Fuzzy Complex Grassmannian Spaces and their Star Products

a Xi :hep- h/0111020 2 7 Jan 2003 DIAS-STP-01-16 hep- h/0111020 Oc obe 2002 Fuzzy Complex G assmannian Spaces and hei S a P oduc s B ian P. Dolan 1,a and Oli e Jahn 2,b aDepa men o Ma hema ical Physics NUI Maynoo h, Maynoo h, I eland bDublin Ins i u e o Ad anced S udies 10 Bu ling on Road, Dublin 4, I eland Abs ac We de i e an explici exp ession o an associa i e s a p oduc on non- commu a i e e sions o complex G assmannian spaces, in pa icula o he case o complex 2-planes. Ou exp ession is in e ms o a ini e sum o de i a i es. This gene alises p e ious esul s o complex p ojec i e spaces and gi es a disc e e app oxima ion o he G assmannians in e ms o a non-commu a i e algeb a, ep esen ed by ma ix mul iplica ion in a ini e- dimensional ma ix algeb a. The ma ices a e es ic ed o ha e a dimension which is p ecisely de e mined by he ha monic expansion o unc ions on he commu a i e G assmannian, unca ed a a ini e le el. In he limi o in ini e- dimensional ma ices we eco e he commu a i e algeb a o unc ions on he complex G assmannians. 1bdolan@ hphys.may.ie 2[email p o ec ed] 1 In oduc ion The e has ecen ly been much in e es in non-commu a i e geome y, [1, 2], bo h as a no el di ec ion in s ing heo y [3] and as a new ool in quan um ield heo y [4]. In he la e app oach he heo y is o mula ed on a “ uzzy” space, a se ies o disc e e app oxima ions o a con inuous space- ime mani old M. The space o ields on each app oxima ion is ini e-dimensional, so he uzzy space se es as a egula o , simila o a la ice. In con as o he la e , he unca ion enjoys all symme ies o he app oxima ed con inuous mani old. This has se e al use ul implica ions including he absence o a doubling p oblem o chi al e mions [5, 6]. Topologically non- i ial ield con igu a ions can be included and he chi al anomaly eme ges na u ally in his app oach [7, 8]. In mo e de ail, a uzzy space is gi en by a se ies o ini e-dimensional algeb as AL ha app oxima e he commu a i e algeb a o unc ions on Min he ollowing way: each ALis iden i ied wi h a subse o unc ions on Mand he p oduc in AL induces a (non-commu a i e) p oduc o unc ions in his subse , he so-called s a p oduc ; in he limi L→ ∞, he subse should exhaus he se o all unc ions and he s a p oduc go o e o he (commu a i e) poin wise p oduc o unc ions. In he p esen case, he algeb as ALa e ull ma ix algeb as. The uzzy spaces a e hus ma ix geome ies, which go o e o he usual con inuous mani old as he size o he ma ix is aken o in ini y. Non-commu a i e s a p oduc s a e known o exis o e e y Poisson mani old, in pa icula o symplec ic mani olds [9]. S a p oduc s ealised by ini e-dimensional ma ix algeb as ha e been cons uc ed in [10] o all homogeneous K¨ahle mani olds p o ided a ce ain quan isa ion condi ion on he me ic is sa is ied. These algeb as can also be ob ained using gene alised cohe en s a es [11] o he me hod o o bi s o i educible ep esen a ions o Lie g oups [12, 13]. The ela ion be ween hese app oaches has been discussed in [14] and ha o de o ma ion quan isa ion in [15, 16]. Fo he ela ion o Fou ie ans o ma ion on g oup space see [17]. In e ms o unc ions on he mani old, he abo e o mula ions p o ide an exp ession o he s a p oduc as an in eg al o e he analy ical con inua ion o he unc ions. This is no e y con enien o explici calcula ions in non-commu a i e ield heo y. An explici , local o mula in e ms o a ini e numbe o de i a i es is so a only known o complex p ojec i e spaces [18] including he case o he uzzy 2-sphe e [19] al eady ea ed in [20]. In his pape , we de i e an analogous o mula o uzzy complex G assmanni- ans, ha is ini e ma ix geome y app oxima ions o he usual G assmannians, GN k∼ =U(N)/[U(k)×U(N−k)], which a e homogeneous spaces isomo phic o he space o complex k-planes in CN. The ma ix geome ies consis o ma ices ac - ing on he i educible ep esen a ion o SU(N) which is gi en by he L- old Young p oduc o he ep esen a ion on an isymme ic k- enso s. As Lis inc eased he con inuous G assmannian is eco e ed. The special case k= 1 equi es he L- old symme ic p oduc o he undamen al ep esen a ion o SU(N), as in [18]. The s a p oduc o he case k= 2 is cons uc ed explici ly, using globally well-de ined