Fermions Coupled to Emergent Noncommutative Gravity

0.13 MB·English

by  Daniela Klammer

Fermions Coupled to Emergent Noncommutative Gravity
Daniela KLAMMER1 and Harold STEINACKER2
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0 Fakulta¨tfu¨rPhysik,Universita¨tWien
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Boltzmanngasse5,A-1090Wien,Austria

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h Abstract
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We study the coupling of fermions to Yang-Mills matrix models in the framework of emergent
e
h noncommutativegravity. Itisshownthatthematrixmodelactionprovidesanappropriatecoupling
[ forfermionstogravity,albeitwithanon-standardspin-connection.Integratingoutthefermionsina

nontrivialgeometricalbackgroundinducesindeedtheEinstein-Hilbertactionforon-shellgeometries
1
v plusa dilaton-liketerm. ThisresultexplainsUV/IRmixingasa gravityeffect. Italso illuminates
2 whyUV/IRmixingremainseveninsupersymmetricmodels,exceptintheN=4case.
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[email protected],TalkgivenattheWorkshoponBlackHolesinGeneralRelativityandStringTheory,24-30August,2008,
VeliLosˆinj,Croatia.
[email protected]
Contents
1 Introduction 1
2 Matrixmodelsandeffectivegeometry 2
3 Quantizationandinducedgravity 4
4 RelationwithgaugetheoryonR4 7
q
5 ComparisonwithUV/IRmixing 9
6 Discussionandoutlook 11
1 Introduction
Noncommmutativity& gravity. Heisenberg’suncertaintyprincipletogetherwith Einsteins’generaltheoryof
relativity lead to the conclusion that the classical concept of spacetime loses its meaning in the small. When
measuringa spacetime coordinatewith greataccuracya, there is an uncertaintyin momentumof the order1/a.
Thatistosaymeasuringsmalldistancesrequireshighenergies,whichwillcurvelocallytheregionofspacetime
youwanttomeasure. Whenthegravitationalfieldbecomessostrongastopreventanysignalfromescapingthat
regiontheoperationalmeaningofthislocalizationgetslost. Theprocessofmeasuringaspacetimecoordinateto
infiniteaccuracyisthusasamatterofprinciplenotpossible.
IthasbeenshowninafundamentalpaperbyDoplicher,FredenhagenandRoberts[1]thattheaboveargument
leadstouncertaintyrelationsforthespacetimecoordinateswhichcanbederivedfromnoncommutingspacetime
coordinates,suchas
[xm ,xn ]=iq mn . (1)
One can then study so called called “noncommutative (NC) quantum field theories” on spaces with such non-
commutingcoordinates;forbasicreviewsseee.g.[2,3]. InNCfieldtheoriesquantumfluctuationsofspacetime
coordinatesoccurnaturally.Thusitisbelievedthatthesetheoriescouldplayanimportantroˆleonthewaytowards
aquantumtheoryofgravity. Recently,aspecificrealizationofthisideawaspublishedunderthenameof“emer-
gentnoncommutativegravity,”see [4]and [5–8].There,matrixmodelsofnoncommutativegaugetheorydescribe
dynamicalnoncommutativespaces. Themainlessonlearnedisthatgravityisalreadycontainedinnoncommuta-
1
tivegaugetheories. Thereisnoneedtoaddnewconcepts. Herewediscussspecificresultsofthisapproach: We
studythesuccessfulcouplingoffermionstotheframeworkofemergentnoncommutativegravity.
2 Matrix models and effective geometry
Considerthematrixmodelaction
SYM=−Tr[Ya,Yb][Ya′,Yb′]gaa′gbb′ (2)
for
gaa′ =d aa′ or gaa′ =h aa′ (3)
in the Euclidean resp. Minkowski case. g should not be interpreted as a fixed, physical background metric,
aa′
butratherasaprescriptiontofixthesignature. Herethe“covariantcoordinates”Ya fora=1,2,3,arehermitian
matricesoroperatorsactingonsomeHilbertspaceH. Wewilldenotethecommutatoroftwomatricesas
[Ya,Yb]=iq ab (4)
sothatq ab∈L(H )isanantihermitianmatrix,whichisnotassumedtobeconstanthere.Westudyconfigurations
Ya (not necessarily solutions of the equation of motion) which can be interpreted as quantizationsof a Poisson
manifold (M,q ab(y)) with general Poisson structure q ab(y). This defines the geometrical background under
consideration,andessentiallyany(local)PoissonmanifoldisapossiblebackgroundYa. Inparticular,weassume
that q ab is small and well approximatedby the classical Poisson structure q ab(y) in a semi-classical expansion.
