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
4Conclusions
Derivativeexpansionsofopen-stringe ectiveactionsmaybeobtainedfromalgebraicrulesappliedto uctuationsofworld-sheetcoordinates.TheserulesembodythepresenceofD-branesasboundarystates,andaredictatedbytheclosed-stringbackground.WhenthelattercontainsalargeconstantB- eld,noncommutativityintroducessomemorestructureintotheformofthee ectiveaction,underthedisguiseofstar-products.
Theseproductsarethensubjecttodeformationscomingfromshort-distancesingularitiesintheoperatorproductexpansions.Wehaveseenthatthisprogramcanbecompletedinthecaseoftachyoninsertionsontheboundaryontheworld-sheet,aswellasinthecaseofphotoninsertions,whereitwasoriginallyworkedout.Moreover,objectssuchask-aryproducts kor kwithk≥5,whichwerede nedonlyformallyinthecontextgaugetheory,contributetothe
partitionfunctioncomputedabove.EventhoughtheSeiberg–Wittenmapisintimatelytiedtogaugeinvarianceandgaverisetothede nitionoftheundeformedproducts[7,8],alltheseproductsandtheirdeformations ndahomeinthee ectivetheoryoftachyons.Thismakesthemmorerelevantforopen-stringe ectiveactionsinthebackgroundofaB- eld.
Weobtainedaclosedexpressionforthepartitionfunction,whichdressestheformerresultswithanomalousdimensionsonlythroughdeformationsofthestar-products.Moreover,ourexpressiongivesbacktheexactsolutioninthecaseofaquadratictachyon eld.Asforthee ectiveaction,takinganomalousdimensionsintoaccountwillalwaysgenerateakineticterm.Thee ectiveactionisindeedrelatedtothepartitionfunctionthrough[16]
δS(T)=1 d2xβ(T(x))
a tru