Summary Talk: Field Theory
J. Barelos-Neto
Instituto deFsia
UniversidadeFederaldoRiodeJaneiro,
Riode Janeiro, RJ,21945-970, Brazil
Reeivedon9Marh,2001
WepresentasummaryoftheontributionsonFieldTheoryattheXXIIBrazilianNationalMeeting
onPartilesandFields.
In the meeting of this year, there were a total of
165 ontributions in the form of panel and oral
se-tions, besidestwoplenaryandthree paralleltalks. As
ourred in thepreviousmeetings, thesetalks referto
reentand/orsigniativesubjetsintheFieldTheory.
Atthisyear,onetalkwasontheAnti-deSitter(AdS){
ConformalFieldTheory(CFT)orrespondene,twoon
nononommutativetheories,oneonappliationofeld
theory in ondensed matter and oneon derivative
ex-pansionsand nitetemperature. I ndalsoimportant
to mention that there were 14 ontributions from the
CosmologyandGravitationsetorwhihalsoonerns
toeld theory. Thequantumandsemilassialaspets
ofgravitationaswellasquantum eldsin lassialand
urvedbakgroundaresubjetswhoseinteresthasbeen
inreasingyearafteryear.
It might be tedious and uninteresting to make an
analysis of eah ontribution (or group of them)
iso-lated from the general view of where this work is in
theontextoftheFieldTheory asawhole. Themain
reasonis that oneof thepurposesof thistalk isto let
peopleofotherareasknowwhathasbeendoneinField
Theory.
Inmyopinion,anotherimportantpointwouldbeto
situate theworkswithin thedevelopmentofthe
orre-spondingresearhgroupsin Brazilin ordertohavean
ideaofwhathasbeendoneineahgroupduringthelast
years. Conerningthislastpart,werefertotheareful
analysis of Prof. MareloGomes in thesummary talk
of the last meeting. I am goingto onentrateon the
rstpart.
We may say that the suess of the Field Theory
startsfromthequantizationoftheeletromagnetiand
fermion elds, desribing the eletromagneti
intera-tion(QED), wherethereis afantastiagreementwith
experiments. QED isagaugetheorywhose symmetry
group is the U(1). In the year 1954, Yang and Mills
proposedanextensionforthisgaugetheoryinorderto
inlude non-Abelianelds. This wasahievedby
on-sideringmoregeneralgroupsSU(N),wherethenumber
ofgaugeeldsisN 2
1. Eventhoughonsideredavery
nietheoretialidea,therst suessfulappliation of
theYang-Mills eldsjust ourred almostfteenyears
later(1968), in the onsistentdesription of theweak
interationinauniedtheoryalsoinvolvingQED.The
orrespondingsymmetrygroupistheSU(2)
L U(1)
Y
(where\L"means\leftdoublets"and\Y"referstothe
hyperharge)anditsdevelopmentwasduetoGlashow,
SalamandWeinberg. It isopportuneto mentionthat
theexperimentalobservationofweakgaugeeldstook
fteen years more, after the onstrution of powerful
aelerators.
Lateron,thestronginterationwasalsoformulated
asa Yang-Mills theory, whose symmetry group is the
SU(3) and the basi ingredients are not protons and
neutrons, as in the old Nulear Physis, but quarks
and gluons. Nowadays, the gaugetheory whose
sym-metrygroupis SU(3)SU(2)
L
U(1)
Y
, alsoknown
asstandardmodel,orretly explainsalltheeventswe
know involving weak, eletromagneti and strong
in-terations. The standardmodel and problems related
with onnement, vauum QCD et. are always an
interesting area of researh in quantum eld theory
[1, 2, 3, 4℄ 1
. We also mention that problems related
to theCasimir Eet haveinreased of interest in
re-entyears[5,6,7,8,9,10,11,12℄.
The path integral formalism in Field Theory has
a very high resemblane with the partition funtion
in Statistial Physis. The intersetion of these
ini-tially distint subjets leads to a fruitful line of
re-searh beause the knowledge of eah one ould be
used into the other. The so alled Quantum Field
Theory at Finite Temperature, orThermal Field
The-ory, has always been an interesting researh subjet
1
[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24℄. In the
meeting of this year, it was devotedaparallel talk on
this subjet, involving the Derivative expansion
teh-niqueandChern-Simonstheories [25℄.
