Duality Anomaly Cancellation, Minimal String Unication and the Eective Low-Energy Lagrangian of 4-D Strings

Transcription

1 CERN-TH.68/9 Dulty Anomly Cncellton, nml Strng Uncton nd the Eectve Low-Energy Lgrngn of 4-D Strngs Lus E. Ib~nez nd? Deter Lust CERN, Genev, Swtzerlnd Abstrct We present systemtc study of the constrnts comng from trgetspce dulty nd the ssocted dulty nomly cncelltons on orbfoldlke 4-D strngs. A promnent role s plyed by the modulr weghts of the mssless elds. We present generl clsscton of ll possble modulr weghts of mssless elds n Abeln orbfolds. We show tht the cncellton of modulr nomles strongly constrns the mssless fermon content of the theory, n close nlogy wth the stndrd ABJ nomles. We emphsze the vldty of ths pproch not only for (,) orbfolds but for (,) models wth nd wthout Wlson lnes. As n pplcton one cn show tht one cnnot buld Z or Z7 orbfold whose mssless chrged sector wth respect to the (level one) guge group SU() SU() U() s tht of the mnml supersymmetrc stndrd model, snce ny such model would necessrly hve dulty nomles. A generl study of those constrnts for Abeln orbfolds s presented. Dulty nomles re lso relted to the computton of strng threshold correctons to guge couplng constnts. We present n nlyss of the possble relevnce of those threshold correctons to the computton of sn W nd for ll Abeln orbfolds. Some prtculr mnml scenros, nmely those bsed on ll ZN orbfolds except Z6 nd Z8, re ruled out on the bss of these constrnts. Fnlly we dscuss the explct dependence of the SUSY-brekng soft terms on the modulr weghts of the physcl elds. We nd tht those terms re n generl not unversl. In some cses specc reltonshps for gugno nd sclr msses re found.? Hesenberg Fellow Jnury 99

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3 . Introducton Four-dmensonl superstrngs [] consttute t the moment the best cnddtes for uncton of ll nterctons. In tryng to use these theores to descrbe the observed physcs, t s of fundmentl mportnce to obtn the eectve low-energy eld theory of ech gven 4-D strng. uch progress long these lnes hs been cheved n the lst few yers nd there s t present good knowledge of the form of the eectve Lgrngn for the cse of orbfold-lke 4-D strngs. One of the generl fetures found s tht the knetc term of the mtter elds s not cnoncl but presents -model structure, whch depends on the modul elds of the orbfold vrety. Furthermore, the eectve Lgrngn s n generl nvrnt under certn dulty symmetres ctng on the modul spce. In mny cses, n prtculr orbfolds, these symmetres lwys contn the subgroup SL (; Z) [],.e. three copes of the modulr group ctng on the modul of ech of the three orbfold complex plnes. The requred trget spce modulr nvrnce provdes connecton between the eectve Lgrngn nd the theory of modulr functons []. The eectve treelevel Lgrngn of orbfolds hs been found to be nvrnt under these symmetres nd the Yukw couplngs to trnsform s modulr functons of dente modulr weght [4,5]. The sme nvrnce hs been checked n one-loop strng computtons [6,7] nd ndeed t s expected on generl grounds tht these modulr symmetres wll survve non-perturbtve eects nd tht they my be broken only spontneously, but not explctly [8]. Ths s mportnt becuse t mples constrnts on the possble form of non-perturbtve strng eects, s remrked n refs.[,8,9,]. In the present pper we nlyze severl derent contexts n whch the SL (; Z) symmetry of orbfold 4-D strngs constrns the low-energy eectve eld theory. The crucl ngredent n these constrnts s the cncellton of \dulty nomles". The structure of the pper s s follows. In secton two, fter dscussng the dulty-nvrnt structure of the eectve low-energy Lgrngn of 4-D strngs, we concentrte on the orbfold cse. We clssfy ll the possble modulr weghts for mtter elds n rbtrry Abeln Z [,] nd Z Z [] orbfolds. We lso N N show how the vlble rnges for modulr weghts depend on the Kc{oody level of the guge lgebrs. In secton three we dscuss the constrnts on the mssless prtcle content of 4-D strngs comng from cncellton of dulty nomles [4{7]. Indeed, the SL (; Z)

4 dulty trnsformtons nduce chrl U() rottons on the mssless fermons. The ssocted current hs to be nomly-free n order to preserve the dulty symmetres. The nomly my be cncelled n two possble wys: ether through sort of Green{Schwrz mechnsm [8] or through contrbutons comng from the superhevy strng spectrum. In some prtculr cses only the rst mechnsm s vlble nd the prtculr unversl (guge group ndependent) structure of the Green{Schwrz mechnsm forces the mssless fermon content of the theory to be remrkbly constrned. Ths would hve very lmted pplctons f t were only true for the few exstng Abeln (,) orbfolds. In fct we emphsze tht t lso pples to the mllons of possble (,) Abeln orbfolds one cn construct wth or wthout Wlson lnes. Ths we hve explctely checked n lrge number of such (,) models. We present generl study of such constrnts n Abeln orbfolds. As n exmple of the power of the dulty nomly cncellton condtons we re ble to show tht one cnnot possbly buld Z or Z 4-D strng whose mssless chrged sector wth respect to 7 SU() SU() U() s tht of the mnml supersymmetrc stndrd model. We lso dscuss constrnts from the cncellton of mxed dulty-grvttonl nomles. The mxed dulty-guge nomly s relted through supersymmetry to one-loop modul-dependent correctons [9,6] to the guge couplng constnts of the eectve eld theory. In mny models the one-loop threshold eects, comng from mssve strng modes, provde n ddtonl modul dependence to the runnng of the guge couplng constnts. In secton four we study the possble role of these threshold eects n renderng vlues for sn nd consstent wth the expermentl results. W Indeed, f we restrct ourselves to the mnml prtcle content of the supersymmetrc stndrd model, we nd tht there s no possble (,) Z orbfold prt from Z, Z yeldng the correct vlues for those qunttes unless, the guge groups re relzed t hgher Kc{oody level. On the other hnd, greement cn be obtned n Z Z orbfolds (even wth lowest Kc{oody levels). Ths s generlzton of the nlyss of ref.[] to nclude the three orbfold complex plnes nd mkng use of the generl clsscton of modulr weghts presented n secton two. For completeness we lso dscuss the cse n whch one modes the mnml prtcle content of the stndrd model to nclude extr mssless multplets. In secton ve we dscuss some constrnts on SUSY-brekng soft terms from dulty-nvrnt ctons. We do not n fct rely on ny specc ssumpton bout how N N 6 8

