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
1Introduction
WhenSΣistakentobethecouplingofΣtothe atspace-timemetric,aswellastoaconstantB- eld,andS Σtobetheintegralofthegauge eldcarriedbyopenstrings,namely
SΣ[X]=(δµνdXµ∧ dXν+B),
ΣOpenstringspropagatinginade niteclosed-stringbackgroundgiverisetoapathintegralwhoseclassicalactionconsistsoftwoterms.The rstoneisanintegraloverthewholeworld-sheetΣencodingtheclosed-stringbackground,whilethesecondoneisanintegralovertheboundaryencodingthecouplingofopenstrings: Z=DXexp SΣ[X] S Σ[X].
S Σ[X]=
thesaddle-pointapproximationtothepathintegralaboveallowsforastringyderivationoftheBorn–Infeldaction[1,2].WewilldwellonthecaseoffullyNeumannboundaryconditionsandAbeliangauge eld(asinglespace- llingD-braneispresent).
Goingbeyondthisapproximationamountstotakingderivativesofthe eldstrengthintoaccount.AndreevandTseytlin[3]workedoutthe rstderivativecorrections,andWyllardbroughtmoresystematics[4]intothecomputationofthosecorrectionsathigherorders,bothintheBorn–InfeldandintheWess–Zuminocouplings.TheSeiberg–Wittenlimit[5]ofstringtheory,wherenoncommutative eldtheorybecomesavaliddescriptionofthegaugetheoryalongthebrane,subsequentlyprovidedlotsofpredictionsforcorrectionsfromtheverystruc-tureofstar-products[6].Thosepredictions,encoded[7,8]inn-aryproducts nwereproventobeconsistentwiththeworkofWyllardatlowordersinderivatives,andthenderivedfromcommutativestringtheory[9,10].
TheseencouragingresultsobtainedintheSeiberg–Wittenlimitinvitedtogobeyondthislimit,investigatingthecontributionofanomalousdimensions.Thisledtothede nitionofdeformedstar-products nfromthecomputationofpathintegrals,thatinturninduceddefor-mationsofnoncommutativegaugetheory[11,12].
Anotherphysicalsectorcarriedbyopenstringscomesfromthetachyon eldcoupledtotheboundary, S Σ[X]=TX| Σ(τ)dτ.
Σ A, Σ
Thisisadi erentphysicalframeworkfromtheoneofthegaugesector,withimportantnon-perturbativeoutputsregardingthevacuumofstringtheorythroughthetachyoncondensation