1
AnAnalysisofAnomalyCancellation
forTheoriesinD=10
AndreaAntonelli1,2
1DepartmentofPhysics,FacultyofScience,
UniversityofTokyo,Bunkyo‐ku,Tokyo133‐0022,Japan
2DepartmentofPhysics,King’sCollegeLondon,
TheStrand,LondonWC2R2LS,UnitedKingdom
Abstract
We prove that the swampland for D=10
1SUGRA coupled to D=10
1
SYM is only populated by 1
and
1
. With this goal in mind, we
review the anomalies for classical and exceptional groups, retrieving trace
identities up to the sixth power of the curvature for each class of groups. We
expand this idea for low‐dimensional groups, for which the trace of the sixth
power is known to factorize, and we retrieve such factorization. We obtain the
total anomaly polynomials for individual low dimensional groups and
combinationsofthemandfinallyweinvestigatetheirnon‐factorization,insucha
way that 1
and
1
are non‐trivially shown to be the only
anomaly‐free theories allowed in D=10. Using the method developed for
checking the factorization of gauge theories, we retrieve the Green‐Schwarz
termsforthetwotheoriespopulatingtheswampland.
2
Contents
1.Introduction3
1.1Motivationsbehindthispaperandorganizationofthematerial3
1.2TheGreen‐SchwarzMechanism4
1.3Anomalycancellationin SO(32) and E8xE86
2.AnomalycancellationinLiegroupswith496generators7
2.1Aimofthepaper7
2.2CartanClassificationofSimpleLieGroups8
2.3AnomaliesinClassicalGroups9
2.3.1
9
2.3.2
11
2.3.3
12
2.4AnomaliesinExceptionalgroups13
2.4.1G213
2.4.2F416
2.4.3E617
2.4.4E718
2.4.5E819
2.5Some“strange“low‐dimensionalgroups20
2.5.1
2 20
2.5.2
2 22
2.5.3
3 23
2.5.4
4 and
5 24
2.5.4.1
4 24
2.5.4.2
5 25
3.Combinationsofanomalousgroups25
3.1Outlineofthechapter25
3.2Combinationsof
2 and 1 27
3.3Combinationsof
2 and 1 30
3.4Combinationsof
2 ,
2 and 1 32
Appendix34
A.ConstructingaGreen‐Schwarztermfor
B.ConstructingaGreen‐Schwarztermfor
34
34
3
1.Introduction
1.1Motivationsbehindthispaperandorganizationofthematerial
Inrecentyears,followingtheworkofVafa,the“swamplandprogram”hasaimed
atgivingaboundarytotheswamplandoftheeffectivetheoriesthatarenotfully
embeddedintheoriesofquantumgravity[11,12].Oneimportanttoolathandfor
physicistsistheconceptofquantumanomaly,andconsequentlyanomalieshave
beenstudiedindetail.
In D=10 it has long been known [3] that
32 ,
, 1
and
1
are theories where quantum anomalies are cancelled. The former two
arealsolowenergylimitsofstringtheoriesanddonotthereforepertaintothe
swampland.
and
1
, conversely, do live in the swampland and it was not
1
knownuntilrecentlywhetherornottheywerealsolowenergylimitsofstring
theories.FollowingtheworkofFiol[10]andthejointworkofAdams,DeWolfe
and Taylor [4] it was shown that there are no theories of gravity that can be
coupled to them, thus confining both theories to the swampland, without
possibilities to be upgraded to string theories. However, there is still one
unanswered question: are
32 ,
, 1
and
1
the only
theorieswheretheanomaliesarecancelled?
Hintsforthisstatementcanbetracedbackintheliterature,mostlyfollowingthe
original statement found in chapter 13 of [3], but no proof has explicitly been
carried out. Therefore answering this question might help understand the
structureoftheswamplandinD=10.
In this paper it is shown that indeed there are no other theories in the
swampland in D=10: in the process of proving, we devise some non‐trivial
machinery that is worthwhile presenting and that might be useful in anomaly
cancellation in other dimensions; also, we deal with low‐dimensional theories
thatareoftenoverlookedintheanalysisofanomalycancellation.
Thereaderisencouragedtoreadthefirstpartofthispaper,astheremostofthe
material and notation is presented: more in detail, the Green‐Schwarz
MechanismisrevisitedinthenotationofBilal[5]andsomeimportantpointsfor
theupcomingsectionsareexplained.Plus,proofwithappropriatereferencesare
givenforthenon‐factorizationofthetraceofthesixthpowerofthecurvaturein
classicalandexceptionalgroups,followingtheCartanclassificationofsimpleLie
groups.
Theexpertreadermightwanttoskiptheclassificationandnon‐factorizationof
classical and exceptional groups and dive into the last part involving low‐
dimensionalgroups.There,theconceptoffactorizationistreatedindetailandit
isshownwhyanomaliesofindividualgroupsarecarriedalongwhencombining
them and specifically what non‐canceling terms of the polynomial contain the
informationabouttheanomalyofthetheory.
This paper might be of interest also to readers looking for the factorization or
non‐factorizationofthetraceofthesixthpowerofthecurvatureinallclassical,
exceptional and low‐dimensional groups, although a fairly more rigorous
treatmentformost(butnotall)ofthemisfoundin[6]and[8].
4
1.2TheGreen‐SchwarzMechanism
Itisinstructiveatthispointtoreviewthemechanismdevisedin[2]thatallows
anomaly cancellation. In doing so, the reader will be reminded of numerous
concepts that will be fundamental for understanding the upcoming sections.
Moreover,severaloftheideaspresentedhereinwillbeusefulwhentheGreen‐
Schwarzmechanismisusedfornon‐trivialgroupsinthefinalsections.
Wewillbeginsayingthatalltheinformationabouttheanomalyofasystemcan
be encapsulated in the total anomaly polynomial [5]: in principle, then, such a
polynomialcanvanishifacountertermdependingonthechoiceofaparticular
gaugetransformationisadded.Ifthishappens,thegaugetheorybecomesnon‐
anomalous(or,equivalently,anomaly‐free).
TheGreen‐Schwarzmechanismdoesexactlythis;thegroundbreakingideawas
to devise an “inflow” method to construct a counterterm and then recast it in
terms of a 12‐form polynomial, in such a way to be able to add it to the total
anomalypolynomialandcancelit.
We will now explain the method in much more detail referring to a ten
dimensionalgaugegroupGthatisa =1Supergravitycoupledtoa =1Super
Yang‐Mills theory. As a remainder, the matter content of SUGRA is a multiplet
comprisingapositivechiralityMajorana‐Weylgravitinoandanegativechirality
spin ½ fermion. Moreover, SUGRA contains the graviton, a scalar renamed
dilatonandatwo‐formB.
Ontheotherhand,theSuperYangMillscontainsamultipletofgaugefields andgauginos thatliveinthesameadjointrepresentation[5].