5 supersymmetry s broken but smply ssume tht the uxlry elds of the mrgnl deformton elds (the dlton S nd the three complex untwsted modul T) re non-vnshng nd provde the seed for SUSY-brekng. Ths s ndeed wht hppens n ll supersymmetry-brekng scenros dscussed up to now. If ths s the cse one cn wrte down certn constrnts mongst the physcl gugno msses dependng on the modulr weghts of the mssless elds. Smlr though more model-dependent expressons my be found for the soft SUSY-brekng sclr msses. One mportnt pont s the lck of unverslty of these soft terms, unlke the usul ssumptons mde n the stndrd mnml low-energy supersymmetry models. Some nl comments nd conclusons re left for the sxth secton.. Dulty-Invrnt Eectve Feld Theory.. Generl supersymmetrc 4-D strng compctctons Let us dene the bsc strng degrees of freedom whch pper n the low-energy eectve N = supergrvty cton [] of four-dmensonl strngs nd whch re needed for our dscusson. In the followng we wll consder compctctons of the ten-dmensonl heterotc strng [] on sx-dmensonl (smooth) Clb{Yu spce [] or on sx-dmensonl orbfold. (We wll n generl not ssume tht we del wth (,) compctctons.) Frst, there re modul elds whch descrbe the llowed deformtons of gven bckground. In the context of the underlyng conforml eld theory the modul re the couplng constnts relted to the truly mrgnl opertors whch cn be dded to the two-dmensonl world-sheet cton wthout destroyng the conforml nvrnce of the theory. These modul prmeters correspond n the eectve low-energy theory to chrl mtter elds ( x) (we wll denote both the chrl superelds nd ther sclr components by the sme symbols) wth completely t potentl,.e. wth undetermned vcuum expectton vlues ( vev s) (t lest f one dsregrds possble non-perturbtve (nnte genus) strng eects). The modul tke ther vlues n the so-clled modul spce,, whose metrc s determned by the two-pont functon of the ssocted mrgnl opertors. Loclly, the modul spce s smooth Khler mnfold. However t s very mportnt to relze tht there my exst dscrete reprmetrztons of the form! ( ) (: ) f

6 whch chnge the geometry of the nternl spce but leve the mssve strng spectrum nd the underlyng conforml eld theory nvrnt. These re the fmous trget spce dulty trnsformtons [4,,5,6] whch form the dscrete dulty group. The dulty nvrnce s n entrely strngy phenomenon nd s not present n the eld theory compctctons. The most promnent exmple s the R =R dulty symmetry [4] for closed strng compctcton on crcle orgntng from the exstence of topologclly stble wndng sttes. The trget spce dulty nvrnce mples tht certn ponts n hve to be dented, whch mens tht the modul spce of the conforml eld theory s Khler orbfold nd not smooth Khler mnfold: = =. CFT In ddton to the modul elds there re n generl chrl mtter elds A ( x) wth non-t potentl such tht A =. Furthermore there re N = vector Q supermultplets wth guge bosons correspondng to guge group G = G. In generl, both the mtter elds Am s well s the modul elds re guge nonsnglets nd buld representtons R, R respectvely. It follows tht t P P lest some prt of the guge group s generclly spontneously broken by the modul vev s. Only t specl ponts n the modul spce the unbroken prt of the guge group gets enhnced. (Very often these ponts re gven by orbfold ponts of.) Fnlly there s the grvttonl supermultplet together wth the chrl dlton{xon eld commonly denoted by S. Ths eld cn be equvlently descrbed by lner supermultplet. Let us now descrbe the strng tree-level low-energy cton nvolvng the modul elds. Snce ll S-mtrx elements computed n the underlyng conforml eld theory re nvrnt under the dscrete trget spce dulty trnsformtons, t follows tht lso the correspondng eectve cton of the modul elds must be dulty nvrnt. (For recent summres on these ssues see refs.[7{9].) The knetc energy of the modul s gven by 4-dmensonl supersymmetrc non lner -model wth the modul spce eld theory s trget spce nd s therefore determned by the Khler potentl K( ; ) ledng to the Khler metrc K ( ; : L kn Under the dscrete trget spce dulty trnsformtons, beng just dscrete symmetres of h, the Khler metrc trnsforms s K ( ; ) f ( ) f ( ) K ( ; ) (: ) 4 j j j j! j k jl kl!

7 @ where the holomorphc Jcobn s gven s f ( ) = e. On the other hnd, the Khler potentl s n generl not nvrnt under the dscrete dulty trnsformtons. They wll ct s (U()) Khler trnsformtons on K,.e. K; ( ) K; ( ) + g ( )+ g ( ); (: ) where g ( ) s holomorphc functon of the modul elds. The knetc energy of the mtter elds s n generl modul dependent functon. Expndng round the clsscl vcuum elds n lowest order n A looks lke: =, the Khler potentl for the mtter mtter mtter K = K ( ; ) A A + ::: (: 4) The dscrete reprmetrztons (.) n generl nduce chnge of the mtter Khler potentl K ( ; ) of the form K h ( ) h ( ) K : (: 5) It follows tht lso the mtter elds possess non-trvl trnsformton behvour under trnsformtons n order to obtn dulty-nvrnt knetc energy terms for the mtter elds: A h ( ) A : (: 6) Some of the modul re relted to the geometrcl nd topologcl dt of the sxdmensonl complex spce. Specclly, these modul elds splt nto two clsses: rst there re the modul T ( = ;:::;h ), whch re ssocted wth the deformtons (; ) A of the Khler clss. Second, we del wth deformtons of the complex structure of the compct spce prmetrzed by modul U, ( m = ;:::;h ). ( h nd h re the non-trvl Hodge numbers of the compct spce.) The modul spce fctorzes nto two dstnct subspces: =. Thus the modul Khler potentl splts nto the sum K = K( T ;T ) + K( U ;U ). It follows tht the chnge of the Khler potentl under dulty trnsformtons s gven by two derent holomorphc functons gt ( ) nd gu ( ). There s however n mportnt derence between the dscrete m reprmetrztons of the T nd U elds. Specclly, the dscrete trnsformtons ctng on the complex structure modul U m (; ) (; ) (; ) 5 T m m h U do not chnge t ll the geometry of the

8 underlyng compct spce; these nvrnces re of eld-theoretcl nture. On the other hnd, the modul elds T re relted wth the sze of the nternl compct spce. The exstence of possble dscrete reprmetrzton nvrnces n the T{eld sector s n entrely strngy phenomenon nd cn be vewed s the generlzton of the R =R dulty symmetry. Thus the strngy dulty group s gven by nd CFT = =. Consderng only the overll modulus T whose rel prt determnes the overll sze of the compct spce, the Khler potentl for T becomes n the lrge rdus lmt [] K = log( T + T ): (: 7) However, for smll vlues of the T{elds, formul (.7) gets n generl corrected by world-sheet nstnton eects. These correctons determne the explct form of the dulty group. Consderng the prtculr cse of (,) compctctons, the mtter elds re n one-to-one correspondence wth the modul. Specclly, there exst h mtter elds A n the 7 representton nd h mtter elds A n the 7 representton of E. (Note tht some of these \mtter" elds my ctully hve completely t 6! potentl such tht they re lso (Wlson) lne modul whose non-zero vev s brek the guge group E eld theory (; ) m 6 T T (; ) nd destroy the (,) superconforml symmetry, see the dscusson n secton.6.) For the compctctons on smooth Clb{Yu spce, the Khler metrc of these mtter elds s relted to the modul metrc n specl wy []: T mtter ( KT ( ) ( )) K = e KU m = j Kj( T ;T ) ; mtter ( KU ( ) ( )) K = e m KT = K ( U ;U ): mn mn m m (: 8) It follows tht the mtter metrcs trnsform under trget spce dulty trnsformtons wth combned reprmetrzton nd Khler trnsformton,.e. mtter gt ( ) mtter : K e = f ( T ) f ( T ) K (: 9) T j!j j k jl kl nd lkewse for trnsform lke K mtter mn. The trnsformton eq.(.9) mples tht the mtter elds gt ( ) : A e = f ( T ) A : (: ) T! j j 6

