The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields

JHEP05(2003)018 Published by Ins i u e o Physics Publishing o SISSA/ISAS Recei ed: Ap il 10, 2003 Accep ed: May 9, 2003 The spec um o he Di ac ope a o on cose spaces wi h homogeneous gauge ields B ian P. Dolan Dep . o Ma hema ical Physics, NUI, Maynoo h, I eland and 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 E-mail: [email p o ec ed]e Abs ac : The spec um and degene acies o he Di ac ope a o a e analysed on com- pac cose spaces when he e is a non-ze o homogeneous backg ound gauge ield which is compa ible wi h he symme ies o he space, in pa icula when he gauge ield is de i ed om he spin-connec ion. I is shown how he degene acy o he lowes Landau le el in he ecen ly p oposed highe dimensional quan um Hall e ec is ela ed o he A iyah-Singe index heo em o he Di ac ope a o on a compac cose space. Keywo ds: Field Theo ies in Highe Dimensions, Di e en ial and Algeb aic Geome y. c °SISSA/ISAS 2003 h p://jhep.sissa.i /a chi e/pape s/jhep052003018 /jhep052003018 .pd JHEP05(2003)018 Con en s 1. In oduc ion 1 2. Symme ic spaces 3 3. Non-symme ic spaces 11 4. Conclusions 13 A. Me ic and connec ion on G/H 14 B. Spec um o he Di ac ope a o on CP 216 C. Index heo em on SU(3)/U(1) ×U(1) 18 1. In oduc ion The e has long been a ui ul in e play be ween condensed ma e physics and ield heo y in pa icle physics, many concep s ha we e i s de eloped in he o me la e being applied o he la e and ice e sa. The quan um Hall e ec [1] has a ac ed he in e es o many high ene gy heo is s, no leas because he ac ional QHE exhibi s collec i e exci a ions which mimic a ac ional elec ic cha ge, bu also because he e a e deepe connec ions be ween he Hall e ec and s ing heo y [2]. Recen ly Zhang and Hu p oposed a highe dimensional analogue o he quan um Hall e ec , on S4[3], based on Haldane’s desc ip ion o he Hall e ec on S2wi h a magne ic monopole a he cen e, [4]. Zhang and Hu’s idea was de eloped u he in [5] and ex ended o complex p ojec i e spaces in [6]. The connec ion be ween he highe dimensional quan um Hall e ec and s ing heo y was analysed in [7]. The highe dimensional quan um Hall e ec in ol es a gene alisa ion o he Landau p oblem o pa icles mo ing on a compac cose space G/H, in he p esence o a backg ound gauge ield: such as a U(1) monopole on CP no a homogeneous SU(2) ins an on ield on S4. A common ing edien o he hese analyses is he calcula ion o he degene acy o he g ound s a e o pa icles mo ing in a homogeneous backg ound, i.e. a backg ound ield which has he symme y o he isome y g oup G. In p e ious wo ks, and in his pape also, he gauge g oup will be es ic ed o be he holonomy g oup H(o a ac o g oup o same i His a p oduc o smalle g oups). In [3, 6] he degene acy o he g ound s a e was calcula ed using g oup heo y: he non- ela i is ic hamil onian o a spinless pa icle mo ing in a homogeneous backg ound ield in ol es he quad a ic Casimi s o he g oups Gand Hand he allowed s a es in ol e – 1 – JHEP05(2003)018 i educible ep esen a ions o G ha con ain p e-o dained ep esen a ions o H. The di- mension o he ep esen a ion o Gco esponding o he g ound s a e is iden i ied wi h he degene acy o he g ound s a e. I is common in discussions o he quan um Hall e ec o igno e he elec on’s spin. Zhang and Hu ea ed scala pa icles sa is ying