The requirement for a consistent coupling is given by the following expression
forthefieldstrength:
,
,
≡
where
, isthegaugeChern‐Simonsformand
, isthegravitational
one [5]. The invariance of H when acted upon with a gauge and Lorentz
transformationisgivenasfollows:
,
,
Given the above ingredients, it is now possible to choose a counterterm with an
retrieved from the characteristic classes:
appropriate 8-form
ΔΓ
∧
0
When we act upon the counterterm with a gauge and Lorentz transformation we
obtain:
ΔΓ
,
,
∧
,
,
∧
5
∧
(1.2.1)
Using the descent equations, (1.2.1) can be recast in terms of 12-form polynomial:
Δ
∧
The 12-form polynomial tells us that the total anomaly polynomial is cancelled by
adding the counterterm if and only if it does not vanish and takes a factorized form.
This idea is so important that we can safely say that the rest of this report will be
centered on the concept of factorization: specifically we will be interested in if and
how the total anomaly polynomial factorizes and therefore vanishes upon adding a
counterterm ΔΓ.
Consider again the gauge group G. Its matter content is such that the various
contributions to the anomaly are [5]:
(1.2.2.a)
1
64 2
1
32 2
1
1
5670
1
360
1
4320
1
288
1
10368
1
1152 2
(1.2.2.b)
The total anomaly polynomial is given by the contribution of all the components of
the matter content and can be expressed as follows:
|
1
64 2
1
32 2
496
5670
64
10368
224
4320
|
1
360
1
288
1
1152 2
(1.2.3)
Where Tr and tr represents the trace in the adjoint and fundamental representations
respectively and n is the dimensionality of the gauge group.
From (1.2.3) two conditions emerge in order for the anomaly to be cancelled. Firstly,
the number of the dimensions of the gauge group must be 496 in order for
to
disappear. A second, equally important condition is given by the factorization of the
last term
. Indeed, if this does not factorize the Green-Schwarz mechanism
cannot be used. This idea will be explained in detail later: for the sake of
6
completeness, however, it is now anticipated that the total anomaly polynomial must
factorize in such a way that only 8-form and 4-form terms appear, since such is the
coupling of the electric and magnetic currents. Clearly,
is a 12-form and does
not pertain to either factor: factorizing this term in 8-forms and 4-forms allows
recovering the shape of the coupling of currents, thus giving us a chance to cancel the
anomaly via the Green-Schwarz mechanism.
In the next paragraph we explore further the latter condition and we shall see how
(1.2.3) works for two choices of the gauge group G.
1.3 Anomaly Cancellation in SO(32) and E8xE8
Let us first focus on G ≡ SO(32): such a Lie group has 496 generators, so its
dimensions are just enough to cancel the first term of (1.2.3). The remaining problem
is related to the trace of the sixth order of the curvature F, which should factorize. We
will now anticipate some useful relations between the traces of the adjoint and those
of the fundamental representations. The proof will be found in chapter 2, which deals
with the more general SO(n) group.
30
24
15
3
(1.3.1)
The reason why such trace identities are looked for is simple: in D=10 the SYM
contains vector gauge fields and Majorana-Weyl spinors that live in the adjoint
representation. Moreover the anomaly is found to be [3] proportional to a term T
constructed out of the trace of the elements of the gauge algebra t that live in the
adjoint representation. Such elements of the gauge algebra can sometimes be daunting
to evaluate and for this reason relations between traces in the adjoint and fundamental
representations are looked for: it is in general easier to deal with generators of the
gauge algebra than with its elements.
Coming back to the point of the discussion, the relations (1.3.1) are inserted in the
total anomaly polynomial to obtain a factorized form:
1
384 2
1
4
8
In order to cancel this it is clear that the counterterm ΔΓ must be constructed from the
following chern 8-form:
Δ
1
384 2
1
384 2
1
4
8
1
4
8
The counterterm is now nothing more than the opposite of the total anomaly
polynomial so that when the two are added, the anomaly vanishes via G-S
mechanism:
7
Δ
0
A second candidate for a consistent theory is
. The dimensions of
is 248,
thereforethedimensionsofthecombinationareconsistentwiththefirsttermof
theanomalypolynomial.Againtraceidentitiescanberetrieved[5]:
75
75
9
9
,
20
20
where the indices are used to distinguish between the curvatures of the two
groupsE8 involved. The total anomaly polynomial again factorizes:
1
384 2
1
4
2
TheGreenSchwarztermisthenchosentobe:
1
1
384 2
4
2
2
2
2
2
So that, following the same reasoning as above, the total anomaly polynomial
cancels.
Itturnsout,toconclude,thattherearetwomoregroupswheretheanomalyis
[3]. These groups, however, do not
cancelled, namely 1
and
1
give rise to string theories [4,10]; both groups will be treated conceptually in
section2.4.5andalgebraicallyintheAppendix.
2.AnomalycancellationinLiegroupswith496generators
2.1Aimofthepaper
Itisoftensaidthatfourgroupscanceltheanomaly:
32 ,
, 1
and
1
.Forthefirsttwoexplicitcalculationshavebeencarriedoutinthe
Introductionsection,forthelattertwoitisrecommendedtoreadtheAppendix
ofthispaper.
Nonetheless the set of Lie groups with 496 dimensions contains many more
groups.Thesituationlooksthenlikethefollowing:
8
A
LieGroups
with496
generators
B
1
1
32 Fig1.SetofalltheLieGroupsin496dimensionsanditsinternalclassification.
WherethesetAcontainsthetheoriesinwhichtheanomalyiscancelledandthe
subsetBoftheoriesembeddedinatheoryofquantumgravity.
This paper aims at checking that the above‐mentioned quartet of groups
containsallthegaugegroupsforwhichtheanomalyiscancelledandtherefore
that|A|=4.
2.2CartanClassificationofSimpleLieGroups
The Cartan classification of Lie groups into classical and exceptional groups
comes in very handy for our purposes. All the Lie groups can be reduced to 4
classicaland5exceptionalgroups[6]:
ClassicalGroups
ExceptionalGroups
An (n ≥ 1) compact
E6
Bn (n ≥ 2) compact
E7
Cn (n ≥ 3) compact
E8
Dn (n ≥ 4) compact
F4
G2
Table1.CartanClassificationofSimpleLieGroups.
Themainideaistocheckthatgaugetheorieswith496dimensionsconstructed
out of classical groups, exceptional groups and mixtures of the two (with the
exception of
) inevitably carry anomalies that cannot be canceled. We
9
willseethatsuchstatementisintrinsicallyrelatedtothenon‐factorizationofthe
traceintheadjointrepresentationofthesixthordercurvature
.
Once the classical and exceptional groups are found to be anomalous, it is
obtainedasacorollarythatnoanomaly‐freetheoriescanberetrievedcoupling
anyofthegroupto 1 foraparticularchoiceofn(seesection3.1),againthe
onlyexceptionbeingthetheoryconstructedvia ,e.g.
1
.
In the next paragraph we will show how the anomaly is preserved in some
representatives of the classical groups, that nonetheless carry as much
information as the above‐mentioned An, Bn, Cn, Dn. In doing so, we will derive
someusefultraceidentitiesthatarecrucialinourstudyofnon‐trivialgroupsin
thelastsection.
2.3 Anomalies in Classical Groups
Throughoutthischapterweassumethat issufficientlylarge,e.g.
