Supersymmetric Standard Models, Flux Compactification and Moduli Stabilization

0.18 MB·English

by  Mirjam Cvetic

UPR-1088-T/rev, hep-th/0409032
Supersymmetric Standard Models, Flux Compactification and Moduli Stabilization
Mirjam Cvetiˇc and Tao Liu
∗
Department of Physics and Astronomy,
University of Pennsylvania,
Philadelphia, PA 19104-6396, USA
(Dated: February 1, 2008)
Based on the T-dual constructions of supersymmetric intersecting D6-models on Z2 ×Z2 ori-
entifolds, whose electroweak sector is parallel with the orientifold planes with Sp(2f)L×Sp(2f)R
gauge symmetry (hep-th/0407178), we deriveand classify Standard Model-like vacua with RR and
5 NSNSfluxes,whichstabilizetoroidalcomplexstructuremoduliandthedilaton. Wefindconsistent
0 four-family (f = 4) and two-family (f = 2) models with one- and two-units of the quantized flux,
0 respectively. Suchmodels typicallypossess additionalgauge group factors with negativebetafunc-
2 tions and may lead, via gaugino condensation, to stabilization of toroidal K¨ahler moduli. These
models havechiral exotics.
n
a
PACSnumbers: 11.25.-w,11.25.Mj,12.25.Wx
J

1
2 Introduction Constructions of explicit supersym- ifications is still awaiting further study.

metric Standard Models with supergravityfluxes turned
4 On the Type IIB side the intersecting D6-brane con-
v on is an outstanding technical problem in string theory. structions correspondto models with magnetized branes
2 Supersymmetric semi-realistic constructions with inter-
whichhavethe roleof intersectingangle being playedby
3 secting D6-branes provide a beautiful, geometric frame-
the magnetic fluxes on the branes. The dictionary for
0
work to engineer Standard Models with three families.
9 consistency and supersymmetry conditions between the
The first such supersymmetric models were based on
0 twoT-dualconstructionsisstraightforward,seee.g.,[14].
4 Z2×Z2 orientifolds[1,2][Non-supersymmetricconstruc- ThesupersymmetricTypeIIBfluxcompactificationsare
0 tions were givenin [3, 4, 5, 6] (see also[7] and for earlier alsobetterunderstood. Inparticular,examplesofsuper-
/ work [8, 9]).] Turning on supergravity fluxes will intro-
h symmetricfluxesandtheinternalspace,whichis confor-
t duce a supergravitypotential, helping stabilize the com- mallyCalabi-Yau,arewellknown(seee.g. [15]andrefer-
-
p pactification moduli fields by lifting continuous moduli encestherein). Theprototypeexampleisaself-dualcom-
e spaceofthestringvacuaintheeffectivefour-dimensional
bination of the NSNS H and RR F three-forms, corre-
h theory (see e.g. [10]). However, introducing the super- 3 3
spondingtotheprimitive(2,1)formonCalabi-Yauspace.
:
v gravity fluxes further restricts the constructions, since Since the back-reaction of such flux configurations is
i such fluxes modify the global conditions on the tadpole
X mild,i.e. theinternalspaceremainsconformaltoCalabi-
cancellations. The fluxes will typically generate a back
Yau,theseTypeIIBfluxcompactificationsareespecially
r
a reaction on the original geometry of the internal space, suitable for adding the probe magnetized D-branes in
thus changing the nature of the internal space.
this background. However, the quantization conditions
On the Type IIA side the supersymmetry conditions
on fluxes and the modified tadpole conditions constrain
of flux compactifications are less understood. Neverthe-
the possible D-brane configurations severely. In Refs.
less recent work [11, 12] revealedthe existence of unique
[14, 16] techniques for consistent chiral flux compactifi-
fluxvacuaformassiveTypeIIAstringtheorywithSU(3)
cations on orbifolds were developed, however no explicit
structure,whosegeometryoftheinternalsix-dimensional
supersymmetric chiral Standard Model flux compactifi-
space is nearly-K¨ahler and four-dimensional space anti-
cations were obtained. Most recently in [17] an exam-
deSitter. One such example is the SSUU((22))3 ≃ S3 ×S3 ple of a one-family Standard Model with the supersym-
coset space that has three supersymmetric three-cycles metric (three units of quantized) flux was constructed.