he exclusion p inciple, as did Ka abali and Nai : his is pe ec ly jus i ied when he Zeeman spli ing is la ge enough ha ansi ions be ween spin s a es can be igno ed and he g ound s a e is e ec i ely isola ed om he nex highes spin s a e. Ne e heless one is emp ed o ask wha is he ˆole o elec on spin in he highe dimensional quan um Hall e ec , and i will be a gued he e ha he e is an impo an quan i a i e elic o he Fe mionic na u e o he pa icles in he degene acy o he g ound s a e, o e and abo e he i ial consequences o he exclusion p inciple. I is shown in sec ion 2 ha he degene acies calcula ed in [3, 4, 6], o S4,S2and CP n espec i ely, a e ela ed o he index o he Di ac ope a o o Fe mions mo ing in he app op ia e backg ound ield: in ac he degene acy is he numbe o ze o-modes o he Di ac ope a o and, gene ically, his is he modulus o he index. Fu he mo e he g ound s a e wa e- unc ions, he highe dimensional analogues o he Laughlin wa e- unc ions, a e p ecisely he ze o-modes o he Di ac ope a o . We do no ha e o look a o disco e he eason o his — he squa e o he Di ac ope a o is no hing o he han he hamil onian o a non- ela i is ic Fe mion mo ing in a s a ic backg ound ield, (i D)2=−DαDα+R 41−i 2Fαβγαβ ,(1.1) whe e Ris he Ricci scala and he las e m ep esen s he Zeeman spli ing. (The e is an ex a e m on he igh hand side o (1.1) i he spin connec ion in ol es o sion, his equa- ion mus he e o e be modi ied o non-symme ic cose spaces wi h o sion as conside ed in sec ion 3.) Thus a non- ela i is ic pa icle mo ing in a s a ic backg ound magne ic ield is an example o supe symme ic quan um mechanics wi h a sel -dual p e-po en ial [8]. Since he Di ac ope a o is he mi ean, he eigen alues o he hamil onian (1.1) a e posi i e semi- de ini e and a ze o-mode equi es exac cancella ion o all h ee e ms on he igh hand side. I is shown in sec ions 2 and 3 ha , o speci ic homogeneous backg ound ields (ana- logues o monopole and ins an on ields on S2and S4) all h ee e ms on he igh hand side o (1.1) a e mu ually commu ing and so can be simul aneously diagonalised so ha all spin componen s decouple om each o he . The Di ac laplacian ∆ = −DαDαis i sel a posi i e ope a o on a compac space wi h posi i e cu a u e, so a ze o-mode o he Di ac ope a o , i one exis s, equi es a cancella ion o he lowes eigen alue o he laplacian wi h he lowes eigen alue o he sum o cu a u e and Zeeman e ms on he igh hand side o (1.1). In ac , o he homogeneous backg ound ields ha a e conside ed he e, he eigen alues o (1.1) can be de e mined pu ely in e ms o ce ain quad a ic Casimi s o he isome y g oup G and he holonomy g oup Hand a e gi en by equa ion (2.15) o , mo e gene ally, (3.9). E en wi hou calcula ing he ull spec um i is possible o ind he ep esen a ion o G wi h he lowes eigen alue o any Fe mion in a gi en ep esen a ion o he gauge g oup. I he chosen ep esen a ion o he gauge g oup allows o a ze o-mode o he Di ac ope a o – 2 – JHEP05(2003)018 hen he dimension o he ep esen a ion co esponding o he lowes eigen alue gene ically gi es he numbe o ze o-modes. Since his calcula ion in ol es only he lowes eigen alue o he Zeeman and he cu a u e e ms, which a e ixed in ad ance, he Fe mionic na u e o he pa icles can be igno ed and he p oblem educes o choosing he co ec ep esen a ions o G o scan in minimising he laplacian. This