5forAn,
4forDnand
3forBn, and Cn. Itwillbesoonexplainedthat,forlarge ,
doesnotfactorize,whereasitdoeswhenlowdimensionalclassicalgroups
areconsidered.Thelattercaseisthoroughlydiscussedinsection2.5.
2.3.1
Totacklethisclassofgroupsweneedtofindtherelationbetweenthetraceof
theadjointrepresentationandtheoneinthefundamentalanddemonstratefor
whichgroups
canorcannotfactorize.
Thestartingpointofthediscussionistheactionofthegroup:
indeedacts
on an anti‐symmetric carrier space denoted by that, being anti‐symmetric,
.
satisfies
Thetransformationoperatedbytheactionofthegaugegroupis[7]:
→
Ω
with
∑
Ω
(2.3.1.1)
Ω
The indices k and lrun over the dimension of the group, e,g k,l=1,…., ½ n(n‐1),
wherewerename
1 forfuturereference.
What has been found so far is an action via an adjoint representation of the
gauge group. It is however possible to rewrite the adjoint transformation in
termsofthefundamentalrepresentationO,sotolinkadjointandfundamentalin
afirst,unpolished,relation.Thisisdoneasfollows(confrontappendixFof[7]):
∑ ′, ′ ′′ ′′ ′ ′ 2 ∑ ′ ′ ′′ ′′ ′ ′ (2.3.1.2)
Ω
Bycomparing(2.3.1.1)and(2.3.1.2)arelationbetweentheactionoftheadjoint
andfundamentalrepresentationsisfound:
Ω
(2.3.1.3)
10
wheretheminussignisduetotheasymmetryofthecarrierspaceandplaysa
crucialroleindetermining
32 astheonlypossibleconsistenttheoryfor
.Letusseehow.
Therelation(2.3.1.3)hasbeenworkedouttothepointwherethevectorspace
doesnotappearanymore.Sinceweeventuallywantarelationbetween and wecalculatetheformeratthisverystage:
1
Ω Ω
Ω
2
,
∑ ,
Ω
(2.3.1.4)
Withequation(2.3.1.4)thetraceoftheadjointhasbeenrelatedtothetraceof
thefundamentalrepresentation.However,theactionastheargumentforbothis
not as useful as a general element of the Lie Algebra: as far as anomalies are
concerned,indeed,thetraceisusuallycalculatedoverthecurvature ,whichis
anelementoftheLiealgebra.
Theeasiestwaytoproceednowistorecasttheactiononthevectorspaceasan
exponentialrepresentation:
Ω
∈
∈
ItisalwayspossibletoTaylorexpandanelementoftheLieAlgebraaroundthe
identity:
1
2
1
Ω
1
2!
3!
2!
..
1
3!
2
.. 2
2!
2
3!
..
wherethepreviousidentity(2.3.1.4)hasbeenusedinthesecondstep.Itisnow
aneasytasktoretrievethetraceidentities,sincewhatislefttodoiscomparing
thetermswithsameorder:
Table2:TraceIdentitiesforSO(n)
1
1
2
0
1
2
8
32
1
1
2
3
15
11
wherewehaveused 1
.Forsimplicity,wehavedroppedthesymbol ;itis
intended,however,thattheelementintheLHSofwhichwecalculatethetrace
alwayslivesintheadjointrepresentation.
TheboxoftraceidentitiesforSO(n)allowsustoansweramajorquestionthat
wasleftopenacoupleofparagraphsago.TheonlySO(n)groupthatisallowedis
SO(32) because of the last identity; moreover the identities used in the
introduction are retrieved from the above ones. For
32, indeed,
does
not factorize and therefore remains unchanged in (1.2.3), which consequently
cannotbefactorizedin8‐formsand4‐forms.
Ifthetotalanomalypolynomialisnotfactorized,theanomalycannotbecanceled
viatheGreen‐Schwarzmechanism.
SO(32)alsocontains496generatorsandiscoupledtoaknownstringtheory.All
theseconditionsguaranteeaspecialplaceinsubsetBofFig.1tothisveryspecial
gaugegroup.AsfortheremainingSO(n)groups,theycannotevenbeincludedin
subsetAof“groupswheretheanomalyiscancelled”forthereasonsjustcited.
2.3.2Sp(n)
isaclassofgroupsthatsharesseveralpropertieswith
.Asfarasour
topic is concerned, the major and crucial difference between the two gauge
groupsliesinthecarrierspace:itisanti‐symmetricfor
andsymmetricfor
.Theanalysisofthelatterclassgoeshandinhandwiththatoftheformer,
withthe,atfirst,harmless‐lookingdifferenceinthesignoftherelationbetween
thetracesinadjointandfundamentalrepresentations.For
wehave:
Ω
whichleadsto:
1
Ω
2
At this point the expression can be Taylor expanded in much the same way as
before:
Ω
1
1
2
1
2!
2!
3!
3!
..
Sothat,finally,thetracerelationsfor
Table3:TraceIdentitiesforSp(n)
.. 1
2
canbefound.
2
2!
2
3!
..
12
1
2
0
1
1
2
8
32
1
1
2
3
15
Andhereisthecaveat.Therecanbenogaugegroupforwhichncancelsthefirst
term
of the last identity. Therefore, as the term survives, the anomaly
cannotbecanceledviaG.S.mechanism.
2.3.3
Inthecaseof
theactiononthecarrierspaceisgivenby[7]:
;
∗
whereas the relation between adjoint and fundamental actions is obtained by
takingthedirectproductofnandn*in
minusthetrace:
1
;
∗
Uponcalculatingthetracesweobtain:
1(2.3.3.1)
∗
since
.Therelation(2.3.3.1)isquitewhatwewanted.Analogouslyto
theactionsarerecastintermsofelementsofthegaugealgebra.
∈
∈
ExpandingLHSandRHSofrelation(2.3.3.1)thefollowingisobtained:
1
2!
3!
1
..
∗
2!
3!
..
1
2!
3!
..
1
ItisimportanttonoteatthispointthatFisanti‐hermitianandthereforesatisfies
∗
. Expanding the RHS in light of this property and then comparing the
termswithsamepowerleadstothetraceidentitiesforSU(n):
13
Table4:Traceidentititesfor
1
1
0
2
2
2
20
6
30
Weseethatthetraceofthesixthordercurvatureisnotcancelledandtherefore
all
theoriesareanomalous.
2.4AnomaliesinExceptionalgroups
2.4.1G2
G2hasbeenwidelystudied,thereforemanyofitspropertiesarewell‐known.Itis
known, for example, that the group possesses 14 generators and that the
fundamentalrepis7[7].
Forourpurposes,wewillonlybeconcernedwiththemaximalsubgroup
2 .
∗
The action of the maximal subgroup on the 7=
, ̅ , , where
̅ represent hermitian spinors and y represents a scalar, defines transformations
closed under a Lie Algebra with generators (see [7] for more details). The
decompositionofadjointandfundamentalrepsunderSU(3)yields:
∗
⟶
,
∗
⟶
,
G2hasonlytwoCasimiroperators,C2andC6:thereforebothofthemplayarolein
. Specifically, the identity of the sixth power must satisfy the
determining
followingrelation:
(2.4.1.1)
Let us have a look at this equation. As repeatedly emphasized, anomaly
cancellationcanonlyoccurwhen
factorizes.Inthemethodreportedinthis
section, such statement is equivalent to saying that is vanishing: if this is the
case, indeed,
can be expressed in terms of traces of lower power in the
fundamentalrepresentation.Conversely,if
0thetheoryisanomalous.
