial gauge variations of L
PQ +L
GCS
. Can ellation of these
remaining ontributions an only be a hieved with thehelp of mixed gauge anomalies, but
we will dis uss this example in more detail in se tion 5.4, where we will expli itly larify
therelationof thesymmetri oeÆ ientsC (s)
tothequantumanomalyand showhowmixed
In theearly 1960s, Gell-Mann and Ne'eman,proposeda way to arrangethe known hadrons
into a uni edframework and, inthis way, brought some order into a whole zoo of parti les
that had beenfounduntilthen [63℄. The su ess of their model is based on a globalSU(3)
symmetry whi h puts parti les of the same spininto SU(3)-multiplets. This model aused
a lotof enthusiasm, ande orts weremade to uniteparti les ofdi erent spinaswell. Inthe
non-relativisti regime this ould be a hieved by an SU(6) model, whi h made predi tions
that were quite well approximated by experimental data [64{ 66℄. Unfortunately, further
attemptsto onstru ttherelativisti versionsofsu hmodels,inwhi htheinternalsymmetry
group is nontrivially entangled with the Poin are group to form a so- alled Master group,
failed. Allthese e orts to reate aMastergroup didnotsu eedbe ausethe Mastergroups
alwayshad nonphysi alproperties su h asanin nitenumberofparti les inea h irredu ible
representation or ontinuous mass spe tra. After Coleman and Mandula proved a no-go
theorem, that stated that every nontrivial union of the Poin are group with an internal
symmetry group within the framework of ordinary Lie algebras would yield an essentially
trivialS-matrix[67℄, all these e ortsseemed to be leadingnowhere.
In1971,anewsymmetrywasfoundfromtheNeveu-S hwarz-Ramondsuperstring[68{ 72℄
that Wess and Zumino extended to quantum eld theories in four dimensions [73℄. 13
As a
novel feature, some of the generators of the symmetry algebra satisfy anti ommuting
rela-tions instead of ommutation relations. This, however, evaded theColeman-Mandula
theo-rem be ause the assumptions made in its proof onsidered only symmetry generators with
ommutationrelations. Thisnewsymmetry, alledsupersymmetry,doesnotonlyrepresenta
mathemati aloddity,butprovidedthegroundsfornontriviallyentanglingthePoin aregroup
withinternalsymmetry groups. To date, there is no dire t experimentalhint for
supersym-metry beingrealized innature butit hasmanypropertiesthat justify furtherinvestigation.
It is for example the onlyknown symmetry, that an prote t fundamental s alars, su h as
theHiggs eld,fromobtaininghugeradiative orre tions up to veryhigh energys ales(this
13
Unknownto Wess and Zumino at that time, this symmetry had already appeared in a pairof papers
publishedinthe SovietUnion. In1971, Gol'fand and Likhtmanhad extended the algebraof the Poin are
grouptoasuperalgebraandhadeven onstru tedsupersymmetri eldtheoriesinfourdimensions[74℄. The
is the so- alled \hierar hy problem") where more fundamental theories like grand unifying
theoriesorsuperstringtheory ouldsupersedethestandardmodel.
Another feature of supersymmetry is the improved renormalization evolution of the three
gauge oupling onstants of the standard model. These oupling onstants do not exa tly
meet ata ommonenergys aleifweusetherenormalizationgroupequationsobtainedfrom
thestandardmodel. Withtheaddition ofsupersymmetry,gauge ouplinguni ation anbe
a hieved in onsisten ywithphenomenologi al onstraints.
There is extensive observational eviden e for an additional omponent of the matter
den-sity in the universe that goes under the name dark matter. Dark matter parti les must
be ele tri ally neutral,otherwise they would s atter light and, thus, be dire tly observable.
Thelightestof theadditionalhypotheti alparti lesfoundinsupersymmetri models( alled
\lightest supersymmetri parti le") is apossible andidatefordark matter.
Inse tion4.1weintrodu eglobalN =1supersymmetryanddis ussbrie y
supersymme-try in thegauge se tor. We willsee that a nontrivialgaugekineti fun tionindu es several
newintera tionsinthegaugese tor. Forfuturereferen ewequotethesupersymmetri gauge
se torand thene essary supersymmetrytransformations.
In se tion4.2we brie ymotivateN =1 supergravityandwepresent thegaugese torof
N =1supergravitytogetherwiththe supergravitytransformations.
