a Xi :hep- h/0207007 2 24 Sep 2002
P ep in ypese in JHEP s yle - HYPER VERSION DIAS-STP-02=08
Chi al Fe mions and SpincS uc u es on
Ma ix App oxima ions o Mani olds
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 : The A iyah-Singe index heo em is in es iga ed on a ious compac
mani olds which admi ini e ma ix app oxima ions (“ uzzy spaces”) wi h a iew o
applica ions in a modi ied Kaluza-Klein ype app oach in which he in e nal space
consis s o a ini e numbe o poin s. Mo i a ed by he chi al na u e o he s anda d
model spec um we in es iga e mani olds ha do no admi spino s bu do admi
Spincs uc u es. I is shown ha , by wis ing wi h app op ia e bundles, one gen-
e a ion o he elec oweak sec o o he s anda d model, including a igh -handed
neu ino, can be ob ained in his way om he complex p ojec i e space CP2. The
uni a y G assmannian U(5)/(U(3) ×U(2)) yields a spec um ha con ains he co -
ec cha ges o he Fe mions o he s anda d model, wi h a ying mul iplici ies o
he di e en pa icle s a es.
Keywo ds: Non-Commu a i e Geome y, Field Theo ies in Highe Dimensions,
Di e en ial and Algeb aic Geome y.
Con en s
1. In oduc ion 1
2. Chi al Fe mions on CP25
3. Chi al Fe mions on Sp(2)/U(2) 7
4. Uni a y G assmannians 8
5. Conclusions 10
A. Spin and spincs uc u es on a mani old 12
B. Cohomology and spincon CP216
C. Cohomology and spincin six dimensions 17
D. Cohomology and gene alised spino s o a 12 dimensional G ass-
mannian. 18
1. In oduc ion
Non-commu a i e geome y has ecen ly come o he o e as con ende o a possible
modi ica ion o physics wi h applica ions in a emp s o uni y g a i y and gauge he-
o ies ( o e iews see [1]). Long be o e he cu en su ge o in e es ia supe s ings
i was sugges ed by Connes and Lo ha he s anda d model o pa icle physics
could be de i ed om non-commu a i e geome y, [2] [3]. A ela ed concep is ha
o “ma ix mani olds”— hese a e a e sion o non-commu a i e geome y in which
con inuous spaces wi h an in ini e numbe o deg ees o eedom a e eplaced wi h
ini e dimensional non-commu a i e ma ix algeb as app oxima ing he con inuum
space. As he size o he ma ices is aken o in ini y he algeb a becomes commu-
a i e and he con inuum space is eco e ed. These algeb as a e o en called “ uzzy
spaces” bu we shall e e o hem as “ma ix mani olds” in o de o a oid he neg-
a i e conno a ions o he wo d “ uzzy”. The ma ix mani old app oach has much in
–1–
common wi h gene alised cohe en s a es in quan um mechanics, [4] [5] and exam-
ples o mani olds which admi a ini e ma ix app oxima ion a e S2[6] [7], and mo e
gene ally CPn, [9] as well as uni a y G assmannians
U(n)
U(k)×U(n−k)(1.1)
[10] (s a p oduc s on con inuous complex p ojec i e spaces and uni a y G assmanni-
ans we e cons uc ed in [11] [12]). One o he a ac i e ea u es o ma ix mani olds
is ha hey ha e he same symme ies as he con inuum space, so a ma ix e sion
(G/H)Mo a cose space G/H has all he same symme ies o i s con inuous pa en ,
despi e being a ini e app oxima ion. Ma ix mani olds a e also closely ela ed o ha -
monic expansions o unc ions on cose spaces, indeed he ma ix algeb as a e no hing
mo e han cunning ea angemen s o he expansion coe icien s in o a ma ix, and
i is na u al o ask i ma ix mani olds migh ha e a ˆole o play in Kaluza–Klein
heo y. This ques ion was in es iga ed in [13] and is one o he mo i a ions o he
p esen wo k. Ano he mo i a ion is he calcula ion o he spec um o he Di ac
ope a o on CP2
Min [14] [15], he calcula ion in he la e e e ence being buil on a
cons uc ion which bea s a ema kable esemblance o he elec oweak sec o o he
s anda d model. This na u ally leads one o ask i he e migh be a la ge ma ix
mani old which could inco po a e he whole s anda d model in i s spec um.
