a Xi :hep- h/0207078 2 14 No 2002
P ep in ypese in JHEP s yle - HYPER VERSION DIAS-STP-02-07
The S anda d Model Fe mion Spec um
F om Complex P ojec i e Spaces
B ian P. Dolan and C. Nash
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
bdolan@ hphys.may.ie, cnash@ hphys.may.ie
Abs ac : I is shown ha he qua ks and lep ons o he s anda d model, including
a igh -handed neu ino, can be ob ained by gauging he holonomy g oups o complex
p ojec i e spaces o complex dimensions wo and h ee. The spec um eme ges as
chi al ze o modes o he Di ac ope a o coupled o gauge ields and he demons a ion
in ol es an index heo em analysis on a gene al complex p ojec i e space in he
p esence o opologically non- i ial SU(n)×U(1) gauge ields. The cons uc ion
may ha e applica ions in ype IIA s ing heo y and non-commu a i e geome y.
Keywo ds: Field Theo ies in Highe Dimensions, Di e en ial and Algeb aic
Geome y, Non-Commu a i e Geome y.
Con en s
1. In oduc ion 1
2. The Pa icle Spec um F om CPn3
3. Conclusions 6
A. The Index Theo em Fo CPn8
1. In oduc ion
I has been a long s anding goal o heo e ical pa icle physics o uni y space- ime
symme ies wi h he in e nal SU(3)×SU(2)×U(1) gauge symme y o he s anda d
model. Mos cu en esea ch in his di ec ion elies on s ing heo y in highe
dimensions, hoping o de i e g and uni ied heo ies as low ene gy limi s o he ull
heo y, and a c ucial aspec o his p og amme is he ˆole o compac in e nal spaces.
Compac in e nal spaces we e i s in oduced in o physics in Kaluza-Klein models
whe e a cose space G/H wi h isome y g oup Gand holonomy g oup Hcan gi e ise
o a gauge g oup Gin 4-dimensional space- ime. A pu e Kaluza-Klein app oach was
la gely abandoned in he 80s due in pa o he ealisa ion ha i was di icul , i no
impossible, o ob ain chi al Fe mions his way [1]. In his pape we ake a di e en
app oach o in e nal cose spaces, ocusing on he holonomy g oup H a he han G.
We shall show ha a single gene a ion o he s anda d model spec um, including a
igh handed neu ino, can a ise om he complex p ojec i e spaces CP2and CP3.
The app oach adop ed he e has he a ac i e ea u e ha i is somewha mo e
in une wi h he spi i o gene al ela i i y han s anda d Kaluza-Klein heo y. In he
s anda d app oach one assumes ha he cose space has a speci ic me ic which has
isome y g oup G, which is somewha con a y o he philosophy gene al ela i i y
whe e a pa icula me ic is me ely one solu ion o Eins ein’s ield equa ions and
he e may be many o he s wi h smalle isome ies, indeed a gene ic solu ion has
none. In he cons uc ion used he e i is he holonomy g oup ha is impo an and
o a complex mani old wi h eal dimension 2n his g oup is gene ically U(n). Fo
4-dimensional space ime equipped wi h a Lo en zian me ic he holonomy g oup is
gene ically SO(1,3) and gene al ela i i y can be iewed as a gauge heo y o angen
space o a ions using his g oup. Inco po a ing spino s in o gene al ela i i y equi es
–1–
gauging he double co e o SO(1,3), namely Spin(1,3) ∼
=Sl(2,C). I a compac
complex in e nal space wi h eal dimension 2nis added he ex a holonomy g oup is
gene ically U(n) and so i seems qui e na u al o gauge SU(n)×U(1). Taking wo
in e nal spaces wi h n= 2 and n= 3 would gi e SU(3)×SU(2)×U(1)×U(1) which
is no qui e wha is equi ed, bu is close.
