1

AnAnalysisofAnomalyCancellationforTheoriesinD=10

AndreaAntonelli1,2

1DepartmentofPhysics,FacultyofScience,UniversityofTokyo,Bunkyo‐ku,Tokyo133‐0022,Japan2DepartmentofPhysics,King’sCollegeLondon,TheStrand,LondonWC2R2LS,UnitedKingdom

AbstractWeprove that the swampland forD=10 1SUGRAcoupled toD=10 1SYM is only populated by 1 and 1 .With this goal inmind,wereview the anomalies for classical and exceptional groups, retrieving traceidentities up to the sixth power of the curvature for each class of groups.Weexpand this idea for low‐dimensional groups, for which the trace of the sixthpower isknownto factorize,andweretrievesuch factorization.Weobtain thetotal anomaly polynomials for individual low dimensional groups andcombinationsofthemandfinallyweinvestigatetheirnon‐factorization,insuchaway that 1 and 1 are non‐trivially shown to be the onlyanomaly‐free theories allowed in D=10. Using the method developed forchecking the factorization of gauge theories, we retrieve the Green‐Schwarztermsforthetwotheoriespopulatingtheswampland.

2

Contents1.Introduction31.1Motivationsbehindthispaperandorganizationofthematerial31.2TheGreen‐SchwarzMechanism41.3Anomalycancellationin SO(32) and E8xE862.AnomalycancellationinLiegroupswith496generators72.1Aimofthepaper72.2CartanClassificationofSimpleLieGroups82.3AnomaliesinClassicalGroups9 2.3.1 92.3.2 112.3.3 122.4AnomaliesinExceptionalgroups132.4.1G2132.4.2F4162.4.3E6172.4.4E7182.4.5E8192.5Some“strange“low‐dimensionalgroups202.5.1 2 202.5.2 2 222.5.3 3 232.5.4 4 and 5 242.5.4.1 4 242.5.4.2 5 253.Combinationsofanomalousgroups253.1Outlineofthechapter253.2Combinationsof 2 and 1 273.3Combinationsof 2 and 1 303.4Combinationsof 2 , 2 and 1 32Appendix34A.ConstructingaGreen‐Schwarztermfor 34B.ConstructingaGreen‐Schwarztermfor 34

3

1.Introduction1.1MotivationsbehindthispaperandorganizationofthematerialInrecentyears,followingtheworkofVafa,the“swamplandprogram”hasaimedatgivingaboundarytotheswamplandoftheeffectivetheoriesthatarenotfullyembeddedintheoriesofquantumgravity[11,12].Oneimportanttoolathandforphysicistsistheconceptofquantumanomaly,andconsequentlyanomalieshavebeenstudiedindetail.In D=10 it has long been known [3] that 32 , , 1 and1 are theorieswhere quantum anomalies are cancelled. The former two

arealsolowenergylimitsofstringtheoriesanddonotthereforepertaintotheswampland. 1 and 1 , conversely, do live in the swamplandand itwasnotknownuntilrecentlywhetherornottheywerealso lowenergy limitsofstringtheories.FollowingtheworkofFiol[10]andthejointworkofAdams,DeWolfeand Taylor [4] it was shown that there are no theories of gravity that can becoupled to them, thus confining both theories to the swampland, withoutpossibilities to be upgraded to string theories. However, there is still oneunanswered question: are 32 , , 1 and 1 the onlytheorieswheretheanomaliesarecancelled?Hintsforthisstatementcanbetracedbackintheliterature,mostlyfollowingtheoriginal statement found in chapter 13 of [3], but noproof has explicitly beencarried out. Therefore answering this question might help understand thestructureoftheswamplandinD=10.In this paper it is shown that indeed there are no other theories in theswampland in D=10: in the process of proving, we devise some non‐trivialmachinery that isworthwhile presenting and thatmight be useful in anomalycancellation in other dimensions; also, we deal with low‐dimensional theoriesthatareoftenoverlookedintheanalysisofanomalycancellation.Thereaderisencouragedtoreadthefirstpartofthispaper,astheremostofthematerial and notation is presented: more in detail, the Green‐SchwarzMechanismisrevisitedinthenotationofBilal[5]andsomeimportantpointsfortheupcomingsectionsareexplained.Plus,proofwithappropriatereferencesaregivenforthenon‐factorizationofthetraceofthesixthpowerofthecurvatureinclassicalandexceptionalgroups,followingtheCartanclassificationofsimpleLiegroups.Theexpertreadermightwanttoskiptheclassificationandnon‐factorizationofclassical and exceptional groups and dive into the last part involving low‐dimensionalgroups.There,theconceptoffactorizationistreatedindetailanditisshownwhyanomaliesofindividualgroupsarecarriedalongwhencombiningthem and specificallywhat non‐canceling terms of the polynomial contain theinformationabouttheanomalyofthetheory.Thispapermightbeof interest also to readers looking for the factorizationornon‐factorizationofthetraceofthesixthpowerofthecurvatureinallclassical,exceptional and low‐dimensional groups, although a fairly more rigoroustreatmentformost(butnotall)ofthemisfoundin[6]and[8].

4

1.2TheGreen‐SchwarzMechanism

Itisinstructiveatthispointtoreviewthemechanismdevisedin[2]thatallowsanomaly cancellation. In doing so, the reader will be reminded of numerousconcepts that will be fundamental for understanding the upcoming sections.Moreover,severaloftheideaspresentedhereinwillbeusefulwhentheGreen‐Schwarzmechanismisusedfornon‐trivialgroupsinthefinalsections.Wewillbeginsayingthatalltheinformationabouttheanomalyofasystemcanbe encapsulated in the total anomalypolynomial [5]: inprinciple, then, such apolynomialcanvanishifacountertermdependingonthechoiceofaparticulargaugetransformation isadded. If thishappens, thegaugetheorybecomesnon‐anomalous(or,equivalently,anomaly‐free).TheGreen‐Schwarzmechanismdoesexactlythis; thegroundbreaking ideawasto devise an “inflow”method to construct a counterterm and then recast it interms of a 12‐formpolynomial, in such away to be able to add it to the totalanomalypolynomialandcancelit.We will now explain the method in much more detail referring to a tendimensionalgaugegroupGthatisa =1Supergravitycoupledtoa =1SuperYang‐Mills theory. As a remainder, thematter content of SUGRA is amultipletcomprisingapositivechiralityMajorana‐Weylgravitinoandanegativechiralityspin ½ fermion. Moreover, SUGRA contains the graviton, a scalar renameddilatonandatwo‐formB.Ontheotherhand,theSuperYangMillscontainsamultipletofgaugefields andgauginos thatliveinthesameadjointrepresentation[5].Therequirement foraconsistentcoupling isgivenby the followingexpressionforthefieldstrength:

