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M PI-M IS-70/2003

A nom aly freedom in Seiberg-W itten

noncom m utative gauge theories

Friedem ann Brandta 1,C.P.M art��nb2 and F.RuizRuizc3

a M ax-Planck-Institute forM athem aticsin the Sciences,

Inselstra�e 22-26,D-04103 Leipzig,Germ any

b;c Departam ento de F��sica Te�orica I,Facultad de CienciasF��sicas

Universidad Com plutense de M adrid,28040 M adrid,Spain

W e show that noncom m utative gauge theories with arbitrary com pact

gauge group de�ned by m eansofthe Seiberg-W itten m ap have the sam e

one-loop anom alies as their com m utative counterparts. This is done in

two steps. By explicitly calculating the ��1�2�3�4 part of the renor-

m alized e�ective action, we �rst �nd the would-be one-loop anom aly

of the theory to all orders in the noncom m utativity param eter ��� .

And secondly we isolate in the would-be anom aly radiative corrections

which are not BRS trivial. This gives as the only true anom aly occur-

ring in the theory the standard Bardeen anom aly ofcom m utative space-

tim e,which is set to zero by the usualanom aly cancellation condition.

Keywords:anom aly freedom ,Seiberg-W itten m ap,noncom m utative gaugetheories.

1E-m ail:fbrandt@ m is.m pg.de

2E-m ail:carm elo@ elbereth.�s.ucm .es

3E-m ail:t63@ aeneas.�s.ucm .es

1 Introduction

It is a wellknown fact that not allrelevant gauge groups in particle physics are consistent

with the M oyalproductofnoncom m utative �eld theory. An exam ple ofthisisprovided by

theM oyalproduct A �(x)?A �(x) oftwo SU(N ) Lie algebra valued gauge�elds A �(x) and

A �(x). Itisclearthatsuch productdoesnotlie in the SU(N ) Lie algebra butin a repre-

sentation ofitsenveloping algebra,so A �(x) can notberegarded asa truly noncom m utative

SU(N ) gauge �eld. This m akes it di�cult to form ulate,even classically,noncom m utative

extensions ofsom e physically relevant gauge theories like e.g. the standard m odel. A way

to circum vent this problem [1,2]is to build noncom m utative gauge and m atter �elds from

ordinary ones by m eans ofthe Seiberg-W itten m ap [3]. Using this approach,classicalnon-

com m utative gauge theories have been constructed forarbitrary com pact groups[1,2,4,5]

and noncom m utative gauge theorieswith SU(5) and SO (10) gauge groupshave been con-

structed in ref.[13].Furtherm ore,a noncom m utative standard m odelhasbeen form ulated in

ref. [6]and som e ofits phenom enologicalconsequences have been explored in a num ber of

papers[7,8,9,10,11,12]. M any ofthese noncom m utative gauge theories,am ong them the

noncom m utativestandard m odel,involvechiralferm ions,so thecorresponding classicalgauge

sym m etry m ay be broken by quantum corrections. In other words,an anom aly m ay occur

and theresulting quantum theory m ay then becom einconsistent.To study theconsistency of

quantum noncom m utative gauge theoriesde�ned by m eansofthe Seiberg-W itten m ap,itis

therefore necessary to study whethernew typesofanom aliesoccur{i.e. anom alieswhich do

notappearin ordinarycom m utativespacetim eand hencethatm ay requireadditionalanom aly

cancellation conditions.

In refs.[14,15,16]ithasbeen shown thatforYang-M illstypegaugetheorieswith arbitrary

sem isim ple gaugegroupstheonly nontrivialsolution to theanom aly consistency condition is

the usualBardeen anom aly,regardless ofwhether or not the theory is Lorentz invariant or

renorm alizable by powercounting.Thisresultreadily appliesto gaugenoncom m utative �eld

theoriesconstructed by m eansofthe W itten-Seiberg m ap,since,asfarasthese m attersare

concerned,thepresenceofa noncom m utativem atrix param eter ��� with m assdim ension �2

only precludes Lorentz invariance and power-counting renorm alizability. Thus,for noncom -

m utativegaugetheorieswith sem isim ple gaugegroups,thereareno ���-dependentanom alies

and any ���-dependentbreaking oftheBRS identity,being cohom ologically trivial,can beset

to zero by adding appropriatecounterterm sto thee�ective action.Notethatthe addition of

these ���-dependentcounterterm sto thee�ectiveaction m akessensewithin thefram ework of

2

e�ective �eld theory,butthisagreeswith theobservation thatnoncom m utative �eld theories

de�ned by m eans ofthe Seiberg-W itten m ap should be considered ase�ective �eld theories

[13,17]. Allthe above im plies thatno anom alous ���-dependent term s should occur in the

Green functions ofnoncom m utative theories with sem isim ple gauge groups,a fact that has

been proved to hold true atorderone in ��� forthe three-pointfunction ofthe gauge �eld

and a sim plegaugegroup by explicitcom putation oftheappropriateFeynm an diagram s[18].

The situation is very di�erent ifthe gauge group is not sem isim ple. In this case, the

consistency condition for gauge anom alies has other nontrivialsolutions besides Bardeen’s

anom aly. In particular,in fourdim ensionsand ifthe gauge group is G � U(1)Y ,with G a

sem isim ple gaugegroup,theadditionalnontrivialsolutionsareoftheformZ

d4x cIinv[f��;G ��]: (1.1)

Here m atter�eldshave been integrated out, c isthe U(1)Y ghost�eld and Iinv[f��;G ��] is

a gauge invariant function ofthe U(1)Y �eld strength f�� ,the G �eld strength G �� and

their covariant derivatives. Note that there are in�nitely m any candidate anom alies ofthis

typesinceneitherpowercounting norLorentzinvarianceareavailabletoreducethenum berof

invariants Iinv[f��;G ��].Furtherm ore,when thegaugegroup containsm orethan oneabelian

factor,thereareadditionalcandidateanom aliesofyetanothertype[14,15,16].Thepurpose

ofthispaperistoinvestigatewhetheranom aliesofthesetypesoccurin noncom m utativegauge

theorieswith nonsem isim ple gauge groupsde�ned through the Seiberg-W itten m ap. Thisis

not a trivialquestion and has farreaching im plications. Indeed,did solutions oftype (1.1)

occurin perturbation theory,the corresponding quantum gauge theory would be anom alous,

theanom aly being ���-dependent.To rem ove theresulting anom aly and renderthequantum

theory consistent,one would then have to im pose constraints on the ferm ionshypercharges.

A conspicuous instance ofa m odelwith such a gauge group forwhich this point should be

cleared isthenoncom m utativestandard m odel[6].

In thispaperwewillprovethat,foranoncom m utative�eld theory with arbitrary com pact

gauge group de�ned by m eans of the Seiberg-W itten m ap, the only anom aly that occurs

at one loop (hence,to allorders in perturbation theory,ifone assum es the existence ofa

nonrenorm alization theorem for the anom aly) is the usualBardeen anom aly. The paper is

organized asfollows. In Section 2 we �x the notation,de�ne the chiralBRS transform ations

and use the Seiberg-W itten m ap to classically de�ne the noncom m utative m odel. Section 3

uses dim ensionalregularization to explicitly com pute the ��1�2�3�4 partofthe renorm alized

e�ective action. This yields a com plicated power series in the noncom m utativity param eter

3

���; of which the term of order zero is the usualBardeen anom aly of com m utative �eld

theory. In Section 4 we show thatallterm sin this series oforderone orhigherin ��� are

cohom ologically trivialwith respectto thechiralBRS operatorand �nd thecounterterm that

rem ovesthem from therenorm alized e�ective action.Section 5 containsourconclusions.W e

postpone to two appendices som e very technicalpointsofourargum ents. Letusem phasize

thatin thispaperwewillonly discussgaugeanom alies{see refs.[19,20]forrelated work on

therigid axialanom aly.

2 T he m odel,notation and conventions

Letusconsidera com pactnonsem isim plegaugegroup G = G 1 � � � � � GN ,with G i a sim ple

com pactgroup if i= 1;:::;s and an abelian group if i= s+ 1;:::;N . W e m ay assum e

withoutlossofgenerality thattheabelian factorscom ewith irreduciblerepresentations,which

ofcourse are one-dim ensional. Letusdenote by i1���is a Dirac �eld on ordinary M inkowski

spacetim e carrying an arbitrary unitary irreducible representation ofthe Lie algebra of G .

