The Moyal Sphere

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by  Michał Eckstein

The Moyal Sphere
Michał Eckstein1, Andrzej Sitarz1,2, Raimar Wulkenhaar3
1InstituteofPhysics,JagiellonianUniversity,
Łojasiewicza11,30-348Krako´w,Poland.
2 InstituteofMathematicsofthePolishAcademyofSciences,
S´niadeckich8,00-950Warszawa,Poland.
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1 3MathematischesInstitutderWestfa¨lischenWilhelms-Universita¨t
0
Einsteinstraße62,D-48149Mu¨nster,Germany
2

n
a
Abstract
J

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WeconstructafamilyofconstantcurvaturemetricsontheMoyalplane
2
andcomputetheGauss–Bonnettermforeachofthem. Theyarisefromthe

] conformal rescaling of the metric in the orthonormal frame approach. We
A
findaparticularsolution,whichcorrespondstotheFubini–Studymetricand
Q
which equips the Moyal algebra with the geometry of a noncommutative
.
h sphere.
t
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m Keywords: Moyaldeformation,noncommutativemetricspace,spacesofconstant
[ curvature

1 PACS:02.40.Gh,02.40.Ky
v
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Contents
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5
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1 Introduction 2
.
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0
6 2 ClassicalFubini–Studymetricontheplane 2
1
:
v 3 TheflatgeometryoftheMoyalplane 4
i
X 3.1 MatrixbasisfortheMoyalalgebra . . . . . . . . . . . . . . . . . 4
r 3.2 Radialfunctionsinthematrixbasis . . . . . . . . . . . . . . . . . 5
a
4 ConformallyrescaledmetricontheMoyalplane 6
4.1 Theorthormalframeapproach . . . . . . . . . . . . . . . . . . . 7
4.2 Solutioninthematrixbasis . . . . . . . . . . . . . . . . . . . . . 7
4.2.1 AsymptoticbehaviorandGauss–Bonnetterm . . . . . . . 8
4.3 TheMoyal–Fubini–Studymetric . . . . . . . . . . . . . . . . . . 10
4.4 Curvaturea` laRosenberg . . . . . . . . . . . . . . . . . . . . . . 11
1
5 PerturbativesolutionfortheMoyal–Fubini–Studymetric 11
5.1 PerturbativeexpansionoftheMoyalproduct . . . . . . . . . . . . 12
5.2 TheMoyal–Fubini–Studymetricuptoorderθ2 . . . . . . . . . . 12
6 Conclusionsanddiscussion 13
1 Introduction
Noncommutative geometry provides a unified framework to describe classical,
discrete as well as singular or deformed spaces [3,9]. Most of the examples stud-
iedsofarwereconstructedwithafixedmetric,allowingonlysmallpertubationsof
the gauge type. Only recently a class of models with conformally modified non-
commutative metrics was constructed for the noncommutative tori [5] allowing
forthecomputationofthescalarcurvatureusingdifferentapproaches[4,6,7,14].
The Moyal deformation of a plane [1,13] is one of the oldest and best-studied
noncommutative spaces. Motivated by the appearence in the string-theory target
space [15] it is often used as model of noncommutative space-time (see [16] for
a review) and a background for a noncommutative field theory [17]. However, so
far the only geometry considered is the flat geometry, which corresponds to the
constant metric on the plane. In this paper we introduce a class of conformally
rescaledmetricsonthetwo-dimensionalMoyalplaneusingtheorthonormalframe
formalism adapted to the noncommutative setting. We compute the scalar curva-
tureandlookforthesolutionsoftheconstantcurvaturecondition,thusfindingthe
Moyal–Fubini–Studymetrics.
The paper is organized as follows: first, we recall the classical (commutative)
constant curvature solutions, then we briefly review the noncommutative Moyal
plane and compute the scalar curvature using conformally rescaled orthonormal
frames. We discuss explicit solutions in the matrix basis for the Moyal algebra
as well the first order perturbative correction to the Fubini–Study metric using
smoothfunctionsontheplane.
