Tachyon Potential in a Magnetic Field with Anomalous Dimensi(3)

Products defined in the context of noncommutative gauge theory allow for an interpolation between exact results on tachyon potentials at zero and large background B-fields. Techniques for computations of effective actions are transposed from the framework

picture[13].Boundarystring eldtheory[14,15,16]gaverisetosuccessfulchecksoftachyoncondensation[17,18].Investigationsofnoncommutativetachyonswereperformedinthelarge-Blimit[19,20].Fromthetechnicalpointofview,similaritiesmaybenotedwiththeresultsonthegaugesector,asfarasnoncommutative eldtheoryisconcerned.Star-productshaveindeedbeenproventoberelevantfortheexpressionofthetachyone ectiveactionsforthebosonicstringinworksbyCornalbaandOkuyama[21,22].Themodelwasinfactexactlysolvedforaquadratictachyon eld,andshowntobeconsistentwithtachyoncondensation.

However,thestepofthedeformationofstar-productsstillhastobecompleted,inordertoreallyparalleltheinsightswegotonthegaugesectorfromnoncommutative eldtheory.Thisisthepurposeofthepresentnote.Ishall rstperformcomputationswithageneraltachyon eld,thusrecognizingthepatternofAbeliangaugetheory.Di erencescomingfromtheab-senceoffermionsandinducedformgradingswillbeoutlined.Theresultwillbeanalogoustothepreviouslyknownnoncommutativetachyone ectiveaction,exceptforthedeformationofstar-products,andtherestorationofthekineticterm.Asacheck,Ishallrecovertheexactsolutionforaquadratictachyonpotential,justbyboundingtheorderofallowedderivativecorrections.

2Deformingthepartitionfunction

LetΣbetheworld-sheetwiththetopologyofadisk.Theboundaryiscoupledtothetachyon eldT.ExpandingthefactorcontainingtheboundaryinteractioninpowersofTinthepartitionfunction Z(T)=DXe SΣ[X] ΣT(X(τ))dτ

naturallyleadstointegralsovervariouswaysofinsertinganynumberoftachyonsalongtheboundaryofthedisk.Infactwearecomputingageneratingfunctionforsuchamplitudes,correspondingtosmearingtachyonsovertheboundary.

ThecomputationofthepathintegralZ(T)isformallysimilartotheoneperformedtoobtainstar-productsfromordinarystringtheory.Ofcoursewedonothavefermioniczeromodesinthepresentcontexttobuildthegradingofdi erentialformscoupledtoRamond–Ramondform elds,buttheexpansionofthespatially-varyingtachyon eldmakes uctuationsofscalarsappearatvariouspowers,http://www.77cn.com.cnly,letussplitthescalar eldintozeromodeand uctuatingpart:

Xµ(σ)=xµ+ξµ(σ),

andwritesubsequently

T(X(σ))=T(x)+ 1

p≥1

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