Since theabelian factorscom ewith one-dim ensionalrepresentations,theindicesin theDirac

�eld i1���is correspond to the sim ple factors. From now on we willcollectively denote the

\sim ple" indices (i1� � � is) by the m ulti-index I. The corresponding vectorpotential v� on

M inkowskispacetim e in the representation ofthe Lie algebra carried by I willhave the

following decom position in term softhe gauge �elds ak� and al� associated to the factorsof

thegroup G

v� =

sX

k= 1

gk (ak�)

a (Tk)a +

NX

l= s+ 1

glal� T

l:

Here gk and gl arethecoupling constantsand f(Tk)a;Tlg,with a = 1;:::;dim G k forevery

k = 1;:::;s and l= s+ 1;:::;N ,stand forthegeneratorsofthe G Liealgebrain theunitary

irreducible representation under consideration. Asusual,a sum over a is understood. The

m atrix elem ents IJ ofthesegeneratorsarealwaysoftheform

(Tk)aIJ = �i1j1 � � � (Tk)aikjk � � � �isjs

TlIJ = �i1j1 � � � �isjsY

l ;

where (Tk)aikjk arethem atrix elem entsofthegenerator (Tk)a oftheLiealgebra ofthefactor

G k in som e given irreducible representation. Given any two generators (Tk)aIJ and (Tk0)a0

IJ

4

asabovewede�nethetraceoperation Tr as

Tr(Tk)a (Tk0)a0

= (Tk)aIJ (Tk0)a

0

JI

= �i1j1 � � � (Tk)aikjk � � � �isjs�j1i1 � � � (Tk0)a0

jk0ik0� � � �jsis :

Theghost�eld � associated to v� ,also in therepresentation furnished by I,is

�=

sX

k= 1

gk (�k)a (Tk)a +

NX

l= s+ 1

gl�lTl;

with (�k)a and �l being the ghost�eldsforthe factorsin G . Now we considerthe theory

thatarises from chirally coupling,say left-handedly,the ferm ion �eld I to the gauge �eld

v� .Theferm ionicpartofthecorresponding classicalaction reads

Sferm ion =

Z

d4x � IiD (v)IJ J ; (2.1)

with � I = y

I 0 and

D (v)IJ J = �IJ @= J + v=IJP� J :

Here P� istheleft-handed chiralprojector,given by

P� =1

2(1� 5) 5 = �i

0 1 2 3;

the gam m a m atrices � being de�ned by f �; �g = 2��� and the convention for the

M inkowskim etric ��� being ��� = diag(+;�;�;�). This action is invariant under the

chiralBRS transform ations

sv� = @��+ [v�;�] s = ��P � s� = � �P + s�= ���: (2.2)

Asusual,theBRS operator s com m uteswith @� ,satis�estheanti-Leibnizruleand isnilpo-

tent,i.e. s2 = 0.

To construct the noncom m utative extension ofthe ordinary gauge theory de�ned by the

classicalaction Sferm ion,we use the form alism developed in refs. [1,2,4,5]. To thisend,we

�rstde�nethenoncom m utativegauge�eld V � ,thenoncom m utativespinor�eld I and the

noncom m utativeghost�eld � in term softheirordinary counterparts v� , I and � by using

theSeiberg-W itten m ap [3].Thisisdoneasfollows.The�elds V� = V� [v;�], I = I[ ;v;�]

and �= �[�;v;�]areform alpowerseriesin � ��,with coe�cientsdepending on theordinary

�elds and their derivatives,that take values in the representation ofthe enveloping algebra

5

ofthe Lie algebra ofthe group G furnished by the ordinary Dirac �eld I and solve the

Seiberg-W itten equations

s?V� = sV� s?= s s ?�= s� (2.3)

subjectto theboundary conditions

V� [v;�= 0]= v� I[v; ;�= 0]= I �[�;v;�= 0]= �: (2.4)

In eq.(2.3) s istheordinary BRS operatorofeq.(2.2),while s? denotesthenoncom m utative

BRS chiraloperator,whoseaction on thenoncom m utative�eldsisgiven by

s?V� = @��+ [V �;�]? s?= ��?P � s ?�= � �?� : (2.5)

Thecom m utator [f;g]? standsfor

[f;g]? = f ?g� g?f ;

with f ?g the M oyalproductoffunctionson M inkowskispacetim e,de�ned forarbitrary f

and g by

(f ?g)(x)=

Zd4p

(2�)4

Zd4q

(2�)4e�i(p+ q)x

e�

i

2��� p� q� ~f(p)~q(q);

~f(p) and ~g(q) being the Fouriertransform sof f and g. Forthe noncom m utative �eld I

wefurtherdem and itto belinearin I,so that

�I =

�

�IJ ��� + M [v;@; ; 5;�]�� IJ

�

�J ; (2.6)

where � and � areDiracindices.Notethat,in accordancewith theboundary condition for

I[ ;v;�],thedi�erentialoperator M [v;@; ; 5;�]�� IJ vanishesat �= 0.Taking i linear

in I,asin eq. (2.6),isalwayspossible [21]and isthe naturalchoice within the fram ework

ofnoncom m utative geom etry [17]. Once the noncom m utative �elds have been de�ned,one

considersthefollowing noncom m utative classicalaction

Sferm ionnc =

Z

d4x � I ?iD (V )IJ J ; (2.7)

where � I = �y

I 0 and

D (V )IJ J = �IJ@= J + V=IJ?P� J :

6

W e stress that the noncom m utative �elds are functions ofordinary �elds as given by the

Seiberg-W itten m ap and hence thenoncom m utative action isalso a functionalofthese.Fur-

therm ore, the noncom m utative action Sferm ionnc is invariant under the ordinary chiralBRS

transform ationsin eq.(2.2)since,by de�nition oftheSeiberg-W itten m ap,

sSferm ionnc = s?S

ferm ionnc

and,by construction,

s?Sferm ionnc = 0 :

Thee�ectiveaction �[v;�]ofthenoncom m utativetheory isform ally de�ned by

�[v;�]= �ilnZ[v;�]

Z[v;�]= N

Z

[d� ][d ]exp

�

iSferm ionnc

�

;(2.8)

with N a norm alization constantchosen so that Z[v= 0;�]= 1,i.e.

N�1 =

Z

[d� ][d ]exp

�Z

d4x � i@=

�

;

and [d� ][d ]them easureforordinary ferm ion �elds.Also form ally,theinvarianceofSferm ionnc

under s leadsto the invariance of �[v;�] underordinary gauge transform ationsof v� . The

problem isthatallthisisform alsince de�ning the e�ective action requiresrenorm alization.

Thequestion thatshould really beaddressed iswhetheritispossibleto de�nea renorm alized

e�ectiveaction �ren[v;�]invariantunder s.W erethisthecase,thetheory would beanom aly

free. In thispaperwe provide an answerin the negative and show thatthe anom aly hasthe

sam eform asfortheordinary,i.e.com m utative theory.

3 Form ofthe noncom m utative anom aly

In this section we use dim ensionalregularization to de�ne a renorm alized e�ective action

�ren[v;�] and �nd a closed expression for the anom aly s�ren[v;�] in term s ofthe noncom -

m utative �elds V� and �. To dim ensionally regularize the theory,we consider the action

Sregnc =

Z

d2!x � I ?i

�

�IJ @= J + � �V� IJ ?P� J

�

; (3.1)

�rstintroduced in thecontextofnoncom m utativegaugetheoriesin ref.[18]for U(N ) theories

and theories with sim ple groups. Here we use dim ensionalregularization �a la Breitenlohner

7

and M aison [22]. W e willuse the notation in thatreference,in which 4-dim ensionalobjects

aredenoted with bars (�g�� = 4) and evanescentor (2!�4)-dim ensionalquantitiesaredenoted

with hats (g�� = 2!� 4).Thedim ensionally regularized partition function Z reg[v;�]isde�ned

asthesum ofthedim ensionally regularized Feynm an diagram sgenerated by thepath integral

Zreg[v;�]= N

Z

[d� ][d ]exp

�

iSregnc

�

: (3.2)

In the regularized partition function we perform the change of variables �J;� �J !

