On new theta identities of fermion correlation functions on genus g Riemann surfaces

Transcription

1 O w thta dtts o rmo corrlato uctos o us Rma suracs A.G. Tsucha. Oct. 7 Last rvsd o ov. 7 Abstract Thta dtts o us Rma suracs whch dcompos smpl products o rmo corrlato uctos wth a costrat o thr varabls ar cosdrd. Ths tp o thta dtts s a ss dual to Fa s ormula b whch t s possbl to sum ovr sp structurs o crta part o suprstr ampltuds SR ormalsm wthout us Fa s ormula or Rma s thta ormula much smplr mor traspart wa. Also such dtts wll hlp to cast corrlato uctos amo arbtrar umbrs o ac-ood currts a closd orm. As or us th dtts ar rportd bor [] []. Basd o som ots o us cas whch wr ot rportd [] [] ad rlat thos to th rsults o th Dola Goddard mthod [] o dscrb ac-ood currts a closd orm w propos a da o ral us dtts to th cas o us suracs. Ths s ot a complt drvato o th hhr us ormula du to dcults o vstat sular part o drvatvs o us Wrstrass uctos. athmatcal ssus rmad usolvd or us > ar dscrbd th tt.

2 . Itroducto Coclusos ad otatos What w cosdr ths documt s a smpl product o rmo corrlato uctos S S... S wth a costrat.. [ ] I w [ ] I w r S w s th So rl E w s th [ ] E w h h w rm orm o us Rma suracs ad C w I w rprsts Abl map.... C. Th varabls... ar th srt pots o tral partcls bosos o somtms aturall occurs Rma suracs. Th costrat calculat cotractos o corrlato uctos us Wc thorm ad ths costrat s a our vstatos blow. Bascall calculat suprstr ampltuds or ampl ach o th rmo ld cotractos s multpld wth complcatd momtum ad polarato vctors o th tral partcls ad w ot ac to cop wth ths tp o corrlato uctos th dscoctd part o part-v ampltuds. I summ ovr sp structurs SR ormalsm b appl Fa s ormula o usuall acs hard paul ad utraspart calculatos bcaus th ormula has a dtrmat orm o rmo corrlato uctos. Th ormula ar sutabl to calculat r rmo cotractos but ot som othr cass. Our am s to v aothr tp o dtts whch dcompos smpl products o So rls to th ollow orm: S modular varat uctos o modul costat trms. so that th rht had sd plctl shows that sp structur dpdc o th product s cludd ol th modul costat trms ad to s that th orm lads to smplr ad mor traspart calculatos. I ral cas th actor th modul costat trms s us Wrstrass ucto at o-sular ad v hal prods o Rma suracs as w wll s latr. Also aothr purpos ths documt s to tr to obta plct orm o modular varat uctos o us whch wll b rmad uactd th procss o sp sum ad whch wll b cotad th al orm o str ampltuds. Th mthod adoptd s basd o rard Dola-Goddard rat ucto mthod v r. [] as a paso b th Wrstrass ucto. Ths mthod lads to a smpld dcomposto ormula o us cas as wll b show qs....5 ad also.7. W

3 th that th ormula o us > wll also b v alo th sam da whch lads to qs....7 but thr ar mathmatcal dcults rmad usolvd mal o structurs o us sma ucto ad uctos. I ths ar d th rsults ma also hlp dscrb ac-ood currts a closd orm o us Rma suracs. Th author hops that somo wll accomplsh rorous drvatos or ral us alo th das dscrbd hr utur. I ths documt w ot us matrcs ad dtrmats whch th sam ucto s ld up ach colum ad th sam varabl o th uctos ar ld up ach row:. Ths dtrmat wll b dotd b dt.. W also us th ollow otato whr all lmts o th -th colum rom th lt ar rplacd wth... ;... dt th =. A -th drvatv o a ucto s alwas dotd as. otatos o us rods: m m W st th ollow. al prods dotd b ad wll pla mportat rols. From sma-ucto o torus w d ucto as ollows:

4 d l d d d Th valus o ucto at hal prods ar th brach pots o th curv:.5 Th brach pots ad ar rlatd to thta costats as Also w us..6.7 Th ar classcal otatos o modular orms whch ar rlatd to Esst srs G as 6 m m 6 m m m m G.8 W ma us o th ollow rlatos th blow as wll as.5: d polomal o o dr d *[ polomal o o dr ].9. ODD. I partcular th polomal o.9 s dotd as Q th ollow wa. Q Q A w ampls o Q ar: Q Q Q 6 8 Th Q s coscutvl costructd b drtat... { } { } Th coct o th hhst dr trm Q s!..5 Two ormula o th rmo corrlato ucto: S ad S p.6

5 . Gus O th torus w dscuss S whr w S w ad w. corrspods to R S S rspctvl. W d.... ad hc.. As was show [] orall [] start rom th Frobus Stclbrr ormula s Wrstrass ucto F alwas mas th drtato o th ucto F. : dt [... ]... ar ts drvatvs. I ths documt!!!. t s possbl to prov th ollow dcomposto ormula o product o So rls udr th codto : S...! whr Q Q or ach o. s th polomal dd. whch appars wh r-wrt b tsl ad hc Th costats costats as.6 ad Q. ar th brach pots o th us curv whch rlats to thta ar th hal prods whch rlats to as.5. Th ar mastl modular varat uctos o wrtt ol b ucto dd as ollows: dt dt [ [ ]... ] th ;.... 5

6 dt dt [ [ ] th ; ] th ; Th summato. s utl th tr whch dos ot cd. Each o has th orm o a rato o two dtrmats com rom a act that ths s a rato o two roots o lar quatos obtad b Cramr s ormula. ot th drc o th ovrall s. ad.5. Th drvato o q.. s dscrbd Appd A to ma ths documt sl-cotad. I th dtrmats ucto ad ts drvatvs ar ld up as... rom th lt. I th domators th last rht d colum th th colum s rplacd wth. For latr covc w dscrb o mor dcomposto ormula. W start rom aothr orm o Frobus Stclbrr ormula : dt [... ] r th dtrmat compard wth q.. v umbr o drvatvs o ucto s rplacd wth moomals ad odd umbr o drvatvs o ucto s rplacd wth moomals... rom th lt th sam as q.... Th ordr o th pols s Th b th smlar arumt to drv. abov w hav aothr dcomposto ormula: S V... or ach o.8 whr V... ar obtad b rplac th cotts o dtrmats 6

7 wth.... utl s v s odd. Th hhst ordr o pols ths srs s or both cass. Also.8 V..9 Th q.. ad q..8 ca b wrtt b us hal prods as S....! whr ad V... S. ots: For th cas both o. ad.8 bcoms S S S. wth. That s q.. ad.8 or. ad. ar drct ralatos o a classcal ormula S. For. ad.8 s show to b quvalt to Fa s ormula o thr varabls cas. For t s mor covt to calculat th sp sum tha us Fa s ormula at us as plad [][]. masurs ca b wrtt as Sc th suprstr ampltud D D D whr D th ral orm o rmo corrlato part bcoms wh w us q..8 7