that add up to zero in homology [12, 13]. Therefore the Thisconstructionalsoyields three-andtwo-familymod-
total charge of the D6-branes wrapping such cycles is els with one- and two-units of the quantized flux, thus
zero and no introduction of orientifold planes on such providing the first examples of semi-realistic Standard
spaces is needed. Moreover, since the three-cycles inter- Model flux compactifications. The D-brane sector con-
sectpair-wise,themasslesschiralmatterappearsatthese struction is T-dual to models of intersecting D6-branes
intersections. This construction [12] therefore provides ontheZ ×Z orientifoldwiththeSp(2) ×Sp(2) sym-
2 2 L R
anexplicitexampleofsupersymmetricfluxcompactifica- metryintheelectroweaksector[18,19]. [Withoutfluxes,
tions with intersecting D6-branes. Further progress has the first models of that type were toroidal models with
also been made in the constructions of N=1 supersym- intersecting D6-branes [18] where the RR tadpoles were
metric Type IIA flux vacua with SU(2) structures [12], not explicitly cancelled. The Z × Z orientifold con-
2 2
leading to examples with the internal space comformally struction in [19] cancelled the RR-tadpoles by introduc-
Calabi-Yau. However, explicit constructions of models inganadditionalstackofbraneswithunitarysymmetry.
withintersectingprobeD6-branesforsuchfluxcompact- However, the original additional stack had U(1) gauge
2
symmetry,whichintroducedthe discrete globalanomaly introduce,ontheTypeIIBside(i.e. afterT-dualizingthe
[20], as pointed out in [21]. The anomaly is due to an Re(z ), Re(z ), and Re(z ) toroidal directions), 64 O3-
1 2 3
odd-numberofintersectionsofU(1)braneswithSp(2N) planesand4O7 -planes,eachofthemsittingonZ fixed
i 2
branes, thus resulting in an odd-number of chiral super- points of the ith T2 and wrapping the other two.
fields in the fundamental representation of Sp(2N); the TocompensateforthenegativeRRchargesfromthese
non-anomalous models can be easily constructed by in- O-planes, we need to introduce D(3+2n)-branes in our
troducing analogous U(2) branes instead, as in the re- models which are filling up D = 4 Minkowski space
vised version of [19].] andwrapping2n-cyclesonthe compactmanifold. These
The aim of this paper is to study the construction D(3+2n)-branescanbemagnetized(Thedetaileddiscus-
of supersymmetric Standard Models with fluxes turned sion for toroidal/orbifold compactifications with magne-
on. We shall employ the specific Z × Z orientifold tized branes is given in [14].) Concretely, for one stack
2 2
constructions that are T-dual to intersecting D6-brane ofNa D-braneswrappedmia times onthe ith 2-torusTi2,
constructions whose electroweak sector arises from D6- we turn on nia units of magnetic fluxes Fa for the U(1)a
branesoriginallyresidingontopoftheorientifoldplanes. gaugefactor(associatedwiththe D-branecenterofmass
These constructions (without fluxes) were discussed in motion) on each T2, such that
i
detail in [19]. The first class is based on one-family
1
oUb(s4e)rv×abSlep(s2efct)oLr)×, wShpi(c2hf)yRielgdasugtheesfy-mfammeiltyrySt(ainndathrde mia 2π ZT2Fai = nia. (1)
i
Model gauge symmetry (with additional U(1)’s) by em-
Hence the topological information of this stack of D-
ploying the brane-splitting mechanism. [For f = 4 and
branes is encoded in N -number of D-branes and the
f = 3 such constructions (without fluxes) yield [19] a
co-prime number pairs (ni,mi).
anomaly free models without chiralexotics.] The second a a
The chiral massless spectrum can be generated from
class is based on constructions with Sp(2) ×Sp(2) as
L R
the “intersection” of two stacks of D-branes a and b de-
the starting electroweak symmetry [17, 18, 19]. Within
noted by the intersection number
this framework we first present two classes of construc-
tions with supersymmetric fluxes (three-units of quan-
3
tizedflux)which,however,sufferfromthediscreteglobal I = (nimi −nimi) (2)
ab a b b a
anomaly [20]. iY=1
We further classify anomally-free flux compactifica-
Thepointforthespectraisthattheyshouldbeinvariant
tions based on the brane constructions with Sp(2f) ×
L
under the full orientifold symmetry group. A detailed
Sp(2f) electroweak sector: we find three- and four-
R
discussion of Type IIA has been given in [1, 2], which
family models (f = 3,4) with one-unit of quantized
can be easily interpreted into type IIB (see Table I).
