A Fuzzy Three Sphere and Fuzzy Tori

DIAS-STP-03-5 A Fuzzy Th ee Sphe e and Fuzzy To i B ian P. Dolana),b)∗and Denjoe O’Conno b)† a)Dep . o Ma hema ical Physics, NUI, Maynoo h, I eland b)School o Theo e ical Physics, Dublin Ins i u e o Ad anced S udies, 10 Bu ling on Rd., Dublin 8, I eland Sep embe 23, 2005 Abs ac A uzzy ci cle and a uzzy 3-sphe e a e cons uc ed as subspaces o uzzy complex p ojec i e spaces, o complex dimension one and h ee, by modi ying he Laplacians on he la e so as o gi e un- wan ed s a es la ge eigen alues. This lea es only s a es co esponding o uzzy sphe es in he low ene gy spec um ( his allows he commu a- i e algeb a o unc ions on he con inuous sphe e o be app oxima ed o any equi ed deg ee o accu acy). The cons uc ion o a uzzy ci cle opens he way o uzzy o i o any dimension, hus ci cum en ing he p oblem o powe law co ec ions in possible nume ical simula ions on hese spaces. 1 In oduc ion One o he p incipal goals o he s udy o ield heo ies on uzzy spaces is o de elop an al e na i e non-pe u ba i e echnique o he amilia la ice one [1]. To da e, his new app oach in he case o ou dimensional ield heo ies has been limi ed o s udies o Euclidean ield heo y on S2×S2 [2], CP2[3] and S4[4]. All bu S2×S2ha e addi ional complica ions. Fo ∗[email p o ec ed] †[email p o ec ed] 1 example, CP2is no spin bu spincand S4is eally a squashed CP3and includes many unwan ed massi e Kaluza-Klein ype modes. E en S2×S2 is no ideal since i has cu a u e e ec s ha d op o as powe co ec ions a he han exponen ially as in he case o o oidal geome ies. The uzzy app oach does, howe e , ha e he ad an age o p ese ing con- inuous symme ies such as he SU(2) symme y o a ound S2and does no su e om e mion doubling [5]. The ad an ages a e gained a he cos o in- oducing a non-locali y associa ed wi h he non-commu a i i y o he uzzy sphe e. The e is he e o e a balance o ad an ages and disad an ages asso- cia ed wi h he uzzy app oach. The inal decision on whe he he app oach has eal ad an ages o e he s anda d la ice app oach should be de e mined by doing genuine simula ions. Fo his eason Mon e Ca lo simula ions o he uzzy app oach a e now in p og ess. In he la ice app oach non-locali y is also a p oblem when e mions a e included. So ou expec a ion is ha as a as Mon e Ca lo simula ions a e conce ned he uzzy app oach will no be compe i i e wi h he la ice one un il e mions a e included. The app oach will gain u he ad an ages in si ua ions whe e symme ies a e mo e impo - an . I also ex ends na u ally o allow o supe symme y. (see [6] whe e a uzzy supe sphe e was cons uc ed). So we expec he ue powe o he app oach o eme ge when supe symme y and chi al symme y a e p esen in a model. A adically di e en al e na i e o he Euclidean Mon e Ca lo app oach becomes a ailable once one has a uzzy h ee-dimensional space. Such a space has he ad an age ha i allows one o de elop e y di e en non- pe u ba i e me hods, since now one can add ess he non-pe u ba i e ques- ions om a Hamil onian poin o iew. The pu pose o his a icle is o in oduce p ecisely such uzzy h ee- dimensional spaces. We will begin by p esen ing a uzzy e sion o he ci cle S1 F, om which one can ob ain o i o a bi a y dimension. We will hen p esen a uzzy app oxima ion o he h ee-sphe e, S3 F. Un o una ely, bo h o hese spaces a e s ill no ideal in ha hey in ol e many unwan ed addi- ional deg ees o eedom which we supp ess so ha hey do no con ibu e o he low ene gy physics. The p esence o addi ional