DIAS-STP-03-09
Fuzzy Complex Quad ics and Sphe es
B ian P. Dolan,a),b)∗Denjoe O’Conno b)†and P. P eˇsnajde c)‡
a)Dep . o Ma hema ical Physics, NUI, Maynoo h, I eland
b)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
c)Dep . o Theo e ical Physics, Comenius Uni e si y,
Mlynsk´a dolina, SK-84248 B a isla a, Slo akia
Janua y 17, 2004
Abs ac
A ma ix algeb a is cons uc ed which consis s o he necessa y
deg ees o eedom o a ini e app oxima ion o he algeb a o unc-
ions on he amily o o hogonal G assmannians o eal dimension
2N, known as complex quad ics. These ma ix algeb as con ain he
ele an deg ees o eedom o desc ibing unca ions o ha monic ex-
pansions o unc ions on N-sphe es. An In¨on¨u-Wigne con ac ion o
he quad ic gi es he co- angen bundle o he commu a i e sphe e in
he con inuum limi . I is shown how he deg ees o eedom o he
sphe e can be p ojec ed ou o a ini e dimensional unc ional in e-
g al, using second-o de Casimi s, gi ing a well-de ined p ocedu e o
cons uc ion unc ional in eg als o e uzzy sphe es o any dimension.
∗[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
1
1 In oduc ion
Non-commu a i e geome y has slowly been inc easing in impo ance in
physics o e he las 20 yea s and has ecen ly ecei ed a s ong impe us
h ough wo k in s ing heo y. An impo an concep in he non-commu a i e
p og amme is ha o a “ uzzy” space — his is pe haps mo e co ec ly de-
sc ibed as a ini e, non-commu ing, ma ix app oxima ion o he algeb a o
unc ions on a con inuous mani old which can ep oduce he commu a i e
algeb a in he limi o he ma ices becoming in ini e in size. I has been
sugges ed ha uzzy spaces could p o ide a egula isa ion echnique o nu-
me ical calcula ions in quan um ield heo y which would be an al e na i e o
la ice gauge heo y, [1]-[4]. The p o o ypical example o a uzzy space is he
uzzy wo-sphe e, [5], bu he e a e many mo e examples, and indeed much
o he wo k on gene alised cohe en s a es in quan um mechanics [6], when
es ic ed o compac g oups, can be ela ed o uzzy spaces. The geome y
o uzzy CPNand uzzy uni a y G assmannians has been examined in he
li e a u e in some de ail [7]-[10].
Clea ly, om he poin o iew o a egula isa ion echnique in ield heo y
as well as o he heo y o D-b anes in s ing heo y, i would be desi able o
ha e an explici cons uc ion o he uzzy sphe e SN
Fin dimensions o he han
N= 2. Un o una ely, despi e he elegan simplici y o he uzzy wo-sphe e
S2
F, highe dimensional sphe es a e no so amenable o a uzzy desc ip ion. To
ou knowledge he e is no closed ini e ma ix app oxima ion o he algeb a o
unc ions on a sphe e o dimensions g ea e han wo, hough he p oblem
was ackled in [11] and [12] (non-commu a i e sphe es in he con inuum
we e analysed in [13]). I appea s ha , while one can ep esen unca ed
ha monic expansions o unc ions on sphe es by squa e ma ices, he p oduc
o wo such ma ices akes one ou o he equi ed space and a p ojec ion
back in o he space o unc ions is necessa y a e e e y mul iplica ion — his
ende s he p oduc non-associa i e. In o he wo ds a s a p oduc canno
be de ined on he uzzy sphe e SN
F o N > 2 ( his is ela ed o he ac
ha , o N6= 2, SNcanno be ob ained as he co-adjoin o bi o a compac
g oup). Ne e heless he cons uc ion in [11] does associa e a squa e ma ix
wi h he unca ion o a ha monic expansion on SNand so does, in a sense,
cons i u e a uzzy sphe e, e en hough he e is no associa i e p oduc .
Fo S4
Fal e na i e desc ip ions a e possible, [14] [15]. The cons uc ion
in [15] is in e ms on CP3
Fand uses he ac ha he con inuum CP3is an
S2bundle o e S4[15]. This app oach has he ad an age ha i is designed
o be used in a unc ional in eg al and he echnique was ex ended o S3
F
and S1
Fin [16]. The me hod o [11] would be pa icula ly cumbe some o
implemen in unc ional in eg als o e S3
Fand S1
F, o indeed any odd sphe e.