4.1 Global Supersymmetry
Supersymmetryisa symmetryrelatingbosons and fermionsand, therefore,we an make an
ansatz forin nitesimalsupersymmetrytransformationswithparameter"to behave roughly
as
Æf = "b; (4.1)
Æb = "f : (4.2)
Thesetransformationlawsareonlys hemati andbosonsarerepresentedbyb,whilef stands
for fermions. Although, equations (4.1) and (4.2) are of a rather symboli nature, we an
already drawseveral important on lusions fromthem. The rst is,that thetransformation
be ausethelefthandsideof(4.1),andthereforealsotherighthandside,hastobefermioni ,
i.e. anti ommuting. The parameter" arriesspin 1
2
insupersymmetry[75℄.
In natural units (~ = = 1) the a tion be omes dimensionless and the dimension of
massand length areinverse to one another. The derivative operatorhasthen positivemass
dimension(inverselength),i.e. [
℄=1. FromtheDira a tionforthefermionandthe
Klein-Gordona tionforthes alar wetherefore obtainthe anoni al massdimensionforfermioni
and bosoni elds in four spa etime dimensions: [f℄ = 3
2
and [b℄ = 1. The transformation
lawforbosons (4.2) wouldthenlead usto ["℄= 1
2
,whi hwouldbein onsistent with(4.1).
The simplest way to obtain an algebra linear in the elementary elds without introdu ing
new dimensionfulparametersis to assume
Æf =
"
b; (4.3)
whi htogetherwith(4.2)is onsistentwith["℄= 1
2
. Thus,already fordimensionalreasons,
transformation laws for a symmetry relating fermionsand bosons must have theform (4.1)
and (4.3), and the derivative in (4.3) an be understood as the mismat h in derivatives
betweentheDira and theKlein-Gordonequation. The lastimpli ationofthis on ernsthe
ommutator of two transformations,whi hwe an expe tto have theform
[Æ(" 1 );Æ(" 2 )℄b / (" 2 " 1 ) b (4.4)
forbosonsandequivalentlyforfermions. The ommutatoroftwosupersymmetry
transforma-tions ausesatranslationinspa etimeandthisresultisfoundinanygloballysupersymmetri
model.
Nowlet us onstru ta globallysupersymmetri modelwithgauge elds, asthisplaysan
important role in se tion 5. The Abelian ase is onvenient to begin with, and it leads to
resultsthatare straightforwardlygeneralized to thenonabelian ase.
Supersymmetry relates fermions and bosons, and, onsequently, the gauge elds ome
together with fermioni partners, so- alled gaugini 14
. A rst ansatz for a supersymmetri
gaugekineti a tionis
L gk = 1 4 F F 1 2 (4.5) 14
Thegauginiareparti lesofspin 1
2 .
wherewein orporatedthegaugino,,bymeansofakineti term. Notationsand onventions
aresummarizedinappendixB. The rsttermrepresentstheusualMaxwellLagrangian. Let
usde nethetransformation laws of the eldsina ordan ewith (4.1) and (4.3) by
Æ = " A = 1 2 "F (4.6) Æ = 1 2 " F (4.7) ÆA = 1 2 " : (4.8) Here, := 1 4 [ ;
℄ are the generators of SO(1,3) in the spinor representation. The
transformation behaviour ofthe eld strength an be reado from (4.24) to be
ÆF = " [ ℄ : (4.9)
Usingthis,the variationof theMaxwelltermin(4.5) is theneasilywritten down
Æ( 1 4 F F ) = 1 2 F " : (4.10)
The variationof these ond termof (4.5) isa little bit more involved, and relations su h as
(B.4) and (B.5) are onvenient for the relevant omputations. The variation of the se ond
termof (4.5) isfoundto be Æ( 1 2 ) = 1 2 F " i 8 " F " 5 : (4.11)
Altogether, thevariationof(4.5) gives
ÆL gk = i 8 " F " 5 : (4.12)
Observe, that (4.12) a tually vanishes,be ause after a partialintegration the variation
be- omesproportionalto" F
whi hisidenti allyzerodueto theBian hi-identity. Thus,
we haveproventhat(4.5) isinvariantunderthetransformations(4.6)and (4.8). Wearenot
nished yet be ause ounting the degrees of freedom, we nd for the fermion 4 degrees of
freedom,whiletheve tor eldonlyprovides3degreesoffreedomo -shell. On-shell,however,
thenumberofdegreesoffreedomforthegauginois2,justasfortheve tor eld. Soon-shell
thedegreesoffreedomareequalforfermionsandbosons. Tobalan ethedegreesoffreedom,
we introdu eanother real s alar eld D 15
that hasalgebrai equations of motionand, thus,
15
an be eliminated on-shell. The additional termin theLagrangian ontaining theauxiliary eld is 1 2 D 2
. Thisauxiliary eld hasto transforminto thegaugino, and the transformation
law forthefermionhasto beextended byaterm ontaining D. Note, thatD isa real eld.
