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

TL;DR

This paper explores the Higgs branch structure of N=2 supersymmetric gauge theories using hyperKahler quotients, analyzing singularities, dualities, and their relation to Seiberg-Witten theory.

Contribution

It provides a pedagogical review of Higgs branch construction via hyperKahler quotients and demonstrates new dualities between different gauge theories using geometric interpretations.

Findings

Higgs branch singularity matches small instanton moduli space.

Explicit metric evaluation confirms duality between U(N_c) and U(N_f-N_c) theories.

Higgs phase duality offers insights into Seiberg's N=1 duality conjecture.

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.