Composite Operators and Topological Contributions in Gauge T(4)

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

G(Pi,Pf)=

Wp(Pi,Pf)Wq(Pi,Pf) e Sp Sq<XE(Wp,Wq)>,(2)

wherethesumisoverallp-dimensionalworldvolumeWpandq-dimensionalworldvolumeWqinterpolatingbetweentheinitialandthe nalcon gurationsPiandPfofthe(p 1,q 1)-branecomposite.ThepurelygeometricalactionSpfora(p 1)-branewithaclassicaltrajectory,parametrizedby

xµ=xµ(ξ0,...ξp 1),(3)

isthep-dimensionworld-volumeinducedonthetrajectorybytheexternalDdi-mensionalmetrictimes(p 1)-branetensionTp

Sp=Tp Wpd(Vol),(4)

wheretheRiemannianvolumeelementisgivenintermsofthedeterminantoftheworld-volumemetrich=det(hij),by

d(Vol)=dξp 1∧...∧dξ0

×exp( i

2(p+1)!Wq F)exp(i 2dDxFp+1)WpAp),(8)