1.36fedora15

arXiv:1110.
5229v1[hep-th]24Oct2011LMU-ASC51/11TUM-HEP818/11DESY-11-187LPSC-11220TheOrbifolder:ATooltostudytheLowEnergyEectiveTheoryofHeteroticOrbifoldsH.
P.
Nillesa,S.
Ramos-Sanchezb,P.
K.
S.
Vaudrevangec,d,e,A.
WingerterfaBetheCenterforTheoreticalPhysicsandPhysikalischesInstitutderUniversit¨atBonn,Nussallee12,53115Bonn,GermanybDepartmentofTheoreticalPhysics,PhysicsInstitute,UNAM,MexicoD.
F.
04510,MexicocDeutschesElektronen-SynchrotronDESY,Notkestrae85,22607Hamburg,GermanydPhysik-DepartmentT30,TechnischeUniversit¨atM¨unchen,James-Franck-Strae,85748Garching,GermanyeArnold-Sommerfeld-CenterforTheoreticalPhysics,Theresienstrae37,80333M¨unchen,GermanyfLaboratoiredePhysiqueSubatomiqueetdeCosmologie,UJFGrenoble1,CNRS/IN2P3,INPG,53AvenuedesMartyrs,F-38026Grenoble,FranceAbstractTheorbifolderisaprogramdevelopedinC++thatcomputesandanalyzesthelow-energyeectivetheoryofheteroticorbifoldcompactications.
Theprogramincludesroutinestocomputethemasslessspectrum,toidentifytheallowedcouplingsinthesuperpotential,toautomaticallygeneratelargesetsoforbifoldmodels,toidentifyphenomenologicallyinterestingmodels(e.
g.
MSSM-likemodels)andtoanalyzetheirvacuum-congurations.
Keywords:Orbifold,stringcompactication,extradimensions,particlespectrum,MSSMProgramSummaryProgramtitle:orbifolderProgramobtainablefrom:http://projects.
hepforge.
org/orbifolder/Distributionformat:tar.
gzProgramminglanguage:C++Computer:PersonalcomputerOperatingsystem:TestedonLinux(Fedora15,Ubuntu11,SuSE11)Wordsize:32bitsor64bitsExternalroutines:NoneDependencies:Boost,GSLTypicalrunningtime:Lessthanasecondpermodel.
Natureofproblem:Calculatingthelowenergyspectrumofheteroticorbifoldcompactications.
Solutionmethod:Quadraticequationsonalattice;representationtheory;polynomialalgebra.
PreprintsubmittedtoComputerPhysicsCommunicationsSeptember24,20181.
IntroductionStringtheoryisacandidateforaconsistentuniedquantumtheoryofgravityandgaugeinteractionsandcouldthusprovideuswithanultravioletcompletionformodelsofparticlephysics.
Thesearchfor4-dimensionalstringvacuaresemblingthestandardmodel(SM)(oritssupersymmetricextension(MSSM))isthereforeoneofthecentralquestionsinstringtheoryresearch.
Overtheyears,awidelandscapeof4-dimensionalstringmodelshasemergedanditremainstobeseenhowparticlephysicsphenomenacanbeincorporatedwithintheschemeofstringtheory.
Somehintspointtoaunicationofgaugecouplingswithintheframeworkofexceptionalgroups(ase.
g.
E8)butadirectroadfromstringstoparticlephysicshasnotyetbeenidentied.
Itisperhapsthetimetostepback,collectandclassifyexistingmodelconstructionsandtrytoidentifypropertiesrelevantforadescriptionofnature.
Herewepresentandanalyzeaspecicapproachthatwasstudiedalreadyinthe1980sandhasledtointerestingresultssincethen:orbifoldcompactication[1,2,3]oftheheteroticstrings[4,5].
Thereasonforthesuccessofthisapproachis"computability"pairedwithgeometricintuition.
Exacttoolsofconformaleldtheoryarehereatourdisposal[6,7]thatareusuallynotavailableinapproachesbasedoncompacticationonsmoothmanifolds.
Besides,inparticularE8*E8asagaugegroupallowsaperturbativeinclusionofthestandardmodelgaugegroup(aswellaspossiblygrandunication).
Atoroidalorbifoldisatwiththeexceptionofanumberofxedpointsorxedtori.
Itgivesrisetoapicturecalledtheheteroticbraneworldscenario[8,9,10]:eldscaneitherliveinthe10-dimensionalbulk(untwistedsector)orcanbelocalizedatthesexedpointsorxedtori(twistedsectors).
Therelativelocationoftheseeldsaswellasthelocalgaugestructuredeterminesmanypropertiesofthe4-dimensionalstringvacuaandisthesourceofgeometricintuitionformodelbuilding.
Theorbifoldpointisapointofenhancedsymmetries(inthemodulispaceofcompactication)andthosesymmetriesmightberelevantforadescriptionofnature.
Modelsofparticlephysicsseemtorequiremany(discrete)symmetries,e.
g.
avoursymmetriesorsymmetriestopreventtoofastprotondecay.
Someofthesesymmetriescouldbeslightlybrokentoprovideuswithsmallparametersrelevantforthedescriptionofhierarchiesase.
g.
observedinthespectrumofquarkandleptonmasses.
Thissupportsourhopethatorbifoldcompacticationisnotjustanapproximationwithincreased"computability",butthatitprovidesarealisticcompactication:naturemighthavechosentoliveclosetotheorbifoldpointwithenhancedsymmetries.
Explicitorbifoldmodelconstructionsinthelast5to10yearshavebeenextremelysuccessful[11,12,13,14](see[15,16]forearlierreviews).
Manyofthepropertiesofparticlephysicscanbeincorporatedinthescheme.
Thisincludesgrandunication,gauge-Yukawaunication,satisfactoryYukawatextures,solutionstothe-problem,thecreationofhierarchiesandasuccessfulincorporationofneutrinoMajoranamasses.
Discretesymmetriesareidentiedtosolvetheavourproblemandtoavoidtoofastprotondecay.
ThesepropertieshavebeenidentiedinthesocalledMinilandscape(basedontheZ6-IIorbifold)[17,18,19]andsubsequentwork[20,21,22,23,24,25,26,27,28,29,30,31,32,33].
Thesearchforsuchmodelsrequiresacomputerassistedscanthatincorporatestherulesforaconsistentstringtheoryconstruction(ase.
g.
modularinvariance).
Thepurposeofthisworkistomakethetoolsandtechniquesavailabletothepublic.
Wehopethatthiswillgivemorepeopletheopportunitytocontributetothisexcitingeldofmodelconstructions.
Wepresenttheorbifolder,aprogramdevelopedinC++thatallowsthecalculationofthelow-energyspectrumofheteroticorbifoldconstructions.
Theprogramincludesroutinestocomputethemasslessspec-trumandtoidentifytheallowedcouplingsofthesuperpotential.
Itallowstheconstructionofarbitraryorbifoldmodels,theidenticationofphenomenologicallyinterestingmodelsandaclassicationoftheirvacuumcongurations.
TheorbifoldercanbeconsideredinsomeaspectsasthestringyanalogofprogramssuchasSoftSusy,2SuSpectandSPheno:Thelatteraredevotedtothedetailedcomputationofparticlespectra,interactionsandphenomenologicalquantities,usingasinputahighscalesupersymmetricmodelandimposinglow-energyconstraints.
Analogously,theorbifoldertakesasastartingpointthe10Dheteroticstringsand,providedsomegeometricinputdescribingthefeaturesofthesixcompactieddimensions,computesthe(massless)spectrum,interactionsandsymmetriesoftheresultinglower-energyeective4Deldtheory.
AfterashortintroductiontoheteroticorbifoldcompacticationinSection2,weexplainhowtodown-loadandinstalltheC++programinSection3.
Section4discussestheexplicitrecipetoruntheprogram,whileinSection5weconcludeandgiveanoutlookforfutureresearch.
Technicaldetailsarerelegatedtotheappendicesandtothewebpage[39,§Complementarynotes].
2.
HeteroticOrbifoldCompacticationsInthissectionwegiveabriefintroductiontoheteroticorbifolds,inordertointroduceournotationandconventionsusedintheorbifolder.
Formoredetailsonorbifoldcompactications,werefertothereviews[15,16,11,13,14].
Inthecontextofheteroticstringcompactications,wedeneanorbifoldasthequotientofsix-dimen-sionalrealspace6dividedbytheso-calledspacegroupS,wherethequotientistakenusingtheequiva-lencerelationXgXforallg∈SandX∈6.
Morespecically,thespacegroupischosentoconsistoftwoparts:discreterotationsthatformtheso-calledpointgroupP.
Forsimplicity,wechoosePtobeAbelian.
ToobtainN=1supersymmetryin4D,PiseitherMorM*Ngeneratedbyrotationmatricesθandω,whereweuseω=1forM.
Thesematricescanberepresentedbyso-calledtwistvectorsv1andv2thatgivethethreerotationalphasesinthethreecomplexplanesandthesumoverallentriesintegertoensureN=1.
Forexample,v1=0,13,13,23andv2=0forthe3orbifold.
translationsgeneratedbythevectorseα∈6,forα=1,6.
Theyforma6DlatticedenotedbyΓandhencedeneasix-torus.
ElementsofPmustmapthelatticeΓtoitself.
Indetail,anelementofthespacegroupisoftheformg=θkωl,nαeα∈S,wherek,l∈,nα∈(orinsomecasesnα∈)andsummationoverα=1,6.
ItactsonX∈6asgX=θkωlX+nαeα.
Usingthesedenitions,wecandealwithall6DAbelianandtoroidalorbifoldsincludingthecasesofroto-translationsandfreelyactinginvolutions(seee.
g.
Ref.
[34]).
Duetomodularinvariance,thegeometricactionofthespacegroupShastobeembeddedintothegaugedegreesoffreedomoftheheteroticstring,denotedbyXI,I=1,16inthebosonicformulation.