8 S V. Th modular varat uctos V rma uactd udr th sp sum ad th modul dpdt trms ca b wrtt b lmtar smmtrc polomals o ad hc b modular orms b.7. I V ol th trms mod appars as s th ODD procss o th proo rlct th act that. Th V ar v b a rato o dtrmats o -matr wth umbr o varabls out o varabls. W ca choos a st o varabls out o varabls o ; a choc vs th sam V. ols ad rsdus D th vrs o our varabls o. as :. I w tract out th most sular part o uctos rom th rht had sd o q...5 w hav a rato L G or... whr G B. Thr ar two was to s ths act: Frst s that th orm o V coms rom th rato o two roots o umbr o lar quatos o varabls as plad blow q.a.. Th scod s that start rom a umbrs vstat pol structurs o th rst o varabl aturall appars ad vs th sam rsult as dscrbd at th d o Appd B. 8

9 L B. I Appd B t s provd that wh w dot W as th lmtar smmtrc polomal o dr m amo varabls L G m W. Th lt had sd cluds varabls but o th rht had sd cluds varabls du to th codto. Go bac to th otatos o stad o W smultaous sl pol structur o w ca sa that ths pol part o ucto has umbr o varabls b b... b out o umbr o varabls.... W dot th llptc ucto whch has ths structur as [ W ]. ol structurs B chc th dtrmat rato..5 carull t ca b show that... A [ W ] A [ W ]. A [ W ] or mod. I ral th costat actors A... cota th A Esst srs. S subscto -6- ad th last commt o Appd D. ot that... has ot ol th rst trm A [ ] but also othr trms. Th scod thrd trms appar pottall or 5. W Rsdus I th Appd B t s also show b cosdr th rato o two dtrmats. that w rard... as th ar dpdt th rsdu o... satss th ollow quato: Rs O ths pot r.[] has mslad dscrptos o th pol structurs o th rsults althouh all thta ucto dtts r.[] ar corrct. 9

10 Ths s alrad show bor r.[]. R-wrt V trms o drvatvs o sma ucto Th q..5 mas that th pol o s o ordr as o varabls... udr th costrat. Ths atur matchs wth th act that th pols o S th product s.... Utl ths sl pol structur t s possbl to r-wrt th ormula o V trms o drvatvs o sma-ucto as ollows: V.7 V.8 V.9 c cv. c cv. r c c c ar umrcal costats whch wll b plctl dtrmd c subscto -6- pa 8. Th mus ss o th rht had sds o.. ar rom th dto o ucto. All o th qs..9.. hav th pol structurs o.5 or =. I vstat suprstr ampltuds ths orm s somtms mor covt tha th dtrmat-rato ormula It s rportd r.[] that av tsos o th rsults o.9-. to hhr valus o or ar ot wor wll. Ths act closl rlats to th stc o th scod thrd trms o th rht had sd o q..5 o th pol structurs o ral. owvr thr s a rmd o ths ad w ca drv a ormula or a. S subscto -6- o ths documt.

11 Altratvl r.[] a wa o costruct llptc uctos as drvatvs o sma ucto whch hav smultaous sl pols th varabls... or arbtrar s rportd th cott o llptc multpl ta valus. Sc such uctos ar sstall dtrmd b th pol structurs thos uctos would b th sam as or V up to modul dpdt costat trms. It s crta S that th product o So rls ca b padd b V ZV.... orovr th mthod s appld to o-mamal supr smmtrc cas r.[5]. O dscussos rlatd to th structurs o suprstr ampltud structurs s also [6][7]. [ ot addd: Th rat ucto mthod whch wll b dscrbd scto -6- blow ad th llptc multpl ta valus mthod or th us o sp sum wr oud out to b quvalt. Appd E s addd or pla t th last vrso o ths documt. ] 5 Zros Th v thta uctos at us hav ro pots at. That s bcoms ro quals to hal prods. rspctvl. Sc th rmo corrlato ucto has th orm w S w th product S bcoms ro a o th w varabls s qual to. Thror th rht had sd o..8 should also has ths atur. Ths ca b chcd ral th procss o prov.. r w mpl o ampl o =. S

12 . Wh sc ad umrator = =. I o o th varabls o varabls s st qual to.7-. or ral ormula.5 t lads to o-trval thta dtts but thos ma ot b so trst. 6 Grat ucto mthod r. [] : ts modcato ad applcatos I r.[] Dola-Goddard troducd a rat ucto DG so that cocts DG hav th dsrd rsdu structurs whch s th sam as.6 to obta pot o loop corrlato uctos or th currts o a arbtrar a ac-ood albra a closd orm: DG DG v.5 Th DG ar dd ths quato as cocts o th paso o a ucto. Ths orm o quato rmds us a classcal ormula o rmo corrlato ucto o torus: p S.6 Th S.7 bcaus th p actor cacls b th codto. To pursu ths smlart urthr w cosdr match th sular trms o.5 ad ucto:

13 m d cm! c... d!.8 m r c s rlatd to Esst srs as c m G. m Wrt th rht had sd o q..5 as m m DG DG DG v....9 ad compar wth th sular part o drvatvs o... w hav v v d DG DG DG d d... Ths match ds at t tm; all sular trms ad th costat trm match th th holomorphc part wll match automatcall. Thror w hav a paso b th ucto as d DG ad.. wth DG DG sc thr s o sl pol ad. ot that d DG o whch o sular ucto o s multpld s dd. ths documt. odd Sc w st ths bcoms S d DG.. That s DG rlats to our q...5 whch wr orall obtad as a closd orm as a rato o two dtrmats b a trc plad Appd A wthout cosdr a pasos l.. W hav DG...!... d. or th trms mod. Th rat ucto mthod s quvalt to th paso b th drvatvs o ucto as!..

14 Two ots whch w wll cosdr aa latr wh w tr to ral th dtts to us cas chaptr : -6-: Cosdr a paso o a ucto L whos varabls ar vrt srt pots ad a sub-varabl cocts as b th drvatvs o ucto wth som L ad.5 L should hav sular part o ad s som umbr dtrmd b th hhst ordr o th pol. Also cosdr o mor paso o th ucto b rst drvatv o ucto ad moomals o as L V V V V V It s alwas possbl to do ths pasos.5.6 as lo as L s llptc sc th rst drvatv o ucto ad moomals o whch rlat to hhr drvatvs o ucto ar th bass o th ucto spac at us. As ca b s rom.5 ad.6 w cosdr som obcts whch hav th orm ODD L supr str ampltuds sc th sp structur dpdc o L s ol throuh o d o costats whch us o quals to th brach pot. O mportat codto to b mposd o L s that t should b modular varat so that th paso cocts ar also modular varat. S Th cotracto s o o such cass b th codto whr th ucto L s. Th dpdt ucto th domator s cssar to hav sularts ad b padd b ucto. Ths S s a atural plaato wh w ca dcompos to th orm. or.8 whr sp structur dpdc s ol throuh. It would b also atural to td ths obsrvato to th cas o ral us.

