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
Asdiscussedabove,theonlycontractionsinthecomputationsof[4]thatcontributeinourbosonicframeworkarethosebetweenscalars.WearegoingtomapthecomputationoftheWess–Zuminocouplingstoourcontext,
dτdθDφµAµ(φ)→T(X(τ))dτ,
byreplacingthe eldstrengthsFµνinthosecouplingswithtachyon eldsT,andcancellinganytermthatdoesnotcarrytwozero-modesforeach eldstrength(sincethoseinvolvefermions).Finally,wehavetoreplaceanyproductoffermioniczero-modesby1,thusdestroyingthegradedstructureandallowingfortermsofarbitrarilyhighdegreeinT.
Thesimpli cationthatoccurs,duetothequadraticnatureofthetachyon eld,comesfromthelownumberofderivativesthatcanactonit.Acorrectionoforderninuµνnamelyhastocomefromtheexpansionatorderninthetachyon eld,withtwoderivativesactingoneachofthefactors,as:
p 1dσi
p≥2i=1