bu o e -comple e co-o dina es on he G assmannian. The esul is exp essed as a ini e sum o e mul iple de i a i es o he unc ions, which a e decomposed in o 1 i educible ep esen a ions o he s abili y g oup S[U(k)×U(N−k)] ac ing on he angen space. I is shown ha he s a p oduc educes o he usual commu a i e p oduc as L→ ∞. The pa icula case G4 2may be o some in e es in ield heo y as i is a symplec ic mani old which has S4as a Lag angian sub-mani old. The co - esponding ma ix geome ies can be iewed as non-commu a i e e sions o T∗S4, he co- angen bundle o S4[21]. A s a p oduc on he complex G assmannians as an in ini e sum o e de i a i es is known [22, 23]. Howe e , his o mula canno be es ic ed o ini e-dimensional sub-algeb as and he e o e canno se e as a s a p oduc on a uzzy app oxima ion o he mani old. The layou o he pape is as ollows: in sec ion 2 we desc ibe he complex G assmannians, GN k, in e ms o p ojec ion ope a o s ac ing in CN, we in oduce a se o global co-o dina es which a e o e -comple e and sa is y a se o quad a ic cons ain s which ensu es ha hey indeed desc ibe GN k; in sec ion 3 we analyse he algeb a o unc ions in e ms o ep esen a ions o SU(N); in sec ion 4 we desc ibe he ini e ma ix geome ies ha de ine he uzzy G assmannians GN k,F; sec ion 5 gi es he cons uc ion o he s a p oduc o he special case k= 2, GN 2,F, while sec ion 6 p esen s a conjec u e on he possible o m o k > 2 and some obse a ions on he cons uc ion; sec ion 7 gi es a summa y and conclusions and some echnical de ails a e elega ed o he appendices. 2 Complex G assmannian spaces The complex G assmannian space GN k∼ =U(N)/[U(k)×U(N−k)] = SU(N)/S[U(k)× U(N−k)] can be ep esen ed as he space o all He mi ean ank-kp ojec o s Pac - ing on CN. This is easily seen since any such p ojec o can be diagonalised by an elemen o U(N) and he e is s ill a esidual adjoin ac ion o U(k)×U(N−k) which lea es i in a ian . I will be con enien o desc ibe GN kusing a edundan se o globally well-de ined co-o dina es, ξA,A= 1,...,N2−1. Fi s we in oduce o hono mal He mi ean gene a o s Ao he Lie algeb a o SU(N) sa is ying A B=1 NδAB +1 √2(dABC+i ABC) C,(1) whe e ABCa e he s uc u e cons an s o SU(N) and dABC he componen s o he usual symme ic aceless enso . The Aa e no malised so ha ( A B) = δAB. This allows us o pa ame e ise Pin e ms o N2−1 eal pa ame e s ξA, P=k N+ξA A.(2) The condi ion ha Pis as p ojec o ansla es in o ξAξA=k(N−k) Nand 1 √2dABCξAξB=N−2k NξC,(3) and ξAnow pa ame e ise GN kwhen hese condi ions a e imposed. This cons uc ion embeds he G assmannian in he space o aceless He mi ian ma ices, which we iden i y wi h RN2−1pa ame e ised by he un es ic ed ξA. 2 As discussed in de ail in e e ence [18] o he case o CPN−1≡GN 1, he complex s uc u e and me ic a e encoded in a He mi ean p ojec o KAB≡ P A(1 −P) B=1 2PAB+iJAB(4) whe e JAB=√2 ABC ξCis he complex s uc u e on GN kand PAB=PBA=−(J2)AB (indices a e aised and lowe ed wi h he la me ic δA Bo RN2−1). The ma ix KAB p ojec s de i a i es wi h espec o ξAon o he holomo phic angen space o he G assmannian when ac ing on he igh and on o he an i-holomo phic angen space when ac ing on he le ; so ∇A≡KAB∂/∂ξBis a holomo phic de i a i e and ¯ ∇B≡KAB∂/∂ξAan an i-holomo phic one. To see ha KABis a p ojec o we use he comple eness ela ion o he gene a o s, ( A)i j( A)k l=δilδkj−1 Nδijδkl,(5) o show ha KAB B=P A(1 −P) and BKBA= (1 −P) AP.(6) (These ela ions will p o e e y use ul in he ensuing analysis.) Examining he eal and imagina y pa s o KABsepa a ely e eals ha P=−J2and PJ =JP =J. This means ha Pi sel is a p ojec o on o he angen space o GN k, wi h ank 2k(N−k). An impo an obse a ion o he ollowing analysis is ha he di e en ial ope a o s KAB∂/∂ξBcommu e wi h he cons ain s (3), as hey mus do, since K p ojec s on o he angen space. This can also be p o en using (6). I is his ac ha allows us o use he global co-o dina es, a he han local co-o dina es, in he inal di e en ial exp ession o he s a p oduc . Co a ian de i a i es can be cons uc ed by p ojec ing de i a i es wi h espec o he la coo dina es ξA o he angen space. Mul iple co a ian holomo phic de i a i es a e hus de ined as ∇A1···∇An (ξ)≡KA1 B1···KAn Bn∂B1KB2 C2∂C2···KBn Cn∂Cn (ξ)(7) whe e ∂A=∂/∂ξA. In ou case he e is a simpli ica ion because KABKCD(∂BKDE) = 0 (8) which ollows om he de ini ion (4) o Kand he comple eness ela ion (5). I implies ∇A1···∇An (ξ) = KA1 B1···KAn Bn∂B1···∂Bn (ξ).