�I;� �I ,with

�I =��IJ ��� + M [v;@; ; 5;�]�� IJ

� �J

� �I =��IJ ��� + �M [v;@; ; 5;�]�� IJ

�� �J

[d� ][d ]= det�I+ �M

�det

�I+ M

�[d�][d];

(3.3)

where the determ inantsare de�ned by theirdiagram m atic expansion in dim ensionalregular-

ization in powersof �.Now,in dim ensionalregularization wehave

det�I+ M

�= det

�I+ �M

�= 1 : (3.4)

To seethis,takee.g.thedeterm inant det�I+ M

�and writeitasthepartition function

det�I+ M

�=

Z

[d� ][d ]eiS[M ]

ofa ferm ion theory with classicalaction

S[M ]=

Z

d2!x �

�I+ M

� :

The propagator ofsuch a theory is the identity and the interaction vertices com e from the

operator M [v;@; ; 5;�],so theFeynm an integralsthatenterthediagram m aticexpansion of

det�I+ M

�areoftheform

Zd2!q

(2�)2!q�1 � � � q�n :

Since thisintegralvanishes in dim ensionalregularization,eq. (3.4)holds and the change of

variables(3.3)givesforthepath integralin (3.2)

Zreg[v;�]= N

Z

[d�][d]exp

�

iSregnc [

�;;V �]

�

: (3.5)

8

Hence Z reg[v;�]isa functionalof V� ,and so istheregularized e�ective action

�reg[v;�]= �ilnZ[v;�]reg = �reg[V ]: (3.6)

In otherwords,theregularized e�ective action dependson v� through V� .

Eq.(3.6)for �reg[V ]isto beunderstood in a diagram m aticsense asthegenerating func-

tionalof1PIGreen functionsforthe�eld V� .Thatisto say,

i�reg[V ]=

1X

n= 1

1

n!

Z

d2!x1:::

Z

d2!xn V�1I1J1(x1):::V�n In Jn(xn)�

�1:::�nI1J1:::In Jn

(x1;:::;xn); (3.7)

with

��1:::�nI1J1 :::In Jn

(x1;:::;xn)= hJ�1I1J1

(x1):::J�nInJn

(xn)iconn

0(3.8)

and

J�iIiJi

(xi)= ( �iJi ?� �iIi)(xi)(�

�i 5)�i�i : (3.9)

Herethesym bolhO iconn

0standsfortheconnected com ponentofthecorrelation function hO i

0

de�ned by

hO i0=

Z

[d�][d]O exp

�

i

Z

d2!x � I@= I

�

: (3.10)

Notethateqs.(3.8)and (3.10)de�ne ��1:::�nI1J1:::In Jn

(x1;:::;xn) astheresultofapplying W ick’s

theorem to J�1I1J1

(x1):::J�nIn Jn

(xn) with regard to thecontraction

h �J(y)� �I(x)i0 = �JI

Zd2!q

(2�)2!e�iq(y�x) iq=��

q2 + i0+: (3.11)

Itisnotdi�culttoseethatin eq.(3.8)thereare (n� 1)!di�erentcontractionsand that,upon

com bination with the V 0s in eq.(3.7),they allyield the sam e contribution.The regularized

e�ective action then takestheform

i�reg[V ]= �

1X

n= 1

(�1)n

n

Z

d2!x1:::

Z

d2!xn Tr[V�1(x1):::V�n(xn)]�

�1:::�n(x1;:::;xn);(3.12)

where

Tr�V�1(x1):::V�n(xn)

�= V�1I1I2(x1)V�2I2I3(x2):::V�n� 1In� 1In(xn�1 )V�n In I1(xn);

the1PIGreen function ��1:::�n(x1;:::;xn) reads

��1:::�n(x1;:::;xn)= in

Z nY

i= 1

d2!pi

(2�)2!(2�)2!�(p1 + :::+ pn)e

inP

i= 1

pixi

e�

i

2

P

1� i< j< n

��� pi� pj�

�

Zd2!q

(2�)2!

tr�(q=+ p=1)�

�1P� q=� �2 P� (q=� p=2):::

�q=�

P n�1

i= 2p=i�� �n P�

�

(q+ p1)2q2(q� p2)

2� � � (q�P n�1

i= 2pi)

2

(3.13)

9

and thesym boltr denotestraceoverDiracm atrices.Forcom pletenesswepresentvery brie y

an alternativederivation of(3.12).Integrating over [d�]and [d]in eq.( 3.5)and using eq.

(3.6),weobtain

i�reg[V ]= Trln�1+ (@=)�1 � �V�P� ?

�= �

nX

n= 1

(�1)n

nTr�(@=)�1 � �V�P� ?

�n; (3.14)

where Tr isto be interpreted as

Z

d2!x forthe continuousindicesofthe operatoron which

Tr actsand (@=)�1 hasm atrix elem ents hyj(@=)�1 jxi given by theright-hand-sideofeq.(3.11).

Clearly theright-hand sideofeq.(3.14)hasaneatdiagram m aticrepresentation which readily

leadsto eq.(3.12).

W e stress the fact that the noncom m utative �eld V�(x) in eqs. (3.12) and (3.14) is a

m ere spectatorin the sense thatthese equations hold whatever the algebra on which V�(x)

takesvaluesbe,provided theoperation Tr m akesense.Eqs.(3.12)and (3.14)arethusvalid

fornoncom m utative U(N ); sim ple,sem isim ple and non-sem isim ple gauge groups. One then

expects thatfornonsem isim ple gauge groupsa renorm alized e�ective action �ren[V ] can be

de�ned so thatthenoncom m utativegaugeanom aly hasthesam eform asfornoncom m utative

U(N ) group,i.e.such that

s?�ren[V ]= A ? ; (3.15)

with

A ? = �i

24�2

Z

d4x �

�1�2�3�4 Tr�?@�1

�

V�2 ?@�3V�4 +1

2V�2 ?V�3 ?V�4

�

: (3.16)

In therem ainderofthesection weprovethatisindeed so.

To dem onstrateeqs.(3.15)and (3.16)weproceed asfollows.Since theintegralover d2!q

in eq.(3.12)doesnotinvolveany nonplanarfactor eiq� ��� pi� ,thee�ectiveaction in eq.(3.12)

isgiven by asum overdim ensionally regularized planardiagram s.Hence,theQuantum action

principle[22]holdsforthise�ectiveaction and thefollowing equation isvalid

s?�reg[V ]= �� �reg[V ]: (3.17)

Here �� �reg[V ]istheinsertion in �reg[V ]oftheevanescentoperator � de�ned by

� = s ?Sregnc =

Z

d2!x

h� I ?�IJ ?i@=P+ J � � I ?i@=(�IJ ?P� J)

i

:

10

Substituting thisin eq.(3.17),weobtain foritsright-hand side

�� �reg[V ]= �

1X

n= 1

(�1)nZ

d2!x

Z

d2!x1:::

Z

d2!xn

� Tr��(x)V �1(x1):::V�n(xn)

���1:::�n(x;x1;:::;xn j�);

(3.18)

where

��1:::�n(x;x1;:::;xn j�)= in+ 1

Zd2!p

(2�)2!

Z nY

i= 1

d2!pi

(2�)2!ei

�px+

nP

i= 1

pixi

�

� e�

i

2

P

1� i< j< n

��� pi� pj�

(2�)2! �(p+ p1 + � � � pn)��1:::�n(p;p1;:::;pn j�);

(3.19)

and the1PIGreen function ��1:::�n(p;p1;:::;pn j�) with theinsertion reads

��1:::�n(p;p1;:::;pn j�)=

Zd2!q

(2�)2!

