Exact discretness and mass gap of N=1 symplectic Yang-Mills from M-theory

0.06 MB·English

by  L. Boulton

DFTT-01/2007
Exact discreteness and mass gap of N = 1 symplectic Yang-Mills from
M-theory.
7
0 L.Boulton1,M.P.Garc´ıadelMoral2,A.Restuccia3
0
2
1DepartmentofMathematicsandtheMaxwellInstituteforMathematicalSciences,Heriot-Watt
n
University,EdinburghEH142AS,UK
a
J
2 Dipartamento de Fisica Teorica, Universita di Torino and I.N.F.N., Sezione di Torino, Via P.
0
Giuria1,I-10125,TorinoItaly
1

3 Departamento de F´ısica, Universidad Simo´n Bol´ıvar, Apartado 89000, Caracas 1080-A,
1
v Venezuela.
0
0
1
Abstract. In this note we summarize some of the results found recently in [1]. We show the
1
0 pure discreteness of the non-perturbativequantum spectrum of a symplectic Yang-Mills theory
7 definedonaRiemannsurfaceofpositivegenus,livinginatargetspacethat,inparticular,canbe
0 4D. Thistheorycorrespondsto the membranewith centralcharges. The presenceof the central
/
h chargeinducesaconfinementinthephaseatzerotemperature. Whentheenergyrises,thecenter
t of the group breaks and the theory enters in a quark-plasmaphase after a topologicaltransition
-
p correspondingtotheN=4wrappedsupermembrane.
e
h
:
v
i
X
r
a 1. Introduction
Thenon-perturbativequantizationofStringTheoryisstillanopenproblemwhichreceivesmuchoftheattention
of specialists. It can be reformulatedin terms of the quantizationof the M-theoryin 11 dimensionswhich, in
turns,reducestofindingthequantizationofthebasicingredients:M2-braneorsupermembraneandM5brane.A
furtherimportantopenproblem,notnecessaryconnectedwithStringTheory,isthenon-perturbativequantization
ofYang-Millstheories. AttemptsalongthisdirectionincludelatticeQCD, twistors, gauge-gravityduality,spin
chains,largeNmatrixmodelsandcanonicalquantization.
The aim of this note is to draw to the attention of specialists, the fact that the membrane with central
charges,whichisthequantumequivalentofasymplecticnoncommutativeYang-Millstheory[2],[3],hasapurely
discretespectrumattheexactlevelofthetheory.Thisisanextensionofpreviousresultsfoundfortheregularized
supermembranewithcentralcharges,([4]-[9]).
ThecorrespondencewithanN=1Yang-Millstheorydefinedona(2+1)DRiemannsurfaceofpositive
genusgthatcanliveinatargetspaceof4D,allowsthistheorytobeofinterestalsooutsidethescopeofString
Theory. AdmittinganinterpretationintermsofSQCD,itconsistsoftwodifferentphasesatzerotemperature: a
confinedphasecomprisingglueballsin the bosonicsector anda quark-gluonplasma phasewhich hasa micro-
scopicoriginintheM-theory. Thequantumconsistencyofsupermembraneswithfixedcentralchargesprovides
thenanindirectproofofconsistencyofallthesenoncommutativegaugetheories.
1E-mail:[email protected]
2E-mail:[email protected]
3E-mail:[email protected]
1
2. Thesupermembranewithnon-trivialcentralcharge
Supermembranesareextendedobjectsdefinedintermsofabasemanifold,aRiemannsurfaceS ,whichliveina
Minkowskitargetspace. ThecanonicallyreducedHamiltonianinthelightconegauge[10]hastheexpression
1 P 2 1
√W M + XM,XN 2+ Fermionicterms (1)
ZS 2(cid:18)√W(cid:19) 4{ } !
where
e ab
XM,XN 2= ¶ XM¶ XN. (2)
{ } W(s ) a b
restrictedbythefirstclassconstraint,
p
P
M XM=0. (3)
C √W
I
which generates area preserving diffeomorphismsof S for C any given closed path. Here and below M,N =
1,...,9. The continuityof the spectrum of the aboveHamiltonian at the SU(N) regularizedlevel was demon-
stratedin[11].Thispropertyreliesontwobasicfacts:supersymmetryandthepresenceofsingularconfigurations
withzeroenergyataclassicallevel.
