Symplecticstructureof N =1supergravity ... · iImfABFA µν F˜µν B, (1.1) where the gauge...

arX

iv0

705

4216

v2 [

hep-

th]

11

Sep

2007

KUL-TF-0711MPP-2007-61

arXiv07054216

Symplectic structure of N = 1 supergravity

with anomalies and Chern-Simons terms

Jan De Rydt 1 Jan Rosseel 1 Torsten T Schmidt 2

Antoine Van Proeyen 1 and Marco Zagermann 2

1 Instituut voor Theoretische Fysica Katholieke Universiteit LeuvenCelestijnenlaan 200D B-3001 Leuven Belgium

2 Max-Planck-Institut fur Physik Fohringer Ring 680805 Munchen Germany

Abstract

The general actions of matter-coupled N = 1 supergravity have Peccei-Quinn terms that may vio-

late gauge and supersymmetry invariance In addition N = 1 supergravity with vector multiplets

may also contain generalized Chern-Simons terms These have often been neglected in the liter-

ature despite their importance for gauge and supersymmetry invariance We clarify the interplay

of Peccei-Quinn terms generalized Chern-Simons terms and quantum anomalies in the context of

N = 1 supergravity and exhibit conditions that have to be satisfied for their mutual consistency

This extension of the previously known N = 1 matter-coupled supergravity actions follows natu-

rally from the embedding of the gauge group into the group of symplectic duality transformations

Our results regarding this extension provide the supersymmetric framework for studies of string

compactifications with axionic shift symmetries generalized Chern-Simons terms and quantum

anomalies

e-mails JanDeRydt JanRosseel AntoineVanProeyenfyskuleuvenbe

schto zagermanmppmumpgde

Contents

1 Introduction 3

2 Symplectic transformations in N = 1 supersymmetry 5

3 Kinetic terms of the vector multiplet 7

31 The action 732 Gauge and supersymmetry transformations 11

4 Chern-Simons action 12


5 Anomalies and their cancellation 14

51 The consistent anomaly 1552 The cancellation 16

6 Supergravity corrections 17

7 Specializing to Abelian times semisimple gauge groups 19

8 Conclusions 22

A Notation 24

2

1 Introduction

Matter couplings in low energy effective actions of string compactifications generically dependon scalar fields such as the moduli An important example of such a scalar field dependenceis provided by non-minimal kinetic terms for gauge fields1 (here enumerated by an index A)

eminus1L1 = minus14Re fABFA

microνFmicroν B + 14i Im fABFA

microνFmicroν B (11)

where the gauge kinetic function fAB(z) is a nontrivial function of the scalar fields zi whichin N = 1 supersymmetry has to be holomorphic The second term in (11) is often referredto as the Peccei-Quinn term

If under a gauge transformation with gauge parameter ΛA(x) some of the zi transformnontrivially this may induce a corresponding gauge transformation of fAB(z) If this trans-formation is of the form of a symmetric product of two adjoint representations of the gaugegroup

δ(Λ)fAB = ΛCδCfAB δCfAB = fCADfBD + fCB

DfAD (12)

with fCAB the structure constants of the gauge group the kinetic term (11) is obviously

gauge invariant This is what was assumed in the action of general matter-coupled super-gravity in [1]2

If one takes into account also other terms in the (quantum) effective action however amore general transformation rule for fAB(z) may be allowed

δCfAB = iCABC + fCADfBD + fCB

DfAD (13)

Here CABC is a constant real tensor symmetric in the first two indices which we willrecognize as a natural generalization in the context of symplectic duality transformations

If CABC is non-zero this leads to a non-gauge invariance of the Peccei-Quinn term inL1

δ(Λ)eminus1L1 =14iCABCΛ

CFAmicroνFmicroν B (14)

For rigid parameters ΛA = const this is just a total derivative but for local gauge pa-rameters ΛA(x) it is obviously not If (11) is part of a supersymmetric action the gaugenon-invariance (14) also induces a non-invariance of the action under supersymmetry as wewill recall in section 3

In order to understand how this broken gauge and supersymmetry invariance can berestored it is convenient to split the coefficients CABC into a sum

CABC = C(s)ABC + C

(m)ABC C

(s)ABC = C(ABC) C

(m)(ABC) = 0 (15)

where C(s)ABC is completely symmetric and C

(m)ABC denotes the part of mixed symmetry 3

Terms of the form (14) may then in principle be cancelled by the following two mechanismsor a combination thereof

1We use the non-Abelian field strength FAmicroν = FA

microν +WBmicro WC

ν fBCA where FA

microν = 2part[microWAν] is the Abelian

part The tilde denotes the Hodge dual as is further specified in the appendix2This construction of general matter-couplings has been reviewed in [2] There the possibility (13) was

already mentioned but the extra terms necessary for its consistency were not considered3This corresponds to the decomposition otimes = oplus

3

(i) As was first realized in a similar context in N = 2 supergravity in [3] (see also the sys-

tematic analysis [4]) the gauge variation due to a non-vanishing mixed part C(m)ABC 6= 0

may be cancelled by adding a generalized Chern-Simons term (GCS term) that containsa cubic and a quartic part in the vector fields

LCS = 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(16)

This term depends on a constant tensor C(CS)ABC which has also a mixed symmetry

structure The cancellation occurs provided the tensors C(m)ABC and C

(CS)ABC are the same

It has been shown in [5] that such a term exists as well in rigid N = 1 supersymmetry

(ii) If the chiral fermion spectrum is anomalous under the gauge group the anomaloustriangle diagrams lead to a non-gauge invariance of the quantum effective action of theform dABCΛ

CFAmicroνFmicroνB with a symmetric4 tensor dABC prop Tr(TA TBTC) If C(s)

ABC =dABC this quantum anomaly cancels the symmetric part of (14) This is the Green-Schwarz mechanism

As has recently been emphasized in [6] both the Green-Schwarz mechanism and the GCSterms are generically needed to cancel the anomalies in orientifold models with intersectingD-branes Moreover it is argued in [6] that non-vanishing GCS terms might have observableconsequences for certain variants of Z prime bosons On the other hand as described in [5]GCS terms may also arise in certain flux and generalized Scherk-Schwarz compactificationsFinally they also play a role in the manifestly symplectic formulation of gauged supergravitywith electric and magnetic potentials and tensor fields introduced in [7]

In view of these applications it is surprising that the full interplay between gauge in-variance and (local) supersymmetry in the presence of GCS terms and quantum anomaliesis still only partially understood In fact before the work of [5] supersymmetric GCS termswere only studied in the context of extended supersymmetry [389101112] We would liketo point out however that there is an important qualitative difference between N = 1 andN ge 2 supersymmetry In extended supersymmetry the CABC of (13) have no symmetricpart This was already pointed out in [3] for the vector multiplets in N = 2 supergravity

at least in the presence of a prepotential The equation C(s)ABC = 0 is also the basis of

the manifestly symplectic formulation [7] where it is motivated by constraints known fromN = 8 supergravity In N = 1 supergravity by contrast we find that the symmetric part ofCABC may be present and could in principle cancel quantum anomalies This is consistentwith the above-mentioned results on extended supergravity theories because only N = 1supergravity has the chiral fermions that could possibly produce these quantum anomalies

It is the purpose of this paper to give a systematic discussion of the structure of generalN = 1 supersymmetry with anomaly cancellation and GCS terms We will do this for ageneral gauge kinetic function and an arbitrary gauge group with quantum anomalies We

4More precisely the anomalies have a scheme dependence As reviewed in [6] one can choose a schemein which the anomaly is proportional to dABC Choosing a different scheme is equivalent to the choice ofanother GCS term (see item (i)) We will always work with a renormalization scheme in which the quantumanomaly is indeed proportional to dABC

4

also consider the full coupling to supergravity and discuss its embedding into the frame-work of the symplectic duality transformations This generalizes the work of [5] which wasrestricted to linear gauge kinetic functions of theories without quantum anomalies and torigid supersymmetry As far as supersymmetry is concerned the quantum anomalies of thegauge symmetries are as important as a classical violation of gauge invariance because thequantum anomalies of the gauge symmetries also lead to supersymmetry anomalies as a con-sequence of the supersymmetry algebra The consistent gauge and supersymmetry anomalieshave been found for supergravity in [13] Our result for the non-invariance of the sum of thekinetic terms and GCS terms in the classical action matches with the results of [13]