ThegeneratorsoftheLieAlgebraarethestartingpointofthediscussionforG2;
thisisagainduetotheirrelativeeasetousecomparedtothegeneralelementsof
the Lie Algebra such as the curvature F. Throughout the next paragraphs, the
calculationsthatwillbeperformedusingthegenerators.Indeed,theyareeasier
to use and a non‐vanishing trace of the sixth power of the generator in the
adjointrepresentationbearsthesameconsequencesofanon‐vanishing .
Inourcasethegeneratorsofthemaximalsubalgebraare:
14
0 1 0
1 0 0
2
0 0 0
1 0 0
0
1 0
2
0 0 0
0
0
0
0
2
0
0 0
0 0
2
0
0
0
0
0
0
0 0
0 0 1
0 0 0
2
1 0 0
2
0
0
2
0
1
0
2√3 0
0
0
0 0 0
0 1
1 0
0 0
1 0
0
2
0
Wecanthenretrieveanexpressionforthesecond,fourthandsixthpowerofa
generatorandcalculatetherelevanttraces.Thetraceoftheadjointiscalculated
usingtheidentitiesof
;thechoiceofcoursebeingrelatedtothemaximal
subalgebra
3 .
Since relation (2.4.1.1) contains two constants, the procedure described above
mustbecarriedfortwogenerators,sotoobtainasystemoftwoequationsintwo
variables:thegeneratorsofinterestforusare and .Webeginbycalculating
afirstexpressionintwovariablesusingtheformergenerator:
1 1 0 0
1 1 0 0
0 1 0 0 1 0 4
64
0 0 0
0 0 0
andtherefore,
1
1
2
64 64
64
1
1
2
∗
64 64
64
Thelasttracetoobtainbeforecalculatingthetotaltraceoftheadjointis
.
Upon using the trace identities of SU(n) the following result is obtained:
1
2
16
andinconclusionthetraceoftheadjointrepresentationisgivenby:
2
(2.4.1.2)
Asfarasthetracesofthefundamentalrepresentationgo,wehave:
∗
∗
,
1(2.4.1.3)
15
Substitutingequations(2.4.1.3)and(2.4.1.2)intotherelation(2.4.1.1)thefirst
expressionuptotwoconstantsisobtained:
1 (2.4.1.4)
Thisisclearlynotenoughtodetermine and ,butitshowsthepathtofollow
toobtainasecondexpressionofsuchtype.
Nowletuscalculatethesquare,fourthpowerandsixthpowerof .
1 0 0
1 1 0 0
0 1 0
0 1 0
12
24
3
√
0 0 4
0 0
8 1
0
0
1
0
0
1
1
0 1 0
0 1 0
144
12
0 0 16
0 0 64
Hence,thetracesofthefundamentalrepresentationaregivenby
1,
66 66
whereasthetraceoftheadjointis:
Theexpressionobtainedinthiscaseis:
1 (2.4.1.5)
Itisclearthatthisisincompatiblewith(2.4.1.4).Afirst,naïve,thoughtwouldbe
toconsidertheonlycaseofinterestforus,namely
0.Suchparticularresult
leads to two different values of in the system; clearly a contradiction.
.Relation
Tobemoreprecise,thesystemofequationsyields
26and
(2.4.1.1)thenbecomes:
26
(2.4.1.6)
anddoesnotfactorize.Inconclusion,G2isnotanomalous.
Incidentally, one might wonder if the trace
of (2.4.1.1) further factorizes
into
; in other words, one might wonder if
is a primitive Casimir.
The above discussion rules out this possibility since and are determined
uniquely.Suppose,conversely,thatitispossibletofactorize
asfollows:
(2.4.1.7)
Theninserting(2.4.1.7)into(2.4.1.6)weobtain:
16
26
15
4
Sothatinthiscasethemostgeneral(2.4.1.1)hassolutions:
15
,
0, 26
4
These values are clearly differing from the unique constants obtained with the
generalmethod,sothattheinitialassumptionisnotvalidand
isaprimitive
Casimirbycontradiction.
aredetermineduniquely.
Thisproofholdswhenevertheconstantsof
is a
As we shall see this is the case for all the exceptional groups, so that
primitiveCasimirinallthegroupsthatwewillbeinterestedin.
2.4.2F4
F4 has 52 generators and
9 as the maximal subalgebra. The fundamental
representationis26.Thedecompositionunder
9 is:
⟶
,
⟶
,
F4hasalsorank4,thereforeitpossesses4Casimiroperators,whichare:C2,C6,C8,
C12.[8]Inlightofthis,theexpressionforthetraceisstill(2.4.1.1).Theproblemis
stilltocheckif
0.Thetworepresentationsofthegeneratorschosenare:
0 1 0
0 1 1
1 0 0 …
1 0 0 …
0 0 0
1 0 0
⋮
⋱
⋮
⋱
[7].
Thespinorrepresentationisgivenby
Thetraceofthefundamentalrepresentationisthesumofthecontributionfrom
thegeneratorandthatofthespinorrepresentation.Thereforewehave:
16
2
6
4
16
9
2
64
4
andfollowingthetraceidentitiesfor
weobtainastheadjointtrace:
16
57
14
64
4
whichleadstoanexpressionforthetraceintwoconstants:
17
57
4
9
4
216 The same procedure is repeated for the second generator. The spinor
.Onethenobtains:
representationisinthiscase
4 8
12(2.4.2.1)
10 2
12
132(2.4.2.2)
Where the last trace has been found with the usual relation for the trace
identitiesof
.Thesecondtraceidentityuptotwoconstantsisthen:
132
12
1728 and it is clear that is non‐vanishing. The system of equations results in
3,
,sothatthetraceidentity(2.4.1.1)becomes:
7
3
72
ThereforeF4isanomalous.
2.4.3E6
Thisgrouphasadjointrepresentation78andfundamental27.Thereareseveral
subgroupsthatonecouldusetodecomposesuchrepresentations.
Two examples are
10
1 and
8 , under which the decomposition
canbefoundin[9].AnothersuitablesubgroupisF4andwewilluseitsincemost
ofthecalculationsinvolvedhavealreadybeencarriedoutinsection2.4.2.Under
F4thedecompositionis:
⟶
,
⟶
,
As far as the Casimir operators are concerned, the relevant ones are C2and C6.
