  We give a gauge-invariant definition of the vortex surface in SU(N)
Yang-Mills theory without using the gauge fixing procedure. In this
construction, gauge-invariant magnetic monopoles with fractional magnetic
charges emerge in the boundary of the non-oriented vortex surface such that the
asymptotic string tension reproduces the correct $N$-ality dependence. We show
that gauge-invariant magnetic monopoles and vortices are simultaneously
responsible for quark confinement in four dimensional spacetime based on the
Wilson criterion. These results are extracted from a non-Abelian Stokes theorem
derived in the previous paper.
