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
ThereforetheexpectationvalueofcompositeoperatorcangivethelinkingnumberidenticaltothatintopologicalBFtheory.
Inthistheory,sincethelinkingnumberisnotself-linkingone,inanyspacetimedimensionsthesetopologicalphasescanbeoccurred.InhigherdimensionalChern-Simonstheorythepossibletopologicalcontributionsstemingfromtheselflinkingofthesinglegaugeinvariantoperator,however,arelimitedonlytothespacetimedimensionsD=3,7,11,15,sinceChern-Simonstermcanbeconstructedonlyinodddimensionsandtheself-linkingoftheobjectwithevendimensionalstructurevanish[10][11].
Inconclusions,wehaveshownthetopologicalcontributionsfromtheexpecta-tionvaluesofthecompositeoperatorX(Wq,Wp)inthegaugetheorywithaminimalkinetictermareidenticaltothoseobtainedforWilsonhypersurfacesintopologicalquantum eldtheorywithaBFChern-SimonsterminDdimensions.Thistopolog-icalphasefactorsoriginatefromthedistinctwaysinwhichthe(p 1)-braneworldsurfacemaybraidaroundthe(q 1)-braneworldsurfacewhentheyevolveinDdimensionalspace-time.Thisresultcanbegeneralizedtothenon-intersectingmultip-branehypersurfacesandthenon-Abelianextensiondeservesfutherinvestigations.Sincestatisticshasbarelybeentouchedforobjectssuchasstring,membrane,p-brane,itisverytemptingtoanalyzethegeneralizedstatisticsindi erentcontexts.Acknowledgements
ThisworkwassupportedbytheResearchFundofWoosukUniversity.
References
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