In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen
isavector eldnormaltothehyper-surfaceWp.Thisregularizationcontributesto
RtherenormalizationofthestringTp→Tp.
Indiscussion,theclassicalactionofthisgaugetheoryisnottopological,howdoesitgiverisetothetopologicalcontributionsidenticaltothoseoftopologicalgaugetheorywithaChern-Simonsterm(orBFterminhigherdimensions)?Thoughtheactionisdynamical,theoperatorbeingconsideredisthecompositeformedfromaelectric(p 1)-braneandamagnetic(q 1)-branetowhichap-formgauge eldcouplesinDdimensions.So,whentheyevolveandformaclosedhyper-path,theymaybemutuallylinked.Thatis,iftwocompositesareinterchanged,eachcompositeshouldbecovariantlytransportedinthep-formgaugepotentialoftheother.Thusthetopologicalcontributionsareoriginatedfromtheelectro-magneticdualstructureofthecompositesource.Inmoremathematicaldetails,thepropagatorinthetopologicalChern-Simonstheoryisgivenastheinverseoperatorofthedi erentialoperator” ”,thatisthepropagatorcontainstheantisymmetric tensor,butthegaugeinvariantsourcetermdoesnot.Ontheotherhand,inthepresentgaugetheorywithakineticterm,thepropagatorwhichistheinverseofLaplacianoperatordoesnotcontain tensor,whileoneofthecurrentsourcesJandKtowhichthegauge eldcouplesdoesit.Ifwewritetheineractiontermsindetails,
SI= i
=
where
Jµ2...µp+1(x)= i
y
Kµ1...µp(D)µp+2...µDδ(x y)dy,(11)µ1 Wq( F)D p 1+i WpAp(10)dDx(J+K)µ1µ2...µpAµ1µ2...µp,(x)=i Wpδ(x y)dyµ1...µp.(12)