(9) 3 Ha monic analysis In o de o ob ain a ini e-dimensional unca ion o he space o unc ions on he G assmannian, which is compa ible wi h he symme ies, we decompose he space o unc ions in o i educible ep esen a ions o he isome y g oup G=SU(N). We hink o GN kas he space G/H o ( igh ) cose s in Gwi h espec o H= S[U(k)×U(N−k)], he subse o ma ices in U(k)×U(N−k) wi h uni de e minan . 3 Func ions on G/H can hus be conside ed as unc ions on G ha a e in a ian unde he le ac ion o H. They ans o m unde Gacco ding o he igh ac ion. The ull space o unc ions on Gis spanned by he ma ix elemen s DJ MM′o all i educible uni a y ep esen a ions Jo G. Decomposing in o i educible ep e- sen a ions o H, we may w i e he i s componen index as M= (n, j, m) whe e j labels he i educible ep esen a ions o H,m he co esponding componen s and n dis inguishes copies o equi alen ep esen a ions. The le ac ion o His hen gi en by DJ (n,j,m)M′(h−1g) = X m′′ DJ (n,j,m) (n,j,m′′)(h−1)DJ (n,j,m′′ )M′(g).(10) The H-in a ian ma ix elemen s a e hose o which he i s index co esponds o he i ial ep esen a ion o H, labelled o ins ance by j=m= 0. The space o unc ions on G/H is hus spanned by he ma ix elemen s DJ (n,0,0) M′. Unde he igh ac ion o G, DJ (n,0,0) M′(gg′) = X M′′ DJ (n,0,0) M′′ (g)DJ M′′ M′(g′),(11) so, o ixed Jand n, he DJ (n,0,0) M′span he ec o space o he ep esen a ion J o G. The space o unc ions on GN k hus con ains all i educible ep esen a ions o SU(N) ha con ain he i ial ep esen a ion upon es ic ion o S[U(k)×U(N−k)]. The mul iplici ies a e gi en by he mul iplici y o he i ial ep esen a ion in he es ic ion. We will desc ibe ep esen a ions o SU(N) by Young diag ams. These will be deno ed by hei symbols J= [j1, j2,...,jN−1] whe e jideno es he numbe o columns o heigh io he diag am.3The undamen al ep esen a ion, o ins ance, has J= [1,0,...] and he adjoin one J= [1,0,...,0,1]. No e ha he complex conjuga e ep esen a ion is gi en by J∗= [jN−1,...,j1]. In appendix C, we show ha a ep esen a ion Jo SU(N) con ains he i ial ep esen a ion o S[U(k)× U(N−k)] i and only i i appea s in he di ec p oduc ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (12) o L≥n/N whe e n=Pijiis he numbe o boxes in he diag am Jand 0k−1 s ands o k−1 ze o en ies. The mul iplici y in he es ic ion is he same as ha in he di ec p oduc . The MLsa is y M1⊂M2⊂ ···, so hey p o ide a hie a chy o unca ions o he space o unc ions on he G assmannian. The ep esen a ion [0k−1, L, 0N−k−1] is he Young p oduc o Lan i-symme ic k- enso s, o ins ance o k= 2, L= 5. As an example, o N= 6 and k= 2, he i s unca ion is M1= [0,1,0,0,0] ⊗[0,0,0,1,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0] , ⊗= 1 ⊕ ⊕ (13) 3No e ha his symbol is di e en om he highes weigh ec o some imes used o desc ibe a diag am, he en ies o he la e being he numbe o boxes in each ow. 4 he second is M2= [0,2,0,0,0] ⊗[0,0,0,2,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0] ⊕[2,0,0,0,2] ⊕[1,1,0,1,1] ⊕[0,2,0,2,0] . ⊗= 1 ⊕ ⊕ ⊕ ⊕ ⊕ (14) Al hough no needed in he ollowing, we would like o s a e o illus a ional pu poses ha he ull ha monic analysis on GN k( he “union” o all ML) is gi en by he ep esen a ions J=([m1,...,mk,0,...,0, mk,...,m1] i 2k < N , [m1,...,mk−1,2mk, mk−1,...,m1] i 2k=N , (15) wi h mi= 0,1,..., each ep esen a ion occu ing once. The case 2k > N can be ob ained by eplacing kby N−k, since GN k∼ =GN N−k. The ep esen a ions (15) co espond o Young diag ams which can be ob ained by pu ing a diag am wi h a mos k ows nex o i s conjuga e (which has a leas N−k≥kboxes in any column), as can be e i ied o he examples gi en abo e. This esul is also de i ed in appendix C. 4 Ma ix geome y In he p e ious sec ion, we ha e ob ained a se ies o unca ions o he space o unc- ions on he G assmannian. We will now see ha hese ca y a na u al p oduc . The ep esen a ion con en MLo a gi en unca ion can in ac be ealised as an algeb a o ma ices in a ep esen a ion Jo SU(N): since such ma ices ans o m unde SU(N) by conjuga ion, hey o m he ep esen a ion space o J⊗J∗; so ML, as in oduced in eq. (12), is equi alen o he space o