Fuzzy Complex Quadrics and Spheres

DIAS-STP-03-09 Fuzzy Complex Quad ics and Sphe es B ian P. Dolan,a),b)∗Denjoe O’Conno b)†and P. P eˇsnajde c)‡ 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 c)Dep . o Theo e ical Physics, Comenius Uni e si y, Mlynsk´a dolina, SK-84248 B a isla a, Slo akia Janua y 17, 2004 Abs ac A ma ix algeb a is cons uc ed which consis s o he necessa y deg ees o eedom o a ini e app oxima ion o he algeb a o unc- ions on he amily o o hogonal G assmannians o eal dimension 2N, known as complex quad ics. These ma ix algeb as con ain he ele an deg ees o eedom o desc ibing unca ions o ha monic ex- pansions o unc ions on N-sphe es. An In¨on¨u-Wigne con ac ion o he quad ic gi es he co- angen bundle o he commu a i e sphe e in he con inuum limi . I is shown how he deg ees o eedom o he sphe e can be p ojec ed ou o a ini e dimensional unc ional in e- g al, using second-o de Casimi s, gi ing a well-de ined p ocedu e o cons uc ion unc ional in eg als o e uzzy sphe es o any dimension. ∗[email p o ec ed] †[email p o ec ed] ‡[email p o ec ed] 1 1 In oduc ion Non-commu a i e geome y has slowly been inc easing in impo ance in physics o e he las 20 yea s and has ecen ly ecei ed a s ong impe us h ough wo k in s ing heo y. An impo an concep in he non-commu a i e p og amme is ha o a “ uzzy” space — his is pe haps mo e co ec ly de- sc ibed as a ini e, non-commu ing, ma ix app oxima ion o he algeb a o unc ions on a con inuous mani old which can ep oduce he commu a i e algeb a in he limi o he ma ices becoming in ini e in size. I has been sugges ed ha uzzy spaces could p o ide a egula isa ion echnique o nu- me ical calcula ions in quan um ield heo y which would be an al e na i e o la ice gauge heo y, [1]-[4]. The p o o ypical example o a uzzy space is he uzzy wo-sphe e, [5], bu he e a e many mo e examples, and indeed much o he wo k on gene alised cohe en s a es in quan um mechanics [6], when es ic ed o compac g oups, can be ela ed o uzzy spaces. The geome y o uzzy CPNand uzzy uni a y G assmannians has been examined in he li e a u e in some de ail [7]-[10]. Clea ly, om he poin o iew o a egula isa ion echnique in ield heo y as well as o he heo y o D-b anes in s ing heo y, i would be desi able o ha e an explici cons uc ion o he uzzy sphe e SN Fin dimensions o he han N= 2. Un o una ely, despi e he elegan simplici y o he uzzy wo-sphe e S2 F, highe dimensional sphe es a e no so amenable o a uzzy desc ip ion. To ou knowledge he e is no closed ini e ma ix app oxima ion o he algeb a o unc ions on a sphe e o dimensions g ea e han wo, hough he p oblem was ackled in [11] and [12] (non-commu a i e sphe es in he con inuum we e analysed in [13]). I appea s ha , while one can ep esen unca ed ha monic expansions o unc ions on sphe es by squa e ma ices, he p oduc o wo such ma ices akes one ou o he equi ed space and a p ojec ion back in o he space o unc ions is necessa y a e e e y mul iplica ion — his ende s he p oduc non-associa i e. In o he wo ds a s a p oduc canno be de ined on he uzzy sphe e SN F o N > 2 ( his is ela ed o he ac ha , o N6= 2, SNcanno be ob ained as he co-adjoin o bi o a compac g oup). Ne e heless he cons uc ion in [11] does associa e a squa e ma ix wi h he unca ion o a ha monic expansion on SNand so does, in a sense, cons i u e a uzzy sphe e, e en hough he e is no associa i e p oduc . Fo S4 Fal e na i e desc ip ions a e possible, [14] [15]. The cons uc ion in [15] is in e ms on CP3 Fand uses he ac ha he con inuum CP3is