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PublishedforSISSAbySpringerReceived:April10,2018Accepted:May21,2018Published:May30,2018D6R4curvaturecorrections,modulargraphfunctionsandPoincareseriesOlofAhlenaandAxelKleinschmidta,baMax-Planck-Institutf¨urGravitationsphysik(Albert-Einstein-Institut),AmM¨uhlenberg1,DE-14476Potsdam,GermanybInternationalSolvayInstitutes,ULB-CampusPlaineCP231,BE-1050Brussels,BelgiumE-mail:olof.
ahlen@aei.
mpg.
de,axel.
kleinschmidt@aei.
mpg.
deAbstract:InthisnotewestudytheU-dualityinvariantcoecientfunctionsofhighercurvaturecorrectionstothefour-gravitonscatteringamplitudeintypeIIBstringtheorycompactiedonatorus.
ThemainfocusisontheD6R4termthatisknowntosatisfyaninhomogeneousLaplaceequation.
WeexhibitanovelmethodforsolvingthisequationintermsofaPoincareseriesansatzandrecoverknownresultsinD=10dimensionsandndnewresultsinD0}=SL(2,R)/SO(2)is[41](12)E(0,1)=E(0,0)2.
(3.
1)TheR4functionE(0,0)onHisgivenbyE(0,0)(z)=2ζ(3)E3/2(z),(3.
2)–7–withE3/2(z)beingthes=32caseofthenon-holomorphicEisensteinseriesEs(z)=γ∈B(Z)\SL(2,Z)[Im(γz)]s=ys+ξ(2(1s))ξ(2s)y1s+n=0Fn,s(y)e2πinx(3.
3)withthenon-zeroFouriercoecientsFn,s(y)=2ξ(2s)y1/2|n|s1/2σ12s(|n|)Ks1/2(2π|n|y)(3.
4)andξ(s)=πs/2Γ(s/2)ζ(s).
Anelementγ=abcd∈SL(2)actsonz∈Hbyz→γz=az+bcz+d.
Kt(y)intheaboveisamodiedBesselfunctionofthesecondkindandσt(k)=d|kdtisthe(positive)divisorsumfork∈Z>0.
TheEisensteinseriesEs(z)isinducedfromthecharacterχs(z)=ys.
TheLaplaceoperatoronHis=y22x+2y.
Thephysicalinterpretationofthemodulusz∈HhereissuchthatitsrealpartxcorrespondstotheRRaxionoftypeIIBstringtheorywhileitsimaginarypartyistheinversestringcouplingg1s.
TheFourierexpansionofE3/2(z)givenin(3.
3)thencontainstwozeromodeterms,y3/2=g3/2sandy1/2=g1/2s.
Thesecorrespondtotheperturbativetree-levelandone-loopcontributionstothefour-gravitonscatteringamplitudeexpressedinEinsteinframe;thefactthattherearenofurtherzeromodesisastrongperturbativenon-renormalisationtheorem[2].
MakingthePoincareseriesansatz(2.
7)forE(0,1)(z),viz.
E(0,1)(z)=γ∈B(Z)\SL(2,Z)σ(γz)(3.
5)theinhomogeneousLaplaceequation(3.
1)leadstothe'folded'equation(12)σ(z)=4ζ(3)2y3/2E3/2(z).
(3.
6)Thisequationcorrespondsto(2.
10)inthegeneraldiscussionandwecanfurtheranalyseitbyFourierexpandingbothsides.
TheabelianunipotentinvariancegrouphereisgivenbyU(Z)=N(Z)=1k1k∈Z.
WritingtheFourierexpansionofσ(z)asσ(z)=c0(y)+n=0cn(y)e2πinx(3.
7)leadstothefollowingtwoequationsforthezeroandnon-zeroFouriermodes(y22y12)c0(y)=4ζ(3)2y343π2ζ(3)y,(3.
8a)(y22y4π2n2y212)cn(y)=16πζ(3)y2|n|σ2(|n|)K1(2π|n|y),(3.
8b)wherewehaveusedtheexplicitformoftheFouriercoecientsforE3/2(z)givenin(3.
3).
Theseequationscorrespondto(2.
15)inthegeneraldiscussion.
Thegeneralsolutiontoequation(3.
8a)forthezeromodec0(y)issimpletoobtain:c0(y)=23ζ(3)2y3+19π2ζ(3)y+αy3+βy4.
(3.
9)–8–Here,αandβareaprioriundeterminedintegrationconstantsthathavetobexedbyboundaryconditions.
WenotethatperformingthePoincaresumofanytermoftheformysproducestheEisensteinseriesEs(z)withperturbativetermsysandy1s,cf.
(3.
3).
Havingeitherαorβnon-zerowillthereforenecessarilyleadtothetwozeromodetermsy3andy4inthesummedE(0,1).
Whilethetermy3=g3scorrespondstoathree-loopcontribution(afterchangingtostringframe),thetermy4=g4swouldbeacontributionoflooporder1/2,somethingthatisincompatiblewithstringperturbationtheory.
Sincebothhomogeneoussolutionsparametrisedbyαandβwouldleadtothisinconsistentbehaviour,weareledtosetα=β=0c0(y)=23ζ(3)2y3+23ζ(2)ζ(3)y.
(3.
10)Thesolutionofequation(3.
8b)forthenon-zeromodescn(y)ismorecomplicated.
ThehomogeneousequationcanberecastinBesselformandhasonlyonesolutionthatfallsoattheweakcouplingcuspy=g1s→∞.