deg ees o eedom is p obably una oidable as i seems o be he p ice one pays o he classical space no being a phase space. The h ee-sphe e is also cu ed and hence he esul s ob ained om s udies o ield heo ies on his space will app oach hose o a la h ee-dimensional space wi h polynomial co ec ions. I has, howe e , he ad an ages o p ese ing he ull SO(4) symme y o a ound S3. F om he cons uc ion i seems clea ha bo h o hese spaces will also be ee o e mion doubling p oblems. We will es ic ou ocus he e o scala ield heo ies and demons a e how 2 he unwan ed deg ees o eedom can be supp essed so ha he limi ing la ge ma ix heo y o a scala ield heo y eco e s ield heo y on he commu a i e spaces. We will a gue ha he da a speci ying he geome ies can be cleanly speci ied by gi ing a sui able Laplace- ype ope a o o he scala ield, which oge he wi h he ma ix algeb a and i s Hilbe space s uc u e gi es a spec al iple. Aside om ou pe sonal mo i a ions, non-commu a i e geome y has e- cen ly become a e y popula a ea o esea ch om bo h he poin o iew o possible new physics in s ing heo y and D-b ane heo y, [7, 8], and as a new egula isa ion echnique in o dina y quan um ield heo y, [2]-[4] and [9]-[12]. In bo h hese endea ou s “ uzzy” spaces play an impo an ˆole. Roughly speaking a uzzy space is a ini e ma ix app oxima ion o he algeb a o unc ions on a con inuous mani old, he seminal example being he uzzy wo-sphe e, [13]. I has he impo an p ope y o p ese ing he isome ies o he space ha i is app oxima ing. As such he idea can se e as a sou ce o examples ela ed o ma ix models in s ing heo y and as a egula isa ion echnique o o dina y quan um ield heo y. As a egula isa ion me hod i p o ides one ha p ese es he unde lying space- ime symme ies and is amenable o nume ical compu a ion. Fuzzy sphe es in dimensions o he han wo we e analysed in [14]-[17], bu he cons uc ion he e was incomple e. They also ad oca e p ojec ing ou he unwan ed modes and wo king wi h a non-associa i e algeb a which we conside unsa is ac o y. Also he case o odd sphe es wo ks e y di e en ly o ha o e en sphe es. An al e na i e app oach o he uzzy ou -sphe e, S4 F, was gi en in [4], based on he ac ha uzzy CP3and CP1∼ =S2a e well unde s ood [18], and, in he con inuum limi , CP3is an S2bundle o e S4. In his pape we show how he odd-dimensional uzzy sphe es S1 Fand S3 F can be ex ac ed om he ma ix algeb as associa ed wi h he uzzy complex p ojec i e spaces CP1 Fand CP3 F. An al e na i e app oach o ob aining a ini e app oxima ion o S3∼ =SU(2), based on con o mal ield heo y, was p esen ed in [19], howe e , in his app oach i is unclea how he unwan ed modes a e o be supp essed. Ou me hod uses a simila supp ession mecha- nism o ha used o S4 Fin [4]. Al hough he e is no closed ini e dimensional ma ix algeb a o SN Funless N= 2, he ele an deg ees o eedom when N= 1 and N= 3 a e con ained in he ma ix algeb as o CP1and CP3 espec i ely. One can he e o e ob ain unc ional in eg als o ield heo ies o e S1 Fand S3 Fby s a ing wi h unc ional in eg als o e CP1 Fand CP3 F and hen supp essing he unwan ed modes so ha hey do no con ibu e o he unc ional in eg al. Because o he high deg ee o symme y inhe en in he cons uc ion, he unwan ed modes can be supp essed simply by using app op ia e quad a ic Casimi s in he Laplacian. In his way we by-pass 3 he p oblems associa ed wi h he ac ha he algeb a o ma ices associa ed wi h unc ions on he sphe e does no close on he sphe e, bu necessa ily li s in o he en eloping