2
In his pape we gene alise he cons uc ion in [16] o uzzy sphe es o any
dimension. The analysis elies on p ope ies o he o hogonal G assmannian,
SO(N+ 2)/[SO(N)×SO(2)], which is a co-adjoin o bi o dimension 2N.
This o hogonal G assmannian can also be ob ained as he complex quad a ic
zaza= 0 in CPN+1, whe e zaa e na u al complex co-o dina es, [17]. The no-
a ion in Kobayashi and Nomizu is QN∼
=SO(N+ 2)/[SO(N)×SO(2)] and
we shall use his as a sho hand. In¨on¨u-Wigne con ac ion o SO(N+ 2)
o he Euclidean g oup o RN+1 ela es QN o he co- angen bundle T∗SN.
In a sense, elabo a ed on in sec ion 2, QNcan be hough o as he com-
pac i ied co- angen bundle o a non-commu a i e sphe e. The e is a ini e
dimensional, non-commu a i e, ma ix algeb a app oxima ion o he algeb a
o unc ions QNwhich con ains ha monic expansions on SN(desc ibed in
sec ion 3). While i is s ill he case ha he esul ing ma ix algeb a akes
one ou o he space o ha monic expansions on SNupon mul iplica ion (so
one does no ha e a closed ma ix app oxima ion o he algeb a o unc ions
on SN o N6= 2) i is ne e heless e y easy o supp ess unwan ed modes in
a unc ional in eg al o e a uzzy complex quad ic, QN, in a manne which
lends i sel na u ally o nume ical compu a ion o ield heo y on SN
F, as
desc ibed in sec ion 4.
2 The de o med co- angen bundle
In his sec ion i is shown how he co- angen bundle T∗SNcan be de o med
o a e sion ela ed o a non-commu a i e sphe e and compac i ied o com-
plex quad ic, QN.
The cons uc ion s a s wi h Ca esian co-o dina es Xain RN+1, whe e
a= 1, . . . , N + 1. Conside he sphe e SN∼
=SO(N+ 1)/SO(N) o adius R
de ined by XaXa=R2. The isome y g oup is SO(N+ 1) wi h algeb a
[Lab, Lcd] = i(δbcLad +δadLbc −δacLbd −δbdLac),(1)
whe e Lab =−Lba ( he e is no dis inc ion be ween uppe and lowe Euclidean
indices in RN+1,Xa=Xa). This can be ex ended o a ep esen a ion o he
Euclidean g oup, EN+1 ac ing on RN+1,
[Lab, Lcd] = i(δbcLad +δadLbc −δacLbd −δbdLac) (2)
[Lab, Xc] = i(δbcXa−δacXb) (3)
[Xa, Xb]=0.(4)
An explici ealisa ion o his algeb a in he con inuum is
Lab =i Xa
∂
∂Xb
−Xb
∂
∂Xa!(5)
3
ac ing on unc ions and Xabeing commu a i e mul iplica ion by he co-
o dina es.
The algeb a will now be de o med, essen ially using he in e se o In¨on¨u-
Wigne con ac ion. Le Xa:= µLa,N+2, wi h µ eal, and eplace he com-
mu a o (4) abo e wi h
[Xa, Xb] = −iµ2Lab,(6)
lea ing he o he commu a o s unchanged. The Euclidean g oup EN+1 is
hus de o med o SO(N+ 2)
[LAB, LCD] = i(δBCLAD +δADLBC −δACLBD −δBDLAC),(7)
whe e A, B, C, D = 1, . . . , N + 2 and La,N+2 =−LN+2,a. In e ms o he
quad a ic Casimi s1CN+1
2= (1/2)LabLab and CN+2
2= (1/2)LABLAB we see
ha
XaXa=µ2CN+2
2−CN+1
2.(8)
While i may be emp ing o hink o Xain (8) as co-o dina es on a uzzy
sphe e we mus be ca e ul: a gene al i educible ep esen a ion o SO(N+2)
will decompose in o a sum o di e en i educible ep esen a ions o SO(N+ 1),
wi h di e en alues o CN+1
2, so he igh -hand side o (8) will no be cen-
al. Howe e XaXais cen al in he undamen al spino ep esen a ion o
Spin(N+ 2). To see his conside he e en and odd cases sepa a ely:
•E en N: choose one chi ali y o spino wi h 2N/2componen s. Unde
Spin(N+ 2) →Spin(N+ 1) his educes uniquely o he single spino
ep esen a ion o Spin(N+ 1), which has he same dimension. In his
case he igh -hand side o (8) is a mul iple o he iden i y.