Formodenumbernitisgivenbyy1/2K7/2(2π|n|y)andhasinfactaniteasymptoticexpansionaroundweakcoupling.
Aparticularsolutionof(3.
8b)canbeextractedfromtheanalysis[41]6andcombinedwiththehomogeneoussolutionweobtaincn(y)=8ζ(3)σ2(|n|)|n||n|y+10π2y|n|K0(2π|n|y)+6π+10π3y2|n|2K1(2π|n|y)+αny1/2K7/2(2π|n|y),(3.
11)whereαnistheintegrationconstantassociatedwiththehomogeneoussolution.
Again,thisintegrationconstanthastobexedfromasymptoticconsiderations.
Inthiscase,weconsiderthestrongcouplingregiony→0wheretheseinstantoniccontributionsdominate.
ByS-dualitythisregionisalsorelatedtotheperturbativeregimey→∞.
Theabsenceofanysingulartermsinthelimity→0determinesthevalueofαntobeαn=128ζ(3)σ2(|n|)3π|n|,(3.
12)leadingtothenalexpressionaftersomerearrangementscn(y)=8ζ(3)σ2(|n|)y1+40(2π|n|y)2K0(2π|n|y)+122π|n|y+80(2π|n|y)3K1(2π|n|y)163π(|n|y)1/2K7/2(2π|n|y).
(3.
13)Thiswaycn(y)isoforderyasy→0anddoesnotcontainanysingularterms.
7Theresultingexpressionagreespreciselywiththeresultfoundin[41]butnowobtaineddirectlyfromaPoincareseriesansatz.
6InappendixA,wepresentanalgebraicformalismthatalsoleadstothissolution.
7ThisconditionappearsslightlystrongerthantherequirementO(y2)asy→0forthesummedsolutionE(0,1)(z)thatwasfoundbyS-dualityinLemma2.
9in[41].
–9–Wenotethattheseedfunctionσ(z)=c0(y)+n=0cn(y)e2πinxdependsnon-triviallyonbothBorelcoordinatesxandyandinparticularisnotacharacteronB.
ThisleadstocomplicationswhencarryingoutthePoincaresum.
Infact,thefullexpressionisnotknown.
LetusmakeafewcommentsaboutconvergenceofthePoincaresumovertheseedwejustdetermined.
Ifthesumwereabsolutelyconvergentonecouldperformthesumoverallthetermsseparately.
ThiscannotbetrueasitiswellknownthatthePoincaresumforthelinearterminyinc0(y)doesnotconverge:itrepresentsthelimitingvalueofthenon-holomorphicEisensteinseriesforSL(2).
Itisthereforedesirabletointroducearegularisedversionofthesolutiondependingonaparameterwithalimitthatisrelatedtotheabovesolution.
8InappendixB,wepresentaregularisedversionoftheequationandthesolutionwherethisproblemdoesnotarise.
ThephysicalcontentofthesolutionisobtainedbyperformingaFourierexpansionofE(0,1)(z)=γσ(γz)withrespecttotheperiodicrealpartxthatrepresentstheRamond-RamondaxionintypeIIBtheory.
ThezeromodepieceinE(0,1)(z)correspondstotheperturbativetermsings,togetherwithnon-perturbativecontributionsofvanishingnetinstantoncharge.
Theseshouldbeinterpretedasinstanton-anti-instantoncontributionsandtheyareexponentiallysuppressedbye4π|n|y[10].
WehavenotmanagedtoderivetheFourierexpansionfromthePoincaresumformandheremerelyquotetheresultfortheperturbativezeromodesobtainedin[10,41]9E(10B,pert.
)(0,1)=23ζ(3)2y3+43ζ(2)ζ(3)y+4ζ(4)y+4ζ(6)27y3.
(3.
14)Theperturbativetermsthusobtainedcorrespondtocontributionsfromtree-leveluptothree-loops.
ThenumericalvaluesobtainedbyusingSL(2,Z)invarianceandthedierentialequationhavebeenconrmedbydirectstringtheorycalculations[10,43,51].
3.
2ToroidalcompacticationtoD=7InthecaseofD=7,theU-dualitygroupisSL(5,Z)andthemodulispaceisSL(5)/SO(5).
Equation(1.
3c)becomes425E(7)(0,1)=E(7)(0,0)2,(3.
15)wheretheR4-functionE(7)(0,0)(g)=2ζ(3)E(3Λ1ρ,g)satises+125E(7)(0,0)=0.
(3.
16)Wewillanalyseequation(3.
15)overthemirabolicP1,i.
e.
themaximalparabolicsubgroupassociatedwithnode1.
Thisistheparabolicthatisassociatedwiththestringperturbation8IntermsoftheEisensteinseriesEs(z)inducedbythecharacterysoneobtainstheso-called(rst)Kroneckerlimitformula[35]fors→1.
9Theexponentiallysuppressedtermsinthezeroandnon-zeromodesarenotknownexplicitly.
–10–theoryexpansion;theLeviisSL(4)*GL(1)=SO(3,3)*R+(locally)givingtheT-dualitymodulispaceofthestringtheorythree-torusandthestringcoupling.
Explicitly,weparametrisethegroupelementg∈SL(5)asg=ulk=1Q01r4/500r1/5e4k(3.
17)whereQisafour-componentrowvectorande4isageneralelementofSL(4)andk∈SO(5).
Thevariablerequalstheinversestringcoupling10inD=7,QcorrespondstothefouraxionsthatBPS-instantonscoupleto11ande4∈SL(4)=SO(3,3)isassociatedwiththemodulispaceofT3.