is p ecisely wha was done in [3] and [6]. The ne esul is ha he numbe o ze o-modes o he Di ac ope a o , o Fe mions in a gi en ep esen a ion o he gauge g oup, can be calcula ed simply om a knowledge o he quad a ic Casimi s o Gand he decomposi ions o i s ep esen a ions unde H7→ G. This gene alises he esul s o [3] and [6] o he quan um hall e ec on any cose space G/H wi h compac Lie g oups Gand H. O cou se an analysis o he spec um o he Di ac ope a o is o in insic in e es , e en wi hou e e ence o he quan um Hall e ec . In pa icula i is o ob ious impo ance in Kaluza-Klein heo ies and and s ing heo y. The layou o he pape is as ollows. In sec ion 2 he case o symme ic spaces G/H is ea ed in de ail and i is shown how he eigen alues o he Di ac ope a o in he p esence o a homogeneous backg ound gauge ield can be exp essed in e ms o quad a ic Casimi s C2(G) and C2(H). The examples o S2,S4and CP 2a e wo ked ou and compa ed o known esul s. Sec ion 3 ex ends he analysis o non-symme ic spaces wi h o sion and he example o SU(3)/U(1)×U(1) is ea ed in de ail — his space is o in e es in s ing heo y whe e i is a ises in he con ex o se en dimensional spaces wi h G2holonomy and hei conical singula i ies [9]. Sec ion 4 gi es a summa y o he esul s. Some echnical de ails a e elega ed o h ee appendices: appendix A e iews aspec s o he geome y o homogeneous spaces used in he ex . Appendix B p esen s he spec um o he Di ac ope a o on CP 2, in he p esence o a homogeneous backg ound SU(2) ×U(1) gauge ield. Appendix C p esen s a s anda d analysis o he A iyah-Singe index heo em on SU(3)/U(1) ×U(1) o compa ison wi h he esul s o sec ion 3. 2. Symme ic spaces Conside he Di ac ope a o o a Fe mion mo ing on a d-dimensional compac space, wi hou bounda y, in he p esence o a backg ound gauge ield: i D=iγαDα=iemαγαµ∂m+1 4ωm,αβγαβ +iAi m i¶,(2.1) whe e ωαβ =ωαβ,mdxmis he spin connec ion, Ai=Ai mdxm he gauge connec ion and ia e gene a o s o he gauge g oup (α, β = 1,2,···, d a e o hono mal indices and m= 1,2,···, d is a co-o dina e index). The γ-ma ices sa is y he usual Cli o d algeb a, nγα, γβo= 2δαβ ,wi h γαβ := 1 2hγα, γβi,(2.2) and emαa e d-beins o he me ic. The cu a u e and ield s eng h ollow om [Dα, Dβ] = iF i αβ i+1 4Rαβγδγγδ.(2.3) – 3 – JHEP05(2003)018 Fo a o sion ee connec ion squa ing he Di ac ope a o gi es (i D)2= ∆ + R 41−i 2Fαβγαβ ,(2.4) wi h ∆ = −DαDα he Di ac laplacian. We shall e e o ∆ + R 41as he kine ic ene gy and −i 2Fαβγαβ =−i 2Fi αβ iγαβ as he Zeeman ene gy. The laplacian he e is he laplacian ac ing on spino s, including he spin and he gauge connec ion, so i s spec um depends on bo h he me ic and he backg ound ield. On a cose space G/H, wi h Gand Hcompac g oups, i is na u al o use he G- in a ian me ic, o which he gene a o s o Ga e Killing ec o s and he holonomy g oup is H⊆SO(d). Fu he mo e we shall conside backg ound gauge ields which a e compa ible wi h he isome ies, in he sense ha Lie anspo o he ield s eng h Fby a Killing ec o Kgene a es a gauge ans o ma ion, LKF=g−1Fg , (2.5) whe e g∈ G, he g oup o gauge ans o ma ions. In pa icula his will be he case i we iden i y he gauge g oup wi h he holonomy g oup and he gauge connec ion wi h he spin connec ion — he de ails o his iden i ica ion a e gi en in appendix A. (A a ia ion on his is i he holonomy g oup ac o ises in o simple g oups