15 -6- O prss b polomals o drvatvs o sma ucto Th ucto ca b usd as a rat ucto to obta th orm o as ollows. Frst w drtat th ucto U dd as U p[ l l ]..7 wth rspct to up to tms ad st qual to ro th th drtatos wth rspct to cha to drtato wth rspct to bcaus th combato o ad s lar. W ca obta polomals o drvatvs o sma ucto whch hav smultaous sl pol structur o umbr o varabls out o varabls... or a. That s w choos varabls out o th varabls... th th pols o U ar all combatos o th vrs o th products o such varabls. Ths ca b provd wthout dcult b mathmatcal ducto; s Appd D. It s strahtorward to s that ths procdur rproducs all o or =. owvr ths aïv procdur dos ot wor or to obta ral. Th U s qual to rom b th dto.7 whch s drt. Th procdur hr s alrad dscrbd r.[] chaptr a drt cott; th U s th sam as _hat o q..6 r.[]. I that artcl th corrct orm o rat ucto was oud out rom th rsdu pot o vw about th modular varat uctos. W saw that t rlats to th product o So rls b th ormula.6 wth th costrat. Obta ths polomals or arbtrar rom U ths wa s stll vr usul as w wll s blow but ot ouh to drv ral ormula o or to drv a ral procdur o sp sums. 5

16 Istad w adopt to do as ollows. Eprss a ucto [ ] th ollow orm b us a ormula t product rprstato o sma ucto as p[ ].8 m m m m [ ] p[ l l [ l m m m.9 m ] ] I w drtat [ l ] m m m m o two thr tms t bcoms ro atr stt. I drtat our tms or mor t alwas vs o moomal o Esst srs as. Ths s oth m m but th sam as Laurt paso procss o -ucto as a srs o clud ts sular part. Wh drtat tms or mor th trms m m ar ot mportat bcaus th vash. I w alwas drtat ths trms t tms wth rspct to ad th put b us l m! th trms [ l ] m m m m m ca b rplacd wth G whr G m s th m Esst srs. W wrt. b multpl [ ]! to both sds as Du to smpl pol structur o ucto:.. 6

17 7. O d! d. thr ar o sular trms o both sds o.. To t th orm o or arbtrar ad w ol hav to drtat both sds o. tms ad st. For stad o drtat q..7 w hav ] l l p[!!! G or. Th stc o th last trm th p ucto whch cotas Esst srs acts th polomal orm o th drvatvs o sma ucto. Th pols o ar ot ol th combatos o vrs o actl umbr o varabls rom varabls but also thos o lss umbrs or. Ths act rlats to th structur o q..5 ad also rlcts what was potd out r.[] o th rsdus. Wh bor drtat tms w should pa attto bcaus th rht had sd o. has plural trms proportoal to. W hav as dscrbd Appd D ] l l p[!!! G G }! { D. Th Esst srs G ar ro s odd. W mod D. as F. whr l l p[!!! F ]! {! G G. Eq.. s quvalt to th dtrmat-rato ormula..5 ad rprsts th ral orm o modular varat uctos o vrt srt pots. Th actor ] [ coms rom th domator o ad t dos ot dpd o th drcs o vrt srt pots or so ths actor cotrbuts to th

18 modular varat ucto th rsult should b costat modular orms. O umrcal actors:!. ad. coms rom th procss o proo o.; ths s th cocts o hhst dr o wh a polomal o. Th actor!. s rom actor S!. s rom d!.. Th b. or. or b stt!.... F!!! to drv. s wrtt as.5 As otd all o th drtatos wth rspct to cha to drtatos wth rspct to b stt. Ths rsult.5 as wll as. cluds thta dtts w d practcall at us atr all. Th drvatvs o sma ucto ar prssd b drvatvs o l whr s uqu odd thta ucto o torus. A tra costat appars ol wh w drtat l two tms. Eq.. also sas that th ratos o two dtrmats..5.6 hav paso ormula as th rht had sd o. wh. I w adopt.5 th ollow s th closd orm o th cotractos o rmo ld o loop pot suprstr ampltuds o part cosrv part atr th sp structur sum :! S Q Q F Q.6 Th whr w r-wrot b tsl ad hc Q 8

19 . As show r[] or plctl show r.[] or arbtrar th sp structur sum rducs to lmtar albras o brach pots th Q dpdt actor Q Q Q o q..6. B th cocrt orms o polomals o Q v... trms or = ar ro. I ral th actor ca b rprstd b polomals o lmtar smmtrc uctos o ad cosqutl b th Esst srs G ad G va q..7ad q For a hpr llptc cass th sp structur sum wll b do th sam mar ad sp structur dpdt trms wll b alwas rprstd b th smmtrc polomals o brach pots ad hc b modular orms atr th summatos. Ths hhr us act s qut plausbl but ot t provd ral []. ot that th Esst srs appar [6][7] rom two placs. O s th ucto o th potal actor o F com rom llptc uctos o vrt srt pots. Th othr s rom th Q dpdt actor. Th ormr cotas G at th =6 lvl whch appars ac-ood currts closd orm as o r.[] q.5 but str ampltuds ths trm dsappars atr th sp structur sum bcaus t corrspods to =6 = trm.6. Also ot that sc th trms o = do ot cotrbut to th rsult th rsults o th summato o q..6 do ot cota. I partcular sc 6 dos ot cotrbut th last trms. proportoal to ca b dsrardd or quvaltl w ca us q.. or th prsso o th sp sum. Sc th cotractos o th boso lds th vrt oprators do ot v partcl pols suprstr ampltuds ths uctos rprst th pol structurs o th drcs o vrt srt pots o loop ampltuds ral. For albras o dpdt trms q..6 v ol o-ro rsult wh ad. That o-ro rsult s. Ths act cluds o-rormalato thorms or ad our pot ampltud rsults o 9

20 suprstrs. For 7 th Q dpdt actor alwas vs or umrcal costats or a o. At 8 Q dpdt actor vs rst o-trval modular orm as plctl dscrbd r.[]. Ths s rom ad sc vs ro or Q dpdt actor. Ad also or 8 Esst srs th actor F cotrbuts o-ro rsults str ampltuds. Ths s rom ad. O th othr had th uctos o drcs o vrt srt pots ar alwas smpl com rom th drtatos o sma uctos o th potal actor F. Ths act ma b a ht to cosdr hhr us cass bcaus v hhr us th pol structurs as or th drcs o vrt srt pots ar th sam. Cosdr all abov th q..6 has a prsso as E S F!! whr.7 E F! Q Q Q p[ l l G ].8 whch s th sam as. cpt th ovr-all umrcal actor. Th last trm.7 whch th summato bs at s o-ro ol wh 8. As commtd abov q..8 w do ot hav to us th ull orm o F o q.. bcaus s ot cludd. Thr s o ambut spcall o add or multpl modular varat costat trms or actors th procss o drv.7.8 start rom q.. or a valu o.

21 Th rst trm o th rht had sd o q..7 F! rproducs all o sp sum rsults up to 7 r.[]. Th 5 6 trms th scod trm o q..7 wth th rst trm should match wth th rsults o r.[] up to. Th sp sum mthod ad rsults wr dscrbd r.[] whr th modular varat uctos o wr prssd b dtrmat-rato ormula o. or.. I ths documt thos dpdt trms ar r-wrtt b th drvatvs o sma ucto ad th Esst srs drctl as F. Dscrptos o th pol structurs o dpdt trms r.[] wr corrctd whl th dtrmat-rato ormula [] ar corrct. Th whol o th modul dpdt actor E s o a tp o Schur polomal. Its polomal dr s bcaus th dr o s ad Q so ts modular wht s clud cotrbutos rom all trms o Q. Sc t s ow that th dmso o th modular spac o th wht L s o or L = 68 th whol o ths actor s proportoal to G G G G G or = 5679 rspctvl. 6 8 I ths subscto w saw that th Dola-Godard rat ucto mthod wth a aular varabl was qut ct; ths mad our problm rmarabl asr. Start rom th paso orm. w ca obta th cocts b utl th smplct o th pol structurs o ucto.. Oc S thos ar obtad w ca also drv th dcomposto ormula o b stt. Ths two could b do almost sparatl. It s pctd that th sam mthod ca b appld or hhr us cass. I hhr us structurs o sular part o us ucto ad structurs o sma ucto ar ot t wll ow ad w ca t do th smlar calculatos rorousl at prst. Oc ths udamtal ssus ar clard th mthod wll b drctl rald.