TheLaplacian=SL(5)onSL(5)/SO(5)decomposesinthesecoordinatesasSL(5)=58r22r158rr+r2||e14Q||2+SL(4).
(3.
18)3.
2.
1PerturbativetermsinthestringcouplingBeforesolvingtheinhomogeneousequation(3.
15)usingaPoincaresum,werstconsidertheperturbativepiecessimilarto(3.
14)byconsideringthezeroFouriermodesinanex-pansionofE(7)(0,1)inthedecomposition(3.
17).
Asimilaranalysiscanbefoundin[14].
MakingtheansatzfortheperturbativetermsuptothreeloopswithFhdenotingtheh-loopperturbativepieceas12E(7,pert.
)(0,1)=r14/5r2F0+r0F1+r2F2+r4F3=r24/5F0+r14/5F1+r4/5F2+r6/5F3(3.
19)leadstothefourequationsSL(4)6F0=4ζ(3)2,(3.
20a)SL(4)212F1=16ζ(2)ζ(3)ESL(4)(2Λ1ρ,e4),(3.
20b)SL(4)10F2=16ζ(2)2ESL(4)(2Λ1ρ,e4)2,(3.
20c)SL(4)92F3=0.
(3.
20d)Theequationsnotinvolvingasquaredsourceontheright-handsidearesolvedbyF0=23ζ(3)2,(3.
21a)F1=43ζ(2)ζ(3)ESL(4)(2Λ1ρ,e4)+5π756ζ(7)ESL(4)(7Λ2ρ,e4),(3.
21b)F3=4ζ(6)ESL(4)(6Λ1ρ,e4)+ESL(4)(6Λ3ρ,e4).
(3.
21c)10WeparametrisethestringcouplinginDdimensionsgDsuchthatdierentordersinperturbationtheorydierbyg2D;thisisdierentfromtheconventionusedin[17].
11IntypeIIAlanguage,therearethreeD0-instantons(wrappingoneofthethreecyclesofT3)andoneD2-instanton(wrappingthefulltorus).
IntypeIIBlanguage,thereisone(point-like)D(1)-instantonandthreeD1-instantons(wrappingtwooutofthreecycles).
12Theoverallpre-factorcomesfromrelatingthestringscaletothePlanckscaleinD=7space-timedimensions.
–11–Theconstanttree-levelcontributionfollowsbyexpandingtheknownamplitudewhichalsorulesoutanycontributionsfromthekernelofthedierentialoperator.
Theone-looppieceisathetaliftoftheNarainpartitionfunction[13,52]F1=π3SL(2,Z)\Hd2ττ22Γ(3,3)(e4)ξ(3)E3(τ)+ζ(3),(3.
22)whereτisthecomplexstructureofthestringone-looptorusandthecombinationξ(3)E3(τ)+ζ(3)canalsobeobtainedusingmodulargraphfunctions[37].
Thesecondtermin(3.
20b)isinthekernelofthedierentialoperatorbutisneededforobtainingtherightdecompacticationlimit.
Thethree-loopequationishomogeneousandtheparticularcom-binationofthehomogeneoussolutionsisxedbyhavingtherightdecompacticationlimitconsistentwith(3.
14).
Itisalsogivenbythegenus-threethetaliftoftheconstantfunc-tion[14].
Thetwo-looptermisasalwaysthehardestasitsatisesasimilarinhomogeneousequationtotheoriginalfunction.
ItisconnectedtoanintegralovertheKawazumi-Zhanginvariant[43,44]andconstrainedbyhavingtherightdecompacticationlimit.
3.
2.
2SolutionusingaPoincaresumLetusassumethatE(7)(0,1)isaPoincareserieswithrespecttothesamemaximalparabolicP1asin(2.
7),i.
e.
E(7)(0,1)(g)=γ∈P1(Z)\SL(5,Z)σ(γ·g).
(3.
23)Wenotethatthisparticulartypeofparaboliccosetsumisanassumption.
Ourmotivationforthischoiceisthatisadaptedtoastringperturbationformulationofthesolution,i.
e.
theseedwillbedecomposedintotermsatxedorderinstringperturbationtheoryplusSO(3,3,Z)T-dualityinvariantfunctionscoupledtoinstantons.
Ofcourse,thisansatzfortheseedgetsspreadoutbytheSL(5,Z)orbitsumintoamorecomplicatedU-dualityinvariantfunction.
Weshallndasolutiontothedierentialequationwithourparabolicansatz;itislikelythatotherformsusingotherparabolicssumsexistanditwouldbeveryinterestingtostudytheirrelationandfunctionalequations.
WiththeassumptionoftheparabolicPoincareseries(3.
23),theLaplaceequation(3.
15)thenunfoldsinto425σ(g)=4ζ(3)2r12/5E(3Λ1ρ,g).
(3.
24)SinceweassumeσtobeleftP1(Z)-invariant,itcanbeexpressedastheFourierseriesσ(g)=N∈Z4cN(r,e4)e2πiQN.
(3.
25)AstheunipotentofamaximalparabolicsubgroupofSL(n)isabelian,thisexpressioncapturesthewholeofσwithoutneedfornon-abeliancoecients.
Here,Nisafour-componentcolumn-vector.
TheEisensteinseriesontheright-handsideof(3.
24)canalsobewrittenasaFourierseriesovertheunipotentasE(3Λ1ρ,g)=N∈Z4fN(r,e4)e2πiQN(3.
26)–12–withfN(r,e4)=2ζ(3)r7/5σ2(k)K1(2πr||e14N||)||e14N||forN=0and(3.