, , ≡ where , isthegaugeChern‐Simonsformand , isthegravitationalone [5]. The invariance of H when acted upon with a gauge and Lorentztransformationisgivenasfollows:

, , Given the above ingredients, it is now possible to choose a counterterm with an appropriate 8-form retrieved from the characteristic classes:

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)

164 2 1

15670

14320

110368

132 21360

1288

11152 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:

| |

164 2

4965670

2244320

6410368

132 21360

1288

11152 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 3

15 (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:

1384 2

14 8

In order to cancel this it is clear that the counterterm ΔΓ must be constructed from the following chern 8-form:

1384 214 8

Δ 1384 2

14 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,thereforethedimensionsofthecombinationareconsistentwiththefirsttermoftheanomalypolynomial.Againtraceidentitiescanberetrieved[5]:

9 9 , 75207520

where the indices are used to distinguish between the curvatures of the twogroupsE8 involved. The total anomaly polynomial again factorizes:

1384 2

14 2 2 2

TheGreenSchwarztermisthenchosentobe:

1384 2

14 2

2 2 So that, following the same reasoning as above, the total anomaly polynomialcancels.Itturnsout,toconclude,thattherearetwomoregroupswheretheanomalyiscancelled, namely 1 and 1 [3]. These groups, however, do notgive rise to string theories [4,10]; both groupswill be treated conceptually insection2.4.5andalgebraicallyintheAppendix.2.AnomalycancellationinLiegroupswith496generators2.1AimofthepaperItisoftensaidthatfourgroupscanceltheanomaly: 32 , , 1 and

1 .ForthefirsttwoexplicitcalculationshavebeencarriedoutintheIntroductionsection,forthelattertwoitisrecommendedtoreadtheAppendixofthispaper.Nonetheless the set of Lie groups with 496 dimensions contains many moregroups.Thesituationlooksthenlikethefollowing:

8

Fig1.SetofalltheLieGroupsin496dimensionsanditsinternalclassification.WherethesetAcontainsthetheoriesinwhichtheanomalyiscancelledandthesubsetBoftheoriesembeddedinatheoryofquantumgravity.This paper aims at checking that the above‐mentioned quartet of groupscontainsall thegaugegroups forwhichtheanomaly iscancelledandthereforethat|A|=4.2.2CartanClassificationofSimpleLieGroupsThe Cartan classification of Lie groups into classical and exceptional groupscomes in very handy for our purposes. All the Lie groups canbe reduced to4classicaland5exceptionalgroups[6]:

ClassicalGroups

ExceptionalGroups

An (n ≥ 1) compact Bn (n ≥ 2) compact Cn (n ≥ 3) compact Dn (n ≥ 4) compact

E6 E7 E8 F4 G2

Table1.CartanClassificationofSimpleLieGroups.Themainideaistocheckthatgaugetheorieswith496dimensionsconstructedout of classical groups, exceptional groups andmixtures of the two (with theexception of ) inevitably carry anomalies that cannot be canceled. We

32

1 1

BA

LieGroupswith496generators

9

willseethatsuchstatementisintrinsicallyrelatedtothenon‐factorizationofthetraceintheadjointrepresentationofthesixthordercurvature .Once the classical and exceptional groups are found to be anomalous, it isobtainedasacorollarythatnoanomaly‐freetheoriescanberetrievedcouplinganyofthegroupto 1 foraparticularchoiceofn(seesection3.1),againtheonlyexceptionbeingthetheoryconstructedvia ,e.g. 1 .In the next paragraph we will show how the anomaly is preserved in somerepresentatives of the classical groups, that nonetheless carry as muchinformation as the above‐mentioned An, Bn, Cn, Dn. In doing so, we will derivesomeusefultraceidentitiesthatarecrucialinourstudyofnon‐trivialgroupsinthelastsection. 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 Totacklethisclassofgroupsweneedtofindtherelationbetweenthetraceoftheadjointrepresentationandtheoneinthefundamentalanddemonstrateforwhichgroups canorcannotfactorize.Thestartingpointofthediscussionistheactionofthegroup: indeedactson an anti‐symmetric carrier space denoted by that, being anti‐symmetric,satisfies .Thetransformationoperatedbytheactionofthegaugegroupis[7]:

→ Ω with

Ω ∑ Ω (2.3.1.1)The indicesk and lrunover thedimensionof thegroup,e,gk,l=1,….,½n(n‐1),wherewerename 1 forfuturereference.What has been found so far is an action via an adjoint representation of thegauge group. It is however possible to rewrite the adjoint transformation intermsofthefundamentalrepresentationO,sotolinkadjointandfundamentalinafirst,unpolished,relation.Thisisdoneasfollows(confrontappendixFof[7]): Ω ∑ ′′ ′′ ′ ′′, ′ 2∑ ′′ ′′ ′ ′′ ′ (2.3.1.2)Bycomparing(2.3.1.1)and(2.3.1.2)arelationbetweentheactionoftheadjointandfundamentalrepresentationsisfound:Ω (2.3.1.3)

10

wheretheminussign isduetotheasymmetryof thecarrierspaceandplaysacrucialroleindetermining 32 astheonlypossibleconsistenttheoryfor

.Letusseehow.Therelation(2.3.1.3)hasbeenworkedouttothepointwherethevectorspacedoesnotappearanymore.Sinceweeventuallywantarelationbetween and wecalculatetheformeratthisverystage:

Ω Ω 12 Ω, Ω ∑ , (2.3.1.4)Withequation(2.3.1.4) thetraceof theadjointhasbeenrelatedtothetraceofthefundamentalrepresentation.However,theactionastheargumentforbothisnot as useful as a general element of the Lie Algebra: as far as anomalies areconcerned,indeed,thetraceisusuallycalculatedoverthecurvature ,whichisanelementoftheLiealgebra.Theeasiestwaytoproceednowistorecasttheactiononthevectorspaceasanexponentialrepresentation:

Ω ∈ ∈

ItisalwayspossibletoTaylorexpandanelementoftheLieAlgebraaroundtheidentity: Ω 1 2! 3! . .