The organization of the paper is as follows In section 2 we explain how symplectictransformations act in N = 1 supersymmetry and how this leads to the generalized trans-formation (13) of the gauge kinetic function fAB

In the subsequent three sections we first consider rigid supersymmetry More concretelyin section 3 we explore the non-invariance of the kinetic terms of the vector multiplets undergauge and supersymmetry transformations caused by (13) In section 4 the GCS action andits role in the restauration of gauge and supersymmetry invariance are discussed Thirdlyin section 5 we consider the quantum anomaly as obtained in [13 14] Finally we analysethe complete cancellation of the gauge and supersymmetry anomalies by using the resultsof the two previous sections

The generalization to supergravity is considered in section 6 It turns out that theGCS terms obtained before can just be added to the general actions of matter-coupledsupergravity

To show how this works in practice it is useful to look at a gauge group that is the productof an Abelian and a semisimple group This setup was also considered in [615] and [1617]Our discussion in section 7 is close to the last reference where it is mentioned that localcounterterms turn the consistent mixed anomalies into a covariant mixed anomaly This isthe form of the anomaly that appears as variation of the vector multiplet kinetic terms TheGCS terms that we consider are precisely the counterterms that are mentioned in [17]

We finish with conclusions and remarks in section 8 and some notational issues aresummarized in the appendix

2 Symplectic transformations in N = 1 supersymmetry

In this section we derive the general form (13) of the gauge transformation of the gaugekinetic function from the viewpoint of symplectic duality transformations We begin byrecalling the essential elements of the duality transformations in four dimensions [18192021] The general form of kinetic terms for vector fields can be written in several ways5

eminus1L1 = minus14Re fABF

AmicroνF

microν B + 14i Im fABF

AmicroνF

microν B

= minus12Re

(

fABFminusAmicroν F microνminusB

)

= minus12Im

(

FminusAmicroν Gmicroνminus

A

)

(21)

5The duality transformations and hence the formulae in the first part of this section apply to theungauged action

5

where the dual field strength is defined as

Gmicroν minusA = minus2i

parteminus1L1

partFminusAmicroν

= ifABFmicroνminusB (22)

This shows that the Bianchi identities and field equations can be written as

partmicro ImFAminusmicroν = 0 Bianchi identities

partmicro ImGmicroνminusA = 0 Equations of motion (23)

The set (23) is invariant under the duality transformations

(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)

where the real matrices A B C and D satisfy

ATC minus CTA = 0 BTD minusDTB = 0 ATD minus CTB = (25)

This guarantees that S is a symplectic matrix In order to have Gprime of the form (22) thekinetic matrix fAB is transformed into f prime

AB where

if prime = (C +Dif)(A+Bif)minus1 (26)

Symmetries of the action (21) correspond to symplectic transformations with B = 0 forwhich the Lagrangian (21) transforms into itself plus a total derivative if C 6= 0

eminus1Lprime1 = minus1

2Im(F primeminusA

microν GprimemicroνminusA )

= minus12Im(FminusA

microν GmicroνminusA + FminusA

microν (CTA)ABFBmicroνminus) (27)

Not all of these rigid symmetries of the action can be promoted to gauge symmetries Forthis to be possible the field strengths FA

microν have to transform in the adjoint representationof the prospective gauge group This determines the upper line of the transformation (24)We do not know a priori the transformation rule of fAB and hence of Gmicroν A The conditions(25) however restrict further the corresponding symplectic matrices to a form which atthe infinitesimal level reads

S = minus ΛCSC SC =

(

fCBA 0

CABC minusfCAB

)

(28)

where CABC is a real undetermined tensor symmetric in its first two indices According to(26) the kinetic matrix should then transform under the gauge transformations as

δ(Λ)fAB = ΛCδCfAB δCfAB = iCABC + fCADfBD + fCB

DfAD (29)

The last two terms state that fAB transforms in the symmetric product of two adjointrepresentations The first term is the correction to this and corresponds to the possible

6

generalization by axionic shift symmetries mentioned in the introduction Note that thegauge kinetic function might now transform nontrivially also under Abelian symmetries

The algebra of gauge transformations is

[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)

In order that this algebra is realized by the symplectic transformations (28) the commuta-tors of the matrices SA should be of the form

[SASB] = fABCSC (211)

Written in full this includes the equation

CABEfCDE minus 2CAE[CfD]B

E minus 2CBE[CfD]AE = 0 (212)

which is the consistency condition that can be obtained by acting with δD on (29) andantisymmetrizing in [CD]

Whether or not the CABC can really be non-zero in a gauge theory and to what extentthis could be consistent with N = 1 supersymmetry is the subject of the remainder of thispaper

We finally note that in this section we considered only the vector kinetic terms Thesymplectic formulation gives also insight into other terms of the action which has beenexplored in [22] The additional terms to the action that we will discuss in this paper donot modify this analysis This is due to the fact that these new terms do not involve theauxiliary fields D while the analysis of [22] is essentially dependent on the terms that resultfrom the elimination of these auxiliary fields

3 Kinetic terms of the vector multiplet

Allowing for a nonvanishing shift iCABC in δCfAB breaks both the gauge and supersymme-try invariance In this section we make this statement more precise and begin our discussionwith some subtleties associated with the superspace formulation in the Wess-Zumino gauge

31 The action

The vector multiplet in the N = 1 superspace formulation is described by a real superfieldThe latter has many more components than the physical fields describing an on-shell vectormultiplet which consists of one vector field and one fermion The advantage of this redun-dancy is that one can easily construct manifestly supersymmetric actions as integrals overfull or chiral superspace As an example consider the expression

Sf =

int

d4xd2θ fAB(X)WAα W

Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV

A or a generalization thereof for the non-Abelian case where V A isthe real superfield describing the vector multiplets labelled by an index A The fAB arearbitrary holomorphic functions of a set of chiral superfields denoted by X

The integrand of (31) is itself a chiral superfield As we integrate over a chiral superspacethe Lagrangian transforms into a total derivative under supersymmetry Formally thisconclusion holds independently of the gauge symmetry properties of the functions fAB(X)For the action (31) to be gauge invariant we should have the condition [1]

δCfAB minus fCADfDB minus fADfCB

D = 0 (32)

where δC denotes the gauge transformation under the gauge symmetry related to the vectormultiplet denoted by the index C as in (29)

Due to the large number of fields in the superspace formulation the gauge parametersare not just real numbers but are themselves full chiral superfields To describe the phys-ical theory one wants to get rid of these extra gauge transformations and thereby also ofmany spurious components of the vector superfields This is done by going to the so-calledWess-Zumino gauge [23] in which these extra gauge transformations are fixed and manyspurious components of the real superfields are eliminated Unfortunately the Wess-Zuminogauge also breaks the manifest supersymmetry of the superspace formalism However acombination of this original ldquosuperspace supersymmetryrdquo and the gauge symmetries sur-vives and becomes the preserved supersymmetry after the gauge fixing The law that givesthe preserved supersymmetry as a combination of these different symmetries is called thelsquodecomposition lawrsquo see eg eq (228) in [1] Notice however that this preservation re-quires the gauge invariance of the original action (31) Thus though (31) was invariantunder the superspace supersymmetry for any choice of fAB we now need (32) for this actionto be invariant under supersymmetry after the Wess-Zumino gauge

This important consequence of the Wess-Zumino gauge can also be understood from thesupersymmetry algebra The superspace operator Qα satisfies the anticommutation relation

Qα Qdaggerα

= σmicroααpartmicro (33)

This equation shows no mixing between supersymmetry and gauge symmetries Howeverafter the Wess-Zumino gauge the right-hand side is changed to [24]

Qα Qdaggerα

= σmicroααDmicro = σmicro

αα

(

partmicro minusWAmicro δA

)

(34)

where δA denotes the gauge transformation Equation (34) implies that if an action isinvariant under supersymmetry it should also be gauge invariant

As mentioned before the preservation of the Wess-Zumino gauges implies that the effec-tive supersymmetry transformations are different from the ones in the original superspaceformulation It is shown in [24] that the resulting supersymmetry transformations of a chiral

8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL

δ(ǫ)hi = ǫR DχiL + ǫRλ

ARδAz

i (35)

where we have denoted the scalar fields of the chiral multiplets as zi the left-chiral com-ponents of the corresponding fermions as χi