To see how he s anda d model spec um migh a ise in his way conside i s a
Di ac spino on a gene al compac complex mani old o eal dimension 4. This has
4 componen s and decomposes as
(2,1) + (1,2)−→ 20+ (11+1−1) (1.1)
unde Spin(4) ∼
=SU(2) ×SU(2) →SU(2) ×U(1). Shi ing he U(1) cha ge by −1
and escaling he esul by 1/2 gi es
2−1/2+ (10+1−1).(1.2)
On he o he hand a compac complex mani old o eal dimension 6 has 8 componen
Di ac spino s which decompose as
4+4−→ (31+1−3) + (3−1+13) (1.3)
unde Spin(6) ∼
=SU(4) →SU(3) ×U(1). Taking a single chi al spino in he 4o
SU(4), shi ing he U(1) cha ge by 3 and escaling he esul by 1/6, gi es
(32/3+10).(1.4)
A single gene a ion o he s anda d model spec um, including a igh -handed neu-
ino, can be ob ained by enso ing (1.2) wi h (1.4) and simply adding he U(1)
cha ges. These g oup heo y decomposi ions a e alid on any complex mani olds
o eal dimension 4 and 6 espec i ely and he shi in he U(1) cha ges may be
achie ed by coupling he Fe mions o U(1) gauge ields wi h app op ia e opological
(monopole) cha ges. Ma hema ically his equi es enso ing he spin bundle wi h
an app op ia e line bundle, which may o may no be possible on any gi en man-
i old depending on whe he o no app op ia e line bundles exis . I is shown in
sec ion 2 ha o CP2and CP3 he equi ed line bundles do indeed exis (indeed
o CP2spino s canno e en be de ined unless such a line bundle is in oduced).
Howe e ob aining he s anda d model spec um equi es in oducing some na u al
highe ank bundles as well. Fu he mo e, an index heo em analysis shows ha he
ep esen a ions (1.2) and (1.4) can be ealised as ze o modes o he Di ac ope a o
wi h p ecisely he co ec handedness o one gene a ion o he s anda d model, in-
cluding a igh -handed neu ino. This is a opological s a emen , i is comple ely
independen o he choice o me ic on hese spaces.
O cou se he e is he ques ion o gene a ions— he cons uc ion p esen ed he e
only p oduces a single gene a ion o he s anda d model. This poin is elabo a ed
–2–
on in he concluding sec ion 3, along wi h some sugges ions o how he cons uc ion
migh i in o iable models such as ype IIA supe s ing heo y and non-commu a i e
geome y. De ails o he index heo em on CPn, equi ed o he analysis in he ex ,
a e con ained in an appendix, whe e closed o m exp essions o he index o he Di ac
ope a o coupled o opologically non- i ial SU(n)×U(1) gauge ields a e de i ed.
2. The Pa icle Spec um F om CPn
In his sec ion i will be a gued ha he spec um o a single gene a ion o he
s anda d model can be ob ained om CP2×CP3by gauging he holonomy g oup,
U(2)×U(3), wi h he wo U(1) ac o s iden i ied in one pa icula combina ion. This
gene alises he case o CP2which gi es he elec oweak sec o [2].
The analysis is based on he index heo em o he Di ac ope a o on
CPn∼
=SU(n+ 1)
U(n)(2.1)
de i ed in he appendix. We a e in e es ed in ze o modes o he Di ac ope a o ,
o Fe mions which a e ei he single s o ans o m unde he undamen al n ep-
esen a ion o SU(n), when opologically non- i ial gauge ields a e p esen (CPn
analogues o he monopole ield on S2and he SU(2) BPST ins an on on S4). The
gauge g oup is aken o be he holonomy g oup U(n) o CPn. The index is he
di e ence be ween he numbe o posi i e chi ali y ze o modes o he Di ac ope a o
and he numbe o nega i e chi ali y ze o modes and, o con enience, he impo an
o mulae om he appendix a e ep oduced he e. Fo a single wi h U(1) cha ge
Y(n)=q, whe e qis an in ege , he index is
νq=1
n!(q+ 1) ···(q+n), q ∈Z(2.2)
co esponding o a line bundle o e CPnwi h gauge g oup U(1). A Fe mion in he
undamen al ep esen a ion o SU(n), wi h a U(1) cha ge Y(n)=q+1
n, has index
νq,n=(q+ 1) ···(q+ (n−1))(q+n+ 1)
(n−1)! .(2.3)
We a e ee o escale he U(1) cha ges di e en ly o di e en alues o n, bu o a
gi en n he a io o he single cha ge q o he undamen al cha ge q+1
nis ixed.