27a)f0(r,e4)=r12/5+2ζ(2)ζ(3)r2/5ESL(4)(2Λ1ρ,e4).
(3.
27b)OnehasSL(4)ESL(4)(2Λ1ρ,e4)=32ESL(4)(2Λ1ρ,e4)andk=gcd(N).
ThenotationhereissuchthatΛ1forSL(4)denotesthenodeadjacenttotheoneuseddeningthestringperturbationlimitforSL(5)andthusisoneoftheouternotesofSL(4).
(FromthepointofviewofSO(3,3),thisisaspinornode.
)WenowobtaindierentialequationsfortheFouriercoecientscNofσin(3.
25).
Forthezeromodec0wehavetheequation58r22r158rr+SL(4)425c0(r,e4)=4ζ(3)2r24/58ζ(2)ζ(3)r14/5ESL(4)(2Λ1ρ,e4).
(3.
28)Startingwiththehomogeneousequation,onecanmakeaseparatedansatzoftheformc(h)0(r,e4)=rαFα(e4).
ThisleadstoSL(4)+58α252α425Fα(e4)=0.
(3.
29)Inorderforthistoproducetermsconsistentwithstringperturbationtheoryonlythevaluesα∈{245,145,45,65}areallowed,leadingtotheeigenvalues{6,212,10,92}asin(3.
20).
TheremainingequationforFα(e4)isthensolvedbyappropriateSL(4)Eisensteinseries.
Lookingforaparticularsolutionoftheformα1r24/5+α2r14/5ESL(4)(2Λ1ρ,e4)weareledtotheparticularsolutionc(p)0(r,e4)=23ζ(3)2r24/5+23ζ(2)ζ(3)r14/5E(2Λ1ρ,e4).
(3.
30)Thefullsolutionforthezeromodeisnowc0=c(h)0+c(p)0.
ComparisonwiththeIIBcasesug-geststhatoneshouldchoosethehomogeneoussolutionsc(h)0=5π1512ζ(7)r14/5ESL(4)(7Λ2ρ,e4)butwithoutcomputingtheFourierexpansionofthePoincaresum(3.
23)andcom-paringwith(3.
20)wecannotxthehomogeneoustermdenitively.
Forthenon-zeromodescN(r,e4),wehavetheequation58r22r158rr4π2r2||e14N||2+SL(4)425cN=16πζ(3)σ2(k)r19/5K1(2πr||e14N||)||e14N||.
(3.
31)WeshowinappendixAthatc(p)N=32π2ζ(3)σ2(k)r24/5K02πr||e1N||(2πr||e1N||)2+12K12πr||e1N||(2πr||e1N||)3+40K22πr||e1N||(2πr||e1N||)4(3.
32)isaparticularsolutionofthisequation.
TheproofreliesonwritingouttheSL(4)LaplacianandusingpropertiesoftheBesselfunctionsinawaysimilartodemonstrating(3.
11).
As–13–alsoexplainedintheappendix,therearesolutionstothehomogeneousequationgivenby(see(A.
22))13c(h)N=r24/5K7/2(2πr||e14N||)(2πr||e14N||)5/2andc(h)N=r16/5K7/2(2πr||e14N||)(2πr||e14N||)3/2.
(3.
33)IfweimposethesameconstraintasforSL(2),namelythatthestrongcouplingbehaviourr→0isregular,thisrulesoutthesecondsolutionandselectsthecombinationcN(r,e4)=32π2ζ(3)σ2(k)r24/5K02πr||e1N||(2πr||e1N||)2+12K12πr||e1N||(2πr||e1N||)3+40K22πr||e1N||(2πr||e1N||)42√215√πK7/2(2πr||e14N||)(2πr||e14N||)5/2.
(3.
34)4ModulargraphfunctionsAnotherfamilyofautomorphicfunctionssatisfyinginhomogeneousLaplaceequationisprovidedbymodulargraphfunctions[37].
Thesearefunctionsthatareinvariantun-derSL(2,Z)andhaveanexplicitlatticesumdescription.
Moreover,theysatisfytypi-callyinhomogeneousLaplaceequations.
WenotethatforanyPoincaresumoftheform(z)=γ∈B(Z)\SL(2,Z)σ(γz),wherez=x+iyandtheperiodicseedhastheexpansionσ(x+iy)=n∈Zcn(y)e2πinx,theFouriermodesof(z)=n∈Zfn(y)e2πinxaregivenby[35,53]fn(y)=cn(y)+d>0m∈ZS(m,n;d)Rexp2πiωn2πimωd2(ω2+y2)cmyd2(ω2+y2)dω,(4.
1)involvingtheKloostermansumsS(m,n;d)=q∈(Z/dZ)*e2πi(qm+q1n)/d.
(4.
2)Thiscanbeshownbywritingoutexplicitlythecosetsum.
Weshalltrytoapplythisformalismtore-derivesomeresultsonmodulargraphfunctions.
AsanexampleweconsiderthefunctionC3,1,1(z)inthenotationof[37].
ItcanbedenedexplicitlyfromamultiplelatticesumasC3,1,1(z)=(m1,n1),(m2,n2)∈Z2(mi,ni)=(0,0)(m1+m2,n1+n2)=(0,0)y5π5|m1z+n1|6|m2z+n2|2|(m1+m2)z+(n1+n2)|2.
(4.
3)13WenotethatthesehomogeneoussolutionsappearnottocorrespondtotheFouriermodesofanSL(5)Eisensteinseriesassociatedwiththeminimalseriesonnode1.
