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$SU(N)$ BPS Monopoles in $\mathcal{M}^2\times S^2$

We extend the investigation of BPS saturated t'Hooft-Polyakov monopoles in $\mathcal{M}^{2}\times S^{2}$ to the general case of $SU(N)$ gauge symmetry. This geometry causes the resulting $N-1$ coupled non-linear ordinary differential equations for the $N-1$ monopole profiles to become autonomous. One can also define a flat limit in which the curvature of the background metric is arbitrarily small but the simplifications brought in by the geometry remain. We prove analytically that non-trivial solutions in which the profiles are not proportional can be found. Moreover, we construct numerical solutions for $N=2,3$ and 4. The presence of the parameter $N$ allows one to take a smooth large $N$ limit which greatly simplifies the treatment of the infinite number of profile function equations. We show that, in this limit, the system of infinitely many coupled ordinary differential equations for the monopole profiles reduces to a single two-dimensional non-linear partial differential equation.