22 7 smmtrato I q.6.8 o r.[8] a addto ormula s rportd I... I l[ ]... I I I... I S l l.9 whr s a coctd ortd loop whch passs throuh th pots... oc ad or l : S S. l I aothr word th summato s costructd as ollows. W prpar vrtcs... ad start rom w choos th t vrt sa out o - vrtcs. Th choos t vrt out o th rst - vrtcs ad so o. At th d o vrt rmas ad w ma o product S S S... S..5 Th summato.9 s all ovr possbl prmutatos o! o such products. W do ot dvdd b th actor! ad w dot ths summato th rht had sd o.9 as << Th q..9 sas that atr th smmtrato << costats cpt th cas =. d o v thta ucto at us. >> ad call ths smmtrato tmporall. >> th.5bcoms thta I q..9 w udrstad hr that s th Th spcal cass ad o th ormula.9 clud two corollars dscrbd q.9 ad q. Fa s boo[6]. I s odd th smmtrato vs ro. ad >. I th ollow w assum that s v Lt us calculat << >>. or. b borrow th rsult r.[]. Th smmtrato s th sam as S r.[]. S....! r th varabls o vrt srt pots It s show th Appd B o r.[] that cpt ar usd stad o....5

23 or th cas that s v ad Th wh smmtrd ol th trm w hav S!... vs o-ro valu ad! O th othr had at us rom q..9 S l or thr v thta uctos. l [ ] Th qualt ca b s us as ollows..5.5 From sma ucto w d th ollow thr uctos ot calld co-sma uctos: p.5 B us ths otato w ca wrt.6 as S.55 O th othr had thr s a ormula wh I.... I th otatos hr q.b r[] ma ma that th paso o S has ol o trm! or whch mas ol S s o-ro ad or othrs ar ro S m m. I S S S partcular. For ampl C.9 r[] bcaus ad S [ S S S S ] S S S S 6 6 as C. ad cacls wth th last costat trm C.9.

24 p.56 Th drtat th both sd o th lo o th ollow dtt tms > v p p.57 w hav or > l l.58 Th quato.9 s th rsult o us ad all othrs ar o us. W wll show latr th us vrso o hpr llptc cas o.58. Eq..9 susts that thr wll st us - aalo o th dcompos ormula. wrtt uctos whch has mor ormato tha q..9 ad ol wh t s smmtrd such rald quato rducs to.9. Thr wll also st all othr ormulas o us sma ucto aaloous to or ral suracs. or o V Th cotts o ths subscto ar ot usd latr. Assum aa that s v. Th ucto... V dd as th blow o q..8 whch was obtad start rom aothr prsso o Frobus Stclbrr ormula.7 cotas ol th ucto ad rst drvatv o ucto. Evrth s prssd wthout hhr drvatv trms. Eq..8 also has smpl cocts. I th drvatvs o ucto. ad. ar padd as c c c Th t vs a paso b V. Thror pad Q as a polomal o...! Q S

25 w ma sa that th coct o ach dr o should b th sam as V. Wh Q s padd l.59 w wrt C.6 Q whr mas th dr. Th S... C!.6 R-arra th trms w hav!... C.6 ot that th scod summatos b at as wll as th scod d o s -. Th V C.6! Thror V... also has sl pol structur th varabls... althouh RsV... s ot qual to V.... Calculat smmtrato << >> as V C!... C C!.6 th Q C V.65 Ths mas a trst atur: Wh th drvatvs o ucto or v s padd b th ucto tsl ts cocts ar V. 5

26 That s V orall dd as a rato o dtrmats th dscrpto blow.8 ad ar uctos o... bcoms such cocts whch ar modular varat costats wrtt b Esst srs atr summ up all -! trms o r-shuld varabls.... Sc... s o-ro ol w ca wrt.66! ad V.67. Gus > - Wh cosdr th product o So rls S S... S wth a costrat at us rom th arumt subscto 6. w ma s a ucto o th ralato o us cas ad pad t b us ucto. Such a ucto should b qual to th product o So rls C quals to us hal prods C. r w rstrct to th cas that rls ar v. s o-sular ad v so that th thta uctos th So A atural prhaps almost uqu caddat o such ucto s I A. E whr s th us sma ucto ad C s a aular varabl vctor. Ths s probabl how us sma ucto s cludd ral suprstr ampltuds or currt albras wthout mpos artcal assumptos. A dto o sma ucto us s show th t subscto hpr llptc cas but thr sts mor ral dto mathmatcal ltraturs. Th actor A has th ollow quadratc orm 6

27 A p ui IJ u J. I J whr u s th I-th compot o th Abl map o I that s u. I I I Ths actor coms rom th act that th domator o. cluds prm orm ot sma ucto as was th cas us. Ths actor A cacls th vrs o th quadratc actor sma ucto. las do ot cous holomorphc o orms wth hal prods. Usuall th hal prods ar wrtt as but th smbol s usd as o o prod matrcs ths documt. As dscrbd abov th ollow w cop wth Rma suracs whr th ollow quato s vald : I A E S S... S. wth ad o-sular v hal prods. As or hpr llptc cass hav plct ad smpl orm as. ad a drvato o. s plad th t subscto. It s pctd that q.. s vald or mor ral curvs. Th prm orm. has holomorphc o orms. B th codto thos hav th orm h h... h whr h [ ].. I I I ad s a su o o odd thta uctos usd to d prm orm. Sc or ach d valu o I I... I ach o I vars rom to... th ucto ca b padd b ucto ad so th I I I ucto. wll hav th ollow paso orm: I A E [... I... I I I I I I I I... I I I... I prm... 7

28 ... I I I I I... I I I... I ] I I... I... I I I I I... I prm [ I I... I ] I I... I I... I Th us uctos ad ts drvatvs ar dd as.5 J u u J l u. I I... I II u I u... I u I.6 I.5 a summato prm s troducd so that umbr o dcs I I... I o ad should attach to th vrt srt pots... quall. Th coct. I I... I rot o. wll b ro but w do ot aru t I hr. Sc w hav adoptd a cocrt ucto cocts ar dtrmd b ths paso. I A E th Ths s quvalt to th rat ucto mthod o us cas. Th ar th vr modular varat uctos o vrt srt pots whch rma uactd th procss o sp structur sum ad wll b cludd th al orm o ral -loop -pot suprstr ampltuds. It s dsrabl to hav plct orm o. Ths procss ma b do as th cas o us b multpl a ucto o o th both sds o th paso.5 so that th both sds do ot hav sular part o ad th drtat wth rspct to... whch ar compots o. It s ow that th us sma ucto has a lad trm so calld Schur-Wrstrass polomal S whr... ar compots o as ollows. 8