Ofourse,quantum eldtheorieswereinitially
for-mulated for the spaetime dimension where we live,
D = 4. However, a great deal of interest emerged
for quantum elds at lower dimensions, D = 3 and
D = 2 (even at D = 1). These researh lines were
initially onsideredasjust atheoretial laboratoryfor
eldtheoriesatD=4. There aremanyinteresting
re-searhareasatthispart,involvingChern-Simons
theo-ries[26,27,28,29,30,31,32,33,34,35,36,37,38,39℄,
Anyonsand Fratons[18, 40℄, Nonlinear Sigma-model
[41,42,43,44℄,ShwingerandChiral-Shwingermodels
[45, 46℄ et. Nowadays,eld theories at spaetime
di-mensionslowerthanD=4stillhaveaplentyofinterest
bytheirownrights[47,48,49,50,51,52,53,54,55,56,
57,58,59,60, 61,62,63,64,65,66, 67, 68,69,70,71℄
andnotonlyasatheoretiallaboratoryforD=4. F
ur-ther,theyhavealsoonnetionwiththerealworld,ina
researhareainvolvingondensedmatterwithmany
in-terestingappliations[72,28,73,74,75,76,77,78,79℄.
Itwasalsodevoted aparalleltalkonthissubjet[80℄.
After the suess of the eld quantization method
in the weak, eletromagneti and strong interations,
themostnaturalstepwouldbetousethesame
quan-tizationrulesinto theEinstein theoryof gravity. This
did notwork! The main reason is that quantum eld
theoriesdealwithmanyinnitequantitieswhihare
ei-thersimpledisarded,asthevauumenergy,orare
in-telligent irumvented(the renormalizationprogram).
Boththeseproeduresannotbeappliedtothegravity
theory. Firstbeausesouresofenergyannotbe
sim-pledisardedinpreseneofgravityandseondbeause
therenormalizationprogramsimplydoesnotwork.
Arst attempt to irumventthis problem wasto
follow the same idea before the advent of QED, that
is, just quantum matter elds were quantized and
in-terating witha lassialeletromagnetibakground.
The orresponding researh area of quantum matter
and quantum gauge elds propagating in a lassial
gravitational bakground, and the problems related
with geometrization, lead to a very interesting
devel-opments and onstitute a very fruitful researh area
[81,82,83,84,85, 86,87,88,89,90,91℄.
Theurrentideaonthissubjetisthat theremust
exist some more general formulation for the
gravita-tionaltheorywherethequantizationproeduresanbe
applied. Therstonsistentattemptwasbasedonthe
supersymmetry,thatisanextensionoftheLorentz
sym-metrywherethespaetimealsohasfermionidegreesof
freedom. Theorrespondingsupersymmetritheoryof
gravity,alledsupergravity,wasthenformulatedandit
ontainstheEinsteingravitationaltheoryasa
partiu-larase. Bothsupersymmetryandsupergravityalways
[92,93, 94, 95,96,97,98,30,90,99,100, 101,102℄.
However,theproblemsrelatedwiththeinnitiesof
quantum gravity were not ompletely solved with
su-pergravity. An importantstepwasdonebasedon the
ideathateldsouldnotdependonpoints(onsidered
tobeamathematialidealization),butonextended
ob-jets. Stringsare thesimplest extended objets.
How-ever,aeld theorywhere eldsare funtions(or
fun-tionals)ofstringsisverydiÆulttobehandled. What
remainedisthestringideaitself,whereelementary
par-tiles are not points, but vibrationalstates of strings.
More general extended objets, the branes, were also
onsidered, but it is opportune to say that they are
notexatlythesameofthemodernbranes. Theseare
relatedtoboundarysurfaesdesribedbystrings.
Any-way,stringandbranes(withtheirsupersymmetri
ver-sion) are a very fruitful researh area [103, 104, 105,
106,107,87, 108, 109, 110, 111, 112, 113,114℄. There
was a plenary talk involving string theory (and
non-ommutativeelds) [115℄.
Themathematialstrutures ofstringsand branes
are muh moreinvolved and their quantization led to
agreat developmentin the quantizationmethods and
inthestudyofanomalies[116,117,118,119,120,121,
122, 123, 124, 125, 126, 127, 128, 129, 130, 131℄. The
treatment of onstrained systems (Dira, sympleti,
Senjanoviet. [132, 133, 134, 135, 136, 112, 38, 137℄
were niely improved, based in the BRST symmetry.
