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

TL;DR

The paper develops a coherent HK-quotient framework to describe Higgs branches of N=2 SUSY Yang–Mills theories and uses it to analyze singularities, notably the Seiberg–Witten small instanton point. It demonstrates Higgs-branch equivalences between different gauge theories (e.g., $SU(2)$ with $N_f$ flavors and certain $U(1)$ theories) and proves a duality between Higgs branches of $U(N_c)$ and $U(N_f-N_c)$ theories via geometric realizations as cotangent bundles of Grassmannians. Through Kronheimer–Nakajima quiver constructions and Grassmannian geometry, the work links ADE-type singularities, Higgs-phase dualities, and potential connections to Seiberg duality in N=1 theories. The results provide a geometric lens for understanding nonperturbative structure, gauge symmetry restoration, and dualities in supersymmetric gauge theories with broad implications for string dualities and beyond.

Abstract

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 \hk target space, classically obtained as a \hk 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 $\RR^4$. 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 $N_c \leftrightarrow N_f-N_c$ in N=1 SUSY $SU(N_c)$ gauge theories.