ma ices in he ep esen a- ion [0k−1, L, 0N−k−1]. Since he ma ix p oduc espec s he ac ion o SU(N), he algeb a MLhas he same symme ies as GN k. In o de o ob ain he co esponding p oduc o ( unca ed) unc ions, we shall now cons uc an injec i e map om ML o he space o unc ions on he G ass- mannian which also espec s he g oup ac ion (an equi a ian map). This map au oma ically p o ides he no ions o di e en ia ion and in eg a ion needed o he cons uc ion o ac ions: one jus has o map he co esponding no ions o unc ions back o ma ices. Equi a iance gua an ees ha hey a e compa ible wi h he un- ca ion. The map will also p o ide a non-commu a i e p oduc o unc ions in he image o ML, he s a p oduc . I he s a p oduc ends o he poin -wise p oduc in he limi L→ ∞, we ha e succeeded in cons uc ing a uzzy GN k. Since we will es ic ou sel es o k= 2 in he ollowing sec ions, we p esen he map only o his case. The gene alisa ion o o he alues o kshould be ob ious. 5 Fo GN 2 he basic building block will be he an i-symme ic ep esen a ion , co esponding o L= 1. The i s non- i ial unca ion o unc ions he e o e equi es using [N(N−1)/2]×[N(N−1)/2] ma ices. A unc ion on GN 2is associa ed wi h such a ma ix ˆ Fby es ic ing he enso p oduc o he undamen al p ojec o (2) o he an i-symme ic ep esen a ion , ρ≡(P ⊗P)a,(16) and cons uc ing F1(ξ) = [ρ(ξ)ˆ F].(17) Since Phas ank 2, ρhas ank 1: le he plane on o which Pp ojec s be spanned by he ec o s ~ and ~w;ρ hen p ojec s on o he 1-dimensional subspace o he ep esen- a ion space o spanned by he an i-symme ic p oduc o ~ and ~w (an explici p oo is gi en in appendix B). A gene al unca ion equi es aking he L- old (Young) p oduc [0, L, 0,...] = ··· ··· o , which has dimension nN L=(N+L−1)!(N+L−2)! (N−1)!L!(N−2)!(L+1)! , and using nN L×nN Lma ices. Equa ions (13) and (14), o ins ance, show he decom- posi ion o 15 ×15 and 105 ×105 ma ices as ha monics o G6 2. In he ollowing we shall d op ailing ze os in symbols o ep esen a ions, so he abo e ep esen a ions will be deno ed by [0, L]. The Young p oduc can be ob ained as a componen o he symme ic enso p oduc , so a p ojec o can be cons uc ed by es ic ing he L- old enso p oduc o ρ o he ep esen a ion [0, L], ρL= ( L imes z}| { ρ⊗···⊗ρ)[0,L],(18) (o cou se ρ1=ρ) and a unc ion can be associa ed wi h any nN L×nN Lma ix ˆ Fby FL(ξ) = [ρL(ξ)ˆ F].(19) Appendix B con ains a p oo ha he map (19) is injec i e. The ma ix geome ies in oduced he e coincide wi h hose ob ained om com- plex line bundles [10] o gene alised cohe en s a es [11], see [14, 15]. FLis usually called he co a ian symbol o he ope a o ˆ Fin hese o mula ions, and injec i i y is well known and ollows om an analy ici y a gumen . The ela ion wi h cohe en s a es will be discussed in some mo e de ail in appendix B. 5 S a p oduc on GN 2 Mul iplica ion o unca ed unc ions on he G assmannian can now be de ined using ma ix mul iplica ion. The s a p oduc o wo unc ions, FL= (ρLˆ F) and GL= (ρLˆ G), is ob ained om he ma ix p oduc h ough he map (FL⋆ GL)(ξ) = ρL(ξ)ˆ Fˆ G.(20) By cons uc ion his is an associa i e p oduc and i keeps wi hin he class o unc- ions unca ed a le el L. Ou aim is o ind an explici exp ession o his s a 6 p oduc , pu ely in e ms o FLand GLand hei de i a i es, hus elimina ing he explici e e ence o ma ices. By o hono mali y o he ma ix elemen s in he ep esen a ion [0, L], Rdµ(g)D[0,L] M1M2(g−1)D[0,L] M3M4(g) = (1/nN L)δM1M4δM2M3,ˆ Fcan be expanded as ˆ F=Zdµ(g)˜ F(g)D[0,L](g) (21) wi h ˜ F(g)≡nN L D[0,L](g−1)ˆ F.(22) Inse ing his in o (19), we ob ain FL(ξ) = Zdµ(g)ωL(ξ, g)˜ F(g) (23) wi h ωL(ξ, g)≡ ρL(ξ)D[0,L](g)(24) and he s a p oduc can be exp essed as (FL⋆ GL)(ξ) = Zdµ(g)Zdµ(g′)ωL(ξ, gg′)˜ F(g)˜ G(g′).