Moreformally,thismeansthatthereisanisomorphismofvectorspaces
C(M)→A ⊂L(H ) (5)
whereC(M)denotesthespaceoffunctionsonM,andA isthealgebrageneratedbyYa,interpretedasquantized
algebraoffunctions. Inparticular,Ya correspondstoaclassicalcoordinatefunction1ya onM. Thiscanbeused
todefineastarproductonC(M). Moreover,Ya definesaderivationonA via
[Ya,f]∼iq ab(y)¶ f(y). (6)
b
Inthispaper,werestrictourselvestothe“irreducible”case,i.e. weassumethatA isinsomesensedensein
L(H ). Then any matrix (“function”)in L(H ) can be expressed as a functionofYa resp. ya. From the gauge
theorypointofviewinSection4,itmeansthatwerestrictourselvestotheU(1)case. Thisismostinterestingfor
ussincetheUV/IRmixing(seeSect. 5)happensinthetrace-U(1)sector. Forthegeneralcasesee[4,8].
Scalars. To begin with, we consider the case of scalar fields i.e. hermitian matrices F coupled to the matrix
model(2). Thereitisseenmosteasilyhowtheeffectivemetricappears.Theonlypossibilitytowritedownkinetic
termsformatterfieldsisthroughcommutators[Ya,F ]∼iq ab(y) ¶ F 2,andoneisleadtotheaction
¶ yb
¶ ¶
S[F ]=(2p )2Trgaa′[Ya,F ][Ya′,F ]∼ d4yr (y)Gab(y)¶ yaF (y)¶ ybF (y). (7)
Z
1Thecoordinatesyaarepreferredonessinceintheirframegaa′equalsdaa′resp.h aa′.Inotherframesgaa′willnotbeconstant.
2Throughoutthispaper,∼indicatestheleadingcontributioninasemi-classicalexpansioninpowersofq ab.
2
Here
Gab(y)=q ac(y)q bd(y)g (8)
cd
is the effective metric for the scalar field F . The Poisson manifold naturally acquires a metric structure
(M,q ab(y),Gab(y)) determined by the Poisson structure. The metric is thus no fundamentalbuilding block of
thetheory.WealsousedTr∼ d4yr (y),where
r (yR)=(detq ab(y))−1/2=|G (y)|1/4≡e−s (≡L 4 (y)) (9)
ab NC
isthesymplecticmeasureon(M,q ab(y))whichcanbenaturallyinterpretedasnon-commutativescaleL 4 .After
NC
appropriaterescalingofGab(y),thiscanberewrittenincovariantform
S[F ]= d4yG˜ab(y)¶ yaF ¶ ybF (10)
Z
withtheeffectivemetric
G˜ab=|G |1/4Gab=r (y)Gab, |G˜ab|=1 (11)
ab
beingunimodularinthepreferredya coordinates.
Fermions. Themostobviousactionforaspinorwhichcanbewrittendowninthematrixmodelframework3is
S=(2p )2TrY g [Ya,Y ] ∼ d4yr (y)Y ig q ab(y)¶ Y (12)
a a b
Z
ignoringpossiblenonabeliangaugefieldsheretosimplifythenotation. ThisiswrittenforthecaseofMinkowski
signature,theEuclideanversioninvolvestheobviousreplacementY¯ →Y †. Thisdefinesthe(matrix)Diracoper-
ator
6DY =g [Ya,Y ] ∼ ig q ab(y)¶ Y . (13)
a a b
Wecancomparethiswiththestandardcovariantderivativeforspinors
6DcommY =ig aeam ¶ m +S abw mab Y (14)
(cid:16) (cid:17)
where
1
w mab=i ean (cid:209) m ebn (15)
2
isthespinconnection,andS =i/4[g ,g ]istherepresenta(cid:16)tionof(cid:17)theLorentzalgebra.Comparing(13)with(14),
ab a b
weobserveagainthatinthegeometrydefinedby(8),
em (y):=q m c(y)g (16)
b cb
playstheroˆleofapreferredvielbein. Howeverthismustbeusedwithgreatcare,becausethedistinctionbetween
thecoordinateindexm andtheLorentzindexaislostinthespecial“gauge”inherentin(16).