an S2bundle o e S4[15]. This app oach has he ad an age ha i is designed o be used in a unc ional in eg al and he echnique was ex ended o S3 F and S1 Fin [16]. The me hod o [11] would be pa icula ly cumbe some o implemen in unc ional in eg als o e S3 Fand S1 F, o indeed any odd sphe e. 2 In his pape we gene alise he cons uc ion in [16] o uzzy sphe es o any dimension. The analysis elies on p ope ies o he o hogonal G assmannian, SO(N+ 2)/[SO(N)×SO(2)], which is a co-adjoin o bi o dimension 2N. This o hogonal G assmannian can also be ob ained as he complex quad a ic zaza= 0 in CPN+1, whe e zaa e na u al complex co-o dina es, [17]. The no- a ion in Kobayashi and Nomizu is QN∼ =SO(N+ 2)/[SO(N)×SO(2)] and we shall use his as a sho hand. In¨on¨u-Wigne con ac ion o SO(N+ 2) o he Euclidean g oup o RN+1 ela es QN o he co- angen bundle T∗SN. In a sense, elabo a ed on in sec ion 2, QNcan be hough o as he com- pac i ied co- angen bundle o a non-commu a i e sphe e. The e is a ini e dimensional, non-commu a i e, ma ix algeb a app oxima ion o he algeb a o unc ions QNwhich con ains ha monic expansions on SN(desc ibed in sec ion 3). While i is s ill he case ha he esul ing ma ix algeb a akes one ou o he space o ha monic expansions on SNupon mul iplica ion (so one does no ha e a closed ma ix app oxima ion o he algeb a o unc ions on SN o N6= 2) i is ne e heless e y easy o supp ess unwan ed modes in a unc ional in eg al o e a uzzy complex quad ic, QN, in a manne which lends i sel na u ally o nume ical compu a ion o ield heo y on SN F, as desc ibed in sec ion 4. 2 The de o med co- angen bundle In his sec ion i is shown how he co- angen bundle T∗SNcan be de o med o a e sion ela ed o a non-commu a i e sphe e and compac i ied o com- plex quad ic, QN. The cons uc ion s a s wi h Ca esian co-o dina es Xain RN+1, whe e a= 1, . . . , N + 1. Conside he sphe e SN∼ =SO(N+ 1)/SO(N) o adius R de ined by XaXa=R2. The isome y g oup is SO(N+ 1) wi h algeb a [Lab, Lcd] = i(δbcLad +δadLbc −δacLbd −δbdLac),(1) whe e Lab =−Lba ( he e is no dis inc ion be ween uppe and lowe Euclidean indices in RN+1,Xa=Xa). This can be ex ended o a ep esen a ion o he Euclidean g oup, EN+1 ac ing on RN+1, [Lab, Lcd] = i(δbcLad +δadLbc −δacLbd −δbdLac) (2) [Lab, Xc] = i(δbcXa−δacXb) (3) [Xa, Xb]=0.(4) An explici ealisa ion o his algeb a in he con inuum is Lab =i Xa ∂ ∂Xb −Xb ∂ ∂Xa!(5) 3 ac ing on unc ions and Xabeing commu a i e mul iplica ion by he co- o dina es. The algeb a will now be de o med, essen ially using he in e se o In¨on¨u- Wigne con ac ion. Le Xa:= µLa,N+2, wi h µ eal, and eplace he com- mu a o (4) abo e wi h [Xa, Xb] = −iµ2Lab,(6) lea ing he o he commu a o s unchanged. The Euclidean g oup EN+1 is hus de o med o SO(N+ 2) [LAB, LCD] = i(δBCLAD +δADLBC −δACLBD −δBDLAC),(7) whe e A, B, C, D = 1, . . . , N + 2 and La,N+2 =−LN+2,a. In e ms o he quad a ic Casimi s1CN+1 2= (1/2)LabLab and CN+2 2= (1/2)LABLAB we see ha XaXa=µ2CN+2 2−CN+1 2.(8) While i may be emp ing o hink o Xain (8) as co-o dina es on a uzzy sphe e we mus be ca e ul: a gene al i educible ep esen a ion o SO(N+2) will decompose in o a sum o di e en i educible ep esen a ions o SO(N+ 1), wi h di e en alues o CN+1 2, so he igh -hand side o (8) will no be cen- al. Howe e XaXais cen al in he undamen al spino ep esen a ion o Spin(N+ 2). To see his conside he e en and odd cases sepa a ely: •E en N: choose one chi ali y o spino wi h 2N/2componen s. Unde Spin(N+ 2) →Spin(N+ 1) his educes uniquely o he single spino ep