One o he p oblems wi h he Kaluza–Klein p og amme was he ealisa ion ha
i was unlikely o gene a e a chi al gauge heo y in 4-dimensions wi hou some mod-
i ica ion, [16]. To ob ain a chi al gauge heo y i seems necessa y o in oduce un-
damen al gauge ields and hen one is aced wi h he di icul y o anomaly cancella-
ion, which is mo e di icul in highe dimensions because he e a e mo e po en ially
anomalous g aphs o wo y abou . The in oduc ion o undamen al gauge ields also
nega es he whole Kaluza–Klein philosophy whose aim is o de i e he gauge ields
pu ely om a me ic. I he in e nal space is a ma ix mani old howe e undamen-
al gauge ields a e mo e na u al, as a ma ix mani old has no simple de ini ion o a
me ic, bu i does ha e symme ies. To call a heo y wi h a ma ix mani old as an
in e nal space a Kaluza–Klein heo y is eally a misnome as all i has in common
wi h he usual Kaluza–Klein app oach is an ‘in e nal’ space wi h a symme y—i
a me ic is no de ined he e a e no induced gauge ields so hey mus be added
by hand. Ne e heless he e is a symme y, he symme y o he isome ies and
holonomy o he cose space a e he e e en a he ini e le el— like he g in o he
Cheshi e ca he symme ies emain e en hough he me ic has gone.1Fo his
eason we shall con inue o e e o ma ix Kaluza–Klein heo y because he concep
has much in common wi h con inuum Kaluza–Klein heo y, hough he e a e also
s ong di e ences.
1Fo b e i y we shall e e o Gand Has he isome y and he holonomy g oup, e en when no
me ic o connec ion a e de ined.
–2–
Fundamen al gauge ields a e he e o e na u al in ma ix Kaluza–Klein heo ies,
bu we mus s ill wo y abou anomalies. To ou knowledge he ques ion o gauge
anomalies on ma ix mani olds has no ye been in es iga ed, hough chi al anomalies
ha e been [7] [8]. I one akes a model consis ing o 4-dimensional Minkowski space-
ime wi h a ma ix in e nal space one can hope ha i may be su icien o he
4-dimensional gauge anomalies o cancel, wi hou wo ying abou g aphs wi h mo e
ex e nal legs ha would be impo an i he in e nal space we e con inuous. To a
la ge ex en his is a ques ion o dynamics on he in e nal space, i he dynamics
ep oduces ha o he con inuum in he con inuum limi ( hough i doesn’ ha e
o i we don’ wan o ake ha limi ) hen hese o he g aphs would ha e o be
impo an in he limi . Bu as long as he in e nal space consis s o a (small) numbe
o ini e deg ees o eedom i seems no un easonable o assume ha only he usual
4-dimensional g aphs con ibu e o a po en ial anomaly. Fo example he ma ix
mani old ep esen ing he 2-sphe e S2
Mhas an app oxima ion consis ing o only 2
poin s. A ma ix Kaluza–Klein heo y based on S2
Mwould look like wo copies o
Minkowski space wi h an SU(2) ac ion on he 2 poin s (much like he Higgs sec o
in Connes’ e sion o he s anda d model). The e seems no compelling eason o
belie e ha such a model would exhibi a six-dimensional gauge anomaly.
Fo he easons ou lined abo e i seems wo hwhile in es iga ing he possibili y
o ob aining chi al gauge heo ies in 4-dimensions om a ma ix Kaluza–Klein heo y
wi h in e nal space (G/H)Mand undamen al gauge ields. The ool ha we use will
be s anda d di e en ial geome y and he A iyah–Singe index heo em o he Di ac
ope a o on con inuous mani olds. Though he aim is o apply he concep s o ini e
ma ix geome ies i is no un easonable o expec ha he usual index heo em
applies since i makes s a emen s abou opology by coun ing ini e da a. Indeed
a Di ac ope a o can be de ined on ma ix mani olds, e en hough hey a e ini e
dimensional.
The spec um o he Di ac ope a o on some speci ic ma ix mani olds has al-
eady been in es iga ed, no ably S2
M[7] [8] and CP2
M, [14] [15]. The cons uc ion
o he spec um on CP2in [15] is buil on a 4-dimensional educible ep esen a ion
o SU(2) ×U(1) which is ha o elec oweak sec o o he s anda d model o pa i-
cle physics, including a s a e wi h he quan um numbe s o a igh -handed neu ino
which is a chi al ze o-mode. As is well known CP2∼
=SU(3)/U(2) is no a spin man-
i old, i has an obs uc ion o he global de ini ion o spino s, bu coupling spino s
o an app op ia e backg ound U(1) gauge ield allows spino s o be de ined—a con-
s uc ion which is called a spincs uc u e in he ma hema ical li e a u e—and his
gi es ise o he igh -handed neu ino in [15]. Since CP2
Mis a ini e ma ix algeb a
app oxima ion o con inuum CP2, which cap u es all he opological ea u es o he
con inuum mani old, he opology is e lec ed a he ma ix le el and he eme gence
o chi al spino s on CP2
Mis a di ec consequence o he A iyah-Singe index heo em
o spino s on CP2.