Werestrictourselvestothecaseofshiftembedding,whereθ→V1,ω→V2andeα→Wαforα=1,6.
Then,theactionofSonX∈6inducesanactiononXIasgX=θkωlX+nαeαgXI=XI+kVI1+lVI2+nαWIα,(1)whereI=1,16.
ThevectorsV1,V2andWαarecalledshiftsandWilsonlines,respectively.
Theyareconstrainedbymodularinvariance[2,35,36],e.
g.
M(V21v21)=0mod2andNαW2α=0mod2,(2)whereMistheorderofMandNαtheorderofWα.
Thecombinedgroupformedbythespacegroupanditsactiononthegaugedegreesoffreedomiscalledtheorbifoldgroup.
3Fromamodel-buildingstandpoint,wehavenowintroducedalltheinputdatatodeneaheteroticorbifoldmodel:thespacegroupS(consistingofthepointgroupPandthelatticeΓ)anditsembeddingasshiftsV1,V2andWilsonlinesWα.
Weclosethissectionwithaverybriefsummaryoftheconstructionofmasslessspectraofheteroticorbifolds.
Giventheinputdata,onedistinguishesbetweentwokindsofclosed(andmassless)stringsontheorbifold:rstofallordinaryclosedstrings,alsocalleduntwistedstrings,beingtheremnantsofthe10dE8*E8orSO(32)vectormultipletandthegravitymultiplet.
Secondly,thereareclosedstringsfromthetwistedsectorswhichcloseonlyuptotheactionoftheorbifoldgroup.
Toconstructthem,oneneedstoidentifytheinequivalentnon-trivialspacegroupelementsastheconstructingelementsoftwistedstrings:i.
e.
forg∈SonecandeneatwistedboundaryconditionX(τ,σ+π)=gX(τ,σ)onthestringworld-sheet.
Then,usingstandardCFTtechniques,theequationsformasslessright-andleft-moverswithboundaryconditiongreadq+vg2212+δc=0andp+Vg22+N1+δc=0(3)wherepisfromtheE8*E8orSpin(32)/2weightlattice,qfromthevectororspinorweightlatticeofSO(8),δcdenotestheshiftinthezero-pointenergyandNthenumberoperatorofleft-movingoscillators.
Furthermore,wedenethelocaltwistvg=kv1+lv2andthelocalshiftVg=kV1+lV2+nαWα.
Inthenalstep,onebuildsmasslessstatesastensorproductsofmasslessright-andleft-moverssuchthattheyareinvariantunderthefullorbifoldaction,i.
e.
|q+vgR|p+VgLor|q+vgRα|p+VgL,(4)whereαdenotespossibleoscillatorexcitations.
Wedenetheshiftedmomentaqsh=q+vgandpsh=p+Vg,wherepshdescribesthetransformationpropertiesundergaugetransformations.
ThestatesEq.
(4)correspondtomasslesseldsofthe4Deectiveeldtheory.
Theycarrygaugecharges(frompsh),discreteRcharges,modularweights(fromqshandpossibleoscillatorexcitations)anddiscretenon-Rcharges(fromtheconstructingelementg∈S).
3.
DownloadandInstallationTheminimalrequirementsforcompilingtheorbifolderaretheBoostC++Librariesversion≥1.
0[37]andtheGNUScienticLibrary(GSL)version≥1.
9[38].
AllcomponentsshouldcomepreinstalledonastandardLinuxdistribution.
Ifthisshouldnotbethecase,theycaneasilybeinstalled.
Onayum-baseddistributionlikee.
g.
Fedora,thecommand"yum-yinstallgslgsl-develboostboost-devel"willinstallthecorrespondinglibraries(recommended).
Alternatively,onecanalsodirectlyinstallfromsourceorusetheotherdownloadoptionsavailable[37,38].
TheorbifolderisfreesoftwareunderthecopyleftoftheGNUGeneralPublicLicenseandcanbedownloadedfrom[39]:http://projects.
hepforge.
org/orbifolder/Toinstalltheprogram,downloadtheleorbifolder-1.
0.
tgztoadirectoryofyourchoice,openashellandenterthefollowingcommandsattheprompt:1tarxfvzorbifolder-1.
0.
tar.
gz2cdorbifolder-1.
03.
/configure44make5makeinstallNotethattheversionnumber"1.
0"maychangeovertimeandshouldbesubstitutedaccordingly.
Therstlineunpacksthetar-ballandcreatesasubdirectorystructurewiththesourcecode.
Thesecondlinechangestotheinstallationdirectory.
Thethirdlinestartsthecongurationscriptthatcheckssystemre-quirementsandgeneratestheMakefile.
Thefourthlinecompilesthecode,andnallythefthlineinstallsitonyoursystem.
Aftersuccessfulcompilationandinstallation,themainprogram(namedorbifolder)willbeavailableinthecurrentdirectory.
Wehavedisabledcustominstallationusingthe--prefixswitchinthecongurationscript.
Themainprogramorbifoldercansimplybecopiedtothedirectoryoftheuser'schoicebythestandardshellcommands.
DetailedinstallationinstructionscanalsobefoundintheREADMEleintheinstallationdirectoryandonthewebsite[39].
Wehavetestedtheinstallationprocessona32-bitsystemrunningLinuxUbuntu11.
04withBoost1.
42andGSL1.
14,a64-bitsystemrunningLinuxSuSE11withBoost1.
36andGSL1.
11,a64-bitsystemrunningLinuxFedora15withBoost1.
46andGSL1.
14,andascertainedthatourcodecompilescorrectly.
Shouldthereariseanyproblemsduringtheinstalla-tion,werequestthattheusersendustheleconfig.
logandtheoutputofthemakecommandbyemail(orbifolder@projects.
hepforge.
org).
4.
HowtoruntheprogramTherearethreemainwaystogainaccesstotheorbifolder:throughtheprompt,throughawebinter-faceanddirectlythroughtheC++sourcecode.
Inthefollowingwewillpresentthemindetail.
4.
1.
ThepromptTheorbifoldercanbecontrolledusingaLinux-stylecommandlinecalledtheprompt.
Thepromptoersaninteractiveaccesstoalmostallvariablesandfunctionsoftheorbifolder.
Ithasthestructureofalesystemwhereorbifoldmodelsappearasdirectories.
Inthefollowingweexplainhowtostartandusetheprompt.
4.
1.
1.
HowtogetstartedWebeginwithasmalltutorial,seeTab.
4intheadditionalmaterial[39,§Complementarynotes]forasampleinputandoutput.
Ingeneral,thepromptcanbestartedusingthecommand.
/orbifolderor.
/orbifolder[modelfile].
Intheformercase,nomodelisloadedautomatically.
Inthelattercase,thedetailsofamodelcontainedintheplain-text-basedmodelfileareloaded(FurtherpropertiesofmodellesareexplainedinSection4.
4.
2).
Asanexample,runtheprogramusingthecommand.
/orbifoldermodelZ3.
txt(5)withparametermodelZ3.
txtinordertoloadthestandardembeddingmodelofthe3orbifoldfromthisle.
Havingstartedtheprogram,oneentersthepromptinitsmaindirectory/>.
Thecommanddir(6)5listsallcommandsandsubdirectoriesofthecurrentdirectory.
Inourexample,thereisonesubdirectoryinthemaindirectory/>calledZ3StandardEmbeddingwhichcorrespondstothe3modelloaded.
Ingeneral,a(sub-)directoryAcanbeaccessedusingthecommandcdA.
Inourexample,cdZ3StandardEmbedding(7)resultsintheoutput/Z3StandardEmbedding>fromwhereonecanaccess,analyzeorchangethedetailsofthe3standardembeddingmodel.
Again,typeinthecommanddirtoseeallcommandsandsubdirectoriesofthecurrentdirectory.
Forallorbifoldmodeldirectoriestherearevesubdirectories,/model>,/gaugegroup>,/spectrum>,/couplings>and/vev-config>,(8)containingcommandsoftherespectivecategory:/model>:Printandchangetheinputdataofthecurrentorbifoldmodel.
SeeAppendixB.
2.
2.
/gaugegroup>:Printdetailsonthegaugegroup,changetheU(1)basisandndaccidentalU(1)symmetries.
SeeAppendixB.
2.
3.
/spectrum>:Printdetailsonthespectrumofmasslesselds.
SeeAppendixB.
2.
4.
/couplings>:Createandanalyzethesuperpotentialandtheresultingeectivemassmatrices.
SeeAppendixB.
2.
5.
/vev-config>:Deneandanalyzevariousvev-congurations.
SeeAppendixB.
2.
6.
Eachcongu-rationisspeciedbythedistinctionbetweenobservableandhiddensectorofthegaugegroupandtheassignmentoflabelsandvacuumexpectationvaluestotheelds(labelsareassignedinasubdirectorycalled/labels>,seeAppendixB.
2.
7).
Again,onecanaccessthesedirectoriesusingthecdcommand.
Forexample,trycdgaugegrouptoenterthesubdirectory/gaugegroup>andusethecommandsprintgaugegroupandprintsimplerootstoseethegaugegroup(E6*SU(3)*E8)and(achoiceof)thecorrespondingsimpleroots.
Inordertogobackonedirectoryoneusesthecommandcd.
.
attheprompt.
Next,trythesubdirectory/spectrum>andusethecommandprintsummarytogetasummarytableofallmasslessmatterelds.
Thecommandcdisusedtogobacktothemaindirectory/>.
FurtherstandardcommandsofthepromptaredescribedinTab.
1;seealsoAppendixBforaglossaryofcommands.
commanddescriptiondirdisplaycommandsandsubdirectoriesofthecurrentdirectorycdAchangethecurrentdirectorytoA(ifAexists)cd.
