Tachyon Potential in a Magnetic Field with Anomalous Dimensi(8)

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

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