–3–
Since he holonomy g oup o CP2is U(2) he spec um o he Di ac ope a o can
be decomposed in o ep esen a ions o SU(2) ×U(1) and he ep esen a ions in [15]
a e buil on ha o he elec oweak sec o o he s anda d model wi h he addi ion
o a igh -handed neu ino
10=VR1−2=eR2−1=VL
eL.(1.2)
The e is a ze o-mode s a e in he cons uc ion o [15], he 10, and i s exis ence
equi es a backg ound Abelian ‘monopole’ ield on CP2. In ac , as we shall see,
coupling Fe mions o monopole ields o highe cha ge and non-Abelian backg ound
ields as well allows e e y s a e in (1.2) o be ealised as a ze o-mode o he Di ac
ope a o on CP2. The ac ha a igh -handed neu ino appea s na u ally in he
cons uc ion is pa icula ly appealing in iew o he ecen e idence o sola neu ino
oscilla ions [17] [18] whose simples in e p e a ion equi es a igh -handed neu ino.
Ano he mani old which has holonomy g oup U(2) and does no admi spino s
is Sp(2)
U(2) ∼
=SO(5)
SO(3) ×SO(2).(1.3)
This space has a ini e ma ix app oxima ion and has been p oposed as a ma ix
e sion o he co angen bundle o S3[19]. One migh wonde i he spec um in
(1.2) is gene ic o spincs uc u es on mani olds wi h holonomy U(2) and his space is
a coun e -example. We shall see ha a spec um eme ges which con ains he co ec
cha ges o he elec oweak sec o bu he Di ac ope a o o elec on-neu ino double
has ze o index. Ne e heless i is use ul o include his as an example o a space o
dimension 2 mod 4 which is no spin ( he signi icance o chi al spino s o a dis inc ion
be ween spaces o dimension 0 mod 4 and dimension 2 mod 4 was emphasised in [16]).
The las space which we shall examine is he ma ix e sion o he uni a y G ass-
mannian U(5)
U(3) ×U(2).(1.4)
This space has a ini e ma ix app oxima ion and an explici local o mula o a
s a -p oduc , in e ms o a ini e sum o de i a i es, was de i ed in [10]. I is no a
spin mani old bu admi s a spincs uc u e, and so seems a good candida e o chi al
spino s. Fu he mo e he holonomy g oup is exac ly igh o he s anda d model,
since U(5)
U(3) ×U(2) ∼
=SU(5)
S[U(3) ×U(2)] (1.5)
and he pa icle spec um o he s anda d model is eally such ha he Fe mions
all in o a ep esen a ion whose ue g oup is p ecisely S[U(3) ×U(2)], [20]. Fo his
space he spec um con ains one gene a ion o he ull s anda d model, including
a igh -handed neu ino, hough he mul iplici ies a e di e en o di e en s a es,
some o hem ha ing ze o index.
–4–

Sec ion 2 con ains an index heo em analysis o spino s on CP2and ep oduces
he ze o-mode spec um (1.2). Sec ion 3 con ains a discussion o he 6-dimensional
mani old Sp(2)/U(2). Sec ion 4 analyses spincs uc u es, and hei non-Abelian
gene alisa ions, on he uni a y G assmannian U(5)/(U(3) ×U(2)) and i s ela ion
o he s anda d model spec um. Ou esul s a e summa ised in sec ion 5. The
analysis elies on he index heo em o he Di ac ope a o o a ious bundles o e
hese h ee spaces. The de i a ion o he ele an index o he cases unde s udy is
gi en in ou appendices, whe e a gene al discussion o spincs uc u es is also gi en
as an aid o hose who may no be amilia wi h he cons uc ion.
2. Chi al Fe mions on CP2
The complex p ojec i e space CP2∼
=SU(3)/U(2) was ac i ely in es iga ed in he
1980s as an in e es ing candida e o an Euclidean g a i a ional ins an on [21]. The
Eule cha ac e is ic o CP2is χ= 3 and he signa u e is τ= 1. I is no a spin man-
i old, he e is a global obs uc ion o pu ing spino s on his space, bu one can pu
spino s on i p o ided undamen al gauge ields a e in oduced and an app op ia e
opologically non- i ial backg ound gauge ield is in oduced. This ac was used
in [21] o cons uc a “gene alised spin s uc u e”, whe e spino s wi h an Abelian
cha ge mo e in he ield o he K¨ahle 2- o m on CP2, which is somewha analogous
o a monopole ield on CP1∼
=S2.