.
gobackonedirectorycdgobacktothemaindirectory/>exitexittheprogramifnoprocessisrunning;usetheparameterorbifoldertoenforceexit(alsoinsideascript)Table1:Somestandardcommandsintheprompt.
64.
1.
2.
HowtocreateneworbifoldmodelsBeinginthemaindirectory/>ofthepromptneworbifoldmodelscanbeaccessedbasicallyinthreeways:loadorbifolds(Filename):Loadorbifoldmodelsfromamodelle.
Forexample,loadthe6–IIorbifoldMSSMof[40]usingthecommandloadorbifolds(modelBHLR.
txt).
createorbifold(A)withpointgroup(M,N):CreateaMorM*NorbifoldnamedAbyspec-ifyingMandN(setN=1forM).
AfterenteringthenewdirectoryusingcdA,oneisaskedtospecifymoredetailsonthemodellikeshiftsandWilsonlines,seeAppendixB.
2.
2.
createrandomorbifoldfrom(A).
Seebelow.
Orbifoldmodelscanbecreatedrandomlybyusingthecommandcreaterandomorbifoldfrom(A)(9)inthemaindirectory/>.
TheparameterAmustbeeitherthenameofanexisting(loadedorpreviouslycreated)orbifoldmodelor*foranyexistingorbifoldmodelintheorbifolder.
Thecommandstartsanewprocessthatrunsinthebackgroundsothatonecancontinuetoworkwiththeprompt(seeAppendixB.
1.
5formoredetailsonprocesses).
Onecanspecifyseveraloptionalparameters:if(.
.
.
):Specifythedesiredpropertiesofthemodel:inequivalentinordertochooseonlymod-elswithinequivalentspectraandSM,PSorSU5formodelswitha(net)numberofXgenerationsofStandardModel(SM),Pati-Salam(PS)orSU(5)gaugegroupplusvector-likeexotics,whereXis3bydefaultandcanbechangedusingtheparameterXgenerations;c.
f.
thecommandanalyzeconfiginAppendixB.
2.
6.
saveto(Filename):Savethemodelswiththedesiredpropertiestoamodelle.
use(1,1,0,1,.
.
.
):EightdigitsfortwoshiftsplussixWilsonlines;either1ifthecorrespondingshift/WilsonlineshallbetakenfrommodelA,or0ifitshallbecreatedrandomly.
#models(X):Denehowmanymodels(withthedesiredproperties,ifspecied)shallbecreatedrandomly.
Use#models(all)tocreateasmanymodelsaspossible.
If#models(X)isnotused,onlyonemodelshallbecreated.
printinfo:Printashortsummaryofthespectrumimmediatelywhenanewmodelwiththedesiredpropertieshasbeenfound.
loadwhendone:Loadthecreatedmodelsintotheorbifolderaftertheprocesshasnished.
donotcheckanomalies:Usethisparametertospeeduptheprocess.
compare#couplingsoforder(X):Ifonlyinequivalentmodelsaresaved,thisparameterrenesthecomparisonbetweentwomodels:compareinadditionthenumberofallsuperpotentialcouplingsuptothespeciedorderX.
Slowsdowntheprocessconsiderably.
7Examples.
Atypicalexampleofhowtousethiscommandlookslikecreaterandomorbifoldfrom(Z3StandardEmbedding)if(inequivalent)(10)saveto(Z3NewModels.
txt)use(1,1,0,0,0,0,0,0)#models(10)printinfoexecutedfromthemaindirectory/>.
Inthiscaseanewprocessisstartedthatconstructsnew3orbifoldmodelsusingbothshifts(i.
e.
V2=0)butnoWilsonlinesfrommodelZ3StandardEmbedding,savesonlyinequivalentmodelstoamodellenamedZ3NewModels.
txt,printsabriefsummaryofeachnewmodelandstopsaftercreatingteninequivalentmodels.
Inasecondexample,tenrandommodelsofSM(StandardModel)typearecreatedstartingfromthe6–IIorbifoldMSSMof[40]byusingthecommandcreaterandomorbifoldfrom(Z6IIOrbifoldBHLR)if(inequivalentSM)(11)saveto(Z6IINewMSSMModels.
txt)use(1,1,1,1,1,1,0,0)#models(10)printinfoloadwhendoneinthemaindirectory/>.
Theparameteruse(1,1,1,1,1,1,0,0)speciesthatonlytheWilsonlinesW5andW6arecreatedrandomly,i.
e.
theshiftsandotherWilsonlinesaretakenfromtheoriginalmodel.
Furthermore,onlyinequivalentstandardmodels(SM)areprinted,savedtoleZ6IINewMSSMModels.
txtandnallyloadedintotheorbifolderaftertheprocesshasnished.
NotethattheseMSSMmodelsshouldbepartoftheMini-Landscape[17,19].
4.
2.
ThewebinterfaceTheorbifoldercanalsobeaccessedthroughauser-friendlywebinterface.
Theorbifolderon-linemakesextensiveuseofthepromptexplainedinSection4.
1renderingitavailabletoanyuserwithaccesstoaninternetbrowser,suchasMozillaFirefoxversion≥3.
6,GoogleChromeversion≥2.
1,andInternetExplorer≥8.
0.
Consequently,theprogramisalsoavailabletousersofallkindsofsmartphones.
Oneoftheadvantagesofthewebinterfaceisthatonedoesnotneedtoinstallanyprogramonthelocalcomputertobeabletousemostofthefunctionsoftheorbifolder.
Anotheradvantageisthatitcanbeexecutedfromplatformsthatworkwiththemostpopularoperatingsystems(withoutfurtherauxiliaryapplications):Windows,LinuxandMac.
Onthelessbrightside,oneshortcomingofthisversionisthatitisnotrecommendedtoexecutetime-consuminginstructionssinceshortinterruptionsintheinternetservicemayaecttheresults.
Furthermore,acommandrunningduringmorethan60minutesisdisabledautomaticallytoavoidoverloadoftheserver.
Finally,forsecurityreasons,commandsinvolvinglemanipulationareextremelylimited.
Specically,theparametersandcommands@begin/@endprinttofile(Filename),saveto(Filename),load/savecouplings(Filename),load/savelabels(Filename)aredisabled.
Theorbifolderon-linecanbeusedonourmainpage[39]http://projects.
hepforge.
org/orbifolder/whichredirectstoanyofourmirrorservers.
Togainaccess,youmustclickonthelinkorbifolderon-line.
Thisstartsanorbifoldersessionontheselectedserver.
Thenewpageconsistsofthreeparts:History:Theresultofthelatestinstructionsisshown.
ThebuttondownloadhistoryprovidesanRTFlecontainingthefullhistory,i.
e.
notonlytheresultofthelatestinstructions,buttheresultofallthecommandsusedduringthecurrentsession.
Theusercanresizethiswindow.
ListofCommands:Thisistheinputarea,wherethecommandsofthepromptaretyped.
Toexecutethem,itisnecessarytoclickonthebuttonexecutecommands.
Anadditionalhelpinthissectionisthe8buttonuploadcommands.
Thisbuttoncanbeusedwhenles(notlargerthan100Kb)inplain-textfor-matcontaining(listsof)admissiblecommandshavebeenprepared.
ThebuttondownloadcommandsprovidesanRTFlecontainingthelistofallthecommandstypedduringtheactiveorbifoldersession.
ListsofusefulcommandsandtheiruseisprovidedinSection4.
1andAppendixB.
Help:Thebottompartcontainsalistofallavailablecommandsforthecurrentdirectory.
Eachcommandinthedisplayedlistisalinktoamoreprecisedescriptionofitsuse.
Toterminateanorbifolderon-linesession,itsucestoclickontheupperbuttonEXIT.
Thebuttonsprovidedontheresultingpageallowtheusertodownloadthefullhistoryofthecurrentsessionandthecompletelistofsuccessfullyexecutedcommands.
Occasionally,errorsintheinputdatamaycausetheorbifoldertocrash.
Inthosecases,thewebinterfaceshallcloseyoursession,givingyoutheopportunitytodownloadthelistofinstructions(buttondownloadcommands)tobeusedifyourestarttheorbifolder.
Please,makesurethatthedownloadedlistofcommandsdoesnotcontaintheinstructionsthatledtothefailureoftheprogram.
Weencourageuserstocontacttheauthorsreportinganyfailureintheprogram,preferablybyusingthelinkcontactusonthemainpageofthewebinterface.
4.
3.
TheC++sourcecodeTheorbifolderiswritteninC++,distributedoverseveralles.
Manyphysicalquantities,asbrieyintroducedinSection2,havebeenencapsulatedintoclasses,forexampleCSpaceGroupforthespacegroupSandCSpaceGroupElementforitselementsg∈S,CTwistVectorforthetwistsviandCShiftVectorforthecorrespondingshiftsVi,CWilsonLinesforthesetofsixWilsonlinesWα,COrbifoldGroupfortheorbifoldgroup,CMasslessHalfStateforndingmasslessleft-andright-movers,i.
e.
solutionstoEq.
(3),CHalfStatefortheweightsofCMasslessHalfStatesortedwithrespecttotheirtransformationprop-ertiesundertheelementsofthecentralizer,CStatefororbifold-invarianttensorproductsofmasslessleft-andright-movers,seeEq.
(4),COrbifoldforthefullorbifoldcompactication,CFieldforaeldoftheeective4-dimensionaltheory,CMonomialfor(gaugeinvariant)monomialsofeldscorrespondingtoD=0solutions,andmanymore.
Inadditiontherearetechnicalclasses.
Forexample,thereareseveralclassesdevotedtogrouptheory(likedynkin,freudenthal,gaugeGroupFactorandgaugeGroup),theclassCAnalyseModelcontainsfunctionstoanalyzemodelsfortheirphenomenologicalproperties,theclassCPrintcontainsallprintingcommandsandtheclassCPromptcontainsthesourcecodeoftheprompt.