1

q2(q� p1)2(q� p1 � p2)

2 :::(q�P n

i= 1pi)

2

� tr�q=P+ �

�q=�

nX

i= 1

p=i

�P�

�q=� �1P� (q=� p=1)�

�2 P� (q=� p=1 � p=2):::� �nP�

�q=�

nX

i= 1

p=i�:

(3.20)

Asbefore, tr denotestrace overDirac m atrices. For n � 5 the integralin eq. (3.20)isUV

�niteby powercounting at 2! = 4.Hence,

��1:::�n(p;p1;:::;pn j�)= O (") n � 5 ; (3.21)

where "= ! � 2.Asconcerns n � 4,using theresultsin Appendix A,itisstraightforward

to com pute the contribution ��1:::�neps (p;p1;:::;pn j�) to � �1:::�n(p;p1;:::;pn j�) involving

��1�2�3�4 .Aftersom ecalculations,weobtain

��1eps(p;p1 j�)= 0

��1�2eps (p;p1;p2 j�)=1

24�2���1��2 p1�p2� + O (")

��1�2�3eps (p;p1;p2;p3 j�)= �1

2

1

24�2���1�2�3 (p1 + p2 + p3)� + O (")

��1�2�3�4eps (p;p1;p2;p3;p4 j�)= O ("):

(3.22)

Substituting eqs.(3.21)and (3.22)in eq.(3.19),and the resultso obtained in eq.(3.18),we

havethatthecontribution to theright-hand sideofeq.(3.17)which contains ��1�2�3�4 reads

�� �reg[V ]

����eps

= Areg? ; (3.23)

11

where

Areg? = �

i

24�2

Z

d2!x�

�1�2�3�4 Tr�?@�1

�

V�2 ?@�3V�4 +1

2V�2 ?V�3 ?V�4

�

+ O ("):

Hence,if�regeps[V ]denotesthecontribution totheregularized e�ectiveaction �reg[V ]involving

��1�2�3�4 ,eqs.(3.17)and (3.23)im ply

s?�regeps[V ]= �� �reg[V ]

����eps

= Areg? : (3.24)

Itisnotdi�cultto show thatthepolepartof� reg[V ]at "= 0 doesnotdepend on ��1�2�3�4 .

This,togetherwith theobservation thatany vector-like contribution to theregularized e�ec-

tive action {i.e notinvolving ��1�2�3�4 { can be regularized in a gauge invariant way within

the fram ework ofdim ensionalregularization,im plies that it is always possible to de�ne a

renorm alized e�ective action

�ren[V ]= �renvec�like[V ]+ �reneps[V ]

such that

s?�renvec�like[V ]= 0

and

s?�reneps[V ]= lim

"! 0A

reg)? = A ? ;

with A ? asin eq.(3.16).Henceeqs.(3.15)and (3.16)follow.

Using �nally that s?V� = sV� and that V� isa function of v� and ��� weconcludethat

s�ren[v;�]= A ? : (3.25)

Thisequation givesa sim pleexpression fortheanom aly ifwritten in term softhenoncom m u-

tative �elds V� and �. In fact, A Bardeennc in eq. (3.16)is nothing butthe noncom m utative

counterpart ofBardeen’s ordinary anom aly. However,in term s ofthe �elds v� and �,the

anom aly isa com plicated powerseriesin ��� with coe�cientsdepending on such �elds.The

�rstterm ofsuch seriesisthestandard Bardeen anom aly A Bardeen ofordinary spacetim e,

A ? = A Bardeen + O (�) (3.26)

ABardeen = A ?

���= 0

= �i

24�2

Z

d4x �

�1�2�3�4 Tr�@�1

�

v�2@�3v�4 +1

2v�2v�3v�4

�

: (3.27)

12

4 B R S triviality of �-dependent contributions

ThefunctionalA ? ineqs.(3.15)and(3.16)hasbeenfoundbyexplicitlycom putingtoallorders

in ��� the ��1�2�3�4 partoftheone-loop radiativecorrectionstoallthe1PIGreen functionsof

the�eld V� .Asiswellknown,only radiativecorrectionswhich arecohom ologically nontrivial

with respectto theordinary chiralBRS operator s,thatisto say,thatcan notbewritten as

the s ofsom ething,yield a trueanom alouscontribution.To �nd thetrueanom aly,we m ust

thereforeidentify in A ? thecohom ologically nontrivialcontributionswith respectto s.This

wedo next.

Ifin sections2 and 3 we takeasnoncom m utativity m atrix t���,with t a realparam eter,

we end up with a noncom m utative BRS chiraloperator s(t�)? and an anom aly A

(t�)

? whose

expressionsareobtained from thosein sections2 and 3 by replacing ��� with t���.Notethat

thedependenceon t ofs(t�)? isonly through theM oyalproduct,which now iswith respectto

t���;butthatno explicit t-dependenceisintroduced (seeref.[21]).In Appendix B weprove

thatthe logarithm ic di�erentialwith respectto t of A(t�)? is s

(t�)? trivial,orin otherwords,

thatthereexistsa functionalB[V (t�);t�]such that

td

dtA

(t�)? = s

(t�)? B

�V(t�);t�

�: (4.1)

Letusrem ark thatweusethelogarithm icderivative tddt;and nottheordinary derivative d

dtasin refs. [23,21],to be able to write everything in term softhe noncom m utativity m atrix

t��� and to avoid having to useboth ��� and t��� .Integrating eq.(4.1)over t from 0 to 1

and using that{by de�nition oftheSeiberg-W itten m ap{ s(t�)? V

(t�)� = sV

(t�)� [v;t�],wehave

Z 1

0

dtdA

(t�)?

dt=

Z 1

0

dt

ts(t�)? B

�V(t�);t�

�=

Z 1

0

dt

tsB

�V(t�)[v;t�];t�

�:

Recalling now that A(t�)? = A ? if t= 1 and A

(t�)? = A Bardeen if t= 0,and noting thatthe

ordinary BRS chiraloperator s doesnotdepend on t,weobtain

A ? = ABardeen

� s

Z 1

0

dt

tB?[V

(t�)[v;t�];t�]: (4.2)

HencethefunctionalA ? found in section 3consistsoftwocontributions:thestandard Bardeen

anom aly A Bardeen ofcom m utativespacetim e,and acontribution {given by thesecond term in

eq. (4.2){ which iscohom ologically trivialwith respectto the ordinary chiralBRS operator

s.Com paring with eq.(3.26),weconcludethatallcontributionsto A ? oforderoneorhigher

13

in ��� are cohom ologically trivial,hence harm less, since they can be absorbed by adding

�nite counterterm sto therenorm alized e�ective action.Indeed,considera new renorm alized

e�ective action �00ren[v;�]de�ned by

�0ren[v;�]= �ren[v;�]�

Z1

0

dt

tB�V(t�)[v;t�];t�

�: (4.3)

According to ourdiscussion above,itfollowsthat

s�0ren[v;�]= ABardeen

:

W ethusconcludethattheanom aly is ���-independentand hasBardeen’sform .

5 C onclusion

In thispaperwehavecalculated thechiralone-loop anom alyin 4-dim ensionalnoncom m utative

gaugetheorieswith arbitrary com pactgaugegroup de�ned through theSeiberg-W itten m ap.

Our m ain result is that for allthese theories the chiralanom aly is the sam e as for their

com m utative counterparts. Hence any noncom m utative chiralgauge theory ofthis type is

anom aly free to one-loop order if,and only if,its ordinary counterpart is. This im plies in

particularthatthe anom aly cancellation conditionsforthe noncom m utative standard m odel

[6]and thenoncom m utative SU(5) and SO (10) m odels[13]arethesam easfortheordinary

ones[27].W ewould liketoem phasizethatwehavenotfound anom alycandidatesbutactually

com puted theanom aly,sincewehavecalculated therelevantFeynm an diagram sthatproduce

theanom aly.

There isone key ingredientin ourproof,nam ely thatcounterterm swith m assdim ension

greaterthan fourshould be allowed in therenorm alized e�ective action.Thisisnecessary to

cancelradiative corrections which,on the one hand,do not satisfy the equation s�ren = 0

but, on the other, are cohom ologically trivialwith respect to s. This indicates that the

proper fram ework for these theories is the e�ective �eld theory form alism ,a proposalthat

hasalready been m ade by a num berofauthors[13,17,18]. Ifone insistson power-counting

renorm alizability,then the \safe" representations and the safe \groups" ofordinary gauge

theories[24]aretotallyunsafefornoncom m utativegaugetheories,sincetheyleadtoanom alous

theories[18].

14

A cknow ledgm ents

CPM and FRR aregratefultoCICyT,Spain forpartialsupportthrough grantNo.BFM 2002-

00950.