2 Classical Fubini–Study metric on the plane
Considertheconformallyrescaledmetricontheplane:
k(x,y)2(dx2 +dy2),
Weassumethatk = k(r),thenthescalarcurvatureis:
(cid:18) (cid:19)
k(r)
R(k) = 2k(r)−4 k(cid:48)(r)k(cid:48)(r)− k(cid:48)(r)−k(r)k(cid:48)(cid:48)(r)
r
2
Now, let us look for k such that R(k) = C = const. We obtain the differential
equation:
k(r)
k(r)k(cid:48)(cid:48)(r)+ k(cid:48)(r)−k(cid:48)(r)k(cid:48)(r) = Ck4(r),
r
whichhasafamilyofnondegeneratesolutionsforA > 0,a ≥ 1andb > 0:
Ara−1
k(r) = ,
b+r2a
sothatthescalarcurvatureis:
8a2b
R(k) = ,
A2
thevolume:
A2
V(k) = π
ba
andtheGauss–Bonnetterm:
(cid:90)
√
gR = 8πa.
More generally, assume that we have a class of conformally rescaled metrics for
whichtheasymptoticsofk(r)atinfinityisgivenby:
k(r) ∼ r−α +C r−α−1 +C r−α−2
1 2
Onecaneasilyverifythatthescalarcurvatureisregular,thatis:
lim R(r) < ∞,
r→∞
ifC = 0andα ≤ 2.
1
WecannowcomputetheGauss–Bonnettermforsuchmetrics:
(cid:90) √ (cid:90) ∞ (cid:18) k(r) (cid:19)
gR(g) = 4π rdr k(r)−2 k(cid:48)(r)k(cid:48)(r)− k(cid:48)(r)−k(r)k(cid:48)(cid:48)(r)
r
R2 0
(cid:90) ∞
(cid:0) (cid:1)
= 4π dr rk(r)−2k(cid:48)(r)k(cid:48)(r)−k(r)−1k(cid:48)(r)−rk(r)−1k(cid:48)(cid:48)(r)
0
(cid:90) ∞
(cid:0) (cid:1)
= 4π dr(−rk(cid:48)(r)k(r)−1)(cid:48) = 4π (−rk(cid:48)(r)k(r)−1) −(−rk(cid:48)(r)k(r)−1) .
∞ 0
0
Assuming that k(r) and its derivatives are regular at r = 0 and that k(r) behaves
liker−α atr → ∞weobtain:
(cid:90)
√
gR(g) = 4πα.
R2
3
Note that the special case, where α = 2, which is the Fubini–Study metric yields
thecorrectGauss–Bonnettermforthesphere.
Ontheotherhand,ifwerequirethatthemetricaloneremainsboundedatr = ∞,
wehave,afterchangeofvariablesρ = 1,that
r
(cid:0) (cid:1)
limk2 ρ−1 ρ−4 < ∞.
ρ→0
Thishappensiftheasymptoticsofk(r)isr−α forα ≥ 2. Moreover,ifwerequire
that the metric does not vanish at r = ∞ then α = 2 is the only solution for the
asymptoticbehaviorofk(r).
3 The flat geometry of the Moyal plane
WebeginbyreviewinghereshortlybasicresultsonMoyalgeometry. Wetakethe
algebra of the Moyal plane, A , as a vector space (S(R2),∗), (S is the Schwartz
θ
space),equippedwiththeMoyalproduct∗definedasfollowsthroughtheoscilla-
toryintegrals[13]:
(cid:90)
(f ∗g)(x) := (2π)−2 eiξ(x−y)f(x− 1Θξ) g(y) dny dnξ. (1)
2
R2×R2
(cid:18) (cid:19)
0 θ
where Θ = , θ ∈ R. With the product defined in (1) it is easy to see
−θ 0
that
(cid:90)
f (cid:55)→ f(x)dnx
Rn
isatraceontheMoyalalgebraandthestandardpartialderivations∂ ,∂ remain
x1 x2
derivationsonthedeformedalgebra.
The algebra can be faithfully represented on the Hilbert space of L2-sections of
theusualspinorbundleoverR2 (whichisH = L2(R2)⊗C2)actingbyMoyalleft
multiplicationLΘ(a). Itwasdemonstratedfirstin[8]thattheMoyalplanealgebra
with one of its preferred unitizations yield nonunital, real spectral triples for the
standardDiracoperatorontheplanearisingfromtheflatEuclideanmetric.