Hence, the trace to find is still (2.4.1.1). The trace of the fundamental rep for
generator1isthesameas(2.4.2.1),sincethefactor1doesnotcontributetothe
trace.Weseealsothatthetraceoftheadjointisthesumoftraces(2.4.2.1)and
(2.4.2.2),alreadycalculatedpreviously.Thefirstrelationtheniseasilyobtained:
144
12
1728 Similary,thetracesfundamentalandadjointrepresentationarecalculatedusing
and
fromtheprevioussectiontoobtain:
18
33
2
9
4
9
4
,
.Anomalycancellation,evenin
Thesystemofequationsgives
thiscase,isnotpossible.Thetraceidentitybecomes:
5308
224
741
6669
2.4.4E7
Thenextstructureencounteredinourjourneythroughtheexceptionalgroupsis
E7. The group has 133 generators and
8 as a maximal subgroup. It is also
importanttoknowthatthefundamentalrepresentationis 56.Under
8 the
decompositionisasfollows:
⟶
,
∗
⟶
,
The relevant Casimir are unchanged, hence there is no need to modify the
structure of
. Decomposing under the maximal subgroup and taking into
accountthedimensionalityoftherepresentationsweobtain:
2
2
where we have exploited the fact that the fundamental representation is real.
The method used for this exceptional group differs slightly from the previous
ones. To check that
0we calculate the coefficients of the sixth power
elementoftheLieAlgebraforbothfirstandsecondtermintheLHS.Assuming
0such coefficient should vanish: checking that this is or is not the case is
equivalenttocheckinganon‐vanishing andthereforethefactorizationornon‐
factorizationoftheanomalypolynomial.
As generators we choose any linear combination of the seven independent
generators of
8 , labeled for i=1,2,…8 [7]. is therefore an element of the
LieAlgebraandwecanusethelasttraceidentityintable4toobtain:
8 ∙ 2∑ 15 ∑ ∑ 15 ∑ ∑ 20 ∑ 16 ∑ 30 ∑ ∑ 20 ∑ (2.4.4.1)
A similar argument is used for
. In this case, however, the states are
labeled , , , with
. Let us focus on the sixth power term. The
restrictedsumofthestatescanberecastasacombinationofunrestrictedsums
[7].
19
∑,
3∑
,
6∑
, ,
2
2
8∑
,
, ,
2
6∑
3
4
(2.4.4.2)
Using(2.4.4.1)and(2.4.4.2)wecanfinallycollectthetwocoefficientsofthesixth
powerterms.
∝
,
∝
isreadilyobtainedbyinspectionof(2.4.4.1).Clearly,
16.Amoreinvolved
calculationisneededfor .
1
1
1
1
1
6 2
1
1
3 2
2 24
1
2 8 3
6 4
3
Theappropriatecaseforusis
0.Assumingthisisthecaseandconsidering
only the sixth powers, the RHS of relation (2.4.1.1) vanishes: the first term is
indeedzerobyassumption,thesecondisnotproportionalto .
Theconditionthenbecomes:
0∙
0
But clearly
0as it has been calculated just above. We conclude that must be different than zero and that consequently the factorization of
is
notpossible.
2.4.5E8
Weendourdiscussiononexceptionalgroupswiththeonerelevanttoanomaly‐
freegaugetheories.Insection1.2oftheIntroductionitwasdiscussedhow
cancels the anomaly via Green‐Schwarz mechanism. It was also anticipated
that 1
and 1
cancel the anomalies. It is now time to spare a
thought on the last statement. Why does behave differently from its
exceptionalsiblings?
The answer lies in the Casimir operators of
. These are
for
i=2,8,12,14,18,20,24,30.Thereforethegroupdoesnotpossess .Consequently,
thefactorizationof
ispossiblebecausethereisnotsuchfactoratall.Inthe
and
arenon‐vanishingandfactorizable
literature[2]onecanseethat
[seethetraceidentitiesin(1.3.1)].
Fortheabovereasons providesagoodbasetodevelopanomaly‐freemodels.
In the introduction
was considered, however the reader should also
remember that the group 1 behaves differently from its class
. Indeed,
thisgaugegroupcancelstheanomaly(agoodreferenceischapter12of[5]).Asa
corollaryany496‐dimensionaltheoryconstructedsolelyuponmultiplicationof
20
1 isanomaly‐free.Clearly,theonlypossiblewayistoconsidertheanomaly‐
free 1
theory.
Finally, we could combine E8 and 1 in a 496‐dimensional theory that is
anomaly‐freeforthereasoningjustcited.Again,thereisonlypossiblewaytodo
this, e.g. 1
. For an algebraic treatment of these theories, e.g.
factorization via the Green‐Schwarz mechanism, consult the appendix of this
paper.There,theGreenSchwarztermsareretrievedforboththegroupsinthe
swampland,inamannerthatissimilartotheanalysiscarriedoutfor
and
32 intheintroduction.
2.5Some“strange“low‐dimensionalgroups
Itseemsthatthemethodusedaboveprovidesawaytoanswerpositivelytothe
initial question: are the groups where the anomaly is cancelled only four?
However,caremustbetakeninanalyzingthemethod.
Indeed, we start our discussion assuming that
factorizes if the first term
vanishes. This is checked demonstrating that
0for any class of
groups. However, a vanishing does not logically preclude the idea that
itselfmightfactorizeinlowerorderpowersofF.Suchfactorizationisaproperty
ofsomelow‐dimensionalgroupsthat,nicely,canallbetracedtosome
for
5,
2 andproductsthereof.
The method presented previously, therefore, does not stand for them as
formulatedaboveandanomalycancellationmustbecheckeddifferently,insuch
awaytoaccountforthefactorizationof
.
Fortunatelyenough,theGreen‐Schwarzmechanismholdsforanygroupandthus
also for low dimensional
groups and
2 . It seems a good idea to start
fromthisreassuringmechanismandwewilldoitemployingitfor
2 first.
2.5.1
Thegoalhereistofactorizethetotalanomalypolynomial(1.2.3),inlinewiththe
general idea of the mechanism explained in section 1.3.The three terms of the
,
and
.
polynomialthatfactorizeinlowerorderonesare
Specifically,thestructureofthefactorizationissuchthat:
∙ ∙
∙ ∙
∙ Thegenerators usedforchecking
2 are:
0
0
0
0
21
Only one generator is needed for
, as it contains one constant, and the
factorizationistriviallyfoundusingtheresultsinTable3:
6 (2.5.1.1)
needstwosetsofcalculationsperformedonthetwogenerators.Again,the
LHSiscalculatedusingthegeneraltraceidentityofthefourthpowerfor
with
2.Thetwogeneratorsyieldasystemofequationswithvariables and
that,oncesolved,allowsustowritedownthetraceidentityfor
:
3 12 (2.5.1.2)
Asimilarargumentisbroughtuponforthetraceofthesixthpower,whichreads:
42 (2.5.1.3)
For simplicity we now set
496in (1.2.3), so to obtain the total anomaly
polynomialofinterestforthefactorization(weadapttothenotationof[2]here):
1
1
1
|
4
5
15
24
960
(2.5.1.4)
Relations(2.5.1.1),(2.5.1.2)and(2.5.1.3)arenowpluggedinto(2.5.1.4):
9
42
1
1
30
15
8
2
1
1
1
1
40
32
8
32
(2.5.1.5)
Thequestionisnowwhetherorthepolynomialabovefactorizes.
Beforegoinganyfurther,itiscompellingtoreportthattheGreen‐Schwarzinits
originalsenseistheapplicationofthecountertermΔΓinheteroticsupergravity:
this theory has a 12‐degree Chern class that factorizes as ∧ ; that is
preciselyhowthephysicalelectricandmagneticcurrentsrespectivelyfactorize.