L and the auxiliary fields as hi while λA is thegaugino of the vector multiplet V A These transformations are valid for any chiral multipletin particular they can be applied to the full integrand of (31) itself We will make use ofthis in section 32

Compared to the standard superspace transformations there are two modifications in(35) The first modification is that the derivatives of zi and χi

L are covariantized withrespect to gauge transformations This covariant derivative acts on the chiral fermions χi

L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)

Here the gauge variation of the chiral fermions δAχiL can be expressed in terms of the

gauge variation δAzi of the scalar fields using the fact that supersymmetry and gauge

transformations commute

δ(ǫ)δAzi = δAδ(ǫ)z

i = δAǫLχiL = ǫLδAχ

iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)

The second modification is the additional last term in the transformation of the auxiliaryfields hi The origin of this term lies in the contribution of the decomposition law for oneof the gauge symmetries contained in the chiral superfield of transformations Λ after theWess-Zumino gauge is fixed

To avoid the above-mentioned subtleties associated with the Wess-Zumino gauge wewill use component field expressions in the remainder of this text Therefore we reconsiderthe action (31) and in particular its integrand The components of this composite chiralmultiplet are [1]

z(fW 2) = minus12fABλ

ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)

λBL minus 12partifABχ

iLλ

ALλ

BL

h(fW 2) = fAB

(

minusλAL DλBR minus 12FminusA

microν FmicroνminusB + 12DADB

)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)

where we used the notation parti =partpartzi

The superspace integral in (31) means that the realpart of h(fW 2) is (proportional to) the Lagrangian

Sf =

int

d4x Reh(fW 2) (310)

9

From (39) and (310) we read off the kinetic terms of Sf

Sfkin =

int

d4x[

minus 14Re fABFA

microνFmicroνB minus 12Re fABλ

A DλB

+ 14i Im fABFA

microνFmicroνB + 14i(Dmicro Im fAB)λ

Aγ5γmicroλB]

(311)

In comparison to [1] we have used a partial integration to shift the derivative from thegaugini to (Im fAB) and rearranged the structure constants in the last term so as to obtaina ldquocovariantrdquo derivative acting on (Im fAB) More precisely we define

DmicrofAB = partmicrofAB minus 2WCmicro fC(A

DfB)D (312)

In the case that the gauge kinetic matrix transforms without a shift as in (32) the derivativedefined in (312) is fully gauge covariant

In section 2 we motivated a more general gauge transformation rule for fAB in whichaxionic shifts proportional to CABC are allowed6 as in (29) Then (312) is no longer thefull covariant derivative The full covariant derivative has instead the new form

DmicrofAB equiv partmicrofAB minusWCmicro δCfAB = DmicrofAB minus iWC

micro CABC (313)

The last term in (311) is therefore not gauge covariant for non-vanishing CABC Hence in

presence of the new term in the transformation of fAB we replace the action Sf with Sf in

which we use the full covariant derivative Dmicro instead of Dmicro More precisely we define

Sf = Sf + Sextra Sextra =

int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)

Note that we did not use any superspace expression to derive Sextra but simply added Sextra

by hand in order to fully covariantize the last term of (311) As we will further discuss inthe next section Sextra can in fact only be partially understood from superspace expressionswhich motivates our procedure to introduce it here by hand We should also stress thatthe covariantization with Sextra does not yet mean that the entire action Sf is now fully

gauge invariant The gauge and supersymmetry transformations of Sf will be discussed insection 32

We would finally like to emphasize that in the context of N = 1 supersymmetry thereis a priori no further restriction on the symmetry of CABC apart from its symmetry in thefirst two indices This however is different in extended supersymmetry as is most easilydemonstrated for N = 2 supersymmetry where the gauge kinetic matrix depends on thecomplex scalars XA of the vector multiplets These transform themselves in the adjointrepresentation which implies

δ(Λ)fAB(X) = XEΛCfECDpartDfAB(X) (315)

6We should remark here that [5] restrict their work to the case in which fAB is at most linear in scalarsand these scalars undergo a shift This is the most relevant way in which (29) can be realized

10

Hence this gives from (29)

iCABC = XEfECDpartDfAB(X)minus fCA

DfBD minus fCBDfAD (316)

which leads to CABCXAXBXC = 0 As the scalars XA are independent in rigid supersym-

metry7 this implies that C(ABC) = 0

32 Gauge and supersymmetry transformations

The action Sf is gauge invariant before the modification of the transformation of fAB In

the presence of the CABC terms the action Sf is not gauge invariant However the non-

invariance comes only from one term Indeed terms in Sf that are proportional to derivativesof fAB do not feel the constant shift δCfAB = iCABC + They are therefore automaticallygauge invariant Also the full covariant derivative (313) has no gauge transformation pro-portional to CABC and also Re fAB is invariant Hence the gauge non-invariance originatesonly from the third term in (311) We are thus left with

δ(Λ)Sf = 14iCABC

int

d4xΛCFAmicroνFmicroν B (317)

This expression vanishes for constant Λ but it spoils the local gauge invariance

We started to construct Sf as a superspace integral and as such it would automaticallybe supersymmetric However we saw that when fAB transforms with a shift as in (29) thegauge symmetry is broken which is then communicated to the supersymmetry transforma-tions by the Wess-Zumino gauge fixing The CABC tensors then express the non-invarianceof Sf under both gauge transformations and supersymmetry

To determine these supersymmetry transformations we consider the last line of (35) forzi χi hi replaced by z(fW 2) χ(fW 2) h(fW 2) and find

δ(ǫ)Sf =

int

d4xRe[

ǫRpartχL(fW2)minus ǫRγ

microWAmicro δAχL(fW

2) + ǫRλARδAz(fW

2)]

(318)

The first term in the transformation of h(fW 2) is the one that was already present in thesuperspace supersymmetry before going to Wess-Zumino gauge It is a total derivative aswe would expect from the superspace rules The other two terms are due to the mixing ofsupersymmetry with gauge symmetries They vanish if z(fW 2) is invariant under the gaugesymmetry as this implies by (37) that χ(fW 2) is also gauge invariant

7The same argument can be made for supergravity in the symplectic bases in which there is a prepotentialHowever that is not the case in all symplectic bases Bases that allow a prepotential are those were XA canbe considered as independent [2225] An analogous argument for other symplectic bases is missing This isremarkable in view of the fact that spontaneous breaking to N = 1 needs a symplectic basis that allows noprepotential [26] Hence for the N = 2 models that allow such a breaking to the N = 1 theories that we areconsidering in this paper there is also no similar argument for the absence of a totally symmetric part inCABC except that for N = 2 there are no anomalies that could cancel the corresponding gauge variationdue to the non-chiral nature of the interactions

11

Using (39) and (29) however one sees that z(fW 2) is not gauge invariant and (318)becomes using also (38)

δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)

Note that this expression contains only fields of the vector multiplets and none of the chiralmultiplets

It remains to determine the contribution of Sextra to the supersymmetry variation whichturns out to be

δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)

By combining this with (319) we obtain after some reordering

δ(ǫ)Sf =

int

d4xRe(

12CABCε

microνρσWCmicro FA

νρǫRγσλBL minus 3

2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)

In sections 4 and 5 we describe how the addition of GCS terms and quantum anomaliescan cancel the left-over gauge and supersymmetry non-invariances of equations (317) and(321)

4 Chern-Simons action

41 The action

Due to the gauged shift symmetry of fAB terms proportional to CABC remain in the gauge

and supersymmetry variation of the action Sf To re-establish the gauge symmetry andsupersymmetry invariance we need two ingredients GCS terms and quantum anomaliesThe former were in part already discussed in [3 4 5] They are of the form

SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)

The GCS terms are proportional to a tensor C(CS)ABC that is symmetric in (AB) Note that

a completely symmetric part in C(CS)ABC would drop out of SCS and we can therefore restrict

C(CS)ABC to be a tensor of mixed symmetry structure ie with

C(CS)(ABC) = 0 (42)

A priori the constants C(CS)ABC need not be the same as the CABC introduced in the

previous section For N = 2 supergravity [3] one needs them to be the same but we willfor N = 1 establish another relation between both which follows from supersymmetry andgauge invariance requirements

12

As was described in [5] the GCS terms can be obtained from a superfield expression

S primeCS = C

(CS)ABC

int

d4x d4θ[

minus23V CΩAB(V ) +

(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]