flux, as well as one- and two-family models (f = 1,2)
withtwo-unitsofquantizedflux. [Notehowever,thatfor
f =3 the breaking pattern of Sp(6) ×Sp(6) down to
L R TABLEI:GeneralspectrumonmagnetizedD-branesintype
Sp(2)L×Sp(2)Rbreakssupersymmetry[19].] Forallcon- IIBT6/(Z2×Z2)orientifold. Therepresentationsinthetable
structedmodels,theturnedonfluxeswillfixthetoroidal refertoU(Na/2),theresultinggaugesymmetryduetoZ2×Z2
complexstructuremoduli[24]. Inaddition,thesemodels orbifold projection. In our convention, positive intersection
typically possess a “hidden sector” group factor with a numbersimplies left-hand chiral supermultiplets.
negativebetafunction;ifsuchagaugefactorisassociated
Sector Representation
with D7-branes, the non-perturbative infrared dynamics
ontheseD7-branemayleadtogauginocondensationand aa U(Na/2)vector multiplet
stabilization of a toroidal K¨ahler modulus. The addi- 3adjointchiralmultiplets
tional toroidal K¨ahler moduli may be fixed due to the ab+ba Iab ( a, b)fermions
supersymmetry constraints. [Note that the open string ab′+b′a Iab′ ( a, b)fermions
moduli and the K¨ahler moduli can form combined flat aa′+a′a 12(Iaa′− 12Ia,O) fermions
directions, allowing for brane recombinations (see [2]). 12(Iaa′+ 21Ia,O) fermions
However,duetothefluxback-reactionitisexpectedthat
the open string moduli become massive [14]; in this case
the supersymmetry conditions fix K¨ahler moduli.] D(3+2n)-branes can induce D-brane charges of lower
Model The generators θ and ω for the orbifold odd dimension due to world-volume couplings. Explic-
group Z2×Z2, act on the complex coordinates of T6 as itly, for one stack of Na D-branes with wrapping num-
θ : (z ,z ,z ) → (−z ,−z ,z ) and ω : (z ,z ,z ) → ber (ni,mi), it carries D3-, D5-, D7 and D9-brane RR
1 2 3 1 2 3 1 2 3 a a
(z ,−z ,−z ) and the projection ΩR, where Ω is the charges
1 2 3
world-sheet parity projection and R (acting on Type
IIA as the holomorphic Z2 involution): (z1,z2,z3) → Q3a =Nan1an2an3a, (Q5i)a =Namianjanka,
(−z ,−z ,−z ). TheactionsofΩR,ΩRω,Ωθω andΩRθ (Q7 ) =N nimjmk, Q9 =N m1m2m3, (3)
1 2 3 i a a a a a a a a a a
3
where i6=j 6=k and a permutation is implied for (Q5 ) K¨ahler modulus and θ = arctan[(Fi) 1] is the ”angle”
i a i −
and(Q7 ) . BesidestheD-braneandO-plane,fluxescan between the D-brane and the O-plane in the ith T2.
i a
also contribute RR tadpole. A consistent string model Frameworks We choose to construct the Standard-
requiresthattheRRsourcessatisfytheGausslawinthe likeModelsasdescendantsofthePati-Salammodelbased
compact space, i.e, RR tadpole must be cancelled. For onSU(4) ×SU(2) ×SU(2) . Thehyperchargeisthus:
C L R
D3- and D7-branes, thus we have
Q
B L
1 QY =QI3R + 2− , (8)
−N(0)+ Q3 − N =−16
a flux
2 where the non-anomalous U(1) is obtained from the
Xa B−L
splitting oftheU(4) branes→U(3) ×U(1) . Sim-
−N(i)+ (Q7i)a =−16,i6=j 6=k, (4) ilarly, anomaly-freeCU(1) is part oCf the noBn−-ALbelian
Xa part of U(2) or Sp(2) I3gRauge symmetry. Within this
R R
where N(0) and N(i) with i = 1,2,3 and 4 respectively framework we found two classes of constructions which
may yield Standard models with fluxes turned on [22]:
denotes the number of filler branes, i.e., D-branes which
(i) The starting symmetry is one-family U(4) ×
wrap along O3- and O7 -planes and only contribute one
i
Sp(2f) ×Sp(2f) , (f = 4) which can be broken down
of the four kinds of D3- and D7-brane charges. As for L R
tothefour-familyU(4)×U(2) ×U(2) byparallelsplit-
D5- and D9-brane RR tadpoles, their cancellation is au- L R
ting the D-branes,originallypositioned onthe O-planes,
tomatic since D-branes and their ΩR images carry the
in some two-tori directions. [Both the string theory and
same number charges with different signs.