29 S hhr trms o.7 Eampls o S ar S S.8 6 S 5.9 It would b ood th both sd o.5 hav o sular trms o atr a powr o Schur-Wrstrass polomal s multpld but sms t s ot so as to prov ths act. r w assum as ollows: Thr sts a ucto X C C such that th product [ X ] I I... I has o sular trms th paso: [ X ] I I... I = cost hhr dr trms o..... I us X p[ ] X m m m m ad X vs sular part o drvatvs o ucto. I. s assumd as w saw or th = cas subscto 6. th cocts II... I ar bascall obtad b drtat approprat tms o I [ X ] A wth rspct to th varabls... ad th E stt all o thm qual to us q..5. W wrt X I I... I I A E Ap l I l E l X p l I l E l I... I X whr. 9

30 E [ ] I [ ] I [ ] [ ]... [ ] I I I l A. or a d st o I I... I. Th d F as F p l I l E l. X ad pad ths trms o... to obta plct orm o II... I. I thr ar a tms o I... I I a tms o I... I I a tms o I... I I w hav I I... I cost F cost F a a a I I I whr. a a... a..5 Sc ral s odd ad I I... I ca b prssd as a polomal o s v.6 I I... I th ollow dtt hold b. : S [ I I... I ] I I... I I... I I I... I prm IJ S.7 ad th sp structur dpdc o th product s totall prssd b o d o costats ucto valus at th o sular v hal IJ prods. Ths s a atural tso o th cas o us. Th statmts.6 ar vald at last or hpr llptc curvs. It s pctd that ths holds or mor ral cass too. I t s ot th cas w us.5 wth. Sc... ar all st qual to ro atr th drtatos th dtals o th structur o X acts ol umrcal costat actors.. Th costats

31 rot o th rht had sd o. ar dtrmd oc X s clard. Th ampls o plct orm o umrcal costats. II... I wll b as ollows up to ovr all cost.8 F l I I I I.9 I I F I I = [ I l I ][ I l I ] II I W assumd that th cas o us. l X dos ot cotrbut to At us th actor l X II... I. or small valus o as ca b rplacd wth Esst srs. It sms that th us vrso o th ormula o.8 s ot ow ral cas but probabl t sts ad modular orms ad v l X wll b rplacd wth us dpdt cotrbuto o th prms o stt... atr th drtatos.. Th actor th domator o th oral ucto I A s obvousl dpdt o. E Ths actor s th last trm o F. ad ths cotrbuts to modular varat ucto II... I b drtatos o F th ts rsult wll b vtabl costat modular orms o us as us. I ths actor cotrbuts to th orm o II... I or t ma cotradct th act us cas wh pch th Rma surac

32 to tor. Also atr stt... atr drtatos a ovr-all actor I E appars th rsults. Ths dos ot hav pols o ad wll b costats but urthr cosdratos tothr wth l should b dd. X Th prmutato summato prm.5 ad.7 mas that w also hav to cosdr... I I I.9 ad all combatos o I I.. I us ad our-pot suprstr ampltuds v o-ro rsult ol rom. All othr II... I ar or hhr pot ampltuds. Wh w calculat smmtrato o.7 th rsult wll b.9. I ths scto w mad two substatal assumptos to obta plct orm o II... I du to th dcult o vstat dtald structur o sma ucto ad structurs o sular part o drvatvs o ucto o th sd. O s that th ucto. ca b padd as I... I I I... I.5 whr summato s ol up to ad ucto has smpl sular structur as o.. I th structur o sular part o I I... I s clard ma o ths wll b clard. Th othr assumpto s.. B ths as was th cas us th uctos o vrt srt pots com ol rom drtatos o sma ucto o th potal actor F.. As a cosquc polomal orms o drtals o l I ar qut smlar as thos us th drc s that hhr us thr ar umbrs o varabls to drtat stad o o varabl. Ths s a c smplct sc th product o rmo corrlato uctos us has th sam pol structur as that us or sd smultaous sl pol structur th varabls ad th rht had sd o.7 should hav such pol structurs. Th orms o th uctos ar bascall dtrmd althouh umrcal costat actors ad cotrbutos rom th us modular orms ar ot dtrmd. Strctl spa th possblt o paso.5 wth trms o th sd o ds to b arud mathmatcall hhr us. Cosdr th pols o

33 th sd o such paso as wll as th assumptos w mad loo atural. Also q..9 whch was obtad lo tm ao susts th valdt o th paso orm o.5 as ts ralato. Ths pols o wll b partcl pols o loop pot suprstr ampltuds th sam mar as us. I rorous ormula clud umrcal actors about th dcomposto dtts ar obtad th rsults o wll b usd to dscrb ac-ood currts a closd orm o us. - pr llptc cas It s structv to cosdr hpr llptc cass bcaus w ca s plct orms o hal prods corrspod to o-sular v sp structurs as trals v. blow ad ca s how arumts th prvous subscto wor cocrt calculatos. W also would l to pot out that hpr llptc cas thr s a mthod to calculat plctl th costats hlpd b a lat IJ classcal thor rlatd to Jacob vrso problm[][]. As dscrbd th prvous subscto sp structur dpdc o ampltuds ar trl dtrmd ol b ths o d o costats or a ad at last or hpr llptc cass. IJ I II... I ar actuall v as.5 tp ormula all tools to calculat th sp structur sum o S S... S wth a costrat or arbtrar loop pots ca b prpard as ollows alo a scaro ussd r.[]. Cosdr th brach pots o th curv... ad th valu o at. As lo as o-sular v sp structurs ar cocrd thr s o to o corrspodc btw o sp structur ad o choc o umbr o brach pots sa out o... pots.... Ths ca b s as ollows. Frst thr ar possbl chocs ad ths umbr s qual to whch s th wa o roup brach pots to two parts ad s qual to th umbr o o-sular v sp structurs.

34 Cosdr th abla ma o ach o brach pots as U C. Ths tral ca b do plctl. For all o holomorphc o orms th rsults o th trals hav a orm... U E E. whr both o E ad E or ach ar dmso vctors whos all lmts ar ro or /. Obvousl = + all compots o E ad E ar ro. S th cocrt ampl o = th Appd C. Th U thmslvs ar hal prods ad th summatos ovr th ollow umbr o brach pots chos out o + umbr ar also hal prods: m C. m Dlta dots o choc o brach pots. Ths s th plct dto o hal prods usd ths documt th curv s hpr llptc ad ths corrspods to o sular v sp structurs as w wll s blow. It s ow that th vctor o Rma costats has also ths tp o summato: U.. W d th compot vctor dcs a b corrspod to th Rma costat b th ollow quato: a b.5 Th us hpr llptc sma ucto s dd us a thta ucto whos dcs ar th Rma costat