•Odd N: in his case he 2N/2dimensional spino ep esen a ion o
Spin(N+ 2) decomposes in o wo spino ep esen a ions o opposi e
chi ali y unde SO(N+ 2) →SO(N+ 1), bo h o dimension 2(N/2)−1.
Bu he second o de Casimi CN+1
2has he same alue on he wo
chi ali ies, i does no dis inguish be ween hem, [18]. So again he
igh -hand side o (8) is a mul iple o he iden i y.
The undamen al spino ep esen a ions o Spin(k) ha e quad a ic Casimi
Ck
2=k(k−1)/4 o bo h e en and odd k, see [18] o example. Fo a
undamen al spino ep esen a ion o Spin(N+ 2) equa ion (8) he e o e
gi es
XaXa=µ2(N+ 1)
21,(9)
1Rela i e o he s anda d con en ions o SU(n) ou no malisa ion he e is such ha
CSU(2)
2=1
2C3
2and CSU(4)
2=1
2C6
2.
4
whe e 1is he iden i y ope a o , and we migh in e p e qN+1
2µas he adius
o a uzzy sphe e. Howe e , unlike S2
F, he ma ix algeb a gene a ed by Xa
will no close in gene al e en when XaXais cen al. Also one canno ge
highe dimensional ep esen a ions o SN
Fby aking enso p oduc s. This
e lec s he ac ha he e is no closed ini e ma ix app oxima ion o SN
F o
N6= 2.
Le us now in es iga e he geome y o he space gene a ed by he adjoin
ac ion o Spin(N+ 2) on a iducial di ec ion XN+1 ( he “no h pole” o
SN
F). Clea ly [Lαβ, XN+1] = 0 o α, β = 1, . . . , N, so Spin(N) lea es XN+1
in a ian . Also LN+1,N+2 commu es wi h XN+1, since hey a e jus mul iples
o one ano he , so he SO(2) gene a ed by LN+1,N+2 also lea es Xain a ian .
The upsho o his is ha he mani old ha D(g)∈Spin(N+ 2) gene a es
wi h he ac ion
D−1(g)XN+1D(g) (10)
is he o hogonal G assmannian QN, which has dimension 2N. Fo N≤4
hese spaces ha e he ollowing s uc u es:
•Q1∼
=SO(3)/SO(2) ∼
=S2;
•Q2∼
=SO(4)/[SO(2) ×SO(2)] ∼
=S2×S2(S2×S2was used in a
conside a ion o uzzy S3/Z2in [19]);
•Q3∼
=SO(5)/[SO(3) ×SO(2)] ∼
=CP3/Z2( his iden i ica ion is de-
sc ibed in [16]);
•Q4∼
=SO(6)/[SO(4) ×SO(2)] ∼
=SU(4)/[S(U(2) ×U(2))] (ma ix ap-
p oxima ions o his space we e desc ibed in [10]).
We p opose o iden i y QNwi h a ‘compac i ied’ co- angen bundle o
a non-commu a i e sphe e. This educes o he usual T∗SNunde In¨on¨u-
Wigne con ac ion µ→0: his limi pe o ms he dual unc ion o en-
de ing he X’s in (6) commu a i e while a he same ime de-compac i ying
SO(N+ 2) o he Euclidean g oup EN+1.
3 Fuzzy Complex Quad ics, QN
F
Any uni a y i educible ep esen a ion To a simple compac Lie g oup is
ini e dimensional and he ex ension o i s en eloping algeb a is a ini e di-
mensional ma ix algeb a. This ma ix algeb a p o ides a uzzy app oxima-
ion o he algeb a o unc ions on he co-adjoin o bi o any Lie algeb a
elemen in T.
5

We shall e e o a ini e ma ix app oxima ion o QNas a uzzy complex
quad ic and deno e i by QN
F. Being a co-adjoin o bi QNis a symplec ic
mani old whose algeb a o unc ions can be app oxima ed by closed ini e
dimensional ma ix algeb as. The ha monic expansion o a unc ion on QN
equi es all ep esen a ions o SO(N+ 2) which con ain he i ial ep esen-
a ion unde SO(N)×SO(2). A spino ep esen a ion o SO(N+ 2) ne e
con ains a single o SO(N)×SO(2) o ei he e en o odd N, so we e-
s ic o ec o ial ep esen a ions. The ep esen a ion space o SO(N+ 2)
is hen he space o ank-n enso s, TA1···An. I educible ep esen a ions can
be cha ac e ised by hei symme ies unde in e change o hei indices and
a e aceless in any wo indices. They can be ep esen ed by Young ableau
wi h nboxes which e lec hei pe mu a ion symme ies.