Asthereareintotalmorethan40classeswecannotexplainthemindetailhere.
WegivefurtherdetailsofsomeofthemoreimportantclassesinAppendixAandashortexampleprograminAppendixA.
1.
94.
4.
FilesdeninganorbifoldmodelTherearetwolesthatdeneanorbifoldmodel:i)Thegeometrylecontains,asthenamesuggests,thegeometricalinformationabouttheorbifold,suchasthespacegroup,andii)themodellecontainsshiftsandWilsonlines,i.
e.
theactionoftheorbifoldonthegaugesectoroftheheteroticstring.
Inthefollowingwegivesomemoredetails.
Examplesaregivenintheadditionalmaterial[39,§Complementarynotes].
4.
4.
1.
ThegeometryleThegeometrylebasicallycontainsinformationaboutthespacegroup,i.
e.
theorderofthetwist(s),thesixlatticevectorsofΓandthegeneratorsofthespacegroup,thediscrete(Randnon-R)symmetriesoftheorbifold(importantforthecomputationofallowedsuperpotentialcouplings),theinequivalentxedpointsspeciedbytheirconstructingelementsθkωl,nαeαandforeachconstructingelementalistofcentralizerelements,i.
e.
elementsh∈Swith[h,g]=0.
Wegivetwoexamplesintheadditionalmaterial[39,§Complementarynotes]:Tab.
1givesadetaileddescriptionofageometryleusingthe3exampleandTab.
2explainshowtocreateanewgeometryleof,forexample,model(1-3)ofRef.
[34].
4.
4.
2.
ThemodelleThemodellecontainsalistoforbifoldmodels,whereeachmodelisspeciedbythenameofthemodel(willbeusedasthenameofthecorrespondingdirectoryintheprompt),thenameofthegeometryle,thetypeofheteroticstring(i.
e.
Spin(32)/2orE8*E8),twoshiftsV1,V2(wheretheV2=0forMorbifolds),sixWilsonlinesWα,optionally,theparametersof(generalized)discretetorsiona,bα,cαanddαβasdenedinRef.
[36],optionally,someoftheU(1)generators,optionally,onecanspecifyascriptthatisexecutedautomaticallyafterthemodelhasbeenloaded.
Anexampleforthecaseofa3orbifoldwithstandardembeddingisgivenintheadditionalmaterial[39,§Complementarynotes].
105.
ConclusionsandoutlookWiththetoolsprovidedhereitshouldbepossibletothoroughlyinvestigatethelandscape(inparticulartheMSSM-likelandscape)oforbifoldcompacticationoftheheteroticstring.
This"heteroticbraneworld"providesacoherentgeometricpictureofMSSM-likemodels.
Crucialpropertiesoftheschemedependonthegeographyofeldsinextradimensions.
Thisleadstotheconceptof"LocalGrandUnication"andageometricunderstandingofe.
g.
Yukawacouplings,the-term,neutrinomassesandprotondecay.
Detailedpropertiesofmodelscanbecomputedreliablywithinthecontextofconformaleldtheory.
Themodelsconstructedhereshouldbecompared(andpossiblyrelated)tootherregionsoftheMSSM-likelandscape,ase.
g.
fermionicformulations[41,42,43],tensoringofconformedeldtheories[44],smoothcompacticationsoftheheteroticstring[45,46,47,48,49,50],typeII(intersecting)brane-models[51,52,53,54,55],M-andF-theoryconstructions[56,57,58,59,60,61,62].
Itwouldbeinterestingtoidentifysimilaritiesanddierencesofthecorrespondingschemes.
Allthesecasesrelyona(sometimeshidden)geometricinterpretationthatdenesthepropertiesofthemodelssuchastheappear-anceofgrandunication,valuesofYukawa-couplingsandtheexistenceofhierarchies.
Oneoftheimportantobservationsintheframeworkoftheheteroticbraneworldconcernsthecrucialroleplayedbydiscrete(gauge)symmetries.
Theyariseasremnantsofthegaugesymmetryaswellassym-metriesduetothespeciallocationofeldsinextradimensions.
Theycontrolpropertiesofthescheme,ase.
g.
avoruniversalityandthequestionofprotonstability.
Attheorbifoldpointweencounteranen-hancementofdiscretesymmetriesandparticlespectra.
Thesesymmetriesareabasicingredientofmodelbuilding.
Slightlybroken,theymightgiveusanexplanationfortheappearanceofhierarchiesinparticlephysics(ase.
g.
theratioofYukawacouplings).
Attheorbifoldpointwecanrelyonexactconformaleldtheorytechniquesthatcouldbeusefultounderstandtheblow-upprocedure[6,7,63,64,65,30,66]oforbifoldsingularitiesinacontrolledwayandthusconnecttosmoothcompactications.
WithabetterknowledgeoftheMSSM-landscapewemighthopetorelatethevariousconstructionsandimprovethecalculationalpowerinthosemodelswherewestillhavetorelyonaneectivelow-energysupergravityand/orlargevolumeapproximation.
Thefutureoftheeldrequiresreliablecalculationaltools(ase.
g.
conformedeldtheorytechniques)whichareuptonowonlyapplicableinsomecornersofthelandscape.
Butwemightbeluckyandnaturemighthavechosentoliveclosetosuchacorner.
AcknowledgmentsWewouldliketothankMichaelBlaszczyk,NanaGeraldineCaboBizet,StefanF¨orste,DavidGrell-scheid,TomaˇsJeˇzo,StefanGrootNibbelink,MichaelRatz,FabianR¨uhleandJesusTorrado-Cachoforusefuldiscussions.
P.
V.
wouldliketothankLMUExcellentforsupport.
ThismaterialispartlybaseduponworkdoneattheAspenCenterforPhysicsandsupportedbytheNationalScienceFoundationunderGrantNo.
1066293.
ThisresearchwassupportedbytheDFGclusterofexcellenceOriginandStruc-tureoftheUniverse,theSFB-TransregioTR33"TheDarkUniverse"andTR27"NeutrinosandBeyond"(DeutscheForschungsgemeinschaft)andtheEuropeanUnion7thnetworkprogram"UnicationintheLHCera"(PITN-GA-2009-237920).
S.
R-S.
waspartiallysupportedbyCONACyTproject82291andDGAPAprojectIA101811.
11AppendixA.
GlossaryofrelevantclassesTheorbifoldermakesextensiveuseoftheclassstructureoeredbyC++.
Theinformationofanorbifoldmodelisdistributedaccordingtothefollowingclasses.
ClassCAnalyseModel.
TheclassCAnalyseModelcontainsseveralfunctionsthatanalyzethephenomenol-ogyoforbifoldmodels,mainlyforthecasesoftheStandardModel,PatiSalamorSU(5)gaugegroup.
ClassCField.
ACFieldobjectcontainsallphysicalinformationaboutamasslesseld,suchastherep-resentation,theU(1)charges,theqshcharges,thelocalizationanditsvev.
Inaddition,itcontainsalistofindiceswhichspecifyitsweightspsh(storedinthemembervariablevectorWeightsoftheassociatedCMasslessHalfStateobject).
ClassCFixedBrane.
ACFixedBraneobjectcontainsallinformationaboutall(untwistedortwisted)stringswithconstructingelementg=θkωl,nαeα∈S.
Theleft-movingpartofthestringiscomputedusingthelocalshiftVg=kV1+lV2+nαWα.
Hence,thesolutionsoftheequationformasslessleft-movers,Eq.
(3),arestoredhereinavectorofCMasslessHalfStateobjects,oneentryforeachdierentchoiceofoscillatorexcitation.
Afterthemasslesssolutionshavebeenidentied,theyaresortedwithrespecttotheircentralizereigenvaluesandstoredinacorrespondingvectorofCHalfStateobjects(oneentryforeachdierentchoiceofNwithdierenteigenvalues).
Inthelaststep,themasslessright-movingCHalfStateobjectsfromCSectoraretensoredtogetherwiththemasslessleft-movingCHalfStateobjectsstoredheretoformorbifold-invariantstringstates.
ThesestatesarestoredinvectorInvariantStates.
ClassCHalfState.
ACHalfStateobjectdescendsfromaCMasslessHalfStateobjectbysortingthemass-lesssolutions(i.
e.
theweightsqshorpshforright-orleft-movers)withrespecttotheircentralizer-eigenvalues.
Theindicesoftheweights(aslistedinvectorWeightsofthecorrespondingCMasslessHalfStateobject)havingthesameeigenvaluesvectorEigenvaluesarestoredinvectorWeights.
ClassCMasslessHalfState.
ACMasslessHalfStateobjectstoresthesolutionsoftheequationformass-lessright-orleft-movers,respectively,seeEq.
(3).
TheconstructorCMasslessHalfState(MoversTypeType,constSOscillatorExcitation&Excitation)needstwoparameters:therstparameterMovers-TypecanbeeitherLeftMoverorRightMoverandthesecondonespeciestheoscillatorexcitation.
Then,onecancallthememberfunctionboolSolveMassEquation(constCVector&constructingElement,constSelfDualLattice&Lattice)tocreatethesolutionsofEq.
(3),whereconstructingElementde-notesthelocaltwistvgorthelocalshiftVgandSelfDualLatticecanbeeitherE8xE8,Spin32orSO8.
ThesolutionsarestoredinvectorWeights.
ClassCOrbifold.
ACOrbifoldobjectcontainsallinformationaboutasingleorbifoldcompactication.
ThemainmembervariableisvectorSectors,i.
e.
avectorofMtimesNCSectorobjects,oneforeachsectorofaM*Norbifold.
Therstelementcorrespondstotheuntwistedsectorandtheresttothevarioustwistedsectors.
ClassCOrbifoldGroup.
ACOrbifoldGroupobjectbasicallycontainsthespacegroupanditsgaugeembed-dingasobjectsofclassCSpaceGroup,CShiftVectorandCWilsonLines.