35 a u cp ui IJ uj u b..6 I J r th Jacob thta ucto s dd a stadard otato: a t u p{ a a a u b} b.7 Z u C Th thta ucto s calld odd or v dpd o whthr th ab s v or odd. Ths ca also b wrtt as a t u u b a p{ a a a u b} b.8 whr u s th stadard thta ucto wth ro d. Suppos that hal prods ar cludd th varabls o sma ucto: u u..9 Thr ar som stuatos whch th p actor.8 t p{ a a a u b} ca b dsrardd. Th lt had sd o q.. I A s o o such ampls wh s st qual to. Ths s E a rato o two sma uctos ad cosdr th codto th potal t actor p{ a a a u b} bcoms. I such cass b s th orm o varabls u b a q..8 th ol has th ct to sht th d o th thta ucto rom to Ad a c act s that t s ow that th thta uctos whch hav dcs. ar v. Actuall ths s th dto o th su whch dots th v thta uctos. Ths s how th lt had sd chas to th product o rmo corrlato uctos whch cluds v thta uctos. t Thr s o mor stuato whr th actor p{ a a a u b} ca b dsrardd. Th drtato o l u wth rspct to th I-th compot o u wll v such a stuato. Ev th drtato s ol o tm du to th codto w ca dsrard that actor. Th o hpr llptc curvs w hav 5

36 l[ ] u... I I II... I u. or > whch s a ralato o th us ormula q..58. Lt us s th rht had sd o th dcomposto ormula.7. w saw that all ormato about o d sp structur s cludd. I IJ ths has ol o lmts or ach sp structur ad s qual to th brach pot lmts as a smmtrc matr o th dcs tsl. I us whch has IJ I J ca also b rprstd b th brach pots thmslvs hlpd b th mthod o solv so calld Jacob vrso problm [][][] as ollows. Frst wh o o th d s qual to that s I ar show to b qual to lmtar smmtrc polomals o th chos brach pots... cpt th ovrall s + -. Th has th hhst dr o udamtal smmtrc polomals ad or.. th dr s dcras as... Th rst umbrs o lmts o ollows. Epad th curv as IJ ar dtrmd as.... whr... ar lmtar smmtrc polomals o.... Costruct a cocrt polomal F as : F { } { }.... 6

37 Th th ollow quatos hold or a choc o two brach pots rom r s th chos umbr o pots... : F. I J r s IJ r s I J r s Thr ar umbr o such quatos ad t matchs th cssar umbr o quatos to dtrm all lmts b brach pots. IJ I suprstr thors ol hpr llptc curv s cocrd th str masur s costructd b brach pots[]. Th summ ovr sp structur s a albrac calculato o brach pots as str masur rrspctv IJ o uctos o vrt srt pots as was th cas us. Th rsult o albrac calculato wll b rprstd b polomals o smmtrc uctos o brach pots whch wll b prssd b thta costats ad b modular orms at th us. As us th modular orms wll appar rom two parts: O s hr ad th othr s rom l q... X To do all calculatos o suprstr ampltuds ths s ot th whol o th stor v th hpr llptc cas. W hav to cosdr coctd parts o th cotractos as has b potd out r.[9] ad thr rlatd artcls. Ths vstatos ar bod th scop o ths documt. Th Dola-Goddard rat ucto mthod mas t possbl to cosdr obta th orms o II... I ad modul costat dpdt parts almost sparatl or hhr us. A lot o arumts ar mad o th lattr For ampl [9][7]-[] clud suprstr masurs. To solv th ormr ma b mor strahtorward oc structurs o us sma ucto ad uctos ar clard ad t ma hav applcatos bod str thors. Acowldmts Th author thas ro. A.oroov or hs d hlp th procss o submtt r.[] ad ths documt to arxv. Ths documt wll b submttd to arxv ol. 7

38 Appd A roo o. B th codto th rht had sd o. bcoms ro bcaus that s dt [ ]... A. Thror or... thr st a... a a whch sats a a a a a A. us ss rot o a a... a ar ol or otatoal covto. Ths ca solvd or a a... a b Cramr s ormula. Thr ar lar quatos or umbr o varabls; ths s a rsult o rdudat rlato. It s possbl to choos a o quatos or varabls to solv. Choos th varabls... th quatos ar mat = [ ]... * a a a a T... A. Th w hav dt [... ] th ;... a A. dt [... ]... T ow w cosdr polomals ad h assum s v. dd as ollows. For a whl w 5 { h } { a a a 5 a } A.5 h a a a a A.6 EVE ODD That s s a polomal o ucto o a orm { } { } ad h s ts v drvatvs part. Th s a dr polomal bcaus t ca b wrtt as : h { } [ polomal o ] A.7 8

39 ad th dr o s. O th othr had w actor th dto o A.5 as { EVE ODD EVE ODD } { } th w ca s that th quato has umbr o solutos at... A.. Thror ca b wrtt a drt wa: c a... A.8 Th costat c s strctl dtrmd so that th hhst dr o should b qual to that o A.5. Sc!... c s dtrmd as c {!]}. S I w compar wth th orm o bcaus o th rlatoshp b S th lattr s oud to b proportoal to th squar root o wth putt. That s [ ] S A.9 / c a I w ot th but sc w d rom A.8 that h { } [ polomal o ] h Thror th squar root A.9 dsappars ol lav h : A. [ ] [ ] h S! a! a! a a! a a a a A. EVE Th all o trms ar rprstd b th polomals o ad 9

40 th a a dpdt trms ar rprstd b actors a. a ad us th solutos or a a a a a. Ths ar th dto o a A.. W hav to b carul that thr s a ovr-all s ambut wh w calculat.... W dtrmd ths so that th ormula matchs wth S wh =. Sc S S S A. w choos th ovrall s b d dt dt [ [ ]... ] th ;... dt dt [ [ ] th ; ] th ; to prss th q.a. as S....! I s odd th w r-d th polomal o A.5 ad A.6 as 5 { h } { a a a 5 a } A. h a a a a A. r th hhst dr trm s cludd th odd drvatv part. Th rpat th sam arumt th last trm o th umrator o A. bcoms a stad o a. Ths mas that w do ot hav to cha ath. ad.5 as lo as th

41 summato. s utl. Th proo o.8 as wll as obta th orm o... V ca b do th smlar wa. For ths w d to cha th dto o ad h A.5 A.6 such that th drvatvs o ucto ar ot cludd but th moomals o ad ar cludd. As s show chaptr ad o ths documt du to th smplct o pol structurs o drvatvs o ucto t s alwas mor covt to aru o rathr tha V. Appd B OLES I th ollow ar compl umbrs. D th dtrmat G whr th st dr o th varabl s abst sc ths rprsts th pols o ucto : G B. D L as ollows: L B. Also w d W as th lmtar smmtrc polomal o dr amo varabls. W d W= wh =. For ampl W W

42 W cossts o trms ad s th sum o th products o - umbrs out o. W cossts o ol o trm whch s all products o varabls. Th w ca prov that : Formula : W G B. Formula : W W W W L B. Th G ad L ar products o Vadr mod tp dtrmat ad W. Th smplst ampl o th ormula s L } { W W W W W as dd abov. Th most sular pol structur o s prssd b th rato G L. B th ormula ad ormula abov W W W W W G L W W W W. B.5 Th rato W W s wh cosdr th oral ma o th umrator ad th domator o bac to th varabls whch s. Thror W W G L B.6 Th ma o th rst trm o rht had sd W s: th sum o th products o umbr o varabls chos out o varabls clud