complex p ojec i e space. In a simila ashion we expec ha when a Hamil onian app oach o ield heo y is de eloped using hese spaces he unwan ed modes will cause no di icul ies since hey can be made a bi a ily di icul o exci e. The pape is o ganized as ollows. In sec ion 2 we summa ize how a gi en geome y is cap u ed in he uzzy app oach. Sec ion 3 hen gi es ou cons uc ion o a uzzy ci cle, S1 F. Sec ion 4 summa ises he cons uc ion o S4 Fp esen ed in [4] and in sec ion 5 we p esen ou uzzy h ee-sphe e, S3 F. Sec ion 6 gi es an al e na i e cons uc ion o S3 Fwhich lends i sel o a gene alisa ion o SN F o any N[20]. 2 Encoding he geome y o a uzzy space F ¨ohlich and Gaw¸edzki [19] ( ollowing Connes, [21]) ha e demons a ed ha he abs ac iple (H, A,∆γ), whe e His he Hilbe space o squa e in e- g able unc ions on he mani old M, wi h Laplace-Bel ami ope a o ∆γ,γ being he me ic, and A=C∞(M) is he algeb a o smoo h bounded unc- ions on M, cap u es a opological space oge he wi h i s me ical geome y. In a simila ashion one can speci y a uzzy space, MF, as he sequence o iples MF:= (HL,Ma dL,∆L) (1) pa ame e ized by L, whe e HL=Cd2 Lis he Hilbe space ac ed o he com- ple e ma ix algeb a Ma dLo dimension d2 Lwi h inne p oduc < M, N >= 1 dLT (M†N) and ∆Lis a sui able Laplacian ac ing on ma ices. One can eadily ex ac in o ma ion such as he dimension o he space om hese da a. The Laplacian comes wi h a cu o and so he dimension can be ead om he g ow h o he numbe o eigen alues. The da a con ained in he iple (H, A,∆γ) a e p ecisely he da a ha go in o he Euclidean ac ion o a scala ield heo y on he space Mand hence speci ying he scala ac ion is a con enien me hod o p esc ibing hese da a. In he uzzy app oach he algeb a will always be a ma ix algeb a and we will e ain he Hilbe space inne p oduc speci ied abo e so he only da a om he iple, (HL,Ma dL,∆L), emaining o be supplied a e he pe mi ed ma ix dimensions, dLand a ealiza ion o he Laplacian, ∆L. Once his in o ma ion is gi en he uzzy geome y is speci ied. Though i may be con enien o gi e a map o unc ions his is no neces- sa y. Once he Laplacian is gi en i s eigenma ices and spec um can be used o p o ide such a map i needed. Suppose o example ha he spec um o 4 ∆Lis iden ical o ha o ∆γup o some cu o and a comple e se o eigen- ma ices is gi en by ˆ Ψλwi h he co esponding commu a i e eigen unc ions being Ψλ, hen he symme ic symbol-map D gi en by D = d2 L X λ Ψλˆ Ψλ(2) p o ides a map o unc ions wi h M=1 dL T (DM) (3) he unc ion co esponding o he ma ix M. By cons uc ion he map has no ke nel and he symbol-map induces a ∗p oduc on unc ions gi en by M∗D N=1 dL T (DMN) (4) which ep esen s ma ix mul iplica ion in e ms o an ope a ion on he image unc ions. The ∗p oduc depends on D, a di e en bu equi alen one could be ob ained by gi ing a nonze o weigh ing cλ(L) o he di e en e ms in he sum (2). In he case o CPNa pa icula choice o he cλ(L) will gi e he diagonal cohe en s a e p esc ip ion1as discussed in [18]. I he symbol-map (2) has he p ope y ha ∆γ M=1 dL T (D∆LM) (5) whe e ∆γis a na u al Laplacian o he space o be app oxima ed, hen he spec um o he uzzy space will be p ecisely a cu o e sion o ha o he commu a i e space M. This is p ecisely wha happens in he case o CPN F, see [18]. Howe e , i is con enien o ex end he de ini ion o uzzy space o he case whe e he spec um coincides o low-lying eigen alues, bu de ia es o a amily o eigen alues ha can be gi en a bi a ily high alue and which co espond o deg ees o