Theworksof Batalin, Fradkin, VilkoviskyandTyutin
are veryrelevant in thequantizationsmethods known
in literatureasBFV (Hamiltonian),BV(Lagrangian),
andBFT(Hamiltonianembedding)[41,138,139,140℄.
Mathematial struturesandtheir algebras,aswellas
problemsrelatedtoLaxpairs,KPhierarhy,integrable
modelsaquiredmuhinterest[141,142,143,144,145,
34,146,147,148℄.
Anotheraspetofextended objetsisthattheyare
onsistentlyquantized justin thespaetimedimension
D=10. Thismightmeansthatthereexistssome
om-patiation proedure whih leads to our spaetime
dimension [28, 149℄. There is another interesting
as-pet in (super)string theories. At rst,it appeared to
exist ve independenttheories for them. Butnowwe
knowthattheyareonnetedbyduality,togetherwith
supergravityat D=11(wearegoingtoexplainbelow
why an extra dimension appears). It is important to
emphasizethat supergravityisagainaveryinteresting
researh area. The onsequene of theduality among
string and supergravity theory is that they must be
dierent manifestation of asome fundamental theory,
alledM-theoryinliterature,thatshouldbeformulated
at D =11. In this line of researh, problems related
to duality [150, 151, 152, 153, 39, 154℄ and topology
[155,98,156,157,158,154, 159, 160,161,162℄havea
greatdealofinterest.
pub-ation and to havea unique theory able to desribe
everythingis again in evidene. As it wassaid, there
were problems to inlude gravitation in the family of
quantum gaugetheories. Ontheotherhand,after the
adventofstrings,theproblemhashangedinareversed
way. Stringtheoriesnaturallyontaingravity,but the
diÆultywastoinlude MaxwellandYang-Mills
theo-ries into this formalism. Maldaena onjetured that
there exists a duality between (super)string theories
orsupergravityinAdS
D+1
-spaesandonformalgauge
theories livingin the boundary at D-dimensions(that
is why the previous ritial dimension D = 10 of
su-perstrings has hanged do D = 11). It is true that
a unied theory of everything might be still far, but
the duality among all the string theories and
super-gravity at D = 11 and the work of Maldaena have
ertainly shed new light on this old dream. This line
ofresearh, inludingthereentworks ofLisaRandall
and Raman Sundrum and the possibility of
nonom-mutativeelds,isthemostreentsubjetsinquantum
eld theory [163, 164, 165, 166, 167, 168℄. There was
aparalleltalk onAdS=CFT orrespondene [169℄and
aplenarytalk onnonommutativeelds [115, 170℄. It
might be opportune to say that it was also planned
a plenarytalk on AdS=CFT with Maldaena himself,
that hadonrmed but delined few weeks before the
meeting byvirtueofpersonalproblems.
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[35℄ A.P.B. Sarpelli and J.A. Helayel-Neto, Vortex lines
and Dira magneti monopoles for the Abelian Higgs
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[36℄ J.A.Helayel-Neto,M.A.Santos,andP.I.Trajtenberg,
Probingthesympletiprojetormethodofquantization
inplanargauge models.
[37℄ P.T.B.Crispim,R.R.Landim,andC.A.S.deAlmeida,
NonminimalChern-Simons-Higgs model:
broken-sym-metryphase.
[38℄ K.C. Mendes, R.R. Landim, and C.A.S. de Almeida,
Dira quantization of a nonminimal Chern-Simons
gaugedO(3)sigma-model.
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and C. Wotzasek, On the equivalene of the
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deOliveiraNeto,EquivalenebetweenDirarst-lass
onstraintsandreduedphasespaequantizationofthe
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[42℄ M.S.Cunha,Vortiesassimetriosnomodelo O(3)
n~aominimamenteaoplado.
[43℄ M.S. Goes-Negr~ao, J.A. Helayel-Neto, and M.R.
Ne-gr~ao, Coupling of a nonlinear -model to a less
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[44℄ M.S. Goes-Negr~ao, J.A. Helayel-Neto, and M.R.
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[45℄ E.M.C.AbreuandA.S.Dutra,Remarksonthephysial
mehanismbehindthesolderingformalism.