9 Also note tht n the lrge rdus lmt both types of mtter metrcs n eq.(.8) dsply prtculrly smple, unversl dependence on the overll T {eld: mtter K ( ) T + T : :.. (,) Abeln orbfolds In the followng we wll dscuss symmetrc Z nd Z ZN orbfolds. Ths dscusson wll be n certn sense lso more generl thn the bove Clb{Yu consdertons snce, t wll not be restrcted to the cse of (,) orbfolds. All our formuls wll be vld lso for (,) models wth non-stndrd guge embeddngs nd/or wth the presence of Wlson lnes. These compctctons nclude some exmples tht re of phenomenologcl nterest snce the guge group cn be derent from E6E8. In fct, there exst models wth stndrd model guge group G = SU() SU() U() nd three genertons plus ddtonl vector-lke mtter elds [{4]. Every orbfold of ths type hs three complex plnes, nd ech orbfold twst ~ = ( ; ;) cts ether smultneously on two or ll three plnes. In the followng we wll consder the dependence of the eectve cton on the untwsted modul elds. The twsted modul elds, whose non-trvl vev s blow up the orbfold sngulrtes ledng to smooth Clb{Yu spce, wll be treted s mtter elds. Furthermore, the twsted modul re generclly only present for the unque (,) embeddng of ech orbfold. The untwsted modul splt nto T nd U elds. All orbfolds hve h (; ) = ; 5 or 9 respectvely, where the three generc T{elds ( = ; ; ) descrbe the sze of the three complex plnes. The possble ddtonl elds correspond to the reltve shpe of the three plnes. On the other hnd, the number of U-elds, wth h (; ) = ; or, depends on whether the orbfold twsts re comptble wth the freedom of vryng the complex structure of the three subtor. Specclly, there re sx derent cses of untwsted orbfold modul spces [5]: " # " # SU (; ) SU (; ) () h (; ) = ; h (; ) =; ; : = U() U() : (: ) () h (; ) = 5 ; h (; ) =; : " # " # h(; ) SU (; ) (; ) SU (; ) = : U() SU() SU() U() U() T U (: ) 7 T h U (; )

10 () h =9 ; h =: = The correspondng Khler potentls re gven by SU (; ) SU() SU() U() : (: 4) () h = ; h =; ; : K = log( T + T ) log( U + U ): () h =5 ; h =; : (: 5) K = log( T + T ) log det( T + T ) log( U + U ): (: 6) () h = 9 ; h = : K = log det( T + T ): (: 7) (In order to keep contct wth the prevous notton we dene T = T ( = ; for cse () nd = ; ; for cse ().) For the overll modulus eld, whch s obtned by dentfyng the three generc T{elds,.e. T = T = T = T, nd settng T = ( = j), the bove formule exctly reproduce K = log( T + T) (cf. eq.(.7)). The correspondence wth the Abeln orbfolds s the followng () h = ; h =: () h =5 ; h =: h =: Z ; Z ; Z Z ; Z Z ; Z =: () h =9 ; h =: (: 8) The correspondng [] twst vectors ~, whch dene ech of the bove ZN or ZN Z orbfold, re shown n the the second column of tble. Note tht necessry (but not sucent) condton for the presence of the complex structure modulus U the correspondng m (; ) (; ) (; ) (; ) X h(; ) X = m= (; ) (; ) h(; ) j j m= (; ) (; ) j j (; ) (; ) (; ) (; ) (; ) 4 (; ) 6 (; ) (; ) th h (; ) =: Z7; Z8; Z; Z6 Z; Z Z; Z6Z; Z4Z4; Z6Z6 h s tht complex plne s left unrotted by t lest one of the orbfold twsts. An nlogous sttement s lso true for the T{elds: for cse () (()), the two (three) complex plnes whch re ssocted wth the four (nne) modul elds Tj 6 Z Z Z Z " # re rotted by ll of the orbfold twsts. Notce tht the bove clsscton of 8 Z X T m m m j m m

11 6 orbfold modul lso pples to the ZN Z orbfolds n the presence of dscrete torson descrbed n ref.[]. The trget spce dulty groups correspondng to the bove sx derent cses hve the followng form (up to possble permutton symmetres): (; ) () h = ; h = ; ; : = [ SL (; Z)] [ SL (; Z)] : (: 9) (; ) (; ) T () h (; ) = 5 ; h (; ) = ; : (; ) = [ SU (; ; Z) SL (; Z)] T [ SL (; Z)] U : (: ) () h = 9 ; h = : = SU (; ; Z) : (: ) (; ) (; ) Consderng only the three generc `dgonl' elds T wth Khler potentl s shown n eq.(.5), the trget spce dulty group (up to permuttons) s smply gven by the product of three modulr groups, T = [ SL (; Z)], ctng on the three modul s T b T! ( ) c T + d ; : wth ;b;c;d Z nd d bc =. The Khler potentl of the mtter elds A (whch nclude, s lredy mentoned, lso possbly exstng twsted modul) hs prtculrly smple dependence on the three generc modul T nd on the three model-dependent elds U :? h(; ) Y n Y = m= mtter K = ( T + T ) ( U + U ) A A (: ) Thus ech mtter eld A s chrcterzed by + h(; ) rtonl numbers whch we wll collect for convenence nto two vectors ~n = ( n ;n ;n ) nd ~ l = l m ( m = ;:::;h(; ) ). Invrnce of the mtter knetc energes under the trget spce h(; ) modulr group = [ SL (; Z)] T[ SL (; Z)] U requres the followng trnsformton behvour of the mtter elds (up to constnt mtrces): Y n h(; ) Y = m= m m l m A! A ( c T + d ) ( c U + d ) : (: 4) m m m l m h U m h? We omt to dscuss the dependence on the o-dgonl modul snce these elds wll not enter the threshold contrbutons dscussed n the next sectons. Alterntvely, we cn regrd the elds Tj ( = j) lso s mtter elds snce the Khler potentls (.6) nd (.7) expnded to lowest order n these elds hve the form of eq.(.4). 9