ΩAB = DαV (AWB)α + DαV

(AW αB) + V (ADαWB)α (43)

The full non-Abelian superspace expression (43) is valid only in the Wess-Zumino gaugewhere it reduces to the bosonic component expression (41) plus a fermionic term [5]

S primeCS = SCS + (S prime

CS)ferm (S primeCS)ferm =

int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)

where we used the restriction C(CS)(ABC) = 0 from (42)

Note that the fermionic term in (44) is of a form similar to Sextra in (314) More precisely

in (44) the fermions appear with the tensor C(CS)ABC which has a mixed symmetry (42) Sextra

in (314) on the other hand is proportional to the tensor C(s)ABC + C

(m)ABC From this we

see that if we identify C(m)ABC = C

(CS)ABC as we will do later we can absorb the mixed part of

Sextra into the superspace expression S primeCS This is however not possible for the symmetric

part of Sextra proportional to C(s)ABC which cannot be obtained in any obvious way from a

superspace expression As we need this symmetric part later it is more convenient to keepthe full Sextra as we did in section 3 as a part of Sf and not include (S prime

CS)ferm here Thus wewill further work with the purely bosonic SCS and omit the fermionic term that is includedin the superspace expression (43)

As an aside we will show in the remainder of this subsection that for semisimple algebrasthe GCS terms do not bring anything new [4] at least in the classical theory By this wemean they can be replaced by a redefinition of the kinetic matrix fAB This argument is notessential for the main result of this paper and the reader can thus skip this part It showshowever that the main application of GCS terms is for non-semisimple gauge algebras

We start with the result [4] that if

C(CS)ABC = 2fC(A

DZB)D (45)

for a constant real symmetric matrix ZAB the action SCS can be reabsorbed in the originalaction Sf using

f primeAB = fAB + iZAB (46)

In fact one easily checks that with the substitution (45) in (29) the C-terms are absorbedby the redefinition (46) The equation (45) can be written as

C(CS)ABC = TCAB

DEZDE TCABDE equiv 2fC(A

(DδE)B) (47)

In the case that the algebra is semisimple one can always construct a ZAB such that thisequation is valid for any C

(CS)ABC

ZAB = C2(T )minus1AB

CDTECDGHgEFC

(CS)GHF (48)

13

where gAB and C2(T )minus1 are the inverses of

gAB = fACDfBD

C C2(T )CDEF = gABTACD

GHTBGHEF (49)

These inverses exist for semisimple groups To show that (48) leads to (47) one needs(212) which leads to

gHDTH middot(

12C

(CS)C fDE

C + T[D middot C(CS)E]

)

= 0 (410)

where we have dropped doublet symmetric indices using the notation middot for contractions ofsuch double indices This further implies

gABTE middot TB middot C(CS)A = C2(T ) middot C(CS)

E (411)

with which the mentioned conclusions can easily be obtained


The GCS term SCS is not gauge invariant Even the superspace expression S primeCS is not gauge

invariant not even in the Abelian case So just as for Sf we expect that S primeCS is not

supersymmetric in the Wess-Zumino gauge despite the fact that it is a superspace integralThis is highlighted in particular by the second term in (43) which involves the structureconstants Its component expression simply gives the non-AbelianW andWandWandW correctionin (41) which as a purely bosonic object cannot be supersymmetric by itself

For the gauge variation of SCS one obtains

δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)

εmicroνρσFAmicroνW

Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]

where we used the Jacobi identity and the property C(CS)(ABC) = 0

A careful calculation finally shows that the supersymmetry variation of SCS is

δ(ǫ)SCS = minus12

int

d4x εmicroνρσ Re[

C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)

5 Anomalies and their cancellation

In this section we combine the classical non-invariances of (Sf+SCS) with the non-invariancesinduced by quantum anomalies

14

51 The consistent anomaly

The physical information of a quantum field theory is contained in the Greenrsquos functionswhich in turn are encoded in an appropriate generating functional Treating the Yang-Millsfields Wmicro as external fields the generating functional (effective action) for proper verticescan be written as a path integral over the other matter fields

eminusΓ[Wmicro] =

int

DφDφeminusS(Wmicroφφ) (51)

The gauge invarianceδAΓ[Wmicro] = 0 (52)

of the effective action encodes the Ward identities and is crucial for the renormalizability ofthe theory Even if the classical action S is gauge invariant a non-invariance of the pathintegral measure may occur and violate (52) leading to a quantum anomaly Even thoughthe functional Γ[Wmicro] is in general neither a local nor a polynomial functional of the Wmicro thequantum anomaly

δ(Λ)Γ[W ] = minusint

d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)

does have this property More explicitly for an arbitrary non-Abelian gauge group theconsistent form of the anomaly AA is given by

AA sim εmicroνρσ Tr(

TApartmicro(

WνpartρWσ +12WνWρWσ

)

)

(54)

where Wmicro = WAmicro TA and TA denotes the generators in the representation space of the chiral

fermions Similarly there are supersymmetry anomalies such that the final non-invarianceof the one-loop effective action is

A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)

This anomaly should satisfy the Wess-Zumino consistency conditions [27] which are thestatement that these variations should satisfy the symmetry algebra Eg for the gaugeanomalies these are

δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)

If the effective action is non-invariant under gauge transformations then also its supersym-metry transformation is non-vanishing As we explained in section 3 this can for examplebe seen from the algebra (34)

A full cohomological analysis of anomalies in supergravity was made by Brandt in [1314]His result (see especially (92) in [14]) is that the total anomaly should be of the form8 (55)

8This result is true up to local counterterms The latter are equivalent to a redefinition of the C(CS)ABC

This is the same as the scheme-dependence mentioned in [6] which is also equivalent to a modification ofthese GCS terms

15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F

microνAǫLγmicroλBR

+38dABCfDE

AεmicroνρσWDmicro W

Eν W

Cσ ǫLγρλ

BR

]

(58)

The coefficients dABC form a totally symmetric tensor that is not fixed by the consistencyconditions Comparison with (54) implies that they are of the form

dABC sim Tr (TA TBTC) (59)

52 The cancellation

Since the anomaly A is a local polynomial in Wmicro one might envisage a cancellation of thequantum anomaly by the classically non-gauge invariant terms in the action in the spirit ofthe Green-Schwarz mechanism

The sum of the variations of the kinetic terms (317) and (321) and of the variationsof the GCS term (412) and (413) simplifies if we set

C(CS)ABC = C

(m)ABC = CABC minus C

(s)ABC (510)

and then use the consistency condition (212) for the tensor CABC The result is

δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)

The integrand of these expressions cancel the gauge anomaly (57) and supersymmetryanomaly (58) if we set

C(s)ABC = dABC (512)

Thus if C(m)ABC = C

(CS)ABC and C

(s)ABC = dABC both gauge and supersymmetry are unbro-

ken in particular anomaly-free Note that this does not mean that any anomaly proportionalto some dABC can be cancelled by a C

(s)ABC A gauge kinetic function with an appropriate

gauge transformation induced by gauge transformations of scalar fields such that (512) holds

may simply not exist Our analysis only shows that if (512) holds and C(m)ABC = C

(CS)ABC is

satisfied the theory is gauge and supersymmetry invariant

16

6 Supergravity corrections

In this section we generalize our treatment to the full N = 1 d = 4 supergravity theoryWe check supersymmetry and gauge invariance of the supergravity action and show that noextra GCS terms (besides those already added in the rigid theory) have to be included toobtain supersymmetry or gauge invariance

The simplest way to go from rigid supersymmetry to supergravity makes use of thesuperconformal tensor calculus [28 29 30 31] A summary in this context is given in [2]Compared to the rigid theory the additional fields reside in a Weyl multiplet ie the gaugemultiplet of the superconformal algebra and a compensating multiplet The Weyl multipletcontains the vierbein the gravitino ψmicro and an auxiliary vector which will not be importantfor us The compensating multiplet enlarges the set of chiral multiplets in the theory byone The full set of fields in the chiral multiplets is now (XI ΩI HI) which denote complexscalars fermions and complex auxiliary fields respectively The physical chiral multiplets(zi χi hi) form a subset of these such that I runs over one more value than i As our finalresults depend only on the vector multiplet this addition will not be very important for usand we do not have to discuss how the physical ones are embedded in the full set of chiralmultiplets