[46℄ E.M.C. Abreu and A.S. Dutra, Bosonized QED with
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[50℄ A.P.C. Malbouisson and J.M.C. Malbouisson
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Quei-roz,Thefermion-bosonmappingappliedtoLagrangian
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[52℄ L.P.Collato,A.L.A.Penna,W.C.dosSantos,C.M.M.
Polito, (1,1) matrix world from a (2n+1,2n+1)
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Moura-Neto,OnDira-likemonopoles inthree
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[54℄ D. Bazeia, A.S.Inaio, andL. Laerio,Some new
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[55℄ S.P.Gavrilov,D.M.Gitman,andA.A.Smirnov,Green
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[56℄ A.N.VaidyaandL.E.S.SouzaAlgebraialulationof
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[57℄ A.N.VaidyaandL.E.S.SouzaAlgebraialulationof
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[58℄ A.N. VaidyaandL.E.S. SouzaGreen funtion forthe
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[59℄ A.W.Smith, J.A.Helayel-Neto,and R.S.Sim~oes,
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[87℄ V.B. Bezerra, L.P.Collato, M.E.X.Guimar~aes, R.M.
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[89℄ R. Aldrovandi and A.L. Barbosa, Yang-Mills eld as
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[90℄ D.H.T.FranoandC.M.M.Polito,Somestrutural
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SantosGalilei-ovariantBhabhaequations.
[92℄ M.A.C.KneippandP.BrokillBPSstringsolutionsin
non-AbelianYang-Mills theoriesandonnement.
[93℄ A.W.SmithandJ.A.Helayel-Neto,Auxiliaryelds in
higher-dimensional supergravities: the r^oleof p-forms
andfermionondensates.
[94℄ A. Petrov, Supereld eetive ationin
nonommuta-tive Wess-Zuminomodel.
[95℄ C.N.Ferreira,M.B.Maia-Porto,and J.A.Helay
el-Ne-to,Cosmistringongurationinthesupersymmetri
CSKR theory.
[96℄ C.N.Ferreira,C.F.L.Godinho,andJ.A.Helayel-Neto,
Gaugeeldmixinginthesupersymmetriosmistring
[97℄ H.R. Christiansen, J.A. Helayel-Neto, A.L.M.A.
No-gueira,andL.R.U.Mansur,N=2superalgebraofa
non-minimumMaxwell-Chern-Simons-Higgs model:
estab-lishingtheentralextension.
[98℄ C.P. Constantinidis, O. Piguet, and F. Gieres, Sobre
a supersimetria vetorial em teorias topologias e de
gravita~ao.
[99℄ M.B. Silka, Modelos integraveis supersimetrios de
Wess-Zumino-Witten.
[100℄ G.S.DiasandJ.A.Helayel-Neto,N=2-Susyofa
spin-1
2
partileinanextended externaleld.
[101℄ M.A. de Andrade and I.V. Vanea, D=4 Eulidean
supergravity: theDiraeigenvaluesanditsobservables.
[102℄ R.L.Rodrigues,Thesusymethodforthespetrumof
therstPoshl-Tellerpotential.
[103℄ N. Berkovits, O. Chandia, and B.C. Vallilo,
quan-tiza~aoovariantedasuperorda.
[104℄ Y.Matsuo,ProjetionoperatorandD-braneinstring
theory.
[105℄ R.Medina,The lowenergy eetiveations for
ele-tromagnetismandgravitythatomefromtheopenand
losedsuperstring theories
[106℄ A. Deriglazov, Separated and irreduible onstraints
forIIBGreen-Shwarzsuperstring.
[107℄ E.V.Gorbar,V.P.Gusynin,andV.A.Miransky,
Dy-namialhiralsymmetrybreakingonabraneinredued
QED.
[108℄ G.N. Santos Filho and W.F. Chagas-Filho,
In-vari^aniade gaugeparapartulas eordas.
[109℄ D.NedelandN.Berkovits,TypeIIBsigmamodeland
onstraints of4DN=2supergravity.
[110℄ M.C.B.Abdalla,A.L.Gadelha,andI.V.Vanea,
Es-tudodaentropiaparaaordabos^oniafehadano
for-malismodeTFD.
[111℄ A.Deriglazov,C.Neves,W.Oliveira,andJ.Ananias
Neto,Ontheanonialquantizationofopenstrings
at-tahed at D
p
-branes in the presene of a onstant
B-eld.
[112℄ D.F.Z. Marhioro, Aplia~ao do metodo de Dira a
a~aodaordabos^onia.