field theory aspects of the brane splitting in this frame-
Four-dimensional N = 1 supersymmetric vacua from
work is discussed in detail in [19].] In the field theory
flux compactificationrequirethat1/4superchargesfrom
picture (“Higgsing”), the four-families (f = 4) are ob-
ten-dimensionalTypeIT-dualbepreservedinbothopen
tained as a decomposition of the original chiral super-
string and closed string sectors. On the type IIB orien-
multiplets(4,8,1)and(4,1,8)intofourcopiesof(4,2,1)
tifold,supersymmetricfluxsolutionshavebeendiscussed
and(4,1,2). TheHiggsing,asdiscussedin[19],preserves
in [15, 24]. The configuration has RR F and NSNS H
3 3
D- and F-flatness and thus symmetry breaking can take
three-form flux turned. The supersymmetry conditions
place at the string scale, generating a 4D N = 1 su-
imply thatthe three-formG =F −τH isaprimitive,
3 3 3
persymmetric four-family Pati-Salam model. Note fur-
self-dual(2,1)form. Here τ =a+i/g is type IIB axion-
s
thermore, that in this case the U(1) and U(1) are
dilatoncoupling. TheG fluxwillcontributeaD3-brane L R
3
not anomalous since they arise from the non-Abelian
RR charge
Sp symmetry. One expects that at least the gauge bo-
1 i son of U(1) will have a mass at the electroweak scale.
L
N = G ∧G , (5)
flux (4π2α)2 2τ Z 3 3 [This analysis can also be applied to two- (f = 2)
′ I X6
and three- (f = 3) family examples. Note however,
whereτ istheimaginarypartofthecomplexcouplingτ. that while for f = 2 the Higgsing Sp(4) ×Sp(4) →
I L R
Dirac quantizationconditions of F andH on T /(Z × Sp(2) ×Sp(2) issupersymmetric,forf =3theHiggs-
3 3 6 2 L R
Z ) orientifoldrequirethat N is a multiple of 64 and ing Sp(6) ×Sp(6) → Sp(2) ×Sp(2) breaks super-
2 flux L R L R
and BPS-like self-duality condition: ∗ G =iG ensures symmetry [19].]
6 3 3
that its contribution to the RR charge is positive. We (ii)The startingsymmetry is Pati-Salam-likeU(4) ×
C
shall employ the following specific solution [14, 24]: Sp(2) ×Sp(2) . Recently, in [17] the constructions of
L R
this type were explored to obtain the first examples of
G3 = √83e−πi/6(dz1dz2dz3 + dz1dz2dz3 + dz1dz2dz3),(6) semi-realistic Standard Model flux compactifications.
The gauge symmetry in the observable sector is
where the additional factor 4 is due to the Z2×Z2 orb- SU(4)×SU(2)L × SU(2)R, with the subsequent sym-
ifoldsymmetry. Thefluxstabilizesthecomplexstructure metry breaking chain [19]:
toroidal moduli at values
SU(4)×SU(2) ×SU(2)
L R
τ1 =τ2 =τ3 =τ =e2πi/3, (7) → SU(3)C ×SU(2)L×SU(2)R×U(1)B L
−
→ SU(3) ×SU(2) ×U(1) . (9)
leadingtotheRRtadpolecontributioninEq(4): N = C L Y
flux
192. Here the first step can be achieved by splitting the ath
FormagnetizedD-braneswithworld-volumemagnetic stack of D-branes at string scale and the second one by
field Fi = ni in the open string sector, the four- giving VEVs to the scalar component of right-handed
miχi
dimensional N = 1 SUSY can be preserved by the ori- neutrino superfield at TeV scale. Therefore at the elec-
entationprojectionif andonly if its rotationangles with troweak scale one would only have the minimal super-
respect to the orientifold-plane are an element of SU(3) symmetric Standard Model content.