When going from rigid supersymmetry to supergravity extra terms appear in the action(310) they are proportional to the gravitino ψmicro The integrand of (310) is replaced bythe so-called density formula which is rather simple due to the use of the superconformalcalculus [32]

Sf =

int

d4x eRe[

h(fW 2) + ψmicroRγmicroχL(fW

2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)

where e is the determinant of the vierbein For completeness we give the component ex-pression of (61) It can be found by plugging in the relations (39) where we replace thefields of the chiral multiplets with an index i by the larger set indexed by I into the densityformula (61) The result is

Sf =

int

d4x e[

Re fAB(X)(

minus14FA

microνFmicroν B minus 12λAγmicroDmicroλ

B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA

microνFmicroνB + 14i(

Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(

HI + ψmicroRγmicroΩI

L

)

λALλBL

]

+ 14partIpartJfAB(X) ΩI

LΩJLλ

ALλ

BL + hc

]

(62)

where the hat denotes full covariantization with respect to gauge and local supersymmetryeg

FAmicroν = FA

microν + ψ[microγν]λA (63)

17

Note that we use already the derivative Dmicro Im fAB(X) covariant with respect to the shift

symmetries as explained around (313) Therefore we denote this action as Sf as we didfor rigid supersymmetry

The kinetic matrix fAB is now a function of the scalars XI We thus have in the super-conformal formulation

δCfAB = partIfABδCXI = iCABC + (64)

Let us first consider the supersymmetry variation of (62) Compared with (321) thesupersymmetry variation of (62) can only get extra contributions that are proportional tothe C-tensor These extra contributions come from the variation of HI and ΩI in covariantobjects that are now also covariantized with respect to the supersymmetry transformationsand from the variation of e and λA in the gauge covariantization of the (Dmicro Im fAB)-termLet us list in more detail the parts of the action that give these extra contributions

First there is a coupling of ΩI with a gravitino and gaugini coming fromminus1

4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ

microνλBL ψ[microγν]λA + hc

]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ

microνγρǫRψmicroγνλA + + hc

]

(65)

We used the expression (63) for FAmicroν and (35) where DmicroX

I is now also covariantized with re-

spect to the supersymmetry transformations ie DmicroXI There is another coupling between

ΩI a gravitino and gaugini that we will treat separately

S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ

ργmicroψmicroRλALλ

BL + + hc

]

(66)

A third contribution comes from the variation of the auxiliary field HI in S3 where

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)

The variation is of the form

δǫHI = ǫRγ

microDmicroΩIL + = minus1

2ǫRγ

microγνDνXIψmicroR + = 1

2δCX

IWCν ǫRγ

microγνψmicroR + (68)

Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ

microγνψmicroRλALλ

BL + + hc

]

(69)

18

Finally we need to consider the variation of the vierbein e and the gaugini in a part of thecovariant derivative on Im fAB

S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(

λARγmicroλBL ǫRγ

ρψmicroL + 14ǫRγ

ργmicroγνψνLλALγmicroλ

BR

+ 14ǫRγ


BL

)

+ 14iCABCW

microCψmicroRǫRλALλ

BL + + hc

]

(610)

It requires some careful manipulations to obtain the given result for δ(ǫ)S4 One needs thevariation of the determinant of the vierbein gamma matrix identities and Fierz relations

In the end we find that δ(ǫ) (S1 + S2 + S3 + S4) = 0 This means that all extra contri-butions that were not present in the supersymmetry variation of the original supergravityaction vanish without the need of extra terms (eg generalizations of the GCS terms) Weshould also remark here that the variation of the GCS terms themselves is not influencedby the transition from rigid supersymmetry to supergravity because it depends only on thevectors WA

micro whose supersymmetry transformations have no gravitino corrections in N = 1Let us check now the gauge invariance of terms proportional to the gravitino Neither

terms involving the real part of the gauge kinetic function Re fAB nor its derivatives violatethe gauge invariance of Sf The only contributions to gauge non-invariance come from thepure imaginary parts Im fAB of the gauge kinetic function On the other hand no extraIm fAB terms appear when one goes from rigid supersymmetry to supergravity and hencethe gauge variation of Sf does not contain any gravitini This is consistent with our earlier

result that neither δ(ǫ)Sf nor SCS contain gravitiniConsequently the general N = 1 action contains just the extra terms (41) and we can

add them to the original action in [1]

7 Specializing to Abelian times semisimple gauge groups

We mentioned at the end of section 41 that simple gauge groups do not lead to non-trivialGCS terms Therefore we consider now a relevant case the product of a (one-dimensional)Abelian factor and a semisimple gauge group This will allow us to clarify the relationbetween our results and previous work in particular [16 17] In these papers the authorsstudy the structure of quantum consistency conditions ofN = 1 supergravity More preciselythey clarify the anomaly cancellation conditions (required by the quantum consistency) fora U(1) times G gauge group where G is semisimple We introduce the notations Fmicroν and Ga

microν

for the Abelian and semisimple field strengths respectivelyIn this case one can look at ldquomixedrdquo anomalies which are the ones proportional to

Tr(QTaTb) where Q is the U(1) charge operator and Ta are the generators of the semisimplealgebra Following [17 section 22] one can add counterterms such that the mixed anomalies

19

proportional to Λa cancel and one remains with those that are of the form Λ0Tr(

QGmicroν Gmicroν)

where Λ0 is the Abelian gauge parameter Schematically it looks like

Anomalies ΛaAamixed con + Λ0A0

mixed con

δ(Λ)Lct minusΛaAamixed con minus Λ0A0

mixed con

+ Λ0A0mixed cov

sum 0 + Λ0A0mixed cov

(71)

where the subscripts lsquoconrsquo and lsquocovrsquo denote the consistent and covariant anomalies respec-tively The counterterms Lct have the following form

Lct =13ZεmicroνρσCmicroTr

[

Q(


)

]

Z =1

4π2 (72)

where Cmicro and Wmicro are the gauge fields for the Abelian and semisimple gauge groups respec-tively The expressions for the anomalies are

Aamixed con = minus1

3Zεmicroνρσ Tr

[

T aQpartmicro(

CνpartρWσ +14CνWρWσ

)

]

A0mixed con = minus1

6Zεmicroνρσ Tr

[

Qpartmicro(


)

]

A0mixed cov = minus1

8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)

The remaining anomaly A0mixed cov is typically cancelled by the Green-Schwarz mechanism

We will compare this now with our results for general non-Abelian gauge groups whichwe reduce to the case Abelian times semisimple The index A is split into 0 for the U(1) anda for the semisimple group generators We expect the GCS terms (41) to be equivalent tothe counterterms in [17] and the role of the Green-Schwarz mechanism is played by a U(1)variation of the kinetic terms fab hence by a C-tensor with non-trivial components Cab0

It follows from the consistency condition (212) that

C0a0 = C00a = 0 (74)

and the Cab0rsquos are proportional to the Cartan-Killing metric in each simple factor We writehere

Cab0 = Z Tr(QTaTb) (75)

where Z could be arbitrary but our results will match the results of [17] for the value of Zin (72)

We will not allow for off-diagonal elements of the gauge kinetic function fAB

f0a = 0 rArr C0ab = 0 (76)

There may be non-zero components C000 and Cabc but we shall be concerned here only withthe mixed ones ie we have only (75) different from zero

20

If we reduce (317) using (74) and (75) we get

[

δ(Λ)Sf

]

mixed=

int

d4x[

18ZΛ0εmicroνρσ Tr (QGmicroνGρσ)

]

(77)

Splitting (75) into a totally symmetric and mixed symmetry part gives

C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)

We learned in section 52 that for a final gauge and supersymmetry invariant theory we haveto take CCS = C(m) and hence the mixed part of the GCS action (41) reads in this case

[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)

Finally we reduce the consistent anomaly (57) using dABC = C(s)ABC We find

A0 = minus16Zεmicroνρσ Tr

[

Qpartmicro(


)]

Aa = minus13ZΛaεmicroνρσ Tr

[

TaQpartmicro(


)]

(710)

where Gmicroν is the Abelian part of the gauge field Gmicroν We can make the following observations

(i) The mixed part of the GCS action (79) is indeed equal to the counterterms (72)introduced in [17]

(ii) The consistent anomalies (710) for which we based our formula on [13 14] matchthose in the first two lines of (73) As we mentioned above the counterterm hasmodified the resulting anomaly to the covariant form in the last line of (73)