3Inparticular,fermionsshouldalsobeintheadjoint,otherwisetheycannotacquireakineticterm.Thisdoesnotruleoutitsapplicabilityin
particlephysics,seee.g.[13].
3
One notices that the spin connection does not appear in (12), which seems very strange at first. In spite of
this strange feature, the action (12) is a good action for a fermion propagatingin the geometrydefined by G˜ .
ab
To see this, recallthatthe spin connectiondetermineshow the spinorsare rotatedunderparalleltransportalong
a trajectory. However, the spin-connectionw mab can always be eliminated (via parallel-transportresp. a suitable
gauge choice) along an open trajectory. Then the conventionalkinetic term (14) boils down to (12). Therefore
in the point-particle limit, the trajectory of a fermion with action (12) will follow properlythe geodesics of the
metric4 G˜ , albeit with a differentrotation of the spin. Furthermore, the induced gravitational action obtained
ab
byintegratingoutthefermionin(12)indeedinducestheexpectedEinstein-Hilbertterm d4yR[G˜]L 2 atleastfor
“on-shell geometries”, albeit with an unusual numericalcoefficientand an extra term dependingon s . All this
R
showsthat(12)definesareasonableactionforfermionsinthebackgrounddefinedbyG˜ .
ab
Equations of motion. So far we considered arbitrary background configurationsYa as long as they admit a
geometricinterpretation.Theequationsofmotionderivedfromtheaction(2)
[Ya,[Ya′,Yb]]gaa′ =0 semi−→cl.limit q ma¶ mq nbgab=0 (17)
selecton-shellgeometriesamongallpossiblebackgrounds,suchastheMoyal-Weylquantumplane(43). However
sinceweareinterestedinthequantizationhere,wewillneedgeneraloff-shellconfigurationsbelow.
3 Quantization and induced gravity
Nextwestudythequantizationofourmatrixmodelcoupledtofermions. Inprinciple,thequantizationisdefined
in terms of a (“path”) integral over all matricesYa and Y . In 4 dimensions, we can only perform perturbative
computationsforthe“gaugesector”Ya,whilethefermionscanbeintegratedoutformallyintermsofadeterminant.
Letusfocushereontheeffectiveactionatoneloop5obtainedbyintegratingoutthefermionicfields,
1
e−G Y = dY dY¯e−S[Y ] with G Y =− Trlog6D2. (18)
2
Z
foranon-interactingfermionicfieldwithactionEq. 12.
SquareoftheDiracoperatorandinducedaction. ThesquareoftheDiracoperatortakesthefollowingform
6D2Y =g g [Ya,[Yb,Y ]]∼−g g q ac¶ (q bd¶ Y )=−Gcd¶ ¶ Y −ad¶ Y , (19)
a b a b c d c d d
with
ad = g g q ma¶ q db=−2iS q ac¶ q bd+g q ac¶ q bd. (20)
a b m ab c ab c
6D2 definesthequadraticform
S =(2p )2TrY †6D2Y ∼ d4yr (y)Y †6D2Y = d4y|G |1/4Y †6D2Y , (21)
square ab
Z Z
4Formasslessparticles,thegeodesicsofG˜abcoincidewiththoseofGab.Massesshouldbegeneratedspontaneously,whichisnotconsidered
here.
5ForthesakeofrigorweworkinEuclideancasenow.
4
whichisverysimilartothescalaraction.IntermsoftheunimodularmetricG˜ Eq. (11),S canbewrittenin
ab square
standardcovariantform
S = d4y |G˜|Y 6D2Y with 6D2Y =− G˜cd¶ ¶ Y +e−s ad¶ Y . (22)
square c d d
Z q (cid:16) (cid:17)
Wenowcomputetheeffectiveactionusifng f
21Tr(log6D2−log6D20) = −12Tr 0¥ da a1 e−a D6 2−e−a D6 20 e−2a1L 2, (23)
Z (cid:18) f f(cid:19) e
f f
where L 2 denotesthe cutoff for 6D2 regularizingthe divergencefor small a . Now we can apply the heat kernel
expansion
e fTre−a D6 2 = (cid:229) a n−24 d4ya y,6D2 (24)
n
2 M
f n≥0(cid:16) (cid:17) Z (cid:16) (cid:17)
f
wheretheSeeley-deWittcoefficientsa (y,6D2)aregivenby[14]
n
1
f
a (y) = tr1l,
0 16p 2
1 R[G]
a (y) = tr 1l+E ,
2 16p 2 6
!