Furthermore,itcontainsavectorofallinequivalentconstructingelements(vector)andacorrespondingvectorofcentralizerelements(vector>).
12ClassCPrint.
TheCPrintclasscontainsallprintingcommands.
FortheconstructorCPrint(OutputTypeoutputtype,ostream*out)oneneedstospecifytheOutputType,beingeitherTstandard,Tmathematica,orTlatex,andaostreamobjecttosetthedestinationoftheoutput,eithertothescreenusing&coutortoaleusinganofstreamobject.
ClassCSector.
ACSectorobjectcontainsallinformationaboutanuntwistedortwistedsector.
Itismainlyspeciedbythelocaltwistvg(orinotherwordsbykandlsincevg=kv1+lv2).
Astheoscillatorexcitationsandtheright-movingpartofthestringonlydependonthelocaltwist,seeEq.
(3),theyareidenticalforallstringsfromagivensector.
Hence,thisdataisstoredinCSector.
ClassCSpaceGroup.
ACSpaceGroupobjectcontainsthedetailsaboutthespacegroupofanorbifoldmodel,asexplainedinSection2.
AllconstructingandcentralizerelementsarestoredinCSpaceGroupElementobjects.
Additionally,itincludesthegeometricalinformationofthecompactspace,suchasthe6Dlattice,itssymmetriesandtheorderoftheassociatedWilsonlines.
ClassCState.
ACStateobjectisbasicallyanorbifold-invariantcombinationofamasslessright-movingCHalfStateobjectandamasslessleft-movingCHalfStateobject.
AppendixA.
1.
ExamplesourcecodeWepresentasampleprogramthatcomputesandanalyzesthespectrumofthe6–IIorbifoldmodelof[9].
Inthesourcecodedistribution,thecorrespondingleissrc/examples/samplemain01.
cpp.
1#include2#include"cprompt.
h"3usingnamespacestd;45intmain(intargc,char*argv[])6{7ifstreamin("modelKRZ_A1.
txt");8if((!
in.
is_open(in.
good()))9exit(1);1011CPrintPrint(Tstandard,&cout);12stringProgramFilename="";1314COrbifoldGroupOrbifoldGroup;15if(OrbifoldGroup.
LoadOrbifoldGroup(in,ProgramFilename))//loadfromfile16{17coutModelfile\"modelKRZ_A1.
txt\"loaded.
"Orbifold\"KRZ_A1\"created.
\n"PrintshiftandWilsonlines:"Printspectrum,firstwithandthenwithoutU(1)charges:"Analyzemodel:"AllVEVConfigs;34boolSM=true;//lookforSM35boolPS=true;//lookforPSmodels36boolSU5=true;//lookforSU(5)models37//analyzetheconfiguration"KRZ_A1.
StandardConfig"of"KRZ_A1"38//andsavetheresultto"AllVEVConfigs"39CAnalyseModelAnalyze;40Analyze.
AnalyseModel(KRZ_A1,KRZ_A1.
StandardConfig,SM,PS,SU5,41AllVEVConfigs,Print);42if(SM||PS||SU5)//ifoneofthethreepossibilitiesistrue43{44coutModelhas3generationsplusvector-likeexotics:",seeAppendixB.
2.
7.
Notethatinagivenvev-congurationonecandeneseverallabelsforeacheld.
Finally,inmathematicatypesetting,forexample,thelabelq1isdisplayedasfldq1.
AppendixB.
1.
2.
SetsofeldsOnecanaccessseveraleldssimultaneouslynotonlybytheireldlabelsbutalsousingsetsofelds.
Thesesetsarestoredinthecurrentlyusedvev-congurationoftheorbifoldmodel.
(Consequently,onecannotaccessasetinadierentvev-congurationthanintheonewhereitwascreated.
)Formoredetailsonvacua,seeAppendixB.
2.
6.
Notethatsetsareonthesamefootingaseldlabels1.
I.
e.
onecanbuildintersectionslike:A-BfortheintersectionoftwosetsAandB,*-Afortheintersectionofallelds*andasetA,A-qfortheintersectionofasetAandalleldsofnameqorq-AfortheintersectionofalleldsofnameqandasetA.
Thecommandstocreateandmanipulatesetsaredisplayedinanyorbifoldmodeldirectoryofthepromptusingthecommandhelpsets.
(B.
1)Thecommandsare:Commandcreateset(SetLabel).
CreateanemptysetwithnameSetLabelandsaveitinthecurrentlyusedvev-conguration.
Optionally,thiscommandallowsfortheparametersfrommonomialsorfrommonomial(MonomialLabel)inwhichcasealleldsfromeitherallmonomialsoronlyfrommonomialMonomialLabelwillbeinsertedintothenewset.
SeeAppendixB.
1.
3formoredetailsonmonomials.
Commanddeleteset(SetLabel).
DeletethesetSetLabelofthecurrentlyusedvev-conguration.
Commanddeletesets.
Deleteallsetsofthecurrentlyusedvev-conguration.
Commandinsert(fields)intoset(SetLabel).
InsertfieldsintothesetSetLabel.
Optionally,theparameterif(condition)canbeusedtoinsertonlythosefieldsintothesetSetLabelthatsatisfythecondition.
Fordetailsonif(condition)seeAppendixB.
1.
4.
1Both,eldsandsetsofelds,willbedenotedasfieldsintheexplanationsofthefollowingsections.
15commanddescriptioncreateset(Test)createanemptysetnamedTestinsert(F1F2F3F4F5)intoset(Test)insertFi,i=1,5intothesetTestremove(*)fromset(Test)if(#osci.
!
=0)removeeldswithnon-zeronumberoperatorN(i.
e.
F5)printsetsprintallsetsTableB.
2:Shortexamplefortheuseofset-commandsinthedirectory/Z3StandardEmbedding>ofthe3standardembeddingmodel(usingthestandardlabelsFiofthevev-congurationTestConfig1).
Commandprintset(SetLabel).
PrintthecontentofthesetSetLabel.
Commandprintsets.
Printallsetsdenedinthecurrentlyusedvev-conguration.
Onecanusetheoptionalparameterifnotemptytoprintonlythenon-emptysets.
Commandremove(fields)fromset(SetLabel).
RemovefieldsfromthesetSetLabel.
Optionally,theparameterif(condition)canbeusedtoremoveonlythoseeldsthatsatisfythecondition.
Command#fieldsinset(SetLabel).
CountthenumberofeldsinthesetSetLabel.
AshortexampleshowingsomeofthebasiccommandsforsetsisgiveninTab.
B.
2.
AppendixB.
1.
3.
Gaugeinvariantmonomials(Holomorphic)gaugeinvariantmonomials(short:monomials)areusedtodescribesolutionstotheD=0supersymmetrycondition[67,68,69,70].
A(sub-)setofsolutionscanbefoundusingthecommandfindD-flat(fields)describedinAppendixB.
2.
6.
Moredetailsandexamplescanbeseenusingthecommandhelpmonomials(B.
2)inanyorbifoldmodeldirectory.
AppendixB.
1.
4.
IfconditionsManycommandsthatdealwitheldsallowfortheparameterif(condition)(orseveralcopiesthereof)sothatonlythoseeldsarechosenthatfulllalltheconditions.
AnexplicitexamplewasalreadygiveninTab.
B.
2.
Ingeneral,aconditionconsistsofthreeparts:thelefthandsidegivesthevariable(e.
g.
Qifortheeldsi-thU(1)charge,vevfortheeldsvacuumexpectationvalueor#osci.
forthenumberofoscillators),themiddlegivesthecomparisonoperator(e.
g.
==forequalor!
=forunequal)andtherighthandsidegivesavalue(e.
g.
arationalnumberor0).
Moredetailsandexamplescanbeseenusingthecommandhelpconditions(B.
3)inanyorbifoldmodeldirectory.
16AppendixB.
1.
5.
ProcessesThefollowingcommandsstartnewprocessesthatruninthebackgroundsothatonecancontinuetoworkwiththeprompt:/>createrandomorbifoldfrom(OrbifoldLabel)/A/couplings>createcoupling(fields)/A/vev-config>findD-flat(fields)/A/gaugegroup>findaccidentalU1sEachprocesshasanID,thesocalledPID.
SimilartotheLinuxcommandlineonecanseeallrunningprocessesusingthecommandpsandterminateaprocesswithPIDAusingkill(A).
Onecanalsokillallactiveprocessesusingthecommandkill(all).
Inascriptthecommandwait(X)mightbeusefulinordertocheckeveryXsecondsifallprocesseshavenishedandtocontinuewiththenextcommandsafterwards.
Moredetailscanbeseenusingthecommandhelpprocesses(B.
4)inanyorbifoldmodeldirectory.
AppendixB.
1.
6.
VectorsManycommandsneedavectorofrationalnumbersasaparameter.
ExamplesincludethecommandssetshiftV(i)=andsettorsionb=.
Inthesecasesthereareseveralpossibilitiesofhowtowritethevector.
Forexample,thefollowingformsofaarepossible:(1/31/10/10/1)=(1/3100)=(1/3,1,0,0)=(1/3,1,02)=1/3(1302)(B.
5)Inaddition,fortherstfourformsoftheexample-vector,onecanleavethebracketsaway.
AppendixB.
1.
7.
OutputformathematicaorinlatexstyleOftenitisusefultotransferdatafromtheorbifoldertomathematica,forexample,inordertouseSTRINGVACUA[71],SINGULAR[72],NonAbelianHilbert[69,70]orDiscreteBreaking[73,74].
Therefore,manycommandsallowfortheparameter@mathematica(B.
6)sothattheoutputofthecommandwillbeprintedinamathematicacompatiblestyle(ifavailable).
Forexample,printlistofcharges@mathematica(B.
7)inthedirectory/spectrum>.
Similarly,theparameter@latexcanbeusedinordertogettheoutputinlatexcode(again,ifavailable).
