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Introducing vortices in the continuum using direct and indirect methods

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and \textquotedblleft center projection,\textquotedblright QCD is reduced to a gauge theory including vortices, which corresponds to the non-trivial first homotopy group $Π_1\left( \text{SO}(3)\right) =Z_2.$ On the other hand, using the indirect method, in addition to the center gauge transformation and \textquotedblleft center projection,\textquotedblright an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.