A A ppendix: U sefulintegrals

To obtain the ��1�2�3�4 contribution to the n-pointfunctions ��1:::�n(p;p1;:::;pn j�) with

oneevanescentinsertion � given in eqs.(3.22)thefollowing integralsareneeded:

Zd2!q

(2�)2!

q2

q2(q� q1)2(q� p2)

2= �

1

2

i

16�2+ O (")

Zd2!q

(2�)2!

q2 �q�

q2(q� q1)2(q� q2)

2= �

1

6

i

16�2(�q1 + �q2)� + O (")

Zd2!q

(2�)2!

q2 �q�1�q�2

q2(q� q1)2(q� q2)

2(q� q3)2= �

1

12

i

16�2�g�1�2 + O (")

Zd2!q

(2�)2!q2 �q�1�q�2�q�3

q2(q� q1)2(q� q2)

2(q� q3)2=

= �1

48

i

16�2

3X

i= 1

(�g�1�2 �qi�3 + �g�1�3 �qi�2 + �g�2�3 �qi�1)+ O (")

Zd2!q

(2�)2!

q2 �q2 �q�1�q�2

q2(q� q1)2(q� q2)

2(q� q3)2(q� q4)

2= �

1

16

i

16�2�g�1�2 + O ("):

Here "= ! � 2.

B A ppendix: Proofofeq. (4.1)

In whatfollowswewilluse !�� for t��� ,denotethe M oyalproductwith respectto !�� by

? and writea sm allcircle � forthelogarithm icdi�erentialwith respectto t,i.e.

!�� = t�

��?= ?!

�

F = tdF

dt: (B.1)

ThefunctionalA(t�)? ,which in thisnotation wewriteas A ?,hasa pieceoforderzero in !��

given by A Bardeen in eq.(3.27)and apiecethatcollectsallthehigherorderterm sin !�� and

15

which precisely givesthe contributionsto�

A ?. W e wantto prove eq. (4.1),which now takes

theform�

A ? = s?B : (B.2)

Using

f�? g =

1

2!��@�f?@�g (B.3)

and [23]�

V � = �i

4!��

fV� ;F�� + @�V�g?

�

�=i

4!��

f@��;V �g? ; (B.4)

thefunctional�

A ? can beexpanded asa sum

�

A ? =�

A ?;3 +�

A ?;4 +�

A ?;5 +�

A ?;6 ; (B.5)

where�

A ?;n collectsallcontributionsin�

A ? ofdegree n in the �elds � and V � (see below

fortheirexplicitexpressions). In turn,the noncom m utative chiralBRS operator s? can be

written asthesum

s? = s?;0 + s?;1 (B.6)

oftwo operators s?;0 and s?;1 whoseaction on the�elds � and V � isgiven by

s?;0V� = @�� s?;0�= 0 (B.7)

s?;1V� = [V�;�]? s?;1�= ��?� : (B.8)

Thesetwo operatorssatisfy

s2?;0 = 0 s?;0s?;1 + s?;1s?;0 = 0

and havetheim portantproperty that s?;0 preservesthedegreein the�eldsand s?;1 increases

itby one.From eqs.(B.5)and (B.6)itfollowsthatto prove eq.(B.2)itissu�cientto take

for B an expansion

B = B3 + B4 + B5 + B6

in thenum berof�eldsand show that�

A ?;3 = s?;0B3 (B.9)

�

A ?;4 � s?;1B3 = s?;0B4 (B.10)

�

A ?;5 � s?;1B4 = s?;0B5 (B.11)

�

A ?;6 � s?;1B5 = s?;0B6 (B.12)

s?;1B6 = 0 : (B.13)

16

Hence,to prove (B.2)allwe have to do is�nding functionals B3;B4;B5 and B6 satisfying

the ladder equations. To do this it is convenient to use di�erentialform s,so let us write

eqs. (B.7) and (B.8) in term s ofdi�erentialform s. Recalling that V = V�dx� and using

fdx�;s?;0g = fdx�;s?;1g = f�;dx �g = 0,wehave

s?;0V = � d� s?;0�= 0 (B.14)

s?;1V = � fV;�g?

s?;1�= ��?� : (B.15)

B .1 C om putation of B3 and B4

Taking thelogarithm icdi�erentialwith respectto t of A ? and using eqs.(B.3)and (B.4),it

isstraightforward to seethat

�

A ?;3 = �i

24�2

Zi

2!��

Tr @��?@�dV ?dV :

Itisclearthat

B3 = �i

24�2

Zi

2!��

Tr [xV� ?@�dV ?dV � (1� x)V� ?dV ?@�dV ]; (B.16)

with x an arbitrary param eter,solveseq.(B.9).Indeed,acting with s?;0 on B3,integrating

by partsthe derivative @� in the second term in eq. (B.16)and neglecting the integralofa

divergence,werecover�

A ?;3.Notethateq.(B.16)providesaone-param eterfam ilyofsolutions

for B3.Furtherm ore,to B3 wecan also add a term

Z

!��Tr @�V� ?dV ?dV

with arbitrary coe�cient,since s ?;0 acting on itvanishes.

Letusm ove now on to eq. (B.10). W e �rstcalculate�

A ?;4 and s?;1B3. Acting with t ddt

on A ?,noting eqs.(B.3)and (B.4),retaining term soforderfourin the�elds,using thecyclic

property ofthetrace Tr and oftheintegralofa M oyalproductoffunctionstopush theghost

�eld � to the farleft,and integrating by partswhateverpartialand/orexteriorderivatives

17

acton �,weobtain aftersom elengthy algebra that

�

A ?;4 = �i

24�2

Zi

4!��Tr �?

h

V� ?dV ?@�dV � V� ?@�dV ?dV + V ?@�V ?@�dV

� V ?@�dV ?@�V � @�V� ?dV ?dV � dV ?dV ?@�V� � 2@�V ?dV� ?dV

+ 2dV� ?dV� ?dV � 2dV� ?@�V ?dV � 2dV ?@aV ?dV� + 2dV ?dV� ?dV�

� 2dV ?dV� ?@�V + dV ?@�V ?@�V � @�V ?@�V ?dV � @�V ?dV ?@�V

� @�dV ?dV ?V� + 2@�dV ?V� ?dV + dV ?@�dV ?V� � 2dV ?V� ?@�dV

+ @�dV ?@�V ?V � @�V ?@�dV ?V + @�dV ?V ?@�V + @�V ?V ?@�dV

i

Proceeding sim ilarly for s?;1B3,and taking forsim plicity x = 1,wehave

s?;1B3 =i

24�2

Zi

2!��Tr�?

h

V� ?@�dV ?dV + V ?dV� ?@�dV

+ V ?dV ?@�dV� + V ?@�dV ?dV� + dV� ?@�dV ?V

+ dV ?@�dV� ?V + @�dV ?dV ?V� + @�dV ?dV� ?V

i

To sim plify theseexpressionsweintroducethenotation

Ai= �

i

24�2

Zi

4!��

Tr �?ai�� ;

with ai�� asin Table 1. Note that !��ai�� isa 4-form with one explicit !�� ,three explicit

derivativesand threenoncom m utativegauge�elds.By \explicit" herewem ean ! 0s and @ 0s

thatare nothidden in the ?-product. In Table 1 we have listed allsuch form sthatcan be

constructed.W ith thisnotation�

A ?;4 � s?;1B reads

�

A ?;4 � s?;1B = A1� A

2 + A3 + A

4� A

5 + A6 + A

7� A

8 + A9� A

10� A

12

� 2A 13� 2A 15

� 2A 16� 2A 18 + A

19 + A20� 2A 21 + 2A 22 + A

23 (B.17)

+ A24 + 2A 26 + 2A 27 + 2A 28 + 2A 29 + 2A 32 + 2A 36 + 2A 37 + 2A 39

:

Now,notallthe4-form s w ��ai�� in Table1 arelinearly independent.To seethis,consider

e.g.the5-form � = !��@�V ?dV ?dV and acton itwith theinnercontraction

i� � i@� =@

@(dx�):

18

a1�� dV ?@�V ?@�V a14�� @�V ?dV ?dV� a27�� V ?dV� ?@�dV

a2�� @�V ?dV ?@�V a15�� @�V ?dV� ?dV a28�� @�dV ?dV� ?V

a3�� @�V ?@�V ?dV a16�� dV ?dV� ?@�V a29�� dV� ?@�dV ?V

a4�� @�dV ?V ?@�V a17�� dV� ?dV ?@�V a30�� dV� ?V ?@�dV

a5�� V ?@�dV ?@�V a18�� dV� ?@�V ?dV a31�� @�dV� ?dV ?V

a6�� V ?@�V ?@�dV a19�� @�dV ?dV ?V� a32�� dV ?@�dV� ?V

a7�� @�dV ?@�V ?V a20�� dV ?@�dV ?V� a33�� dV ?V ?@�dV�

a8�� @�V ?@�dV ?V a21�� dV ?V� ?@�dV a34�� @�dV� ?V ?dV

a9�� @�V ?V ?@�dV a22�� @�dV ?V� ?dV a35�� V ?@�dV� ?dV

a10�� dV ?dV ?@�V� a23�� V� ?@�dV ?dV a36�� V ?dV ?@�dV�

a11�� dV ?@�V� ?dV a24�� V� ?dV ?@�dV a37�� dV ?dV� ?dV�

a12�� @�V� ?dV ?dV a25�� @�dV ?V ?dV� a38�� dV� ?dV ?dV�

a13�� dV ?@�V ?dV� a26�� V ?@�dV ?dV� a39�� dV� ?dV� ?dV

Table 1:All4-form swith three derivativesand three gauge �elds.