12 Therefore, n the followng, we wll cll the numbers n nd l the modulr weghts of the mtter elds. Comprng wth the formule for smooth (,) Clb{Yu spces we recognze tht the orbfold Khler potentl eq.(.) s n generl derent from eqs.(.8),(.) usng the modul Khler potentl eq.(.5). However for the specl cse ~n = ~e ( ~e re the three unt vectors) the orbfold mtter knetc energy reproduces the lrge rdus lmt of Clb{Yu spces eq.(.) upon dentcton of the three T{elds,.e. T = T = T = T. As we wll dscuss now, ths stuton s exctly true for the untwsted mtter elds. Let us now dscuss wht the modulr weght vectors ~n nd ~ l re n the cse of Abeln orbfolds. One cn mmedtely derve those from eq.(.) nd the knowledge of the correspondng metrcs of the mtter elds (the pure guge nd grvttonl elds re modulr nvrnt). The metrc of untwsted nd twsted mtter elds were computed (to rst order n the mtter elds) n ref.[]. For the untwsted mtter th untw elds ssocted to the j complex plne, A = A, one nds n = ; l = : (: 5) Sttes n the twsted sectors re creted by the twst elds ( z;z) whch re ssocted wth n order N twst vector ~ = ( ; ; ) ( <, = ) ctng nontrvlly on two or ll three complex plnes. The knowledge of the metrc of these elds gves the result for the modulr weghts n = ( ); l = ( ); = n = l = ; = : (: 6) In the twsted sector of the theory, there re lso mssless sttes, whch correspond to osclltors ctng on the twsted vcuum. Ths s the cse of the twsted modul n (,) orbfolds, but there re lso plenty of other twsted osclltor sttes both n (,) nd (,) models wthout tht geometrc nterpretton. The ssocted vertces contn excted twsted elds ( z;z) obtned upon opertor product expnsons of ground stte twst opertors wth j j ~ z = m+ z mz z = m z : mz j j j m m 6 X X e m P (: 7)

13 A generc twsted osclltor stte wll then look lke ( + ) ( ) > P ~ + ~ m p m j n n j j > ( : 8) where ~ s the shft vector whch descrbes the embeddng of the twst nto the E E lttce. In ref.[] the metrc for prtculr type of osclltor sttes (the twsted modul ssocted to the reprton of the orbfold sngulrtes n (,) models) ws computed. Exmnng those results nd usng symmetry rguments, one sees tht n th osclltor chnges the modulr weght n the drecton of the twsted ground stte by. (An explct check of ths s gven n ref.[6].) Ths s the cse rrespectve of the order m of the osclltor. Due to the commutton reltonshps [ m+ ; ] = nj jm; n( m+ ) one concludes tht the e osclltors contrbute to the modulr weght wth the opposte sgn. The modulr weghts of osclltor sttes thus re n = ( + p q ); l = ( + q p ); 6= n = l = ; = where p = p ;q = q count the number of osclltors of ech chrlty. (: 9) Sometmes one s speclly nterested n the dependence on the overll modulus T = T = T = T. The overll modulr weghts of mtter elds wth respect to ths dgonl SL (; Z) modulr group tke prtculrly smple forms. It s just gven by P n= n nd from the prevous equtons one obtns n = (untwsted) n = p + q (twsted sttes wth ll plnes rotted) (: ) n = p + q (twsted sttes wth only two plnes rotted) where p = p ;q = q. An mportnt queston s wht the possble modulr weghts of the mssless prtcles re. From the bove dscusson, t s cler tht ths depends on the number of osclltors present n our stte. Importnt constrnts on the mxml possble osclltor number cn be obtned from the mss formul for the (left-movng) twsted sttes: = P P YYYY j m nj 8 m m n n P P L = N + h + E : (: ) osc K e j qj j O e j 8 8

14 Here N s the frctonl osclltor number nd E s the ground stte energy of the twst eld osc ~ gven s X E = jv j( jv j) ; (: ) where ~v = v s slghtly redened N order twst vector wth jv j<, v =. Fnlly h s the contrbuton to the conforml dmenson of the mtter elds from the left-movng (guge) E 8 8 = prt. In the cse of level one non-abeln guge groups one cn represent ths sector n terms of shfted lttce nd one just hs h = ( P ~ + ~ ) = : ( : ) ore generlly one cn construct Abeln orbfolds wth hgher Kc{oody level lgebrs [7]. In the generl cse one cn wrte down generl expresson for h terms of the Csmrs nd levels of the relevnt guge Kc{oody lgebrs nvolved. For prmry eld n the representton R of the guge group G ssocted to Kc{oody lgebr of level k, one hs h =, where CR ( ) ( CG ( )) s the qudrtc Csmr of the R (djont) representton of G; C( R) s relted to T( R) by CR ( )dm( R) = TR ( )dm G( TR ( ) s the ndex of the representton R). In generl there wll be guge group wth severl group fctors G n (s n the phenomenologclly relevnt cse SU() SU() U()) nd ech mssless prtcle trnsformng lke ( R ; R ; ::::) wll hve the conforml dmenson K E K hk = th = X K CR ( ) CG ( )+ k CR ( ) ( 4) CG ( )+ k : : Notce tht the contrbuton of the K sector to the conforml dmenson decreses s the level ncreses. On the contrry t ncreses s the dmenson of the representton ncreses, whch cuses the bsence of mssless prtcles wth lrge dmensonlty n strng models [7,8]. In phenomenologcl pplctons, the conforml dmenson of the stndrd model prtcles wll be relevnt. For convenence of the reder we show n tble the seprte contrbuton of ech group to the stndrd model prtcles K conforml dmenson. We lso show the totl contrbuton to hk for two choces of K levels consstent wth GUT-lke guge couplng boundry condtons t the strng scle. They correspond to level one nd level two non-abeln P K

15 guge fctors. Notce tht n the presence of extr guge nterctons (such s e.g. extr U()'s) there would be ddtonl contrbutons to h nd hence the results n the tble gve lower bound to the K conforml dmenson of ech stndrd model prtcle. For mssless stte wth number s constrned by Snce for gven orbfold the vlue of E s, the bgger N osc = t now follows tht the mxmum osclltor N ( E h ): (: 5) osc s xed, t s cler tht the smller h wll be nd hence the rnge of llowed modulr weghts. In fct, n the K level one cse, the smllest possble contrbuton to hk s obtned n orbfolds for ~P = n whch cse h = ( ~ ) =. In ny gven Abeln orbfold one cn check tht the shortest (world-sheet) modulr nvrnt shft possble hs length ( ~ ) = ( ~v ). Thus rrespectve of the guge group one nds Nosc ( jv j) : (: 6) In twsted sector of order N the lrgest possble (negtve) vlue for the modulr weght wll be obtned when there re p osclltors wth the lowest possble =N frctonl moddng. Ths number p of osclltors wll thus be bounded by p N ( jv j) : (: 7) From ths equton one cn obtn the mxmum number p of osclltors present n ny possble twsted sector of Abeln orbfolds. Ths s shown n the second column of tble. The mxmum possble number q of osclltors s more constrned. Such osclltors only pper n complex plnes n whch =. Only n ths cse ther contrbuton to the mss formul my llow for mssless prtcles. The thrd column of tble shows qmx for the derent twsted sectors. For the Z orbfold sectors no such type of osclltors re present n the mssless sector. For the rest of the twsts wth three rotted plnes, one osclltor my be present long one prtculr complex plne n ech cse. Ech of these osclltors contrbute to the mss formul whch s lwys bgger thn =N. Ths s why only one such osclltor mght K e mx X X K e K mx K