(25) We seek an exp ession o he s a p oduc in e ms o de i a i es ac ing on FL(ξ) and GL(ξ). By eqs. (25) and (23), his can be achie ed by de i ing an exp ession o ωL(ξ, gg′) in e ms o de i a i es o ωL(ξ, g) and ωL(ξ, g′) wi h espec o ξ. The la e is g ea ly acili a ed by he obse a ion ha ωLcan be exp essed in e ms o ω1, ωL(ξ, g) = [ω1(ξ, g)]L,(26) because, by eq. (18), ρL ac o ises in o ank-1 p ojec o s ρand D[0,L]ac s as a di ec p oduc as well. The eason behind his ela ion is ha he ep esen a ion [0, L] when p ojec ed o S[U(2) ×U(N−2)] by ρL ac o ises as [0,1]Lsince all o he i educible componen s o he p oduc in ol e enso s ha a e an i-symme ic in 3 o mo e indices and he e o e anish in SU(2). As a i s s ep, we ha e o ind an exp ession o ω1(ξ, gg′). This is a s aigh - o wa d bu somewha leng hy exe cise. I is de e ed o appendix A and yields ω1(ξ, gg′) = ω1(ξ, g)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂Dω1(ξ, g′) (27) whe e ∂A=∂/∂ξAand Kis he p ojec o on o he holomo phic angen space in oduced in eq. (4). Subs i u ing (27) in (25) and in e changing di e en ia ion wi h espec o ξwi h in eg a ion o e gand g′, we now ha e he s a p oduc a le el one, (F1⋆ G1)(ξ) = F1(ξ)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂DG1(ξ).(28) 7 Fo highe Lwe ha e o conside ωL= (ω1)L. Equa ion (27) implies ωL(ξ, gg′) = X n+m≤L L! n!m! (L−n−m)! (ωω′)L−n−m(∂Aω)KAB(∂Bω′)n ×1 4(∂C∂Dω)KCEKDF (∂E∂Fω′)m (29) whe e we ha e used he abb e ia ions ω≡ω1(ξ, g) and ω′≡ω1(ξ, g′). The igh - hand side o his equa ion has o be exp essed in e ms o mul iple de i a i es ac ing on ωL(ξ, g) and ωL(ξ, g′). I con ains se e al di e en e ms wi h a gi en numbe o de i a i es. This means ha we ha e o dis inguish componen s o mul iple de i a i es o ωL. To his end, we decompose mul iple holomo phic de i a i es ∇A1···∇AnωLas de ined in eq. (9) wi h espec o i educible ep esen a ions o he s abili y g oup Hwhich ac s on he angen space. I will be su icien o conside he subg oup H0=SU(2) ×SU(N−2) o H. Rep esen a ions o H0will be deno ed by (J, J′) whe e Jis a ep esen a ion o he i s ac o and J′one o he second. To ind he ep esen a ion con en o a single holomo phic de i a i e ∇A=KAB∂B, no e ha he undamen al ep esen a ion [1] o SU(N) decomposes as [1]H0= ([1],[0]) ⊕([0],[1]) (30) in o he undamen al ep esen a ions o SU(2) and SU(N−2) upon es ic ion o H0. The wo componen s can be ob ained by p ojec ion wi h Pand 1 −P. Now use eqs. (4) and (5) o w i e ∇Ain e ms o (an i-) undamen al indices, ( A)ij∇A (ξ) = (1 −P) BPi j∂B (ξ).(31) The ma ix B∂B ans o ms like he aceless componen o [1]×[1]∗unde SU(N). Since he index iis p ojec ed by 1 − P o ([0],[1]) while jis p ojec ed by P o ([1],[0])∗, we ind ha he holomo phic de i a i e ans o ms like ([0],[1]) ⊗ ([1],[0])∗= ([1]∗,[1]). No e ha acelessness is gua an eed by he p ojec ions in eq. (31). By he same easoning, an an i-holomo phic de i a i e ¯ ∇A=KBA∂B ans o ms like ([1],[1]∗). A mul iple holomo phic de i a i e ans o ms like he symme ic enso p oduc o ncopies o he ep esen a ion ([1]∗,[1]). In o de o ob ain an explici exp ession o he decomposi ion o his p oduc , i u ns ou o be use ul o i s decompose he enso p oduc o ncopies o he undamen al ep esen a ion o SU(N). This can be done by conside ing he ac ion o he symme ic g oup Sn, whose elemen s pe mu e he ac o s in he enso p oduc [28]. The la e can hen be decomposed in o i educible ep esen a ions o SU(N)×Snwi h he help o cha ac e p ojec ion ope a o s. They p o ide he ollowing decomposi ion o uni y, 1 = X |J|=n PJwhe e PJ≡dJ n!X π∈Sn χJ(π)π . (32) He e, he sum is o e all Young diag ams wi h nboxes, χJis he cha ac e o he symme ic g oup in he ep esen a ion Jand dJ he dimension o ha ep esen a ion. 8 Since ρ1= (P ⊗P)ais a ank-1 p ojec o , he simple p oduc can be w i en as ω(ξ, g)ω(ξ, g′) = (P ⊗P)a(g⊗g)a(P ⊗P)a(g′⊗g′)a.(64) Using 1 = (P ⊗ P)a+ (P ⊗ (1 − P))a+ ((1 − P)⊗ P)a+ ((1 − P)⊗(1 − P))a, eqs. (62), (63) and (64) can be combined o ω(ξ, gg′) = ω(ξ, g)1 + ←− ∂AKAB−→ ∂B+1 4←− ∂A←− ∂BKACKBD−→ ∂C−→ ∂Dω(ξ, g′),(65) which is he desi ed exp ession. B Cohe en s a es We show ha he p ojec o ρLas gi en in eq. (18) has ank 1, and we p esen a simple a gumen o why he map om ma ices o unc ions is injec i e. To his end, we equi e a mo e explici ep esen a ion o ρL. The ec o space o he i educible ep esen a ion o SU(N) wi h symbol J= [0, L] can be ealised as a sub-space wi h ce ain symme y p ope ies o he space o 2L-index enso s. We cons uc i as he image o a Young symme ise . We i s assign enso indices o he boxes in he Young diag am o he ep esen a ion by pu ing he numbe s 1,2,...,2Lin ascending o de in o one column a e he o he , o ins ance 1 3 5 7 2468 o [0,4]. The Young symme ise is now de ined as Y[0,L]=2L L+ 1ALSL, AL= L Y i=1 1 2(1 −τi),SL=1 L!X π1∈R1 π1 1 L!X π2∈R2 π2 (66) whe