and U(1) ac o s. When his is he case he gauge g oup can be aken o be one o he ac o s. Fo example his is he si ua ion o he homogeneous SU(2) ins an on on S4, whe e S4= SO(5)/SO(4) and, a he le el o he algeb as, H= SU(2) ×SU(2) so we can ake he gauge g oup o be jus SU(2).) Le Abe he gene a o s o he isome y g oup G, wi h [ A, B] = i ABC C, and i he gene a o s o he holonomy g oup H. Then he cu a u e 2- o ms o a G-in a ian me ic o a symme ic space can be aken o be (see appendix A), Rαβ=1 2Rαβγδeγ∧eδ=1 2 αβi iγδeγ∧eδ.(2.6) Iden i ying he gauge connec ion wi h he spin connec ion gi es ise o he ield s eng h, Fi=1 2Fi αβeα∧eβ=1 2 iαβeα∧eβ.(2.7) Fo a symme ic space he Riemann enso is co- a ian ly cons an and his means ha he abo e ield s eng h is co- a ian ly cons an , DαFi βγ = 0 .(2.8) In pa icula he laplacian commu es wi h he Zeeman e m in he hamil onian. Wi h his choice o backg ound ield he commu a o (2.3) simpli ies, [Dα, Dβ] = i iαβ ½(1⊗ i)−i 4 iγδ ³γγδ ⊗1´¾.(2.9) Now Ti:= −i 4 iγδγγδ (2.10) – 4 – JHEP05(2003)018 a e a ep esen a ion o he gauge g oup (which may be educible, in gene al), [Ti, Tj] = i ijkTk,(2.11) so [Dα, Dβ] = iαβ Di(2.12) wi h Di:= i{(1⊗ i) + (Ti⊗1)}(2.13) being he gene a o s o Hin he enso p oduc ep esen a ion o iwi h he spino ep- esen a ion Ti. This allows he laplacian o be exp essed as he di e ence o quad a ic Casimi s, ∆ = −DαDα=−DADA+DiDi=C2(G, ·)−C2(H, Di).(2.14) Fo spino s in a gi en ep esen a ion io he gauge g oup C2(H, Di) in his exp ession is always calcula ed in he ixed ep esen a ion (2.13), which in gene al in ol es educible ep esen a ions o G, while he ep esen a ions used in C2(G, ·) ange o e all i educible ep esen a ions o G han con ain (2.13). In pa icula he c oss- e m −2 i⊗Ti om (Di)2 in (2.14) exac ly cancels he Zeeman ene gy in (2.4) and, as desc ibed in appendix A, he second o de Casimi o he ep esen a ion Tiis ela ed o he Ricci scala by C2(H, Ti) = R/8. The eigen alues o he squa e o he Di ac ope a o (2.4) can hen be exp essed pu ely in e ms o quad a ic Casimi s: E=C2(G, ·)−C2(H, i) + R 81.(2.15) This cons uc ion will now be illus a ed wi h some examples. (i) S2∼ =SO(3)/SO(2).This was he geome y o iginally s udied by Haldane in he con ex o he quan um Hall e ec [4]. The isome y g oup is gene a ed by he algeb a o SU(2) [ A, B] = i²ABC C(2.16) and we a e ee o choose 3 o gene a e he U(1) holonomy. Fo mula (A.9) o appendix A gi es Rαβ =1 2²αβ3²3γδeγ∧eδ(2.17) so R12 = e1∧e2(2.18) a e he cu a u e 2- o ms o a sphe e o uni adius. Also F3= e1∧e2(2.19) is he ield s eng h i a magne ic monopole a he cen e o he sphe e. Ac ually his co esponds o a monopole o cha ge 2, since 1 2πZS2 e1∧e2= 2 (2.20) – 5 – JHEP05(2003)018 is he Che n class o he angen bundle (which is equal o he Eule cha ac e is ic). In gene al we can pu a monopole o any in eg al cha ge a he cen e o he sphe e F3=M 2e1∧e2.(2.21) (Al e na i ely we can wo k wi h a monopole o cha ge 2 and conside Fe mions o any hal -in eg al cha ge in his backg ound.) Choosing γ1=σ1and γ2=σ2, wi h σ1and σ2Pauli ma ices, we ha e i 2F3 αβγαβ =iF 3 12(iσ3) = −M 2µ1 0 0−1¶.(2.22) The Ricci scala o a sphe e o uni adius is 2, so equa ion (2.4) gi es (i D)2= ∆ + 1 21+M 2µ1 0 0−1¶.(2.23) Fo posi i e M his indica es ha he e a e spin down ze o-modes o he Di ac ope a o i ∆ + 1 2=M 2(2.24) while o nega i e M he e a e spin up ze o-modes i ∆ + 1 2=−M 2.