We study magnetic flux tubes in the Higgs vacuum of the [Formula: see text] mass deformation of SU(Nc), [Formula: see text] SYM and its large Nc string dual, the Polchinski-Strassler geometry. Choosing equal masses for the three adjoint chiral multiplets, for all Nc we identify a "colour-flavour locked" symmetry, SO(3)C+F which leaves the Higgs vacuum invariant. At weak coupling, we find explicit non-Abelian k-vortex solutions carrying a ℤNc-valued magnetic flux, with topological winding 0 < k < Nc. These k-strings spontaneously break SO(3)C+F to U(1)C+F resulting in an S2 moduli space of solutions. The world-sheet sigma model is a nonsupersymmetric ℂℙ1 model with a theta angle θ1+1 = k(Nc - k)θ3+1 where θ3+1 is the Yang-Mills vacuum angle. We find numerically that k-vortex tensions follow the Casimir scaling law Tk ∝ k(Nc - k) for large Nc. In the large Nc IIB string dual, the SO(3)C+F symmetry is manifest in the geometry interpolating between AdS5 × S5 and the interior metric due to a single D5 -brane carrying D3 -brane charge. We identify candidate k-vortices as expanded probe D3 -branes formed from a collection of k D -strings. The resulting k-vortex tension exhibits precise Casimir scaling, and the effective world-sheet theta angle matches the semiclassical result. S -duality maps the Higgs to the confining phase so that confining string tensions at strong 't Hooft coupling also exhibit Casimir scaling in [Formula: see text] theory in the large Nc limit.
The topological susceptibility in the large-N limit of SU(N) Yang–Mills theory