e
E = −Gmn ¶ W +W W −G k W , (25)
m n m n mn k
1 (cid:16) (cid:17)
W m = Gemn e−s an+G n , e (26)
2
(cid:16) (cid:17)
wheretrdenotesthetraceoverthespinorialmaterices. Theeffeectiveactionistherefore
1 R[G˜]
G Y = 16p 2 d4y 2tr(1l)L 4+tr 6 1l+E L 2+O(logL˜) , (27)
Z (cid:18) (cid:18) (cid:19) (cid:19)
wheretr(1l)=4assumingDiracfermions. EverythinegisexpressedintermseoftheunimodularmetricG˜ ,which
ab
canbewrittenintermsofG using
ab
3
R[G] = r (y) R[G]+3D s − Gab¶ s¶ s ,
G a b
2
(cid:18) (cid:19)
D Ges = Gab¶ a¶ bs −G c¶ cs ,
G a = GbcG a ,
bc
e−s (y) = r (y)=(detG )1/4,
ab
G a = GcdG a = e−s G a−e−s (¶ s )Gba. (28)
cd b
NotetherelativeminussignoftheevariousteermesintheeffectiveactionG Y comparedwiththeinducedactiondue
toascalarfield[5],
1 1
G F = 16p 2 d4y −2L 4−6R[G]L 2+O(logL ) . (29)
Z (cid:18) (cid:19)
e e e e
5
hence
1
G Y +4G F = 16p 2 d4ytrEL 2. (30)
Z
Thisexpressesthecancellationoftheinducedactionsduetofermionsandbosons,apartfromtheE term. Forthe
e
standardcouplingofDiracfermionstogravityoncommutativespaces,onehas[15]
trE =−R (31)
comm
whichoriginatesfromanadditionalconstantterm−1Rin6D2 (Lichnerowicz’sformula). Inourcase,E turns
4 comm
outtobesomewhatmodifiedduetothemissingspinconnection,neverthelessitcontainstheappropriatecurvature
scalarplusanadditionalterm,seeEq. (41).
Ricciscalarintermsofq mn. Thecurvatureisgivenasusualby
R d =¶ G d −¶ G d +G e G d −G e G d . (32)
abc b ac a bc ac eb bc ea
TheRicciscalaristhen
R=GacRb =Gac ¶ G b −¶ G b +G e G b −G e G b . (33)
abc b ac a bc ac eb bc ea
(cid:16) (cid:17)
Bypluggingintheexplicitformulaformetrictensor,
Gmn(y)=q ma(y)q nb(y)g (34)
ab
onecanexpresstheRicciscalarRintermsofq . BymakinguseoftheJacobiidentity,
¶ q −1+¶ q −1+¶ q −1=0 (35)
a bc c ab b ca
¶ q pq=− ¶ q −1 (q mpq cq−q mqq cp), (36)
a c am
andbyexploitingrelationscomingfromparti(cid:0)alinteg(cid:1)rationseveraltermsappearinginthecomputationoftrE and
R[G] are equivalent6. After a rather lengthy computation, which can be found in [6], one yields the following
compactformfortheRicciscalarintermsoftheunimodularmetricG
ab
e
1 1
d4yR[G]L 2 = e−s 2Gmk ¶ kq n−a1 Gnl ¶ lq m−b1 gabe−2GmnGpq ¶ pq m−a1 ¶ qq n−b1 gab
Z
1n (cid:0) (cid:1) (cid:0) 1 (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
e e − (¶ q pa)Gqk ¶ q −1 + Gmn(¶ s )(¶ s ) L 2. (37)
2 p k qa 2 m n
(cid:0) (cid:1) o
e
EvaluationoftrE. Wealsoneedtoevaluate
trE = −trGab ¶ W +W W −G r W , (38)
a b a b ab r
(cid:16) (cid:17)
where
e e
1
W = G an+G˜n
m mn
2
= 1 eGm(cid:0)negagbq pa(cid:1) ¶ pq nb −Gmn(¶ pGpn)+¶ ms (39)
2
(cid:16) (cid:16) (cid:17) (cid:17)
6Bymeansoftheserelationsonecanalsocheckthattheaction(21)isindeedhermitian.