e τiin e changes he wo boxes o he i h column o he diag am and Rideno es he se o pe mu a ions ha pe mu e he boxes o ow i. So SLsymme ises he ows o he diag ams, while ALan i-symme ises he columns. Bo h a e symme ic p ojec o s. The Young symme ise is a p ojec o , Y2 [0,L]=Y[0,L], bu no symme ic. Ope a o s in he ec o space o [0, L] can be unambiguously desc ibed as ope - a o s ˆ Fon 2L- enso s ha sa is y ˆ FY[0,L]=Y[0,L]ˆ F=ˆ F . (67) Now we can p o e ha ρLhas ank 1 by exp essing he ank-2 p ojec o Pin e ms o an o hono mal basis |ϕi,|ψio he complex plane on o which i p ojec s, P=|ϕihϕ|+|ψihψ|. The le el-1 p ojec o ρwas de ined in (16) as he p ojec ion o he enso p oduc o Pwi h i sel o he an i-symme ic ep esen a ion [0,1]. Since Y[0,1] educes o a single an i-symme isa ion, ρ≡(P ⊗P)a= (P ⊗P)Y[0,1] =|ϕψihϕψ|(68) 15 whe e |ϕψi ≡ 1 √2|ϕi|ψi−|ψi|ϕi.(69) Fo he p ojec o a le el L, we ob ain ρL= (ρ⊗···⊗ρ)Y[0,L]=|ϕψiLhϕψ|LY[0,L](70) The s a e |ϕψicomple ely cha ac e ises he plane ha co esponds o a poin in GN 2. I is he e o e na u al ha i occu s as a undamen al objec in he cons uc ion. The s a es |ϕψiLcoincide, up o a con en ional phase, wi h he gene alised cohe en s a es discussed in [11]. Since ˆ FY[0,L]=Y[0,L]ˆ F, FL(ξ) = hϕψ|Lˆ F|ϕψiL.(71) So FLis he co a ian symbol, as de ined in [11], o he ope a o ˆ F. This exp ession can be used o show ha he map is injec i e. We ha e o show ha ˆ Fcan be econs uc ed om FL. Since |ϕψiL= 2L/2AL(|ϕi|ψi)L, FL(ξ) = 2Lhϕ|hψ|LALˆ FAL|ϕi|ψiL. Due o he an i-symme isa ion be ween |ϕiand |ψi his unc ion can be homo- geneously ex ended o gene al (non-o hono mal) |ϕiand |ψi. We choose |ϕi= PN n=1 an|niand |ψi=PN n=1 bn|niwi h canonical basis ec o s |ni. By di e en- ia ing wi h espec o a,band hei complex conjuga es, all ma ix elemen s o SLALˆ FALSLcan be ob ained. Using he symme y (67), we ob ain SLˆ Fand hus also 2L L+1ALSLˆ F=ˆ F. C Res ic ions and di ec p oduc s We shall de i e he ela ion be ween he es ic ion o ep esen a ions o G=SU(N) o H=S[U(k)×U(N−k)] and he di ec p oduc o ce ain ep esen a ions used in sec ion 3. In his appendix, we will allow o columns o heigh N,J= [j1, j2,...,jN], in diag ams desc ibing ep esen a ions o SU(N). These do no lead o new ep esen a ions, since ep esen a ions di e ing only by jNa e uni a ily equi - alen , bu his gene alisa ion will make o mulas much simple . We embed Hin o SU(N) as ei(N−k)ϕU′0 0 e−ikϕU′′(72) whe e U′∈SU(k) and U′′ ∈SU(N−k) and eiϕ∈U(1). This shows ha H= [SU(k)×SU(N−k)×U(1)]/Znwhe e nis he leas common mul iple o kand N−k. Rep esen a ions o Hcan hus be conside ed as ep esen a ions o SU(k)× SU(N−k)×U(1) ha ep esen Zn i ially. This ixes he cha ge qo he U(1) ac o eiqϕ o he ep esen a ion modulo n. We will deno e hese ep esen a ions as (J′, J′′)qwhe e J′and J′′ a e symbols o SU(k) and SU(N−k) ep esen a ions, espec i ely, and qis he cha ge o he U(1) ep esen a ion. 16 The es ic ion o an SU(N) ep esen a ion o Hcan be w i en as JH=M J′,J′′ mJ J′,J′′ (J′, J′′)(N−k)|J′|−k|J′′|.(73) He e, |J|=Pijiis he numbe o boxes in he diag am Jand we assume ha he diag ams ha e been chosen such ha he o al numbe o boxes in J′and J′′ is he same as in J, |J|=|J′|+|J′′|.(74) No e ha he U(1) ep esen a ion is de e mined by he SU(k)×SU(N−k) ep esen- a ion, so he mul iplici ies mJ J′,J′′ a e he same as o he es ic ion om SU(N) o he la e . I is known ([24, 25], also see [26, 27]) ha hese can be ob ained om he decomposi ion o he di ec p oduc o he SU(N) ep esen a ions wi h diag ams J′and J′′, J′⊗J′′ =M J mJ J′,J′′ Jin SU(N). (75) He e, again, he es ic ion (74) on he numbe o boxes applies. No e ha in eq. (75) J′and J′′ a e in e p e ed as SU(N) ep esen a ions while hey a e in e p e ed as SU(k) espec i ely SU(N−k) ep esen a ions in eq. (73). We a e in e es ed in he case whe e he i ial ep esen a ion o Happea s on he igh -hand side o (73). This means ha J′and J′′ only ha e columns o heigh kand N−k, espec i ely, J′= [0k−1, L′,0N−k−1] and J′′ = [0N−k−1, L′′,0k−1] whe e 0k−1s ands o k−1 ze o en ies, e c. In addi ion, he U(1) cha ge has o anish, (N−k)|J′|=k|J′′|. Since |J′|=L′kand |J′′|=L′′(N−k), his implies L′=L′′ ≡L, so ha J′and J′′ a e complex conjuga e ep esen a ions o SU(N). We conclude ha a ep esen a ion o SU(N) con ains