esen a ion o Spin(N+ 1), which has he same dimension. In his case he igh -hand side o (8) is a mul iple o he iden i y. •Odd N: in his case he 2N/2dimensional spino ep esen a ion o Spin(N+ 2) decomposes in o wo spino ep esen a ions o opposi e chi ali y unde SO(N+ 2) →SO(N+ 1), bo h o dimension 2(N/2)−1. Bu he second o de Casimi CN+1 2has he same alue on he wo chi ali ies, i does no dis inguish be ween hem, [18]. So again he igh -hand side o (8) is a mul iple o he iden i y. The undamen al spino ep esen a ions o Spin(k) ha e quad a ic Casimi Ck 2=k(k−1)/4 o bo h e en and odd k, see [18] o example. Fo a undamen al spino ep esen a ion o Spin(N+ 2) equa ion (8) he e o e gi es XaXa=µ2(N+ 1) 21,(9) 1Rela i e o he s anda d con en ions o SU(n) ou no malisa ion he e is such ha CSU(2) 2=1 2C3 2and CSU(4) 2=1 2C6 2. 4 whe e 1is he iden i y ope a o , and we migh in e p e qN+1 2µas he adius o a uzzy sphe e. Howe e , unlike S2 F, he ma ix algeb a gene a ed by Xa will no close in gene al e en when XaXais cen al. Also one canno ge highe dimensional ep esen a ions o SN Fby aking enso p oduc s. This e lec s he ac ha he e is no closed ini e ma ix app oxima ion o SN F o N6= 2. Le us now in es iga e he geome y o he space gene a ed by he adjoin ac ion o Spin(N+ 2) on a iducial di ec ion XN+1 ( he “no h pole” o SN F). Clea ly [Lαβ, XN+1] = 0 o α, β = 1, . . . , N, so Spin(N) lea es XN+1 in a ian . Also LN+1,N+2 commu es wi h XN+1, since hey a e jus mul iples o one ano he , so he SO(2) gene a ed by LN+1,N+2 also lea es Xain a ian . The upsho o his is ha he mani old ha D(g)∈Spin(N+ 2) gene a es wi h he ac ion D−1(g)XN+1D(g) (10) is he o hogonal G assmannian QN, which has dimension 2N. Fo N≤4 hese spaces ha e he ollowing s uc u es: •Q1∼ =SO(3)/SO(2) ∼ =S2; •Q2∼ =SO(4)/[SO(2) ×SO(2)] ∼ =S2×S2(S2×S2was used in a conside a ion o uzzy S3/Z2in [19]); •Q3∼ =SO(5)/[SO(3) ×SO(2)] ∼ =CP3/Z2( his iden i ica ion is de- sc ibed in [16]); •Q4∼ =SO(6)/[SO(4) ×SO(2)] ∼ =SU(4)/[S(U(2) ×U(2))] (ma ix ap- p oxima ions o his space we e desc ibed in [10]). We p opose o iden i y QNwi h a ‘compac i ied’ co- angen bundle o a non-commu a i e sphe e. This educes o he usual T∗SNunde In¨on¨u- Wigne con ac ion µ→0: his limi pe o ms he dual unc ion o en- de ing he X’s in (6) commu a i e while a he same ime de-compac i ying SO(N+ 2) o he Euclidean g oup EN+1. 3 Fuzzy Complex Quad ics, QN F Any uni a y i educible ep esen a ion To a simple compac Lie g oup is ini e dimensional and he ex ension o i s en eloping algeb a is a ini e di- mensional ma ix algeb a. This ma ix algeb a p o ides a uzzy app oxima- ion o he algeb a o unc ions on he co-adjoin o bi o any Lie algeb a elemen in T. 5 We shall e e o a ini e ma ix app oxima ion o QNas a uzzy complex quad ic and deno e i by QN F. Being a co-adjoin o bi QNis a symplec ic mani old whose algeb a o unc ions can be app oxima ed by closed ini e dimensional ma ix algeb as. The ha monic expansion o a unc ion on QN equi es all ep esen a ions o SO(N+ 2) which con ain he i ial ep esen- a ion unde SO(N)×SO(2). A spino ep esen a ion o SO(N+ 2) ne e con ains a single o SO(N)×SO(2) o ei he e en o odd N, so we e- s ic o ec o ial ep esen a ions. The ep esen a ion space o SO(N+ 2) is hen he space o ank-n enso s, TA1···An. I educible ep esen a ions can be cha ac e ised by hei symme ies unde in e change o hei indices and a e aceless in any wo indices. They can be ep esen ed by Young ableau wi h nboxes which e lec hei pe mu a ion symme ies. Any enso ha is an i-symme ic in h ee o mo e o i s indices mus anish when es ic ed o a ep esen a ion o SO(2) and hence canno con- ibu e o he ha monic expansion o unc ions on QN. Thus we can es ic ou a en ion o enso s ha a e an i-symme ic in pai s o indices only. I TA1···Anhas mpai s o an i-symme ic indices and n−2msymme ic indices hen he co esponding Young ableau is m z }| { ·· ·· n−2m z }| { ·· .