6
and a˜n =e−s an. For the computation of trE we use again the Jacobi identity (35) and relations from partial
integrationandwefind:
trE = e−s GklGmn ¶ q −1 ¶ q −1 gab−Gmk ¶ q −1 Gnl ¶ q −1 gab . (40)
k ma l nb k na l mb
n (cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) o
Comparingwith(37)forL 2 regardedasconstantcutoffofD ,wecanwritethisas
G
terE = −2R[G˜]−(¶ pq pa)Gqk e¶ kq q−a1 +Gmn(¶ ms )(¶ ns )
e=om −2R[G˜]+Gmn¶ ms¶ ns (cid:0), (cid:1) (41)
assumingon-shellgeometries(17)inthelastline. ThisformulaappliesforDiracfermions,andwithanadditional
factor 1 forWeylfermions. ItisremarkablethattrE isessentiallygivenbytheappropriatecurvaturescalarR[G˜],
2
anduptoacontributionfromthedilaton-likescalingfactorr =e−s . Thisisaveryreasonablemodificationofthe
standardresult(31),asdesiredandtellsusthatEinstein-Hilbertactionalsoemergesforfermionsatone-loop.
4 Relation with gauge theory on R4
q
OnemotivationtostudyNCfieldtheoriescomesfromthefactthatNCspacetimecoordinatesinthesmalltendto
cure the UV divergencies. However, the supposedlyremoveddivergenciesreappearin the infraredlimit p→0.
This effect is the notorious UV/IR mixing which spoils renormalizability [17]. In the framework of emergent
noncommutativegravitytheUV/IRmixingproblemofnoncommutativegaugetheoriesisunderstoodintermsof
aninducedgravityaction.Inordertoshowthiswewanttointerpretthefermionicaction(12)asactionforaDirac
fermionontheMoyal-WeylquantumplaneR4 coupledtoaU(1)gaugefieldintheadjoint. Thispointofviewis
q
obtainedbywritingthegeneralcovariantcoordinateresp. matrixYaas
Ya=Xa+Aa. (42)
HereXaaregeneratorsoftheMoyal-Weylquantumplane,whichsatisfy
[Xa,Xb]=iq¯ab, (43)
whereq¯ab isaconstantantisymmetrictensor. Theseareparticularsolutionsoftheequationsofmotion(17). The
effectivegeometryfortheMoyal-Weylplaneisflat,givenby
g¯ab = q¯acq¯bdg
cd
g˜ab = r¯g¯ab, detg˜ab=1
r¯ = (detq¯ab)−1/2=|g¯ |1/4≡L 4 . (44)
ab NC
Considernowthechangeofvariables
Aa(x)=−q¯abA (x) (45)
b
whereA arehermitianmatricesinterpretedassmoothfunctionsonR4. Thuswecanwrite
a q¯
¶
[Ya,f]=[Xa+Aa,f]=iq¯ab f+i[A ,f] ≡iq¯abD f, (46)
¶ xb b b
(cid:18) (cid:19)
7
givingforthequadraticform(21)
S = (2p )2TrY †g g Ya, Yb,Y
square a b
h h ii
= − d4xr¯Y †g g q¯amq¯bnD D Y
a b m n
Z
= d4xY †6D2 Y . (47)
A
Z
ThisisanexactexpressiononR4,where f
q
6D2 =−r¯g g q¯amq¯bnD D =−g˜mg˜nD D , (48)
A a b m n m n
and
f
g˜a=(detg¯ab)81gbq¯ba, {g˜a,g˜b}=2g˜ab. (49)
WenowwanttorewritethegeometricalresultsofSection3intermsofgaugetheoryonR4 inx-coordinates.Todo
q
this,notethatmostformulasofSection3arenotgenerallycovariant,butonlyvalidinthepreferredy-coordinates
definedbythematrixmodelswhereg =d resp. g =h . Eq.(42)definestheleading-orderrelationbetween
ab ab ab ab
yandxcoordinates,
ya=xa−q¯abA¯ +O(q 2). (50)
b
See [6]fordetails ofthis changeof variables. Letus moreoverdenote¶¯ =¶ /¶ xa. ThePoisson tensorcan be
a
writtenintermsoftheu(1)fieldstrengthas
iq ab(y)= Ya,Yb =iq¯ab−iq¯acq¯bdF¯ , (51)
cd
h i
whereF¯ isaranktwotensorinxcoordinatesonR4. Wealsoneedtheeffectivemetric(8)inx-coordinates,
cd q
Gab= q¯ac−q¯aiq¯cjF¯ q¯bd−q¯beq¯dfF¯ g . (52)
ij ef cd
Wefindfortheone-loopinducedacti(cid:0)on (cid:1)(cid:16) (cid:17)