16 be present for such type of twsts. For twsted sectors wth one unrotted plne (lst four cses n the tble), e osclltors re generclly present n one of the rotted plnes. Of course the bove rguments only gurntee tht the relevnt stte cn be mssless, but ths does not men tht t s present n the mssless spectrum snce t hs lso to survve the generlzed GSO projecton of the orbfold symmetres. From the second nd thrd columns of tble nd eq.(.9) one cn obtn the llowed rnge for the modulr weghts n ech of the three complex dmensons. In the cse of the overll modulus T, usng eq.(.), t s esy to nd the llowed vlues of the modulr weghts. Ths s shown n the fourth column of tble. The extreme vlues P for n ssume the mnmum vlue for hk = jv j = correspondng to level one non- Abeln fctors. Notce tht ech prtculr ZN or ZN Z orbfold s constructed by jonng n (world-sheet) modulr nvrnt wy ll the twsts generted by the vectors n the rst column of tble. We thus observe tht the modulr weghts wth respect to the dgonl modulus vry n n bsolute rnge 9 n 4 for Abeln ZN nd ZN Z orbfolds. If the vlue of the Kc{oody conforml dmenson of the relevnt mssless eld s lrger thn the mnml vlue dscussed bove, the rnge of possble modulr weghts s substntlly decresed. Consder n prtculr the phenomenologclly nterestng cse of possble mssless sttes descrbng the stndrd model elds. For the stndrd choce of the K levels, = 5 k = k = k =, we know we hve h K = = 5 for Q;U;E-type sttes nd h K = = 5 for L;D;H;H-type sttes (see tble ). Pluggng those vlues n eq.(.5), one obtns the much more restrcted vlues for the dgonl modulus modulr weghts shown n columns 5 nd 6 of tble. We thus see tht n Abeln orbfolds the rnge of overll modulr weghts s gven by nq;u;e nd 5 n L;D;H;H (for = 5 k = k = k = ). All these constrnts on the modulr weghts wll ply n mportnt role n the dscussons n the followng sectons.. Cncellton of Trget Spce odulr Anomles.. The -model nd dulty nomles n generl 4-D strngs In ths secton we wll dscuss some loop eects n four-dmensonl strng theores. As s well known, guge nd grvttonl nomles destroy the quntum 4

17 consstency of \ordnry" eld theores. The requrement of the bsence of these nomles puts severe constrnts on the form of the mssless fermonc spectrum of the theory. Consderng four-dmensonl strng theores, guge nd grvttonl nomles must lso be bsent, ledng to the sme constrnts on the mssless prtcle spectrum s compred to eld theory. They come s n utomtc consequence of world-sheet modulr nvrnce [9]. An mportnt excepton of ths sttement s gven by n nomlous U() guge group. In fct, the nomlous trngle dgrm wth externl U() guge bosons cn rse n strng theory. The quntum consstency of the theory s stll preserved snce the U() nomly cn be cncelled by the so-clled Green{Schwrz mechnsm [8]. Ths mechnsm leds to mxng [4] between the U() guge boson nd the xon (dlton) eld, such tht the nomlous vrton of the eld theory trngle dgrm s cncelled by proper chnge of the dlton eld under U() trnsformtons. Aprt from these guge nomles, supersymmetrc non-lner -models exhbt nother type of nomles, nmely the so-clled -model nomles [4]. These nvolve, smlr to the locl guge nomles, trngle dgrms where some of the externl elds re gven by composte -model connectons, whch do not correspond to dynmcl, propgtng degrees of freedom. In the strng cse, the contnuous -model symmetres re generclly broken by explct couplngs mong the mssless elds, e.g. by the tree-level Yukw couplngs or by one-loop eects. Ths explct brekdown orgntes from the nuence of the hevy strng spectrum, whch s not nvrnt under contnuous chnges of the modul. Therefore possble -model nomles would not constrn the eld theory spectrum further. However, n strng theory, the mssve spectrum,.e. momentum together wth `wndng' sttes, s nvrnt under the dscrete dulty trnsformtons. In fct, the -model nomles due to the mssless fermons lso nduce non-nvrnce of the correspondng one-loop eectve cton under trget spce dulty trnsformtons. These \strngy" dulty nomles must be cncelled snce one knows tht the dulty symmetres re preserved n ny order of strng perturbton theory. Ths s n shrp contrst wth norml eld theory. As we wll dscuss n the followng, for brod clss of strng compctctons the requrement of the bsence of the trget spce dulty nomles provdes new constrnts on the mssless strng spectrum whch were not dscussed before. In prtculr, we wll show tht for some orbfold compctctons the mnml supersymmetrc stndrd 5

18 model spectrum would led to trget spce dulty nomles nd hs therefore to be dscrded. Specclly, consder the supersymmetrc non-lner -model of the modul wth trget spce coupled to guge nd mtter elds s descrbed n the lst secton. At the one-loop level one encounters two types of trngle dgrms wth two guge bosons of the guge group G = G nd severl modul elds s externl legs nd mssless gugnos nd chrged (fermonc) mtter elds crcultng nsde the loop: rst the couplng of the modul to the chrged elds contns prt descrbed by composte Khler connecton, proportonl to K; ( ), whch couples to gugnos s well s to chrl mtter elds A. (We ssume here tht the modul re guge snglets.) It reects the (tree-level) nvrnce of the theory under Khler trnsformtons. Second, there s couplng between the modul nd the A 's by the composte curvture (holonomy) connecton. It orgntes from the non-cnoncl knetc energy K mtter R Q R, eq.(.4), of the mtter elds A. These two nomlous contrbutons led, together wth the tree-level prt whch s gven by the dlton/xon eld S, to the followng (non-locl) one-loop eectve supersymmetrc Lgrngn [4,5,6]:? R Lnl = d W W S DDDD X Z ( CG ( ) X Ã X ) R R mtter T( R ) K( ; ) + T( R ) log det K ( ; ) + h: c: (: ) Here W s the Yng{lls supereld. Wrtng ths expresson n components t leds to non-locl contrbuton to the CP odd term F F nd to locl contrbuton to the guge couplng constnt. Now, t s esy to recognze tht, eq.(.), s not nvrnt under Khler trnsformton nd reprmetrztons whch ct non- mtter trvlly on K; ( ), K respectvely. These non-nvrnces correspond to the two types of -model nomles, nmely to the Khler nd curvture (holonomy) nomles. It follows tht L nl s not nvrnt under the dscrete trget spce dulty trnsformtons (.) snce these ct exctly s combned Khler trnsformtons e L nl Anlogously, there s lso mxed grvttonl, -model nomly wth two grvtons nd one composte Khler/curvture connecton s externl legs (see secton.5).? 6