(iii) We see that the variation of the kinetic term for the vector fields (77) is able to cancelthis mixed covariant anomaly (this is the Green-Schwarz mechanism)

Combining these remarks our cancellation procedure can schematically be presented asfollows


mixed con

δ(Λ)L(CS) minusΛaAamixed con minus Λ0A0

mixed con

+ Λ0A0mixed cov

δ(Λ)Sf minus Λ0A0mixed cov

sum 0 + 0

(711)

21

8 Conclusions

In this paper we have studied the consistency conditions that ensure the gauge and su-persymmetry invariance of matter coupled N = 1 supergravity theories with Peccei-Quinnterms generalized Chern-Simons terms and quantum anomalies Each of these three ingre-dients defines a constant three index tensor

(i) The gauge non-invariance of the Peccei-Quinn terms is proportional to a constantimaginary shift of the gauge kinetic function parameterized by a tensor CABC Thistensor in general splits into a completely symmetric part and a part of mixed symmetryC

(s)ABC + C

(m)ABC

(ii) Generalized Chern-Simons terms are defined by a tensor C(CS)ABC of mixed symmetry

(iii) Quantum gauge anomalies of chiral fermions are proportional to a tensor dABC whichin the appropriate regularization scheme can be chosen to be completely symmetricdABC prop Tr(TA TBTC)

We find the full quantum effective action to be gauge invariant and supersymmetric if

CABC = C(CS)ABC + dABC (81)

The inclusion of the quantum anomalies encoded in a non-trivial tensor dABC is thekey feature that distinguishes N = 1 theories from theories with extended supersymmetryBecause of their possible presence the Peccei-Quinn shift tensor CABC can now have a

nontrivial symmetric part C(s)ABC In the context of N = 2 supergravity the absence of

such a completely symmetric part can be directly proven for theories for which there existsa prepotential [3]

We performed our analysis first in rigid supersymmetry Using superconformal tech-niques we could then show that only one cancellation had to be checked to extend theresults to supergravity It turns out that the Chern-Simons term does not need any grav-itino corrections and can thus be added as such to the matter-coupled supergravity actionsOur paper provides thus an extension to the general framework of coupled chiral and vectormultiplets in N = 1 supergravity9

Our results are interesting for a number of rather different applications For example inreference [7] a general set-up for treating gauged supergravities in a manifestly symplecticframework was proposed In that work the completely symmetric part of what we call CABC

was assumed to be zero following the guideline of extended supergravity theories As we em-phasized in this paper N = 1 supergravity theories might allow for a non-vanishing C

(s)ABC

and hence a possible extension of the setup of [7] in the presence of quantum anomalies Itmight be interesting to see whether such an extension really exists

9We should emphasize that we only considered anomalies of gauge symmetries that are gauged by el-ementary vector fields The interplay with Kahler anomalies in supergravity theories can be an involvedsubject [16 17] which we did not study

22

In [6] orientifold compactifications with anomalous fermion spectra were studied inwhich the chiral anomalies are cancelled by a mixture of the Green-Schwarz mechanism andgeneralized Chern-Simons terms The analysis in [6] was mainly concerned with the gaugeinvariance of the bosonic part of the action and revealed the generic presence of a completelysymmetric and a mixed part in CABC and the generic necessity of generalized Chern-Simonsterms Our results show how such theories can be embedded into the framework of N =1 supergravity and supplements the phenomenological discussions of [6] by the fermioniccouplings in a supersymmetric setting

The work of [6] raises the general question of the possible higher-dimensional origins ofGCS terms In [5] certain flux and generalized Scherk-Schwarz compactifications [3334] areidentified as another means to generate such terms In [35] it was also shown that N = 2supergravity theories with GCS terms can be obtained by ordinary dimensional reductionof certain 5D N = 2 supergravity theories with tensor multiplets [36 37] It would beinteresting to obtain a more complete picture of the possible origins of GCS-terms in stringtheory and supergravity theories

Acknowledgments

We are grateful to Massimo Bianchi Claudio Caviezel Bernard de Wit Sergio FerraraDan Freedman Elias Kiritsis Simon Kors Henning Samtleben and Kelly Stelle for usefuldiscussions We are also grateful for the hospitality of the MPI in Munchen and of theKU Leuven during various stages of this work JR and AVP thank the Galileo GalileiInstitute for Theoretical Physics for hospitality and the INFN for partial support

This work is supported in part by the European Communityrsquos Human Potential Pro-gramme under contract MRTN-CT-2004-005104 lsquoConstituents fundamental forces and sym-metries of the universersquo JR is Aspirant FWO-Vlaanderen The work of JdR JR andAVP is supported in part by the FWO - Vlaanderen project G023505 and by the FederalOffice for Scientific Technical and Cultural Affairs through the lsquoInteruniversity AttractionPoles Programme ndash Belgian Science Policyrsquo P611-P The work of TS and MZ is supportedby the German Research Foundation (DFG) within the Emmy-Noether Programme (Grantnumber ZA 2791-2)

23

A Notation

We follow closely the notation of [2] but use real ε0123 = 1 The (anti)selfdual field strengthsare the same as there ie

Fplusmnmicroν = 1

2

(

Fmicroν plusmn Fmicroν

)

F microν = minus12ieminus1εmicroνρσFρσ (A1)

One difference is that we use indices AB for gauge indices such that α β can be usedfor 2-component spinors in superspace expressions Square brackets around indices like [AB]denote the antisymmetrization with total weight one thus for two indices it includes a factor12 for each combination

For comparison with Wess and Bagger notations the γm are

γm =

(

0 iσm

iσm 0

)

(A2)

where these sigma matrices are σm

αβor σmαβ Spinors that we use are in their 2-component

notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)

where χ = χTC = iχdaggerγ0 Further translation is obtained by replacing in Wess-Bagger

S rarr S χrarrradic2χ F rarr F

λrarr minusλ D rarr minusD Wmicro rarrWmicro

ǫrarr 12ǫ (A4)

References

[1] E Cremmer S Ferrara L Girardello and A Van Proeyen Yang-Mills theories withlocal supersymmetry Lagrangian transformation laws and superhiggs effect NuclPhys B212 (1983) 413

[2] R Kallosh L Kofman A D Linde and A Van Proeyen Superconformal symmetrysupergravity and cosmology Class Quant Grav 17 (2000) 4269ndash4338hep-th0006179 E 21 (2004) 5017

[3] B de Wit P G Lauwers and A Van Proeyen Lagrangians of N = 2 supergravity -matter systems Nucl Phys B255 (1985) 569

[4] B de Wit C M Hull and M Rocek New topological terms in gauge invariantactions Phys Lett B184 (1987) 233

[5] L Andrianopoli S Ferrara and M A Lledo Axion gauge symmetries and generalizedChern-Simons terms in N = 1 supersymmetric theories JHEP 04 (2004) 005hep-th0402142

24

[6] P Anastasopoulos M Bianchi E Dudas and E Kiritsis Anomalies anomalousU(1)rsquos and generalized Chern-Simons terms JHEP 11 (2006) 057 hep-th0605225

[7] B de Wit H Samtleben and M Trigiante Magnetic charges in local field theoryJHEP 09 (2005) 016 hep-th0507289

[8] B de Wit H Samtleben and M Trigiante On Lagrangians and gaugings of maximalsupergravities Nucl Phys B655 (2003) 93ndash126 hep-th0212239

[9] B de Wit H Samtleben and M Trigiante Gauging maximal supergravities FortschPhys 52 (2004) 489ndash496 hep-th0311225

[10] J Schon and M Weidner Gauged N = 4 supergravities JHEP 05 (2006) 034hep-th0602024

[11] J-P Derendinger P M Petropoulos and N Prezas Axionic symmetry gaugings inN = 4 supergravities and their higher-dimensional origin arXiv07050008 [hep-th]

[12] B de Wit H Samtleben and M Trigiante The maximal D = 4 supergravities JHEP06 (2007) 049 arXiv07052101 [hep-th]

[13] F Brandt Anomaly candidates and invariants of D = 4 N = 1 supergravity theoriesClass Quant Grav 11 (1994) 849ndash864 hep-th9306054

[14] F Brandt Local BRST cohomology in minimal D = 4 N = 1 supergravity AnnalsPhys 259 (1997) 253ndash312 hep-th9609192