G Y = d4y a0L 4+a2L 2+O logL
Z
(cid:16) 1 r (y)(cid:16) (cid:17)(cid:17)
= −4G F −1e6p 2 de4y 2 g¯acg¯ebdF¯ab¶¯2F¯cdL 2. (53)
Z
Finally, thereis a nontrivialrelationbetweenthecutoffL ofthe geometricalactieonandthe cutoffL ofthe u(1)
gaugetheory,whichfollowsfromtheidentity
e
r (y)
S =TrY †g g Ya, Yb,Y = d4yY †6D2 Y = d4y Y †6D2 Y . (54)
square a b G r¯ A
Z Z
h h ii
FortheLapaciansthismeans fe f
r (y)
6D2 = 6D2 . (55)
G r¯ A
SinceweimplementthecutoffsusingSchwingerfpaerameterizfationtheyarerelatedasfollows
r (y)
L 2= L 2. (56)
r¯
e
8
Thismakessense providedr (y)/r¯ variesonlyonlargescales respectivelysmall momenta p≪L , whichis our
workingassumption. Weobtainasafinalresultforthegeometricone-loopeffectiveactionexpressedintermsof
gaugetheoryonR4
q
L 2
G Y = −4G F − d4xr¯ g¯acg¯bdF¯ab¶¯2F¯cd
2
Z
d4p p2 L 2
= −4G F + (2p )4gacgbdF¯ab(p)F¯cd(−p)L 4 2 (57)
Z NC
wherep2=p p gij.Thisagreespreciselywiththeone-loeopecomputationinthegaugetheorypointofviewobtained
i j
below.NotethatthelasttermcorrespondstotrE in(30).
5 Comparison with UV/IR mixing
Inthissection,wecomparethegeometricalformoftheone-loopeffectiveactionobtainedintheprevioussection
with the one-loop effective action obtained from the gauge theory point of view. The strategy is to apply first
theconceptofcovariantcoordinatestoobtainanoncommutativegaugetheorycoupledtofermionsandcompute
thereafter the one-loop effective action. The result is of course the same, which provides not only a nontrivial
check for our geometrical interpretation, but also sheds new light on the conditions to which extent the semi-
classical analysis of the previoussection is valid. This generalizes the results of [5] to the fermionic case. We
findas expectedthatthe UV/IRmixingtermsobtainedbyintegratingoutthe fermionsare givenbythe induced
geometricalresp. gravitationalaction (27), in a suitable IR regime. In particular, we need an explicit, physical
momentumcutoffL .
Usingthe variablesandconventionsoftheprevioussection, theaction(12)canbeexactlyrewrittenasU(1)
gaugetheoryonR4,whichintheEuclideancasetakestheform
q
S[Y ] = (2p )2TrY †g [Ya,Y ]
a
= d4xY˜†ig˜a(¶¯ Y˜ +ig[A ,Y˜]) (58)
a a
Z
Weintroduceanexplictcouplingconstantg,anddefinearescaledfermionicfield
Y˜ =|g¯ab|116Y (59)
in order to obtain the properly normalized effective metric g˜ab; we will omit the tilde on Y henceforth. Recall
also that onlyU(1) gauge fields are considered here, because only those correspond to the nontrivial geometry
consideredintheprevioussection.
WeneedtheO(A2)contributiontotheone-loopeffectiveactionobtainedbyintegratingoutthefermionicfield
Y . While this computationhas been discussed severaltimes in the literature [16–20], the knownresults are not
accurateenoughforourpurpose,i.e. intheregime p2,L 2<L 2 wherethesemiclassicalgeometryisexpectedto
NC
makesense. WeneedtoanalyzecarefullytheIRregimeofthewell-knowneffectivecutoffL (p)(64)fornon-
eff
planargraphsas p→0,keepingL fixed. Inthisregimethenon-planardiagramsalmostcoincidewiththeplanar
diagrams, and the leading IR corrections due to the nonplanar diagrams correspond to the induced geometrical
9