LookingforsuchanEisensteinseriesleadstoirrationalpowersofr.
Thereis,however,awell-knownhomogeneoussolutiongivenbyESL(5)(7Λ2ρ)[25].
–14–Asshownin[37,eq.
(3.
19)]itsatisestheequation(6)C3,1,1(z)=865E54E2E3+ζ(5)10,(4.
4)whereEs=2πsζ(2s)Esisthenon-holomorphicEisensteinseriesEsinadierentnormalisation.
WeshallnowforgetthatwehaveanexplicitsolutionforC3,1,1(z)andtrytoconstructonebysolvingtheLaplaceequation(4.
4)usingaPoincareansatz.
Inordertotreattheniteconstantontheright-handside,wereplaceitbyanEisensteinseriesEandsend→0attheend,usinganalyticcontinuation.
WritingC3,1,1(z)=γ∈B(Z)\SL(2,Z)σ(γz)inPoincareform,wehavetosolvetheequation(6)σ(z)=172π5467775y58π542525y3E2+ζ(5)10y,(4.
5)wherewehaveexplicitlywrittenoutthenormalisingfactorsandyrepresentstheregulatorfortheconstant.
Inwritingtheequationwehavechosento'fold'E3.
Theboundaryconditionforsolvingthisequationisthatthesolutionf(z)shouldnothaveatermgrowingasy3whenapproachingy→∞.
ReducingtheequationtoFouriermodesσ(z)=n∈Zcn(y)e2πinxleadstoy22y6c0(y)=4π522275y58π2945ζ(3)y2+ζ(5)10y,y22y4π2n2y26cn(y)=32π3945|n|3/2σ3(n)y7/2K3/2(2π|n|y)=8π2945σ3(n)y2(1+2π|n|y)e2π|n|y.
(4.
6)Aparticularsolutiontotheseequationsisgivenbyc0(y)=2π5155925y5+2π2ζ(3)945y2+ζ(5)10((1)6)y,cn(y)=2π2945σ3(n)y2e2π|n|y.
(4.
7)(ThehomogeneoussolutionscorrespondtothevariousFouriermodesofE2(z),butwewillnotrequirethemhere.
)UsingthegeneralformulaforFourierexpansionsofPoincaresumsonecannowcon-structthezeromodeoftheC3,1,1(z)=n∈Zfn(y)e2πinxbycomputingf0(y)=c0(y)+d>0m∈ZS(m,0;d)Rexp2πimωd2(ω2+y2)cmyd2(ω2+y2)dω.
(4.
8)–15–Thisexpressioncanbeevaluatedforthepresentcaseasfollows.
Werstrescaletheintegrationvariableandrestricttothetermswithm=0inthesum.
Theyareyd>0m=0S(m,0;d)R2πimy1d2t1+t2cmy1d21+t2dt=4π2945y1m>0d>0S(m,0;d)d4σ3(m)R(1+t2)2exp2πmy1d21+it1+t2dt=4π2945y1m>0k≥01k!
d>0S(m,0;d)d42k=σ32k(m)ζ(4+2k)mkσ3(m)(2πy1)kR(1+t2)k2(1+it)kdt=2k2(k+2)π=π3945y1m>0k≥0k+2k!
σ3(m)σ32k(m)mkζ(4+2k)(πy1)k.
(4.
9)Asisevidentfromthisexpressionthesumovermisdivergentandsothetworemainingsummationscannotbeinterchangedstrictly.
OnecanpartlymakesenseofthesumovermbyanalyticallycontinuingtheRamanujanidentitym>0σa(m)σb(m)ms=ζ(s)ζ(sa)ζ(sb)ζ(sab)ζ(2sab)(4.
10)fromtheconvergentregionwithlargerealpartofstos=k.
Thisleadsto(4.
9)=π3945y1k≥0k+2k!
ζ(k)ζ(3k)ζ(k+3)ζ(k+6)ζ(4+2k)ζ(6)(πy1)k=2ζ(3)221πy1+7ζ(7)16π2y2ζ(3)ζ(5)2π3y3+11ζ(9)32π4y4.
(4.
11)Thek-summationterminatesduetothezeroesoftheRiemannzetafunctionatnegativeevenintegers.
Thecasek=2requirestakingalimit.
Tocompletethecalculationofthezeromodef0(y)wealsohavetotakeintoaccountthecontributionfromc0(y)andthetermswithm=0inthegeneralexpression.
ThesearejusttheusualconstanttermsforEisensteinseries[35].
Thisleadsintotaltof0(y)=2π5155925y5+2π2ζ(3)945y2+ζ(5)10((1)6)y+2π5155925ξ(9)ξ(10)y4+2π2ζ(3)945ξ(3)ξ(4)y1+ζ(5)10((1)6)ξ(21)ξ(2)y12ζ(3)221πy1+7ζ(7)16π2y2ζ(3)ζ(5)2π3y3+11ζ(9)32π4y4→0→2π5155925y5+2π2ζ(3)945y2ζ(5)60+7ζ(7)16π2y2ζ(3)ζ(5)2π3y3+43ζ(9)64π4y4(4.
12)ThisagreeswiththeLaurentpolynomialstatedin[37,eq.
(6.
2)]exceptfortheconstantterminy0.
14ThereasonforthisappearstobethattheconstanttermofE2E3hasa14WenotethatthisLaurentpolynomialcanbededucedbyconsideringthezeromodeofequation(4.
4)andapplyingtheRankin-Selbergmethodtoxthesolutiontothehomogeneoussolution.