The simples case is CP1∼
=S2, whe e he holonomy g oup is U(1). In oducing
U(1) gauge ields and Fe mions wi h cha ge q he e a e νq=q+ 1 ze o modes o
he Di ac ope a o . The case q=−1 gi es no unpai ed ze o modes—gene ically
he e a e none a all bu e en when ze o modes exis le and igh -handed pa icles
wi h cha ge −1 occu in pai s, like an elec on in QED. Fo q6=−1 he heo y is
necessa y chi al. Fo example choosing he con en ion ha posi i e νco esponds o
–3–
igh -handed spino s in 4-dimensions, q=−2 would be in e p e ed as a le -handed
pa icle wi h hype cha ge −2. Howe e he e is no place in his cons uc ion o a
igh -handed pa icle wi h he same cha ge as his would equi e q=−2 and ν > 0.
This is necessa ily a chi al heo y o q6=−1, and would gene a e a le hal gauge
anomaly in 4-dimensions.
To in oduce qua ks we u n o CP3—since he holonomy g oup o CP3is U(3)
qua ks can be in oduced as iple s. Fo example an SU(3) iple wi h q=−3
gi es ν−3,3= 1 and has U(1) cha ge Y(3) =−8/3. Since he o e all no malisa ion o
he cha ge is a ou disposal, his could ep esen ei he :
a igh handed d-qua k ( escale cha ge by 1/8), dR=3−1/3,(2.4)
o
a igh -handed u-qua k ( escale cha ge by −1/4), uR=32/3.(2.5)
Ei he possibili y o ces us o in e p e posi i e chi ali y as co esponding o igh -
handed Fe mions in 4-dimensions in o de o ma ch he s anda d model spec um.
I is ins uc i e o examine he complex conjuga e ep esen a ions in o de o un-
de s and he CPT conjuga e o he spec a. In he appendix i is shown ha un-
de complex conjuga ion o he bundles o e CPn he U(1) cha ges ans o m as
Y(n)→Y(n)=−Y(n)−(n+ 1) and he index as ν→ν= (−1)nν. So o he exam-
ples (2.4) and (2.5) abo e, wi h jus a single SU(3) iple on CP3wi h Y(3) =−8/3,
Y(3) =−4/3 and complex conjuga ion lips he chi ali y so i maps
dR=3−1/3−→ 3−1/6(2.6)
and
uR=32/3−→ 31/3= (d)L.(2.7)
I we s a wi h a igh -handed u-qua k complex conjuga ion o ces us o in oduce
he le -handed an i-dwhile s a ing wi h a igh -handed d-qua k complex conjuga-
ion o ces he in oduc ion o a s a e ha has no place in he s anda d model. This
nicely illus a es he kind o cons ain s ha CPT can place on he possible choices.
Fo weak in e ac ions we b ing in CP2wi h holonomy g oup U(2). An index
heo em analysis hen allows us o ob ain a single gene a ion o he elec oweak
sec o , including a igh -handed neu ino. This is achie ed by aking he ollowing
h ee ep esen a ions:
•An SU(2) single wi h q= 0 gi ing ze o cha ge and index ν0= +1;
•A second single wi h q=−3 gi ing cha ge Y(2) =−3 and index ν−3= +1;
•An SU(2) double wi h q=−2 gi ing cha ge Y(2) =−3/2 and ν−2,2=−1.
–4–

In e p e ing posi i e νas gi ing igh -handed spino s, and escaling he cha ge
by 1/3, his esul s in a single gene a ion o pa icles o he elec oweak sec o o he
s anda d model, including a igh -handed neu ino:
10= (V)R1−1=eR2−1/2=VL
eL(2.8)
( he no malisa ion is such ha he elec ic cha ge is Q=I3+Y). Now o CP2
complex conjuga ion p ese es he chi ali y and sends Y(2) →Y(2) =−Y(2) −3, so
complex conjuga ion maps (2.8) o
1−1=eR10= (V)R2−1/2=eL
VL(2.9)
( o he 2 he elec ic cha ge is Q=−I3+Yand o cou se he 2is equi alen o he
2unde o a ion by he Pauli ma ix iσ2). The cu ious conclusion is ha , con a y
o wha one migh na¨ı ely expec , complex conjuga ion does no change he sign
o he hype cha ge bu ins ead in e changes he elec on and neu ino, lea ing he
elec oweak mul iple in (2.8) in a ian .