The holonomy g oup o CP2is
U(2) ⊂SO(4) ∼
=SU(2) ×SU(2)
Z2
.(2.1)
I spino s could be de ined his would be li ed o SU(2)×U(1) ⊂Spin(4) ∼
=SU(2)×
SU(2), and he wo di e en chi ali ies o Weyl spino s would ans o m unde he
di e en ac o s o SU(2) ×U(1) as, o example,
ψ+=11+1−1and ψ−=20,(2.2)
whe e he subsc ip deno es he U(1) cha ge. Bu since spino s canno be de ined
globally (c . appendix B), he spino bundle does no exis . This can be cu ed by
in oducing a U(1) gauge ield wi h non- i ial opology and co ela ing he cha ge
wi h ha o he U(1) subg oup o Spin(4). Ma hema ically, on a complex mani old
X, we ake he squa e oo o he canonical line bundle K, as desc ibed in appendix
A, and enso i wi h he he spin bundle S(X). Nei he o hese bundles exis s
sepa a ely bu S(X)⊗K−1/2does. In ac , i Lis a gene a ing line bundle (c . he
appendix) wi h RS2c1(L) = −1, whe e S2is a non- i ial wo sphe e embedded in
X,2 hen S(X)⊗Lpis a well de ined bundle o any hal -in eg al p. Fo CP2i is
2This is ambiguous i H2(X;Z) has dimension g ea e han one, bu in all he examples we shall
conside in his pape H2(X;Z) is one dimensional and his in eg al is uniquely de ined.
–5–
shown in appendix B ha
S(X)⊗Lp=∧0,∗TX ⊗K1/2⊗Lp=∧0,∗T X ⊗L−q, X =CP2(2.3)
whe e q=−p−3
2, since he canonical line bundle o CP2is gi en by K=L3.
The ne numbe o ze o modes depends on qand o CP2is gi en in (B.5) o
appendix B as
ν=1
2(q+ 1)(q+ 2).(2.4)
In ac qcan be in e p e ed as he e ec i e U(1) cha ge. The cha ge is no
pbecause he e is a con ibu ion om he angula momen um associa ed wi h he
spino bundle S(X). To e alua e he cha ge we use a gene al a gumen conce ning
spino bundles o e complex mani olds. We de ine he U(1) cha ge, which will be
iden i ied la e wi h he hype cha ge Y, using he Che n cha ac e o he gene a ing
line bundle aised o he app op ia e powe , in his case L−q, by aking a non- i ial
S2embedded in he mani old Xand de ining
q=ZS2
ch(L−q) = −qZS2
ch(L) since ZS2
c1(L) = −1.(2.5)
Fo compa ison wi h he usual cha ge assignmen s o he s anda d model below,
we escale his by 2/3 o Y= 2q/3. Fo q= 0 o example ν= 1 and, iden i ying
posi i e chi ali y wi h igh -handed spino s, his would appea as a neu al igh -
handed pa icle: a igh -handed neu ino VR. A spino wi h q=−3 also has ν= 1,
so would be igh -handed, wi h Y=−2: he igh -handed elec on, eR.
I a undamen al SU(2) gauge ield is added wi h he spino s aken o be SU(2)
double s hen spino s can be ob ained om he bundle ∧0,∗TCP2⊗F⊗L−q, whe e
Fis he ank 2 ec o bundle de ined by F⊕L=I3(I3deno ing a i ial ank 3
bundle). The s uc u e g oup o Fis U(2) co esponding o a SU(2) ×U(1)-gauge
ield. In ac Fis associa ed o he p incipal U(2) bundle induced by he cose
cons uc ion U(2) −→ SU(3)
↓
CP2.
(2.6)
The Di ac index o ∧0,∗TCP2⊗F⊗L−qis de i ed in appendix B and is gi en by
(B.11)
ν= (q+ 1)(q+ 3).(2.7)
Ze o modes would gi e ise o chi al SU(2) double s.
The U(1) cha ge is now calcula ed as he Che n cha ac e ch(F⊗L−q) e alua ed
on a opologically non- i ial S2embedded in CP2, he esul is 2q+1. As he Che n
cha ac e in ol es acing o e a 2 ×2 ma ix he U(1) gene a o is (2q+1)
21, whe e 1
is he 2 ×2 iden i y ma ix, so he indi idual cha ges a e q+1
2. Re-scaling by 2/3,
–6–
as abo e, gi es Y=2q+1
3. In pa icula q=−2 yields Y=−1 wi h ν=−1 and,
iden i ying posi i e chi ali y wi h igh -handed pa icles, we ge a single gene a ion
o a le -handed double wi h cha ge −1, he elec on-neu ino double .
So we can ob ain a single gene a ion o he elec oweak sec o o he s anda d
model om CP2by aking SU(2) single s wi h q= 0 and q=−3 (ν= +1) and a
single SU(2) double wi h q=−2 (ν=−1), ha is
10=VR1−2=eR2−1=VL
eL, ν > 0 igh -handed.(2.8)
3. Chi al Fe mions on Sp(2)/U(2)
As an example o a six-dimensional space which does no admi a spin s uc u e, bu
does admi a Spincs uc u e, conside Sp(2)/U(2). This space has Eule cha ac e -
is ic χ= 4. In ac
Sp(2)
U(2) ∼
=SO(5)
SO(3) ×SO(2) (3.1)
and his space admi s a ma ix app oxima ion. The spino bundle does no exis
bu a Spincs uc u e can be de ined using S(X)⊗Lp, wi h L he gene a ing line
bundle and phal -in eg al. The canonical line bundle is ela ed o he gene a ing
line bundle by K=L3(see appendix C) so ha
S(X)⊗Lp=∧0,∗TX ⊗L−q, X =Sp(2)
U(2) (3.2)
whe e q=−p−3
2.
