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Xiv:hep-th/0005085v2 26 Sep 2000PreprinttypesetinJHEPstyle.-HYPERVERSION
JustinR.DavidDepartmentofPhysicsUniversityofCaliforniaSantaBarbara,CA93106USA.justin@vulcan.physics.ucsb.edu
Abstract:ThenaivelowenergyeffectiveactionofthetachyonandtheU(1)gaugefieldobtainedfromstringfieldtheorydoesnotcorrespondtotheworldvolumeactionofunstablebranesinbosonicstringtheory.Weshowthatthereexistsafieldredefinitionwhichrelatesthegaugefieldandthetachyonofthestringfieldtheoryactiontothefieldsintheworldvolumeactionofunstablebranes.WeidentifyastringgaugesymmetrywhichcorrespondstotheU(1)gaugetransformation.Thisisdonetothefirstnonlinearorderinthefields.Weexaminethevectorfluctuationsatthetachyoncondensatetilllevel(4,8).
Keywords:BosonicStrings,D-branes,ConformalFieldModelsinStringTheory.1.Introduction
Recentworkshaveshownthatstringfieldtheoryisusefultounderstandthephysics
oftachyoncondensation[1].Inbosonicstringtheory,thedecayoftheD25-brane
hasbeenshowntocorrespondtotherollingofthetachyonofopenbosonicstring
1theorydowntoanewstationarypointinthetachyonpotential.Theanalysisof
thetachyonpotentialwascarriedoutusingstringfieldtheory.Itwasalsoseenthat
leveltruncationisagoodapproximationschemeforthecalculationofthevalueof
theminimumofthetachyonpotentialandthephysicsofsolitonsinthepotential[2,3,4,5,6,7,8].Theleveltruncationseemstobeagoodapproximationscheme
insuperstringfieldtheory[9,10,11,12].
AstheD25-branehascompletelydecayedatthenewstationarypointofthe
tachyonpotential,thereshouldbenodynamicalopenstringdegreesoffreedom.In
particulartheU(1)gaugefieldshouldnolongerbedynamical.Itwasconjectured
in[1]thatatthisstationarypointtheeffectiveactiondoesnothaveakineticterm
fortheU(1)gaugefield.Infactthetermmultiplyingthekinetictermwasargued
tobethetachyonpotential[13].Thiswouldrenderthisfieldauxiliary.Itwasalso
suggesteditwouldbeinterestingtoseethisphenomenondirectlyfromstringfield
theory.Thedifficultywiththedirectevaluationofthecoefficientofthekineticterm
ofthegaugefieldusingstringfieldtheoryistheproblemoffieldredefinition.
Wewillnowstatethisproblem.Considereliminatingallthemassivemodesand
theauxiliaryfieldsofthestringfieldusingtheequationsofmotion.Weobtainthe
effectiveactionS(˜Aµ,˜φ)asafunctionoftheU(1)gaugefieldandthetachyon˜φat
thetreelevel.Canweidentifythegaugefield˜Aµandthetachyon˜φwiththatofthe
gaugefieldoftheD25-braneanditstachyon,andthusobtaintheeffectiveactionof
theD25-branefromstringfieldtheory?AstringfieldΦhasthefollowinggauge
transformationinstringfieldtheory
δΦ=QBΛ+Λ∗Ψ−Ψ∗Λ(1.1)
HereΛ,theinfinitesimalgaugeparameterisastringfieldofghostnumberzeroand
‘∗’isthestarproductinstringfieldtheory.LetuscalltheU(1)fieldpresentinthe
expansionofΦas˜Aµ.Then,itisclearthat˜Aµdoesnottransformas
δ˜Aµ=∂µǫ(1.2)
Thetransformationpropertyof˜Ahasextratermsarisingfromthenon-Abelian
liketransformationpropertyofthestringfieldΦ.Thereforethegaugefieldthat
appearsinthelowenergyeffectiveLagrangianAµwhichhastheconventionalgauge
2transformationpropertycannotbeidentifiedwith˜Aµ.Thusthenaivelowenergy
effectiveactionS(˜Aµ,˜φ)cannotbeidentifiedwiththeworldvolumeactionofthe
unstablebrane.
ItmustbepossibletofindarelationshipbetweenAµand˜Aµandφand˜φ.This
willinvolveafieldredefinition
˜Aµ=˜Aµ(Aµ,φ),˜φ=˜φ(Aµ,φ)(1.3)
ItalsoinvolvestheidentificationofthestringfieldΛwhichcorrespondstothe
conventionalgaugetransformationpropertyofAµ.Inthispaperweshowthatit
isindeedpossibletofindsucharelationshipatthefirstnon-linearlevelinthefields.
Todothisweusemethodsdevelopedtoidentifygeneralcoordinatetransformation
andanti-symmetrictensorgaugetransformationasasetofstringfieldtheoretic
symmetriesdevelopedforthenon-polynomialclosedstringfieldtheoryby[15]1.
Thefactthatgaugefieldofthelowenergyeffectiveactioninvolvesnotonly˜Aµbutalsootherfieldsisimportantindeterminingthecoefficientofthekinetictermof
Aµfromstringfieldtheory.Determiningthenaivekinetictermof˜Aµisnotenough.
ThisfieldredefinitioninvolvedinrelatingAµand˜Aµisalsoimportantindetermining
termsofthenon-AbelianDiracBornInfeldactionfromstringfieldtheoryasdone
in[4].Thiswouldhelpinunderstandingthediscrepancyingaugeinvariancewhich
aroseintryingtodeterminingthecoefficientsofcertaintermsinthenon-Abelian
DiracBornInfeldactionfromstringfieldtheory.
ThusitseemsdifficulttocomparetheeffectiveactionS(˜Aµ,˜φ)obtainedfrom
stringfieldtheorywiththeworldvolumeactionofunstablebranesandverifythe
claimsof[13].ItstillmustbepossibletoseeifU(1)gaugefieldisnolongerdynamical
fromstringfieldtheory.Tothisend,itisofinteresttocalculatethetwopoint
functionsofallvectorparticlesatthetachyoncondensateusingleveltruncation.
Itisofinteresttocomparewiththesituationinp-adicstringtheory[14].The
kinetictermsforthetranslationalyzeromodesofthesolitoninp-adicstringtheory
disappearsafterfieldredefinitionatthenewstationarypoint2.
