Higgs branch, hyper-Kahler quotient and duality in SUSY N = 2 Yang-Mills theories

Low-energy limits of N = 2 supersymmetric field theories in the Higgs branch are described in terms of a nonlinear four-dimensional sigma-model on a hyper-Kahler target space, classically obtained as a hyper-Kahler quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low-energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg-Witten SU(2) theory with N-f flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on R-4. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a U(1) N = 2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg-Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of U(N-c) N-f flavors and U(N-f-N-c) N-f flavors theories, by using a geometric interpretation due to Biquard ct al. This duality may be relevant for understanding Seiberg's conjectured duality N-c <-> N-f-N-c in N = 1 SUSY SU(N-c) gauge theories.