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We give a gauge-invariant definition of the vortex surface in SU(N)Yang-Mills theory without using the gauge fixing procedure. In thisconstruction, gauge-invariant magnetic monopoles with fractional magneticcharges emerge in the boundary of the non-oriented vortex surface such that theasymptotic string tension reproduces the correct $N$-ality dependence. We showthat gauge-invariant magnetic monopoles and vortices are simultaneouslyresponsible for quark confinement in four dimensional spacetime based on theWilson criterion. These results are extracted from a non-Abelian Stokes theoremderived in the previous paper.

Magnetic monopoles and center vortices as gauge-invariant topological defects simultaneously responsible for confinement