he i ial ep esen a ion o S[U(k)×U(N−k)] i and only i i appea s in he decomposi ion o he di ec p oduc ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (76) o some L. The mul iplici ies a e gi en by he mul iplici ies mJ [0k−1,L,0N−k−1],[0N−k−1,L,0k−1] in he p oduc . No e ha he mul iplici y does no depend on he numbe jNo columns o heigh Nin he diag am Jchosen o a gi en ep esen a ion. This means ha a ep esen a ion appea s in MLi and only i NL ≥ |J|whe e Jis he minimal diag am (jN= 0) o he ep esen a ion. The e o e ML⊂ML′i L < L′. Decomposi ion o he di ec p oduc Fo illus a ional pu poses, we will explici ly pe o m he decomposi ion o he di ec p oduc (76) in o i educible ep esen a ions. We can assume k≤N−ksince GN k∼ =GN N−k. The decomposi ion is achie ed by Young diag am echniques. Recall he ules o decomposing he di ec p oduc o wo i educible ep esen a ions o SU(N) [28]: 1. Label each box in he second diag am by i s ow numbe . 17 nN−k+1 boxes → nN−k+2 boxes → nN−k+kboxes → 1·· ·· ·· ·· ·· ·· ·· ·· ·· 1 2·· ·· ·· ·· ·· ·· 2 ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· k·· k 1·· 1 2 ·· 2·· ·· k·· k 2·· 2·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· k·· k ←n11’s ←n22’s ←nkk’s Figu e 1: Decomposi ion o he enso p oduc [0N−k−1, L, 0k−1]⊗[0k−1, L, 0N−k−1]. 2. A ach all boxes wi h 1’s o he i s diag am, hen all boxes wi h 2’s and so on, such ha (a) a all s ages he in e media e diag am co esponds o an i educible ep- esen a ion o SU(N), i.e. all columns s a in he i s ow and a e con- nec ed and he heigh o he columns mono onically dec eases om le o igh , (b) no column con ains any numbe mo e han once, and (c) when coun ed om he igh , he n- h idoes no appea be o e he n- h i−1. Applying hese ules o ou case, we ha e o a ach boxes wi h Lcopies o each o he numbe s 1,...,k o a ec angle o heigh N−kand wid h L. This is indica ed in igu e 1. We ha e al eady an icipa ed some ac s abou he esul ing dis ibu ion o boxes, which we will explain now. The numbe o 1’s in he i s ow has been deno ed by n1. The emaining L−n11’s ha e o be in he N−k+ 1-s ow. Deno ing he numbe o 2’s in he second ow by n2, he e can be a mos n1−n2 2’s in ow N−k+ 1, because o ule 2c. The e mus hus be a leas L−n12’s in ow N−k+ 2. Howe e , owing o ule 2b, he numbe o 2’s in ha ow is a mos L−n1. The e o e, he e ha e o be exac ly n1−n22’s in ow N−k+ 1 and L−n1 2’s in ow N−k+ 2. By he same easoning, we ind ha he numbe o i’s in ow N−k+l o l > 0 is equal o he numbe o i−l+ 1’s in he ow N−k+ 1 which is in u n equal o ni−l−ni−l+1 i i≥land 0 i i < l (we ha e pu n0≡L). Adding up, we ind o he numbe o boxes in ow N−k+l, nN−k+l= k X i=l (ni−l−ni−l+1) = L−nk−l+1 o l= 1,...,k. (77) G aphically, his means ha he subdiag am o columns L+ 1, L + 2,...combines, a e a o a ion by π, wi h he emainde o he diag am o a ec angle o wid h Land heigh N. In e ms o symbols J= [j1,...,jN], whe e jiis he numbe o columns o heigh i, his implies J=([m1,...,mk,0,...,0, mk,...,m1] i 2k < N , [m1,...,mk−1,2mk, mk−1,...,m1] i 2k=N(78) 18 wi h mi=ni−ni+1 whe e nk+1 ≡0. All non-nega i e alues o misa is ying Pk i=1 mi=n1≤Loccu . Fu he mo e, each diag am can be ob ained in only one way, since i is de e mined by he numbe s ni(i= 1,...,k). So all mul iplici ies equal 1. D Symme ic g oup In his appendix, we shall p o ide a p oo o he ac o isa ion p ope y (38). On he way, we will ecall some ac s abou ep esen a ions o he symme ic g oup and he associa ed p ojec o s PJused in he ex . These p ojec o s can be conside ed as elemen s o he g oup algeb a R[Sj]≡span Sn={A=Pπ∈SnAππ|Aπ∈R}, he se o o mal linea combina ions o g oup elemen s. The ec o space R[Sj] ca ies a ep esen a ion o he algeb a R[Sj] whose ac ion is gi en by le mul iplica ion. This ep esen a ion es ic s o a ep esen a ion o he subg oup Sno R[Sj]. I is usually called he egula ep esen a ion. Some imes i is con enien o iden i y R[Sj] wi h he algeb a F(Sn) o unc ions on he g oup Snby se ing A(π)≡Aπ. The ac ion o a g oup elemen πis hen gi en by (πA)(σ) = A(π−1σ). F(Sn) can in ac be conside ed as he dual ec o space o R[Sj] since each unc ion on he g oup can be linea ly and uniquely ex ended o a unc ion on R[Sj]. The iden i ica ion o R[Sj] wi h i s dual space can be ob ained om he inne p oduc hA, Bi ≡ X π∈Sn AπBπ=1 n! (ATB) (79) by pu ing A(B) = hA, Bi. The ace in eq. (79) is o e he egula ep esen a ion and we ha e se AT=PπAππ−1. O pa icula impo ance a e he cen al elemen s o R[Sj], ha a e in a ian unde conjuga ion wi h any g oup elemen π∈Sn,A=πAπ−1. They co espond o class unc ions, i.e. unc ions ha depend only on he conjugacy class o hei a gumen . An o hogonal basis in he subspace o class unc ions is gi en by he cha ac e s χJassocia ed wi h he i educible ep esen a ions Jo Sn, hχJ, χJ′i=n!