Inaddition,onecansetthedefaulttypesettingtomathematica,latexorbacktostandardusingthecommands@typesetting(mathematica),@typesetting(latex)or@typesetting(standard),(B.
8)respectively.
Finally,theparameternooutputcanbeusedtosuppresstheoutputofthecurrentcommand.
17AppendixB.
1.
8.
SystemcommandsandvariablesSystemcommandsstartwiththesymbol@andareusedtochangetheoutput'sstyleanddestination.
Moreover,thepromptallowsforsomepre-denedvariableswhichareparticularlyusefullinscripts.
Theystartandendwiththesymbol$.
Command@typesetting(Type).
Changetheoutput'stypesetting,seeAppendixB.
1.
7.
Command@beginprinttofile(A).
StartprintingoutputtoleAandnottothescreen.
Incontrast,onecanusetheparametertofile(A)sothattheoutputofonlythecurrentcommandisprintedtole,e.
g.
printsummarytofile(Summary.
txt).
Command@endprinttofile.
Stopprintingoutputtole.
Command@status.
Displaythedestinationoftheoutput(e.
g.
screen)andthestyleofthetypesetting(i.
e.
standard,latexormathematica).
Variables.
Therearethreepre-denedvariables:$OrbifoldLabel$,$VEVConfigLabel$and$Directory$.
Whenexecuted,avariableisreplacedbyacorrespondingstring,beingthelabelofthecurrentorbifoldmodel,thelabelofthecurrentvev-congurationorthepathofthecurrentdirectory,respectively.
Theyareparticularlyusefullinscripts,e.
g.
usedastofile($OrbifoldLabel$.
txt).
AppendixB.
2.
ThedirectoriesThestructureofthepromptconsistsofamaindirectory/>andsubdirectoriesthatcorrespondtoorbifoldmodels.
Eachorbifoldmodeldirectoryhasfurthersubdirectories/model>,/gaugegroup>,/spectrum>,/couplings>and/vev-config>.
Theyoercommandsoftherespectivecategory.
Inthissectionwegiveanalphabeticallyorderedglossaryofdirectory-commandsandexplaintheiruseindetail.
AppendixB.
2.
1.
Themaindirectory/>Inthemaindirectoryonecanbasicallycreate,loadandsaveorbifoldmodels.
Commandcreateorbifold(OrbifoldLabel)withpointgroup(M,N).
Createanemptyorbifoldmodeldirectoryforanorbifoldofspeciedpointgrouporders(useN=1forMorbifolds).
Commandcreaterandomorbifoldfrom(OrbifoldLabel).
Randomlycreateneworbifoldmodels.
De-tailsaregiveninSection4.
1.
2.
Moredetailsandexamplescanbeseenusingthemaindirectory's/>commandhelpcreaterandom.
(B.
9)Commanddeleteorbifold(OrbifoldLabel).
DeletetheorbifoldmodeldirectoryOrbifoldLabel.
Commanddeleteorbifolds.
Deleteallorbifoldmodeldirectories.
Commandloadorbifolds(Filename).
LoadallorbifoldmodelsfromthemodellenamedFilename.
Commandloadprogram(Filename).
LoadascriptfromleFilenameandexecutethecommandscon-tainedinthatle.
Commandsaveorbifolds(Filename).
SaveallorbifoldmodelsofthemaindirectorytoamodellenamedFilename.
18AppendixB.
2.
2.
Thedirectory/model>Inthedirectory/model>theinputdata(e.
g.
pointgroup,twists,shifts,Wilsonlines,etc.
)ofthecurrentorbifoldmodelcanbedisplayedandchanged.
Commandcreatesuborbifoldwithfactor(i).
StartingfromanorbifoldwithspacegroupS,onecancreateaso-calledsuborbifoldbasedonasubgroupS′S.
ForMorbifoldsthesubgroupS′Sisspeciedbyonenumber,ibeingadivisorofM.
DenotetheMtwistgeneratorofthespacegroupbyg∈S.
Then,thesubgroupS′Sisbasedonthetwistgeneratorgi∈S′SwhichgeneratesM/i.
Onecanusetheoptionalparameterand(j)tochoosegi∈S′Sandgj∈S′S(withi,jcoprimeandi,jdivideM)togeneratorM/i*M/j.
InthecaseofM*Norbifolds(withtwistgeneratorsg1,g2∈S)onehastospecifytwonumberscreatesuborbifoldwithfactor(i,j)(whereidividesMandjdividesN)forthenewgeneratorgi1gj2ofthesubgroupS′S.
Again,onecanusetheoptionalparameterand(k,l)tospecifyasecondgeneratorgk1gl2.
Notethatthiscommandisparticularlyusefultoanalyzethe6DorbifoldGUTlimitofanorbifoldmodel.
Forexample,startwiththe6–IIorbifoldMSSMof[40].
Thenthecommandcreatesuborbifoldwithfactor(2)willproducethe6D3orbifoldGUTlimitasanalyzedin[75].
Commandprintavailablespacegroups.
Printalistofallgeometrylescompatiblewiththespeci-edpointgroup.
Thegeometrylesaresearchedbytheorbifolderinthedirectory/localdirectory/Geometry>(ofthelocalPC).
Formoredetailsonthecontentofgeometryles,seeSection4.
4.
1.
Commandprintdiscretesymmetries.
Printthediscrete(Randnon-R)symmetriesasdenedinthegeometryle.
NotethatRsymmetriesneedtobedenedinthegeometryleinordertobeusedinthecomputationofallowedsuperpotentialcouplings.
Commandprintdiscretetorsion.
Printthe(generalized)discretetorsionparametersa,bα,cαanddαβasdenedinRef.
[36].
CommandprintmasslessA-movers.
whereAcanbeleftorright.
Printthemasslessleft-orrightmoversbeforesomeofthemareprojectedoutbytheactionofthecentralizer.
Commandprintorbifoldlabel.
Printtheorbifoldlabel(i.
e.
thenameofthecurrentorbifolddirectory).
Commandprintpointgroup.
Printthepointgroup.
Theoutputreads,forexample,PointgroupisZ3.
Commandprintshift.
Printtheshift(s)as16Dvector(s).
Theoutputreadse.
g.
V1=(1/31/3-2/300000)(00000000).
(B.
10)Commandprintspacegroup.
First,printthepointgroupandtheroot-lattice.
Next,printthegeneratorsofthespacegroup.
Theoutputreadse.
g.
SpacegroupbasedonZ3pointgroupandroot-latticeofSU(3)^3.
(B.
11)Generatorsare:(1,0)(0,0,0,0,0,0)(0,0)(1,0,0,0,0,0)···19where(k,l)(n1,.
.
.
,n6)correspondstotheelementθkωl,nαeαofthespacegroup.
Notethatroto-translationsandfreely-actinginvolutionsareallowedasgeneratorsofthespacegroup.
Forexam-pleinRef.
[34],oneofthegeneratorsofthe(0-2)modelreads(0,1)(0,0,0,0,1/2,0)correspondingtoω,12e5andoneofthegeneratorsofthe(1-1)modelreads(0,0)(0,1/2,0,1/2,0,1/2)correspondingto11,12(e2+e4+e6).
Commandprinttwist.
Printthetwist(s)asfour-dimensionalvector(s).
Theoutputreadse.
g.
v1=(01/31/3-2/3).
(B.
12)CommandprintWilsonlines.
PrinttherelationsamongtheWilsonlines(e.
g.
W1=W2for3),theirorder(e.
g.
order3for3Wi∈Λfor3)andtheWilsonlinesthemselvesas16Dvectors.
Theoutputreads,forexample,Wilsonlinesidentifiedontheorbifold:(B.
13)W1=W2,W3=W4,W5=W6AllowedordersoftheWilsonlines:333333W1=(00000000)(00000000)···Commandprint#SUSY.
Printthenumberofsupersymmetryin4D.
Theoutputreads,forexample,N=1SUSYin4D.
(B.
14)Commandsetheteroticstringtype(type).
Denethe16Dgaugelatticeoftheheteroticorbifoldmodel.
Here,typecanbeE8xE8orSpin32.
Commandsetshiftstandardembedding.
ChooseV=(v1,v2,v3,013)forMorbifoldsorV1=(v11,v21,v31,013),V2=(v12,v22,v32,013)forM*Norbifolds.
(Thenotation013means13timestheentry"0".
)CommandsetshiftV=orsetshiftV(i)=.
DenetheshiftvectorVofMorbifoldmodelsoroneofthetwoshiftvectorsVi(withi=1,2)ofM*Norbifoldmodelsas16Dvector,seeEq.
(1).
FormoredetailsonvectorsseeAppendixB.
1.
6.
Commandsettorsiona=n/d,b=,c=ord=.
Setthe(generalized)discretetorsionparametersasdenedin[36],i.
e.
a,bα,cαanddαβ(forα,β=1,6;dαβ=dβαhas15components).
Notethattheparametersarenotcheckedformodularinvarianceandhencemightcauseinconsistentspectra.
FormoredetailsonvectorsseeAppendixB.
1.
6.
CommandsetWLW(i)=.
DenetheWilsonlineWiasa16Dvector(withi=1,6),seeEq.
(1).
FormoredetailsonvectorsseeAppendixB.
1.
6.
Commandusespacegroup(i).
withi=1,Loadthespacegroupfromthei-thgeometryle,wheretheindexicorrespondstothepositioninthelistofgeometrylesasdisplayedusingthecommandprintavailablespacegroups.
20AppendixB.
2.
3.
Thedirectory/gaugegroup>Inthisdirectoryonecanprintandchangesomedetailsofthegaugegroupforthecurrentlyusedvev-conguration.
Inmoredetail,onecandisplaytheU(1)generatorsandthesimpleroots,changethebasisofU(1)generators,deneaB-LgeneratorandidentifyaccidentalU(1)symmetriesofthesuperpotential.