The gauge theories that we have been considering in this paper fall in this
category,andthereforewhenwecheckfactorizationwealsohavetocheckthat
thefollowingconstraintissatisfied:
∧ (2.5.1.6)
Of course, nothing prevents the polynomial (2.5.1.4) from factorizing without
condition (2.5.1.6). However, the resulting gauge theories would not be
consistentfromthestringviewpoint.Inlightofthis,wewillbeconcernedonly
withtheoriesthatfactorizeandsatisfytheconstraint.
22
Letusturnourattentionto
2 again.FandRhavebydefinitiondegree2.In
principle, therefore, the 4‐degree and 8‐degree terms can be separated up to
sevenconstants:
∧
∙
(2.5.1.7)
What is not sure, however, is whether or not seven constants that satisfy the
equationsimultaneouslyexist.Thiscanbecheckedbyinspectionsettingoneof
thevariablestobe
.Byconsequentexpansionandcomparisonoftermsin
equations (2.5.1.7) and (2.5.1.5) we obtain:
3,
,
,
and
, which have been computed comparing terms , ,
and
respectively.
Letushavealookatwhatwehavesofar:
1
1
∧
10
24
1
1
1
∙ 3
768
48
192
Theonlyremainingconstantis .However,wecanseethatitcanberetrieved
and
. In fact, the system
comparing two addends, satisfying is overdetermined as there are two equations for one variable. If
theseareinaccordanceauniquesolutionfortheconstantcanbefoundandthus
thetotalanomalypolynomialcanbefactorizedcompletingtheworkdonesofar.
termstheresultreads
0.
However,comparingthe Acontradictionisobtainedwhenthisisinsertedintheequationcomparingthe
, thus proving that the anomaly of
2 cannot be
latter term
cancelledthroughtheG.S.mechanism.
2.5.2
In section 2.5.1 we have developed a method to construct a factorized
polynomialuptosixconstantsandtocheckitforaparticulartheory.Ofcourse
theparticulartheoryneednotbe
2 andwehereinshowthattheargument
standsforthenextgroupconsidered,
2 .
The structures of the factorization of higher order traces is even simpler than
2 asonesingleconstantispresentineachrelation:
∙ ∙
∙
Thegeneratoris
23
0
0
andthetraceidentitiesarethoseoftable4,
withtheparticularchoice
2:
4 (2.5.2.1)
8 (2.5.2.2)
16 (2.5.2.3)
Thetotalanomalypolynomialforsuchatheoryistherefore:
16
1
1
4
5
15
3
240
1
1
8
32
andagainwelookforafactorizationsatisfying(2.5.1.6).Thishastheform:
∙
Let us begin by setting
. By comparing terms with a factor
we
readily obtain
. Then, in much the same way as described above, the two
constants are used as bases to obtain:
,
16and
. Now the
situationlookslikethis:
1
1
1
1
∙ 16
15
2
16
4
is determined by an usual overdetermined system of equations obtain by
and
. Unfortunately,
comparison of the coefficients of
eventhistimethesystemleadstoacontradictionand
2 cannotthereforebe
factorizedandcancelledviaG.S.mechanism.
2.5.3
Thisisacriticalgroup.Thefactorizationofhigherordertracesisasfollows:
∙ ∙
∙
∙
0
0
Upon using
as generator and the trace identities of table 4
0
0
0 0
with
3weget:
24
6 9
33
2
22
Insertingsuchrelationsintotheanomalypolynomialweobtain:
11
22
3
1
10
15
8
40
1
1
1
32
8
32
(2.5.3.1)
Andalreadyitcanbeseenthatthesecondtermisproportionalto having
degree6.Factorizationofthispolynomialmightbepossible,butitisimpossible
forittosatisfycondition(2.5.1.6).Thereforeanon‐vanishing
termleadsto
thenon‐factorizabilityof(2.5.3.1).Thetheoryisanomalous.
2.5.4
and
Thetraceidentityofthesixthpowercurvatureisforboththeories:
∙
∙
∙ (2.5.4.1)
Anargumentsimilartothatusedfor
3 canbeexploited.Itisnowclearthat
the problem now reduces to the preliminary question: does the coefficient of
vanish?If
0,indeed,wecannoteventrytocanceltheanomalyviathe
G.S.mechanismaswearepreventedfromdoingitbycondition(2.5.1.6).
Thewaythisisformulatedremindsusofthemethodusedtocheckclassicaland
exceptional groups in sections 2.3 and 2.4, where the question was: does the
vanish?
coefficientof
To proceed, then, we will first the question answer whether
0or not;
consequently,wedemonstratethattheG.S.mechanismcannotbeused.
2.5.4.1
Apartfromtheaboveconstructedtraceofthesixthorder(2.5.4.1),
4 hasthe
followingtracestructures:
and
one generator is sufficient, however
has 3
To obtain
constants and thus requires 3 generators. We use the general traceless matrix
0
0
andsetthreedifferentcombinationsofa,b,c.
0
0
0 0
25
1
0
0
0
0 0
1 0
0 0
0 0
0
0
0
2
2
0
0
0
0
1
0
0
0
0
0
0
0
0
0
3
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
3
(2.5.4.1.1)
The trace identities for
with
4are used. The method is always to
createasystemin3equationsusingthegeneratorsandsolveit.Theresultsare
directlyshownbelow:
8 (2.5.4.1.2)
10 (2.5.4.1.3)
1844
3
804
75
5
25
By inspection
0and therefore the total anomaly polynomial contains a 6‐
degreetermthatdoesnotsatisfy(2.5.1.6).Thetheoryisanomalous.
2.5.4.2
Oncethecalculationsfor
4 areperformedtheonesfor
5 aretrivialand
the group is easily demonstrated to be anomalous. In fact, the generators of
5 are the same as those of
4 . This means that the generators used
before(2.5.4.1.1)canbeusedinthiscaseaswell.
Theonlyconditiontobecarefulaboutisthatthetraceidentitiesof
must
nowbeusedwith setto5.
This of course only affects the trace of the sixth power, so that (2.5.4.1.2) and
(2.5.4.1.3) still hold in this case. After the system of equations is computed we
have:
386
3
327
15
4
10
andthetheoryisanomalousforthesamereasonsexplainedabove.
3.Combinationsofanomaloustheories
3.1Outlineofthechapter
Insection2.5.4weaccomplishedmuchinourstudyof496dimensionalgroups.
It is now known that even the low dimensional groups are individually
anomalous.However,theG.S.mechanismisaglobalpropertyofthetheory:that
26
is, different contributions can combine in such a way that the total anomaly of
the combination factorizes. This is what is done when 1 theories are
combinedwith orwhen iscombinedwithitself.
Hereweshowthatifanyofthecontributionsisanomalous,theinformationof
the anomaly is transmitted to the whole group, thus making it anomalous. In
and
1
cancanceltheanomaly
doingso,wecanconcludethat
onlybecausethebasestheoriesareanomaly‐free.