12 1 2! 3! . . 1 222!

23! . .

wherethepreviousidentity(2.3.1.4)hasbeenusedinthesecondstep.Itisnowaneasytasktoretrievethetraceidentities,sincewhatislefttodoiscomparingthetermswithsameorder:Table2:TraceIdentitiesforSO(n)

1 12 1 1 2 102 8 3 32 15

11

wherewehaveused 1 .Forsimplicity,wehavedroppedthesymbol ;itisintended,however,thattheelement intheLHSofwhichwecalculatethetracealwayslivesintheadjointrepresentation.Theboxof trace identities forSO(n)allowsustoansweramajorquestionthatwasleftopenacoupleofparagraphsago.TheonlySO(n)groupthatisallowedisSO(32) because of the last identity; moreover the identities used in theintroduction are retrieved from the above ones. For 32, indeed, doesnot factorize and therefore remains unchanged in (1.2.3), which consequentlycannotbefactorizedin8‐formsand4‐forms.Ifthetotalanomalypolynomialisnotfactorized,theanomalycannotbecanceledviatheGreen‐Schwarzmechanism.SO(32)alsocontains496generatorsandiscoupledtoaknownstringtheory.AlltheseconditionsguaranteeaspecialplaceinsubsetBofFig.1tothisveryspecialgaugegroup.AsfortheremainingSO(n)groups,theycannotevenbeincludedinsubsetAof“groupswheretheanomalyiscancelled”forthereasonsjustcited.2.3.2Sp(n)

isaclassofgroupsthatsharesseveralpropertieswith .Asfarasourtopic is concerned, the major and crucial difference between the two gaugegroupsliesinthecarrierspace:itisanti‐symmetricfor andsymmetricfor

.Theanalysisofthelatterclassgoeshandinhandwiththatoftheformer,withthe,atfirst,harmless‐lookingdifferenceinthesignoftherelationbetweenthetracesinadjointandfundamentalrepresentations.For wehave:

Ω

whichleadsto:

Ω 12 At thispoint theexpressioncanbeTaylorexpanded inmuch the samewayasbefore: Ω 1 2! 3! . .

12 1 2! 3! . . 1 222!

23! . .

Sothat,finally,thetracerelationsfor canbefound.Table3:TraceIdentitiesforSp(n)

12

1 12 1 1 2 102 8 3 32 15

Andhereisthecaveat.Therecanbenogaugegroupforwhichncancelsthefirstterm of the last identity. Therefore, as the term survives, the anomalycannotbecanceledviaG.S.mechanism.2.3.3 Inthecaseof theactiononthecarrierspaceisgivenby[7]:

; ∗ whereas the relation between adjoint and fundamental actions is obtained bytakingthedirectproductofnandn*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! . .∗

1ItisimportanttonoteatthispointthatFisanti‐hermitianandthereforesatisfies∗ . Expanding the RHS in light of this property and then comparing the

termswithsamepowerleadstothetraceidentitiesforSU(n):

13

Table4:Traceidentititesfor

1 102 2 6 2 30

20 Weseethatthetraceofthesixthordercurvatureisnotcancelledandthereforeall theoriesareanomalous.2.4AnomaliesinExceptionalgroups2.4.1G2G2hasbeenwidelystudied,thereforemanyofitspropertiesarewell‐known.Itisknown, for example, that the group possesses 14 generators and that thefundamentalrepis7[7].Forourpurposes,wewillonlybeconcernedwiththemaximalsubgroup 2 .The action of the maximal subgroup on the 7= , ̅ , , where ∗ ̅ representhermitian spinorsandy representsa scalar,defines transformationsclosed under a Lie Algebra with generators (see [7] for more details). ThedecompositionofadjointandfundamentalrepsunderSU(3)yields:

⟶ ∗, ⟶ ∗ ,

G2hasonlytwoCasimiroperators,C2andC6:thereforebothofthemplayaroleindetermining . Specifically, the identityof thesixthpowermustsatisfy thefollowingrelation: (2.4.1.1)Let us have a look at this equation. As repeatedly emphasized, anomalycancellationcanonlyoccurwhen factorizes.Inthemethodreportedinthissection, suchstatement isequivalent to saying that isvanishing: if this is thecase, indeed, can be expressed in terms of traces of lower power in thefundamentalrepresentation.Conversely,if 0thetheoryisanomalous.ThegeneratorsoftheLieAlgebraarethestartingpointofthediscussionforG2;thisisagainduetotheirrelativeeasetousecomparedtothegeneralelementsofthe Lie Algebra such as the curvatureF. Throughout the next paragraphs, thecalculationsthatwillbeperformedusingthegenerators.Indeed,theyareeasierto use and a non‐vanishing trace of the sixth power of the generator in theadjointrepresentationbearsthesameconsequencesofanon‐vanishing .Inourcasethegeneratorsofthemaximalsubalgebraare:

14

20 1 01 0 00 0 0

20 0

0 00 0 0

21 0 00 1 00 0 0

20 0 10 0 01 0 0

20 00 0 0

0 0 20 0 00 0 10 1 0

20 0 00 00 0 2√3

1 0 00 1 00 0 2

Wecanthenretrieveanexpressionforthesecond,fourthandsixthpowerofageneratorandcalculatetherelevanttraces.Thetraceoftheadjointiscalculatedusingthe identitiesof ; thechoiceofcoursebeingrelatedtothemaximalsubalgebra 3 .Since relation (2.4.1.1) contains two constants, the proceduredescribed abovemustbecarriedfortwogenerators,sotoobtainasystemoftwoequationsintwovariables:thegeneratorsofinterestforusare and .Webeginbycalculatingafirstexpressionintwovariablesusingtheformergenerator:

14

1 0 00 1 00 0 0

1641 0 00 1 00 0 0

andtherefore,

164

164

264

∗ 164

164

264

Thelasttracetoobtainbeforecalculatingthetotaltraceoftheadjointis .Upon using the trace identities of SU(n) the following result is obtained:

2 116andinconclusionthetraceoftheadjointrepresentationisgivenby: 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) the firstexpressionuptotwoconstantsisobtained: 1 (2.4.1.4)Thisisclearlynotenoughtodetermine and ,butitshowsthepathtofollowtoobtainasecondexpressionofsuchtype.Nowletuscalculatethesquare,fourthpowerandsixthpowerof .