eedom ha ha e no coun e pa in he commu a- i e space M. This allows us o ob ain uzzy app oxima ions o addi ional spaces — in pa icula , as we will see, o o i and he h ee sphe e. 1In he case whe e he symbol-map is he p ojec o o cohe en s a es he unc ion M is e e ed o as he co a ian symbol o he ma ix Mand since he coe icien s cλ(L) a e no one i will di e om he co esponding con a a ian symbol, see Be ezin [22]. The symbol-map is e e ed o as symme ic when i s co a ian and con a a ian symbols a e equal and coincides wi h he case o cλ(L) = 1. 5 I one akes he Euclidean quan um ield heo y poin o iew hen he desi ed geome y appea s as ha associa ed wi h he accessible con igu a- ions o he ield heo y and he de ia ions a e supp essed in a p obabilis ic ashion. A success ul me hod o supp essing he unwan ed modes would be o add o he scala ac ion a e m SI[Φ] which is non-nega i e o any Φ, ze o only o ma ices ha co espond o unc ions on M, and posi i e o hose ha do no . The modi ied ac ion would he e o e be o he o m S[Φ] + hSI[Φ]. The pa ame e hshould be chosen o be la ge and posi i e. The p obabili y o any gi en ma ix con igu a ion hen akes he o m P[Φ] = e−S[Φ]−hSI[Φ] Z(6) whe e Z=Zd[Φ]e−S[Φ]−hSI[Φ] (7) is he pa i ion unc ion o he model. I he p esc ip ion is o wo k o ee ield heo ies, hen SI[Φ] should be a mos quad a ic in Φ. This can hen be hough o as a modi ica ion o he Laplacian in he iple (1). Fu he mo e he p oblem o UV/IR mixing in scala heo ies can be e- mo ed by including a highe de i a i e ope a o in he quad a ic e m o he ield heo y such ha i ende s all diag ams ini e when he ma ix size is sen o in ini y. Wi h such a p esc ip ion since each diag am has a limi ing commu a i e alue in he la ge ma ix limi each diag am mus ake his alue and hence no UV/IR mixing can occu . The p esc ip ion o sending he ma ix size o in ini y and sending he coe icien o he i ele an highe de i a i e ope a o o ze o do no commu e. This p esc ip ion o adding an i ele an ope a o o he ac ion is simple han he no mal o de ing p e- sc ip ion p oposed in [23] and wo ks o any dimension. F om he abo e discussion i should be clea ha he en i e p oblem o cons uc ing a uzzy app oxima ion o a space is he p oblem o gi ing a sui able p esc ip ion o he ma ix Laplacian. 3 App oxima ing a ci cle om a uzzy sphe e Conside he ini e ma ix algeb a ep esen a ion o he uzzy sphe e S2 F[13]. The algeb a o (L+ 1) ×(L+ 1) ma ices, which will be deno ed by Ma L+1, has he same dimension as he numbe o deg ees o eedom in a sphe ical 6 ha monic expansion o a unc ion on S2, unca ed a angula momen um L, L(θ, φ) = L X l=0 l X m=−l lmYlm(θ, φ).(8) Tha is L X l=0 (2l+ 1) = (L+ 1)2.(9) The p ecise iden i ica ion be ween a ma ix Φ ∈Ma L+1 and a cu -o unc- ion L(θ, φ), as discussed in he p eceding sec ion, is no unique, bu he pos- sible maps can be gi en in e ms o cohe en s a es o he symme ic symbol- map D o (2), and he esul ing p oduc o unc ions is non-commu a i e o ini e L. I is c ucial o ou cons uc ion ha only maps o which he p od- uc o unc ions becomes commu a i e in he limi L→ ∞ be conside ed. The symbol-map (2) associa es o hono mal (L+ 1) ×(L+ 1) pola isa ion enso s ˆ Ylm wi h sphe ical ha monics Ylm(θ, φ). The con en ions used he e will be ha ˆ Ylm =1 √L+ 1 ˆ Tlm (10) whe e he pola isa ion enso s ˆ Tlm a e hose o [24]. The SO(3) symme ic Laplacian, L2, on he uzzy sphe e ac s on ma ices Φ and is ep esen ed by he second o de Casimi co esponding o he adjoin ac ion o he angula momen um gene a o s Liin he (L+ 1) ×(L+ 1) ep esen a ion: L2Φ = [Li,[Li,Φ]].