Undercompactifications,forexampleregardthetargetspaceasbeingM S1, itisbelieved,[12], that
10
×
the spectrum of the theory remains continuous. Therefore, the compactification procedure by itself, does not
seemenoughtochangethespectralpropertiesofthemodel.
Wenowimposesometopologicalrestrictionsontheconfigurationspace. Thesecompletelycharacterize
theD=11supermembranewithnon-trivialcentralchargegeneratedbythewrappingonthecompactsectorof
thetargetspace. We willassumethatthetargetspaceisM S1 S1anditsbasemanifoldS ofpositivegenus
9
× ×
g,forsimplicity,g=1. Westressthatthesepropertiesremainvalidforanyothertargetspaceofdimensionless
than9inparticularD=4. AllmapsfromthebasespaceS ,mustsatisfy
dXr=2p SrRr, r=1,2; dXm=0 m=3,...,9 (4)
i
ICi ICi
fori=1,2andthetopologicalcondition,
Z= dXr dXs=e rs(2p 2R R )n, (5)
1 2
S ∧
Z
wheren=detSr isfixed,eachentrySr isinteger,andR andR denotetheradiiofthetargetcomponentS1 S1.
Notethat(4)deiscribemapsfromS toiS1 S1withdXm1anon-2trivialclosedone-form.Theonlyrestriction×upon
×
thesemapsistheassumptionthatnisfixed. Thetermontheleftsideof(5)describesthecentralchargeofthe
supersymmetric algebra. Among all the maps from the torus S to the target space satisfying (4),(5) there is a
minimizeroftheHamiltonian. ItcorrespondstoaminimalimmersionfromS tothetargetspacewhichimplies
that,forthecaseofflattargetspaces,theworldvolumeofthesupermembraneisacalibratedsubmanifold.
Minimalimmersionscanalsodescribenon-BPSminimalsolutions,[8]. Thetheoryresultstobeinvariant
underSL(2,Z). ThedegreesoffreedomareexpressedintermsofA andthediscretesetofintegersdescribedby
r
theharmonicone-forms.Wecanalwaysfixthesegaugetransformationsby
Sr=lrd r, l1l2=n. dXr=2p RrlrdXˆr+d rdA . (6)
i i → s s
Afterthegaugefixingthereisaresidualtransformation[1]Z(2)underwhich,
A A A A . (7)
1 2 2 1
→ →−
ThisallowsustorewritetheHamiltonianintermsofXm,m=1,..,7andA ,r=1,2.Theresultingexpressionis:
r
1 1 1
H = √W[P2+P 2+ W Xm,Xn 2+W(D Xm)2+ W(F )2]
S 2 m r 2 { } r 2 rs
Z
1
+ [ √Wn2 L (D P + Xm,P )] (8)
r r m
S 8 − { }
Z
+ √W[ YG G D Y +GY G Xm,Y +L GY ,Y ]
r r m
S − − − { } { − }
Z
2
where([2],[4])D =2D +[A , ],F =D A D A +[A ,A ],andP andP aretheconjugatemomenta
r r r rs r s s r r s m r
toXm andA respe•ctively. D• and•F arethecova−riantderivativeandcurvatureofasymplecticnoncommuta-
r r rs
eab
tivetheory[2,5],constructedfromthesymplecticstructure introducedbythecentralcharge. Thelastterm
√W
representsitssupersymmetricextensionintermsofMajoranaspinors. Therelevantdegreesoffreedominorder
toquantizethetheoryaretheXm andthegaugeinvariantpartofA whicharesinglevaluedoverthebaseman-
r
ifold. ASU(N)regularizationwasobtainedin[4]andthespectralpropertiesofthespectrumwererigourously
demonstratedatclassicallevel,andatquantumlevelinseveralpapers,[5],[6],[9].
3. Onthespectrumoftheexacttheory
According to the results reported in [5], the bosonic regularized Hamiltonian of the D=11 supermembrane
with central charge, HB, relates to its semi-classical approximation,HB , by means of the following operator
N sc,N
inequality:
HB C HB . (9)
N ≥ N sc,N
Here N denotesthe size of the truncationin the Fourierbasis of S andC is a positiveconstant. A seemingly
N
crucialstepintheproofof(9)foundin[5],reliesheavilyonthecompactnessoftheunitballoftheconfiguration
spacewhichhappenstobefinitedimensional. TheexactbosonicHamiltonianhowevercontainsaconfiguration
spacewhichisinfinite-dimensional,sothattheunitballisnotcompact. In[1]weshowthatthe sameoperator
relationholdstruefortheexactbosonicHamiltonians.Weovercomethedifficultyoftheanalysisbycarryingout
adetailedanalysisofeachterminvolvedintheexpansionofthepotentialtermoftheexactbosonicHamiltonian,
HB.
