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
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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
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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)
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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)
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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)
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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)
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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 .
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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)
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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.
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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:
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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 .
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