The Di ac index o his bundle is de i ed in appendix C and is gi en in (C.18):
ν=1
6(2q+ 3)(q+ 1)(q+ 2).(3.3)
The ze o-modes will gi e ise o pa icles in 4-dimensions whose U(1) cha ge is q,
which we e-scale by 2/3 o b ing i line wi h he usual s anda d model con en ions
below. so, o example, q=−3 gi es a single gene a ion o nega i e chi ali y pa icles
wi h cha ge −2 while q= 0 would gi e a single gene a ion o posi i e chi ali y neu al
pa icles.
As be o e we can also couple he Fe mions o a undamen al SU(2) gauge ield
by in oducing a ank 2 ec o bundle Fassocia ed o he p incipal bundle
U(2) −→ Sp(2)
↓
Sp(2)/U(2)
(3.4)
wi h s uc u e g oup U(2). I is shown in appendix C ha he index o ∧0,∗TX ⊗
F⊗L−qis now
ν=2
3q(q+ 1)(q+ 2).(3.5)
–7–
The Che n cha ac e ch(F⊗L−q) e alua es o 2q+ 1 on a non- i ial S2. Again
his is he ace o a 2 ×2 ma ix and he indi idual s a es ha e cha ge q+1
2which
is escaled by 2/3 o gi e he U(1) cha ge as Y=2q+1
3. Fo example q= 1 gi es
Y= 1 and ν= 4 and hus ou copies o posi i e chi ali y double s while q=−2
gi es Y=−1 and ν= 0.
We can y o ge he elec oweak cha ges om his cons uc ion. Fo example
in e p e ing posi i e chi ali y as le -handed he single s would be he igh -handed
elec on eRand a le -handed an i-neu ino (V)L. Bu he double s wi h Y= 1
would ha e o ha e nega i e chi ali y o i wi h he s anda d model ( he igh -
handed posi on and an i-neu ino) and νis posi i e. I we in e p e posi i e chi ali y
as igh -handed, he double could he posi on–an i-neu ino double (V)R
(e)R, bu
hen he single wi h Y=−2 has he w ong chi ali y o be he igh -handed elec on.
On he o he hand choosing a double wi h q=−2 gi ing Y=−1, in addi ion
o he single s abo e, gi es ν= 0 o he double : in gene al he Di ac ope a o
will ha e no ze o modes o his double hough i may ha e o speci ic choices o
he U(2) connec ion, bu e en hen he ze o modes will occu in pai s o opposi e
chi ali y. The spec um con ains one gene a ion o he elec oweak sec o o he
s anda d model, bu he e is an addi ional unwan ed double o he w ong chi ali y.
4. Uni a y G assmannians
The inal sou ce o examples ha we wish o discuss is he uni a y G assmannians
U(n)
U(k)×U(n−k)∼
=SU(n)
S(U(n−k)×U(k)) (4.1)
o which he complex p ojec i e spaces, k= 1, a e special cases. The i s Che n
class o he angen bundle o hese space e alua es o n, [23], and he second
S ie el-Whi ney class is nmod 2—so hese spaces admi a spin s uc u e i and only
i nis e en. We shall ocus on he pa icula case o n= 5 and k= 2, his is an
in e es ing case because he holonomy g oup o SU(5)/S(U(3) ×U(2)) is p ecisely
ha o he s anda d model, [20]. This condi ion dic a es ha he Fe mions ac ually
si in ep esen a ions o SU(3)×SU(2)×U(1) in which he gene a o s a e aceless—
whence S(U(3) ×U(2)). As a ma ix mani old SU(5)/S(U(3) ×U(2)) was s udied
in [10], whe e a s a p oduc was explici ly cons uc ed in e ms o de i a i es.
The G assmannian SU(5)/S(U(3) ×U(2)) has Eule cha ac e is ic χ= 10 and
signa u e τ= 2. I is no a spin mani old bu a Spincs uc u e exis s. Taking he
bundle ∧0,∗T X ⊗L−q, wi h X he G assmannian and L he gene a ing line bundle,
he Di ac index is calcula ed in appendix D as (D.27),
ν{q,1,1}=1
144(q+ 1)(q+ 2)2(q+ 3)2(q+ 4),(4.2)
–8–
We shall call L he gene a ing line bundle and, in each case, Kwill be some powe
o L— his powe will be odd i Xis no a spin mani old—so ha
K=Lm, m ∈Z.(A.22)
Hence a gene al spincs uc u e will ha e he spincbundle
∧0,∗TX ⊗L−q=Sc(X)⊗L−q, q ∈Z(A.23)
( he minus sign in he exponen is o la e con enience). When q= 0 we ha e he
canonical spincs uc u e; he e is also a dependence o he spincs uc u e on an
elemen o H1(X;Z2) bu ou examples ha e H1(X;Z) = 0 so we do no need o
conside his.