43 . Thr s o or Th ma o th scod trm th rst trm. W s th such sum o product o varabls whch o o th varabls s alwas I total o ths two th ma o clud o mor varabl Summato o all possbl sl pol trms L G s actl W whch was ot cotad at th b that s b b... b whr b b... b ar chos umbr o varabls out o varabls... W cosdrd th cas that th scod varabl o L th umrator o th rato L... s v but th sam rsult holds or a o. G... Two mor obsrvatos: Suppos that w start wth a st o umbrs o varabls whch satss L ad calculat th rato o dtrmats. Th as abov G th last varabl sa total th valu o aturall appars rom th actor L G W W. Thror dos ot dpd o th choc o varabls.. A choc o varabls vs th sam ucto B loo at th scod trm o rht had sd o q.b.6 W w ca coclud that wh w rard as ar dpdt varabls... Rs atur as was plad r.[]. whch s a dsrabl

44 Appd C hr us Lt A... A A ad B B... B b a caocal homolo bass. W choos caocal holomorphc drtals o th rst d... ad assocatd mromorphc drtals o th scod d r... r r. Th prods ar v as A I J IJ C. B I J IJ C. r A I J IJ C. r B I J IJ C. Th ollows ar ampls o us cas. ampl. D trals For dtals plas s [][] or U C C.5 Th or = sc thr ar + = 6 brach pots U U U U U U E E E E E E E E E E E E E E E E E 5 E 5 5 E5 E5 E 6 E 6 6 E6 E6 ot that + =. a b a E E b E E

45 5 Id o 6 odd hal prods U =.6 Id o v hal prods U U 5 D th curv as... R C.6 whr a varabl stad o s o q.. s usd ad o o th brach pots s d at. A polomal F s dd as ollows: F... } { } { C.7 Th ollow thorms ar ow rlatd to Jacob s vrso problm Lt I J I u or a umbr o pots... o th curv. Th th ollow rlatos hold or a r :... r r r r C.8 J r J J r u C.9 s r s r s r J s I r IJ J I F u C. or a choc o two pots s s r r From ths t ca b drvd that... J ordr o o ucto smmtrc udamtal u J J. I u s qual to hal prod th th pot r r s r. As a ampl at = th curv s

46 R C. Th ucto F s v b 5 } { } { F C. Suppos that w adopt two pots m out o v pots For a d sp structur th quato C. vs ol o rlatoshp: m m m m F C. O th othr had q.c.9 vs } { m m } { C. Th w hav th ollow soluto at a d sp structur m m r q p m r q p m m F C.5 whr a o r q p s drt rom m Appd D O. ad th ucto U.7 Ths appd s mal to avod possbl cous about th dto o. I r.[] th cocts ar dd q..67 ad.68 that papr as ollows: ] [ D. B drtat tms or both sd o ths accord to th loc th tt w hav ] l l p[! G D. or all valus o clud. Ths wll match th rsdu rqurmt r.[]. s v b ust putt = D.. B ths dto ad

47 7 prsso has a cotrbuto o G or ampl. Appartl ] [ s v b a t sum. O th othr had ths documt ] [ s wrtt as a t sum as. or. paso b uctos:.! ] [. r or th cocts ar sstall th sam as D. cpt th act that th ovr-all actor! s rplacd wth!!! du to smpl otatoal drcs as show.: ] l l p[!!! G or. W hav to pa attto or bor drtat tms. B s th paso o ucto m m m c m m G m c ad ts drvatv orms thr ar ma trms th rht had sd o. whch ar proportoal to. All o c.!! D. whr s v ar such trms. Atr th drtato ad r-wrt th orm o D. wll b th last l o th ollow quato or arbtrar : ] l l p[!!! G G }! { D.

48 W cludd ths last trm ucto as.:! F p[!! G {! G } th p l l {! G ].! Th last trm o. vs o-ro rsults ad ol = ad all drtatos wth rspct to ar o ths trm. Trms o or < ar obtad b q.. bor calculat b q... ot that th Esst srs wth hhst d G dos ot appar a o bcaus at = th cotrbutos whch clud G th last two trms. cacl ach othr. That s or ampl dos ot clud G dos ot clud G tc t w dscrb th pol structurs o th drvatvs o th ucto U dd.7 whr U D.5 U p[ l l ]...7 Th pol structur o D.5 s th sam as that o.. I w slct umbr o varabls b b... b out o umbr o varabls... th th pols o D.5 ar b b... b All combatos that s smultaous sl pol structur o umbr o varabls b b... b out o umbr o varabls... approprat ovr-all umrcal costats whch dpd o ar multpld. Th proo s do b mathmatcal ducto. For = t ca b s that ths s vald rom.9... Suppos ths s 8

49 tru or. Th or o slcto o b b... b th corrspod trm wll hav th ollow orm o pols bor stt : b U b. B drctl drtat ths trm wth b rspct to w hav or th pol part [ L b L That s rom [ ] b b ] o mor actor b U. whr s drt rom a o b b... b appars. I w start rom aothr st o b th duplcatd trms wll appar. Thos trms wll cha th umrcal actors o ach ductvl ad stll th sam arumt ca b appld. O commt o th pols o sp sum rsult q..7 ad.8 whr w ca rstrct to th cas. Th pol structur o D.5 rlats to thos o a rato o two dtrmats.. Th rato o. s mad rom tract out th most sular part o q..5. whol o th structur o.5 also has lss sular pols Th b th stc o holomorphc trms th ucto ad ts drvatvs th dtrmats. Th stc o th Esst srs G th p actor o. or.8 vs such lss sular parts atr th drtato wth rspct to. For ampl th trm smultaous sl pols o F q..7 or =.5 cotas ot ol umbr o varabls but also thos o smallr umbr o varabls as ca b s rom.8 ral wh S >. Th pol o s th vrs o product o varabls.... Atr summ ovr sp structurs all pols ar stll sl ordr or a varabl o. Th umbr o varabls o pol s qual to or lss tha as s appart orm q..7. 9

50 Appd E Equvalc o Ellptc ultpl Zta valu mthod ad th mthod scto -6- Addd th ov. 7 vrso I Appd E w clar th mthod dscrbd scto -6- o o loop sp sum s quvalt to th rsults basd o llptc multpl ta valu ZV mthod dscrbd [] b som smpl obsrvatos. Ths also vs a plct proo that th rsult o sp structur sum [] or part v part s vald or arbtrar pot uctos clud umrcal actors. It ma b usul or o-prts o ZV mthods l m. I th ollow t s also dscrbd th sp sum how th otatos ZV ar rlatd to th classcal otatos whch sma ucto ad ts drvatvs ar usd. Th otatos o ZV cssar hr ar as ollows. Th Esst ucto ad th rocr-esst srs ar dd b E E. m ad m F E. Eq.E. satss th ollow rlatos: l E Eq.E. has th paso orm as : E E E. F p E G E. Som othr uctos ar dd b Im p F E.5 Im E.6 E.7 V p E.8 p ot that V... s totall drt rom th quatt V... p 5

51 5 dd.8 th tt. B troduc th drcs o srt pots o vrt oprators.... whch satss w cosdr th mult-varabl ucto o E. F to cosdr th sp sum or th pot cas. Ths quals to p G E F E.9 A obsrvato s that b us a stadard ormula p E. as wll as F s qual to F. E. That s t s qual to Dola-Goddard rat ucto.5 rplacd th aular varabl wth. Thror all arumts scto -6- ca b appld atrwards. Or mor drctl w mod th sl F q.e. b us E l! E. as: l! p p G G E F l l p G l l p G E. I th last qualt w usd th = trm th p ucto G to prss l b th sma ucto. Thror w hav