[113℄ G.F. Hidalgo, Extendon (p-brane) solutions in loal
n-formeldtheories.
[114℄ M.C.LeiteandW.Siegel,Constru~aodeum
formal-ismoovariantepara asuperorda aberta em4=d(big
piture).
[115℄ Y.Matsuo,Tahyon ondensation and
nonommuta-tivegeometryinstringtheory.
[116℄ F.Toppan,Anomalous eets in lassialdynamial
systems.
[117℄ V.E.R.Lemes,S.P.Sorella,O.S.Ventura,L.C.Q.
[118℄ M.V.Cougo-Pinto,C.Farina, A.C.Tort, andJ.F.M.
Mendes, Vauum polarization in salar QED under
external eletri eld and deformed periodi boundary
ondition.
[119℄ A.S.DutraandM. Hott,Mapeamenton~ao-loalpara
ateoriaBCS.
[120℄ J.L. Boldo and C.A.G. Sasaki, Symmetry aspets of
fermionsoupledtotorsionandeletromagnetields.
[121℄ A.T.Suzuki, A.G.M. Shmidt, and E.S.dos Santos,
Tri^angulodefermionsporintegra~aoemdimens~ao
neg-ativa.
[122℄ O.A.Battistel,O.L.Battistel,andG.Dallabona,
Me-sonpropertiesintheSU(2)NJLmodel.
[123℄ R.BentinandA.Suzuki,Equival^eniaentrea
ovari-antiza~ao eaintegra~aoemdimens~aonegativa.
[124℄ C.A.M.Melo,B.M.Pimentel,andP.J.Pompeia,
Sh-winger'sprinipleandtheB-eldformalismforthefree
eletromagneti eld.
[125℄ A.T.SuzukiandA.G.M.Shmidt,Epsilon-expansion
fornon-planardouble-boxes.
[126℄ J.T.T.Lunardi,B.M.P.Esobar,J.S.S.Valverde,and
L.A.M. Vieira Junior, Metodo ausal apliado a
ele-trodin^amia esalar via equa~ao de
DuÆn-Kemmer-Petiau: resultadosaum\loop".
[127℄ R.B. Rodrigues and N.F. Svaiter, Vauum
utua-tionsofasalareldinaretangularwaveguide.
[128℄ M.J.SalvayandC.M.Naon,OnaCFTpreditionin
thesine-Gordonmodel.
[129℄ C.A.Iui,C.M.Naon,andK.Li, NonloalThirring
modelwithspinippinginterations.
[130℄ V.G.C.S.dosSantosandW.G.Pereira,Teoriada
per-turba~aode Rayleigh-Shroedinger versus ambig uidade
deordenamento.
[131℄ S.V.L. Pinheiro, V.S. Alves, and F.P. Campos, A
renormalization group study and the
Nambu-Jona-Lassiniomodel.
[132℄ M.A.deAndrade,M.A.dosSantos,andI.V.Vanea,
Metododosprojetoressimpletiosemteoriasdeampos
degauge.
[133℄ A.C.R.Mendes,W. Oliveira, C.Neves, and F.I.
Ta-kakura,Metauiddynamisasagaugetheory.
[134℄ A.C.R. Mendes, J. Ananias Neto, W. Oliveira, C.
Neves,andD.C.Rodrigues,Gaugingseondlass
sys-temsviasympleti gauge-invariantformalism.
[135℄ E.M.C. Abreu, D.Dalmazi, and E.A. Silva,The
Ja-obiidentity forDira-likebrakets.
[136℄ V.G.C.S. dos Santosand A.S.Dutra, Partula
ar-regada na presena de um ampo mangnetio
ho-mog^eneosobondi~oesdeontornon~aotriviais.
[137℄ C.F.L. Godinho, C.N. Ferreira, and M.B.D.S. Maia
Porto, Sympleti quantization of the Maxwell and
Kalb-Ramondeldinthelight-one.
[138℄ A.R. Fazio, V.E.R. Lemes, M.S. Sarandy, and S.P.
Sorella,Thediagonalghost equation Wardidentityfor
[139℄ C.N. Ferreira and C.F.L. Godinho, Tripleti gauge
xing forN=1superYang-Mills.
[140℄ J. Ananias Neto, C. Neves, and W. Oliveira,
Con-vers~ao de sistemas de segunda-lasse em primeira
baseadaemsimetriasdostermosdeWess-Zumino.
