Tachyon Potential in a Magnetic Field with Anomalous Dimensi(4)

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

TheintegrandsplitsintoaBorn–Infeldcontributionandanexpectationvalueoftheboundaryinsertion. Z(T)=dx

k! 2ξµ1(σ)...ξµk(σ1)ξν1(σ2)...ξνk(σ2) µ1... µkT(x) ν1... νkT(x).

Oneofthetwointegrationsisdealtwithusingtranslationinvariance,oneofthe1/k!factorsiscompensatedbythenumberofpossiblecontractionswiththetwo-pointfunctionofscalar eldsonthedisk(regularizedbyasmallparameter ):

µν θµν +iσ2Dµν(σ)=α′+Glog|1 e|.1 e iσ

Thispropagatorwasusedin[11,12]inordertodressnoncommutative eldtheorywiththecontributionofanomalousdimensions.Thesimplernoncommutative eldtheoryobtainedintheSeiberg–Wittenlimitcorrespondstokeepingonlythe rstterm.Asfarascombinatorialproblemsareconcerned,thecountingofcontractionsisperformedinthesameway,whetherornotthesecondtermisincluded.Itisexplainedin[10]howtorecursivelyobtain k-productsforthecontributionoforderkinthe eldstrength.Theseproductsthereforefoundaderiva-tionfromcommutativestringtheoryaftertheir rstappearancefromgeometricconsistencyconditionsinnoncommutativegaugetheory[7,8].Itwasnoticedthattherecursionworkedforarbitrarilyhighorderk,andwasonlystoppedbydimensionality(antisymmetryofthe nitealgebraoffermioniczeromodes).Inthepresentcontextoftachyon elds,thecontributionuptoquadraticordertothepathintegralreads

Z(T)=dxT 2T+...,2!

2denotesthedeformationofthebinarydi erentialoperator 2:where

2=Γ(1+2 G ′)

θ ′/2.

The rstterminZcomesfromthezero-tachyoncontribution,andthe rstorderinTdoesnotreceivecorrectionsbecausethosecouldonlycomefromself-contractionsofscalars.Therecursivecomputationsof[10,12]canbereproducedinthesamefashionasabove,leadingto:

k[Tk(x)], Z(T)=dxk!

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