Tachyon Potential in a Magnetic Field with Anomalous Dimensi(2)

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

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