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!