Being a 5-form in fourdim ensions, � isidentically zero,and so is i� acting on it.Hence

0= i� (!��@�V ?dV ?dV )

= !���@�V� ?dV ?dV � @�V ?(@�V � dV�)?dV � @�V ?dV ?(@�V � dV�)

�

= !���a12�� � a

3�� + a

15�� � a

2�� + a

14��

�;

which im pliestherelation

A12� A

3 + A15� A

2 + A14 = 0 :

Thissuggeststhat,to generateallthelinearrelationsam ong thefunctionals A i;itisenough

toactwith i� on allthe5-form s � with oneexplicit !�� ,threeexplicitderivativesand three

noncom m utative gauge �elds. In listing the form s � ,two restrictions should be observed.

The �rst one is that it is only necessary to consider 5-form s � with at m ost two explicit

19

derivatives acting on the sam e �eld,since in Table 1 there is no ai�� with m ore than two

explicit derivatives on the sam e gauge �eld. The second one is that whenever two explicit

derivatives act on the sam e gauge �eld,they should not be both exterior derivatives. The

reason forthisisthat5-form s � containing an explicit d2 do notprovide,upon acting on

them with i� ,any relation am ong the !��ai��:Indeed,since i�d2 = @�d� d@� ,theaction of

i� on a 5-form containing an explicit d2 yieldsa linearcom bination

i� (5�form with d2)� = 4�form swith d2 + 4�form with @�d� d@�

of4-form s each ofwhich is identically zero. There are twelve di�erent form s � that can

be constructed satisfying these restrictionson the derivatives,nam ely � = !��~a� ,with ~a�

given by

@�V ?dV ?dV dV� ?dV ?dV @�dV ?dV ?V @�dV ?V ?dV

dV ?@�V ?dV dV ?dV� ?dV dV ?@�dV ?V V ?@�dV ?dV

dV ?dV ?@�V dV ?dV ?dV� dV ?V ?@�dV V ?dV ?@�dV :

Ifweactwith i� on thesetwelve 5-form s,weobtain thelinearrelations

A2 + A

3� A

12� A

14� A

15 = 0 A7 + A

19� A

28� A

31 = 0

A1 + A

3� A

11� A

13� A

18 = 0 A8� A

20� A

29 + A32 = 0

A1 + A

2� A

10� A

16� A

17 = 0 A9 + A

21� A

30� A

33 = 0

A12 + A

17 + A18� A

38� A

39 = 0 A4� A

22� A

25 + A34 = 0

A11 + A

15 + A16� A

37� A

39 = 0 A5 + A

23� A

26� A

35 = 0

A10 + A

13 + A14� A

37� A

38 = 0 A6� A

24� A

27 + A36 = 0 :

Solving thissystem ofequationsfor A i (i= 1;:::;12) and substituting the solution in eq.

(B.17),wewrite�

A ?;4� s?;1B3 in term softhefunctionals Ai (i= 13;:::;39),theresultbeing

�

A ?;4 � s?;1B3 = A13 + A

14� 4A 15

� 4A 16 + A17 + A

18� 3A 21 + 3A 22

+ 2A 23 + 2A 24 + A25 + A

26 + 3A 27 + 3A 28 + A29 + A

30 (B.18)

+ A31 + 3A 32 + A

33� A

34� A

35 + A36 + 2A 37

� 3A 38 + 2A 39:

W e have thus obtained the left-hand side ofeq. (B.10)in term s oflinearly independent

functionals A i (i= 13;:::;39),each ofwhich hasoneexplicit !�� and threeexplicitderiva-

tives and has degree three in the noncom m utative gauge �eld. It then follows that,for eq.

20

(B.10)tohaveasolution,B4 ontheright-handsidem ustbealinearcom bination offunctionals

Br = �

i

24�2

Zi

4!��

Trbr�� ; (B.19)

with br�� a 4-form ofordertwo in explicitderivativesand fourin thenoncom m utative gauge

�eld.W ith som epatience,itcan beseen thatthereareforty such functionals B r whose s?;0

variation isnotzero.Thirty ofthem can bewritten aslinearcom binationsofthefunctionals

b1�� V� ?V� ?dV ?dV b6�� @�V� ?V ?dV ?V

b2�� dV� ?V� ?dV ?V b7�� @�V� ?dV ?V ?V

b3�� dV ?dV� ?V ?V� b8�� V� ?dV ?@�V ?V

b4�� dV� ?V� ?V ?dV b9�� V� ?@�V ?V ?dV

b5�� @�V� ?V ?V ?dV b10�� V ?dV ?@�V ?V�

Table 2:All4-form swith two derivativesand fourgauge �elds.

B r whose br�� arecollected in Table2.To illustratethatthisisindeed so,letusconsideras

an exam ple

B = �i

24�2

Zi

4!��

Tr b�� b�� = V ?V� ?@�V ?dV :

Clearly,this b�� in notin Table2.However,using that

(a) both Tr and theintegralofa M oyalproductoffunctionsarecyclic,

(b) that @� = fi�;dg,and

(c) that i�(dV )?dV ?V ?V� = �dV ?i�(dV ?V ?V�);

and integrating by partsand neglecting totalderivatives,wehave

B(a;b)=

i

24�2

Zi

4!��

Tr (i�d+ di�)V ?dV ?V ?V�

(c;d)= �

i

24�2

Zi

4!��

Tr [dV ?i� (dV ?V ?V�)+ i�V ?d(dV ?V ?V�)]

(a)= �

i

24�2

Zi

4!��

Tr (b8�� + b3�� + b

2�� )

= B8 + B

3 + B2:

Sim ilarly,any otherfunctional B whose b�� isnotin Table 2 can be expressed asa linear

com bination offunctionals B r with br�� in Table2.Itthen followsthatitisenough to write

21

for B4

B4 =

10X

r= 1

crBr: (B.20)

To solveeq.(B.10)weneed the s?;0 variation of B4.Acting with s?;0 on (B.20)and writing

theresultin term softhelinearlyindependentfunctionals A i,correspondingto i= 13;:::;39,

weobtain

s?;0B4 = (�c1 � c2 + c5 + c6 + 2c9)A13

+ (�c1 � c4 + c5 + c6 + c8 + 3c9)A14

+ (�2c3 � c5 + c7 + c8 + c10)(A15 + A

16)

+ (c1 � 2c2 � c4 � c6 � c7 + 2c8 � c9 + c10)A17

+ (c1 � c2 � 2c4 � c6 � c7 + c8 + c10)A18

+ (c1 + c8 � c9)(A19 + A

20)

+ (c3 � c8)(�A21 + A

22 + A27 + A

28)

+ (c1 � c2 � c4)A23

+ (c1 � c2 � c4 + c8 + c10)A24

� c2(A25 + A

29)

+ (�c4 + c9 + c10)(A26 + A

30)

+ (c2 + c7 � c8)A31

+ (c3 + c5)A32

+ (c2 � c6 � c9)A33

+ (c4 + c6 + c10)A34

+ (c3 � c7 � c8 � c10)A35

+ (c4 � c5 � c9)A36

+ (c1 + c3 � c4 � c6 � c7 � c9)A37

+ (c2 � c3 + c4 � c5 + c7 � c9)A38

+ (�c1 + c34+ c4 + c5 + c6 + c8 � c9 + c10)A39: (B.21)

Substituting now eqs. (B.18) and (B.21) in eq. (B.10) and equating the coe�cients of A i

(i= 13;:::;39) on both sides,weobtain a system of21 equationswith unknowns c1;:::;c10:

22

Itssolution is

c1 = y+ z c2 = �1 c3 = 3� z c4 = �1+ y+ z c5 = z

c6 = �y c7 = 2� z c8 = �z c9 = y c10 = z ;

where y and z arearbitrary param eters.Thisprovidesatwo-param eterfam ily offunctionals

B4 forwhich eq.(B.10)holds.Notethatifwetake y = z = 0,then B4 only hasfourterm s.

