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

Original · EN

Low--energy limits of N=2 supersymmetric field theories in the Higgs branch are described in terms of a non--linear 4--dimensional sigma--model on a target space, classically obtained as a 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 Nf flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on 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(Nc) Nf flavors and U(Nf-Nc) Nf flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality Nc ↔ Nf-Nc in N=1 SUSY SU(Nc) gauge theories.