I we use he ac ha K=Lm hen he co esponding Di ac ope a o hen
becomes /∂L−(q+m/2) which we shall nea en up sligh ly by w i ing i as
/∂Lpwhe e p=−q−m/2.(A.24)
The e is also an index o mula o he ze o modes o /∂Lpwhich in ol es he usual
ˆ
Agenus o Xand he ‘Che n class’ o he line bundle Lp. Le /∂Lpdeno e he Di ac
ope a o coupled o Lp hen i s index is gi en by 3
index (/∂Lp) = ch (Lp)ˆ
A(X)[X] (A.25)
= exp [pc1(L)] ˆ
A(X)[X].(A.26)
We will also need he case whe e he Di ac ope a o is u he coupled o a second
ec o bundle Eo ank possibly g ea e han one; in his case he equisi e index
o mula is
index (/∂Lp⊗E) = ch (Lp⊗E)ˆ
A(X)[X] (A.27)
= ch (E) exp [pc1(L)] ˆ
A(X)[X],(p=−q−m/2).(A.28)
In he nex sec ion we ea an ac ual spincexample in ou dimensions.
3We could equally ha e used ins ead he o mula o index (¯
∂Lp) which would ha e in ol ed
ch (L) and he Todd class d(X). In ac his ealisa ion o he Di ac ope a o as ¯
∂Lpenables one
o easily unde s and why index (/∂L−q−m/2) is equal o uni y o q= 0: i is because, when q= 0,
index (¯
∂L−m/2) gi es he a i hme ic genus P(−1)sh0,s o he complex mani old Xwhe e he Hodge
numbe h ,s deno es he dimension o he space o holomo phic o ms o ype ( , s). Now o he
mani olds Xwe conside in his pape he only holomo phic o ms a e o ype (s, s) a ac which
educes he a i hme ic genus o h0,0which is i ially uni y.
– 15 –

B. Cohomology and spincon CP2
On CP2 he Che n class is [25]
c(CP2) = 1 −3c1(L) + 3c2
1(L) (B.1)
whe e he gene a ing line bundle Lhas c(L) = 1 + c1(L) wi h −c1(L) gene a ed by
he K¨ahle 2- o m. The Eule cha ac e is ic is 3 so c2
1(L)[CP2] = 1 and in his case
c1(CP2) = −3c1(L) so m= 3. Since he coe icien o c1(L) is odd w26= 0 and CP2
does no admi a spin s uc u e. The index o he Di ac ope a o coupled o Lpis
index (/∂Lp) = ch (Lp)ˆ
A(X)[X] = exp [pc1(L)] ˆ
A(X)[X] (B.2)
=1 + pc1(L) + 1
2p2c2
1(L)1−p1(X)
24 [X], X =CP2(B.3)
=1
8(4p2−1),(B.4)
whe e p1is he Pon jagin class and p1(X) = c2
1(X)−2c2(X) = 3c2
1(L) on CP2. This
index is in eg al o hal -in eg al pand, se ing p=−q−3/2, we ob ain
index (/∂Lp) = 1
2(q+ 2)(q+ 1).(B.5)
We can de ine a non- i ial ank 2 bundle Fo e CP2wi h s uc u e g oup
U(2) by F⊕L∼
=I3whe e I3is he i ial ank 3 bundle. Then c(F)c(L) = 1 so
c1(F) = −c1(L) and c2(F) = c2
1(L); enso ing his wi h pcopies o he gene a ing
line bundle L hen gi es, o he Che n cha ac e ,
ch(Lp⊗F) = ch(Lp)ch(F) (B.6)
=1 + pc1(L) + 1
2p2c2
1(L) + ···2−c1(L)−1
2c2
1(L) + ···(B.7)
= 2 + (2p−1)c1(L) + p2−p−1
2c2
1(L) + ···,(B.8)
leading o
index (/∂Lp⊗F) = ch (F⊗Lp)ˆ
A(X)[X] = exp [pc1(L)] ch (F)ˆ
A(X)[X] (B.9)
=2 + (2p−1)c1(L) + p2−p−1
2c2
1(L)1−p1
24[X]
=1
4(2p−3)(2p+ 1), X =CP2(B.10)
= (q+ 1)(q+ 3),again using p=−q−3/2.(B.11)
– 16 –
C. Cohomology and spincin six dimensions
In his sec ion Xis he complex mani old gi en by
X=Sp(2)
U(2) (C.1)
whose eal dimension is 6. The cohomology ing o Xis gene a ed by he e en
dimensional classes σ1∈H2(X;Z) and σ2∈H4(X;Z) subjec o he single ela ion
σ2
1= 2σ2.(C.2)
Now Xis no a spincmani old because we can compu e ha
c(X) = 1 + c1(X) + c2(X) + c3(X) (C.3)
= 1 + 3σ1+ 8σ2+ 4σ1σ2(C.4)
⇒c1(X) = 3σ1=−3c1(L).(C.5)
We no e ha σ1gene a es H2(X;Z) and so deduce ha c1(X) is odd and so
w2(X)6= 0 ⇒Xis no spin.(C.6)
We also see ha
K=L3(C.7)
so ha he in ege mo appendix A is equal o 3.