B .2 C alculation of B5 and B6

Onem ay proceed analogously asfor B4 and explicitly com pute B5 and B6.Here,instead,we

presentan alternative m ethod which usescohom ologicaltechniques. To apply them we shall

em ploy the approach ofref. [23]which introduces gauge �elds vA� and ghost�elds �A not

only forthe Lie algebra g ofthe gauge group G butalso forthe whole enveloping algebra

U = fTAg = fT�a;Tig in which V� and � takevalues.Heretheindex �a runsovertheelem ents

of g,so thatin thenotation ofsection 2 onehas fT�ag = f(Tk)a;Tlg,whiletheindex i runs

overthe com plem entary elem entsof U . Asshown in ref. [23],the standard Seiberg-W itten

m ap can beextended to include U -valued �elds v� and � satisfying

svA� = @��

A + fB CAvB� �

C (B.22)

s�A =

1

2�B�CfC B

A; (B.23)

with fA BC the structure constants ofthe Lie algebra U ,given by [TA;TB ]= fA B

CTC . Of

course,g being a subalgebra ofU m eans f�a�bi= 0 and im pliesthattheBRS transform ations

abovearesubjectto thetruncation conditions

svA�

���vi� = �

i= 0

=

(@��

�a + f�b�c�av�b��

�c ifA = �a

f�b�civ�b��

�c = 0 ifA = i(B.24)

s�A

���vi� = �

i= 0

=

(1

2��b��cf�b�c

�aifA = �a

1

2��b��cf�c�b

i= 0 ifA = i:

(B.25)

Theextended Seiberg-W itten m ap isde�ned by dem anding

s?VA� = sV

A� s?�

A = s�A; (B.26)

subjectto theusualboundary conditionsand with s? de�ned by

s?VA� = @��

A + fB CAVB� �

Cs?�

A =1

2�B �C

fC BA: (B.27)

23

By setting in itall�elds vi� and �i to zero,the standard Seiberg-W itten m ap isrecovered.

Furtherm ore,the truncation conditionsim ply thatallform ulasthathold for U-valued �elds

vA� and �A willalso hold for g-valued �elds v�a� and ��a ,and in particulareq.(B.2)thatwe

want to prove. The idea is then to dem onstrate eq. (B.2)forthe extended Seiberg-W itten

m ap.

W estartfrom thefactthat A ? satis�estheanom alyconsistency condition s?A ? = 0 which

followsfrom eq.(B.2)becauseofs2? = 0.In term softhecom m utative�elds vA� and �A ,one

has sA ? = 0.Thisim plies s�

A ? = 0 since s com m uteswith the logarithm ic derivative with

respectto t.Using (B.26)again,oneconcludes s?�

A ? = 0 which decom posesinto

s?;0

�

A ?;3 = 0 (B.28)

s?;0

�

A ?;4 + s?;1

�

A ?;3 = 0 (B.29)

s?;0

�

A ?;5 + s?;1

�

A ?;4 = 0 (B.30)

s?;0

�

A ?;6 + s?;1

�

A ?;5 = 0 (B.31)

s?;1

�

A ?;6 = 0 : (B.32)

In theprevioussubsection wehaveshown byexplicitcom putation that(B.28)and (B.29)im ply�

A ?;3 = s?;0B3 and�

A ?;4 = s?;0B4 + s?;1B3. W e shallnow show by cohom ologicalm eansthat

therem aining equationsim ply�

A ?;5= s?;0B5 + s?;1B4,�

A ?;6= s?;0B6 + s?;1B5 and s?;1B6 = 0,

which willcom pletetheproofofequations(B.9)to (B.13).

Tothatend we�rstderivearesulton thecohom ology ofs?;0 in thespace F ? ofintegrated

?-polynom ialsin the�elds V A� ,�A and theirderivatives.An elem entofthisspaceisa linear

com bination,with coe�cientsthatm ay depend on ! �� ,ofterm softheform

Z

d4x a1 ?a2 ?:::?an;

with n �niteand each ai oneofourbasicvariables(VA� ,�A and theirderivatives),

ai2 fVA� ;�

A;@�V

A� ;@��

A;@�@�V

A� ;@�@��

A;:::g :

It is obvious why this cohom ology is relevant to the present case. Using the result�

A ?;4 =

s?;0B4 + s?;1B3 from Subappendix B.1 in eq. (B.30) and noting that s2?;1 = 0,we obtain

24

s?;0(�

A ?;5 � s?;1B4)= 0,with�

A ?;5 � s?;1B4 obviously in F ?. Ouraim is to show thatthis

im plies�

A ?;5 � s?;1B4 = s?;0B5 for som e B5 2 F ?,or in other words that�

A ?;5 � s?;1B4 is

trivialin the s?;0-cohom ology in F ?.Assum ethatwehaveshown this.Inserting theresultin

(B.31)and proceeding sim ilarly yields s?;0(�

A ?;6 � s?;1B5)= 0.Again,we wantto show that�

A ?;6 � s?;1B5 = s?;0B6 for som e B6 2 F ? and thus that�

A ?;6 � s?;1B5 is also trivialin the

s?;0-cohom ology in F ?.Notethat(B.9)and (B.10)actually expressanalogousresults,nam ely

the triviality of�

A ?;3 and�

A ?;4 � s?;1B3 in the sam e cohom ology.However,asitwillbecom e

clearbelow,they cannotbe proved by m eansofthe resulton the cohom ology for s?;0 in F ?

thatwederivein thesequeland thereforehaveto beshown by otherm ethods.

To exam ine the s?;0-cohom ology in F ? we adaptm ethods developed in ref. [28]forthe

com putation ofthe cohom ology of s0. W e �rst derive a result on the s?;0-cohom ology in

the space P? ofnon-integrated ?-polynom ials. Forthatpurpose we introduce the following

variables u‘,v‘ and w i:

fu‘g= fV

A� ;@(�V

A�);:::;@(�1 :::@�kV

A�k+ 1)

;:::g (B.33)

fv‘g= fs?;0u

‘g = f@��

A;@(�@�)�

A;:::;@(�1 :::@�k+ 1)

�A;:::g (B.34)

fwig= f�A

;@[�VA�];:::;@�1 :::@�k@[�V

A�];:::g : (B.35)

Evidently every ?-polynom ialin the�elds V A� ,�A and theirderivativescan beexpressed as

a ?-polynom ialin thevariables u‘,v‘,w i and viceversa4.On non-integrated ?-m onom ials

in u‘,v‘,w i wede�netheoperation % through

%(a1 ?a2 ?:::?an)=

=1

n

�

u‘ @a1

@v‘

�

?a2 ?:::?an

+1

n

n�1X

i= 2

(�)ja1j+ ja2j+ :::+ jai� 1ja1 ?:::?ai�1 ?

�

u‘ @ai

@v‘

�

?ai+ 1 ?:::?an

+1

n(�)ja1j+ ja2j+ :::+ jan� 1ja1 ?a2 ?:::?an�1 ?

�

u‘ @an

@v‘

�

;

where ai isany ofthevariables u‘,v‘,w i,

ai2 fu‘;v

‘;w

ig ;

4The set of w ’s is actually overcom plete because the w ’s are not alllinearly independent owing to the

identities @[�@�V�]= 0 and theirderivatives.Howeverthisdoesnotm atterto ourargum ents.

