We completely determine the moduli space MN,k of k vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (C×CPN-1)k/Sk. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. The relation to Kähler quotient construction is discussed.
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