Any enso ha is an i-symme ic in h ee o mo e o i s indices mus
anish when es ic ed o a ep esen a ion o SO(2) and hence canno con-
ibu e o he ha monic expansion o unc ions on QN. Thus we can es ic
ou a en ion o enso s ha a e an i-symme ic in pai s o indices only. I
TA1···Anhas mpai s o an i-symme ic indices and n−2msymme ic indices
hen he co esponding Young ableau is
m
z }| {
··
··
n−2m
z }| {
·· .(11)
When SO(N+ 2) is es ic ed o SO(N)×SO(2) he an i-symme ic pai s
all con ain single s o SO(N)×SO(2) (when bo h indices in a pai a e SO(2)
indices, o example). Also i nis e en he n−2msymme ic indices can
con ain single s o SO(N)×SO(2), bu no when nis odd. Thus he ha monic
expansion o a unc ion on QN equi es all SO(N+ 2) enso ep esen a ions
o he o m (11) wi h ne en. The dimension o hese ep esen a ion can be
de e mined, using he ele an o mulae in [18] o example. Wi h n= 2l
hey a e, o N≥3,
dN(2l, m) = (2l+N−1)(2l+ 1 −2m)(4l+N−2m)(N−2+2m)
×(2l+N−2−m)!(N−3 + m)!
N!(N−2)!(2l−m+ 1)!m!(12)
whe e m≤l.2I he ha monic expansion o a unc ion on QNis unca ed
a lmax =L he o al numbe o deg ees o eedom is
L
X
l=0
l
X
m=0
dN(2l, m) = "(2L+N)(L+N−1)!
L!N!#2
= [dN(L, 0)]2.(13)
2In he no a ion o [18] he ep esen a ions (11) ha e highes weigh s
( 1, . . . , [N/2]+1) = (n−m, m, 0, . . . , 0), whe e [N/2] is he in ege pa o N/2.
6
The ac ha his is a pe ec squa e e lec s he ac ha he deg ees o
eedom in a unca ed ha monic expansion can be a anged in o a squa e
ma ix o size dN(L, 0)×dN(L, 0). This can be ep esen ed in e ms o Young
ableau by
L
z }| {
·· ×
L
z }| {
·· = 1 ⊕ ⊕ ⊕ · · · ⊕
L
z }| {
··
·· ⊕ · · · ⊕
2L
z }| {
·· (14)
which ela es he ma ix s uc u e, on he le -hand side, o he ha monic
expansion, on he igh -hand side. Ma ix mul iplica ion hen gi es a closed
associa i e, bu non-commu a i e, algeb a which ep oduces he commu a-
i e algeb a o unc ions on QNas L→ ∞. A he le el o unc ions he
non-commu a i e p oduc a ini e Lcan be ealised as a ∗-p oduc o he
uzzy complex quad ic, QN
F.
4 Func ional In eg als on Fuzzy Sphe es, SN
F
The e is no closed ini e ma ix app oxima ion o he unca ed algeb a o
unc ions on SNknown, excep o he special case N= 2 which was i s
desc ibed in [5]. The e does exis a ma ix app oxima ion o unc ions on
SN, bu in gene al ma ix mul iplica ion does no co espond o he algeb a
o unc ions and he la e can only be eco e ed by p ojec ing back on o
a unc ion on SNa e ma ix mul iplica ion [11]. This esul s in a non-
associa i e algeb a when N6= 2 which, by a sligh abuse o language, is
ne e heless s ill e e ed o as a “ uzzy” sphe e, SN
F.
In he cons uc ion p esen ed he e a simila p ojec ion can be pe o med,
since he unca ed ha monic expansion o a unc ion on SNis bu ied in QN
F.
To see his no e ha unc ions on SNcan be expanded in symme ic enso
ep esen a ions o SO(N+ 1). A unca ion a le el lmax equi es using all
symme ic enso s o SO(N+ 1), Ta1···al, wi h 0 ≤l≤lmax. These a e all
con ained in one symme ic ep esen a ion
lmax
z }| {
·· (15)
o SO(N+ 2) unde SO(N+ 2) →SO(N+ 1). Se ing lmax = 2Lwe see
ha he las ep esen a ion on he igh -hand side o (14) con ains all he
SO(N+1) ep esen a ions necessa y o he ha monic expansion o a unc ion
on SNup o angula momen um 2L. We can hus ob ain he uzzy sphe e
SN
Fby p ojec ing he i educible ep esen a ion
2L
z }| {
·· ou om he ma ix
algeb a o QN
Fin (14).