[141℄ V.B.Bezerra, E.M.F. Curado, and M.A.
Rego-Mon-teiro,Anewlassofquantumeldtheory: perturbative
omputation.
[142℄ H.L.Carrion,M.Rojas,andF.Toppan,AnN=8
su-peraÆneMalevalgebraanditsN=8Sugawara.
[143℄ H.L.Carrion,M.Rojas,andF.Toppan,Division
al-gebrasandextended N=1,2,4,8KdVs.
[144℄ E.M.F.CuradoandM.A.Rego-Monteiro,Heisemberg
type algebras of one-dimensional quantum
Hamiltoni-ans.
[145℄ G.Shieber,Fun~oesdeparti~ao\twistadas"para
teo-ria de ampo onforme om ondi~oes de ontorno e
algebrasdesimetriasqu^antias.
[146℄ E.L.da Graa, H.L.C. Salazar, and R.L. Rodrigues,
Uma representa~ao da algebra de virasoro viatenia
algebria deWigner-Heisemberg emsistemabos^onio.
[147℄ M.A. deAndrade and I.V. Vanea, Spinors in
arbi-trarydimensions.
[148℄ H.QueirozandJ.L.Tomazelli,Oproblemada
unitari-dadedooperadorSnateoriadeDuÆn-Kemmer-Petiau
para as-SED.
[149℄ L.A.FerreiraandE.E.Leite,Algunsaspetos de
teo-riasdeYang-Millsautoduaisreduzidas.
[150℄ V.O. Rivelles, Dualidade o-shell na teoria de
born-infeld.
[151℄ A. Ilha and C. Wotzasek, Equival^enia dual entre
os modelos autodual e topologiamente massivo n~
ao-abelianaos.
[152℄ M.B. Canthe, Hodge-type self(antiself)-duality in
arbitrary dimensionandtensorialrank.
[153℄ D.M.Medeiros,R.R.Landim,andC.A.S.deAlmeida,
Dualidade em um modelo topologio n~ao-abeliano em
3D.
[154℄ A.IlhaandC.Wotzasek,Dualityequivalenebetween
nonlinearself-dualandtopologiallymassivemodels.
[155℄ A.A.BytsenkoandZ.G.Kuznetsova,Field-theoretial
desriptionofohomologies.
[156℄ R.R. Landim and C.A.S. de Almeida, Gera~ao de
massa topologia om termo BF n~ao-abeliano em
D-dimens~oes.
[157℄ E.L. de Graa and R.L. Rodrigues, Defeitos
topologios n~ao BPS assoiados a dois ampos reais
aoplados.
[158℄ A.B.AdibandC.A.S.deAlmeida,Kinkdynamisin
atopologialPhi 4
lattie.
[159℄ V.G. Lima, V.S. Santos, and R.L. Rodrigues,
Con-gura~oestopologiasen~aotopologiasexatamente
so-l uveis.
[160℄ J.L. Strapasson, V.G. Lima, and R.L. Rodrigues,
du-[161℄ V.G.Lima,J.L.Strapasson,andR.L.Rodrigues,
As-petostopologioseestabilidadedesolitonsem1+1
di-mens~oes.
[162℄ O.M. Del Cima, J.M. Grimstrup, and M. Shweda,
On thenitenessofanewtopologialmodelinD=3.
[163℄ G.S.Lozano,E.Moreno,andF.Shaposnik,Solitons
innonommutative spae.
[164℄ P. Mines and V.O. Rivelles, Noether urrents and
AdS/CFTorrespondene.
[165℄ A.A. Bytsenko and A.E. Gonalves, Fun~ao de
parti~aoeentropiade teoriasdeampoonforme.
[166℄ M.A. deAndrade,J.E.F. Milione, and J.L.M. Valle,
Otermo deGauss-Bonnet nomodelode
Randall-Sun-drum.
[167℄ M.A. de Andrade, M.A. dos Santos, and I.V.
Van-ea,Free variables ononstraintsurfae for
non-om-mutativeopenstrings.
[168℄ R. Amorimand J. Barelos-Neto, Embedding of the
massivenonommutative U(1)theory.
[169℄ N.R.F.Braga,Quantumeldsinanti-de-Sitter
spae-time anddegrees of freedom inthe bulk-boundary
or-respondene.