–16–contributiontoy0thatismissedbytheaboveconstructionandthisseemstobeageneralfeaturethatrequiresadditionalstudy.
AfurtherpointthatrequiresinvestigationisthatourmethodoftreatingthedivergentsumovermseemstohavelostalltheexponentiallysuppressedtermsoftheformO(ey)inthezeromodewhileonewouldexpectthemfromtheknownfunctionC3,1,1(z).
However,wenotethatthemethodproducesalsotherightcombinationofsolutionstothehomogeneousequationfoundin[37].
5DiscussionInthepresentpaper,wehaveoutlinedamethodforsolvinginhomogeneousautomorphicdierentialequationsofthetypethatappearinstringtheoryinseveralplaces.
ThismethodreliedonmakinganansatzforthesolutionofaPoincaresumformasin(2.
7).
TheadvantageofthismethodisthattheresultingdierentialequationfortheseedσofthePoincaresumislessinvolvedthantheoriginalequationandissolvedinaFourierexpansion.
Weexempliedthismethodforfourexamples,namelytheD6R4correctionforten-dimensionaltypeIIBstringtheoryandtheD6R4correctioninD=7space-timedimensions.
Therstexamplereproducedaknownresultfrom[10,41]whilethesecondexamplegaveanewproposalforD=7.
ThethirdexampledealtwithamodulargraphfunctionandhowtoreproduceitsLaurentpolynomial.
AlastfamilyofexampleswasageneralisationoftheD6R4functioninD=10presentedinappendixB.
Whilethemethodseemspowerfulandconvenientforproducingformalsolutionstotheequations,thereareanumberofimportantandinterestingpointsthatrequirefurtherinvestigationforbringingthemethodtoitsfullpower.
BesidesthequestionofconvergenceofthePoincaresum,theseare1.
Iftheoriginalautomorphicdierentialequationcomesequippedwithboundarycon-ditionssuchascompatibilitywithperturbationtheory,theseboundaryconditionsmustberephrasedforthenewdierentialequationfortheseedσ.
Thedirecttrans-lationisnotobviousandageneralprocedurewillprobablyrelyonasolutiontothesecondpointbelow.
Intheexamplesinthispaperwehavegivenaheuristicsetofboundaryconditionsbasedonstrongcouplingslimitsattheleveloftheseed.
2.
TheboundaryconditionsandthephysicalcontentofthePoincaresum(g)=γσ(γg)areexpressedthroughtheFourierexpansionof.
EventhoughσwassolvedusingFourierexpansion,thedirecttranslationofthisintotheFourierexpansionofisveryhard.
ForthecaseofSL(2,Z)someintuitioncanbegleanedbyconsideringtheformoftheFourierexpansionofthesolutiongiveningeneralin(4.
1).
Theseexpressions,thoughexplicit,seemimpossibletoevaluatefortheseedsσthatwefoundfortheD6R4solution15andalsointhecaseofmodulargraphfunctionsinsection4wehadtodealwithdivergentsums.
15Plugginginoursolution(3.
10)and(3.
13)thisagreeswith[41,eq.
(B.
13)].
Inbothcases,thereappearstobeaproblemwithabsoluteconvergenceasthena¨veseparateevaluationofthem=0terminthezeromodef0(y)leadstod>0S(0,0;d)d2s=ζ(2s1)/ζ(2s)→∞fors→1fromthelinearterminyinc0(y).
ThisproblemisrelatedtothelackofconvergenceofthePoincareseriesforthistermalonethatwasmen-tionedbelow(3.
13)asthistermnormallyproducesthesecondconstanttermoftheEisensteinseriesEs(z).
–17–However,wecanseefrom(4.
1)thattheidentityelementinthePoincaresumalwaysyieldstheFouriermodecnoftheseedandthisiswhywereliedonthepropertiesofthisinourheuristicanalysisoftheboundaryconditions.
OneobtainsthesametypesofintegralifonedoesthedirectFourierexpansionofthesolutionintheSL(5)case.
Itwouldbeverygoodtodevelopgeneralmethodsforthisandalsoforhigherrankcases.
3.
Automorphicformssolvinghomogeneousequationscanbelongtoprincipalseriesrepresentationsandsatisfyinterestingfunctionalequations[48].
IsthereasimilartheoryunderlyingthesolutionstotheinhomogeneousequationsThereisnoobvioussuchfunctionalrelationfortheone-parameterfamilyofsolutionsinappendixB.
Itisalsonotclearwhattherepresentation-theoreticmeaningofthesolutionsis.
Theymostprobablyrepresentvectors(orpackets)inthetensorproductofprincipalseriesrepresentations.
4.
ThesolutionwefoundforD6R4inD=7space-timedimensionsisverydierentfromtheintegralformulasfoundin[25,29].
SinceaFourierexpansionhasnotbeenachievedinthosecaseseither,itishardtocomparethetworesults.
ThesolutionweconstructedwasbasedontheLaplaceequation.
Forhigherrankgroupsandcurva-turecorrectionsonetypicallyhasmoredierentialequationstosolveandtheycanbeofhigherorderinderivatives[27],eitherofhomogeneousorinhomogeneoustype.
Theyrepresentelementsinthecenteroftheuniversalenvelopingalgebraandgen-eratedtheannihilatoridealforstandardautomorphicforms.
ItwouldbeinterestingtoinvestigatethesetensorialtypeequationsforPoincaresumsolutionsandtheywillconstrainthehomogeneoussolutionsfurther.
AcknowledgmentsWewouldliketothankG.
Bossard,S.
Friedberg,M.
B.