7
This can be achie ed in a unc ional in eg al o e QN
Fin he same manne
as in [16]. Le Φ be a ma ix in he algeb a o QN
F o a gi en L. The
SO(N+ 2) in a ian Laplacian on QN
Fis
L2
(N+2)Φ = −(1/2)[LAB,[LAB,Φ]].(16)
Then he ac ion o a scala ield on QN
Fcan be w i en as
S[Φ] = 1
dN(L, 0)T nֆL2
(N+2)Φ + V(Φ)o.(17)
wi h he scala po en ial V(Φ†) = V(Φ) assumed bounded below. A unc-
ional in eg al hen in ol es
Z=ZDΦe−S[Φ].(18)
We ocus on he SN
Fembedded in QN
Fby penalising all he SO(N+ 2) ep-
esen a ions in Zexcep he las one on he igh -hand o (14). This can
be achie ed by modi ying he kine ic e m in he ac ion. The second o de
Casimi o he ep esen a ion (11), wi h n= 2l, is [18]
CN+2
2(2l, m) = (2l−m)(2l−m+N) + m(m+N−2).(19)
Obse e ha he comple ely symme ic enso s wi h m= 0 ha e he la ges
Casimi o any gi en l,
CN+2
2(2l, 0) = 2l(2l+N).(20)
Hence he ope a o
−1
2[LAB,[LAB,·]−2L(2L+N) = CN+2
2−2L(2L+N) (21)
ac ing on Φ is nega i e o all modes in Φ excep o he op mode, wi h
l=Land m= 0, on which i anishes. The SO(N+ 1) in a ian Laplacian
on SN
Fwould be
L2
(N+1)Φ = −(1/2)[Lab,[Lab,Φ]].(22)
So he ac ion
Sh[Φ] = 1
dN(L, 0)T nֆL2
(N+1)Φ + hΦ†−L2
(N+2) + 2L(2L+N)Φ + V(Φ)o,
(23)
wi h h1, will supp ess all he unwan ed modes in a unc ional in eg al
and lea e he equi ed modes o SN
Funa ec ed. In he limi h→ ∞ all
modes, excep he ones ele an o SN
F, will be supp essed and co ela ion
unc ions calcula ed wi h
Z=ZDΦe−S∞[Φ].(24)
will be hose o he uzzy sphe e, unca ed a le el 2L.
8
5 Conclusions
By cons uc ing uzzy app oxima ions o complex quad ics, QN
F, a p esc ip-
ion o de ining unc ional in eg als o e ini e app oxima ions o sphe es
has been p esen ed. Al hough ini e ma ix app oxima ions o he algeb a
o unc ions on N-dimensional sphe es a e no known o N6= 2, one can
cons uc ma ix app oxima ions o he unc ions, bu ma ix mul iplica ion
hen akes one ou o he space o unc ions on he sphe e.
The cons uc ion p esen ed he e elies in he ac ha he complex quad ics
(which a e o hogonal G assmannians QN∼
=SO(N+ 2)/[SO(N)×SO(2)])
a e co-adjoin o bi s, and hence do ha e ini e ma ix app oxima ions o hei
algeb a o unc ions, QN
F. These spaces a e ela ed o he co- angen bun-
dles T∗SN— hey a e in a sense compac i ied e sions o he co- angen
bundles, compac i ied a he expense o in oducing a non-commu a i i y on
he sphe e. The algeb a o unc ions on QN
Fcon ains he ele an deg ees
o eedom o a unca ed ha monic expansion o a unc ion on SN. The
unc ional in eg al o a ield heo y de ined on QNcan be egula ised in a
manne ha p ese es he isome ies and a oids Fe mion doubling [20] by
de ining i o e he uzzy space QN
F. By modi ying he kine ic e m and using
he ac ion (23) he deg ees o eedom ha a e no ele an o he unde lying
sphe e can be p e en ed om con ibu ing o he unc ional in eg al and he
esul is a well de ined, ini e app oxima ion o he unc ional in eg al o a
quan um ield heo y on SN. The co ec algeb a is ensu ed by es ic ing Φ
o be ma ices o size dN(L, 0) gi en in (13) and he con inuum is eco e ed
in he limi L→ ∞. This cons uc ion is well sui ed o nume ical e alua ion.
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9