112

1 0 00 1 00 0 4 24√3

1 0 00 1 00 0 8

11441 0 00 1 00 0 16

112

1 0 00 1 00 0 64

Hence,thetracesofthefundamentalrepresentationaregivenby 1, 66 66 whereasthetraceoftheadjointis: Theexpressionobtainedinthiscaseis: 1 (2.4.1.5)Itisclearthatthisisincompatiblewith(2.4.1.4).Afirst,naïve,thoughtwouldbetoconsidertheonlycaseofinterestforus,namely 0.Suchparticularresultleads to two different values of in the system; clearly a contradiction.Tobemoreprecise,thesystemofequationsyields 26and .Relation(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 factorizesinto ; in other words, one might wonder if is a primitive Casimir.The above discussion rules out this possibility since and are determineduniquely.Suppose,conversely,thatitispossibletofactorize asfollows: (2.4.1.7)Theninserting(2.4.1.7)into(2.4.1.6)weobtain:

16

26 154

Sothatinthiscasethemostgeneral(2.4.1.1)hassolutions:

, 0, 26 154 Thesevaluesare clearlydiffering from theuniqueconstantsobtainedwith thegeneralmethod,sothattheinitialassumptionisnotvalidand isaprimitiveCasimirbycontradiction.Thisproofholdswhenevertheconstantsof aredetermineduniquely.Aswe shall see this is the case for all the exceptional groups, so that is aprimitiveCasimirinallthegroupsthatwewillbeinterestedin.2.4.2F4

F4 has 52 generators and 9 as the maximal subalgebra. The fundamentalrepresentationis26.Thedecompositionunder 9 is:

⟶ , ⟶ ,

F4hasalsorank4,thereforeitpossesses4Casimiroperators,whichare:C2,C6,C8,C12.[8]Inlightofthis,theexpressionforthetraceisstill(2.4.1.1).Theproblemisstilltocheckif 0.Thetworepresentationsofthegeneratorschosenare:

0 1 01 0 00 0 0

…⋮ ⋱

0 1 11 0 01 0 0

…⋮ ⋱

Thespinorrepresentationisgivenby [7].Thetraceofthefundamentalrepresentationisthesumofthecontributionfromthegeneratorandthatofthespinorrepresentation.Thereforewehave:

2 164 62 1664

94

andfollowingthetraceidentitiesfor weobtainastheadjointtrace:

14 1664574

whichleadstoanexpressionforthetraceintwoconstants:

17

574

94 216

The same procedure is repeated for the second generator. The spinorrepresentationisinthiscase .Onethenobtains: 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 traceidentitiesof .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:

3 772 ThereforeF4isanomalous.2.4.3E6

Thisgrouphasadjointrepresentation78andfundamental27.Thereareseveralsubgroupsthatonecouldusetodecomposesuchrepresentations.Two examples are 10 1 and 8 , underwhich the decompositioncanbefoundin[9].AnothersuitablesubgroupisF4andwewilluseitsincemostofthecalculationsinvolvedhavealreadybeencarriedoutinsection2.4.2.UnderF4thedecompositionis:

⟶ , ⟶ ,

As far as theCasimiroperatorsare concerned, the relevantones areC2andC6.Hence, the trace to find is still (2.4.1.1). The trace of the fundamental rep forgenerator1isthesameas(2.4.2.1),sincethefactor1doesnotcontributetothetrace.Weseealsothatthetraceoftheadjointisthesumoftraces(2.4.2.1)and(2.4.2.2),alreadycalculatedpreviously.Thefirstrelationtheniseasilyobtained:

144 12 1728 Similary,thetracesfundamentalandadjointrepresentationarecalculatedusing

and fromtheprevioussectiontoobtain:

18

332

94

94

Thesystemofequationsgives , .Anomalycancellation,even inthiscase,isnotpossible.Thetraceidentitybecomes:

5308741

2246669

2.4.4E7ThenextstructureencounteredinourjourneythroughtheexceptionalgroupsisE7. The group has 133 generators and 8 as amaximal subgroup. It is alsoimportant toknowthat the fundamentalrepresentation is56.Under 8 thedecompositionisasfollows:

⟶ , ⟶ ∗,

The relevant Casimir are unchanged, hence there is no need to modify thestructure of . Decomposing under the maximal subgroup and taking intoaccountthedimensionalityoftherepresentationsweobtain:

2 2 wherewe have exploited the fact that the fundamental representation is real.Themethod used for this exceptional group differs slightly from the previousones. To check that 0we calculate the coefficients of the sixth powerelementoftheLieAlgebraforbothfirstandsecondtermintheLHS.Assuming

0such coefficient should vanish: checking that this is or is not the case isequivalenttocheckinganon‐vanishing andthereforethefactorizationornon‐factorizationoftheanomalypolynomial.As generators we choose any linear combination of the seven independentgeneratorsof 8 , labeled for i=1,2,…8[7]. is thereforeanelementof theLieAlgebraandwecanusethelasttraceidentityintable4toobtain:

8 ∙ 2∑ 15∑ ∑ 15 ∑ ∑ 20 ∑

16∑ 30 ∑ ∑ 20 ∑ (2.4.4.1)A similar argument is used for . In this case, however, the states arelabeled , , , with . Let us focus on the sixth power term.Therestrictedsumofthestatescanberecastasacombinationofunrestrictedsums[7].

19

∑ , , , 6 ∑ 2, ,3 ∑ 2 2, 8∑ 3, 6∑ 4

(2.4.4.2)Using(2.4.4.1)and(2.4.4.2)wecanfinallycollectthetwocoefficientsofthesixthpowerterms.