(11) When SO(N+ 2) is es ic ed o SO(N)×SO(2) he an i-symme ic pai s all con ain single s o SO(N)×SO(2) (when bo h indices in a pai a e SO(2) indices, o example). Also i nis e en he n−2msymme ic indices can con ain single s o SO(N)×SO(2), bu no when nis odd. Thus he ha monic expansion o a unc ion on QN equi es all SO(N+ 2) enso ep esen a ions o he o m (11) wi h ne en. The dimension o hese ep esen a ion can be de e mined, using he ele an o mulae in [18] o example. Wi h n= 2l hey a e, o N≥3, dN(2l, m) = (2l+N−1)(2l+ 1 −2m)(4l+N−2m)(N−2+2m) ×(2l+N−2−m)!(N−3 + m)! N!(N−2)!(2l−m+ 1)!m!(12) whe e m≤l.2I he ha monic expansion o a unc ion on QNis unca ed a lmax =L he o al numbe o deg ees o eedom is L X l=0 l X m=0 dN(2l, m) = "(2L+N)(L+N−1)! L!N!#2 = [dN(L, 0)]2.(13) 2In he no a ion o [18] he ep esen a ions (11) ha e highes weigh s ( 1, . . . , [N/2]+1) = (n−m, m, 0, . . . , 0), whe e [N/2] is he in ege pa o N/2. 6 The ac ha his is a pe ec squa e e lec s he ac ha he deg ees o eedom in a unca ed ha monic expansion can be a anged in o a squa e ma ix o size dN(L, 0)×dN(L, 0). This can be ep esen ed in e ms o Young ableau by L z }| { ·· × L z }| { ·· = 1 ⊕ ⊕ ⊕ · · · ⊕ L z }| { ·· ·· ⊕ · · · ⊕ 2L z }| { ·· (14) which ela es he ma ix s uc u e, on he le -hand side, o he ha monic expansion, on he igh -hand side. Ma ix mul iplica ion hen gi es a closed associa i e, bu non-commu a i e, algeb a which ep oduces he commu a- i e algeb a o unc ions on QNas L→ ∞. A he le el o unc ions he non-commu a i e p oduc a ini e Lcan be ealised as a ∗-p oduc o he uzzy complex quad ic, QN F. 4 Func ional In eg als on Fuzzy Sphe es, SN F The e is no closed ini e ma ix app oxima ion o he unca ed algeb a o unc ions on SNknown, excep o he special case N= 2 which was i s desc ibed in [5]. The e does exis a ma ix app oxima ion o unc ions on SN, bu in gene al ma ix mul iplica ion does no co espond o he algeb a o unc ions and he la e can only be eco e ed by p ojec ing back on o a unc ion on SNa e ma ix mul iplica ion [11]. This esul s in a non- associa i e algeb a when N6= 2 which, by a sligh abuse o language, is ne e heless s ill e e ed o as a “ uzzy” sphe e, SN F. In he cons uc ion p esen ed he e a simila p ojec ion can be pe o med, since he unca ed ha monic expansion o a unc ion on SNis bu ied in QN F. To see his no e ha unc ions on SNcan be expanded in symme ic enso ep esen a ions o SO(N+ 1). A unca ion a le el lmax equi es using all symme ic enso s o SO(N+ 1), Ta1···al, wi h 0 ≤l≤lmax. These a e all con ained in one symme ic ep esen a ion lmax z }| { ·· (15) o SO(N+ 2) unde SO(N+ 2) →SO(N+ 1). Se ing lmax = 2Lwe see ha he las ep esen a ion on he igh -hand side o (14) con ains all he SO(N+1) ep esen a ions necessa y o he ha monic expansion o a unc ion on SNup o angula momen um 2L. We can hus ob ain he uzzy sphe e SN Fby p ojec ing he i educible ep esen a ion 2L z }| { ·· ou om he ma ix algeb a o QN Fin (14). 7 This can be achie ed in a unc ional in eg al o e QN Fin he same manne as in [16]. Le Φ be a ma ix in he algeb a o QN F o a gi en L. The SO(N+ 2) in a ian Laplacian on QN Fis L2 (N+2)Φ = −(1/2)[LAB,[LAB,Φ]].