rotation, thus implying: θ +θ +θ = 0 mod 2π. Here Within the above frameworks classes four- and three-
1 2 3
χi = RiRi, the area of the ith T2 in α units, is the familysupersymmetricStandardModelsarepresentedin
1 2 ′
4
brane models are not supersymmetric. [Within frame-
TABLE II: D-brane configurations and intersection numbers
work (ii), the solution found in [17] is the only possible
forthefour-familysupersymmetricStandard-likeModelswith
one.]
three-unitsof quantized flux. Here, r=3,4, χi is theK¨ahler
modulusfortheith two-torus,andβjg isthebetafunctionfor (3)Two-families(f =2): (nf)max =2. InTableIVwe
theSpgroupfromthejthstackofbranes. Thesemodelshave present a model with the U(2) symmetry in the hidden
discrete global anomaly.
sector. We also found models with U(1)×U(1) in the
hidden sector. [Witnin framework (ii) the only example
[U(4)C×Sp(8)L×Sp(8)R]o×[U(1)×Sp(32r−88)×Sp(8−2r)]h is given in [17].]
j N (n1,m1)(n2,m2)(n3,m3) n n b c d d′ 1 2 (4) One-family (f = 1): (n ) = 3. In this case
f max
frameworks(i)and(ii)areequivalent. Theonlyexample
a 8 (1,0)(1,1)(1,−1) 0 0 1 -1 15 -15 0 0
b 8 (0,1)(1,0)(0,−1) 0 0 - 0 4r -4r 0 0 with (nf)max = 3 was obtained in [17]. For the case
c 8 (0,1)(0,−1)(1,0) 0 0 - - 4r -4r 0 0 with one tilted two-torus, there are models with n = 2
f
d 2 (−r,−1)(4,1)(4,1) 72r 56r - - - - 1 -16 and U(2) or U(1)×U(1) gauge symmetry in the hidden
1 32r-88 (1,0)(1,0)(1,0) χ1=8rχ2/(χ22−16),χ2=χ3 sector.
2 8-2r (1,0)(0,−1)(0,1) β1g =−11/2
Moduli Stabilization These models typically con-
taina“hiddensector”,associatedwithD-branes,parallel
with the O-planes (“filler” branes), whose gauge group
TableIIandTableIII,i.e.,thosemodelshavethree-units
factors have negative beta functions. If such a gauge
of quantized flux. However, the branes with the U(1)
gauge factor were associated with a stack of D7 -branes,
i
symmetry have an odd number of intersections with the
theinfrarednon-perturbativegaugedynamicscouldgen-
Sp branes, thus introducing an odd number of funda-
erate a superpotential of the form (see, e.g., [26]):
mentalrepresentationsofSp, i.e. the models suffer from
the discrete global anomaly [20]. The conditions for the
absence of discrete globalanomaliescan be derivedfrom β Λ3 8π2
i
W = exp f (T ) +W , (11)
the K-theory [25] and have been given for the Z2 ×Z2 eff 32eπ2 (cid:18) βi W i (cid:19) o
orientifolds in [21]:
Nam1am2am3a =4Z, Naniamjamka =4Z, where the gauge function fW(Ti)=n1im2jm3kTi (i6=j 6=
X N minjnk =4Z, P N n1n2n3 =4Z, (10) k 6= i), Ti is the toroidal K¨ahler modulus for the i-th
a a a a a a a a two-torus, β is the beta function, Λ is the cutoff string
i
X P
scaleandW is a contributionfromthe fluxes [28]which
where Z is an integer. These conditions turn out to be o
fixes the dilaton and complex structure moduli. [Due to
extremely constraining and within the above framework
the flux back-reaction the open string sector moduli are
no anomaly-free four- or three-family models with three
expectedtogetamass[14];inthiscasethesupersymme-
units of quantized flux exist. Thus, explicit construc-
tryconditionsfixtherestoftoroidalK¨ahlermoduli(say,
tionsofsemi-realisticmodelswithsupersymmetricfluxes
ReT ∼χ intermsofT ).] ModelsdesplayedinTable
remain elusive. j,k j,k i
IV possess only the hidden sector D3-branes with con-
Classification We now turn to the classifica-
fining gauge symmetry. However, a generalization [27]
tion of the supersymmetric D-brane constructions with
of the present constructions, does provide Standard-like
Sp(2f) × Sp(2f) electroweak sector (framework (i))
L R flux models with such confining D7 -branes. Note, that
withthemaximalnumberoffluxunitsturnedon[Within i
for supersymmetric flux (n = 3) W = 0, and for the
framework(ii)arelatedanalysiswasdonein[21].] Inthe f o
non-supersymmetricones(n =1,2)W 6=0. Inthe lat-
following n = N /64, and (n ) is the largest al- f o
f flux f max tercasetheresultingW ,alongwiththeK¨ahlerpoten-
lowed unit of the quantized flux: eff
tial for the volume modulus, determine the supergravity
(1) Four-families (f = 4): (n ) = 1. We dis-
f max potentialwhosegroundstatewouldfixalsotheremaining
play a representativeexample in Table IV with the U(2)
toroidal K¨ahler modulus [28]. However, for the models
symmetry in the hidden sector. We also found another
under consideration W = O(M3) and thus the volume
model with hidden sector symmetry U(1) × U(1) and o P
modulus f = O(1). A mechanism to turn on a small
respective brane configurations (−1,−1)(3,1)(2,1) and W
value for W remains an open problem.