19 nd reprmetrztons. Usng eqs.(.) nd (.5), the chnge of trnsformtons s gven by the locl expresson ( ) = 6 d W W CG ( ) TR ( ) g ( ) 4 Lnl R + T( R ) log det h ( ) + h: c: R under dulty (: ) It follows tht the dulty nomles must be cncelled by ddng new terms to the eectve cton. Specclly, there re two wys of cncellng these nomles. In the rst one [4],[6] the S eld my trnsform non-trvlly under dulty trnsformtons nd cncels n ths wy some prt or ll of the dulty non-nvrnce of eq.(.). Ths non-trvl trnsformton behvour of the S{eld s completely nlogous to the Green{Schwrz mechnsm for the cse of n nomlous U() guge group nd leds to mxng between the modul nd the S{eld n the S{eld Khler potentl. Note tht the Green{Schwrz mechnsm does not only cncel the nomles ssocted to the dscrete dulty trnsformtons, but n fct ll contnuous -model nomles. Second, the dulty nomly cn be cncelled by ddng to eq.(.) term, whch s relted to the guge group dependent threshold contrbuton due to the mssve strng sttes. The threshold contrbutons re gven n terms of utomorphc functons of the trget spce dulty group, whch hve the requred trnsformton behvour only under the dscrete dulty trnsformtons but not under ll contnuous reprmetrztons nd Khler trnsformtons. The explct brekng of the contnuous -model symmetres by the one-loop threshold eects follows from the fct tht the hevy mss spectrum s not nvrnt under contnuous reprmetrztons of the modul, but only under the dscrete dulty trnsformtons. It s cler tht the prt of the -model nomly whch s removed by the Green{ Schwrz mechnsm s unversl,.e. guge group ndependent. (Otherwse one would need for ech guge group fctor n extr S{eld whch s not present n strng theory.) Thus for cses where there re no modul dependent threshold contrbutons from the mssve sttes the -model nomly coecents hve to concde for ech guge group fctor G. As we wll dscuss, ths prtculrly nterestng constrnt rses for mny X X Z X Ã orbfold compctctons where the modul dependent threshold contrbutons re bsent becuse of n enlrged N = 4 supersymmetry n the mssve spectrum. 7 Lnl

20 Before we come to the orbfolds, let us brey evlute the one-loop eectve cton eq.(.) for the cse of (,) Clb{Yu compctctons wth guge group G = E E nd wth h mtter elds n the 7 representton of E nd wth h (; ) 8 6 (; ) 6 mtter elds n the 7 representton. Usng the specl relton between the Khler metrcs of the modul nd mtter elds s gven n eq.(.8) one derves [4,4] X Z ( d W W S Lnl = DDDD CG ( ) T( R)( 5 h + (; ) h(; ) ) K( T ;T) à + CG ( ) ( )( 5 + TR h(; ) h(; ) ) KU ( m;um) à ) + T( R) log det K ( T ;T ) + log det K ( U ;U ) + h: c:; j mn m m à (: ) wth = E;E nd CE ( ) =, CE ( ) =, T(7) = T(7) =. Lookng for exmple t the Khler clss modul T one observes tht one needs n prncple two types of utomorphc functons: the rst one provdes dulty nvrnt completon of KT;T ( ), where the second one s needed to cncel the dulty nomly comng from log det K ( T ;T ). These two types of utomorphc functons cn be, t lest j formlly, constructed for ll (,) Clb{Yu compctctons [4]. It s nterestng to compute eq.(.) for the lrge rdus lmt of Clb{Yu compctctons s functon of the overll T{eld. Wth eq.(.7) one obtns d W W S Lnl = DDDD b log( T + T ) + h: c : (: 4) We see tht n the lrge rdus lmt the -model nomly s smply determned by the N = -functon coecent b = C( G ) + ( h + h ) T( R). X Z (; ) (; ).. Trget spce modulr nomles n Abeln orbfolds Let us now return to the (,) nd (,) orbfold compctctons. We wll dscuss the -model nomles s functon of the dgonl modul T (= ; ; ) nd U ( m = ;:::;h ). Recll tht the Khler metrcs of the mtter elds, eq.(.), m (; ) re determned n terms of the modulr weghts n nd l. The modul Khler 8 m

21 potentl s gven n eq.(.5). Then the tree-level Yng{lls cton, together wth one-loop nomlous contrbuton of the mssless gugnos nd mtter fermons, tkes the form X Z Lnl = d W W S DDDD h(; ) = m= X X m m m b log( T + T ) + b log( U + U ) + h: c: (: 5) The nomly coecents b nd b look lke [5],[4] m b = C( G ) + T( R )( + n ); m R b = C( G ) + T( R )( + l ): X X R R m R (: 6) Consderng only the dependence on the overll modulus T = T = T = T the Yng{lls Lgrngn becomes wth X Z Lnl = d W W S DDDDb log( T + T ) + h: c : (: 7) b = C( G ) + T( R )( + n ) = b + T( R )( + n ); (: 8) where n s the overll modulr weght: n= n. We recognze tht the gugnos nd mtter elds wth n = contrbute to the nomly coecent b n the sme wy s to the N = -functon coecent b. Snce ll untwsted mtter elds hve n =, the whole untwsted sector of orbfolds behves lke the lrge rdus lmt of Clb{Yu compctctons (cf. eq.(.4)). The devton from ths behvour s cused by the twsted orbfold elds, wheres world-sheet nstntons modfy the lrge rdus lmt for the cse of Clb{Yu compctctons. As lredy mentoned, the trget spce modulr nomly,.e. the non-nvrnce of the guge couplng constnt eq.(.5) under T R X R X R P 9 = R! R Tb c T + d, cn be cncelled n two

22 6 derent wys. Frst, the unversl, guge group ndependent prt of the nomly s removed by the Green{Schwrz mechnsm, whch nduces the followng modulr trnsformton behvour of the dlton eld [4]: k S! S 8 X = log( c T + d ): (: 9) For modulr trnsformtons on the Umelds, n nlogous trnsformton holds. Here the GS s the guge group ndependent \Green{Schwrz" coecent whch descrbes the one-loop mxng between the S{eld nd the modulus T. Second, the trget spce modulr nomly cn be lterntvely cncelled by locl contrbuton to Lnl, whch s relted to the one-loop threshold contrbutons to the guge couplng constnts. Ths topc wll be dscussed n secton 4. Let us here consder the nterestng cse provded by those orbfold compctctons for whch modulus T does not pper n the threshold correctons. Ths exctly hppens [6] f ll orbfold twsts ~, whch dene prtculr Z or Z ZN orbfold, ct th non-trvlly on the correspondng complex plne of the underlyng sx-torus,.e. = for ll ~. Then the contrbuton from the momentum nd wndng sttes wth T dependent msses to the threshold correctons s bsent, snce ths prt of the mssve spectrum s orgnzed nto N = 4 supermultplets. Furthermore, the mssve spectrum n the twsted sectors does not depend on the untwsted modul. Thus n ths cse the trget spce modulr nomly ssocted to T hs to be completely cncelled by the Green{Schwrz mechnsm. Ths mples tht GS = b =k for ll guge group fctors G. Snce GS s guge group ndependent, t mmedtely follows tht the nomly coecents (.6) must tke the sme vlues for ll group fctors of Q the guge group G= G : b b b b c = = = ::: (: ) k kb kc Ths requrement of the complete cncellton of the -model nomles by the Green{ Schwrz mechnsm wll provde very strong condtons on the possble spectrum of guge non-snglet sttes, specclly on the llowed representtons R n connecton wth the possble modulr weghts n R s we wll dscuss now n the followng. In the next secton we wll return to the threshold eects nd ther mplctons on strng uncton scenros. GS

