Non-Abelian Vortices, Hecke Modifications and Singular Monopoles

In this note, we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-Abelian vortex equations over C. Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I.