Composite Operators and Topological Contributions in Gauge T(3)

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

objects(string,membrane,...,p-braneetc.)asthefundamentalconstituentsoftheuniverse[8].Inthispaper,inordertoseeifthetopologicalcontributionsarisefromthehigherdimensionalgaugetheorieswiththeminimalMaxwellkineticterm,wewillapplyPolyakov’spathintegralarguments[3]tothecompositeofelectric(p 1)-braneandmagnetic(q 1)-braneinD(=p+q+1)spacetimedimensions.Theapplicationto(2+1)or(3+1)dimensionalgaugetheoryshowedthattheexpectationvalueofthecompositeoperatorsinvolvessometopologicalcontributionsassociatedwithlinkinvariants[7][9].

Westartbyintroducingangaugeinvariantoperatorassociatedwiththecom-positeofanelectric(p 1)-braneandamagnetic(q 1)-braneinDspace-timedimensions

X(Wq;Wp)=exp( iWq (Fp+1))exp(iWpAp).(1)

Thisoperatordependonthep-dimensionalworldvolumeWpandq-dimensionalworldvolumeWqinDspacetimedimensions.HereApisap-formgauge eldwhichcouplestoelectric(p 1)-brane,andFp+1isthecorresponding(p+1)-form eldstrength.TheoperatorX(Wq;Wp)shouldbecalledtheextended’tHooft-Wilsonoperator.Infact,the rstfactoristhecovariantversionofthe’tHooftoperatororiginatedfromtheclosedhyper-pathof(q 1)-brane,thusitshouldbesatis edthatD p 1=q,because (Fp+1)is(D p 1)-form eldinDdimensions.Ouraimistocalculatethetransitionamplitudeforthepairofa(p 1)-braneanda(q 1)-brane,andtoinvestigatetheformofthepartitionfunction,andthentoseehowthetopologicalcontributionsarise.Whenthe(p 1)-braneand(q 1)-braneofthecompositeevolvefromtheinitialcon gurationPitothe nalcon gurationPf,wehavethetransitionamplitude