(16) Then he ac ion o a scala ield on QN Fcan be w i en as S[Φ] = 1 dN(L, 0)T nΦ†L2 (N+2)Φ + V(Φ)o.(17) wi h he scala po en ial V(Φ†) = V(Φ) assumed bounded below. A unc- ional in eg al hen in ol es Z=ZDΦe−S[Φ].(18) We ocus on he SN Fembedded in QN Fby penalising all he SO(N+ 2) ep- esen a ions in Zexcep he las one on he igh -hand o (14). This can be achie ed by modi ying he kine ic e m in he ac ion. The second o de Casimi o he ep esen a ion (11), wi h n= 2l, is [18] CN+2 2(2l, m) = (2l−m)(2l−m+N) + m(m+N−2).(19) Obse e ha he comple ely symme ic enso s wi h m= 0 ha e he la ges Casimi o any gi en l, CN+2 2(2l, 0) = 2l(2l+N).(20) Hence he ope a o −1 2[LAB,[LAB,·]−2L(2L+N) = CN+2 2−2L(2L+N) (21) ac ing on Φ is nega i e o all modes in Φ excep o he op mode, wi h l=Land m= 0, on which i anishes. The SO(N+ 1) in a ian Laplacian on SN Fwould be L2 (N+1)Φ = −(1/2)[Lab,[Lab,Φ]].(22) So he ac ion Sh[Φ] = 1 dN(L, 0)T nΦ†L2 (N+1)Φ + hΦ†−L2 (N+2) + 2L(2L+N)Φ + V(Φ)o, (23) wi h h1, will supp ess all he unwan ed modes in a unc ional in eg al and lea e he equi ed modes o SN Funa ec ed. In he limi h→ ∞ all modes, excep he ones ele an o SN F, will be supp essed and co ela ion unc ions calcula ed wi h Z=ZDΦe−S∞[Φ].(24) will be hose o he uzzy sphe e, unca ed a le el 2L. 8 5 Conclusions By cons uc ing uzzy app oxima ions o complex quad ics, QN F, a p esc ip- ion o de ining unc ional in eg als o e ini e app oxima ions o sphe es has been p esen ed. Al hough ini e ma ix app oxima ions o he algeb a o unc ions on N-dimensional sphe es a e no known o N6= 2, one can cons uc ma ix app oxima ions o he unc ions, bu ma ix mul iplica ion hen akes one ou o he space o unc ions on he sphe e. The cons uc ion p esen ed he e elies in he ac ha he complex quad ics (which a e o hogonal G assmannians QN∼ =SO(N+ 2)/[SO(N)×SO(2)]) a e co-adjoin o bi s, and hence do ha e ini e ma ix app oxima ions o hei algeb a o unc ions, QN F. These spaces a e ela ed o he co- angen bun- dles T∗SN— hey a e in a sense compac i ied e sions o he co- angen bundles, compac i ied a he expense o in oducing a non-commu a i i y on he sphe e. The algeb a o unc ions on QN Fcon ains he ele an deg ees o eedom o a unca ed ha monic expansion o a unc ion on SN. The unc ional in eg al o a ield heo y de ined on QNcan be egula ised in a manne ha p ese es he isome ies and a oids Fe mion doubling [20] by de ining i o e he uzzy space QN F. By modi ying he kine ic e m and using he ac ion (23) he deg ees o eedom ha a e no ele an o he unde lying sphe e can be p e en ed om con ibu ing o he unc ional in eg al and he esul is a well de ined, ini e app oxima ion o he unc ional in eg al o a quan um ield heo y on SN. The co ec algeb a is ensu ed by es ic ing Φ o be ma ices o size dN(L, 0) gi en in (13) and he con inuum is eco e ed in he limi L→ ∞. This cons uc ion is well sui ed o nume ical e alua ion. Re e ences [1] H. G osse, C. Klimˇc´ık and P. P eˇsnajde , In . J. Theo . Phys. 35, (1996) 231, [hep- h/9505175]; H.G osse and A.S ohmaie , Le . Ma h. Phys. 48, (1999) 163, [hep- h/9902138] [2] H. G osse, C. Klimˇc´ık and P. P eˇsnajde , Comm. Ma h. Phys. 178, (1996) 507; H. G osse and P. P eˇsnajde , Le . Ma h. Phys. 46, (1998) 61 [3] P. P eˇsnajde , J. Ma h. Phys. 41 (2000) 2789, [hep- h/9912050] [4] A.P. Balachand an and S. Vaidya, In . J. Mod. Phys. A16, (2001) 17, [hep- h/9910129]; A. P. Balachand an, T. R. Go inda ajan and B. Yd i, Mod. Phys. Le A15 (2000) 1279, [hep- h/9911087]; 9