Green,S.
D.
Miller,D.
Persson,B.
Pioline,P.
VanhoveandD.
Zagierfordiscussions.
AKgratefullyacknowledgesthewarmhospitalityoftheMaxPlanckInstituteforMathematicsandtheHausdorInstituteinBonnduringthenalstagesofthiswork.
AParticularsolutionoftheequationforcNforSL(5)Inthisappendix,weconstructtheparticularsolutionforthenon-zeroFouriermodecN(r,e4)thatwasstatedin(3.
31).
Theequationtosolveis(withk=gcd(N)):58r22r158rr4π2r2||e14N||2+SL(4)425DcN=16πζ(3)σ2(k)Akr19/5K1(2πr||e14N||)||e14N||.
(A.
1)A.
1LaplaciansandBesselfunctionsTheSL(4)invariantscalarLaplacianonSL(4)/SO(4)isgivenbySL(4)=12gikgjlijkl18gijij2+52gijij(A.
2)–18–whereg=e4eT4andij≡gijsatiesijgkl=δikδjl+δjkδil(A.
3)asin[7,appendixA]uptoanoverallnormalisationconsistentwithourconventions.
16Weintroducesomehelpfulnotationu=||e14N||=NTg1N=NigijNj,(A.
4a)x=2πr||e14N||=2πru(A.
4b)anddenethefunctionsfαβs=rαuβKs,f′αβs=rαuβK′s,f′′αβs=rαuβK′′s,(A.
5)sothattheprimeonlyaectstheBesselfunction.
AllthesefunctionshavetheBesselpartevaluatedatx=2πru,e.
g.
f′αβs≡rαuβK′s(x).
Notethattheright-handsideofthedierentialequation(A.
1)isoftheformAkf19/5,11andsocontainsafunctionintheclass(A.
5)thatweshallusebelowtoconstructanansatzforthesolution.
Werecordsomehelpfulidentitiesgijiju=u,(A.
6a)gikgjlijklu=9u,(A.
6b)gikgjlijuklu=u2,(A.
6c)whererepeatedindicesaresummedover.
Wealsorecordhowthedierentialoperatorsin(A.
1)actonafunctionfαβsfrom(A.
5).
Byusingtheidentities(A.
6)onecanderivethefollowingrelationsr22rfαβs=α(α1)fαβs+2αxf′αβs+x2f′′αβs,(A.
7a)rrfαβs=αfαβs+xf′αβs,(A.
7b)gikgjlijklfαβs=β2+8βfαβs+(2β+9)xf′αβs+x2f′′αβs,(A.
7c)gijij2fαβs=β2fαβs+(2β+1)xf′αβs+x2f′′αβs,(A.
7d)gijijfαβs=βfαβsxf′αβs.
(A.
7e)ActingwithDfrom(A.
1)onatermfαβsthengivesDfαβs=x2f′′αβs+5α+3β4xf′αβs+58α(α4)+38β(β+4)x2425fαβs.
(A.
8)ThisexpressioncanbefurtherreducedtoamorealgebraicequationbyusingpropertiesoftheBesselfunctionKs(x)=Ks(x).
TherstidentityisthemodiedBesselequationx2K′′s(x)+xK′s(x)(x2+s2)Ks(x)=0orx2f′′αβs+xf′αβs(x2+s2)fαβs=0(A.
9)16Thederivativeijissecretlywithrespecttothematrixwithdiagonalelementsrescaledby2asin[7],butwhatweneedisitscharacteristicpropertywhendierentiatingthesymmetricmetricgkl.
–19–thatcanbeusedtoeliminatethesecondderivativeoftheBesselfunctionwithoutchangingtheordersoftheBesselfunction.
Moreover,wehavetherecursiveBesselrelationxK′s(x)=sKs(x)xKs+1(x)orxf′αβs=sfαβs2πfα+1,β+1s+1(A.
10)thatcanbeusedtoreplacerstderivativesoftheBesselfunctionatthecostofchangingtheorder.
LetusrstapplythemodiedBesselequationto(A.
8).
ThisyieldsDfαβs=5α+3β44xf′αβs+58α(α4)+38β(β+4)+s2425fαβs.
(A.
11)ApplyingthentheBesselrelation(A.
10)tothisleadstoDfαβs=58α(α4)+38β(β+4)+s2+5α+3β44s425fαβs2π5α+3β44fα+1,β+1s+1,(A.
12)whichimplementsthedierentialoperatorDasacompletelyalgebraicoperationonthespaceoffunctions{fαβs}.
Asacheckontherewritingofthedierentialoperator,onecanworkouttheSL(5)Laplacian(i.
e.
removingthe42/5fromD)ontheFouriermode(3.
27a)oftheR4cor-rectiontermwhichcorrespondstoα=7/5,β=1ands=1toobtaintheeigenvalue12/5asneeded.
Thelasttermwithshiftedorderdropsoutinthiscaseasneeded.
WealsonotethatthefollowingfunctionsareinthekernelofD:Df(43β)/5,βs(β)=0(A.
13)withs(β)=(5012β3β2)/5.
Thisisnotnecessarilyacompletedescriptionofthekernel.
ThereisonemorealgebraicrelationthatwerecordforspecialloworderK2(x)=2xK1(x)+K0(x)orfαβ2=1πfα1,β11+fαβ0.
(A.
14)A.
2ParticularsolutionbyrecursionInordertondaparticularsolutionto(3.
31)wemaketheansatzc(p)N=AkiBirαiuβiKsi=AkiBifαiβisiwithBi∈R(A.