23 6 Let us rst gve lst [,6] of those orbfold compctctons for whch there s complete -model nomly cncellton wthout ddtonl threshold eects (see lso the twst vectors ~ n the second column of tble ). Frst for the Z nd Z orbfolds, ll three complex plnes re rotted by ll orbfold twsts. Therefore there s complete modulr nomly cncellton wth respect to T, T nd T, nd the b coecents hve to concde for ll three plnes = ; ;. The Z4, Z6, Z8, Z8, Z nd Z orbfolds hve two completely rotted plnes, such tht the b coecents hve to concde for = ;. Fnlly, for the Z orbfold, the modul T, T nd U pper n the threshold correctons, nd only the modulr nomly ssocted to T s completely cncelled by the Green{Schwrz mechnsm. On the other hnd, for ll Z Z orbfolds ech complex plne s left unrotted by t lest one orbfold N twst []. Therefore one does not nd the complete modulr nomly cncellton for ny of these compctctons. It s nterestng to note tht f there exsts complex structure modulus U ( h = ), the threshold correctons lwys depend m (; ) on ths modulus. Therefore the nomly coecents b re never constrned by the requrement of complete modulr nomly cncellton. The reverse stuton s true for possble complex plnes wth o-dgonl elds T. These elds never pper n the threshold correctons, snce the correspondng complex plnes re lwys rotted by ll orbfold twsts. It s nstructve to check how the modulr nomly cncellton condtons for completely rotted plnes re stsed for lredy known orbfold constructons. Ths check nvolves the determnton of the modulr weghts of the chrged elds s descrbed n secton. For exmple t s esy to show tht for the symmetrc (,) Z orbfold wth stndrd embeddng of the twst nto E, the nomly coecents gree for ll guge group fctors: b = b = b =, = ; ;. The modulr nomles wth respect to the guge groups E nd E of ddtonl (,) orbfolds were consdered n ref.[4]. In the ppendx A we lso dscuss the cse of the (,) Z orbfold n some detl. 7 E 8 6 It s mportnt to stress tht the modulr nomly cncellton hs to work not only for the few (,) cses but lso for the numerous (,) constructons wth nonstndrd embeddngs nd/or ddtonl bckground elds such s dscrete Wlson lnes. In generl, these orbfolds yeld guge groups derent from E6 E8. As non-trvl (,) exmple we explctly dscuss n the ppendx B Z orbfold wth E 7 6 j m E 8 8 SU() 8 6 4

24 guge group G = E SU() U() E SU() [44]. As requred from our generl rguments on modulr nomly cncellton, we nd for the nomly coecents of the second nd thrd plne: b E = b = = = 6, =. We lso 6 SU() b E b 7 SU() ; checked for (,) Z orbfolds wth dscrete Wlson lnes tht the nomly coecents of ll guge group fctors concde for ech of the three complex plnes. Insted of lookng t lredy explctly constructed orbfold compctctons we now wnt to demonstrte tht the requrement of hvng no trget spce modulr nomles leds to very strong, nd so fr undscussed, constrnts on the possble spectrum of mssless elds for orbfolds wth completely rotted plnes. Specclly, we wnt to nvestgte the mportnt queston whether n orbfold wth the stndrd model guge group G = SU() SU() U(), plus exctly the mssless prtcle content of the mnml supersymmetrc stndrd model, cn be free of trget spce modulr nomles nd cn thus descrbe consstent compctcton scheme. The mssless spectrum of the hypothetcl mnml orbfold s chrcterzed by the modulr weghts n of the stndrd model chrl elds where = Q;U;D;L;E;H;H ( g = ; ; ) n n obvous notton. Then the nomly coecents of the mnml model tke the followng form: 6 7 X b =+ ( n + n + n ); g= X b =5+ ( n + n )+ n + n ; g= X =+ ( b nq n n + n + n )+ n + n : g Ug Dg Lg Eg H H g= (: ) The condton of vnshng trget spce modulr nomles of the mnml orbfold compctctons now reds b =k = b =k = b =k, where corresponds to those complex plnes tht re rotted by ll orbfold twsts. For the cse of g g g g g Q Q U L D g g g g g H U(), our normlzton of the U() chrges used n b s chosen n such wy tht k = 5= corresponds to the stndrd grnd uncton vlue of the Kc{oody level. The bove nomly cncellton condton turns nto two ndependent equtons, whch the modulr weghts of the stndrd model elds hve to stsfy. For our purposes, H Y Y

25 t wll be convenent to dene the followng two lner combntons whch hve to vnsh: A k + = = = + k b b ; B b b k k k b = : (: ) Whether these two equtons hve ny solutons cruclly depends on the dstrbuton of the llowed modulr weghts of the stndrd model elds, whch were clssed n secton. Of course, smlr constrnts my be obtned for other extended guge groups nd prtcle contents... The cse of generl (,) orbfolds. The strongest constrnts rse for the Z orbfolds where eq.(.) must be stsed wth respect to ll three complex plnes, nd where the choce of possble vlues for the modulr weghts s very lmted. Indeed there exst [,4,] severl Z exmples wth stndrd model guge group nd three chrl fmles, but ll of these possess n ddton some extr (non-chrl) mtter elds. Thus, cn there exst nomly free Z orbfolds wth just the mnml prtcle content? In the prevous secton we gve n extensve descrpton of how one determnes the llowed modulr weghts for mssless prtcles. Let us gve the result for the stndrd model prtcles for the Z orbfold cse. The rst possblty for llowed modulr weghts s gven by ~n = ( ; ; ) ;(; ; ) ;(; ; ). These three choces correspond to untwsted chrl elds. The twst vector ~ of the Z orbfold hs the form ~ = ( = ; = ; = ) ( ~v = (= ; = ; = )). The conforml dmensons s functon of the Kc{oody levels re gven n tble for the stndrd model prtcles. Then, usng eq.(.5), we obtn tht for k = = 5 k = k = k = there re no osclltors llowed for the stndrd model prtcles. For k = the elds Q nd U stll must not hve ny osclltor, wheres the other elds my possess one twsted osclltor. Fnlly, for k = ll elds except Q my possess one osclltor, nd for k > we obtn p = for ll stndrd model prtcles. Let us rst nvestgte the most common cse of level one Kc{oody lgebrs, k = 5=, k = k =. Then, prt from the \untwsted" modulr weghts, the g only ddtonl \twsted" possblty for the stndrd model elds s to hve ~n = Z (= ; = ; = ). (The correspondng overll modulr weght n= s dsplyed n tble.) Thus, n totl, the rnge of llowed modulr weghts conssts of four g g mx