(2.25) The e a e o cou se no ze o-modes o M= 0 as equi ed by Lichne owicz heo em. In his example io (2.13) is jus a numbe , M/2, and Tiis σ3/2 so D3=iµM+1 20 0M−1 2¶⇒D3D3=−µ¡M+1 2¢20 0¡M−1 2¢2¶.(2.26) The eigen alues o he laplacian (2.14) a e he e o e ∆j=j(j+ 1) −µM±1 2¶2 ,(2.27) as discussed in [3], so eigen alues o (2.23) a e Ej=(2j+ 1)2−M2 4,(2.28) which can also be ob ained di ec ly om (2.15). Fo M= 0 his ep oduces he well-known esul ha he spec um o he Di ac ope a o is linea in angula momen um (see e.g. [10]). Fo M6= 0 he ep esen a ions jo SU(2) ha appea in a ha monic expansion o ∆ a e es ic ed o hose ha con ain he U(1) ep esen a ion o cha ge M±1 2, i.e. j=|M|−1 2+k wi h ka non-nega i e in ege , Ej=k(k+|M|).(2.29) – 6 – JHEP05(2003)018 The e a e ze o-modes o k= 0, and jmin =M−1 2 o posi i e Mo −(M+1 2) o nega i e M. In ei he case he degene acy o he g ound s a e is d(jmin) = 2jmin + 1 = |M|,(2.30) which is he numbe o ze o-modes o he Di ac ope a o . No e he shi o jmin away om |M|by 1/2, due o he in insic spin o he Fe mion. The degene acy (2.30) ela es o he A iyah-Singe index heo em which s a es ha he he index o he Di ac ope a o is minus he i s Che n class [11], ν=ν+−ν−=−1 2πZS2 F3=−M , (2.31) whe e ν+is he numbe o posi i e chi ali y ze o-modes and ν− he numbe o nega i e chi ali y ze o-modes. Indeed he g ound s a e wa e- unc ions in [4] o he in ege quan um Hall e ec , sphe ical analogues o he Laughlin wa e- unc ions, a e p ecisely hese ze o- modes. The case |M|= 1 co esponds o jmin = 0, in his case he gauge connec ion exac ly cancels he spin connec ion o he ele an chi ali y and single ze o-mode o he Di ac ope a o is a cons an spino . The abo e calcula ion can be ep esen ed g aphically using Young ableaux, which will be use ul in mo e complica ed si ua ions o ollow. The undamen al o SU(2) decomposes as SU(2) →U(1) 2→11+1−1.(2.32) Deno ing 11by ×and 1−1by • he (p+1)-dimensional i educible ep esen a ion o SU(2) con ains × ·· × | {z } s • ·· • | {z } ⊂·· | {z } p (2.33) wi h p= +s. Fixing he U(1) cha ge o be Qcons ains s− =Qso p= 2 +Q. The g ound s a e ene gy o a Fe mion in his backg ound can now be ound by minimising ∆, since all he o he e ms in he ene gy a e cons an s o ixed Q, ha is by minimising p 2³p 2+ 1´= ( + 1) + Q 2(2 + 1) + Q2 4.(2.34) I Q>0 his is minimised by = 0, so p=Qand he degene acy o he g ound s a e is Q+1. I Q<0 i is minimised by =pand hen p=−Q, so he degene acy is −Q+1. In ei he case he g ound s a e has p=|Q|and he degene acy is |Q|+ 1. Clea ly |Q|= 2jmin and he U(1) cha ge Qis no jus he monopole cha ge M, bu includes a shi o accoun o he in insic spin o he Fe mion, |Q|=|M|−1. This me hod, using ep esen a ion heo y o desc ibe he kine ic ene gy and calcula e he degene acy o he g ound s a e, was used in [6]: hough in ha e e ence he pa icles we e ea ed as scala s so he e was no in insic spin — he e was he e o e no Zeeman ene gy o make he o al g ound s a e ene gy anish and no shi in he cha ge o accoun o he in insic spin o he pa icles. The echnique is howe e applicable o bo h he laplacian o scala s and he squa e o he Di ac ope a o because, o a gi en gauge backg ound and ep esen a ion, hey only di e by cons an s. I has he ad an age o a oiding an explici calcula ion o he ull eigen alue spec um o