3.1 Matrix basis for the Moyal algebra
It will be convenient to work with the matrix basis for the Moyal algebra [8]. We
definefirst:
f0,0 = 2e−θ1(x22+x22),
4
andthealgebraA hasanaturalbasisconsistingof:
θ
1
f = √ (a∗)m ∗f ∗(a)n,
m,n 00
m!n!θm+n
ormoreexplicitly:
fm,n(r,φ) = 2(−1)m(cid:114)mn!!eiφ(m−n)(cid:32)(cid:114)2θr(cid:33)(n−m)Lmn−m(cid:18)2θr2(cid:19)e−rθ2. (2)
Wehaveforeachm,n,k,l ≥ 0:
f ∗f = δ f , f∗ = f .
m,n k,l kn m,l m,n n,m
Inparticularforallk ≥ 0allf arepairwiseorthogonalprojectionsofrankone.
k,k
Thenatural(notnormalized)traceontheMoyalalgebrais:
(cid:90)
τ(f) = d2xf(x), τ(f ) = 2πθδ .
m,n mn
R2
Itisconvenienttoworkwiththelinearcombinationsofderivations:
1 1
∂ = √ (∂ −i∂ ), ∂¯= √ (∂ +i∂ ).
2 x1 x2 2 x1 x2
Then:
(cid:114) (cid:114)
n m+1
∂f = f − f ,
m,n m,n−1 m+1,n
θ θ
(cid:114) (cid:114)
m n+1
¯
∂f = f − f ,
m,n m−1,n m,n+1
θ θ
Furthermore,
1
∂∂¯= (∂2 +∂2 ).
2 x1 x2
3.2 Radial functions in the matrix basis
WedefineradialfunctionsontheMoyalplaneasthosewhichintheirpresentation
inthematrixbasishaveonlydiagonalelementsf ,so:
n,n
∞
(cid:88)
h = h f .
n n,n
n=0
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Each function F applied to h is easily computable, since f are projections, we
n,n
have:
∞
(cid:88)
F(h) = F(h )f .
n n,n
n=0
From the orthogonality of matrix basis f , the explicit representation (2) and
n,n
knownintegralsofLaguerrepolynomialsonededuces
∞
(cid:88)
ra = θa F (−n,−a;1;2)f (r).
2 2 1 2 n,n
n=0
Inparticular,
(cid:88)∞ (cid:88)∞ 1(cid:16)r2 (cid:17)
f = 1, nf = −1 .
n,n n,n
2 θ
n=0 n=0
Therefore the inverse of the conformal factor for the Fubini–Study metric, which
isalinearfunctionofr2,k(r)−1 = 1(b+r2)hasthefollowingexpansion:
A
∞
1 1 (cid:88)
(b+r2) = (2θn+b+θ)f .
n,n
A A
n=0
4 Conformally rescaled metric on the Moyal plane
There are several approaches to the conformal rescaling of the metric and com-
puting the curvature for the noncommutative torus and – by analogy – the Moyal
plane. Note that each of them uses a different notation and they do not give com-
patibleresultsinthecaseofthenoncommutativetorus.
Firstofall,onecantakeaconformalrescalingoftheflatLaplaceoperator: ∆ =
h
h∆h, which in the case of the noncommutative torus was studied by Connes–
Tretkoff[5]. Thiscorrespondstotherescalingofthemetricbyh−2
A second approach uses the language of orthonormal frames [6]. It replaces the
standard (flat) orthonormal frames, understood as derivations on the algebra, by
the conformally rescaled ones, δ → hδ , where h is from the algebra (or, more
i i
generally, from its multiplier). Applying the classical formula for the scalar cur-
vature, which easily adapts to the noncommutative case, one obtains a noncom-
mutativeversionofthescalarcurvature. Thiscorrespondsalsototherescalingof
themetricbyh−2.
Finally, extending the computations of Jonathan Rosenberg for the Levi-Civita
connection [14] on the noncommutative torus one might obtain a similar general-
izationoftheexpressionforthescalarcurvature.
We shall concentrate on the case of orthonormal frames and try to determine
whether there exists a radial conformal rescaling for which the scalar of curva-
tureisconstant.
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4.1 The orthormal frame approach
Let the orthonormal basis of frames be e = hδ for h in the Moyal algebra or
i i
its multiplier.1 We assume that h is positive and invertible, by h−1 we denote the
∗
inverse with respect to the Moyal product, similarly all powers are also Moyal
powers.
Wehave:
[e ,e ] = h∗δ (h)∗h−1e −h∗δ (h)∗h−1e .