∝ , ∝ isreadilyobtainedbyinspectionof(2.4.4.1).Clearly, 16.Amoreinvolved

calculationisneededfor .

124 1 1 1 1 6 2 1 1 3 2 2

8 3 1 6 4 3 2 Theappropriatecaseforus is 0.Assumingthis is thecaseandconsideringonly the sixth powers, the RHS of relation (2.4.1.1) vanishes: the first term isindeedzerobyassumption,thesecondisnotproportionalto .Theconditionthenbecomes:

0 ∙ 0But clearly 0as ithasbeencalculated just above.Weconclude that must be different than zero and that consequently the factorization of isnotpossible.2.4.5E8Weendourdiscussiononexceptionalgroupswiththeonerelevanttoanomaly‐freegaugetheories.Insection1.2oftheIntroductionitwasdiscussedhow

cancels the anomaly via Green‐Schwarzmechanism. It was also anticipatedthat 1 and 1 cancel the anomalies. It is now time to spare athought on the last statement. Why does behave differently from itsexceptionalsiblings?The answer lies in the Casimir operators of . These are fori=2,8,12,14,18,20,24,30.Thereforethegroupdoesnotpossess .Consequently,thefactorizationof ispossiblebecausethereisnotsuchfactoratall.Intheliterature[2]onecanseethat and arenon‐vanishingandfactorizable[seethetraceidentitiesin(1.3.1)].Fortheabovereasons providesagoodbasetodevelopanomaly‐freemodels.In the introduction was considered, however the reader should alsoremember that the group 1 behaves differently from its class . Indeed,thisgaugegroupcancelstheanomaly(agoodreferenceischapter12of[5]).Asacorollaryany496‐dimensionaltheoryconstructedsolelyuponmultiplicationof

20

1 isanomaly‐free.Clearly,theonlypossiblewayistoconsidertheanomaly‐free 1 theory.Finally, we could combine E8and 1 in a 496‐dimensional theory that isanomaly‐freeforthereasoningjustcited.Again,thereisonlypossiblewaytodothis, e.g. 1 . For an algebraic treatment of these theories, e.g.factorization via the Green‐Schwarz mechanism, consult the appendix of thispaper.There,theGreenSchwarztermsareretrievedforboththegroupsintheswampland,inamannerthatissimilartotheanalysiscarriedoutfor and

32 intheintroduction.2.5Some“strange“low‐dimensionalgroupsItseemsthatthemethodusedaboveprovidesawaytoanswerpositivelytotheinitial 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.Suchfactorizationisapropertyofsomelow‐dimensionalgroupsthat,nicely,canallbetracedtosome for

5, 2 andproductsthereof.The method presented previously, therefore, does not stand for them asformulatedaboveandanomalycancellationmustbecheckeddifferently,insuchawaytoaccountforthefactorizationof .Fortunatelyenough,theGreen‐Schwarzmechanismholdsforanygroupandthusalso for lowdimensional groupsand 2 . Itseemsagood idea tostartfromthisreassuringmechanismandwewilldoitemployingitfor 2 first.2.5.1 Thegoalhereistofactorizethetotalanomalypolynomial(1.2.3),inlinewiththegeneral ideaof themechanismexplained insection1.3.The three termsof thepolynomialthatfactorizeinlowerorderonesare , and .Specifically,thestructureofthefactorizationissuchthat:

∙

∙ ∙

∙ ∙

Thegenerators usedforchecking 2 are:

00

00

21

Only one generator is needed for , as it contains one constant, and thefactorizationistriviallyfoundusingtheresultsinTable3: 6 (2.5.1.1)

needstwosetsofcalculationsperformedonthetwogenerators.Again,theLHSiscalculatedusingthegeneraltraceidentityofthefourthpowerfor with 2.Thetwogeneratorsyieldasystemofequationswithvariables andthat,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 anomalypolynomialofinterestforthefactorization(weadapttothenotationof[2]here):

| 115124

1960 4 5

(2.5.1.4)Relations(2.5.1.1),(2.5.1.2)and(2.5.1.3)arenowpluggedinto(2.5.1.4):

930

4215

18

121

40132

18

132

(2.5.1.5)Thequestionisnowwhetherorthepolynomialabovefactorizes.Beforegoinganyfurther,itiscompellingtoreportthattheGreen‐SchwarzinitsoriginalsenseistheapplicationofthecountertermΔΓinheteroticsupergravity:this theory has a 12‐degree Chern class that factorizes as ∧ ; that ispreciselyhowthephysicalelectricandmagneticcurrentsrespectivelyfactorize.The gauge theories that we have been considering in this paper fall in thiscategory,andthereforewhenwecheckfactorizationwealsohavetocheckthatthefollowingconstraintissatisfied: ∧ (2.5.1.6)Of course, nothing prevents the polynomial (2.5.1.4) from factorizing withoutcondition (2.5.1.6). However, the resulting gauge theories would not beconsistentfromthestringviewpoint. In lightofthis,wewillbeconcernedonlywiththeoriesthatfactorizeandsatisfytheconstraint.

22

Letusturnourattentionto 2 again.FandRhavebydefinitiondegree2.Inprinciple, therefore, the 4‐degree and 8‐degree terms can be separated up tosevenconstants:

∧ ∙

(2.5.1.7)

What is not sure, however, is whether or not seven constants that satisfy theequationsimultaneouslyexist.Thiscanbecheckedbyinspectionsettingoneofthevariablestobe .Byconsequentexpansionandcomparisonoftermsinequations (2.5.1.7) and (2.5.1.5)we obtain: 3, , , and

, which have been computed comparing terms , ,and respectively.