(−3,−1)(1,1)(2,1). [Within framework (ii) there are no o
models with n 6=0.] We thank Tianjun Li and Gary Shiu for discussions,
f
(2) Three-families (f =3): (n ) =1. The solution and are especially grateful to Paul Langacker for many
f max
possessesthehiddensectorgaugesymmetrywithunitary discussionsandAngelUrangaforusefulcommunications.
gauge sector U(1)×U(1) and the respective brane con- Research is supported in part by DOE grant DOE-EY-
figurations (−2,−1)(2,1)(3,1) and (−2,−1)(3,1)(2,1). 76-02-3071(M.C.,T.L.), NSF grantINT02-03585(M.C.)
However, since the Higgsing of Sp(6) × Sp(6) → and Fay R. and Eugene L. Langberg endowed Chair
L R
Sp(2) × Sp(2) breaks supersymmetry [19] these D- (M.C.).
L R
5
TABLE III:D-braneconfigurations and intersection numbersfor three-and four-family supersymmetricStandard-likeModels
with three units nf = 3 of the quantized flux. f is the family number, and 2≤r ≤ 4 for f = 3 and 3 ≤r ≤4 for f = 4. χi
is the K¨ahler modulus for the ith two-torus, and βg is thebeta function for the Sp group from the jth stack of branes. These
j
models havediscrete global anomaly.
[U(4)C×Sp(2)L×Sp(2)R]o×[U(1)×Sp(98r−8f2−80)×Sp(8−2r)]h
j N (n1,m1)(n2,m2)(n3,m3) n n b c d d′ 1 2
a 8 (1,0)(f,1)(f,−1) 0 0 f -f 49−f2 f2−49 0 0
b 2 (0,1)(1,0)(0,−1) 0 0 - 0 7r -7r 0 0
c 2 (0,1)(0,−1)(1,0) 0 0 - - 7r -7r 0 0
d 2 (−r,−1)(7,1)(7,1) 210r+33 182r-33 - - - - 1 -49
1 98r−8f2−80 (1,0)(1,0)(1,0) χ1=14rχ2/(χ22−49),χ2=χ3
2 8−2r (1,0)(0,−1)(0,1) β1g =−11/2
TABLE IV: D-brane configurations and intersection numbers for the consistent f-family Standard-like Models with n -units
f
of quantized flux. χi is the K¨ahler modulus for the ith two-torus, βjg is the beta function for the Sp group from the jth stack
of branes. The allowed models have f =2,4 with (nf)max =2,1, respectively.
[U(4)C×Sp(2f)L×Sp(2f)R]o×[U(2)×Sp(8(4− f2)2+16−32nf)]h
j N (n1,m1)(n2,m2)(n3,m3) n n b c d d′ 1
a 8 (1,0)(1,1)(1,−1) 0 0 1 -1 (4− f)2−1 −(4− f)2+1 0
2 2
b 8 (0,1)(1,0)(0,−1) 0 0 - 0 2(4− f) −2(4− f) 0
2 2
c 8 (0,1)(0,−1)(1,0) 0 0 - - 2(4− f) −2(4− f) 0
2 2
d 4 (−2,−1)(4− f2,1)(4− f2,1) χ1=(16−2f)χ3/(χ23−(4− f2)2)
1 8(4− f2)2+16−32nf (1,0)(1,0)(1,0) χ2=χ3,β1g =−5
hep-th/0308046.
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∗
Electronic address: [email protected]; Elec-
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