1 2 1 ∗ 2 2 ∗ 1
sothat
c = h∗δ (h)∗h−1, c = −h∗δ (h)∗h−1.
122 1 ∗ 121 2 ∗
and
c = −h∗δ (h)∗h−1, c = h∗δ (h)∗h−1.
212 1 ∗ 211 2 ∗
Wecomputethescalarcurvatureasin[6,(2.1)]:
1 1
R = 2h∗δ (h∗δ (h)∗h−1)−(h∗δ (h)∗h−1)2− (h∗δ (h)∗h−1)2− (h∗δ (h)∗h−1)2
i i ∗ i ∗ 2 i ∗ 2 i ∗
whichaftersimplificationsyields:
R = 2h2 ∗(δ h)∗h−1 +2h∗δ (h)∗δ (h)∗h−1
ii ∗ i i ∗
−2h2 ∗δ (h)∗h−1 ∗δ (h)∗h−1 −2hδ (h)∗δ (h)∗h−1 (3)
i ∗ i ∗ i i ∗
= 2h2 ∗(δ h)∗h−1 −2h2 ∗δ (h)∗h−1 ∗δ (h)∗h−1.
ii ∗ i ∗ i ∗
So,tosolvetheequationR(h) = C = constweneedtosolve:
(∆h)−δ (h)∗h−1 ∗δ (h) = Ch−1,
i ∗ i ∗
where∆isthestandardflatLaplaceoperator,∆ = δ2 +δ2.
1 2
We shall present the proof of the existence of the solution in the matrix basis
as well as compute explicitly the first term of the perturbative expansion for the
Fubini–Studymetric.
4.2 Solution in the matrix basis
Ansatz: Firstwelookforaradialsolution:
∞
(cid:88)
h = φ f .
n n,n
n=0
1Notethattherescaledframesarenolongerderivations,however,ifhistakenfromthecom-
mutant of the algebra (or its multiplier), e will be derivations from the Moyal algebra into the
i
algebraofboundedoperatorsontheHilbertspace[6]. Sincethisdoesnotchangeanythinginthe
computations,forthesakeofsimplicityweworkwithhfromthealgebraitself.
7
UsingtheactionofpartialderivativesandtheLaplaceoperatoronthebasis,
2 (cid:16) (cid:112) √ (cid:17)
∆f = −(m+n+1)f + (m+1)(n+1)f + mnf ,
m,n m,n m+1,n+1 m−1,n−1
θ
weobtainthefollowingequation:
(cid:88)(cid:0) (cid:1)
−(2n+1)φ −nφ2φ−1 −(n+1)φ2φ−1 f
n n n−1 n n+1 n,n
n
(cid:88) (cid:88)
+ (n+1)(φ +φ )f + n(φ +φ )f
n+1 n n+1,n+1 n−1 n n−1,n−1
n n
= R·φ−1f ,
n n,n
wherewehavesetR = Cθ. Ityieldsthefollowingrecurrencerelation:
n+1 n
(φ2 −φ2)+ (φ2 −φ2) = R·φ−1, (4)
φ n+1 n φ n−1 n n
n+1 n−1
for n ≥ 1. Note that although (4) is of the second order, it has only one degree of
freedom,sinceforn = 0wehave
φ
φ2 −φ2 = R 1. (5)
1 0 φ
0
Therecurrencerelationisaquadraticone,solvingitforx = φ wehave:
n+1
(cid:18) (cid:19)
n R
(n+1)x2 +x (φ2 −φ2)− −(n+1)φ2 = 0,
φ n−1 n φ n
n−1 n
and it is easy to see that it has only one positive root. It can also be easily seen
that the sum of roots is positive and since their products is −φ2 then the positive
n
root must be bigger than φ . Hence the solution will be an increasing positive
n
sequence. Since h needs to be a positive operator, we should start with an initial
value φ > 0 and take the positive root at each step to have φ > 0 for every
0 n
n ∈ N. Notethatφ = 0isnotallowedin(4).