Letushavealookatwhatwehavesofar:

∧ 110 124

∙ 3 1768148

1192

Theonlyremainingconstantis .However,wecanseethatitcanberetrievedcomparing two addends, and . In fact, the systemsatisfying is overdetermined as there are two equations for one variable. Iftheseareinaccordanceauniquesolutionfortheconstantcanbefoundandthusthetotalanomalypolynomialcanbefactorizedcompletingtheworkdonesofar.However,comparingthe termstheresultreads 0.Acontradictionisobtainedwhenthisisinsertedintheequationcomparingthelatter term , thus proving that the anomaly of 2 cannot becancelledthroughtheG.S.mechanism.2.5.2 In section 2.5.1 we have developed a method to construct a factorizedpolynomialuptosixconstantsandtocheckitforaparticulartheory.Ofcoursetheparticulartheoryneednotbe 2 andwehereinshowthattheargumentstandsforthenextgroupconsidered, 2 .The structures of the factorization of higher order traces is even simpler than

2 asonesingleconstantispresentineachrelation:

∙

∙

∙

23

Thegenerator is 00 andthetrace identitiesarethoseof table4,withtheparticularchoice 2: 4 (2.5.2.1) 8 (2.5.2.2) 16 (2.5.2.3)Thetotalanomalypolynomialforsuchatheoryistherefore:

1615

13

1240 4 51

8132

andagainwelookforafactorizationsatisfying(2.5.1.6).Thishastheform:

∙ Let us begin by setting . By comparing terms with a factor wereadilyobtain .Then, inmuchthesamewayasdescribedabove, the twoconstants are used as bases to obtain: , 16and . Now thesituationlookslikethis:

115

12 ∙ 16

116

14

is determined by an usual overdetermined system of equations obtain bycomparison of the coefficients of and . Unfortunately,eventhistimethesystemleadstoacontradictionand 2 cannotthereforebefactorizedandcancelledviaG.S.mechanism.2.5.3 Thisisacriticalgroup.Thefactorizationofhigherordertracesisasfollows:

∙

∙

∙ ∙ Uponusing

0 00 00 0

as generator and the trace identitiesof table4with 3weget:

24

6

9

332 22

Insertingsuchrelationsintotheanomalypolynomialweobtain:

1110

2215

38

1401

3218

132 (2.5.3.1)

Andalreadyitcanbeseenthatthesecondtermisproportionalto havingdegree6.Factorizationofthispolynomialmightbepossible,butitisimpossibleforittosatisfycondition(2.5.1.6).Thereforeanon‐vanishing termleadstothenon‐factorizabilityof(2.5.3.1).Thetheoryisanomalous.2.5.4 and Thetraceidentityofthesixthpowercurvatureisforboththeories: ∙ ∙ ∙ (2.5.4.1)Anargumentsimilartothatusedfor 3 canbeexploited.Itisnowclearthatthe problem now reduces to the preliminary question: does the coefficient of

vanish?If 0, indeed,wecannoteventrytocanceltheanomalyviatheG.S.mechanismaswearepreventedfromdoingitbycondition(2.5.1.6).Thewaythisisformulatedremindsusofthemethodusedtocheckclassicalandexceptional groups in sections 2.3 and 2.4, where the question was: does thecoefficientof vanish?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 hasthefollowingtracestructures:

To obtain and one generator is sufficient, however has 3constants and thus requires 3 generators.Weuse the general tracelessmatrix

0 00 00 0

andsetthreedifferentcombinationsofa,b,c.

25

1 0 0 0000

100

000

002

2 0 0 0000

100

000

003

1 0 0 0000

100

010

003

(2.5.4.1.1)

The trace identities for with 4are used. The method is always tocreateasystemin3equationsusingthegeneratorsandsolveit.Theresultsaredirectlyshownbelow: 8 (2.5.4.1.2) 10 (2.5.4.1.3)

184475

35

80425

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 aretrivialandthe 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 usedbefore(2.5.4.1.1)canbeusedinthiscaseaswell.Theonlyconditiontobecarefulaboutisthatthetraceidentitiesof mustnowbeusedwith 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) stillhold in this case.After thesystemofequations is computedwehave:

38615

34

32710

andthetheoryisanomalousforthesamereasonsexplainedabove.3.Combinationsofanomaloustheories3.1OutlineofthechapterInsection2.5.4weaccomplishedmuchinourstudyof496dimensionalgroups.It is now known that even the low dimensional groups are individuallyanomalous.However,theG.S.mechanismisaglobalpropertyofthetheory:that

26

is,different contributionscancombine in suchaway that the totalanomalyofthe combination factorizes. This is what is done when 1 theories arecombinedwith orwhen iscombinedwithitself.Hereweshowthat ifanyof thecontributions isanomalous, the informationofthe anomaly is transmitted to the whole group, thusmaking it anomalous. Indoingso,wecanconcludethat and 1 cancanceltheanomalyonlybecausethebasestheoriesareanomaly‐free.To analyze this statement, combinations of classical and exceptional groups,

for 2,3,4,5and 2 must be checked. In general, if two theoriescombinethetotaltraceintheadjointrepresentationissuchthat:

This helps us rule out combinations involving for 3,4,5as well asanomalousclassicalandexceptionalgroupsfromtheverybeginning:inthecaseof 3 , 4 and 5 the 6‐degree term survives for whatever thechoiceofthelattertheoryis.Inthecaseofanomalousexceptionalandclassicalgroups, 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 ispossible toconstruct three typesofgaugegroupusingsuch two bases and the anomaly‐free 1 . In full generality, the three groupsare:

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 totalanomalypolynomial.Infact,wewillstartfromthesimplestcaseforeachofthethreegroups,e.g.wewill startbyacombinationofonly twogroupsand 1 forsomesuitable .By induction it will be clear that adding more groups will not cancel theanomalouscontributionsandthatcombinationsinvolvingtheabovegroupsandareanomalous.Thisproofwillworkinawayanalogoustofallingdominoes:

tomakeallthedominoesfall,oneneedsastartingpieceandalittlethrust;inthesameway, to rule out all the combinations of anomalous groups one needs astarting anomalous theory, e.g. the ones found in the previous chapter, and a

27

little thrust,e.g.acheck thateventhesimplestcombinationwillnotcancel thepre‐existinganomaly.To construct the total anomaly polynomial we will be as general as possible:specifically,wewill not start from the anomalypolynomial forn=496 thathasbeenusedtofindthoseofthelowdimensionalgroups[see(2.5.1.4)],butratherwewillbeinterestedindevelopingatotalanomalypolynomialstartingfromthemattercontentofeachtheory.We also check in the Appendix that starting from the matter content allowsrecovering the factorization of groups that are known to factorize, henceallowingustoconstructGreenSchwarztermsforthem.3.2Combinationsof and Asanticipated,weconsiderthesimplestcase 2 2 1 .TheindividualcontributionstothetotalanomalypolynomialaretheanomaliesofsupergravityandoftheYang‐Millstheory.Toremindthereader,theanomalyarisingfromsupergravityis: | wheretheindividualcontributionsaretheonesusedalreadyinsection1.2: (3.2.1)