One comple e gene a ion o he qua k sec o o he s anda d model can be ob-
ained by combining (2.8) wi h he SU(3) iple in (2.5), p o ided he o al hype -
cha ge is de ined as a pa icula linea combina ion o he CP2and he CP3cha ges,
Y=−1
4Y(3) +1
3Y(2). Taking he o al chi ali y o be he p oduc o he wo indi id-
ual chi ali ies, and in e p e ing posi i e chi ali y as igh -handed, gi es he pa icle
spec um o he s ong sec o o he s anda d model:
(3,1)2/3=uR(3,1)−1/3=dR(3,2)1/6=uL
dL.(2.10)
The elec oweak sec o can be included by combining an SU(3) single on CP3,
wi h ze o cha ge and ν0= 1, wi h (2.9):
(1,1)0=VR(1,1)−1=eR(1,2)−1/2=VL
eL.(2.11)
Equa ions (2.10) and (2.11) cons i u e a single gene a ion o he s anda d model.
Now obse e ha he combina ion Y=1
4Y(3) −1
3Y(2) does simply change sign
unde complex conjuga ion, Y→Y=−Y. In addi ion he CP3chi ali y changes
while ha o CP2does no so he o e all chi ali y, which is he p oduc o he wo,
lips. The ne esul is ha complex conjuga ion o (2.10) and (2.11) does indeed
ep oduce he an i-pa icles:
(3,1)2/3=uR,(3,1)−1/3=dR,(3,2)1/6=uL
dL(2.12)
7−→
c.c. (3,1)−2/3= (u)L,(3,1)1/3= (d)L,(3,2)−1/6=(u)R
(d)R
–5–
and
(1,1)0=VR,(1,1)−1=eR,(1,2)−1/2=VL
eL(2.13)
7−→
c.c. (1,1)0= (V)L,(1,1)1= (e)L,(1,2)1/2=(V)R
(e)R.
I is in e es ing ha he cons uc ion p esen ed he e necessa ily equi es he
in oduc ion o a igh -handed neu ino, as ecen expe imen al e idence o neu ino
oscilla ions equi es jus such a s a e o he simples explana ion o he esul s [3],
[4].
In summa y a single comple e gene a ion o he s anda d model, (2.10) and
(2.11), a ises om a selec ion o i e di e en bundles, h ee o e CP2and wo o e
CP3:
•Two SU(2) single s on CP2wi h q= 0 and q=−3 and an SU(2) double
wi h q=−2;
•An SU(3) single on CP3wi h q= 0 and an SU(3) iple wi h q=−3.
In e ms o he Spincs uc u es desc ibed in he appendix hese co espond o
he wo bundles
on CP2∧0,∗TCP2⊗1⊕L3
(2) ⊕F(2) ⊗L2
(2) (2.14)
on CP3∧0,∗TCP3⊗1⊕F(3) ⊗L3
(3) (2.15)
whe e, on CP2,L(2) is he gene a ing line bundle and F(2) he ank 2 bundle sa is ying
L(2) ⊕F(2) =I3, while, on CP3,L(3) is he gene a ing line bundle and F(3) he ank
3 bundle sa is ying L(3) ⊕F(3) =I4, wi h I3and I4deno ing i ial bundles o ank
3 and 4 espec i ely.
3. Conclusions
I has been shown ha a single gene a ion o he s anda d model Fe mion spec um,
including a igh -handed neu ino, can be ob ained om he holonomy g oups o CP2
and CP3by enso ing wi h app op ia e bundles. Physically his means in oducing
backg ound SU(3) ×SU(2) ×U(1) gauge ields which a e opologically non- i ial,
con aining analogues o monopoles and ins an ons. A numbe o ques ions p esen
hemsel es.
Fi s ly he e is no ob ious sign o h ee gene a ions. O cou se one can ob ain
mo e gene a ions by aking copies, bu he e seems no compelling eason o ake
h ee such copies and no some o he numbe . This may be ela ed o he ques ion
o wha possible ˆole he isome y g oup may play. In he in oduc ion a i ue
–6–
was made o he ac ha he cons uc ion is no ied down o a speci ic me ic on
CPn, bu i one in oduces he Fubini-S udy me ic one hen has isome y g oup
SU(n+ 1). On CP2one has SU(3) and, using his as a ho izon al gene a ion g oup,
he undamen al ep esen a ion would gi e h ee gene a ions. Bu hen i is no
clea wha he ˆole o he SU(4) om CP3would be. Al e na i ely i is possible
o manu ac u e h ee copies by including CP1wi h q= 2, gi ing ν2= 3, and hen
simply igno ing he U(1) cha ge on CP1. Fo he momen we ha e no compelling
sugges ion as o how he gene a ions migh appea and we lea e his as an open
ques ion.