The index o he Di ac ope a o /∂Lpcan now be compu ed om he expansions
o ch (Lp) and ˆ
A(X) gi ing us he o mula
index (/∂Lp) = 1 + pc1(L) + 1
2p2c2
1(L) + ···1−p1(X)
24 +···[X] (C.8)
=−pc1(L)p1(X)
24 +1
3!p3c3
1(L)[X].(C.9)
Bu we can calcula e ha
p1(X) = c2
1(X)−2c2(X) (C.10)
= 9σ2
1−16σ2,(C.11)
wi h σ1=−c1(L). Hence we ind ha
index (/∂Lp) = p
24σ1(9σ2
1−16σ2)−1
3!p3σ3
1[X] (C.12)
=−(4p3−p)σ3
1
24[X] (C.13)
=−1
12(4p3−p) = −1
12p(2p−1)(2p+ 1),(C.14)
– 17 –
whe e we ha e used he Gauss–Bonne heo em which says ha
c3(X)[X] = χ(X) (C.15)
= 4 = 2σ3
1[X] (C.16)
o deduce ha σ3
1[X] = 2. Be o e inishing we should check ha he index is in eg al.
Recall ha
p=−q−m/2, q ∈Z, m = 3 (C.17)
This ac immedia ely gi es us he o mula
index (/∂Lp) = 1
6(2q+ 3)(q+ 1)(q+ 2), q ∈Z(C.18)
and his is easily checked o gi e an in ege index o in eg al qas i should.
I we enso p oduc wi h a u he ank 2 bundle F, wi h c1(F) = σ1 hen we
ind ha
index (/∂Lp⊗F) = ch(Lp⊗F)ˆ
A(X)[X] (C.19)
=ch(Lp)ch(F)ˆ
A(X)[X] (C.20)
=−1
12(2p+ 3)(2p−1)(2p+ 1) (C.21)
=2
3q(q+ 1)(q+ 2), p =−q−3/2, q ∈Z(C.22)
and again his gi es an in eg al index.
D. Cohomology and gene alised spino s o a 12 dimensional
G assmannian.
In his sec ion Xis he 12 dimensional G assmannian gi en by
X=U(5)
U(3) ×U(2).(D.1)
Xis a pe ec ly s anda d complex mani old (o complex dimension 6) and i s coho-
mology ing H∗(X;Z) has 3 gene a o s
σi∈H2i(X;Z), i = 1,2,3 (D.2)
which obey he single ela ion
σ3= 2σ1σ2−σ3
1.(D.3)
I s Che n class is gi en by
c(X) = (1 + c1(X) + c2(X) + c3(X) + c4(X) + c5(X) + c6(X)) (D.4)
= (1 −5σ1+ 12σ2
1−15σ3
1+ 8σ4
1+ 2σ2
1σ2+ 7σ2
2+ 4σ5
1−25σ1σ2
2(D.5)
−29σ6
1+ 7σ2
1σ2
2+ 56σ4
1σ2−27σ3
2) (D.6)
– 18 –
om which we see ha
c1(X) = −5σ1(D.7)
and hence we deduce, as we did in he p e ious sec ion, ha
w2(X)6= 0 (D.8)
and so Xis no spin.
