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arX
iv:h
ep-t
h/03
0729
2v1
30
Jul 2
003
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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