Notethatallgauge-group-editingcommandsarenotavailableinthevev-congurationStandardConfig1.
CommanddeleteaccidentalU1s.
DeletetheaccidentalU(1)chargesofallelds.
CommandfindaccidentalU1s.
Takethesuperpotential(asfarasithasbeencreatedinthedirectory/couplings>,seeAppendixB.
2.
5)andidentifyitsaccidentalU(1)symmetries.
Thecommandstartsanewprocessthatrunsinthebackoftheprompt.
Theresultsaresavedinthecurrentlyusedvev-conguration.
Presumably,theaccidentalU(1)symmetrieswillbebrokenexplicitlybyhigherorderterms,butneverthelessmightbeofphenomenologicalrelevance,e.
g.
forthestrongCPproblem[27]andprotondecay[31].
Optionally,onecanusetheparameterfieldswithzerocharge(fields)inordertondonlythoseaccidentalU(1)symmetriesunderwhichtheeldsoffieldsareuncharged.
Onatechnicallevel,thisisachievedbyinserting,duringthisanalysis,eacheldoffieldsasalineartermintothesuperpotential.
CommandloadaccidentalU1s(Filename).
LoadaccidentalU(1)chargesfromalenamedFilename.
Commandprintanomalousspacegroupelement.
Printdetailsondiscreteanomaliesusingthediscretesymmetriesdenedinthegeometryleandidentifytheso-calledanomalousspacegroupelement[76].
Commandprintanomalyinfo.
Printdetailsongaugeandgravitationalanomaliesandchecktheiruni-versalityrelations.
Indetail,inthecaseofN=1SUSYin4D,besidethepurenon-Abeliananomalies,therelations2124trQi=16|ti|2trQ3i=12trQi=12|tj|2trQ2jQi=const.
0ifi=1,i.
e.
i=anom0otherwise(B.
15)(withij)areveried,wheretiisa16Dvectorcorrespondingtothei-thU(1)generatorsothataeldwithshiftedleft-movingmomentumpshcarriesthechargeQi=psh·tiandtrsumsoverthecontributionsfromallmasslessleft-chiralelds.
CommandprintB-Lgenerator.
PrinttheU(1)BLgeneratorasa16Dvector.
CommandprintFIterm.
PrinttheFayet-Iliopoulosterm(i.
e.
trQanomasinEq.
(B.
15)),ifthereisananomalousU(1).
Commandprintgaugegroup.
Printtheobservableandhiddenpartofthegaugegroupforthecurrentlyusedvev-conguration.
Commandprintsimpleroot(i).
Printthei-thsimplerootas16Dvector.
Commandprintsimpleroots.
Print(achoiceof)simplerootsofallnon-Abeliangaugegroupfactorsas16Dvectors.
2Intheorbifolder,theconventionforthequadraticDynkinindexissuchthat=1forthefundamentalrepresentationofSU(N)groups.
21CommandprintU1generator(i).
Printthei-thU(1)generatoras16Dvector.
CommandprintU1generators.
PrintallU(1)generatorsas16Dvectors.
CommandsaveaccidentalU1s(Filename).
SavetheaccidentalU(1)chargestoalenamedFilename.
CommandsetB-L=.
DeneU(1)BLasa16Dvector.
SinceintheorbifolderallU(1)generatorsaredemandedtobeorthogonaltoeachother,butU(1)BLisingeneralnotorthogonaltohyper-charge,B-Lisstoredasanadditionalvector.
OnecanusetheoptionalparameterallowforanomalousB-LifU(1)BLisallowedtomixwiththeanomalousU(1).
FormoredetailsonvectorsseeAppendixB.
1.
6.
CommandsetU1(i)=.
ChangethebasisofU(1)generatorsbyspecifyinga16Dvectorasthei-thgenerator.
Thenewgeneratormustbeorthogonaltoallsimplerootsandtothej-thU(1)generator,forjiwillbechangedautomaticallysuchthat,attheend,allgeneratorsareorthogonaltoeachother.
FormoredetailsonvectorsseeAppendixB.
1.
6.
AppendixB.
2.
4.
Thedirectory/spectrum>Thisdirectoryoersaccesstoallinformationaboutthemasslessspectrum.
Indetail,foreachmass-lesseldonecanobtaintheSUSYmultiplettype(i.
e.
forN=1supersymmetryin4D:left-chiral,right-chiral,vectorandmodulus),thelocalization(correspondingtoitsconstructingspacegroupele-ment),theshiftedleft-movingmomentumpsh,thenon-Abelianrepresentation,theU(1)charges,theB-Lcharge(ifdened),theshiftedright-movingmomentumqsh,theoscillatorexcitations,theRcharges,mod-ularweights(ifdened),thelabeloftheeldandnallyitsvev.
Notethatcomplexconjugaterepresenta-tionsareprintedasnegativeintegers;forexample,-3denotestheconjugatefundamentalrepresentation3ofSU(3).
Commandfindpotentialblowupmodes(fields).
Printalistofpotentialblow-upmodesconsideringfieldsonly,i.
e.
printallfieldsforeachxedbrane/point.
Commandfindrandomblowupmodes(fields).
printarandomlistofblow-upmodesfields,oneperxedbrane/point.
TheresultcanbesavedtoasetSetLabelusingtheoptionalparametersavetoset(SetLabel).
Commandprint(fields).
Printsomedetailsoffields.
Thereisoneoptionalparameter,withinternalinformation,thatdisplayssomeinternalinformationabouthowtheelds'datacanbeaccessedintheC++sourcecodeoftheorbifolder.
Commandprintallstates.
Printdetailsofallelds(includingleft-chiralsuperelds,vectorsupereldsofthe(non-Abelian)gaugebosons,moduliandtheirCPT-partners).
Commandprintlistofemptyfixedbranes.
Printalistofallxedbranes/pointsthatpotentiallycouldcarryleft-chiraleldsbutareemptyforthecurrentorbifoldmodel.
22Commandprintlistofcharges(fields).
Printgaugeanddiscretechargesoffields.
Forexample,(-1/3-1/32/300000)(00000000)(-1/32/3-1/3)(2001)"F10"(B.
16)(-1/32/3-1/300000)(00000000)(-1/32/3-1/3)(2001)"F10"(2/3-1/3-1/300000)(00000000)(-1/32/3-1/3)(2001)"F10"fortheeldF10ofthe3orbifoldwithstandardembedding.
Eachlineconsistsofthreeparts:rstthe16Dshiftedleft-movingmomentumpsh(printedastwoeight-dimensionalvectorsinthecaseofE8*E8),theRcharges:(R1,R2,R3)=(13,23,13)intheexample,thechargeswithrespecttothediscretenon-Rsymmetries:(k,n1+n2,n3+n4,n5+n6,)=(2,0,0,1)intheexample,thelabelofthecorrespondingeld,where,ingeneral,theRandnon-Rsymmetriesmustbespeciedinthegeometryle,seeSection4.
4.
1.
Notethatitcanbeveryhelpfultousetheoptionalparameter@mathematicaforthiscommandinordertotransfertheinformationabouttheeldstomathematica,seeAppendixB.
1.
7.
Commandprintsummary.
Printthegaugegroupandasummarytableofthemasslessmatterelds.
Im-portantoptionalparametersare:ofsectorsoffixedpointsoffixedpoint(X)wherethexedpointXcanbespeciedinthreeways:i)usingk,l,n1,n2,n3,n4,n5,n6,ii)usinglocofFiwhereFiisthelabelofatwistedeldandiii)byaxedpointlabelasspeciedinthedirectory/vev-config/labels>.
ofsectorT(k,l)wherekandllabeltheθkωltwistedsector.
UseofsectorT(0,0)fortheuntwistedsector.
Inallthesecasesonecanuseinadditionthefollowing,optionalparameters:withlabels:printthecurrentlyusedlabelsoftheeldsasspeciedinthedirectory/vev-config/labels>.
noU1s:donotprinttheU(1)chargesoftheelds.
typeofSUSYmultiplet,e.
g.
vector,gravity,modulusoranykindforalltypes;ifnotspeciedtheleft-chiraleldsareprintedonly.
Moredetailsandexamplescanbeseenusingthecommandhelpprintsummary(B.
17)inthedirectory/spectrum>.
23Commandtextable(fields).
Printalatextablewithinformationaboutfields.
Thetablecontainsthe(gauge)chargeswithrespecttotheobservablesectorofthecurrentlyusedvev-congurationandthediscretechargesasspeciedinthegeometryle(seeSection4.
4.
1).
Onecanusetheoptionalparameterprintlabels(i,j,.
.
)inordertolistthei-th,j-th.
.
.
label(s)oftheelds.
AppendixB.
2.
5.
Thedirectory/couplings>Thedirectory/couplings>allowstoidentifyandanalyzeallowedtermsofthesuperpotential(i.
e.
termsthatareinvariantunderallgaugeanddiscretesymmetries).
The(ingeneralmoduli-dependent)coecientsarenotcomputed.
Furthermore,onecananalyzemassmatrices(e.
g.
ofvector-likeexotics).
Notethatcouplingsandmassmatricesarestoredinthecurrentlyusedvev-conguration.
Hence,theycanonlybeaccessedinthevev-congurationwheretheyhavebeendened.
Forsimplicity,theabbrevia-tionsmmformassmatrixandmmsformassmatricesapplytoallcommands.
Commandautocreatemassmatrix(AB).
CreatethecouplingsrelevantfortheeectivemassmatrixMijAiBj.
Optionally,onecanspecifythethelabelofthoseeldswhosevevsgenerateMijusingthepa-rametersinglet(N)(withdefaultvalueN=n)andthemaximalorderXinsingletsNusingtheparametermaxorder(X).
Commandcreatecoupling(fields).
Findtheallowedsuperpotential-couplingsbetweenfieldsandstoretheresultinthecurrentlyusedvev-conguration.