0
4.2.1 AsymptoticbehaviorandGauss–Bonnetterm
As we have shown, there exists a family of solutions yielding a positive constant
scalarcurvaturefortheMoyalplane. Weshallnowlookforsomespecialsolutions
and their asymptotic behavior. First of all, observe that in the orthonormal frame
√
formalism we have g = h−2, so we can take as the noncommutative volume
elementh−2. Inordertohaveafinitevolume,thegrowthofthesequenceφ must
∗ √ n
befasterthan nsothatφ−2 givesasummableseries.
n
8
Foreachsolutionoftherecurrencerelation(4)weshallcomputenowtheGauss–
Bonnetterm:
(cid:90)
√ √
τ( g ∗R) = gR,
R2
√
where, again, g is the noncommutative Moyal volume element. Note that the
expression has no ambiguity because of the trace property of τ and we need to
compute:
(cid:90)
√
(cid:0) (cid:0) (cid:1) (cid:1)
gR = τ 2 (∆h)−δ (h)∗h−1 ∗δ (h) ∗h−1
i ∗ i
R2
2π (cid:90) ∞ (cid:88)∞ (cid:8)(cid:2) (cid:3)
= rdr −(2n+1)−nφ φ−1 −(n+1)φ φ−1 f
θ n n−1 n n+1 n,n
0 n=0
(cid:9)
+(n+1)(φ φ−1 +1)f +n(φ φ−1 +1)f .
n+1 n n+1,n+1 n−1 n n−1,n−1
Now recall that (cid:82)∞rdrf (r) = θ for all n ∈ N. However, to compute the
0 n,n
integralwithneedtointroduceacut-offintheseries. Wethushave
N N+1
(cid:88)(cid:2) (cid:3) (cid:88)
−(2n+1)−nφ φ−1 −(n+1)φ φ−1 + n(φ φ−1 +1)+
n n−1 n n+1 n n−1
n=0 n=0
N−1
(cid:88)
+ (n+1)(φ φ−1 +1) = (N +1)(φ φ−1 −φ φ−1 ).
n n+1 N+1 N N N+1
n=0
The expression above has a finite and nonvanishing limit as N → ∞ if the se-
quence h = φ φ−1 has a limit 1 and N(h − 1) has a finite limit. This
N N N−1 N
requirement alone is not sufficient to determine the asymptotic form of the solu-
tion. As the recurrence relation is highly nonlinear we can only check that some
asymptotics are compatible with the relations as well as the above requirement
fortheGauss–Bonnetterm. Inparticular,possibleasymptoticsincludethepower-
growing sequences φ ∼ Ana as well as power-growing sequences modified by
n
logarithms.
Tohaveaninsightintotheentirefamilyofpossiblesolutionswehavecarriedout
anumericalstudyofthesolutions(seefig.1),whichconfirmsthattheasymptotics
isofthattypeandgivesaglimpseoftherelationa = a(φ ).
0
Assumingthatasymptoticallyφ ∼ Ana asntendsto∞forsomeA,a > 0,then
n
lim (N +1)(φ φ−1 −φ φ−1 ) = 2a,
N+1 N N N+1
N→∞
9
a a
1.0
35
30 0.9
25
0.8
20
15 0.7
10
0.6
5
0.0 0.5 1.0 1.5 2.0Φ0 0 10 20 30 40 50Φ0
Figure 1: The above plots were obtained by a numerical computation of φ up
n
to n = 5000 starting from a given value of φ . The asymptotic exponent a is
0
then computed as logφ /logN at N = 5000. The error bars are obtained by
N
performing the numerical computation for N = 6000 (upper) and N = 4000
(lower). Theyshowthestabilityofthenumericalalgorithm.
and
(cid:90)
√
gR = 8πa,
R2
whichis,infact,quitesimilartotheclassicalcase.
4.3 The Moyal–Fubini–Study metric
It is obvious from the above computations that in the case of linear asymptotics
φ ∼ n, that is a = 1, we obtain the same Gauss–Bonnet term as for the classical
n
sphere and therefore the solution to (4) with φ ∼ n could be understood as the
n
Moyal–Fubini–Studysolution.
Although we cannot solve exactly the recurrence relation even in this particular
case, one can systematically find the asymptotics of φ . Explicit computations
n
giveuptoo( 1 ):
n3
(cid:18) (cid:19)
1 11 13R+9 1 1 1 26 29 1
φ = n+ (R+1)+ − + − + R+ R2 +··· .
n 2 8n 144 n2 32 4 9 18 n3
We remark as well that the case a = 1 corresponds exactly to the asymptotic
behavior of the coefficients of the classical Fubini–Study metric. Therefore the
Moyal–Fubini–Studyis,infact,aperturbationoftheclassicalFubini–Studymet-
ric.
10