To analyze this statement, combinations of classical and exceptional groups,
for
2,3,4,5and
2 must be checked. In general, if two theories
combinethetotaltraceintheadjointrepresentationissuchthat:
This helps us rule out combinations involving
for
3,4,5as well as
anomalousclassicalandexceptionalgroupsfromtheverybeginning:inthecase
of
3 ,
4 and
5 the 6‐degree term survives for whatever the
choiceofthelattertheoryis.Inthecaseofanomalousexceptionalandclassical
groups, the 12‐degree term is the contribution preventing the G.S.
mechanism.
Therefore, to investigate how the information of the anomaly is transmitted,
anomaly‐free theories constructed out of
2 and
2 must be studied:
generally speaking, it is possible to construct three types of gauge group using
such two bases and the anomaly‐free 1 . In full generality, the three groups
are:
2
1
2
1
2
2
1
Here we have used the fact that
2 and
2 have 3 and 10 dimensions,
respectively.
We will show by induction that these groups possess a non‐factorizable total
anomalypolynomial.Infact,wewillstartfromthesimplestcaseforeachofthe
three groups, e.g. wewill start by a combination of only two groups and 1 forsomesuitable .
By induction it will be clear that adding more groups will not cancel the
anomalouscontributionsandthatcombinationsinvolvingtheabovegroupsand
areanomalous.Thisproofwillworkinawayanalogoustofallingdominoes:
tomakeallthedominoesfall,oneneedsastartingpieceandalittlethrust;inthe
same way, to rule out all the combinations of anomalous groups one needs a
starting anomalous theory, e.g. the ones found in the previous chapter, and a
27
little thrust, e.g. a check that even the simplest combination will not cancel the
pre‐existinganomaly.
To construct the total anomaly polynomial we will be as general as possible:
specifically, we will not start from the anomaly polynomial for n=496 that has
beenusedtofindthoseofthelowdimensionalgroups[see(2.5.1.4)],butrather
wewillbeinterestedindevelopingatotalanomalypolynomialstartingfromthe
mattercontentofeachtheory.
We also check in the Appendix that starting from the matter content allows
recovering the factorization of groups that are known to factorize, hence
allowingustoconstructGreenSchwarztermsforthem.
3.2Combinationsof
and
Asanticipated,weconsiderthesimplestcase
2
2
1
.The
individualcontributionstothetotalanomalypolynomialaretheanomaliesof
supergravityandoftheYang‐Millstheory.Toremindthereader,theanomaly
arisingfromsupergravityis:
| wheretheindividualcontributionsaretheonesusedalreadyinsection1.2:
(3.2.1)
1
64 2
1
32 2
1
1
5670
1
360
1
4320
1
288
1
10368
1
1152 2
(3.2.2)
As far as the Yang‐Mills theory is concerned, we need to calculate an anomaly
polynomialfor
2 andonefor 1
.
provides a general basis for both theories: in the case of 1
, the
anomalyofthespiniscalculatedsetting 1
1
≡ and
0.
In the case of
2 the trace relations (2.5.2.1), (2.5.2.2) and (2.5.2.3) are
insertedin(3.2.2),with 1
2 ≡ .Theresultsare:
1
64 2
1
8 2
(3.2.3)
5670
1
360
4320
1
288
10368
1
144 2
28
2 3
(3.2.4)
Starting from (3.2.4) it is trivial to obtain an expression for the anomaly of
2
2 .Infact,theonlyrelevantconsiderationistodistinguishafactorR
commontoboththeoriesandafactorFthatisdifferentandlabeledaccordingly.
1
64 2
5670
4320
10368
1
1
1
360
288
8 2
1
144 2
2 3
2 3
(3.2.5)
1
2
2 theory.
where denotesthedimensionsoftheadditional
ThetotalanomalyoftheYang‐Millstheoryissimplythesumofcontributionsto
the gauge theory, namely (3.2.3), (3.2.4) and (3.2.5). We use the fact that
496towrite:
1
496
496
496
64 2
5670
4320
10368
1
1
1
8 2
360
288
1
144 2
2 3
2 3
(3.2.6)
1
2
Inconclusion,theYang‐Millsiscoupledtothesupergravityinordertoobtainthe
totalanomalypolynomialof
2
2
1
.
This, of course, means that the contributions to the total anomaly are
,
and 12
1
2
|
1
64 2
1
8 2
1 .
1
6
1
24
1
360
1
144 2
1
288
1
2 3
2
2 3
(3.2.7)
29
Oncethetotalanomalyhasbeenobtained,afinalcheckofitsfactorizationmust
becarriedout:asusual,thefactorizationmustsatisfy
∧ .
Inprinciple,then,thefactorizationhasthefollowingform:
∙
Here we simply use the method of expanding and comparing that we have
alreadyexploitedtocalculatethefactorizationoflowdimensionalgroups.
1andproceedtoobtain
,
,
,
Weset
and
upon comparison with, respectively,
,
,
,
and
.
Thesenewlycalculatedcoefficientsallowustotackle .Infact, isrepresented
byanoverdeterminedsystemofequations,alongthelinesofthesystemsfound
insection2.5.Specifically,twoequationsdescribe :oneisobtainedcomparing
thetermsproportionalto
,theothercomparingthoseproportional
to
.
Allinall,thesystemyieldstwodifferentvalues,withtheobviousconclusionthat
2
2
1
is indeed an anomalous theory. We can think of the
overdeterminedsystemastheanomalouspartof
2 .Onecanalsocheckthat
if the calculations are repeated for , will be analogously impossible to
retrieve:thereasonisofcoursethefactthat carriestheanomalyofthesecond
2 theory, which is related to the curvature
from which the
overdeterminedsystemarises.
As a corollary, the above result sets constraints to theories in 496 dimensions
constructed increasing the number of 2 theories and simultaneously
decreasingtheorderof 1 .
Letuselaborateonthisthought. isapropertyofthetheoryasacombinationof
supergravity and Yang‐Mills, therefore it does not change in the domain of
theorieswith496dimensions:infact,itwillbeclearinthenextexamplesthat
thefirsttermoftheanomalyisalways
1
1
1
∝
64 2
6
24
Consequently,thistermisinvariantunderadditionof
2 theories.
As far as the curvature F is concerned, adding more theories increases the
number of labeled curvatures, since these depend on the individual
contributions.
However, the system of equations that invalidates only depends on
and
.Thelattertermsurvivesuponadditionof
2 theories,sincethesecond
termofthetotalanomalypolynomialtriviallybecomesproportionalto:
30
∝
…
willcancelthepre‐existing
.
anditisclearthatnoterm
Byinduction,therefore,theanomalyiscarriedalonginallgroupsoftheform
2
1
sothattheyareallanomalous.
Accidentally,combinationswiththeanomaly‐free arenotpermittedaslongas
thereisa
2 theory,sincethiswillcarryananomalythatcannotbecancelled
upon addition of . One way to think about this is to consider that, as far as
anomaliesareconcerned,theproofusedfor isinvariantuponpermutationof
1 and .
3.3Combinationsof
and
The simplest theory in this case is 2
4
1
. The YM
contribution is obtained summing two anomalies arising from
2 and one
.Asusualforthelatterweset 1
1
476and
0.
from 1
Asfartheformerisconcerned,weinsertthetraceidentitiesfor
2 [(2.5.1.1),
(2.5.1.2),(2.5.1.3)]in(3.2.2).