(11) Hence he ac ion can be aken o be S[Φ] = 1 L+ 1T 1 2Φ†L2Φ + V(Φ)(12) o some scala po en ial V(Φ)†=V(Φ), which is assumed o be bounded below. This ac ion can hen be used in a pa i ion unc ion which in ol es o dina y in eg a ion o e (L+ 1)2deg ees o eedom Z=ZDΦe−S[Φ].(13) The p obabili y dis ibu ion o ield con igu a ions is hen P[Φ] = e−S[Φ] Z(14) 7 whe e S[Φ] gi en is by (12). This p obabili y dis ibu ion is associa ed wi h he geome y (HL,Ma L+1,L2) which speci ies a ound uzzy sphe e. The ield heo y wi h quad a ic po en ial, howe e , su e s om UV/IR mixing p oblems [23, 25]. I we add he e m aL4 o he Laplacian and use he iple (HL,Ma L+1,L2+aL4) he UV/IR mixing p oblem is emo ed and we eco e a ield heo y on he commu a i e S2in he in ini e ma ix size limi . The pa ame e acan inally be sen o ze o wi h he esul ha he c i ical alue o he mass pa ame e will be sen o in ini y. The p ocess o aking he la ge ma ix limi and sending a o ze o do no commu e. To ob ain he commu a i e heo y on he sphe e he ma ix size mus be sen o in ini y o non-ze o a. The e is no ini e ma ix app oxima ion o he algeb a o unc ions on S1. Ne e heless, he deg ees o eedom ele an o a ci cle a e ce ainly con ained in Ma L+1. Focusing on he op ha monic in (8), wi h l=L, he YLm con ain all −L≤m≤Land hus ep oduce unc ions on he ci cle as m→ ∞. This implies ha he pa i ion unc ion and co ela ion unc ions o a ield heo y on a ci cle can be ex ac ed om ha o he uzzy sphe e by supp essing all he modes wi h l < L in (13). One way o achie ing his is o penalise modes wi h l < L by gi ing hem a la ge posi i e weigh in he ac ion. To his end we modi y he ac ion (12) o Sh[Φ] = 1 L+ 1T 1 2Φ†[L3,[L3,Φ]] + h 2Φ†L(L+ 1) −L2Φ + V(Φ). (15) All modes wi h l < L now ha e he w ong sign o L2and, when his e y la ge, a e hea ily penalised in he pa i ion unc ion (13), con ibu ing no hing as h→ ∞. In his limi only he modes wi h l=L emain and hese ha e he co ec sign o hei kine ic ene gy, because he e m linea in h anishes on hese and only hese modes. The ‘w ong sign’ o he L2 con ibu ion o he kine ic ene gy he e is analogous o an an i- e omagne ic coupling in a la ice heo y and jus as in he la ice heo y wi h an an i- e omagne ic coupling he ac ion he e is also bounded below. Tha he ac ion emains bounded om below is in ima ely ela ed o he ac ha he e is an ul a iole cu o in he model and he e o e a maximum eigen alue o he Laplacian o equi alen ly a sho es wa eleng h. To see ha he commu a i e algeb a o unc ions on S1is eco e ed in he l=Lsec o o he uzzy sphe e as L→ ∞, we i s decompose he ma ix Φ using he basis o pola isa ion enso s: Φ = L X l=0 l X m=−l Φlm ˆ Ylm.(16) 8 In ou con en ions (10) he commu a o o he pola isa ion enso s is gi en by (see e.g. [24] page 191, equa ion (46)) [ˆ Yl1m1,ˆ Yl2m2] = (2l1+ 1)(2l2+ 1) L+ 1 L X l=0 (−1)L−l1−(−1)l1+l2+l ×l1l2l L/2L/2L/2Clm l1m1,l2m2 ˆ Ylm, (17) whe e l1l2l L/2L/2L/2a e 6j-symbols and Clm l1m1,l2m2a e Clebsch-Go don co-e icien s. Now o la ge L l1l2l L/2L/2L/2≈1 √L+ 1Cl0 l10,l20(18) and Cl0 l10,l20= 0 when l1+l2+lis odd. Thus [ˆ Yl1m1,ˆ Yl2m2]→0 (19) and he algeb a is commu a i e when L→0 as p omised. In pa icula [ˆ YLm1,ˆ YLm2]→0 (20) and he op ha monic alone ep oduces he commu a i e algeb a o unc ions on S1in he con inuum. To summa ize we can encode he geome y speci ying a uzzy ci cle by he iple S1 F:= HL,Ma L+1,L2 3+hL(L+ 1) −L2.