Secondly he e is he ques ion o he ex a U(1). The holonomy g oup o CP2×
CP3is U(3)×U(2) ∼
=SU(3)×SU(2)×U(1)×U(1) and we ha e aken a U(1) which
is only one pa icula combina ion o he wo U(1) ac o s, igno ing he o he one. In
ac he s anda d model spec um has ue g oup S(U(3) ×U(2)), [5], and he e y
ac ha i is ep oduced he e means ha he g oup being used is S(U(3) ×U(2))
a he han he ull holonomy g oup U(3) ×U(2). The e may be a deepe eason o
his, bu o he momen we con ine ou sel es o obse ing ha i wo ks empi ically.
How migh he cons uc ion i in o ealis ic models? I would ce ainly seem
oo na¨ı e o ake CP2×CP3as an in e nal space—i has eal dimension 10 which
is oo la ge o s ing heo y and simply adding i on o 4-dimensional space- ime
p oduces a 14-dimensional space- ime which would ha e anomalies. I may be ha
one could ealise CP2and CP3as a b ane wi hin a b ane in ype IIA s ing heo y,
which has an i-symme ic enso ields o ank 2 and 4 in i s R-R sec o as well as
hei hodge duals, hough IIA s ing s ing heo y is a non-chi al heo y so he Weyl
Fe mion on CP3would ha e o be pu in by hand. Any such in e p e a ion would
necessa ily be a he di e en o he s anda d app oach, as i would no in ol e
g and uni ied heo ies di ec ly. Al e na i ely a “ uzzy” Kaluza-Klein app oach may
be o in e es whe e he con inuum mani olds o CP2and CP3a e eplaced by
non-commu a i e ini e dimensional ma ix app oxima ions wi h a ini e numbe o
deg ees o eedom [2]. I would hen be mo e app op ia e o hink o mul iple copies
o 4-dimensional space- ime a he han an in e nal con inuous mani old, somewha
analogous o Connes’ non-commu a i e geome y app oach o he s anda d model
wi h wo copies o space- ime [6], [7]. S a -p oduc s on CPnwe e s udied in [8] and
[9], and he spec um o he Di ac ope a o was in es iga ed in [10] and [11]. The
smalles ec o space ha could be used o a non- i ial ma ix ep esen a ion is
n+ 1. Fo CP2we ge 3, which ela es o he discussion o he gene a ion p oblem
abo e, while CP2×CP3would equi e 3 ×4 = 12 copies.
While hese a e all in e es ing and impo an p oblems we lea e hem open o
u he wo k.
–7–
A. The Index Theo em Fo CPn
Fo a gene al complex mani old Xo dimension n he o al Che n class is he sum
o he indi idual Che n classes
c(X) = 1 + c1(X) + c2(X) + ···+cn(X).(A.1)
In line wi h common usage we w i e ck(X) o ck(TX)— he k h Che n class o he
angen bundle. In pa icula cn(X) is he op o m and he Gauss–Bonne he-
o em s a es ha e alua ing cn(X) on Xgi es he Eule cha ac e is ic o X: i.e.
cn(X)[X] = χ(X).
Fo CPn he Che n classes a e all gene a ed by a single 2 dimensional class x
[12]
c(CPn) = (1 + x)n+1 = 1 + (n+ 1)x+n(n+ 1)
2x2+···+ (n+ 1)xn.(A.2)
No e ha xn+1 is a (2n+2) dimensional class and so xn+1 = 0—when xis ep esen ed
by a 2 o m ω, say, his co esponds o he ac ha ωn+1 = 0 on a 2n-dimensional
mani old.
The no malisa ion is such ha
ZCPn
ωn= 1 (A.3)
and so, since cn(X) = (n+ 1)xn≡(n+ 1)ωn, we ha e
χ(CPn) = n+ 1,(A.4)
which is he Eule cha ac e is ic o CPnas equi ed by he Gauss-Bonne heo em.
The o m ωis he cu a u e o a line bundle Lwhose complex conjuga e Lwe shall
e e o as he gene a ing line bundle o e CPnwi h Che n class
c(L) = 1 −x. (A.5)
Ano he line bundle ha will be impo an is he canonical line bundle Kwhich
is he maximum ex e io powe o he co angen bundle T∗X: i.e.
K=∧nT∗X. (A.6)
Khas Che n class gi en by
c(K) = 1 −(n+ 1)x. (A.7)
The minus sign appea s because
c1(K) = c1(T∗X) = −c1(TX).(A.8)
–8–