[15] P Anastasopoulos Anomalous U(1)rsquos Chern-Simons couplings and the standardmodel hep-th0701114 To appear in the proceedings of 2nd RTN Network Workshopand Midterm Meeting Constituents Fundamental Forces and Symmetries of theUniverse Naples Italy 9-13 Oct 2006

[16] D Z Freedman and B Kors Kahler anomalies Fayet-Iliopoulos couplings and fluxvacua JHEP 11 (2006) 067 hep-th0509217

[17] H Elvang D Z Freedman and B Kors Anomaly cancellation in supergravity withFayet-Iliopoulos couplings JHEP 11 (2006) 068 hep-th0606012

[18] S Ferrara J Scherk and B Zumino Algebraic properties of extended supergravitytheories Nucl Phys B121 (1977) 393

[19] B de Wit Properties of SO(8) extended supergravity Nucl Phys B158 (1979) 189

[20] E Cremmer and B Julia The SO(8) supergravity Nucl Phys B159 (1979) 141

[21] M K Gaillard and B Zumino Duality rotations for interacting fields Nucl PhysB193 (1981) 221

25

[22] A Ceresole R DrsquoAuria S Ferrara and A Van Proeyen Duality transformations insupersymmetric YangndashMills theories coupled to supergravity Nucl Phys B444 (1995)92ndash124 hep-th9502072

[23] J Wess and B Zumino Supergauge invariant extension of quantum electrodynamicsNucl Phys B78 (1974) 1

[24] B de Wit and D Z Freedman Combined supersymmetric and gauge-invariant fieldtheories Phys Rev D12 (1975) 2286

[25] B Craps F Roose W Troost and A Van Proeyen What is special Kahlergeometry Nucl Phys B503 (1997) 565ndash613 hep-th9703082

[26] S Ferrara L Girardello and M Porrati Minimal Higgs Branch for the Breaking ofHalf of the Supersymmetries in N = 2 Supergravity Phys Lett B366 (1996) 155ndash159hep-th9510074

[27] J Wess and B Zumino Consequences of anomalous Ward identities Phys Lett B37

(1971) 95

[28] S Ferrara M Kaku P K Townsend and P van Nieuwenhuizen Gauging the gradedconformal group with unitary internal symmetries Nucl Phys B129 (1977) 125

[29] M Kaku P K Townsend and P van Nieuwenhuizen Properties of conformalsupergravity Phys Rev D17 (1978) 3179

[30] M Kaku and P K Townsend Poincare supergravity as broken superconformal gravityPhys Lett B76 (1978) 54

[31] A Van Proeyen Superconformal tensor calculus in N = 1 and N = 2 supergravity inSupersymmetry and Supergravity 1983 XIXth winter school and workshop oftheoretical physics Karpacz Poland ed B Milewski (World Scientific Singapore1983)

[32] S Ferrara and P van Nieuwenhuizen Tensor Calculus for Supergravity Phys LettB76 (1978) 404

[33] R DrsquoAuria S Ferrara F Gargiulo M Trigiante and S Vaula N = 4 supergravityLagrangian for type IIB on T 6Z2 in presence of fluxes and D3-branes JHEP 06

(2003) 045 hep-th0303049

[34] C Angelantonj S Ferrara and M Trigiante New D = 4 gauged supergravities fromN = 4 orientifolds with fluxes JHEP 10 (2003) 015 hep-th0306185

[35] M Gunaydin S McReynolds and M Zagermann The R-map and the coupling ofN = 2 tensor multiplets in 5 and 4 dimensions JHEP 01 (2006) 168 hep-th0511025

26

[36] M Gunaydin and M Zagermann The gauging of five-dimensional N = 2MaxwellndashEinstein supergravity theories coupled to tensor multiplets Nucl Phys B572

(2000) 131ndash150 hep-th9912027

[37] E Bergshoeff S Cucu T de Wit J Gheerardyn S Vandoren and A Van ProeyenN = 2 supergravity in five dimensions revisited Class Quant Grav 21 (2004)3015ndash3041 hep-th0403045 E 23 (2006) 7149

27

Introduction

Symplectic transformations in N=1 supersymmetry

Kinetic terms of the vector multiplet

The action

Gauge and supersymmetry transformations

Chern-Simons action

The action


Anomalies and their cancellation

The consistent anomaly

The cancellation

Supergravity corrections

Specializing to Abelian semisimple gauge groups

Conclusions

Notation

Contents

1 Introduction 3

2 Symplectic transformations in N = 1 supersymmetry 5

3 Kinetic terms of the vector multiplet 7


4 Chern-Simons action 12


5 Anomalies and their cancellation 14

51 The consistent anomaly 1552 The cancellation 16

6 Supergravity corrections 17

7 Specializing to Abelian times semisimple gauge groups 19

8 Conclusions 22

A Notation 24

2

1 Introduction








DfAD (12)





DfAD (13)







CABC = C(s)ABC + C

(m)ABC C

(s)ABC = C(ABC) C

(m)(ABC) = 0 (15)





microν +WBmicro WC

ν fBCA where FA




3




LCS = 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(16)














4










AmicroνF


AmicroνF

microν B

= minus12Re

(


)

= minus12Im

(


A

)

(21)


5



parteminus1L1

partFminusAmicroν






(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)




AB where




2Im(F primeminusA


= minus12Im(FminusA






(

fCBA 0

CABC minusfCAB

)

(28)



DfAD (29)


6



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

1 Introduction








DfAD (12)





DfAD (13)







CABC = C(s)ABC + C

(m)ABC C

(s)ABC = C(ABC) C

(m)(ABC) = 0 (15)





microν +WBmicro WC

ν fBCA where FA




3




LCS = 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(16)














4










AmicroνF


AmicroνF

microν B

= minus12Re

(


)

= minus12Im

(


A

)

(21)


5



parteminus1L1

partFminusAmicroν






(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)




AB where




2Im(F primeminusA


= minus12Im(FminusA






(

fCBA 0

CABC minusfCAB

)

(28)



DfAD (29)


6



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation




LCS = 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(16)














4










AmicroνF


AmicroνF

microν B

= minus12Re

(


)

= minus12Im

(


A

)

(21)


5



parteminus1L1

partFminusAmicroν






(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)




AB where




2Im(F primeminusA


= minus12Im(FminusA






(

fCBA 0

CABC minusfCAB

)

(28)



DfAD (29)


6



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation










AmicroνF


AmicroνF

microν B

= minus12Re

(


)

= minus12Im

(


A

)

(21)


5



parteminus1L1

partFminusAmicroν






(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)




AB where




2Im(F primeminusA


= minus12Im(FminusA






(

fCBA 0

CABC minusfCAB

)

(28)



DfAD (29)


6



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation



parteminus1L1

partFminusAmicroν






(

F primeminus

Gprimeminus

)

= S(

Fminus

Gminus

)

=

(

A BC D

)(

Fminus

Gminus

)

(24)




AB where




2Im(F primeminusA


= minus12Im(FminusA






(

fCBA 0

CABC minusfCAB

)

(28)



DfAD (29)


6



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation



[δ(Λ1) δ(Λ2)] = δ(ΛC3 = ΛB

2 ΛA1 fAB

C) (210)











31 The action


Sf =

int


Bβ ε

αβ + cc (31)

7

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

Here WAα = 1

4D2DαV




D = 0 (32)




Qα Qdaggerα



Qα Qdaggerα


αα

(


)

(34)



8

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

multiplet are

δ(ǫ)zi = ǫLχiL

δ(ǫ)χiL = 1

2γmicroǫRDmicroz

i + 12hiǫL


ARδAz

i (35)





L

asDmicroχ

iL = partmicroχ

iL minusWA

micro δAχiL (36)






iL (37)

This leads to

δAχi =

partδAzi

partzjχj (38)




ALλ

BL

χL(fW2) = 1

2fAB

(

12γmicroνFA

microν minus iDA)


iLλ

ALλ

BL

h(fW 2) = fAB

(



)

+ partifABχiL

(

minus12γmicroνFA

microν + iDA)

λBL

minus12partifABh

iλALλBL + 1

2part2ijfABχ

iLχ

jLλ

ALλ

BL (39)



Sf =

int

d4x Reh(fW 2) (310)

9


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


Sfkin =

int

d4x[

minus 14Re fABFA


A DλB

+ 14i Im fABFA


Aγ5γmicroλB]

