Higgs Branch , HyperKähler 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 non–linear 4–dimensional σ– model on a hyperKähler target space, classically obtained as a hyper-Kähler 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 sin-gularities 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 et 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.