We now pass o he spincbundle Sc(X) and o he calcula ion o he index o i s
Di ac ope a o /∂Lpwhe e Lis he gene a ing line bundle as i was in he p e ious
sec ion. Bu his ime we need he ac ha σ1is ac ually a nega i e gene a o o
H2(X, Z) wi h ou o ien a ion con en ions and so we ha e
c1(X) = −5σ1, σ1nega i e, σ1=c1(L) (D.9)
⇒K=L5(m= 5) (D.10)
ˆ
A(X) = 1−p1(X)
24 +1
5760(7p2
1(X)−4p2(X)) (D.11)
−1
210 ·945(16p3(X)−44p1(X)p2(X) + 31p3
1(X)) + ···(D.12)
as well as
p1(X) = c2
1(X)−2c2(X) = σ2
1+ 2σ2(D.13)
p2(X) = −2c1(X)c3(X) + c2
2(X) + 2c4(X) = 10σ4
1−20σ2
1σ2+ 15σ2
2(D.14)
p3(X) = 2c1(X)c5(X)−2c2(X)c4(X) + c2
3(X)−2c6(X) (D.15)
= 51σ6
1+ 72σ2
1σ2
2−144σ4
1σ2+ 68σ3
2.(D.16)
This in o ma ion allows o compu e ha
index (/∂Lp) = exp [pc1(L)] ˆ
A(X)[X] (D.17)
=−1
60480σ3
2[X]−41
15120 +1
360p2σ2
1σ2
2[X] (D.18)
+353
161280 +3
320p2−1
288p4σ4
1σ2[X] (D.19)
+−407
967680 −11
3840p2−1
576p4+1
720p6σ6
1[X].(D.20)
Now use he cohomology gene a o s and he ac ha Xclea ly has Eule cha -
ac e is ic 10 we disco e ha
σ3
2[X] = 1 (D.21)
σ2
1σ2
2[X] = 2 (D.22)
σ4
1σ2[X] = 3 (D.23)
σ6
1[X] = 5.(D.24)
– 19 –
This all gi es he o mulae
index (/∂Lp) = −1
1024 +19
2304p2−11
576p4+1
144p6(D.25)
=1
9.210 (4p2−9)(4p2−1)2(D.26)
=1
144(q+ 1)(q+ 2)2(q+ 3)2(q+ 4),using p=−q−5/2,(D.27)
and his index is an in ege o in eg al qas equi ed.
We shall inish by calcula ing he index when we couple he Di ac ope a o o
some highe ank bundles. We shall gi e he esul s o wo bundles Eand Fwhich
a e na u ally associa ed o Xand also o he enso p oduc E⊗F.
Le Ebe he ank 3 ec o bundle o e
X=U(5)
U(3) ×U(2) (D.28)
whose ib e o e a poin x∈Xis he 3-plane xi sel . This desc ibes he bundle E.
Now conside he p oduc ank 5 bundle X×C5 hen Fis he ank 2 bundle c ea ed
by o ming he quo ien
X×C5
E.(D.29)
The bundles Eand Fsa is y
E⊕F∼
=I5(D.30)
whe e I5is a i ial ank 5 bundle and i is no di icul o wo k ou ha
c(E)c(F) = 1 (D.31)
i.e. (1 + c1(E) + c2(E) + c3(E))(1 + c1(F) + c2(F)) = 1 (D.32)
ch(E) + ch(F) = 5.(D.33)
In ac equa ion (D.33) can be used o de i e he ela ion (D.3) since he classes σi
a e jus he classes ci(E) and so his allows all o c(E) and c(F) o be exp essed in
e ms o he σi.
The Che n cha ac e s o Eand Fa e gi en by
ch(E) = 3 + c1(E) + 1
2c2
1(E)−2c2(E)+1
3! c3
1(E)−3c1(E)c2(E) + 3c3(E)+1
4! c4
1(E)
−4c2
1(E)c2(E) + 4c1(E)c3(E) + 2c2
2(E)+1
5! c5
1(E)−5c3
1(E)c2(E) + 5c2
1(E)c3(E)
+5c1(E)c2
2(E)−5c2(E)c3(E)+1
6! c6
1(E)−6c4
1(E)c2(E) + 6c3
1(E)c3(E)
+9c2
1(E)c2
2(E)−12c1(E)c2(E)c3(E)−2c3
2(E) + 3c2
3(E)
ch(F) = 5 −ch(E).
– 20 –

Now we can calcula e he index o he app op ia e Di ac ope a o s: Fo ming he
p oduc Lp⊗Ewe ha e
index (/∂Lp⊗E) = ch (Lp⊗E)ˆ
A(X)[X] (D.34)
= ch (E) exp [pc1(L)] ˆ
A(X)[X],(D.35)
and we ind ha
index (/∂Lp⊗E) = −15
1024 +3
128p+59
768p2−5
48p3−5
64p4+1
24p5+1
48p6
=1
3.210 (2p+ 5)(2p−1)(4p2−9)(4p2−1) (D.36)
=1
48q(q+ 1)(q+ 2)(q+ 3)2(q+ 4),(p=−q−5/2) (D.37)
and o he p oduc Lp⊗F
index (/∂Lp⊗F) = 5
512 −3
128p−41
1152p2+5
48p3−5
288p4−1
24p5+1
72p6
=1
9.29(2p−5)(2p−1)(4p2−9)(4p2−1) (D.38)
=1
72(q+ 1)(q+ 2)(q+ 3)2(q+ 4)(q+ 5),(p=−q−5
2).(D.39)
Finally o he bundle Lp⊗E⊗Fwe ha e
index (/∂Lp⊗E⊗F) = −25
512 −25
384p+103
384p2+13
48p3−29
96p4−1
24p5+1
24p6
=1
3.29(4p2−25)(2p−3)(4p2−1)(2p+ 1) (D.40)
=1
24q(q+ 2)2(q+ 3)(q+ 4)(q+ 5),(p=−q−5/2) (D.41)
and in each case one can e i y ha he index is an in ege .
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– 22 –