164 2 1

15670

14320

110368

132 21360

1288

11152 2

(3.2.2)As far as theYang‐Mills theory is concerned,weneed to calculate an anomalypolynomialfor 2 andonefor 1 .

provides a general basis for both theories: in the case of 1 , theanomalyofthespiniscalculatedsetting 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) areinsertedin(3.2.2),with 1 2 ≡ .Theresultsare: (3.2.3)

164 2 5670 4320 10368

18 21360

1288

1144 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,theonlyrelevantconsiderationistodistinguishafactorRcommontoboththeoriesandafactorFthatisdifferentandlabeledaccordingly.

164 2 5670 4320 10368

18 21360

1288

1144 2

12 3 22 3 (3.2.5) where denotesthedimensionsoftheadditional 2 theory.ThetotalanomalyoftheYang‐Millstheoryissimplythesumofcontributionstothe gauge theory, namely (3.2.3), (3.2.4) and (3.2.5).Weuse the fact that

496towrite:

164 2

4965670

4964320

49610368

18 21360

12881

144 2

12 3 22 3 (3.2.6) Inconclusion,theYang‐Millsiscoupledtothesupergravityinordertoobtainthetotalanomalypolynomialof 2 2 1 .This, of course, means that the contributions to the total anomaly are ,

and 1212| 1.

164 2

16

124

18 21360

12881

144 2

12 3 22 3 (3.2.7)

29

Oncethetotalanomalyhasbeenobtained,afinalcheckofitsfactorizationmustbecarriedout:asusual,thefactorizationmustsatisfy ∧ .Inprinciple,then,thefactorizationhasthefollowingform:

∙

Here we simply use the method of expanding and comparing that we havealreadyexploitedtocalculatethefactorizationoflowdimensionalgroups.Weset 1andproceedtoobtain , , ,

and upon comparison with, respectively, ,, , and .

Thesenewlycalculatedcoefficientsallowustotackle .Infact, isrepresentedbyanoverdeterminedsystemofequations,alongthelinesofthesystemsfoundinsection2.5.Specifically,twoequationsdescribe :oneisobtainedcomparingthetermsproportionalto ,theothercomparingthoseproportionalto .Allinall,thesystemyieldstwodifferentvalues,withtheobviousconclusionthat

2 2 1 is indeed an anomalous theory. We can think of theoverdeterminedsystemastheanomalouspartof 2 .Onecanalsocheckthatif the calculations are repeated for , will be analogously impossible toretrieve:thereasonisofcoursethefactthat carriestheanomalyofthesecond

2 theory, which is related to the curvature from which theoverdeterminedsystemarises.As a corollary, the above result sets constraints to theories in 496dimensionsconstructed increasing the number of 2 theories and simultaneouslydecreasingtheorderof 1 .Letuselaborateonthisthought. isapropertyofthetheoryasacombinationofsupergravity and Yang‐Mills, therefore it does not change in the domain oftheorieswith496dimensions: in fact, itwillbeclear inthenextexamplesthatthefirsttermoftheanomalyisalways

∝ 164 216

124

Consequently,thistermisinvariantunderadditionof 2 theories.As far as the curvature F is concerned, adding more theories increases thenumber of labeled curvatures, since these depend on the individualcontributions.However, thesystemofequations that invalidates onlydependson and

.Thelattertermsurvivesuponadditionof 2 theories,sincethesecondtermofthetotalanomalypolynomialtriviallybecomesproportionalto:

30

∝ … anditisclearthatnoterm willcancelthepre‐existing .Byinduction,therefore,theanomalyiscarriedalonginallgroupsoftheform

2 1 sothattheyareallanomalous.Accidentally,combinationswiththeanomaly‐free arenotpermittedaslongasthereisa 2 theory,sincethiswillcarryananomalythatcannotbecancelledupon addition of . Oneway to think about this is to consider that, as far asanomaliesareconcerned,theproofusedfor isinvariantuponpermutationof1 and .

3.3Combinationsof and The simplest theory in this case is 2 4 1 . The YMcontribution is obtained summing two anomalies arising from 2 and onefrom 1 .Asusualforthelatterweset 1 1 476and 0.Asfartheformerisconcerned,weinsertthetraceidentitiesfor 2 [(2.5.1.1),(2.5.1.2),(2.5.1.3)]in(3.2.2). (3.3.1)

164 2

105670

104320

1010368

316 21360

12881

1152 2 52 3 2 2 12 2

— 42 (3.3.2)Starting from (3.3.2) we trivially generalize the anomaly to include a secondcontributionof 2 :

164 2

205670

204320

2010368

316 21360

12881

1152 2 52 3 2 12 3 2 12

31

— 1720 292 1

2 42 14 12 92 22 42 24 22

(3.3.3)TheYang‐Millsistheneasilyobtainedfromtheindividualcontributions:

164 2

4965670

4964320

49610368

316 21360

12881

1152 2 52 3 2 12 3 2 12

— 1720 292 1

2 42 14 12 92 22 42 24 22

(3.3.4)Finally, the total anomaly polynomial is retrieved coupling the anomalies ofYang‐Millstheoryandsupergravity,exactlyasdoneinsection3.2.