1
64 2
3
16 2
—
(3.3.1)
10
5670
1
1152 2
1
360
5
10
4320
2
3
10
10368
1
288
2 2
42
12
2
(3.3.2)
Starting from (3.3.2) we trivially generalize the anomaly to include a second
contributionof
2 :
1
20
20
20
64 2
5670
4320
10368
3
1
1
16 2
360
288
1
2
2
2
3
12
3
12
1152 2 5
31
—
1
720 2
9
2
1
2
42
1
4
1
2
9
2
2
2
42
2
4
TheYang‐Millsistheneasilyobtainedfromtheindividualcontributions:
1
496
496
496
64 2
4320
10368
5670
3
1
1
16 2
360
288
1
2
2
2
3
12
3
12
5
1152 2
1
9
9
2
2
—
42 1 4 1 2
42 2 4
1
2
720 2
2
2
2
2
(3.3.3)
2
2
(3.3.4)
Finally, the total anomaly polynomial is retrieved coupling the anomalies of
Yang‐Millstheoryandsupergravity,exactlyasdoneinsection3.2.
1
1
1
64 2
6
24
3
1
1
16 2
360
288
1
2
2
3
12 1 2 3
12 2 2 1
2
1152 2
1
9
9
2
2
—
42 1 4 1 2
42 2 4 2 2 1
2
720 2
2
2
(3.3.5)
It is worthwhile pointing out that, as expected, the first term is the same as
(3.2.7).Theanomalypolynomialmust,fortheusualreasons,factorizeinsucha
waythat:
∙ As a starting point we set
1. Then we obtain
,
,
,
,
,
and
.
,
These values have been obtained comparing terms proportional to
,
,
,
,
and
,respectively.
Suchconstantsleaveuswithanusualoverdeterminedsystemofequationsfor obtainedcomparing
and
.
A trivial inspection shows that the system is impossible and that therefore
2
2
1
isanomalous.Inconclusion,
32
2
1
isfoundtobeanomalousforthesamereasonsasthepreviousgeneralizedgroup
. It is not even possible to combine with
2 in such a way to obtain an
anomaly‐freetheoryin496dimensions.Thisisquicklyruledoutbytheanomaly
carried by
2 , which survives upon addition of anomaly‐free as well as
anomaloustheories.
3.4Combinationsof
,
and
Weexpectalltheoriesofthisformtobeanomalous,thereasonbeingthatthey
areconstructedoutofanomalouscontribution.
Asacheck,however,wewilltrytofactorize
2
2
1
.Thiswill
allowustomakemoregeneralconclusionsforthegroup:
2
2
1
(3.4.1)
ThetwoanomaliesthatadduptotheanomalyoftheYang‐Millstheoryare:
(3.4.2)
1
16 2
1
64 2
13
5670
2
1
1152 2
1
—
16
720 2
13
4320
1
360
3
2
5
1
2
9
2
SothattheanomalyoftheYang‐Millsis:
1
496
496
64 2
5670
4320
1
1
2
3
16 2
360
1
2
8
1152 2 5
2
2
—
16
1
2
1
288
2
8
13
10368
2
2
3
2
42
12
2
4
2
2
(3.4.3)
496
10368
1
288
2
2
3
42
2
4
12
2
2
(3.4.4)
andfinallyweretrievethetotalanomalypolynomial:
1
1
1
64 2
24
6
1
1
1
2
3
360
288
16 2
1
2
2
8
3
1152 2 5
2
2
—
16
42 2 4
1
2
33
2
12
2
2
(3.4.5)
Theproposedfactorizationis:
∙
However,followingthesamemethoddevelopedaboveandcomparingtheusual
termsinF ,wecheckthattheoverdeterminedsystemofequationsisexactlythe
. Comparing the terms inF , we instead
one found for
2
2
1
obtainthesameequationsof
2
2
1
.Thisreinforcesourideas
that the overdetermined systems of equations carry the information of the
anomalyforthegroupoutofwhichtheyareconstructed.
Eitheranomalyisnotcancelleduponadditionofanyterm.and isanomalous.
This last section concludes our analysis of theories in D=10. It is found that,
indeed,
32 ,
, 1
and
1
aretheonlygroupswherethe
anomalyiscancelled.
However, the hope is that this explicit analysis could be of any help in
understandingthequitevastliteratureinanomalycancellation.Wealsohopeto
haveconveyedaconvincingargumentonhowanomaliesbehavewhentheories
areconsideredindividuallyandincombinationwithotherones.
Acknowledgements
First and foremost I would like to thank professor Yuji Tachikawa, without
whose advice and daily support this project would have never started nor it
couldhaveeverbeencompleted.
I would also like to thank the University of Tokyo for the opportunity to
participate in the UTRIP program for undergraduate students and for funding
mysix‐weekresearchstayinJapan.
Lastbutnotleast,Iwouldliketothankalltheprofessors,postdocsandgraduate
students of the Department of Physics of the University of Tokyo, who have
mademefeelathomefortheentiredurationoftheprogram.
34
Appendix
A.ConstructingaGreen‐Schwarztermfor
TheanomalyoftheYMtheorycoincideswiththeanomalyarisingfrom 1 .
496
1
496
496
64 2
5670
4320
10368
Thetotalanomalybecomes:
1
1
1
6
64 2
24
wherewehaveset,asusual,
0.Fromthetotalanomalypolynomialwecan
trivially see by inspection that the term
is common to both addends.
Therefore,thefactorizationofthepolynomialisreadilyobtained:
1
1
1
64 2
6
24
Thecountertermneededtocancelthepolynomialhasthefollowingform:
Δ
∧ since, of course, ,
0for the theory we are considering. The Green‐Schwarz
termischoseninsuchawaythatitcancelstheanomaly.Specifically:
1
1
1
64 2
6
24
sothat
1
1
1
Δ
64 2
6
24
Δ
0
e.g.thetotalanomalypolynomialiscancelledviatheGreen‐SchwarzMechanism.
B.ConstructingaGreen‐Schwarztermfor
In order to construct the anomaly of the YM theory we need the individual
:
contributionsfrom and 1
35
For
1
64 2
248
5670
248
4320
248
10368
weusethefollowingtraceidentities[3]:
1
100
1
7200
Thenthecontributiontotheanomalyis:
1
248
248
64 2
4320
5670
1
1
1
32 2
360
288
2 3
1
518400 2
248
10368
1
115200 2
Sothatthetotalanomalypolynomialis:
1
1
1
64 2
6
24
1
1
1
32 2
360
288
2 3
1
115200 2
Thefactorizationofthetheoryhasthefollowingform:
∙
Uponexpansionandcomparisonwiththetermsofthetotalanomalypolynomial
thefollowingcoefficientsareretrieved:
1
1
384 2
30
4
450
30
Wehaveobtainedthefactorizedformthatallowsuscancellingtheanomalyvia
theGreen‐SchwarzMechanism.Theparticulargaugetermis:
36
1
384 2
4
450
30
Sothat:
Δ
1
384 2
1
30
4
450
30
AsrequiredtocanceltheanomalyviatheGreen‐Schwarzmechanism.
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