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