25

and jaij is the Grassm ann parity of ai,which is 0 for V A� and its derivatives,and 1 for

the �A and its derivatives. Extending the de�nition of % by linearity from ?-m onom ials

to ?-polynom ials,we have thatthe anticom m utatorof s?;0 and % evaluated on an arbitrary

?-polynom ialp?(u;v;w)2 P? givesthedi�erence

fs?;0;%gp?(u;v;w)= p?(u;v;w)� p?(0;0;w); (B.36)

where p?(0;0;w) denotes the ?-polynom ialthat arises from p?(u;v;w) by setting to zero

all u‘ and v‘ before evaluating the star-products{forexam ple,for p? = V A� ?V B

� one has

p?(0;0;w)= 0.Applyingnow eq.(B.36)toan s?;0-closed ?-polynom ial,i.e.toa p? satisfying

s?;0p? = 0,and using thatallw i are s?;0-closed,weobtain

s?;0p?(u;v;w)= 0 , p?(u;v;w)= p?(0;0;w)+ s?;0%p?(u;v;w): (B.37)

In particular,an s?;0-closed ?-polynom ialp?(u;v;w) with p?(0;0;w)= 0 isthe s?;0-variation

ofthestar-polynom ial%p?(u;v;w).

Result(B.37)cannotbeused directlyforourpurposessinceitappliesonlyto ?-polynom ials

butnotto integrated ?-polynom ials,which iswhatwe had initially. Thism akesa di�erence

because,by de�nition,an integrated ?-polynom ialis s?;0-closed when the s?;0-transform ation

ofitsintegrand isa totaldivergence:

s?;0f? = 0 with f? =

Z

d4xp? , s?;0p? = @�!

� forsom e !�:

Since % doesnotcom m utewith @� wecannotdirectly apply theresultaboveto thiscase.To

escape thisproblem we consider the variationalderivatives ofthe equation s?;0f? = 0 with

respectto V A� and �A .Thisyields

s?;0f? = 0; f? 2 F ? ) s?;0�f?

�V A�

= 0; s?;0�f?

��A+ @�

�f?

�V A�

= 0 : (B.38)

Itcan bereadilychecked thatthevariationalderivativeofanyelem ent f? 2 F ? with respectto

V A� or �A isa ?-polynom ialin P?.Supposenow that �f?=�V

A� vanishesat u‘ = v‘ = 0 in

thesenseexplained above.Using the�rstequation in (B.38)and eq.(B.37)wethen conclude

that �f?=�VA� isthe s?;0-variation of %(�f?=�V

A� ):

��f?

�V A�

�

(0;0;w)= 0 )�f?

�V A�

= s?;0%�f?

�V A�

:

26

Using thisin thesecond equation in (B.38)weobtain

s?;0

��f?

��A+ @� %

�f?

�V A�

�

= 0: (B.39)

Applying (B.37)once again we conclude thatthe term in parenthesesis s?;0%(:::) provided

it vanishes at u‘ = v‘ = 0 in the sense above. Note that here %(:::) has ghost num ber

gh(f?)� 2,with gh(f?) the ghostnum berof f? and gh(V )= 0 and gh(�)= 1. Since ?-

polynom ials p?(u;v;w) havenon-negativeghostnum bers,%(:::) vanisheswhen f? hasghost

num ber1,which isthecaseweareinterested in.W ethusconcludethat

��f?

��A+ @� %

�f?

�V A�

�

(0;0;w)= 0; gh(f?)= 1 )�f?

��A= �@� %

�f?

�V A�

: (B.40)

Finally we reconstruct f? from itsvariationalderivatives,neglecting integrated divergences,

using thegeneralform ula

f?[V;�]=

Z

d4x

Z 1

0

d�

�

�

VA� ?

�f?

�V A�

+ �A?�f?

��A

�

[�V;��]; (B.41)

valid forevery functionalf?.Using eqs.(B.39)and (B.40)in (B.41)weobtain

f?[V;�]=

Z

d4x

Z 1

0

d�

�

�

VA� ?s?;0%

�f?

�V A�

� �A?@� %

�f?

�V A�

�

[�V;��]

=

Z

d4x

Z 1

0

d�

�

�

VA� ?s?;0%

�f?

�V A�

+ (s?;0VA� )?%

�f?

�V A�

�

[�V;��]

= s?;0

Z

d4x

Z 1

0

d�

�

�

VA� ?%

�f?

�V A�

�

[�V;��];

wherewehaveused integration by partsand s?;0VA� = @��

A .W ehavethusshown that

s?;0f? = 0; gh(f?)= 1;

��f?

�V A�

�

(0;0;w)=

��f?

��A+ @� %

�f?

�V A�

�

(0;0;w)= 0

) f? = s?;0

Z

d4x

Z 1

0

d�

�

�

VA� ?%

�f?

�V A�

�

[�V;��];

(B.42)

which istheresultforthe s?;0-cohom ology in F ? wewilluseto proveeqs.(B.30)-(B.32).

Considernow�

A ?;5� s?;1B4.Itisan s?;0-closed integrated ?-polynom ialwith ghostnum ber

1 whose integrand is order 5 in the �elds V A� and �A ,has m ass dim ension 4 {recallthat

27

dim (V�)= dim (@�)= 1, dim (�)= 0, dim (! ��)= �2{ and contains one explicit !�� . It

followsthatthe integrand isa linearcom bination of ?-m onom ials !��a1 ?:::?a5,where it

can beassum ed thatoneofthe ai isan undi�erentiated � (foronecan rem oveallderivatives

from � using integrationsby parts,ifnecessary)while therem aining ai0s areeitheroftype

fV;V;@V;@V g or fV;V;V;@@V g.Itiseasy to verify thatthisin turn im plies

��(

�

A ?;5 � s?;1B4)

�V A�

�

(0;0;w)= 0 (B.43)

��(

�

A ?;5 � s?;1B4)

��A+ @� %

�(�

A ?;5 � s?;1B4)

�V A�

�

(0;0;w)= 0 : (B.44)

Eq.(B.42)can then beused and yields�

A ?;5 � s?;1B4 = s?;0B5,with

B5 =

Z

d4x

Z 1

0

d�

�

�

VA� ?%

�(�

A ?;5 � s?;1B4)

�V A�

�

[�V;��]:

Thisproveseq. (B.11)for U-valued �elds,hence for g-valued �elds,aswe wanted to show.

Thefunctional�

A ?;6� s?;1B5 can betreated analogously.Itsintegrand isorder6 in the�elds,

hasm assdim ension 4,ghostnum ber1 and one explicit !�� .Itisthusa linearcom bination

of?-m onom ials !��a1 ?:::?a6,whereitcan beassum ed thatthesetofai hasthestructure

f�;V;V;V;V;@V g.Thism akesitobviousthat�

A ?;6 � s?;1B5 satis�es

��(

�

A ?;6 � s?;1B5)

�V A�

�

(0;0;w)= 0 (B.45)

��(

�

A ?;6 � s?;1B5)

��A+ @� %

�(�

A ?;6 � s?;1B5)

�V A�

�

(0;0;w)= 0 : (B.46)

Eq.(B.42)then im plies�

A ?;6 � s?;1B5 = s?;0B6,with

B6 =

Z

d4x

Z 1

0

d�

�

�

VA� ?%

�(�

A ?;6 � s?;1B5)

�V A�

�

[�V;��];

which proveseq.(B.12).Finally wehavetoshow thateq.(B.13)holds.Thisisvery easy.The

integrand ofB6 isa ?-polynom ialoforder6in the�elds,hasm assdim ension 4,ghostnum ber

0 and oneexplicit !�� .Itisthusa linearcom bination of?-m onom ials !��a1?:::?a6,where

all ai are undi�erentiated V 0s. Furtherm ore,by construction,itcan be written asa trace

Tr.Thelatterim pliesalready s?;1B6 = 0,since

s?;1Tr

�

V�1 ?:::?V�6

�

= Tr

h

V�1 ?:::?V�6;�

i

?

28

isa divergence.

W ecloseby rem arkingthateq.(B.42)cannotbeused toprovethat�

A ?;3 and�

A ?;4� s?;1B3

aretrivialin the s?;0-cohom ology in F ? because the �

�V A�

and �

��A+ @�%

�

�V A�

acting on them

do not vanish at u‘ = v‘ = 0 in the sense explained above,contrary to what happens for�

A ?;5 � s?;1B4 and�

A ?;6 � s?;1B5 {seeeqs.(B.43),(B.44),(B.45)and (B.46).

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