The Lagrangian 16 L gk = 1 4 F F 1 2 + 1 2 D 2 (4.13)
isindeedinvariant underthe variations
Æ = 1 2 "F + i 2 5 "D (4.14) ÆD = i 2 " 5 (4.15)
and (4.8) be ause theextravariationof theDira a tionproportionalto Dpre isely an els
against thevariationof theauxiliaryLagrangian.
The a tion (4.5) an be generalized by means of a gauge kineti fun tion f(z). The
gauge kineti fun tiondependson a set of s alar elds and if then again supersymmetry is
demanded, the superpartners of these s alars must be taken into a ount, too. So let there
be s alar elds z i
and their orrespondingsuperpartners i
. In omplete analogy, one nds
thatthe Lagrangian
L matter = X i z i z i +2 i L i F i F i (4.16)
whi h onsistsof omplexs alar eldsz i
andtheir orrespondingfermioni superpartners i
.
The matterLagrangian is invariantunder thefollowingsupersymmetrytransformations
Æz i = " L i L ; (4.17) Æ i L = 1 2 " R z i + 1 2 F i " L : (4.18) ÆF i = " R i L : (4.19)
We used the hiral proje tions i L = 1 2 (1+ 5 ) i and " R = 1 2 (1 5 )". The supermultiplet
ontainingthiss alarand thisfermionisa ompaniedbya omplexauxiliary eld,F i
,that
16
TheLagrangian(4.13) anbeobtainedbysuperspa emethods,too.Superspa eisintrodu edinappendix
balan es the o -shell degrees of freedom. It is important to note that, a ording to (4.17)
and the hainrule, thegaugekineti fun tionwilltransformunder supersymmetry,i.e.,
Æf(z) =
i f(z)"
i
: (4.20)
Observethatthegaugekineti fun tionisimpli itlyspa etimedependentthroughits
depen-den eons alar elds. Atseveralstepsthatledto(4.11) weusedapartialintegration,whi h
in presen e of a nontrivial gauge kineti fun tion will produ e new terms in (4.12)
propor-tional to f(z) = i f(z) z i where i ==z i
. Observe thatespe iallythe term(4.12) will
not vanish anymore, but will ontribute with i 8 " Ref(z)" 5 F to the
supersym-metry variation. In addition to these ontributions, one has to take Æ[Ref(z)℄F
F
into
a ount, whi h has to be an elled, too. Adding ounterterms that an el these variations
and taking the variations of the ounterterms into a ount, one is led indu tivelyto an
in-variant Lagrangian after a nite number of steps. 17
The omputation is standard and will
not be repeated here but instead let us give the nal result as given in, e.g., [76,77℄. The
supersymmetri Lagrangian ontainingn
V ve tormultiplets(F A ; A ;D A ),A=1:::n V ,and
a nontrivialgauge kineti fun tionf
AB isgiven by L gk = 1 4 Ref(z) AB F A F B 1 2 Ref(z) AB A D B + 1 2 Ref(z) AB D A D B + + 1 8 Imf(z) AB " F A F B + i 4 (D Imf(z) AB ) A 5 B + + i 2 i f(z) AB i L A L D B 1 2 i f(z) AB F A i L B L 1 4 F i i f(z) AB A L B L + 1 4 i L j L i j f(z) AB A L B L +h. . (4.21)
wherewede ned the ovariant derivatives
D Imf AB = Imf AB 2A C f C(A D f B)D ; (4.22) D A = A A B C f BC A : (4.23)
The Lagrangian (4.21) is invariant under the supersymmetry transformations of the gauge
17
Notethatthesuper eldformalismasintrodu edintheappendixCleadsalsoto theresultthatwill be