(311)



DfB)D (312)




micro CABC (313)





int

d4x(

minus14iWC

micro CABC λAγ5γ

microλB)

(314)







10










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation










δ(Λ)Sf = 14iCABC

int





δ(ǫ)Sf =

int

d4xRe[



2) + ǫRλARδAz(fW

2)]

(318)



11


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


δ(ǫ)Sf =

int

d4xRe

iCABC

[

minus ǫRγmicroWC

micro

(

14γρσFA

ρσ minus 12iDA

)

λBL minus 12ǫRλ

CRλ

ALλ

BL

]

(319)



δ(ǫ)Sextra =

int

d4xRe iCABC

[

minus 12WC

micro λBLγ

micro(

12γνρFA

νρ minus iDA)

ǫR minus ǫRλBRλ

CLλ

AL

]

(320)


δ(ǫ)Sf =

int

d4xRe(

12CABCε



2iC(ABC)ǫRλ

CRλ

ALλ

BL

)

(321)



41 The action



SCS =

int

d4x 12C

(CS)ABCε

microνρσ(

13WC

micro WAν F

Bρσ +

14fDE

AWDmicro W

Eν W

Cρ W

Bσ

)

(41)




C(CS)(ABC) = 0 (42)



12


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


S primeCS = C

(CS)ABC

int

d4x d4θ[


(

fDEBV CDαV AD2

(

DαVDV E

)

+ cc)]






int

d4x(

minus14iC

(CS)ABCW

Cmicro λ

Aγ5γmicroλB

)

(44)





(m)ABC From this we









C(CS)ABC = 2fC(A

DZB)D (45)




C(CS)ABC = TCAB


(DδE)B) (47)


(CS)ABC


CDTECDGHgEFC

(CS)GHF (48)

13


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


gAB = fACDfBD


GHTBGHEF (49)


gHDTH middot(

12C

(CS)C fDE


)

= 0 (410)



E (411)







δ(Λ)SCS =int

d4x[

minus 14iC

(CS)ABCΛ

CFAmicroνF

microνB (412)

minus 18ΛC

(

2C(CS)ABDfCE

B minus C(CS)DABfCE

B + C(CS)BEDfCA

B minus C(CS)BDCfAE

B

+ C(CS)BCDfAE

B + C(CS)ABCfDE

B + 12C

(CS)ACBfDE

B)


Dρ W

Eσ

minus 18ΛC

(

C(CS)BGFfCA

B + C(CS)AGBfCF

B + C(CS)ABFfCG

B)

fDEAεmicroνρσWD

micro WEν W

Fρ W

Gσ

]



δ(ǫ)SCS = minus12

int


C(CS)ABCW

Cmicro F

Aνρ + C

(CS)A[BC

fDE]AWE

micro WCν W

Dρ

]

ǫLγσλBR (413)



14



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation



eminusΓ[Wmicro] =

int





d4xΛA

(

Dmicro

δΓ[W ]

δWmicro

)

A

equivint

d4xΛAAA (53)



TApartmicro(


)

)

(54)



A = δΓ(W ) = δ(Λ)Γ[W ] + δ(ǫ)Γ[W ] =

int

d4x(

ΛAAA + ǫAǫ

)

(55)


δ(Λ1)(

ΛA2AA

)

minus δ(Λ2)(

ΛA1 AA

)

= ΛB1 Λ

C2 fBC

AAA (56)





15

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

with

AC = minus14i[

dABCFBmicroν +

(

dABDfCEB + 3

2dABCfDE

B)

WDmicro W

Eν

]

F microνA (57)

ǫAǫ = Re[

32idABC ǫRλ

CRλ

ALλ

BL + idABCW

Cν F


+38dABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(58)



52 The cancellation



C(CS)ABC = C


(s)ABC (510)


δ(Λ)(

Sf + SCS

)

=

14i

int

d4xΛC[

C(s)ABCF

Bmicroν +

(

C(s)ABDfCE

B + 32C

(s)ABCfDE

B)

WDmicro W

Eν

]

F microνA

δ(ǫ)(

Sf + SCS

)

=int

d4x Re[

minus 32iC

(s)ABC ǫRλ

CRλ

ALλ

BL minus iC

(s)ABCW

Cν F


minus 38C

(s)ABCfDE


Eν W

Cσ ǫLγρλ

BR

]

(511)



Thus if C(m)ABC = C

(CS)ABC and C






(CS)ABC is


16





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation





Sf =

int

d4x eRe[


2) + 12ψmicroRγ

microνψνRz(fW2)]

(61)


Sf =

int

d4x e[

Re fAB(X)(

minus14FA


B + 12DADB + 1

8ψmicroγ

νρ(

FAνρ + FA

νρ

)

γmicroλB)

+ 14i Im fAB(X)FA


Dmicro Im fAB(X))

λAγ5γmicroλB

+

12partIfAB(X)

[

ΩIL

(

minus12γmicroνFA

microν + iDA)

λBL minus 12

(


L

)

λALλBL

]


LΩJLλ

ALλ

BL + hc

]

(62)


FAmicroν = FA


17







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation







4epartIfABΩ

ILγ

microνFAmicroνλ

BL

S1 =

int

d4x e[

minus 14partIfABΩ

ILγ


]

rarr δ(ǫ)S1 =

int

d4x e[

minus 18iCABCW

Cρ λ

BLγ


]

(65)





S2 =

int

d4x e[

14partIfABΩ

ILγ

microψmicroRλALλ

BL + hc

]

rarr δ(ǫ)S2 =

int

d4x e[

18iCABCW

Cρ ǫRγ


BL + + hc

]

(66)


S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

(67)


δǫHI = ǫRγ


2ǫRγ


2δCX

IWCν ǫRγ


Therefore we obtain

S3 =

int

d4x e[

minus 14partIfABH

I λALλBL + hc

]

rarr δ(ǫ)S3 =

int

d4x e[

minus 18iCABCW

Cν ǫRγ


BL + + hc

]

(69)

18


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


S4 =

int

d4x e[

14iCABCW

Cmicro λ

Aγmicroγ5λB]

rarr δ(ǫ)S4 =

int

d4x e[

minus 14iCABCW

Cρ

(




BR

+ 14ǫRγ


BL

)

+ 14iCABCW


BL + + hc

]

(610)









microν



19


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


QGmicroν Gmicroν)



mixed con


mixed con

+ Λ0A0mixed cov


(71)



[

Q(


)

]

Z =1

4π2 (72)



3Zεmicroνρσ Tr

[

T aQpartmicro(


)

]


6Zεmicroνρσ Tr

[

Qpartmicro(


)

]


8εmicroνρσ Tr

[

QGmicroνGρσ

]

(73)




C0a0 = C00a = 0 (74)





f0a = 0 rArr C0ab = 0 (76)


20


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


[

δ(Λ)Sf

]

mixed=

int

d4x[


]

(77)


C(s)ab0 = C

(s)0ab =

13Cab0 =

13Z Tr(QTaTb)

C(m)ab0 =

23Cab0 =

23Z Tr(QTaTb) C

(m)0ab = minus1

3Cab0 = minus1

3Z Tr(QTaTb) (78)


[SCS]mixed =

int

d4x[

13ZCmicroε

microνρσ Tr[

Q(


)]

]

(79)



[

Qpartmicro(


)]


[

TaQpartmicro(


)]

(710)







mixed con


mixed con

+ Λ0A0mixed cov


sum 0 + 0

(711)

21

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

8 Conclusions



(s)ABC + C

(m)ABC











(s)ABC



22



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation



Acknowledgments



23

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

A Notation


Fplusmnmicroν = 1

2

(


)




γm =

(

0 iσm

iσm 0

)

(A2)



notation

χ =

(

χα

χα

)

χ =(

χα χα

)

(A3)




ǫrarr 12ǫ (A4)

References






24

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

















25







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation







(1971) 95







(2003) 045 hep-th0303049



26


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation


(2000) 131ndash150 hep-th9912027


27

Introduction



The action


Chern-Simons action

The action




The cancellation



Conclusions

Notation

Symplecticstructureof N =1supergravity ... · iImfABFA µν F˜µν B, (1.1) where the gauge...

Documents

Transcript of Symplecticstructureof N =1supergravity ... · iImfABFA µν F˜µν B, (1.1) where the gauge...