15)wherethenumberoftermsinthesumistobedeterminedandwetakeouttheoverallnumericalfactorAk.
FromtheSL(2)examplein(3.
13)weexpectthatasmallandnitenumbersuces.
Pluggingtheansatz(A.
15)intothedierentialequation(A.
1),wegetuponuseof(A.
12)iBi58αi(αi4)+38βi(βi+4)+s2i+5αi+3βi44si425fαiβisi2π5αi+3βi44fαi+1,βi+1si+1=f19/5,11.
(A.
16)–20–Ourstrategywillbetosolvethisrecursivelysuchthatwealwaysgeneratetheright-handsidefromthenon-orderpreservingtermin(A.
12).
Theright-handsidef19/5,11canbegeneratedbyactingonf14/5,20fromthelasttermin(A.
12).
Since12πDf14/5,20=122πf14/5,20+f19/5,11,(A.
17)westarttherecursionwith12πf14/5,20togeneratetheoriginalrighthandside.
Thisproducesanewright-handsideinvolving122πf14/5,20thatwenowcancelbythesamemethod.
SinceDf9/5,31=10f9/5,31+2πf14/5,20(A.
18a)Df4/5,42=6πf9/5,31,(A.
18b)wecanndalinearcombinationofthetwofunctionsthatproducesexactlythenewright-handside,viz.
D12(2π)2f9/5,3140(2π)3f4/5,42=122πf14/5,20(A.
19)Here,itiscrucialthatin(A.
18b)noeigenvaluetermisproducedandourmethodterminates.
Thus,altogetherweobtaintherelationD12πf14/5,2012(2π)2f9/5,3140(2π)3f4/5,42=f19/5,11,(A.
20)leadingtotheparticularsolutionfortheFouriermodeoftheseedc(p)N(r,e4)=32π2ζ(3)σ2(k)r24/5K0(2πr||e14N||)(2πr||e14N||)2+12K1(2πr||e14N||)(2πr||e14N||)3+40K2(2πr||e14N||)(2πr||e14N||)4=32π2ζ(3)σ2(k)r24/51+40(2πr||e14N||)2K0(2πr||e14N||)(2πr||e14N||)2+12+80(2πr||e14N||)2K1(2πr||e14N||)(2πr||e14N||)3(A.
21)uponsubstitutingbacktheconstantsanddenitions,togetherwiththesymmetryKs=Ks.
Inthenalrewriting,wehaveusedtheidentity(A.
14)toeliminatetheK2Besselfunctionandmakethesolutionmoresimilarto(3.
11)thataroseintheten-dimensionaltypeIIBcase.
SimilartothesolutionofthehomogeneousequationinthetypeIIBcasewecanexpecttohaveatleastahomogeneoussolutioninvolvingK7/2.
Thiscanbemanufacturedusingoneofthefunctionsin(A.
13)usingβ=52orβ=32,i.
e.
thefunctionsf23/10,5/27/2andf17/10,3/27/2(A.
22)arehomogeneoussolutionsto(A.
1).
TheyareusedinthemaintexttoproposetheseedoftheD6R4thresholdfunctionforD=7space-timedimensions.
–21–BRegularisationofinhomogeneousLaplaceequationsinD=10Inthisappendix,wegeneralisethesolutiontotheinhomogeneousLaplaceequation(3.
1)foundinsection3.
1.
Thegeneralisationconsistsindeformingtheinhomogeneousequationto((3)(4))f(z)=4ζ(3)ζ(3+2)E3/2(z)E3/2+(z)(B.
1)Thevalue=0correspondstotheactualD6R4correctioninD=10.
ThereasonforthisgeneralisationisthatwewouldliketoavoidtheproblemwiththeapparentsingularbehaviourwhencarryingoutthePoincaresum.
Notethatthegeneralisationisexactin.
Thisequationcanbetreatedbythesamemethodasinsection3.
1.
WefoldthesumonthedeformedEisensteinseriesontheright-handsidetoreplaceitbyy3/2+andobtainthefollowingequationsforthezeromodeandnon-zeromodesoff(z)=d0(y)+n=0dn(y)e2πinx:y22y(3)(4)d0(y)=4ζ(3)ζ(3+2)y3+43π2ζ(3+2)y1+,(B.
2a)y22y4π2n2y2(3)(4)dn(y)=16πζ(3+2)y2+|n|σ2(|n|)K1(2π|n|y).
(B.
2b)Thesolutiontothezeromodeequation(B.
2a)isgivenbyd0(y)=236ζ(3)ζ(3+2)y3++π29(16)ζ(3+2)y1+.
(B.
3)Here,wehavexedthesolutionofthehomogeneousequationtozerointhesamewayasinsection3.
1.
Thehomogeneousformof(B.
2b)forthenon-zeromodeshasthesolutiony1/2K7/2(2π|n|y).
Combiningitwithaparticularsolutionwenddn(y)=8ζ(3+2)σ2(|n|)y1+14212+104π2(|n|y)2K0(2π|n|y)+6π|n|y+104π3|n|3y3K1(2π|n|y)104Γ72(π|n|y)1/2+K7/2(2π|n|y),(B.
4)wherewehavexedthehomogeneoussolutionsuchthatthemostsingulartermsatstrongcoupling(y→0)areabsent.
Thisisincorrespondencewiththechoiceford0(y)atweakcoupling(y→∞).
Onecancheckthattheabovesolutiontendsto(3.
13)when→0.
TheadvantageofthissolutionisthatthePoincaresumofthezeromode(B.
3)con-vergesfor>0.
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