Followingthe standard notation Lp Lp(s ) denotesthe Banach space of all fields u, such that u =
p
up 1/p<¥ .Let ≡ k k
h i
u =( D u 4+ D D u 4)1/4. (10)
4,2 r r s
k k k k k k
ThefieldsXm,A lieontheconfigurationspaceH4,2offunctionsu H1suchthat u <¥ . Notethattheleft
r 4,2
handsideof(10)isawelldefinednorminH4,2,thelaterisalinea∈rspace,butwekdoknotmakeanyassumption
aboutcompleteness. Thepotential,V, ofthebosonicsector ofthesupermembranewith centralchargesiswell
definedinH4,2as
1
V = D XmD Xm+ F F . (11)
r r rs rs
h 4 i
V isnotwelldefinedinH1butinH4,2.
Byimposingthegaugefixingconditions,D A =0andD A =0,thepotentialcanbere-writtenas,
1 1 1 2
V =r 2+2B+A2 (12)
wherer 2isthesemiclassicalpotentialtermofHB,Letr 2bethepotentialtermofHB,sothat
sc sc
r 2= D XmD Xm+(D A )2+(D A )2 .
r r 1 2 2 1
h i
and
B= D Xm A ,Xm +D A A ,A (13)
r r 1 2 1 2
h { } { }i
A= A ,Xm 2+ A ,Xm 2+ A ,A 2+ Xm,Xn 2 . (14)
1 2 1 2
h{ } { } { } { } i
Thisallowsustoshowthefollowingcrucialidentity,[1]: thereexistsaconstant0<C 1,suchthat
≤
V Cr 2, Xm,A H2. (15)
r
≥ ∀ ∈
Thelatterisaconsequenceoftheparticularexpressionofthepotential,propertiesasthecompactnessofthebase
manifold(notoftheconfigurationspace)andtheCauchy-Schwarzinequality.
3
InordertodefinerigorouslytheLaplacianinthenon-compactinfinitedimensionalconfigurationspacewe
haveintroduced,theHamiltonianisexpressedas
HB=[V +V +(1 C)V ]+[ D +CV ]
quartic cubic quadratic quadratic
− −
where the first bracket acts multiplicatively on the Hilbert space of states, while the operator on the second
bracketmaybeexpressedintermsofcreationandannihilationoperatorsintheusualway. Alongsidewith(15),
thisexpressionensurestheoperatoridentity
HB CHB. (16)
≥ sc
analogousto(9).
4. Confinementofthetheory
ItwasanoriginalideaofG.’tHooft,[13]thatpermanentquarkconfinementoccursinagaugetheoryifitsvacuum
condensesintoastatewhichresemblesasuperconductor.Hisproposalwastoconsidertheconfinementofquarks
as dual of the Meissner effect, where the role of magnetism and electricity are interchanged. In his approach
he considered a nonabelian gauge theory that were seen as an abelian theory enriched with Dirac magnetic
monopoles, see also [14]. The symplectic Yang-Mills naturally creates this effect. The mass contribution of
thecentralcharge,or,analogously,itscorrelatedresidualZ(2)symmetryoftheHamiltonian,canbedescribedin
termsofthequadraticderivativesoftheconfigurationfieldsXmandA . Thederivativesofthesefieldscorrespond
r
tomappingsofthetargetspaceintoS .TheseareinducedbytheminimalimmersionwhichrealisesbyX ,r=1,2,
r
andtheharmonicfieldsoverS ,
b
D Y = X ,Y =l BY =l Y
r A { r A} rA B rA A
where
b
l B = d2s √w X ,Y YB.
rA { r A}
Z
They correspondto a particular subset of the structure consbtants that mixes the harmonic and the exact forms,
gC .
rA
Forthecaseofatorus,anexplicitrelationwasfoundin[4]. Thequadratictermsonthederivativesofthe
configurationvariables,defineastrictlypositivefunctionwhosecontributiontotheoverallHamiltoniangivesrise
toabasinshapedpotential.Thelattereliminatesthestring-likespikesandprovidesadiscretespectrum,evenfor
thesupersymmetricmodel. Thecentrecreatedbyadiscretesymmetryisamechanismforprovidingmasstothe
monopoles[15]. Thesupermembranetheorywhencompactifiedin4Dcanbeinterpretedasatheorymodelling
susyQCDinthespiritof[16]. Itexhibitsconfinementinthephaseatzerotemperaturesincethetheorybecomes
naturallythesupermembranewithcentralcharges,whichhasminimalenergy.