52 5 l l p G F E. whch s th sam as th.9 b rplac wth : ] ] l [ l l p[ ] [ m m m m.9 ] l l p[ G O th othr had rom E.5 E.7 E.8 as wll as w hav ] [!... p F p V E.5 whch s th coct o Talor paso o a holomorphc ucto F b E.8. For covc wthout ar o rptto w dscrb th ollow how ths obsrvatos ar rlatd to th arumt o summ ovr sp structurs v -6-. W rard th product F as a llptc ucto o ad pad b th drvatvs o ucto as h F E.6 whr h ar paso cocts dd hr. Cosdr th pols o : m m G m m...! O G m d m! d E.7 ad multpl o both sds o E.6 drtat tms wth rspct to

53 5 w hav th plct orm o h : ] [! F d h E.8 whch s vald or. Oc th cocts h ar obtad w ca hav th product o rmo corrlato uctos S udr th codto b stt quals to hal prod b.7. ad E.: h F S. F ] [!! E.9 B compar E.5 ad E.8 w hav V S...! E. ad urthr sc ODD ad Q as....! Q V S. E. Ad l l p!... p p p G p V or p. E. To hav sp sum o v part w rwrt th thta costats b brach pots as D D D whr D ad as.7 w calculat to hav

54 ... S E V E. wth E E E E b th albra o. r E s ot th Esst ucto but th polomal dd.8: E Q Q Q.8! ad! s qual to th umrcal coct o th polomal Q whch s th hhst dr trm o th polomal. Th whol o E ca alwas b rprstd b th lmtar smmtrc uctos o as s asl provd ad thror b th Esst srs. ot aa that E surs that w do ot d th orm o V... or p. p Som ots: I ZV th basc buld blocs o th thor ar uctos dd th paso orm o E.6. Th uctos V... dd E.8 ar p prssd b ad v th rsults o sp sum b E.. Ths s basd o a wd ramwor dscrbd [] ad s covt or ral thortcal cosdratos o th rsults. V... ar also rlatd to th classcal p otatos whch sma ucto ad ts drvatvs ar usd as E.. I E. th Esst srs appar bor cosdr sp structurs bcaus aturall has such modular orms sd. For ampl Dola Goddard potd out th addtoal trm q.5 []. Ths s th G G ad t s rrspctv o sp structurs that s t s ot rom o E. E actors E.. W hav to do total p tms o drtatos E. wth rspct to th aular varabl.w ma do 68 tms o drtatos to v th trms proportoal 5

55 to th Esst srs o th trms G.Th rst p L L... tms o drtatos ar o p l l. Ths vs smultaous sl pol structur o p L umbr o varabls chos out o umbr o varabls... atr stt as dscrbd Appd D. Ths mas th pol structurs o th sp sum rsults clar. Th total o V... s th summato o all such pols o drt ordrs p o p p p 6.. Th udamtal raso wh V... s rlatd to th sp structurs p coms rom th ollow rlatos udr th codto : S. F h E.9 as wll as th comparso o V... E.5 wth h p E.8. As o mor ampl w show a sp sum calculato o quartr-mamal suprsmmtrc cas. ot that ths rsult s alrad obtad [5] a ral mar ZV laua ad th calculato blow s ol r-wrt thr rsult classcal otatos. W calculat S S E. or ad th otatos o [5]. r stad o calculat + actors o So rls w valuat S rst b E. ad E. S V V 55

56 56 } { 6. E.5 ot that E. s vald ol or p but V s th cas p. Go bac to D. ad D. w ca sa that or th addtoal trm th last l o D. s ot cssar. B us or th thr varabls cas wth w hav S 6 E.6 Thror B A S S E.7 whr... } 6 { V E A E.8... V E B E.9 Th rsult o A s proportoal to th mamal suprsmmtrc cas E. wth a actor E dd.8 whras B cotas a modd actor:! Q Q Q E. E. Ths ca also b rprstd b th Esst srs ral. As ca b calculatd asl E E E b th albra o.... V ar drvatvs o th rat ucto as E. oth

57 chad rom th mamal suprsmmtrc cas. Th sam wa ca b appld to th hal mamal cas. W valuat S v S v S or. Sc v v w hav S v S v S A B E. whr A v E V... E. B E V... E. Rrcs []A.G. Tsucha hs.rv.d [] A.G.Tsucha O th pol structurs o th dscoctd part o hpr llptc loop pot suprstr ampltuds arxv:9.67 [] L.Dola.Goddard Currt Albra o th Torus Commu.ath.hs arxv:7.7[hp-th] [] J. Brodl C.R.ara.atths O.Schlottrr Ellptc multpl ta valus ad o-loop suprstr ampltuds arxv:.555[hp-th] JE 57 5 [5]. Br I.Buchbrr. O.Schlottrr From mamal to mmal supr smmtr str loop ampltuds arxv:6.56[hp-th] JE [6] S. ohr S.Stbrr oodrom Rlatos hr Loop Str Ampltuds arxv:7.96 [7] S.Stbrr ad T.Talor oabla Bor-Ild acto ad tp.-htrotc dualt : orormalato thorms ucl.hs. B68hp-th/96 [8].awamoto t.al Gomtrc Ralato o Coormal Fld Thor o Rma Suracs Commu ath.hs [9]E.D or ad D.ho Two loop Suprstrs. VI ulc.hs.b755-9 hp-th/597; 57

58 []Chrs oor ad Davd S. Yu Bar Forms ad th hpr llptc suprstr aat arxv:9.55v ov 9 []V.Z.Eols ad..rchtr rods o hprllptc trals prssd trms o thta costats b mas o Thoma ormula hlos Tras A ath hs E Sc ar 8 [] V. Eols B.artma V.aramaova J.u C.Lammrahl.Srmacha Ivrso o a ral hprllptc tral ad partcl moto orava-lsht Blac hol spac-tms J.ath.hs. 5 5 arxv:6.8v[r^qc] Dc [].F.Bar Abla Fuctos Cambrd athmatcal Lbrar 897 rprtd 995 [] Y.Osh Lcturs o hpr llptc curvs Japas 超楕円関数論 ' [5]..Faras ad I.ra Rma Suracs Sprr w Yor 98 [6] J.D.Fa Thta uctos o Rma suracs Lctur ots athmatcs vol.5 Sprr 97 [7] R.Doa ad E.Wtt Suprmodul Spac Is ot roctd arxv:.7798 [hp-th] roc.smp.ur ath [8].ato ad R.Volpato hr us suprstr ampltuds rom th omtr o modul spac ucl.hs. B76 - hp-th/56; [9].ato ad R.Volpato Suprstr masur ad o-rormalato o th thr-pot ampltud ucl.hs. B []R.Salvat a Rmars o suprstr ampltuds hhr us ucl.hs.b886-7; arxv:8.5 [].ato ad R.Volpato Gtt suprstr ampltuds b drat Rma suracs ucl.hs.b89-5 arxv:.5[hp-th] [] A. oroov SR Suprstr asurs Rvstd JE arxv:8.67 [].Br.aac J.U.a ad S.Sors Towards th o loop ahlr mtrc o Calab-Yau ortolds JE 77 arxv:7.7 [hp-th] A.G. Tsucha E-mal Addrss colasum@tha.oc..p 58