he Di ac ope a o . – 7 – JHEP05(2003)018 (ii) S4∼ =SO(5)/SO(4).The nex example, S4, was he case s udied in he i s pape on he highe dimensional quan um Hall e ec , [3]. In his case he algeb a o he holonomy g oup is SU(2) ×SU(2) and we can ake he gauge g oup o be jus one SU(2) ac o . The Riemann enso can be spli in o sel -dual and an i-sel -dual pa s and hese co espond o he cu a u es a ising om he wo SU(2) ac o s o he holonomy g oup. Choosing, o example, he sel -dual SU(2) ac o he esul ing SU(2) backg ound gauge ield is he homogeneous ins an on o cha ge one, which has SO(5) symme y on S4[12] ( his pape was published a li le a e he BPST ins an on [13], bu he echniques a e e y enligh ening and highligh he analogy wi h he Wu-Yang monopole — Yang calls his homogeneous ins an on con igu a ion a non-abelian monopole). Rep esen a ions o SO(5) can be labelled by wo in ege s pand qwi h p≥q. The second o de Casimi and dimension a e gi en by C2(p, q) = p2+q2 2+ 2p+q(2.35) and d(p, q) = 1 6(p+q+ 3)(p−q+ 1)(p+ 2)(q+ 1) (2.36) espec i ely. Now suppose we ha e a pa icle on S4in he ep esen a ion Io SU(2) in he back- g ound o a homogeneous ins an on. Demanding ha an SO(5) i educible ep esen a ion con ains he Io SU(2) implies [12] p−q= 2I , (2.37) and so C2(q+ 2I, q) = q2+q(2I+ 3) + 2I2+ 4I . (2.38) The Ricci scala o he uni ou -sphe e is R= 12 so he eigen alues (2.15) o (i D)2 o a Fe mion in he ep esen a ion Jo he gauge g oup a e hus E=q2+q(2I+ 3) + 2I2+ 4I−2J(J+ 1) + 3 2.(2.39) (The ac o o wo in on o he gauge Casimi J(J+ 1) he e is due o he ac ha he Di ac ope a o is non-chi al, he holonomy g oup is SU(2) ×SU(2), and bo h chi ali ies couple o he gauge g oup in he same way.) The o al isospin Iis a combina ion o he gauge isospin Jand he in insic spin o he Fe mion, I=J±1/2, so he ene gy le els a e labelled by he in ege qand E+(q) = q2+q(2I+ 3) + 2(2I+ 1) E−(q) = q2+q(2I+ 3) (2.40) bo h wi h degene acies d(2I+q,q) = 1 6(2q+ 2I+ 3)(2I+ 1)(q+ 2I+ 2)(q+ 1) .(2.41) – 8 – JHEP05(2003)018 Then he subse eαa e o hono mal 1- o ms o a G-in a ian me ic on G/H and he emaining 1- o ms eican be expanded on G/H as ei= Πiαeα. The o sion ee H- alued2connec ion ωαβis hen de ined by deα+ωαβ∧eβ= 0 (A.5) and e alua es o ωαβ=µ1 2 αβγ + αβiΠiγ¶eγ.(A.6) The cu a u e 2- o ms can hen be calcula ed om Rαβ=dωαβ+ωαγ∧ωγβ(A.7) esul ing in Rαβ=1 4¡2 αβi iγδ + αβ² ²γδ − αγ² ²βδ¢eγ∧eδ.(A.8) On a symme ic space hese educe o he simple o m Rαβ=1 2¡ αβi iγδ¢eγ∧eδ,(A.9) so he Riemann enso has componen s Rαβγδ = αβi iγδ .(A.10) On a non-symme ic space he e is a second, e y use ul, connec ion ha comes om in oducing a o sion enso which is iden i ied wi h he non-symme ic s uc u e cons an s: Tαβγ = αβγ (A.11) gi ing o sion 2- o ms Tα=1 2 αβγeβ∧eγ.(A.12) Then he connec ion wi h o sion is de ined ia deα+ωαβ∧eβ=Tα(A.13) which leads o ωαβ=1 2 αβiΠiγeγ.(A.14) The esul ing cu a u e 2- o ms a e Rαβ=1 2 αβi iγδeγ∧eδ,(A.15) gi ing cu a u e enso Rαβγδ = αβi iγδ .(A.16) 2Fo no a ional simplici y we do no dis inguish be ween he g oup and he algeb a he e. – 15 – JHEP05(2003)018 The Ricci scala o he connec ion wi h o sion is hen easily e alua ed as R=Rαβαβ = αβi iαβ = ABi iAB − jki ijk ,(A.17) which can be de e mined using he app op ia e quad a ic Casimi s o H. A pa icula ins ance o his is when His i ial so G/H ∼ =G. Then ωαβ= 0 and Rαβ= 0, all co- a ian de i a i es a e i ial and Tαis called he pa allelising o sion o G. On a symme ic space, o cou se, (A.8) and (A.15) a e iden ical because αβγ = 0. In ac i is no di icul o show, using (A.14), (A.15) and he Jacobi iden i y, ha Rαβγδ in (A.16) is co- a ian ly cons an , ∇²Rαβγδ = 0 .