The mass terms are determined by the elements of the centre m(z) associated to the correlation length
of the particles [17]. By rising the energy, the theory enter in the phase of asymptotic freedom described by
the supermembranewithoutcentralcharges. The phase transitionis describedby the breakingof the centerof
the groupthatbecomestrivial. The phase transitionof topologicalnature[18]-[20]. The particlesbehaveas if
theywhereinaquark-gluonplasma.Thesequarks-gluonsdonotfeelthetopologicaleffects,sincethecorrelation
lengthbecomesinfiniteandtheeffectivevolumeiszero.Alongthecommutativedirectionsthequarksexperiment
noforce.
5. Conclusions
Inthisnotewehavesumarizedtheresultsoftherecentmanuscript[1]. TheD=11supermembranewithcentral
chargesisquantumequivalenttotheN=12+1Symplecticnon-commutativeSuperYang-MillsTheorydefinedin
targetspacesofdimensionD 9.ThespectrumofthebosonicsectoroftheD=11theoryhasbeendemonstrated
≤
atexactlevelofthe theoryto bepurelydiscrete,hencecontaininga massgap. Thetheoryexhibitconfinement
inthesupermembranewithcentralchargephase. Itentersintheasymptoticfreephasethroughthespontaneous
breakingofthecenter. ThisphasecorrespondstotheN=4wrappedsupermembrane.
4
Acknowledgements
We thank M. Banados, J. Zanelli, for helpful conversations. M.P.G.M thanks the organizersof the Workshop
“’ForcesUniverse2006”forthe opportunityto presentthis work, and to CECS, (Valdivia, Chile) for kindhos-
pitalityandsupport. M.P.G.M.ispartiallysupportedbytheEuropeanComunity’sHumanPotentialProgramme
undercontractMRTN-CT-2004-005104andbytheItalianMURundercontractsPRIN-2005023102andPRIN-
2005024045.
References
[1] BoultonL,Garc´ıadelMoralMandRestucciaA(2006).Preprinthep-th/0609054.
[2] Mart´ınI,OvalleJandRestucciaA2001Phys.Rev.D64040061
[3] Mart´ınI,RestucciaA2002Nucl.Phys.B622240–256.
[4] Garc´ıadelMoralMandRestucciaA2002Phys.Rev.D66045023
[5] BoultonL,Garc´ıadelMoralM,Mart´ınIandRestucciaA2002Class.QuantumGrav.192951-2959.
[6] BoultonL,Garc´ıadelMoralMandRestucciaA2003Nucl.Phys.B671343–358.
[7] M.P.GarciadelMoral,A.Restuccia.InstituteofPhysicsConferenceSeries2005,Vol43,151
[8] BellorinJandRestucciaANucl.Phys.B737(2006)190-208
[9] BoultonLandRestucciaANucl.Phys.B724380-396,2005
[10] DeWitB,HoppejandNicolaiH1988Nucl.Phys.B305545
[11] DeWitB,Lu¨sherandNicolaiH1989Nucl.Phys.B320135–159.
[12] DeWitB,PeetersKandPlefkaJ1998Nucl.Phys.ProcSuppl62405–411.
[13] G.t’HooftNucl.Phys.B138:1,1978;Nucl.Phys.B190:455,1981
[14] E.WittenNucl.Phys.B149(1979)285-320
[15] M.PepeNucl.Phys.Proc.Suppl.153:207-214,2006;
[16] A.Chodos,R.L.Jaffe,K.Johnson,CharlesB.Thorn,V.F.WeisskopfPhys.Rev.D9:3471-3495,1974
[17] K.Holland,M.Pepe,U.J.WieseNucl.Phys.B694:35-58,2004
[18] J.Preskill,A.VilenkinPhys.Rev.D47:2324-2342,1993
[19] A.S.Kronfeld,G.Schierholz,U.J.WieseNucl.Phys.B293:461,1987
[20] A.P.Balachandran,E.Batista,I.P.CostaeSilva,P.Teotonio-SobrinhoInt.J.Mod.Phys.A15:1629-1660,2000
5