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
withallordersnallowed(theyareonlyformallyde nedforn>5inthegauge-theorycontextbecauseofthedimensionoftheworld-volumeandthegradinginducedbyfermioniczeromodes).Withthesenotationsthe rstfewtermsare:
W4=ζ(2)
3
W8=ζ(4)
2tr[(hu)3], ζ(2)
ntr[(hu)n].
Sinceuµνissymmetric,thetracescanbeexpressedeitherascontractionswithhorht,sothatwecanmakecontactwiththemoresymmetricnotationsof[22]:
tr[((G 1+θ)u)n]=tr[(hu)n]=tr[(utht)n]=tr[(u(G 1 θ))n].
WearereadytousetheΓ-functionidentity
e γzΓ(1 z)=exp ζ(n)
n≥2
2 n≥2tr[((G 1+θ)u)n]exp
detΓ(1 (G 1 1 +θ)u)