(A.18) On a gene ic d-dimensional mani old he cu a u e 2- o ms (A.7) a e SO(d) Lie algeb a alued 2- o ms, bu on G/H bo h (A.8) and (A.15) a e H alued 2- o ms, whe e H⊆SO(d). This means ha we can ake linea combina ions o (A.15) ha lie in H wi hou losing any in o ma ion. Fo example, i His semi-simple, aking he combina ion iαβRαβ =1 2³C2(G, adj)−C2(H, adj)´ iγδeγ∧eδ(A.19) sugges s de ining Fi:= 1 2 iγδeγ∧eδ(A.20) and hen Fia e H- alued 2- o ms which a e equi alen o (A.15) ( his o mula is easily adap ed o he case whe e Hcon ains U(1) ac o s). B. Spec um o he Di ac ope a o on CP 2 The calcula ion o he ull spec um o he Di ac ope a o on CP 2p oceeds as ollows ( he spec um on CP n, wi h nodd and no backg ound gauge ield, has been conside ed in [17]). Fo SU(3) he second o de Casimi and dimension a e C2(p, ¯p) = 1 3³p(p+ 3) + ¯p(¯p+ 3) + p¯p´(B.1) and d(p, ¯p) = 1 2(p+ ¯p+ 2)(p+ 1)(¯p+ 1) (B.2) espec i ely. Wi h p= +sand ¯p= ¯ + ¯s, as in he ex , he cons ain s ead (s−¯s)−2( −¯ ) = Yand s+ ¯s 2=I , (B.3) whe e Yis e en (odd) o Iin eg al (hal -in eg al). Now he spec um depends on whe he |Y| ≥ 2Io |Y| ≤ 2I: – 16 – JHEP05(2003)018 •I Y≥2I hen ¯ ≥ : in his case le n= ¯s, so n= 0,...,2I, =kand ¯ = k+n−I+Y 2, o ka non-nega i e in ege . I Y≤ −2I hen ≥¯ : in his case le n=s, so n= 0,...,2I, =k+n−I−Y 2and ¯ =k, o ka non-nega i e in ege . In ei he case C2(p, q) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2+Y2 12 +I(I+ 1) . (B.4) Fo C2(H, i) in (2.15) ake he U(1) backg ound o ha e ixed cha ge Mand he SU(2) backg ound o ha e isospin J, so C2(H, i) = M2 12 +J(J+ 1) ,(B.5) ( he 1 12 he e is because he U(1) gauge ield is a mul iple o 1 2√3 o con o m wi h he no malisa ion o 8in appendix C). Finally he Ricci scala o CP 2can be e alua ed om (A.17) and he s uc u e cons an s in appendix C o be R= 6 so, pu ing all his oge he , he eigen alues o (2.15) a e E(k, n) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+ +(|Y|+ 3)2 12 −M2 12 +I(I+ 1) −J(J+ 1) ,(B.6) while he degene acies a e d(k, n) = 1 2µ2k+n+I+ 2 + |Y| 2¶µk+ 2n−I+ 1 + |Y| 2¶(k−n+ 2I+ 1) , (B.7) wi h n= 0,...,2Iand k≥0 an in ege . I is impo an o unde s and how he gauge cha ges Mand Ja e ela ed o he o al cha ges Yand I(which include he spin connec ion). The e a e ou cases o conside : 1. M=Y±3 and I=J, hese a e s a es ha couple o he U(1) pa o he spin connec ion and no he SU(2) pa ( he ±3 ela es o he ac ha he i s Che n class o he angen bundle o CP 2is 3). The spec um is E(k, n) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y|∓Y 2; (B.8) Fo SU(2) single s I=J= 0, so n= 0, his spec um ag ees wi h he esul s o [19] (in he no a ion o ha e e ence M= 2m+ 3, so Y/2 = mo m+ 3). 