26 possbltes. Wth the help of computer progrm one cn now check whether the two equtons (.), together wth eq.(.), hve ny smultneous solutons usng the four derent llowed vlues for the modulr weght vectors of ech of the stndrd model prtcles. In dong ths we nd tht there re no solutons t ll. It s nterestng to note tht lso the sngle equton b b = cnnot be stsed usng the llowed modulr weghts. In ths wy the uncertnty of the U() normlzton fctor k s elmnted. Thus we hve obtned the strkng result tht for the Z orbfold wth level one guge groups the trget spce modulr nomles cnnot be cncelled by the stndrd model prtcles. In ths wy we hve ruled out the mnml level one Z compctctons by generl consstency rguments. The requrement of trget spce nomly freedom forces us to ntroduce ddtonl elds. At hgher levels the rnge of llowed modulr weghts s broder. Specclly, f there cn be one twsted osclltor one obtns the followng ddtonl llowed modulr weght vectors: ~n = (5= ; = ; = ) ; (= ; 5= ; = ) ; (= ; = ; 5= ). In ths cse we nd tht for k the nomly cncellton condton eq.(.) cn be stsed n vrous wys. Therefore we cnnot prove mnml Z models wth hgher level Kc{oody lgebrs to be nconsstent. Orbfold models wth guge groups relzed t hgher Kc{oody level were constructed n ref.[7] (n exmple s brey dscussed n secton.6). It s however complcted to explctly construct models wth hgher level stndrd model guge group. In ddton we wll show n the next secton on threshold correctons, tht the mnml Z compctctons cnnot meet the phenomenologcl requrement of proper couplng constnt uncton, nd re therefore ruled out on physcl grounds. Let us nvestgte more wht the spectrum of ddtonl mssless sttes hs to look lke n order to stsfy the nomly mtchng condtons. Generl clsses of (,) Z orbfold models wth guge group SU() SU() U() n G nd three qurk hdden lepton genertons plus extr vector-lke mtter were constructed n refs.[,]. Ths ws done by ddng two quntzed Wlson lnes on the four derent bsc embeddngs of the Z orbfold. For exmple, n ref.[] three generton model wth guge group 8 G= SU() SU() U() SO() ws constructed. It s strghtforwrd hdden to show tht the condtons of modulr nomly cncellton re stsed n ths model. Specclly, the untwsted sttes wth non-trvl SU() SU() SO() quntum numbers re gven by [(; ; ) + (; ; ) + (; ; ) + (; ; 6)], wheres the 4

27 twsted sttes re of the form [5(; ; ) + 4(; ; ) + (; ; )]. Then one obtns b = b = b = 8. SU() SU() SO() ore generlly, t ws shown tht one cn construct, just wthn the Z orbfold, of order 5 such three generton models but only nne derent clsses re nequvlent [45]. Furthermore, these nne clsses dvde nto two types, ccordng to the lepton nd qurk content of the untwsted sector. Fve of the clsses hve n tht sector three copes of prtcles wth quntum numbers (; ) + (; ) (: ) wheres the other four models hve three copes of elds trnsformng lke (; ) + (; ) + (; ) (: 4) under SU() SU(). odels belongng two these two clsses lwys hd some specc propertes. In prtculr, rrespectve of the structure of the model nd of whch other guge fctors re present, the number of SU() doublets exceeds by the number of SU() trplets. Ths ment n prctce the mpossblty of gettng the mnml Hggs content snce n the mnml model the number of SU() doublets exceeds the number of SU() trplets only n the number of Hggs prs. Dulty nomly cncellton rguments llow us to understnd the orgn of these peculr propertes of these three generton models. Consder modulr nvrnce wth respect to the overll modulus T = T = T = T. In the type of models of refs.[,4,], the twsted osclltor sttes re lwys SU() SU() snglets nd hence the possble modulr weghts of qurks, lepton-doublets nd Hggses re n = (untwsted) nd n = (twsted). Cncellton of modulr nomles requres = 9 6 ( ) + U U T T b b N N ( N N ) = ; (: 5) T U where N nd N re the numbers of trplets n the twsted nd untwsted sectors respectvely (nd equvlently for the doublets). In the models wth untwsted content U U s n eq.(.) one hs N = 6 nd N = 9 wheres n the second type one hs N U U = 9 nd N =. Thus from eq.(.5) one gets n both cses T T N N = 9; 5

28 nd there re ltogether 9 + = more doublets thn trplets n ll models. Ths gves n understndng of ths pttern prevously found n model-by-model bss. Smlr rguments should be pplcble to other Z three-generton models obtned through the ddton of three Wlson lnes nsted of two (for exmples see ref.[46]). The determnton of the complete mssless spectrum of these models s extremely pnful nd usng dulty nomly cncellton rguments one cn obtn mmedtely nformton bout the twsted spectrum by smply knowng the untwsted sector (whch s trvl to obtn) nd usng equtons nlogous to eq.(.5). Z.4. nd other orbfolds 7 Now let us dscuss the 7 compctctons [,47] whch provde the second clss of orbfolds wth three completely rotted plnes. Agn we wll nd tht the trget spce modulr nomles cnnot be cncelled, ssumng mnml prtcle content together wth the stndrd model guge group SU() SU() U(). The dstrbuton of llowed modulr weght vectors for the stndrd model elds s s follows. For untwsted elds there re gn the three possbltes ~n = ( ; ; ) + perm. The Z twst vector s gven s ~ = (= 7; = 7; 4= 7). Then, for sttes wthout 7 osclltors, one obtns n the twsted sectors three derent llowed modulr weght vectors: ~n = (6= 7; 5= 7; = 7) + perm. For smplcty, we restrct the dscusson from now on to the level one cse = 5 k = k = k =. Ths mples (see tble ) tht the elds Q;U;E g g g re not llowed to hve ny twsted osclltor contrbuton. Thus, the rnge of llowed modulr weghts conssts of the sx bove mentoned choces. On the other hnd, lookng t the rst twsted sector, the elds D;L;H;H mxmlly hve two osclltors ssocted to the rst complex plne or one osclltor n the drecton of the second plne. For the remnng twsted sectors n nlogous sttement s true. Thus we obtn nne ddtonl llowed modulr weght vectors for these type of elds: ~n = (= 7; 5= 7; = 7) ; (= 7; 5= 7; = 7) ; (6= 7; = 7; = 7) + perm. Wth these 5 llowed choces for D;L;H;H g g nd the 6 possbltes for Q;U;E eq.(.) hs not sngle soluton. Ths gn rules out ny mnml Z g g g Z compctcton t the lowest Kc{oody level. However, t hgher level we expect to nd vrety of consstent solutons. 6 g g Y my 7