164 2

16

124

316 21360

1288 1

1152 2 3 12 12 12 3 22 12 22

— 1720 292 1

2 42 14 12 92 22 42 24 22

(3.3.5)It is worthwhile pointing out that, as expected, the first term is the same as(3.2.7).Theanomalypolynomialmust,fortheusualreasons,factorizeinsuchawaythat:

∙

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 isnotevenpossible tocombine with 2 in suchaway toobtainan

anomaly‐freetheoryin496dimensions.Thisisquicklyruledoutbytheanomalycarried by 2 , which survives upon addition of anomaly‐free as well asanomaloustheories.3.4Combinationsof , and Weexpectalltheoriesofthisformtobeanomalous,thereasonbeingthattheyareconstructedoutofanomalouscontribution.Asacheck,however,wewilltrytofactorize 2 2 1 .Thiswillallowustomakemoregeneralconclusionsforthegroup:

2 2 1 (3.4.1)

ThetwoanomaliesthatadduptotheanomalyoftheYang‐Millstheoryare: (3.4.2)

164 2

135670

134320

1310368

116 2 2 31360

12881

1152 2 52 8 2 3 2 12

— 1720 2 16 12 9

2 22 42 24 22

(3.4.3)SothattheanomalyoftheYang‐Millsis:

164 2

4965670

4964320

49610368

116 2 2 31360

12881

1152 2 52 8 2 3 2 12

— 16 12 22 42 24 22 (3.4.4)

33

andfinallyweretrievethetotalanomalypolynomial:

164 2

16

124

116 2 2 31360

12881

1152 2 52 8 2 3 2 12

— 16 12 22 42 24 22 (3.4.5)Theproposedfactorizationis:

∙

However,followingthesamemethoddevelopedaboveandcomparingtheusualtermsinF ,wecheckthattheoverdeterminedsystemofequationsisexactlytheone found for 2 2 1 .Comparing the terms inF ,we insteadobtainthesameequationsof 2 2 1 .Thisreinforcesourideasthat the overdetermined systems of equations carry the information of theanomalyforthegroupoutofwhichtheyareconstructed.Eitheranomalyisnotcancelleduponadditionofanyterm.and isanomalous.This last section concludes our analysis of theories in D=10. It is found that,indeed, 32 , , 1 and 1 aretheonlygroupswheretheanomalyiscancelled.However, the hope is that this explicit analysis could be of any help inunderstandingthequitevastliteratureinanomalycancellation.Wealsohopetohaveconveyedaconvincingargumentonhowanomaliesbehavewhentheoriesareconsideredindividuallyandincombinationwithotherones.AcknowledgementsFirst and foremost I would like to thank professor Yuji Tachikawa, withoutwhose advice and daily support this project would have never started nor itcouldhaveeverbeencompleted.I would also like to thank the University of Tokyo for the opportunity toparticipate in the UTRIP program for undergraduate students and for fundingmysix‐weekresearchstayinJapan.Lastbutnotleast,Iwouldliketothankalltheprofessors,postdocsandgraduatestudents of the Department of Physics of the University of Tokyo, who havemademefeelathomefortheentiredurationoftheprogram.

34

AppendixA.ConstructingaGreen‐Schwarztermfor TheanomalyoftheYMtheorycoincideswiththeanomalyarisingfrom 1 .

164 24965670

4964320

49610368

Thetotalanomalybecomes:

164 2

16

124

wherewehaveset,asusual, 0.Fromthetotalanomalypolynomialwecantrivially see by inspection that the term is common to both addends.Therefore,thefactorizationofthepolynomialisreadilyobtained:

164 2

16

124

Thecountertermneededtocancelthepolynomialhasthefollowingform:

Δ ∧ since, of course, , 0for the theorywe are considering. TheGreen‐Schwarztermischoseninsuchawaythatitcancelstheanomaly.Specifically:

164 2

16

124

sothat

Δ 164 216

124

Δ 0

e.g.thetotalanomalypolynomialiscancelledviatheGreen‐SchwarzMechanism.B.ConstructingaGreen‐Schwarztermfor In order to construct the anomaly of the YM theory we need the individualcontributionsfrom and 1 :

35

164 2

2485670

2484320

24810368

For weusethefollowingtraceidentities[3]:

1100 1

7200 Thenthecontributiontotheanomalyis:

164 2

2485670

2484320

24810368

132 21360

1288

1115200 2

2 3 Sothatthetotalanomalypolynomialis:

164 2

16

124

132 21360

1288

1115200 2

1518400 2

2 3Thefactorizationofthetheoryhasthefollowingform:

∙ Uponexpansionandcomparisonwiththetermsofthetotalanomalypolynomialthefollowingcoefficientsareretrieved:

1384 2

130 4

450 30

WehaveobtainedthefactorizedformthatallowsuscancellingtheanomalyviatheGreen‐SchwarzMechanism.Theparticulargaugetermis:

36

1384 2 4

450 30

Sothat:Δ 1384 2

130 4

450 30

AsrequiredtocanceltheanomalyviatheGreen‐Schwarzmechanism.References[1]Cheng,T.P.; Li, L.F. (1984). “GaugeTheory ofElementaryParticlePhysics”.OxfordSciencePublications.[2]Green,M.B.;Schwarz,J.H.(1984)."AnomalycancellationsinsupersymmetricD=10gaugetheoryandsuperstringtheory".PhysicsLettersB149:117[3] Green, M.B.; Schwarz, J.H; Witten E. (1988). “Superstring theory: volume 2, LoopAmplitudes,anomaliesandphenomenology”.CambridgeUniversityPress.[4] Adams A.; DeWolfe O.; Taylor W. (2014) “String Universality in ten dimensions”,arXiv:1006.1352v2[hep‐th].[5]BilalA.(2008)“LecturesonAnomalies”.arXiv:0802.0634v1[hep‐th][6]Humphreys J.E. (1980) “Introduction to Lie Algebras andRepresentationTheory”.Springer‐Verlag[7]BastianelliF.;VanNieuwenhuizenP.(2006)“PathIntegralsandAnomaliesinCurvedSpace”.CambridgeMonographsonMathematicalPhysics.[8]MacfarlaneA.J.;PfeifferH.(1999)“Oncharacteristicequations, trace identitiesandCasimiroperatorsofsimpleLiealgebras”.arXiv:math‐ph/9907024[math‐ph][9]SlanskyR.,Phys.Rept.79(1981)1.[10]FiolB.;“Populatingtheswampland:TheCaseof 1 and 1 ”,arXiv:0809.1525[hep‐th]. [11]C.VafaC.“Thestringlandscapeandtheswampland”,arXiv:hep‐th/0509212.[12]OoguriH.;VafaC.“Onthegeometryofthestringlandscapeandtheswampland”,arXiv:hep‐th/0605264.