2. M=Yand I=J±1 2, hese a e s a es ha couple o he SU(2) pa o he spin connec ion and no he U(1) pa . The spec um is E(k, n) = kµk+n+2+I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2+I+ 1 ; (B.9) E(k, n) = kµk+n+2+I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2−I . (B.10) – 17 – JHEP05(2003)018 Fo |Y|>2Ionly case 1 abo e allows o ze o-modes (when k=n= 0). Fo |Y|= 2I he e a e ze o-modes in bo h cases. •I 0 ≤Y≤2I, le n=¯s−s+Y 2, so n=−I+Y/2,...,I +Y/2. Then: ei he =k+|n|and ¯ =k; o =kand ¯ =k+|n|. I −2I≤Y≤0, le n=s−¯s−Y 2, so n=−I−Y/2,...,I −Y/2. Then: ei he =k+|n|and ¯ =k; o =kand ¯ =k+|n|. In ei he case: C2(p, q) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+Y2 12 +I2+ 2I . (B.11) The eigen alues o (2.15) a e he e o e E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+ +Y2−M2 12 +I2+ 2I−J(J+ 1) + 3 4,(B.12) wi h degene acies d(k, n)= ½(k+I+ 1)2+|n|(k+I+ 1) −(4n−|Y|)(2n−|Y|) 4¾µk+I+ 1 + |n| 2¶, (B.13) whe e −I+|Y| 2≤n≤I+|Y| 2and k≥0. (The degene acy is always an in ege because o he es ic ion ha Yis odd when Iis hal -in eg al and e en i Iis in eg al.) Again he e a e ou possibili ies: 1. M=Y±3 and I=J, E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n|Y| 2+I∓Y 2; (B.14) 2. M=Yand I=J±1 2, E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+ 2I+ 1 ; (B.15) E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|.(B.16) These eigen alues a e bounded below by ze o and, o |Y|<2I, only (B.16) al- lows o ze o-modes (when k=n= 0). When |Y|= 2I he spec um ag ees wi h equa ions (B.8)-(B.10). C. Index heo em on SU(3)/U(1) ×U(1) In his sec ion we gi e an explici e alua ion o he index o he Di ac ope a o on SU(3)/U(1) ×U(1), using s anda d di e en ial-geome ic echniques. Le λA;A= 1,...,8 be he Gell-Mann ma ices o SU(3), so [ A, B] = i ABC Cwi h A=λA 2(C.1) – 18 – JHEP05(2003)018 and 123 = 1 , 453 =− 673 = 471 =− 561 = 462 = 572 =1 2, 458 = 678 =√3 2. (C.2) In he no a ion o appendix A, i= 3,8 and α= 1,2,4,5,6,7, when 3=1 2  100 0−1 0 000  and 8=1 2√3  1 0 0 0 1 0 0 0 −2  (C.3) a e chosen as he U(1) ×U(1) gene a o s. This space is no symme ic, because some o he αβγ 6= 0. The cu a u e 2- o ms, o he spin connec ion wi h o sion desc ibed in appendix A, a e R12 =1 2¡2 e1∧e2+ e4∧e5−e6∧e7¢ R45 =1 2¡e1∧e2+ 2 e4∧e5+ e6∧e7¢ R67 =1 2¡−e1∧e2+ e4∧e5+ 2 e6∧e7¢. These a e no independen , since R45 =R12 +R67, hey a e associa ed wi h wo U(1) ield s eng hs: F3 3=1 2 αβ3³eα∧eβ´ 3=1 4¡2 e1∧e2+ e4∧e5−e6∧e7¢  1 0 0 0−1 0 0 0 0   F8 8=1 2 αβ8³eα∧eβ´ 8=1 4¡e4∧e5+ e6∧e7¢  1 0 0 0 1 0 0 0 −2  . We ex ac U(1) single s by p ojec ing ou he op le -hand componen s F(3) =1 4¡2 e1∧e2+ e4∧e5−e6∧e7¢ F(8) =1 4¡e4∧e5+ e6∧e7¢. A monopole ield wi h cha ges (M, N) is a linea combina ion o hese F=MF(3) +NF(8) (C.4) om which F∧F∧F=3 16M¡N2−M2¢e124567 (C.5) (we use he sho hand e124567 = e1∧e2∧e4∧e5∧e6∧e7). The index o he Di ac ope a o is [20] ν=1 (2π)3Z½1 6F∧F∧F+1 48F∧ (R∧R)¾.(C.6) – 19 – JHEP05(2003)018 Explici calcula ion e eals ha F∧ (R∧R) = 0, so ν=1 256π3M¡N2−M2¢V, whe e V=Re124567 is he olume o SU(3)/U(1) ×U(1). The no malisa ion can be ixed by using he ac SU(3)/U(1) ×U(1) has Eule cha ac e is ic χ= 6, so χ=1 3! 1 (4π)3Z²α1···α6Rα1α2∧Rα3α4∧Rα5α6= 6 .(C.7) his ixes V= 32π2so ν=1 8M¡N2−M2¢.(C.8) No e ha Mand Nmus be ei he bo h e en o bo h odd o ν o be an in ege . Re e ences [1] S.M. Gi in, The quan um hall e ec : no el exci a ions and b oken symme ies, cond-ma /9907002. [2] B.A. Be ne ig, J.H. B odie, L. Susskind and N. Toumbas, How bob laughlin amed he gian g a i on om aub-nu space,J. High Ene gy Phys. 02 (2001) 003 [hep- h/0010105]. [3] S.-C. Zhang and J.-p. 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