δJJ′.(80) So e e y class unc ion can be expanded as A=X J AJχJwi h AJ=1 n!hχJ, Ai.(81) To each A∈R[Sj], one can associa e a cen al elemen Aby a e aging wi h espec o conjuga ion, A≡1 n!X π∈Sn πAπ−1=1 n!X JhχJ, AiχJ(82) whe e we ha e used (81) and he in a iance o χJunde conjuga ion. The egula ep esen a ion is in gene al educible. I con ains each i educible ep esen a ion Jwi h a mul iplici y ha is gi en by he dimension dJo he ep e- sen a ion. The componen con aining all copies o an i educible ep esen a ion J 19 can be ob ained as he image o he symme ic p ojec ion ope a o in oduced in eq. (32), in he dual pic u e, PJ=dJ n!χJ.(83) The decomposi ion o uni y 1 = X J PJ(84) p o ides a decomposi ion o R[Sj] in o o hogonal subspaces [29]. Now we will show how he a e aged enso p oduc o wo p ojec o s PJ1and PJ2on o i educible ep esen a ions o Sn1and Sn2can be exp essed in e ms o i educible p ojec o s. PJ1⊗PJ2can be ex ended o R[Sj] whe e n=n1+n2. By (82), we ha e PJ1⊗PJ2=X J aJχJ(85) wi h aJ=1 n!hχJ, PJ1⊗PJ2i.(86) The es ic ion o χJ o (π, σ)∈Sn1×Sn2decomposes in o i educible cha ac e s as χJ(π, σ) = X J1,J2 cJ J1J2χJ1(π)χJ2(σ) (87) whe e cJ J1J2∈Za e mul iplici ies o Clebsch-Go dan coe icien s. Wi h (80) we ge aJ=1 n!X J′ 1,J′ 2 cJ J′ 1J′ 2hχJ′ 1, PJ1ihχJ′ 2, PJ2i=dJ1dJ2 n!cJ J1J2(88) and he e o e PJ1⊗PJ2=X J dJ1dJ2 n!cJ J1J2χJ=X J dJ1dJ2 dJ cJ J1J2PJ.(89) By i e a ion, his esul can be gene alised o mul iple p oduc s, 1 dJ1···dJm PJ1⊗···⊗PJm=X J cJ J1...Jm 1 dJ PJ.(90) No e ha symme isa ion wi h espec o Snimplies symme isa ion wi h espec o Sn′⊂Sn, A⊗B⊗C=A⊗B⊗C . (91) Now we can p o e eq. (38). The igh -hand side o his equa ion can be w i en as a single ace like in eq. (36) bu wi h PJ eplaced by P[l]⊗P⊗m [0,1]. Owing o he sym- me ic enso s Sand Tall ac o s in he ace excep P[l]⊗P⊗m [0,1] a e symme ic unde conjuga ion, so P[l]⊗P⊗m [0,1] can be eplaced by i s symme ised e sion P[l]⊗P⊗m [0,1]. Now we can inse eq. (90). The only e m on he igh -hand side ha does no anish when p ojec ed by P⊗n o a ep esen a ion o SU(2) is J= [l, m, 0,...] wi h mul iplici y 1. The dimensions o he ep esen a ions [l] and [0,1] (o he symme ic g oup) a e 1, while he dimension o [l, m] appea ing in he denomina o o (90) jus cancels ha on he igh -hand side o (38), so we ob ain he le -hand side. 20 E P ojec ion o mul iple de i a i es We compu e he p oduc o mul iple (an i-)holomo phic de i a i es o ωLand ω′ L p ojec ed o he ep esen a ion ([l, m]∗,[l, m]) o he s abili y g oup, X(L) l,m ≡(∂A1···∂Al+2mωL)KA1...Al+2m,B1...Bl+2m [l,m](∂B1···∂Bl+2mω′L).(92) By eq. (9) and since K[l,m]con ains he p ojec o K, he de i a i es in his equa ion a e eally co a ian de i a i es, holomo phic ones ac ing on ω′and an i-holomo phic ones on ω. Equa ion (38) implies ha K[l,m]can be eplaced by d[l,m]K[l]⊗K⊗m [0,1], X(L) l,m =d[l,m](∂l+2mωL)K[l]⊗K⊗m [0,1](∂l+2mω′L) (93) whe e we ha e in oduced an index- ee no a ion. Using he second equali y o eq. (41), we ind KAB,CD [0,1] ∂C∂DωL=L(L+ 1) 2ωL−1KAB,CD [0,1] ∂C∂Dω(94) which i e a es o (K[0,1]∂∂)mωL=L!(L+ 1)! (L−m)!(L+ 1 −m)! ωL−m1 2K[0,1]∂∂ωm(95) because he iple de i a i e o ω anishes. The i s equali y o eq. (41) implies ha K[l]∂lωncon ains only single de i a i es o ω, whence K[l]⊗K⊗m [0,1]∂l+2mωL=K[l]⊗K⊗m [0,1]∂l(K[0,1]∂∂)mωL =L!(L+ 1)! (L−l−m)!(L+ 1 −m)! ωL−l−mK[l](∂ω)l1 2K[0,1](∂∂ω)m(96) whe e, in he i s s ep, we ha e used eq. (8). Since a simila equali y holds o an i-holomo phic de i a i es, and K[l]and K[0,1] a e p ojec o s, we ind X(L) l,m =d[l,m]L!(L+ 1)! (L−l−m)!(L+ 1 −m)!2 (ωω′)L−l−m ×(∂ω)lK[l](∂ω′)l1 4(∂∂ω)K[0,1](∂∂ω′)m. (97) Re e ences [1] A. Connes, Noncommu a i e Geome y, Academic P ess (1994). [2] J. 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