Forexample,alltrilinearcouplingsarecreatedusingthecommandcreatecoupling(B.
18)Optionally,onecanrestrictfieldsusingtheparameterallowedfields(.
.
.
),e.
g.
thecommandcreatecoupling(nnn)allowedfields(SetA)createsalltrilinearcouplingsofeldsnfromSetA.
Commandfind(fields).
Displaysalistofallowedcouplingsinvolvingtheeldsfields.
Commandfindeffective(fields).
Asfind(fields),butonlytheeectivecouplings,i.
eafterreplac-ingeldswithnon-zerovevbytheirvevs.
Commandloadcouplings(Filename).
LoadcouplingsfromleFilenameintothecurrentlyusedvev-conguration.
Thiscommandisdisabledinthewebinterface.
Commandmassmatrix(AB).
CreatethemassmatrixMijAiBj(fromthecurrentsuperpotential)andsaveitinthecurrentlyusedvev-conguration.
Commandprinteffectivesuperpotential.
Similartothecommandprintsuperpotentialbutprintonlytheeectivecouplings,i.
eafterreplacingeldswithnon-zerovevbytheirvevs.
Commandprintlistofmassmatrices.
Printallmassmatricesofthecurrentlyusedvev-conguration.
Commandprintmassmatrix(i).
Printthei-thmassmatrix.
Theoptionalparametermaxorder(X)speciestheorderXintheeldsuptowhichamatrixentryshallbeprintedexplicitly.
Commandprintsuperpotential.
Printthesuperpotentialofthecurrentlyusedvev-conguration.
Commandprintvanishingcouplings.
PrintallcasesofvanishingcouplingsaslistsofhighestweightsinDynkinlabels.
Seethecommandremovevanishingcouplings.
24Commandremovevanishingcouplings.
Removecouplingsthatvanishbecauseofsymmetry/anti-sym-metryofrepeatedidenticalelds,e.
g.
letbeanSU(2)doublet,thenthegaugeinvariantcoupling=ijij=0vanishes.
Thiscommandrequiresadditionaluserinput.
Commandsavecouplings(Filename).
Saveallcouplingsofthecurrentlyusedvev-congurationtoale.
OnecanoptionallysaveonlycouplingsoforderXusingtheparameteroforder(X).
Thiscommandisdisabledinthewebinterface.
AppendixB.
2.
6.
Thedirectory/vev-config>Inthisdirectoryonecandeneseveralvev-congurations.
Eachofthemischaracterizedbyachoiceofhiddenandobservablegaugegroup,alabelingoftheeldsandbytheirvev.
Inaddition,onecananalyzephenomenologicalpropertiesandsupersymmetriccongurations(F=D=0)inthisdirectoryanddeterminetheunbrokengaugegroupofagivenvev-conguration.
Foreachorbifoldmodeltherearetwostandardvev-congurations:StandardConfig1andTestConfig1.
Therstonecannotbechanged,butthelatteronecanbeandisusedasdefault.
InbothcongurationsthefullgaugegroupisselectedastheobservablesectorandeldsarelabeledF1,F2,F3,allwithzerovev.
Notethatthelabelsoftheelds(seeAppendixB.
1.
1),setsofelds(seeAppendixB.
1.
2),monomials(seeAppendixB.
1.
3),allowedcouplingsandmassmatrices(createdinthedirectory/couplings>)aresavedinavev-conguration.
Hence,thesedatacanonlybeaccessedinthevev-congurationwheretheyhavebeendened.
Inaddition,notethatallconguration-editingcommandsarenotavailableinthevev-congurationStandardConfig1.
Commandanalyzeconfig.
Automaticallycheckwhetherthecurrentvev-congurationoftheorbifoldmodelallowsforvacuawithStandardModel,Pati-SalamorSU(5)gaugegroup,SU(3)C*SU(2)L*U(1)Y,SU(4)*SU(2)L*SU(2)RorSU(5),(B.
19)respectively,threegenerationsofquarksandleptonsandvector-likeexotics.
InthecasetheorbifolderisnotabletoidentifyoneofthesepossibilitiesforthecurrentorbifoldmodeloneobtainstheoutputNovev-configurationidentified.
Otherwise,correspondingnewvacuawillbecreatedandconvenientlabelswillbeassignedtoallmatterelds(e.
g.
q1,q2andq3forthethreegenerationsofquarkdoublets).
Thecommandallowsfortwooptionalparameters:printSU(5)simplerootstoprintthesimplerootsofanintermediateSU(5)groupthathasbeenusedinordertoidentifythehyperchargegeneratorandXgenerationswithX=0,1,2,3,.
.
.
tospecifythe(net)numberofgenerations.
CommandcomputeF-terms.
ComputetheF-termsusingthesuperpotentialthatwascreatedforthecur-rentlyusedvev-conguration(inthedirectory/couplings>.
Theoptionalparametermaxorder(X)allowstosetanupperlimitXontheorderofsuperpotentialcouplings.
Commandcreateconfig(ConfigLabel).
Createanewvev-conguration.
Optionally,onecanspecifytheoriginofthenewvev-congurationusingtheparameterfrom(AnotherConfigLabel).
Ifthisparameterisnotusedtheoriginofthenewvev-congurationisthestandardvev-congurationStandardConfig1.
Commanddeleteconfig(ConfigLabel).
Deletethevev-congurationConfigLabel.
25CommandfindD-flat(fields).
IdentifygaugeinvariantmonomialsoffieldsassolutionstotheD=0supersymmetrycondition,seeAppendixB.
1.
3.
Thiscommandallowsfortwoparameters:i)withFItoallowformonomialswithnon-vanishinganomalousU(1)chargeinordertocanceltheFItermandii)savetoset(SetLabel)tosavethoseeldsinasetcalledSetLabelthatareinvolvedinthenewmonomials.
Commandfindunbrokengaugegroup.
Dependingonthevevassignmentspeciedinthecurrentlyusedvev-conguration,identifybrokenandunbroken(Abelianandnon-Abelian)gaugegroupfactors.
Inaddi-tion,theU(1)chargesofalleldsarere-computedinthenewU(1)basis.
Thereisoneoptionalparameter:printinfotodisplaysomedetails.
Commandprintgaugegroup.
Printthechoiceofobservableandhiddensectorofthecurrentlyusedvev-conguration,wherethehiddensectorgaugegroupfactorsaremarkedbybrackets,e.
g.
[SU(4)].
Commandprintconfigs.
Printanoverviewofallvacuadenedforthisorbifoldmodel.
Thecurrentlyusedvev-congurationishighlightedbyanarrow->infront,e.
g.
->"TestConfig1".
Commandrenameconfig(OldConfigLabel)to(NewConfigLabel).
Changethenameofavev-congura-tionfromOldConfigLabeltoNewConfigLabel.
Commandselectobservablesector:parameters.
Assignachoiceofobservableandhiddengaugegroupsinthecurrentvev-conguration.
Admissibleparametersare:gaugegroup(i,j,.
.
.
),wheretheindicesi,j=1,2,.
.
.
refertothedierentnon-Abeliangaugegroupfactorssortedasdisplayedbyprintgaugegroup.
Theindicesprovidedarechosenaspartoftheobservablesector.
fullgaugegroupAllnon-Abeliangroupfactorsareassignedasobservablesector.
nogaugegroupsNonon-Abeliangroupfactorisassignedaspartoftheobservablesector.
U1s(i,j,.
.
.
),wheretheindicesi,j=1,2,.
.
.
refertothedierentU(1)gaugesymmetries.
Theindicesprovidedarechosenaspartoftheobservablesector.
allU1sAllU(1)sareassignedaspartoftheobservablesector.
noU1sNoU(1)istakenfortheobservablesector.
Forexample,assumingthatthegaugegroupisE6*SU(3)*E8,theinstructionselectobservablesector:gaugegroup(1,2)selectsE6*SU(3)astheobservableandE8asthehiddengaugegroups.
Commanduseconfig(ConfigLabel).
Changethecurrentlyusedvev-congurationtoConfigLabel.
Commandvev(fields)Changethevevsoffieldstonewvalues.
Forexample,vev(*)=0turnsothevevofallelds,vev(SetA)=randassignsrandomvevstotheeldsofthesetSetAandvev(n1)=0.
1setsthevevofn1to0.
1.
AppendixB.
2.
7.
Thedirectory/vev-config/labels>Inthisdirectoryonecandene,foreachvev-conguration,appropriatelabelsfortheelds.
Themaincommandsareprintlabelsandcreatelabels.
Inbothcases,asummarytableofmasslesseldsisprinted,sortedbythoserepresentationsandU(1)chargesthatbelongtotheobservablesectorofthecurrentlyusedvev-conguration.
Theobservablesectorcanbechangedusingthecommandselectobservablesector:.
.
.
inthedirectory/vev-config>,seeAppendixB.
2.
6.
26Commandchangelabel(Ai)to(Bj).
ChangethelabeloftheeldAitoBj.
Commandassignlabel(Label)tofixedpoint(k,l,n1,n2,n3,n4,n5,n6).
AssignLabeltothexedbrane/pointspeciedby(k,l,n1,n2,n3,n4,n5,n6).
Commandcreatelabels.
First,asummarytableofmasslesseldsisdisplayed.
Thentheuserisaskedtospecifyanameforeachlineofthetable.
Commandloadlabels(Filename).
LoadlabelsfromthelenamedFilename.
Thiscommandisdisabledinthewebinterface.
Commandprintlabels.
Printasummarytableofthecurrentlyusedlabelsdisplayingthegaugerepre-sentationswithrespecttotheobservablesectoronly.
Commandsavelabels(Filename).
SavethelabelstothelenamedFilename.
Thiscommandisdisabledinthewebinterface.
Commanduselabel(i).
Changethecurrentlyusedlabelstothei-thlabeling.
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