(21) wi h h >> 1. This picks ou he uzzy ci cle om he op angula momen um pola iza ion enso ˆ YL,m. One could equally pick i ou om a lowe one, ˆ YL0,m by modi ying he e m p opo ional o h o (L0(L0+ 1) −L2)2. This la e choice may ha e ad an ages o he supp ession o UV/IR mixing e ec s in he uzzy con ex . I oughly co esponds o a mix u e o ‘nea es neighbou ’ and nex nea es neighbou e omagne ic and an i- e omagne ic couplings. Ha ing cons uc ed a uzzy ci cle i is now clea ha he e is no obs acle o cons uc ing uzzy o i o a bi a y dimension, simply by aking p oduc s o uzzy ci cles. This has he ob ious ad an age o nume ical simula ion o a oiding powe -law cu a u e e ec s. 9 I we can penalise all modes wi h 2l < L and m6= 0 o 2l=Lin a unc ional in eg al o e SO(5)/SO(3) ×SO(2) hen we will eally be doing a unc ional in eg al o e S3 F. This is easily achie ed since 2l=Land m= 0 has he la ges second o de Casimi , C(5) 2(L, 0) = L(L+ 3),(51) o all he SO(5) ep esen a ions in Ma d0 L. In he now amilia manne he unwan ed modes in he unc ional in eg al o e SO(5)/SO(3)×SO(2)can be supp essed by using he Laplacian L2 h0=1 2[Jαβ,[Jαβ,·]] + h0L(L+ 3) −L2 (5),(52) which ac s on ields Φ ∈Ma d0 Land Le en. The unwan ed modes a e comple ely elimina ed in he limi h0→ ∞, gi ing S3 F unca ed a le el L. The cons ain ha Lis e en does no change he ac ha we ge he ull con inuum S3as L→ ∞. 7 Conclusions By s a ing wi h he known ini e ma ix algeb as o CP3and CP1, he uzzy CP3 Fand he uzzy sphe e CP1 F∼ =S2 F, ini e unc ional in eg als o scala ield heo ies on S3 Fand S1 Fha e been cons uc ed. The geome y o a uzzy space is speci ied by a iple (HL,Ma dL,∆L) and, al hough he e is no known closed associa ed algeb a gi ing a uzzy S1as a iple di ec ly, CP1 Fne e heless con ains he s a es equi ed o a S1 Fplus o he unwan ed s a es. The unwan ed s a es a e gi en la ge eigen alues by modi ying he Laplacian on CP1 F, as in equa ion (21), lea ing only he s a es o S1 Fin he low ene gy spec um o he Laplacian. In a simila way CP3 Fcon ains he s a es necessa y o a uzzy desc ip ion o S3( ia S4 F) and he Laplacian on CP3 Fcan be modi ied, as in equa ion (42), so ha s a es no ela ed o S3 Fa e gi en la ge eigen alues, lea ing only S3 Fs a es in he low ene gy spec um. An al e na i e cons uc ion o S3 F, based on supp essing modes on a uzzy e sion o he o hogonal G assmannian SO(5)/SO(3) ×SO(2), has been p esen ed in sec ion 6. This has he ad an age o ha ing a na u al ex ension o SN F o any N, [20]. Thus S3 Fcan be ob ained ei he in wo s eps, ia he uzzy S4 Fcons uc ed in [4], CP3 F→S4 F→S3 F, o al e na i ely in a single s ep om he uzzy e sion o SO(5)/[SO(3)×SO(2)] as desc ibed in sec ion 6. 16 The cons uc ion o he uzzy ci cle allows uzzy o i o be de ined in an ob ious way, by aking p oduc s o uzzy ci cles, hus opening he way o nume ical simula ions on o i while p ese ing he ull U(1) ×···×U(1) isome y g oup and a oiding he e mion doubling p oblem [5]. One i s w i es down a ini e unc ional in eg al o a ield heo y on S2 F× ··· × S2 F, which con ain he modes ele an o p opaga ion S1 F× ··· × S1 Fin i s spec um, and hen damps he unwan ed modes. This can be done, in a manne ha p ese es he isome ies o he o us, by in oducing app op ia e combina ions o second o de Casimi s in o he p opaga o s. I is pleasu e o acknowledge A. P. Balachand an, Pe e P eˇsnajde , Ha - ald G osse and Daniel Roggenkamp o help ul discussions. Re e ences [1] I. Mon ay and G. 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